Idea Transcript
APPROVAL SHEET
Title of Thesis: System
On the Longitudinal Vibration of a Moving Elevator Cable-Car
Name of Candidate: Yi Chen Master of Science, 2008
Thesis and Abstract Approved:
(
)
Dr. Weidong Zhu Professor Department of Mechanical Engineering
Date Approved: ________________
Curriculum Vitae Name:
Yi Chen
Permanent Address: 126 Nunnery Lane APT C, Catonsville, Maryland, 21228 Degree and date to be conferred:
Master of Science, 2008
Date of Birth:
December 10, 1979
Place of Birth:
Shandong, China
Secondary education: Collegiate institutions attended: M.S., University of Maryland Baltimore County, U.S.A, 2005-2008 M.S., Beijing University of Technology, B.S.,
Jinan University,
Major: Mechanical Engineering. Minor(s): Professional Publications: Professional positions held:
China, 2002-2005
China, 1997-2001
ABSTRACT
Title of Document:
ON THE LONGITUDINAL VIBRATION OF A MOVING ELEVATOR CABLE-CAR SYSTEM Yi Chen, Master of Science, 2008
Directed By:
Professor Weidong Zhu, Department of Mechanical Engineering
In this work, a new mathematical model for the longitudinal vibration of a moving elevator cable-car system with an arbitrary movement profile is developed. The car is modeled as a mass-spring-damper subsystem attached to the lower end of the cable. Both the Newtonian method and Hamilton’s principle are used to derive the governing equations of the moving cable-car system. The assumed modes method and a new spatial discretization method are used to discretize either the governing partial differential equation of the moving cable or the Lagrangian of the system before the application of Lagrange’s equations. The rate of change of the total mechnaical energy is analyzed from the system viewpoint and the control volume viewpoint. Numerical simulation of the responses from three spatial discretization schemes are obtained and compared. The new spatial discretization method can achieve uniform convergence and overcome the drawbacks of the classical assumed modes method.
ON THE LONGITUDINAL VIBRATION OF A MOVING ELEVATOR CABLECAR SYSTEM
By Yi Chen
Thesis submitted to the Faculty of the Graduate School of the University of Maryland, Baltimore County, in partial fulfillment of the requirements for the degree of Master of Science 2008
1460730
2009
1460730
© Copyright by Yi Chen 2008
Acknowledgements I would like to give great gratitude to my advisor Dr. Weidong Zhu for his persistent advising and help during my research. Thank Dr. Christian von Kerczek, Dr. Haijun Su to serve on my thesis defense committee. I want to express my appreciation to Dr. Nengan Zheng, Hui Ren and Dr. Yan Chen for their valuable discussion. My lab mates—Kun He, Chuang Xiao, Ben Emory, Dr. Guangyao Xu for the help and support. Thank all my colleagues in Department of Mechanical Engineering. This material is based on work supported by CAREER Award from the National Science Foundation, and Department of Mechanical Engineering, UMBC.
ii
Table of Contents
Acknowledgements....................................................................................................... ii Table of Contents......................................................................................................... iii List of Tables ................................................................................................................ v List of Figures .............................................................................................................. vi Chapter 1: Introduction ................................................................................................. 1 1.1 Brief Review on Vibration of Translating Media ......................................... 1 1.2 Historical Review on the Vibration of Elevator System............................... 3 1.2.1 Transverse Vibration............................................................................. 4 1.2.2 Longitudinal Vibration.......................................................................... 6 1.3 Objective and Scope ..................................................................................... 7 Chapter 2: Mathematical Modeling .............................................................................. 9 2.1 Modeling Techniques.................................................................................... 9 2.1.1 Modeling Approaches........................................................................... 9 2.1.2 Simplification of a Prototype Elevator System................................... 10 2.2 Newton’s Law............................................................................................. 13 2.2.1 Original System and Reference System.............................................. 13 2.2.2 Newton’s Law..................................................................................... 14 2.3 Hamilton’s Principle ......................................................................................... 18 Chapter 3: Spatial Discretization .......................................................................... 23 3.1 Trial Functions ............................................................................................ 24 3.2 Assumed Modes Method (Method 1) ......................................................... 27 3.3 Modified Novel Discretization Scheme...................................................... 33 3.3.1 Method 2 ............................................................................................. 38 3.3.2 Method 3 ............................................................................................. 43 Chapter 4: Energy Consideration.......................................................................... 44 4.1 Control Volume Viewpoint......................................................................... 44 4.2 System Viewpoint....................................................................................... 49 4.3 Energy Flux................................................................................................. 52 4.4 Energy Discretization for Method 1 ........................................................... 54 4.5 Energy Discretization for Method 2 ........................................................... 59 Chapter 5: Numerical Simulation & Conclusions ..................................................... 64 5.1 Initial Conditions (from Static Equilibrium)............................................... 65 5.2 Equilibrium ................................................................................................. 66 5.3 Example and Discussion ............................................................................. 69 5.3.1 Method 1 vs. Method 2 ....................................................................... 73 5.3.2 Method 2 vs. Method 3 ....................................................................... 80 5.3.3 Dynamic Responses Based on the Equilibrium Position (Method 2). 86 Chapter 6: Conclusions and Future Work ............................................................ 88 Appendix A: Spatial discretization results for assumed modes method by using Lagrange’s Equation ................................................................................................... 90 Appendix B: spatial discretization results for method 3............................................. 94 iii
Appendix C: energy discretization results for method 3 ............................................ 97 Bibliography ............................................................................................................. 101
iv
List of Tables Table 1 The moving profile regions and polynomial coefficients.............................. 72
v
List of Figures Figure 1 Prototype elevator system............................................................................. 12 Figure 2 Systematic of a moving elevator cable-car system: (a) spatial coordinate axis; (b) the reference configuration which is the undeformed moving cable-car system; (c) the current configuration which is the deformed moving cable-car system ..................................................................................................................................... 13 Figure 3 Free body diagram........................................................................................ 15 Figure 4 Vibrating string............................................................................................. 26 Figure 5 The prescribed moving profile: .................................................................... 71 Figure 6 Vibration of the car with details (M1, 2)...................................................... 74 Figure 7 Vibration of the lower end of the cable with details (M1, 2) ....................... 75 Figure 8 Vibration of physical particle at 10 m above the lower end with details (M1, 2) ................................................................................................................................. 76 Figure 9 Axial force and spring force (M1, 2)............................................................ 77 Figure 10 Total mechanical energy profile (M1, 2).................................................... 78 Figure 11 Rate of change of energy with details (M1, 2) ........................................... 79 Figure 12 Vibration of the car with details (M2, 3).................................................... 80 Figure 13 Vibration of the lower end of the cable with details (M2, 3) ..................... 81 Figure 14 Vibration of physical particle at 10 m above the lower end with details (M2, 3) ........................................................................................................................ 82 Figure 15 Axial force at different location of the cable (M2, 3)................................. 83 Figure 16 Total mechanical energy profile (M2, 3).................................................... 84 Figure 17 Rate of change of total energy with details (M2, 3) ................................... 85 Figure 18 Displacement resulted from the translating motion.................................... 86 Figure 19 Displacement of the lower end of the cable from the newly defined equilibrium position .................................................................................................... 87
vi
Chapter 1: Introduction
1.1
Brief Review on Vibration of Translating Media
Vibration is regarded as a subset of dynamics in which a system that is subjected to restoring forces oscillates about an equilibrium position.
Vibration systems can be
defined into such categories as single-degree-of-freedom models, multi-degree-offreedom models and distributed-parameter models, described by ordinary differential equation, sets of ordinary differential equations and partial differential equations, respectively. The vibration of translating media has been the subject of extensive research efforts in the past few decades. The topic encompasses diverse mechanical systems such as high-speed magnetic tapes, band saws, conveyer belts, textile fibers and transport cable. The vast literature in the area was first reviewed by Mote [1], and subsequently by Wickert and Mote [2]. Most studies in the literature are concerned with time-invariant translating media whose motions are described by partial differential equations with timeindependent coefficients.
The prototypical models for the time-invariant translating
media include translating strings, bands, beams, cables, plates, and webs. The dynamics of time-dependent translating media differs significantly from those of time-invariant translating media and is governed by partial differential equations with 1
time-varying coefficients. Comparing to time-invariant translating strings, methods of solution for time-varying translating strings are far more underdeveloped. This category of translating media was first reviewed by Zhu [3], three problem areas were discussed in the reference: translating media transporting payloads, translating media with variable speed and translating media with variable length. As reviewed in above Reference [3], the translating media with variable length include elevator cables, robotic arms through prismatic joints, flexible appendages, satellite tethers, paper sheets through copiers, and crane and mining hoists. Chen [4] recently reviewed research on transverse vibrations of axially moving strings and their control. Historically, Carrier [5] first studied the dynamics of a translating string and beam with parabolically varying length.
Miranker [6] generalized the linear equation for the
transverse vibration of a translating string with constant length and an arbitrary velocity profile using Hamilton’s principle. Schaffers [7] studied the longitudinal vibration of a slowly moving hoist cable and obtained the response of the hoist by using Riemann’s method of characteristic curves. Swope and Ames [8] examined the motion of a moving threadline of constant length and velocity under homogenous boundary conditions using D’Alembert and characteristics method. More recently Zhu and Ni [9], from a broader view, investigated the general stability characteristics of horizontally and vertically translating strings and beams with variable length.
2
While the linear theory of transverse vibration of axially moving media was extensively examined in the past few decades, nonlinear theoretical studies are also essential to better understand the dynamic mechanism. Ames and Zaiser [10] developed a nonlinear model to examine the motion of a moving threadline under planar periodic boundary excitation and achieved two exact solutions followed by a discussion of experimental validation. Mansfield and Simmonds [11] studied the nonlinear governing equations for an elastica emerging from a horizontal guide at constant speed, and Stolte and Benson in Reference [12] extended their work to allow for an arbitrary exit angle and an accelerating feed rate. Wickert [13] analyzed free nonlinear vibration of an axially moving, elastic, tensioned beam and derived a perturbation theory for the near-modal free vibration of a general gyroscopic system with weakly nonlinear stiffness. Moon and Wickert [14] investigated nonlinear vibration of a prototypical power transmission belt system excited by pulleys having slight eccentricity through experimental and analytical methods.
1.2
Historical Review on the Vibration of Elevator System
Among the vibration of translating media, the vibration of elevator systems is one of the most popular subjects in recent years.
Both transverse vibration and longitudinal
vibration are important to acquire an improved understanding of elevator system dynamics and robust control methods to depress the elevator system vibration. Due to small allowable vibration the transverse and longitudinal cable vibrations in elevators can be assumed to be uncoupled. 3
1.2.1
Transverse Vibration
The transverse vibration or lateral vibration has been a major problem affecting the ride comfort in a high-speed elevator and has been widely studied both theoretically and experimentally. To suppress the lateral vibration of the axially moving media with varying lengths such as the elevator cables, mine hoist, and the crane cables, Lee [15] found the optimal damping coefficient to suppress the vibration of the traveling string based on the wave method. Lee [16] also investigated the vibration control of an axially moving string by the boundary transverse motion or the external boundary forces using an optimal feedback gain for the maximum rate of energy dissipation. In [17-19], Otsuki studied a series of controllers using a non-stationary robust control method, and verified the controller experimentally and numerically. In addition, a series of works of the numerical simulation and the experimental validation on the vibration control of the elevator system were implemented in [20-24] by Yamazaki et al. Arakawa and Miyata [25] developed a variable-structure control method to suppress the vertical vibration of a fast elevator by using a H ∞ control method. Kimura [26] numerically analyzed the lateral vibration of elevator ropes for high-rise building induced by wind forces and proposed a new practical method for reducing the rope vibration by using vibration suppressor. The numerical calculation demonstrated the resonant position of the rope with vibration suppressor is greatly depends on the position of the vibration suppressor. Subsequently, Kimura [27-29] performed forced vibration analysis of elevator rope with
4
constant tension and moving velocity. The theoretical solutions to the forced vibration of a rope are presented based on zero damping coefficient or small linear damping coefficient of the rope with different boundary conditions by using virtual sources of waves. More recently, Kimura [30] compared the experimental results of free vibration and forced vibration of the rope with that of the finite difference analyses. The calculated results of the finite difference analyses are in fairly good agreement with those of the theoretical results and experimental results. It is impossible to effectively depress the lateral vibration of translating media such as the elevator cable and the crane cable without discovering a fully understanding of cable dynamics and instantaneous energy mechanism of the translating media.
More
specifically, a phenomenon termed as the unstable shortening cable behavior is discovered during the upward movement of an elevator: a small disturbance in the hoist cable can lead to a greatly amplified vibratory energy.
The amplitude of the
displacement of a cantilever beam decreases during retraction [31] and that of an elevator cable increases first and then decreases during upward movement [32]. The lateral vibration and the associated energy of the moving media with arbitrary varying length were investigated in [9] and [33]. In addition, in [33], the rates of change of energies of translating media were distinguished from system and control volume viewpoints. From the system viewpoint, the rate of change of the total energy describes the instantaneous work done by the non-conservative forces; from the control volume viewpoint, the 5
corresponding rate of change of the energy describes the instantaneous energy change of the moving media between two boundaries.
In reference [34], a control method was
developed to dissipate the vibratory energy of a translating medium with variable length based on the successful analysis of the vibratory energy.
The vibratory energy
approaches zero during extension and retraction. Zhu and Chen [35] proposed three discretization methods to calculate the lateral response of vertically translating media with variable length and tension, subjected to general initial conditions and external excitation such as building sway, pulley eccentricity and guild rail irregularity.
In
reference [36], Zhu and Chen investigated a comprehensive, theoretical and experimental study of the uncontrolled and controlled lateral responses of a moving cable in a high-rise elevator. A novel experimental method was developed to validate the uncontrolled and controlled response and shown good agreement with the theoretical predictions. Most recently, Zhu and Zheng [37] obtained the exact solution of the free and forced vibration of a uniform tensioned moving string by using the wave method and a special characteristic transformation. 1.2.2
Longitudinal Vibration
While extensive studies focus individually on transverse vibration of translating media, the longitudinal vibration of translating media is also a popular subject recently as well as coupled transverse-longitudinal vibration.
Chi and Shu [38] calculated the natural
frequencies associated with the vertical vibration of a stationary cable coupled with an 6
elevator car. Zhang [39] presented a systematic procedure for deriving the model of a cable transporter system with arbitrarily varying cable lengths and proposed a Lyapunov controller to dissipate the vibratory energy. Terumichi and Yoshizawa [40] studied the longitudinal vibration of the hoist rope which is coupled with the vertical vibration of the moving elevator cab by an analytical model composed of three parts the hoist rope, the compensating rope and the cab.
Kaczmarczyk and Andrew [41] investigated the
influence of building lateral vibration on the coupled lateral-longitudinal dynamic response of the hoist ropes and showed the building sway results in a distributed inertial load acting upon the hoist ropes which may lead to resonance and modal interactions. Hiroaki Ito [42] proposed a wave transmission model for the behavior of the lateral vibration of rope coupled with the longitudinal vibration. The validity of proposed model is shown through some experimental results.
1.3
Objective and Scope
The overall objective of the thesis is to develop a general mathematical model for the longitudinal vibration of a moving elevator cable-car system with arbitrarily varying lengths and arbitrarily varying velocity, numerically simulating the dynamic response and energy variation. More specifically, the thesis covers the following topics: Mathematical modeling of the elevator cable-car system by using both Newton’s law and Hamilton’s principle. 7
Three spatial discretization methods to discretize the governing partial differential equation and energy expression. Numerical simulation of the response and energy discussion of a high-rise, high-speed elevator system.
8
Chapter 2: Mathematical Modeling
2.1
Modeling Techniques 2.1.1
Modeling Approaches
To study a variety of engineering problems, an appropriate mathematical model is essential, which usually composes of a set of dynamic equations. Several modeling techniques can be applied for developing governing equations of a given dynamic system, the first technique that is presented is Newton’s Law, which is fundamental to dynamic systems.
The implementation of Newton’s law intuitionally demonstrates
physical meanings. However, direct applications of Newton’s law become difficult for complex dynamic systems such as automobiles and aircrafts. D’ Alembert’s principle is used to derive the principle of virtual work for a system composing of a number of particles acted on by external forces. Hamilton’s principle uses scalar quantities, which include kinetic energy, potential energy, and virtual work, to develop governing equations of the system. While Newton’s law is good for discrete systems, Hamilton’s principle is much more convenient for distributed systems. This thesis will apply both Newton’s law and Hamilton’s principle in deriving governing equations of moving elevator cable-car systems.
9
How to select a method to develop a mathematical model is essentially problemdependent. For extraordinarily complex dynamic systems, finite element modeling (FEM) can be introduced. The other approach is to acquire the mathematical model through experimentations, which is usually acknowledged as system identification. 2.1.2
Simplification of a Prototype Elevator System
In a high-rise building, the elevator configuration, as shown in Figure 1, consists of a hoist rope, a car frame, a passenger cab placed on the platform pad, and a hitch device coupling the car frame to the hoist rope. To better solve the problems, there needs to be further simplifications of the prototype elevator system. The vertically translating hoist cable in elevators can be modeled as a taut string with a distributed damping coefficient
c , as shown in Figure 2(c). The frame, cab and hitch mass are modeled as a springdamper-mass system attached at the lower end of the cable, in which the stiffness and lumped damping coefficients are kc , cc respectively. The following assumptions constrain the analysis of a moving elevator cable-car system: Transverse vibration and longitudinal vibration are uncoupled and only longitudinal vibration is considered here. Material damping is treated as the effect proportional to the longitudinal velocity. The effect of air drag is neglected.
10
Variation of the cross-sectional dimensions during vibration is not considered. The mass per unit length is constant as well. The oscillations are sufficiently small so that only terms of the first order in the derivation need be considered. In this chapter, Newton’s law is applied in section 2.2 to derive governing equations of an elevator cable-car system. Furthermore, in section 2.3 Hamilton’s principle is utilized to yield governing equations of the same model. Boundary conditions are also given for both sections.
11
Figure 1 Prototype elevator system
12
0 x
u ( x, t ) u ( x + dx, t )
dx
Translational Motion v(t )
k x
v(t )
c
c
k
c
c
c
c
m
m
(b)
f (t ) (c )
c
(a)
l (t )
c
u c (t )
Figure 2 Systematic of a moving elevator cable-car system: (a) spatial coordinate axis; (b) the reference configuration which is the undeformed moving cable-car system; (c) the current configuration which is the deformed moving cable-car system
2.2
Newton’s Law 2.2.1
Original System and Reference System
The original system, as shown in Figure 2(a), is defined here to be in an unstretched state. The elastic elevator cable has no deformation. The spring between the cable and the car has no stretch as well.
13
The reference system is defined to be the same cable-car system which translates at a speed v (t ) as a rigid body as depicted in Figure 2(b).
2.2.2 Newton’s Law In this section, free-body diagram is applied to an infinitesimal segment of the model in Figure 2(c). The moving cable-car system is vibrating with respect to the reference system while translating with the reference system. The following variables are defined by u ( x, t ) : Longitudinal displacement of the particle on the cable passing through the
position x with respect to the reference system. uc (t ) : Displacement of the car with respect to the reference system
ρ : Mass per unit length of the cable P ( x, t ) : Axial force at position x
EA : Axial stiffness of the cable f (t ) : External force applied to the car
14
Since the model is a distributed system, longitudinal displacement u ( x, t ) is timedependent and spatial-dependent.
Both u ( x, t ) and uc (t ) are measured from the
unstretched position. An infinitesimal segment dx at position x shown in Figure 3 is considered for free body diagram.
Figure 3 Free body diagram
15
From mechanics of material, the strain-stress relationship still holds here:
σ = Eε
(1)
σ = P ( x, t ) / A
(2)
ε = [u ( x + dx, t ) − u ( x, t )] / dx
(3)
where
are referring to the stress, strain of the moving cable at position x , respectively.
Substituting Eq. (2), (3) into Eq. (1) yields P ( x, t ) u ( x + dx, t ) − u ( x, t ) =E A dx
(4)
Ignoring higher order in Taylor expansion of u ( x + dx, t ) yields P( x, t ) = EAu x ( x, t )
(5)
where the lettered subscript denotes partial differentiation. Measured by a stationary observer, the longitudinal acceleration is Du / Dt + v (t ) .
Applying Newton’s Law yields
ρ[
D 2 u ( x, t ) Dt
2
+ v (t )]dx = Px ( x, t ) dx + ρ gdx − c[
Rewriting Eq. (6) yields 16
Du ( x, t ) Dt
+ v (t )]dx
(6)
ρ
D 2 u ( x, t ) Du ( x, t ) + c[ + v(t )] − EAu xx ( x, t ) + ρ[v(t ) − g ] = 0 2 Dt Dt
where the overdot denotes time differentiation, the operator
(7)
D ∂ ∂ is referred to = +v Dt ∂t ∂x
as the material derivative or differentiation with respect to the motion and the following expression holds: D 2 u ( x , t ) ∂ 2 u ( x, t ) ∂ 2 u ( x, t ) ∂u ( x, t ) 2 ∂ 2u ( x, t ) = + 2v +v +v Dt 2 ∂t 2 ∂x∂t ∂x ∂x 2
(8)
Similar process applying to derive the governing equation for the car yields ⎡ ⎣
mc [uc (t ) + v (t )] = f (t ) + mc g − kc [ uc (t ) − u (l , t ) ] − cc ⎢uc (t ) −
Du (l , t ) ⎤ Dt
⎥⎦
(9)
where f (t ) is the external force applied to the car.
The boundary conditions can be obtained as well u (0, t ) = 0
(10)
⎡ ⎣
EAu x (l , t ) = kc [uc (t ) − u (l , t ) ] + cc ⎢uc (t ) −
17
Du (l , t ) ⎤ Dt
⎥⎦ − f (t )
(11)
2.3 Hamilton’s Principle In this section, the governing equations of the same elevator cable-car system shown in Figure 2 are developed by using Hamilton’s principle. The main advantage of this method is that there is no necessity to explicitly formulate the internal forces. In the following derivations, we omit the temporal and spatial arguments of the variables (such as u ( x, t ) , uc (t ) , v (t ) ) for simplicity purpose except explicitly emphasis. The kinetic energy and potential energy of the cable are given by: T=
1
V =
∫ 2
l (t )
ρ[
0
1
∫ 2
l (t )
Du ( x, t ) Dt
+ v (t )]2 dx
EAu x2 ( x, t ) dx − ∫
0
l (t )
0
(12)
ρ g[u ( x, t ) + x (t )]dx
(13)
The kinetic energy, potential energy and the virtual work done by the distributed and lumped damping forces of the system are given by: ⎡ Du (t ) ⎤ Tc = mc ⎢ c + v (t ) ⎥ 2 ⎣ Dt ⎦ 1
Vc =
2
(14)
1 2 kc [uc (t ) − u (l (t ), t ) ] − mc g[uc (t ) + l (t )] 2
δ Wnc = − ∫
l (t )
0
c[
Du Dt
⎡ ⎣
+ v ]δ udx − cc ⎢uc (t ) −
(15)
Du (l , t ) ⎤
+ f (t )[δ uc − δ u (l , t )]
Dt
⎥⎦ [δ uc (t ) − δ u (l , t ) ]
(16)
18
Note that the virtual work done by the lumped damping force between the end of the cable and the car includes two parts: one part applies to the cable; the other part applies to the car. In order to substitute above expression into extended Hamilton’s principle, variational operations should be formulated first. Because the length of the cable l (t ) changes with time, the domain of integration for the spatial variable is time-dependent. The standard procedure for integration by parts with respect to the temporal variable does not apply. The use of Leibnitz’s rule gives the variations of the kinetic energy and the potential energy:
δT = ∫
l (t )
0
=∫
l (t )
0
ρ (ut + vu x + v)(δ ut + vδ ux )dx ρ (utδ ut + vutδ u x + vu xδ ut + v 2u xδ u x + vδ ut + v 2δ u x )dx
= ρ (vut + v 2u x + v 2 )δ u +∫
l (t )
0
l (t ) 0
l (t )
0
ρ vuxtδ udx − ∫
l (t )
0
ρv 2u xxδ udx
ρ (ut + vu x + v)δ ut dx
= ρ (vut + v 2u x + v 2 )δ u +
−∫
l (t ) 0
−∫
l (t )
0
ρ (vuxt + v 2u xx )δ udx
l (t ) ∂ l (t ) + + − ρ ( u vu v ) δ udx t x ∫0 ρ (utt + vu x + vuxt + v)δ udx ∂t ∫0
− ρ v (ut + vu x + v )δ u
x =l ( t )
= ρ (vut + v 2u x + v 2 )δ u
l (t ) 0
− ρ v (ut + vu x + v)δ u
19
x =l ( t )
+
δV = ∫
l (t ) ∂ l (t ) ρ (ut + vu x + v)δ udx − ∫ ρ (utt + 2vu xt + vu x + v 2u xx + v)δ udx ∫ 0 ∂t 0
l (t )
0
EAu xδ ux dx − ∫
l (t )
0
= EAuxδ u
l (t ) 0
−∫
l (t )
0
(17)
ρ gδ udx
( EAuxx + ρ g )δ udx
(18)
δ Tc = mc (uc + v)δ uc
(19)
δ Vc = kc [uc − u (l , t )][δ uc − δ u (l , t )] − mc gδ uc
(20)
In order to demonstrate our procedure conveniently, Hamilton’s principle is broken down into two parts, the first part of which is the cable. Substituting Eq. (17) and Eq. (18) yields
∫
t2
t1
δ (T − V )dt t2
= − ∫ ⎡⎣ ρ (vut + v 2u x + v 2 ) − EAu x ⎤⎦δ u t1 −∫
t2
t1
∫
l (t )
0
t2
x =0
t2
−∫
t1
∫
l (t )
0
t1
x =l ( t )
dt
⎡⎣ ρ (utt + 2vu xt + vu x + v 2u xx + v) − EAu xx − ρ g ⎤⎦δ udxdt
= − ∫ ⎡⎣ ρ (vut + v 2u x + v 2 ) − EAu x ⎤⎦δ u t1 t2
dt − ∫ EAu xδ u
t2
x =0
dt − ∫ EAu xδ u t1
x =l ( t )
dt
⎡ D 2u ⎤ ⎢ ρ ( 2 + v) − EAu xx − ρ g ⎥δ udxdt ⎣ Dt ⎦
Substituting Eqs. (16), (19), (20) into Hamilton’s Principle yields
∫
t2
t1
(δ Tc − δ Vc + δ Wnc ) dt
20
(21)
= mc (uc + v)δ uc
t2 t1
t2
t2
t1
t1
− ∫ mc (uc + v )δ uc dt − ∫
+ ∫ kc [uc − u (l , t ) ] δ u (l , t ) dt − ∫ {cc [uc − t2
t2
t1
t1
t2
+ ∫ {cc [uc − t1
[ kc (uc − u (l , t )) − mc g ]δ uc dt
Du (l , t ) ] − f (t )}δ uc dt Dt
t2 l ( t ) Du (l , t ) Du ] − f (t )}δ u (l , t )dt − ∫ ∫ c( + v)δ udxdt t 0 1 Dt Dt
t2 ⎡ Du (l , t ) ⎤ ) − mc g − f (t ) ⎥ δ uc dt = − ∫ ⎢ mc (uc + v) + kc (uc − u (l , t )) + cc (uc − t1 Dt ⎣ ⎦
t2
+ ∫ [kc (uc − u (l , t )) + cc (uc − t1
−∫
t2
t1
∫
l (t )
0
c(
Du (l , t ) ) − f (t )]δ u (l , t )dt Dt
Du + v )δ udxdt Dt
(22)
Substituting Eq. (21) and Eq. (22) into extended Hamilton’s principle
∫
t2
t1
(δ T − δ V + δ Tc − δ Vc + δ Wnc )dt t2
= − ∫ ⎡⎣ ρ (vu xt + v 2u x + v 2 ) − EAu x ⎤⎦ δ u t1 t2
− ∫ [ EAu x − kc (uc − u (l , t )) − cc (uc − t1
−∫
t2
t1
x =0
dt
Du (l , t ) ) + f (t )]δ u Dt
x =l ( t )
dt
⎡ D 2u ⎤ Du ∫0 ⎢⎣ ρ ( Dt 2 + v) + c( Dt + v) − EAuxx − ρ g ⎥⎦ δ udxdt l
t2
− ∫ [mc (uc + v) + kc (uc − u (l , t )) + cc (uc − t1
Du (l , t ) ) − mc g − f (t )]δ uc dt = 0 Dt
(23)
Setting the coefficients of δ u and δ uc to zero in Eq. (23) yields the governing equation for the moving elevator cable-car system:
21
ρ(
D 2u Du + v) + c( + v) − EAu xx − ρ g = 0 2 Dt Dt
mc (uc + v) + kc (uc − u (l , t )) + cc (uc −
Du (l , t ) ) − mc g − f (t ) = 0 Dt
(24)
(25)
Meanwhile, the boundary conditions can be obtained directly from Eq. (23) which is another advantage of Hamilton’s principle.
The boundary conditions at x = 0 and
x = l (t ) can be retrieved from Eq. (23), which is the same as given in section 2.2.
As shown in section 2.2 and 2.3, both approaches eventually yield the same governing equations and boundary conditions of the moving elevator cable-car system.
22
Chapter 3: Spatial Discretization
In practical engineering problems, most mathematical models are represented by partial differential equations (PDE), the exact analytical solutions of which are usually difficult to find. Hence, numerical methods are implemented to obtain the approximate solutions of the problems, the first step of which is discretization. Distributed problems must sometimes be replaced by a discrete problem whose solution is known to approximate that of the distributed problem; this process is called discretization.
In this work,
assumed mode method is applied to discretize the elevator cable-car system. Assumed mode method is a class of methods for converting a continuous operator problem (such as a differential equation) to a discrete problem. In principle, it is the equivalent of applying the method of variation to a function space, by converting the equation to a weak formulation. The organization of this chapter is as follows: section 3.2 gives the derivation of trial functions, which is also used as weight function, for the first discretized scheme. In section 3.3, applying assumed mode method yields the discretized ordinary differential equations (ODE) of the elevator cable-car system. In section 3.4, a novel discretization scheme is introduced and the discretization results are also given, respectively.
23
3.1
Trial Functions
The instantaneous eigenfunctions of a vibrating hanging string (fixed-free) with timevarying length l (t ) as shown in Figure 6 are used as the trial functions for the first discretized scheme. We assume static equilibrium as our zero potential energy position. y ( x, t ) is the vertical displacement with respect to static position. It is easy to obtain in any vibration textbook the governing equation of the vibrating string shown in Figure 6
ρ Aytt ( x, t ) = EAyxx ( x, t )
(26)
Eq. (26) can be rewritten as: ∂2 y 1 ∂2 y = ∂x 2 a 2 ∂t 2
(a 2 = EA / ρ )
(27)
Assume the solution to Eq. (27) is y ( x, t ) = Χ ( x )φ (t )
(28)
Following the standard procedure of separations of variables yields d 2 Χ ωn2 + Χ=0 dx 2 a 2
(29)
The solution to Eq. (29) is written as
24
Χ ( x) = C cos
ωn a
x + D sin
ωn a
(30)
x
Applying fixed-free boundary conditions yields the eigenfunctions
Χ( x) = sin
(n − 1/ 2)π x , n = 1, 2, 3 l (t )
(31)
25
y ( x, t )
EA
ρ
Figure 4 Vibrating string
26
3.2
Assumed Modes Method (Method 1)
The assumed mode method is employed to discretize the governing partial differential equation for the moving elevator cable-car system. The spatial discretization scheme was used in [36]. A new independent variable ξ = x /[l (t )] is introduced and the time-varying spatial domain [0, l (t )] for x is converted to a fixed domain [0, 1] for ξ . eigenfunctions obtained in section 3.2 become from sin
The
(n − 1/ 2)π x to sin(n − 1/ 2)πξ as l (t )
well. For explicit purpose, the governing equation of the moving elevator cable-car system is rewritten without distributed cable damping and external force as:
ρ (utt + 2vu xt + vu x + v 2u xx + v) − EAu xx − ρ g = 0 mc (uc + v) + kc [uc − u (l , t )] + cc [uc −
(32)
Du (l , t ) ] − mc g = 0 Dt
(33)
where the boundary Conditions are: x = 0 : u ( x, t ) = 0 ;
(34)
x = l (t ) : EAu x ( x, t ) = kc [uc (t ) − u (l , t )] + cc [uc (t ) −
Du (l , t ) ] Dt
(35)
The solutions of Eq. (32) are assumed in the form: n
u ( x, t ) = ∑ψ j (ξ )q j (t )
(36)
j =1
27
where q j (t ) ( j = 1, 2,3,… , n) is the generalized coordinate and uc (t ) is treated as the ( n + 1)th generalized coordinate, ψ j (ξ ) are the trial functions, and n is the number of
included modes.
The eigenfunctions of a vertically vibrating string with fixed-free
boundaries as shown in Figure 6 are used as the trial functions. These functions satisfy the orthorgnality relation
1
∫ ψ (ξ )ψ 0
i
j
(ξ )dξ = δ ij / 2 , where δ ij is the Kronecker delta
defined by δ ij = 1 if i = j and by δ ij = 0 if i ≠ j . Substituting Eq. (36) into Eq. (32),
ψ i (ξ ) (i = 1, 2,
multiplying the resulting equation by
, n ) , integrating it from x = 0 to l (t ) , and using the boundary
conditions and the orthorgnality relation for ψ j (ξ ) yields the discretized equations for the moving elevator cable-car system. Similarly, the temporal and spatial arguments of the variables (such as u ( x, t ) , uc (t ) , ψ j (ξ ) and q j (t ) ) purpose except explicitly emphasis. Applying the procedure to EAu xx yields
∫
l (t )
0
ψ i EAu xx dx 1
− ∫ EAu xψ i' dξ
= ψ i EAu x
l (t ) 0
= ψ i EAux
x =l ( t )
0
1
− ∫ EAuxψ i' dξ 0
= ψ i (1) [ kc (uc − u (l , t )) + cc (uc − u (l , t )) ] −
1 1 EAψ 'jψ i' q j dξ ∫ 0 l (t )
28
are omitted for simplicity
= ccψ i (1)uc + kcψ i (1)uc − ccψ i (1)ψ j (1) q j − kcψ i (1)ψ j (1) q j −
EA 1 ' ' ψ jψ i dξ q j l (t ) ∫0
(37)
Similarly for the part ρ v 2u xx yields
∫
l (t )
0
=
ψ i ρ v 2u xx dx
ρ v2 l (t )
=−
ψ i (1)ψ ′j (1)q j
ρ v2
1
ψ ψ dξ q l (t ) ∫ 0
' j
' i
1 0
−
ρv2
1
ψ ψ dξ q l (t ) ∫ 0
' j
' i
j
(38)
j
The partial differentiation of u ( x, t ) with respect to x is given as
ux =
1 ' ψ jqj l (t )
(39)
Partial differentiating Eq. (39) with respect to time yields
u xt = −
v vξ 1 ' ψ 'j q j − 2 ψ ''j q j + ψ jq j l (t ) l (t ) l (t ) 2
(40)
Similarly, the partial differentiation of u ( x, t ) with respect to time is
ut = ψ j q j −
xv ' ψ jqj l (t )
(41)
2
Partial differentiating Eq. (41) with respect to time yields
29
utt = −
xv ' xvl 2 (t ) − 2l (t ) xv 2 ' x 2 v 2 '' xv ψ ψ ψ ψ j q j − 2 ψ 'j q j (42) q + q − q + j j j j j j 2 4 4 l (t ) l (t ) l (t ) l (t )
For the part ρ vu x , applying Eq. (39) yields l (t )
1
0
0
∫
ψ i ρ vu x dx = ρ v ∫ ψ iψ 'j dξ q j
(43)
For the part 2 ρ vu xt , applying Eq. (40) yields l (t )
2 ρ v ∫ ψ i u xt dx 0
1 v 1 vξ ⎡ 1 ⎤ ψ iψ 'j q j dξ − ∫ ψ iψ ''j q j dξ ⎥ = 2 ρ v ⎢ ∫ ψ iψ 'j q j dξ − ∫ 0 l (t ) 0 l (t ) ⎣ 0 ⎦
1
1
0
0
= 2 ρ v[q j ∫ ψ iψ 'j d ξ − q j ∫
−
v ψ iψ 'j d ξ l (t )
1 1 v ' ' q j (ξψ iψ 'j 10 − ∫ ψ iψ 'j d ξ − ∫ ξψ iψ j d ξ )] 0 0 l (t )
1
= 2 ρ v ∫ ψ iψ 'j dξ q j + 0
2ρ v2 l (t )
1
∫ ξψ ψ 0
' i
' j
dξ q j
(44)
For the part ρ utt , applying Eq. (42) yields
∫
l (t )
0
ψ i ρ utt dx
2v 2 1 = ρ l (t ) ∫ ψ iψ j dξ q j − 2 ρ v ∫ ξψ iψ dξ q j − ρ (v − ) ∫ ξψ iψ 'j dξ q j 0 0 l (t ) 0 1
1
' j
30
+
ρ v2
1
ξ ψψ l (t ) ∫ 2
0
i
'' j
dξ q j
1
1
0
0
= ρ l (t ) ∫ ψ iψ j dξ q j − 2 ρ v ∫ ξψ iψ 'j dξ q j −
ρ v2
1
ξ ψψ l (t ) ∫ 0
2
i
' j
dξ q j
1
− ρ v ∫ ξψ iψ 'j dξ q j
(45)
0
Reorganizing the coefficients of q j (t ) , q j (t ) , q j (t ) , putting in matrix form yields the discretized equations for the elevator cable: M *q(t ) + C *q(t ) + K *q(t ) = F * (t ) where q = [ q1 , q2 ,
(46)
, qn ]T is the vector of generalized coordinates of the cable and the
entries of the matrices and the force vector are mij* =
1 ρ l (t )δ ij 2
(47)
1
cij* = 2 ρ v ∫ (1 − ξ )ψ iψ 'j dξ + ccψ i (1)ψ j (1)
(48)
0
⎡ 1 ⎤ EA 1 ' ' v2 1 ' ' (1 − ξ ) 2ψ iψ kij* = ρ ⎢v ∫ (1 − ξ )ψ iψ 'j d ξ − j dξ ⎥ + ∫ ∫ ψ iψ j dξ l (t ) 0 ⎣ 0 ⎦ l (t ) 0 + kcψ i (1)ψ j (1)
(49)
1
fi * = l (t ) ∫ ρ ( g − v)ψ i dξ + (ccuc + kc uc )ψ i (1)
(50)
0
From the system viewpoint, the differential equation of the car should be considered as well. Since the elevator cable and the attached car are coupled, q j (t ) ( j = 1, 2,3,… , n) 31
and uc (t ) are both introduced as the generalized coordinates. In this way, the differential equation of the car is treated as the ( n + 1)th discretized ordinary differential equation.
Combining the discretized equations of the cable and the differential equation of the car yields the complete set of discretized equations of motion for the moving elevator cablecar system which are 2nd order ordinary differential equations. Put the resulting equations of motion in matrix form yields M 1qs (t ) + C 1qs (t ) + K 1qs (t ) = F 1 (t )
where qs = [ q1 , q2 ,
(51)
, qn , uc (t )]T is the vector generalized coordinates of the moving
elevator cable-car system, M 1 , C 1 and K 1 are mass, damping, and stiffness matrices, respectively, which are all ( n + 1) × ( n + 1) matrices.
F 1 (t ) is (n + 1) × 1 column matrix
and the entries of the system matrices and the force vector are:
M
C
1 ( n +1)×( n +1)
1 ( n +1)×( n +1)
⎡ mij* n×n =⎢ ⎣ 0 ⎡ cij* =⎢ ⎣⎢ −ccψ j (1)
⎡ kij* K 1( n +1)×( n +1) = ⎢ ⎢⎣ − kcψ j (1)
F
1 ( n +1)×1
0⎤ ⎥ mc ⎦
1× n
(52)
ccψ i (1) cc kcψ i (1)
1×n
kc
n×1
⎤ ⎥ ⎦⎥
(53)
⎤ ⎥ ⎥⎦
(54)
n×1
⎡ ρ l (t )( g − v) ⎤ n×1 ⎥ = ⎢ (i − 1/ 2)π ⎢ ⎥ ⎢⎣ mc ( g − v) ⎥⎦
(55)
32
Theoretically, by substituting internal condition EAu x ( x, t ) = kc Δl + cc Δv into the governing equation, one can better depict the physical internal condition. disadvantage of this discretized scheme is also self-proven.
The
If the base function
sin(i − 1/ 2)πξ is chosen, the axial force of the lower end EAu x (l , t ) always equals to
zero, which is against Newton’s 3rd Law.
This phenomenon will be demonstrated
through the values of the axial force of the lower end of the cable and the spring force between the cable and the car in later numerical simulation. The discretized equations can also be obtained from Lagrange’s Equation by discretizing the Lagrangian. The results are given in Appendix A.
3.3
Modified Novel Discretization Scheme
From section 3.3, the selection of base function is essential to discretize the governing equation. A better base function could not only imply better approximation to the practical physical system but have better convergence. As shown in 3.3, if the base function sin(i − 1/ 2)πξ is chosen, the axial force of the lower end EAu x (l , t ) always equals to zero. To eliminate this disadvantage, this section will compensate the geometric boundary condition without changing any existing boundary conditions.
33
Introducing new variable ξ = x / l (t ) and τ = t , the original coordinate system ( x, t ) can be transformed into new coordinate system (ξ ,τ ) . The solution form of the governing equation of the elevator system can be assumed as: n
u ( x, t ) = uˆ (ξ ,τ ) = ∑ φ p q p + ϑ (ξ )e(τ )
(56)
p =1
Here ϑ (ξ ) must first satisfy ϑ (0) = 0 . Meanwhile, the following operations hold ∂ ∂ ∂τ ∂ ∂ξ ∂ ∂ v = + = − ξ ∂t ∂τ ∂t ∂ξ ∂t ∂τ l (t ) ∂ξ
(57)
∂ ∂ ∂τ ∂ ∂ξ 1 ∂ = + = ∂x ∂τ ∂x ∂ξ ∂x l (t ) ∂ξ
(58)
Similarly, ∂ ∂ ∂ D ∂ v = +v = + (1 − ξ ) ∂x ∂τ l (t ) ∂ξ Dt ∂t
(59)
∂2 1 ∂2 = ∂x 2 l 2 (t ) ∂ξ 2
(60)
2 D2 ∂2 v 2v 2 ∂ 2v ∂2 v2 2 ∂ = +( − )(1 − ξ ) + (1 − ξ ) + (1 − ξ ) Dt 2 ∂τ 2 l (t ) l 2 (t ) ∂ξ l (t ) ∂τ∂ξ l 2 (t ) ∂ξ 2
(61)
Substituting Eqs. (57) through (61) into Eqs. (32), (34), (35) yields:
34
∂ 2uˆ ∂uˆ 2v ∂ 2uˆ v 2v 2 v2 EA ∂ 2uˆ 2 ξ ξ ξ + − − + − + − − = g −v ( )(1 ) (1 ) [ (1 ) ] ρ l 2 (t ) ∂ξ 2 ∂τ 2 l (t ) l 2 (t ) ∂ξ l (t ) ∂τ∂ξ l 2 (t ) (62) uˆ (0,τ ) = 0
(63)
EA ∂uˆ ∂uˆ v ∂uˆ (1,τ ) = kc [uc − uˆ (1,τ )] + cc [uc − (1,τ ) − (1 − ξ ) (1,τ )] ξ =1 l ∂ξ ∂τ l ∂ξ = kc [uc − uˆ (1,τ )] + cc [uc −
∂uˆ (1,τ )] ∂τ
(64)
Multiplying Eq. (62) by φk and integrating from 0 to 1 yields
∂2 ∂τ 2
∫
1
0
v l
ˆ ξ +( − φk ud
∂uˆ 2v 2 1 2v ∂ dξ + ) ∫ (1 − ξ )φk 2 0 l ∂ξ l ∂τ
1 EA v 2 ∂ 2uˆ 2 − ∫ [ 2 − 2 (1 − ξ ) ]φk 2 dξ = ( g − v) ∫ φk d ξ 0 ρl 0 l ∂ξ 1
1
∫ (1 − ξ )φ
k
0
∂uˆ dξ ∂ξ
(65)
Note,
1
−∫ [ 0
EA v 2 ∂ 2uˆ 2 ξ φ − − dξ (1 ) ] k ρl 2 l 2 ∂ξ 2
EA v 2 ∂uˆ = −[ 2 − 2 (1 − ξ )2 ]φk ρl l ∂ξ
=−
1 0
ˆ EA v 2 2v 2 2 ' ∂u + ∫ [ 2 − 2 (1 − ξ ) ]φk dξ + 2 0 ρl l ∂ξ l 1
1 EA ˆ EA ∂uˆ v2 2v 2 2 ' ∂u φ τ ξ φ ξ + − − d + (1) (1, ) [ (1 ) ] k ∫0 ρl 2 l 2 ρ l 2 k ∂ξ ∂ξ l2
35
∫
1
0
1
∫ (1 − ξ )φ
k
0
(1 − ξ )φk
∂uˆ dξ ∂ξ
∂uˆ dξ ∂ξ
(66)
Substituting Eq. (66) into Eq. (65) yields
∂2 ∂τ 2
v
1
1
ˆ ξ + ∫ (1 − ξ )φ ∫ φ ud l k
0
k
0
∂uˆ 2v ∂ dξ + ∂ξ l ∂τ
1
∫ (1 − ξ )φ
k
0
∂uˆ dξ ∂ξ
1 EA v 2 ∂uˆ EA ∂uˆ + ∫ [ 2 − 2 (1 − ξ )2 ]φk' dξ − 2 φk (1) (1,τ ) = ( g − v) ∫ φk dξ 0 ρl 0 ρl l ∂ξ ∂ξ 1
(67)
Substituting Eq. (64) into Eq. (67) yields
∂2 ∂τ 2
v
1
1
ˆ ξ + ∫ (1 − ξ )φ ∫ φ ud l k
0
1
+∫ [ 0
k
0
∂uˆ 2v ∂ dξ + ∂ξ l ∂τ
1
∫ (1 − ξ )φ
k
0
∂uˆ dξ ∂ξ
1 c ∂uˆ k ∂uˆ EA v 2 − 2 (1 − ξ )2 ]φk' dξ + φk (1){ c [ (1,τ ) − uc ] + c [uˆ (1,τ ) − uc ]} = ( g − v) ∫ φk dξ 2 0 ρl l ρ l ∂ξ ρl ∂ξ
(68) Substituting Eq. (56) into Eq. (68) yields
1
2v
1
1
∑ ∫ φ φ dξ q + ∫ ϑφ dξ e + l ∫ (1 − ξ )φ ∑ φ dξ q p
0
k
p
+{∑ ∫ [ 1
p
0
l
0
k
0
k
p
p
p
+
2v 1 (1 − ξ )ϑ (ξ )φk d ξ e l ∫0
1 EA v 2 v − 2 (1 − ξ ) 2 ]φk' φ p' d ξ + ∑ ∫ (1 − ξ )φkφ p' dξ }q p 2 0 ρl l l p
EA v 2 v 1 +{∫ [ 2 − 2 (1 − ξ )2 ]ϑ ′(ξ )φk′ dξ + ∫ (1 − ξ )ϑ ′(ξ )φk dξ }e 0 ρl l l 0 1
+φk (1){
1 cc ∂uˆ k [ (1,τ ) − uc ] + c [uˆ (1,τ ) − uc ]} = ( g − v) ∫ φk d ξ 0 ρ l ∂ξ ρl
36
(69)
Rearranging Eq. (69) yields n
2v
n
∑ ∫ φ φ dξ q + ∫ ϑ (ξ )φ dξ e + l ∑ ∫ (1 − ξ )φ φ dξ q 1
1
k
0
p =1
p
p
0
k
p =1
1
k
0
p
p
+
2v 1 (1 − ξ )ϑ (ξ )φk dξ e l ∫0
EA v 2 v 1 2 ' ' + ∑ {[ ∫ − (1 − ξ ) φ φ d ξ + (1 − ξ )φkφ p' d ξ ]d ξ }q p k p 2 ∫ 0 ρl 2 0 l l p =1 n
1
1
+{∫ [ 0
EA v 2 v 1 − 2 (1 − ξ )2 ]ϑ ′(ξ )φk′ dξ + ∫ (1 − ξ )ϑ ′(ξ )φk dξ }e 2 ρl l l 0
+φk (1){
1 cc ∂uˆ k [ (1,τ ) − uc ] + c [uˆ (1,τ ) − uc ]} = ( g − v) ∫ φk d ξ 0 ρ l ∂ξ ρl
(70)
Following the same procedure, multiplying the governing equation by ϑ (ξ ) yields:
∂2 ∂τ 2
∫
1
0
ˆ ξ+ ϑ (ξ )ud
1
+∫ [ 0
2v ∂ l ∂τ
∫
1
0
(1 − ξ )ϑ (ξ )
∂uˆ ∂uˆ v 1 dξ + ∫ (1 − ξ )ϑ (ξ ) dξ 0 ∂ξ ∂ξ l
1 ∂uˆ ∂uˆ EA v 2 EA − 2 (1 − ξ )2 ]ϑ ′(ξ ) dξ − 2 ϑ (1) (1,τ ) = ( g − v) ∫ ϑ (ξ )d ξ 2 0 ρl l ρl ∂ξ ∂ξ
Substituting Eqs. (56), (64) into Eq. (71) yields
∫
1
0
1
ϑ (ξ )φ p d ξ q p + ∫ ϑ 2 (ξ )dξ e + [
+[
0
c 2v 1 (1 − ξ )ϑ (ξ )φk′ d ξ + c ϑ (1)φ p (1)]q p ∫ 0 ρl l
c c k 2v 1 (1 − ξ )ϑ (ξ )ϑ ′(ξ )d ξ + c ϑ (1)ϑ ′(1)]e − c uc − c uc ∫ ρl ρl ρl l 0
1
+{∫ [ 0
k EA v 2 v 1 − 2 (1 − ξ ) 2 ]ϑ ′(ξ )φ p′ dξ + ∫ (1 − ξ )ϑ (ξ )φ p′ dξ + c ϑ (1)φ p (1)}q p 2 ρl l ρl l 0 37
(71)
1
+{∫ [ 0
k EA v 2 v 1 − 2 (1 − ξ )2 ]ϑ ′(ξ )ϑ ′(ξ )d ξ + ∫ (1 − ξ )ϑ (ξ )ϑ ′(ξ )dξ + c ϑ (1)ϑ (1)}e 2 ρl l ρl l 0 1
= ( g − v) ∫ ϑ (ξ )dξ
(72)
0
Substituting Eq. (56) into the governing equation of the car yields n
n
p =1
p =1
mc uc + cc [uc − ∑ φ p q p − ϑ (1)e] + kc [uc − ∑ φ p q p − ϑ (1)e] = mc ( g − v)
3.3.1
(73)
Method 2
Supposing the displacement at x = l (t ) is e(t ) yields u (l , t ) = u (1,τ ) = e(t )
(74)
Selecting a set of base function φk (ξ ) = sin kπξ , which is the eignefunctions of a fixedfixed rod, such that φk (0) = φk (1) = 0 and expanding u ( x, t ) as n
u ( x, t ) = u (ξ ,τ ) = ∑ φk (ξ )qk (τ ) + ξ e(τ )
(75)
k =1
n
Note this is a special case of u ( x, t ) = uˆ (ξ ,τ ) = ∑ φ p q p + ϑ (ξ )e(τ ) by setting ϑ (ξ ) = ξ . p =1
Instead of discretizing the governing equations term by term as shown in section 3.3, Lagrange’s Equation is applied to obtain the discretized equations. The total kinetic energy from Eq. (12) and Eq. (14) becomes 38
ρl
T=
2
ρl
=
2
1
∂u
∂u
v
1
∫ [ ∂τ + l (1 − ξ ) ∂ξ + v] dξ + 2 m (u 2
c
0
1
∫ [φ q 0
k
k
c
+ v)2
v v 1 + ξ e + (1 − ξ )φk' qk + (1 − ξ )e + v]2 d ξ + mc (uc + v ) 2 l l 2
(76)
Taking partial derivative of three independent variables qk , e , uc yields 1 1 1 ∂T = ρ l ∫ φkφh d ξ qh + ρ l ∫ ξφk d ξ e + ρ v ∫ (1 − ξ )φkφh' d ξ qh 0 0 0 ∂qk
1
1
0
0
+ ρ v ∫ (1 − ξ )φk dξ e + ρ l ∫ φk dξ v
(77)
1 1 ∂T l v vl = ρ l ∫ ξφk d ξ qk + e + ρ v ∫ ξ (1 − ξ )φk' d ξ qk + e + 0 0 ∂e 3 6 2
(78)
∂T = mc uc + mc v ∂uc
(79)
1 1 ∂T ρ v2 = ρ v ∫ (1 − ξ )φk' φh dξ qh + ρ v ∫ ξ (1 − ξ )φk' dξ e + 0 0 ∂qk l
+
ρv2 l
∫
1
0
1
∫ (1 − ξ ) φ φ dξ q 2
0
1
(1 − ξ ) 2 φk' d ξ e + ρ v 2 ∫ (1 − ξ )φk' d ξ
∫
1
0
(1 − ξ ) 2 φk' dξ qk +
ρv2 3l
e+
The total potential energy from Eq. (13) and Eq. (15) becomes 1 EA 1 ∂u 2 1 ( ) dξ + k (uc − e) 2 − ρ gl ∫ (u + lξ )dξ − mc g (uc + l ) ∫ 0 2l 0 ∂ξ 2
39
l
(80)
0
1 ∂T ρv ρ v2 = ρ v ∫ (1 − ξ )φk dξ qk + e+ 0 ∂e 6 l
V=
' ' k h
ρ v2 2
(81)
=
1 EA 1 ' 1 (φk qk + e) 2 d ξ + k (uc − e) 2 − ρ gl ∫ (φk qk + eξ + lξ ) d ξ − mc g (uc + l ) ∫ 0 2l 0 2
(82)
Taking partial derivative of potential energy V yields 1 ∂V EA 1 ' ' EA 1 ' = d q + d e − gl φ φ ξ φ ξ ρ k h h k ∫0 φk dξ ∂qk l ∫0 l ∫0
=
1 EA 1 ' ' d q gl φ φ ξ − ρ k h h ∫0 φk dξ l ∫0
(83)
∂V EA ρ gl =( + k )e − kuc − ∂e l 2
(84)
∂V = kc uc − kc e − mc g ∂uc
(85)
Substituting Eqs. (77)-(85) into Lagrange Equation with non-conservative force d ∂T ∂T ∂V − + = Qi dt ∂qi ∂qi ∂qi
(86)
where Qi is non-conservative generalized force. Since virtual work done by impressed force equals virtual work done by generalized force, Qi can be achieved by
δ W = −cc (uc − e)[δ uc − δ u (l , t )]
= −cc (uc − e)(δ uc − δ e) =
n+2
∑ Qq
i = n +1
i i
= Qe e + Quc uc
Rearranging Lagrange Equation yields 40
(87)
1
1
1
1
0
0
0
0
ρ l ∫ φiφ j dξ q j + ρ l ∫ ξφi dξ e + ρ v{∫ φiφ j dξ + ∫ (1 − ξ )(φiφ 'j − φi'φ j )dξ }q j 1
+2 ρ v ∫ (1 − ξ )φi dξ e + ρ[ 0
+ ρ (v −
1 2v 2 1 ) ∫ (1 − ξ )φi dξ = ρ l ( g − v) ∫ φi dξ 0 l 0
l 3
1
1 EA 1 ' ' v2 ' φ φ ξ + (1 − ξ ) φ φ ξ − d v d i j ∫0 ρ l ∫0 i j l
1
ρv
0
3
ρ l ∫ ξφi d ξ q j + e − 2 ρ v ∫ (1 − 2ξ )φi d ξ q j + ( 0
1
− ρ[v ∫ (1 − 2ξ )φi dξ + 0
2v 2 l
∫
1
0
(1 − ξ )φi dξ ] qi + (
1
∫ (1 − ξ ) φ φ dξ ] q 0
2
' ' i j
j
(88)
+ cc )e − cc uc
EA ρ vl − 2v 2 ρl + kc + ) e − k c uc = ( g − v ) l 6l 2 (89)
mcuc − cc e + ccuc − kc e + kuc = m( g − v)
(90)
Note Eqs. (88), (89), (90) can also be obtained by setting ϑ (ξ ) = ξ in Eqs. (70), (72) and (73) and by applying boundary conditions. The discretized equations of motion in matrix form is [ M ]x + [C ]x + [ K ]x = f
(91)
where x = ( q e uc ) ; T
(
f = fq
fe
f uc
41
)
T
(92)
⎛ W q W qe W quc ⎞ ⎜ ⎟ [W ] = ⎜ W eq W e W euc ⎟ (W = M , C , K ) ⎜ W uc q W uc e W uc ⎟ ⎝ ⎠
(93)
In which 1
M ijq = ρ l ∫ φiφ j dξ 0
Me =
1
,
M ieq = M iqe = ρ l ∫ ξφi dξ 0
ρl 3 ,
M uc = mc , 1
Ciqe = 2 ρ v ∫ (1 − ξ )φi dξ 0
Ciquc = Ciuc q = 0
K ijq =
K
qe i
, M iuc q = M iquc = M iuc e = M ieuc = 0 ,
1
1
0
0
Cijq = ρ v[ ∫ φiφ j dξ + ∫ (1 − ξ )(φiφ 'j − φi'φ j )dξ ] , 1
,
Cieq = −2 ρ v ∫ (1 − 2ξ )φi dξ 0
C euc = C uc e = −cc
,
2v 2 1 ) (1 − ξ )φi dξ = ρ (v − l ∫0 ,
1
K ieq = − ρ v ∫ (1 − 2ξ )φi dξ − 0
K euc = K uc e = − kc 1
,
fi q = ρ l ( g − v) ∫ φi dξ , 0
1
ρv 3
+ cc ,
,
∫ (1 − ξ ) φ φ dξ , 0
2
' ' i j
K iquc = K iuc q = 0
2ρv2 l
K uc = kc
Ce =
C uc = cc
,
1 EA 1 ρ v2 ' d v d φ φ ξ + ρ (1 − ξ ) φ φ ξ − i j i j ∫0 l ∫0 l
,
∫
1
0
(1 − ξ )φi dξ
,
Ke =
EA ρ vl − 2v 2 + kc + l 6l ,
, fi e =
ρ l ( g − v) 2
,
42
f uc = mc ( g − v )
(94)
3.3.2 Method 3 As mentioned in section 3.3, all trial functions have the property that ψ k' (ξ ) = 0 , such that
∂u (1, t ) = 0 , which doesn’t comply to the physical model. ∂ξ
To improve the
discretization scheme in 3.3, suppose
n
u (ξ , t ) = ∑ψ k (ξ ) qk (t ) + θ (ξ ) w(t )
(95)
k =1
Where the base function ψ k (ξ ) is the k th mode shape of a fixed-free rod and w(t ) is the slope, i.e. w(t ) =
∂u (1, t ) . ∂ξ
Meanwhile, the function
θ (ξ ) = ξ 3 − ξ 2
(96)
is chosen to satisfy θ (0) = θ (1) = θ ' (0) = 0 and θ ' (1) = 1 .
The discretization process is similar to that in 3.4.1, which is given in Appendix B. The numerical simulation of this method is also achieved in latter section. Note the discretized equation can also be obtained by setting ϑ (ξ ) = ξ 3 − ξ 2 and
φk (ξ ) = ψ k (ξ ) in Eqs. (70), (72) and (73) and applying boundary conditions.
43
Chapter 4: Energy Consideration
Rate of change of energy of this system can be attacked from system and control volume viewpoints, which are interrelated on physical meanings [9, 37]. Here we employ both control volume viewpoint and system viewpoint to demonstrate the results analytically.
4.1
Control Volume Viewpoint
The energy density (energy per unit length) associated with the longitudinal vibration of the translating cable, at a fixed spatial position x at time t , is
ε ( x, t ) =
1 1 ρ (ut + vu x + v) 2 + EAu x2 ( x, t ) − ρ g[u ( x, t ) + x] 2 2
A control volume at time t is defined as the spatial domain
(97)
{x x ∈ [0, l (t )]} .
The
vibratory energy of the string within the control volume is
Ecv (t ) = ∫
l (t )
0
ε ( x, t )dx + Ec (t )
(98)
where Ec (t ) =
1 1 2 mc [uc (t ) + v]2 + kc [uc (t ) − u (l (t ), t ) ] − mc g[uc (t ) + l (t )] 2 2
(99)
is the kinetic energy and potential energy relating to the car attached to the lower end of the moving cable.
44
Substituting Eqs. (97), (99) into Eq. (98) yields
Ecv (t ) =
l (t ) 1 l (t ) Du ( x, t ) 1 l (t ) ρ∫ ( + v)2 dx + ∫ EAu x2 ( x, t )dx − ∫ ρ gu ( x, t )dx 0 2 0 Dt 2 0
1 1 1 2 + mc [uc (t ) + v]2 + kc [uc (t ) − u (l (t ), t ) ] − mc g[uc (t ) + l (t )] − ρ gl 2 (t ) 2 2 2
(100)
Differentiating Eq. (100) yields l (t ) ⎧ dEcv (t ) ∂ ⎫ = ∫ ⎨ ρ (ut + vu x + v)[ (ut + vu x ) + v] + EAu x u xt − ρ gut ⎬ dx + BOX 0 dt ∂t ⎩ ⎭
(101)
where BOX is defined by BOX =
1 Du v[ ρ ( + v) 2 + EAu x2 − 2 ρ gu ] 2 Dt + kc [uc − u (l , t )][uc −
x =l ( t )
+ mc (uc + v)(uc + v)
Du (l , t ) ] − mc g (uc + v) − ρ gvl (t ) Dt
(102)
Expanding Eq. (101) yields l (t ) dEcv (t ) = ∫ { ρ (ut + vu x + v)[utt + vu x + vu xt + v] + EAu x u xt − ρ gut } dx + BOX 0 dt
=∫
l (t )
0
[ ρ (ut utt + vut u x + vut u xt + vut + vu xutt + vvu x2 + v 2u xu xt + vvu x + vutt + vvu x
+ v 2u xt + vv ) + EAu x u xt − ρ gut ]dx + BOX
=∫
l (t )
0
[ ρ (ut utt + ut vu x + ut vu xt − ut v 2u xx ) − ut EAu xx + ut ρ v − ut ρ g
+ ρ (vu x utt + vvu x2 + 2vvu x + vutt + v 2 u xt + vv )]dx + ( EA + ρ v 2 )u x ut
Applying governing equation of the cable in Eq. (103) yields 45
l (t ) 0
+ BOX
(103)
dEcv (t ) dt =∫
l (t )
0
ρ (−vut u xt − 2v 2ut u xx + vu x utt + vvu x2 + 2vvu x + vutt + v 2u xt − cut (
ρ vvl (t ) + ( EA + ρ v 2 )u x ut 1 = − ρ vut2 2 +∫
l (t )
0
l (t )
0
=∫
l (t )
0
+ BOX
+ ρ vvl (t ) + ( EA + ρ v 2 )u x ut
l (t ) 0
+ BOX
[ ρ (vu x utt − 2v 2ut u xx + vvu x2 + vvu x ) + ρ (vvu x + vutt + v 2u xt ) − cut (
1 = − ρ vut2 2 +∫
l (t ) 0
l (t ) 0
Du + v))dx Dt
l (t ) 0
+ ρ vvl (t ) + ( EA − ρ v 2 )u x ut
l (t ) 0
Du + v)]dx Dt
+ BOX
[ ρ vu x (utt + 2vu xt + vu x + v) + ρ v(utt + vu x + vu xt ) − cut (
[ ρ vu x (utt + 2vu xt + vu x + v) + ρ v(vu x + utt + vu xt ) − cut (
Du + v)]dx Dt
Du + v)]dx + XBOX Dt
(104)
where XBOX is defined by 1 XBOX = − ρ vut2 2
l (t ) 0
+ ρ vvl (t ) + ( EA − ρ v 2 )u x ut
l (t ) 0
+ BOX
(105)
Applying governing equation of the cable again in Eq. (104) yields l (t ) dEcv (t ) Du = ∫ [vu x ( EAu xx + ρ g − ρ v 2u xx − c( + v)) 0 dt Dt
+ v( EAu xx + ρ ( g − v ) − ρ vu xt − ρ v 2u xx − c( 1 = XBOX + ( EAv − ρ v 3 )u x2 2 − ρ v 2ut
l (t ) 0
l (t ) 0
+ ρ v( g − v )l (t ) − ∫
l (t )
0
+ ρ gvu
c(
Du Du + v)) − cut ( + v)]dx + XBOX Dt Dt
l (t ) 0
+ ( EAv − ρ v 3 )u x
Du + v) 2 dx Dt
46
l (t ) 0
(106)
Substituting Eq. (102) and Eq. (105) in Eq. (106) yields dEcv (t ) 1 = − ρ vut2 dt 2
l (t ) 0
+ ρ vvl (t ) + ( EA − ρ v 2 )u x ut
1 Du + v[ ρ ( + v) 2 + EAu x2 − 2 ρ gu ] 2 Dt + kc [uc − u (l , t )][uc − + ρ gvu
1 = − ρ vut2 2 − ρ v 2ut −∫
l (t )
0
c(
l (t ) 0
l (t ) 0
+ mc (uc + v)(uc + v)
Du (l , t ) 1 ] − mc g (uc + v) − ρ gvl (t ) + ( EAv − ρ v 3 )u x2 Dt 2
+ ( EAv − ρ v 3 )u x
l (t ) 0
x =l ( t )
l (t ) 0
l (t ) 0
− ρ v 2ut
+ ( EA − ρ v 2 )u x ut
l (t ) 0
l (t ) 0
+ ρ v( g − v)l (t ) − ∫
l (t )
0
1 + ( EAv − ρ v 3 )u x2 2
l (t ) 0
c(
l (t ) 0
Du + v) 2 dx Dt
+ ( EAv − ρ v3 )u x
1 + v[ ρ (ut2 + v 2u x2 + v 2 + 2vu x ut + 2vut + 2v 2u x ) + EAu x2 ] 2
Du + v) 2 dx + Ec (t ) Dt
l (t ) 0
x =l ( t )
(107)
where Ec (t ) is defined by Ec (t ) = mc (uc + v)(uc + v) + kc [uc − u (l , t )][uc −
Du (l , t ) ] − mc g (uc + v) Dt
(108)
Eliminating similar terms in Eq. (107) yields dEcv (t ) = EAu x ut dt + EAvu x
x =l ( t )
l (t ) 0
+ EAvu x2
+ ρ v 3u x
x =0
x =l ( t )
+
1 − ( EAv − ρ v 3 )u x2 2
1 3 ρ v + Ec (t ) 2
1 = {− EAu x (v + vu x ) + v[ EAu x2 + ρ (v + vu x ) 2 ]} 2 + EAu x (v + ut + vu x )
x =l ( t )
x =0
−∫
l (t )
0
c(
x =0
Du + v) 2 dx + Ec (t ) Dt
47
(109)
Applying governing equation of the car and using internal conditions in Eq. (108) yields Ec (t ) = {mc g − kc [uc − u (l , t )] − cc [uc − +{EAu x (l , t ) − cc [uc −
Du (l , t ) ] + f (t )}(uc + v) Dt
Du (l , t ) Du (l , t ) ] + f (t )}[uc − ] − mc g (uc + v) Dt Dt
= [mc g − EAu x (l , t )](uc + v) +{EAu x (l , t ) − cc [uc −
Du (l , t ) Du (l , t ) ] + f (t )}[uc − ] − mc g (uc + v) Dt Dt
= [mc g − EAu x (l , t )]uc + [mc g − EAu x (l , t )]v +{EAu x (l , t ) − cc [uc −
Du (l , t ) Du (l , t ) Du (l , t ) ]}uc + {EAu x (l , t ) − cc [uc − ]}[ − ] Dt Dt Dt
− mc g (uc + v) + f (t )[uc − = mc guc − cc [uc − + cc [uc − = −cc [uc −
Du (l , t ) ] Dt
Du (l , t ) Du (l , t ) ]uc + mc gv − EAu x (l , t )v − EAu x (l , t ) Dt Dt
Du (l , t ) Du (l , t ) Du (l , t ) ] − mc g (uc + v) + f (t )[uc − ] Dt Dt Dt
Du (l , t ) 2 Du (l , t ) Du (l , t ) ] − EAu x (l , t )[v + ] + f (t )[uc − ] Dt Dt Dt
(110)
Substituting Eq. (110) into Eq. (109) yields dEcv (t ) 1 = {− EAu x (v + vu x ) + v[ EAu x2 + ρ (v + vu x ) 2 ]} dt 2 + f (t )[uc −
l (t ) Du (l , t ) Du ] − ∫ c( + v) 2 dx 0 Dt Dt
48
x =0
− cc [uc (t ) −
Du (l , t ) 2 ] Dt
(111)
Eq. (111) establishes the rate of change of total energy, the first term on the right-hand side of which represents the rate of work done by the longitudinal tension and energy flux due to mass transfer across the boundary at x = 0 . The rest terms on the right-hand side of Eq. (111) represents the rate of work done by the impressed impact, which include the damping effect c , cc and the applied force f (t ) .
4.2
System Viewpoint
A system is defined here as the collection of material particles occupying the spatial domain {x | x ∈ [0, l (t0 )]} at a time t0 . At time t the system occupies the spatial domain t
t
t0
t0
{x | x ∈ [ ∫ v(τ )dτ , l (t0 ) + ∫ v(τ ) dτ ]} and the vibratory energy of the system is l ( t0 ) +
t
∫t v (τ ) dτ
Es (t ) = ∫ t ∫t v (τ ) dτ 0
ε ( x, t )dx + Ec (t )
(112)
0
where ε ( x, t ) and Ec (t ) are defined in Eq. (97) and Eq. (99), respectively. The time rate of change of Es (t ) is ⎤ dE (t ) dEs (t ) d ⎡ l (t ) + ∫t v (τ ) dτ ε ( x, t )dx ⎥ + c = ⎢∫ t dt dt ⎢⎣ ∫t v (τ ) dτ dt ⎥⎦ t
0
0
(113)
0
To move the time derivative inside the integral in Eq. (113), a moving coordinate x that moves with the same speed as the string and coincides with the coordinate x at time t0 is introduced: 49
t
x = x + ∫ v(τ )dτ ≡ χ ( x, t )
(114)
t0
Using Eq. (114) in Eq. (113) yields dEs (t ) d = dt dt
{∫
l ( t0 )
0
}
ε ( χ ( x, t ), t )dx +
dEc (t ) dt
(115)
Since the limits of the integral in Eq. (115) are fixed, the time derivative can be moved inside the integral in Eq. (115):
dEs (t ) l (t ) d dE (t ) =∫ ε ( χ ( x, t ), t )dx + c 0 dt dt dt 0
=∫
l ( t0 )
=∫
l ( t0 )
0
0
[ε x (χ ( x, t ), t ) χt + ε t ( χ ( x, t ), t )]dx + [
dEc (t ) dt
dE (t ) ∂ ∂ + v(t ) ]ε ( χ ( x, t ), t )dx + c ∂t ∂x dt
(116)
where Eq. (114) has been used. Using Eq. (114) in Eq. (116) yields t
l ( t ) + ∫ v (τ ) dτ ∂ dEs (t ) dE (t ) ∂ [ + v(t ) ]ε ( x, t )dx + c =∫ t t dt dt ∂t ∂x ∫t v (τ ) dτ 0
0
(117)
0
The rate of change of the vibratory energy of the system at time t0 is obtained from Eq. (117)
dEs (t0 ) l (t ) ∂ dE (t ) ∂ = ∫ [ + v(t0 ) ]ε ( x, t0 )dx + c 0 0 ∂t ∂x dt dt 0
(118)
Substituting Eqs. (97), (99) into Eq. (118) and applying governing equations at t = t0 yields
50
dEs (t0 ) Du (l (t0 ), t0 ) 2 = − EAu x (0, t0 )[v(t0 )u x (0, t0 ) + v(t0 )] − cc [uc (t0 ) − ] dt Dt −∫
l ( t0 )
0
c[
Du ( x, t0 ) Du (l (t0 ), t0 ) + v(t0 )]dx + f (t0 )[uc (t0 ) − ] Dt Dt
(119)
Eq. (119) establishes the work-energy relation of the system along the longitudinal direction at time t0 . The first term on the right-hand side of Eq. (119), which is the product of the longitudinal tension and the longitudinal velocity of the cable at x = 0 at
t = t0 , representing the rate of work done by the longitudinal tension at x = 0 . The rest terms on the right-hand side of Eq. (119) represents the rate of work done by the external force. Meanwhile, evaluating Eq. (111) at t = t0 yields the rate of change of the total energy of the string within the control volume at time t0 . Comparing the resulting expression with Eq. (119) yields
dEcv (t0 ) dEs (t0 ) = + v(t0 )ε (0, t0 ) dt dt
(120)
The same form of equation is obtained in [37]. The first term on the right-hand side of Eq. (120) represents the rates of work done by nonconservative domain and boundary forces as explained above.
The second term on the right-hand side of Eq. (120)
represents the energy flux due to mass transfer across the boundary x = 0 . A detailed description of the energy flux in the translating cable is given in Section 4.3.
51
4.3
Energy Flux
The energy flux function for a translating cable is defined by [37] S ( x, t ) = − EAu x [
where
Du ( x, t ) + v(t )] + v(t )ε ( x, t ) , x ∈ (0, l (t )) Dt
(121)
D ∂ ∂ is the material derivative and ε ( x, t ) is the vibratory energy = + v(t ) Dt ∂t ∂x
density defined in Eq. (97). The energy flux function for a translating cable differs from that for a stationary string in that it is the sum of the transfer of the rate of work done by the longitudinal tension at a fixed spatial point x along the cable, − EAu x (
Du ( x, t ) + v) , Dt
and a transfer of the vibratory energy per unit time, vε ( x, t ) , that results from the translational motion of the cable. Substituting Eq. (97) into Eq. (121) yields S ( x, t ) = − EAu x [
Du Dt
+ v] +
1 2
v{ρ (ut + vu x + v ) 2 + EAu x2 − 2 ρ g[u ( x, t ) + x ]}
x ∈ (0, l (t ))
(122)
For instance, by use of ut (0, t ) = 0 and u (0, t ) = 0 the energy flux at x = 0 + is 1 S (0+ , t ) = {− EAu x (vu x + v ) + v[ ρ (ut + vu x + v ) 2 + EAu x2 ]} 2
x =0
(123)
Note that the reaction force at the boundary x = 0 produces an energy source at x = 0 , and S (0+ , t ) represents the energy flux from the source x = 0 to x > 0 . Similarly, the energy flux at x = l (t )− can be obtained due to the reaction force at the boundary.
52
Differentiating Eq. (97) with respect to x and Eq. (122) with respect to t, adding the two resulting expressions, and using the governing equation yields the differential relation between the energy flux function and the energy density: ∂ε ( x , t ) ∂S ( x , t ) + = 0, ∂t ∂x
x ∈ (0, l (t ))
(124)
which is of the same form as that for a translating string in Ref. JAM. Integrating Eq. (124) over x ∈ ( x1 , x2 ) , where x1 , x2 ∈ (0, l (t )) , yields −∫
x2 x1
∂ε ( x , t ) x dx = S ( x, t ) x2 1 ∂t
(125)
Eq. (124) is the differential form of energy conservation that describe the energy balance for a particular point and section of the translating string. When x1 or x2 is a function of time, an integral form can be derived. If x1 is fixed and x2 is a function of time, the order of integration and partial differentiation in Eq. (125) cannot be interchanged. Using Leibnitz’s rule yields
∫
x2 x1
dx d ∂ε ( x, t ) dx + 2 ε ( x2 , t ) = dt dt ∂t
(∫
x2 x1
ε ( x, t )dx
)
(126)
Substituting Eq. (126) into Eq. (125) yields the form of the integral relation between the energy flux function and the energy density: −
d dt
(∫
x2 x1
)
ε ( x, t )dx = S ( x, t ) x − x2 1
dx2 ε ( x2 , t ) dt
53
(127)
If the car system at the lower end of the cable is taken into consideration, setting x1 = 0 and x2 = l (t ) and using Eq. (110) yields dEcv (t ) d ⎛ l (t ) = ⎜ ∫ ε ( x, t )dx + Ec (t ) ⎞⎟ dt dt ⎝ 0 ⎠
= {− EAux (vux + v) + vε ( x, t )]} x=0 − cc [uc (t ) − Du (l , t ) ]2 Dt
= S (0+ , t ) − cc [uc (t ) −
Du (l , t ) 2 ] Dt
(128)
Eq. (128) states that the rate of change of the total energy within the control volume is equal to the energy flux into the control volume across the boundary x = 0 , which includes the energy flux due to mass transfer, v(t )ε (0, t ) =
1 Du + v) 2 + EAu x2 ] , and v[ ρ ( 2 Dt
the energy flux due to the rate of work done by the longitudinal tension at the boundary
x = 0 , − EAu x (vu x + v) , and the energy dissipation −cc [uc (t ) − of which build
4.4
Du (l , t ) 2 ] , The latter two Dt
dEs (t ) in Eq. (119). dt
Energy Discretization for Method 1
In this section, the damping effect and external force are not included for simplicity purpose. Rewriting Eq. (100) as Ecv (t ) =
l (t ) 1 l ( t ) Du ( x, t ) 1 l (t ) ρ∫ ( + v) 2 dx + ∫ EAu x2 ( x, t ) dx − ∫ ρ gu ( x, t ) dx 0 Dt 2 0 2 0
54
1 1 1 2 + mc [uc (t ) + v]2 + kc [uc (t ) − u (l (t ), t ) ] − mc g[uc (t ) + l (t )] − ρ gl 2 (t ) 2 2 2 = E1 (t ) + E2 (t ) + E3 (t ) + Ec (t ) −
1 ρ gl 2 (t ) 2
(129)
where E1 (t ) =
1 l ( t ) Du ( x, t ) ρ∫ ( + v) 2 dx 0 Dt 2
(130)
E2 (t ) =
1 l (t ) EAu x2 ( x, t )dx ∫ 0 2
(131)
E3 (t ) = − ∫
l (t )
0
ρ gu ( x, t )dx
(132)
Ec (t ) is given in Eq. (99) . Using Eq. (36) in the material derivative
Du yields Dt
Du ( x, t ) ∂u ( x, t ) ∂u ( x, t ) = +v Dt ∂t ∂x n
n
n
j =1
j =1
j =1
n
n
j =1
j =1
= ∑ψ j (ξ )q j (t ) − ∑ ξ ll −1ψ 'j (ξ )q j (t ) + ∑ ll −1ψ 'j (ξ )q j (t )
= ∑ q j (t )ψ j (ξ ) + ∑ ll −1 (1 − ξ )q j (t )ψ 'j (ξ )
(133)
For simplicity purpose, E1 (t ) , E2 (t ) , E3 (t ) , Ec (t ) will be discretized separately in the following context. Applying Eq. (133) in Eq. (130) yields
55
E1 (t ) =
1 l ( t ) Du ( x, t ) ρ∫ ( + v) 2 dx 0 Dt 2
=
n n n 1 l (t ) n ρ ∫ [∑ q jψ j + ∑ ll −1 (1 − ξ )q jψ 'j + l ] × [∑ qiψ i + ∑ ll −1 (1 − ξ )qiψ i' + l ]dx 2 0 j =1 j =1 i =1 i =1
=
n n n 1 1 n ρ ∫ [∑ lq jψ j + ∑ l (1 − ξ )q jψ 'j + ll ] × [∑ qiψ i + ∑ ll −1 (1 − ξ )qiψ i' + l ]dξ 2 0 j =1 j =1 i =1 i =1
=
n n n 1 1⎧ n n ρ ∫ ⎨∑∑ lq j qiψ jψ i + ∑∑ l (1 − ξ )q j qiψ jψ i' + ∑ llq jψ j 2 0 ⎩ j =1 i =1 j =1 i =1 j =1
n
n
n
n
n
+ ∑∑ l (1 − ξ )q j qiψ 'jψ i + ∑∑ l −1l 2 (1 − ξ )2 q j qiψ 'jψ i' + ∑ l 2 (1 − ξ )q jψ 'j j =1 i =1
j =1 i =1
j =1
n n ⎫ + ∑ llqiψ i + ∑ l 2 (1 − ξ )qiψ i' + ll 2 ⎬ d ξ i =1 i =1 ⎭
=
{
1 1 1 T T ρ [ q ] l ∫ ψ iψ j d ξ [ q ] + [ q ] 2l ∫ (1 − ξ )ψ 'jψ i d ξ [ q ] 0 0 2
+ [ q ] l 2l −1 ∫ (1 − ξ ) 2ψ 'jψ i' dξ [ q ] +2ll ∫ ψ i dξ [ q ] + 2l 2 ∫ (1 − ξ )ψ i' dξ [ q ] + ll 2 T
1
1
1
0
0
0
} (134)
Applying Eq. (133) in Eq. (131) yields E2 (t ) =
l ( t ) ∂u ( x, t ) 1 EA∫ [ ]2 dx 0 2 ∂x
=
n l (t ) n 1 EA∫ [∑ l −1ψ 'j q j ] × [∑ l −1ψ i' qi ]dx 0 2 j =1 i =1
=
n n 1 1 EA∫ l −1 ∑∑ q j qiψ 'jψ i' dξ 0 2 j =1 i =1
56
= [q]
T
1 1 EAl −1 ∫ ψ 'jψ i' d ξ [ q ] 0 2
(135)
Applying Eq. (133) in Eq. (132) yields
E3 (t ) = − ∫
l (t )
0
ρ gudx = − ρ gl ∫ ψ i dξ [ q ] 1
(136)
0
Applying Eq. (133) in Eq. (99) yields 1 1 2 mc [uc (t ) + v]2 + kc [uc (t ) − u (l (t ), t ) ] − mc g[uc (t ) + l (t )] 2 2
Ec (t ) =
=
n n 1 1 kc [uc (t ) − ∑ q jψ j (1)][uc (t ) − ∑ qiψ i (1)] + mc [uc (t ) + v]2 − mc g[uc (t ) + l (t )] 2 2 j =1 i =1
=
1 mc [uc (t ) + v ]2 − mc g[uc (t ) + l (t )] 2
n n n n 1 + kc [uc2 (t ) − uc (t )∑ qiψ i (1) − uc (t )∑ q jψ j (1) + ∑∑ q j qiψ j (1)ψ i (1)] 2 i =1 j =1 j =1 i =1
1 1 mc [uc (t ) + l ]2 − mc g[uc (t ) + l (t )] + kc uc2 (t ) 2 2
=
+ [q]
T
1 kcψ j (1)ψ i (1) [ q ] − kc uc (t )ψ i (1) [ q ] 2
(137)
Substituting Eqs. (134)-(137) into Eq. (129) yields Ecv (t ) =
[q]
T
{
1 1 1 T T [ q ] ρ l ∫0 ψ iψ j dξ [ q ] + [ q ] 2 ρ l ∫0 (1 − ξ )ψ iψ 'j dξ [ q ] + 2
}
' ' 2 −1 2 ' ' [ EAl −1 ∫ ψ iψ j d ξ + ρ l l ∫ (1 − ξ ) ψ iψ j d ξ + kcψ i (1)ψ j (1)] [ q ] 1
1
0
0
+ ρ ll ∫ ψ i dξ [ q ] + [ ρ l 2 ∫ (1 − ξ )ψ i' dξ − ρ gl ∫ ψ i dξ − kc uc (t )ψ i (1)][ q ] 1
1
1
0
0
0
57
1 1 1 + ρ l (l 2 − gl ) + mc [uc (t ) + l ]2 − mc g[uc (t ) + l (t )] + kc uc2 (t ) 2 2 2 =
{
}
1 T T T [ q ] R [ q ] + [ q ] S [ q ] + [ q ] T [ q ] + P [ q ] + D [ q ] + W (t ) 2
(138)
where 1
R = ρ l ∫ ψ iψ j d ξ = 0
1 ρ lδ ij 2
(139)
1
S = 2 ρ l ∫ (1 − ξ )ψ iψ 'j dξ
(140)
0
T=
1 1 1 ' ' ' ' EAl −1 ∫ ψ iψ d ξ + ρ l 2l −1 ∫ (1 − ξ ) 2ψ iψ j j d ξ + k cψ i (1)ψ j (1) 0 0 2
1
P = ρ ll ∫ ψ i dξ
(141)
(142)
0
1
1
0
0
D = ρ l 2 ∫ (1 − ξ )ψ i' dξ − ρ gl ∫ ψ i dξ − kcuc (t )ψ i (1) 1
= ρ (l 2 − gl ) ∫ ψ i dξ − kc uc (t )ψ i (1)
(143)
0
W (t ) =
1 1 1 ρ l (l 2 − gl ) + mc [uc (t ) + l ]2 − mc g[uc (t ) + l (t )] + kcuc2 (t ) 2 2 2
(144)
From theoretical expression of rate of change of energy in control volume: Applying Eq. (133) to u xx ( x, t ) yields n n n n ∂ 2 u ( x, t ) ' ' −1 −1 −2 = [ l q ][ l q ] = l q j qiψ 'jψ i' ψ ψ ∑ ∑ ∑∑ j j i i 2 ∂x j =1 i =1 j =1 i =1
Without considering damping and external force, rewrite Eq. (111) as
58
(145)
dEcv (t ) 1 3 1 ⎤ ⎡ = ρ v + ⎡⎣ ρ v 2 − EA⎤⎦ v ⋅ ⎢u x + u x2 ⎥ dt 2 2 ⎦ ⎣
x =0
(146)
Applying Eq. (133), (145) in Eq. (146) yields dEcv (t ) 1 T ⎧ ⎫ 1 = ( ρ l 3 − EAl ) ⎨l −1ψ 'j (0) [ q ] + l −2 [ q ] ψ 'j (0)ψ i' (0) [ q ]⎬ + ρ v 3 dt 2 ⎩ ⎭ 2 = [ q ] Z1 [ q ] + Z 2 [ q ] + T
1 3 ρl 2
(147)
where Z1 =
1 ( ρ l 2 − EA)ll −2ψ 'j (0)ψ i' (0) 2
(148)
Z 2 = ( ρ l 2 − EA)ll −1ψ 'j (0)
4.5
(149)
Energy Discretization for Method 2
If the modified discretization scheme in section 3.4 is applied, the energy discretization can be investigated accordingly. The following expressions hold from the approximation in Eq. (75): Du v v (ξ , t ) + v = φk qk + ξ e + (1 − ξ )φk' qk + (1 − ξ )e + v Dt l l
59
(150)
∂u 1 ∂u = = φk' qk + e ∂x l ∂ξ
(151)
From Eq. (76), the kinetic energy is
ρl
T=
2
1
∫ [φ q 0
k
k
v v 1 + ξ e + (1 − ξ )φk' qk + (1 − ξ )e + v]2 d ξ + mc (uc + v ) 2 l l 2
(152)
Putting Eq. (152) in matrix form yields 1 T⎡ ˆ⎤ 1 v ⎣ M ⎦ v + mc (uc + v ) 2 2 2
T=
(153)
where
⎡ [ M qq ] ⎢ qe T ⎢ (m ) ⎡ ˆ ⎤ ⎢ qq T ⎣ M ⎦ = ⎢[ M ] qe T ⎢ (m ) ⎢ (m qv )T ⎣
⎛q⎞ ⎜ ⎟ ⎜e⎟ ν = ⎜q⎟ , ⎜ ⎟ ⎜e⎟ ⎜v⎟ ⎝ ⎠
m qe mee
[ M qq ] meq
m qe mee
m qe
[ M qq ]
m qe
ee
(m ) (m qv )T
ee
qe T
m mev
m qv ⎤ ⎥ mev ⎥ m qv ⎥ ⎥ mev ⎥ mvv ⎥⎦
m mev
(154)
In which 1
M ijqq = ρ l ∫ φiφ j dξ 0
1
,
1
miqe = ρ v ∫ (1 − ξ )φi d ξ , 0
1
mieq = ρ v ∫ ξ (1 − ξ )φi' dξ , 0
M ijqq =
ρv2 l
∫
1
0
1
miqe = ρ l ∫ ξφi dξ
(1 − ξ ) 2 φi'φ j' d ξ ,
M ijqq = ρ v ∫ (1 − ξ )φiφ 'j dξ
,
0
0
1
miqv = ρ l ∫ φi d ξ ,
mee =
0
m ee =
miqe =
ρv 6
ρv2 l
m ev =
,
∫
1
0
ρl 3
ρl 2
(1 − ξ ) 2 φi' d ξ ,
60
,
1
miqv = ρ v ∫ (1 − ξ )φi' dξ 0
m ee =
ρ v2 3l
,
m ev =
ρv 2
,
m vv = ρ l
(155)
From Eq. (82) the potential energy is V=
1 EA 1 ' 1 (φk qk + e) 2 d ξ + k (uc − e) 2 − ρ gl ∫ (φk qk + eξ + lξ ) d ξ − mc g (uc + l ) ∫ 0 0 2l 2
(156)
Putting Eq. (156) in matrix form yields V=
1 T ⎡ ˆ⎤ 1 e+l qˆ ⎣ K ⎦ qˆ + k (uc − e) 2 − ρ gl (k T q+ ) − mc g (uc + l ) 2 2 2
(157)
where ⎛q⎞ qˆ = ⎜ ⎟ , ⎝e⎠
⎛ [ K qq ] k qe ⎞ ⎡ Kˆ ⎤ = ⎜ qe ee ⎟ ⎣ ⎦ ⎝ (k ) k ⎠
(158)
In which K ijqq =
EA 1 ' ' φiφ j d ξ , l ∫0
kiqe =
1
EA 1 ' EA , φi d ξ = 0 , k ee = ∫ 0 l l
ki = ∫ φi dξ 0
(159)
Also, the rate of change of kinetic energy is dT 1 = vT ⎡⎣ Mˆ ⎤⎦ v+ vT ⎡ Mˆ ⎤ v+m(uc + v)(uc + v ) dt 2 ⎣⎢ ⎥⎦
(160)
where ⎛q⎞ ⎜ ⎟ ⎜e⎟ ν = ⎜q⎟ , ⎜ ⎟ ⎜e⎟ ⎜v⎟ ⎝ ⎠
⎡ [ M qq ] ⎢ qe T ⎢ (m ) ⎡ Mˆ ⎤ = ⎢[ M qq ]T ⎣⎢ ⎥⎦ ⎢ qe T ⎢ (m ) ⎢ (m qv )T ⎣
m qe mee
[ M qq ] meq
m qe mee
m qe mee mev
[ M qq ]
m qe mee mev
61
(m qe )T (m qv )T
m qv ⎤ ⎥ mev ⎥ m qv ⎥ ⎥ mev ⎥ mvv ⎥⎦
(161)
In which 1
1
M ijqq = ρ v ∫ φiφ j dξ
,
0
miqe = ρ v ∫ ξφi dξ 0
1
1
0
0
1
,
miqe = ρ v ∫ (1 − ξ )φi d ξ , miqv = ρ v ∫ φi d ξ ,
ρv
m = ee
m = ev
6 , ρv
m = qe i
l
(2v −
ρv
mee =
3l
v2 l
(2v −
ρv 2 ,
M
qq ij
2
ρv 3
,
1
mieq = ρ v ∫ ξ (1 − ξ )φi' d ξ 0
ρv
1
0
0
ρv
m ee =
miqv = ρ v ∫ (1 − ξ )φi' dξ
1
m ev =
0
v2 1 = (2v − ) ∫ (1 − ξ ) 2 φi'φ 'j d ξ l l 0
) ∫ (1 − ξ ) 2 φi' d ξ ,
v2 ), l
M ijqq = ρ v ∫ (1 − ξ )φiφ 'j dξ
,
mvv = ρ v
(162)
Similarly, the rate of change of potential energy is dV 1 = qˆ T ⎡⎣ Kˆ ⎤⎦ qˆ + qˆ T dt 2 − ρ gl (k T q+
⎡ Kˆ ⎤ qˆ + k (u − e)(u − e) − ρ gv(k T q+ e + l ) c c ⎢⎣ ⎥⎦ 2
e+v ) − mc g (uc + v) 2
(163)
where ⎛q⎞ qˆ = ⎜ ⎟ , ⎝e⎠
qq qe ⎡ Kˆ ⎤ = ⎛ [ K ] k ⎞ = − v ⎡ Kˆ ⎤ ⎢⎣ ⎥⎦ ⎜ (k qe ) k ee ⎟ l⎣ ⎦ ⎝ ⎠
(164)
From the control volume viewpoint in Eq. (111) without considering damping and external impact dEcv (t ) Du 1 Du = − EAu x (0, t )[ (0, t ) + v] + v{EAu x2 (0, t ) + ρ [ (0, t ) + v]2 } dt Dt Dt 2
62
(165)
Applying the approximation method of Eq. (75) in above Eq. (165) yields
u x (0, t ) =
1 ∂ 1 (φk qk + ξ e)(0,τ ) = [φk' (0)qk + e] l ∂ξ l
(166)
Du (0, t ) = ut (0, t ) + vu x (0, t ) Dt v = vu x (0, t ) = [φk' (0)qk + e] l
(167)
Substituting Eqs. (166), (167) into Eq. (165) yields dEcv (t ) 1 ρ v 2 − EA ' = v{EA + [φk (0)qk + e + l ]2 } 2 dt 2 l
(168)
The energy discretization for method 3 is very similar to that of method 2 and will be given in Appendix C.
63
Chapter 5: Numerical Simulation & Conclusions
Numerical analysis is the study of algorithms for the problems of continuous mathematics, which naturally finds applications in all fields of engineering and the physical sciences. After the advent of modern computers, numerical analysis is widely implemented in different disciplines such as computing values of functions; interpolation, extrapolation and regression;
solving equations and systems of equations, solving
eigenvalue or singular value problems, optimization, evaluating integrals, differential equations. Among software routines for numerical problems, MATLAB is a popular commercial programming language package for numerical scientific calculations. In this work, all numerical analysis is implemented with the aid of MATLAB. The organization of this chapter is as follows: in section 5.2, initial conditions are derived. In section 5.3, the definition of the equilibrium is discussed. In section 5.4, numerical simulation is demonstrated through an example of an elevator cable-car system in a high-rise building, both the responses of the cable-car system and the energy scenarios studied in details. Lastly, in section 5.5, conclusions are achieved and possible future work is also proposed.
64
5.1
Initial Conditions (from Static Equilibrium)
In practical engineering problems, arbitrary initial conditions could emerge, such as passengers walking into the cab, torque ripple.
In this section, for instance, general
initial conditions will be derived for method 2 from two different aspects. Considering the system starts to move from static equilibrium, the axial force at position
x at t = 0 is: EAu x ( x, 0) = mc g + ρ g (l − x) = ρ gl (mc / ρ l + 1 − ξ )
(169)
Integrating both sides of Eq. (169) yields
u (ξ , 0) =
ρ gl 2 EA
∫
ξ
0
(mc / ρ l + 1 − ξ )dξ
ρ gl 2 mc 1 [( + 1)ξ − ξ 2 ) = 2 EA ρ l
(170)
The displacement of the lower end of the cable can be achieved from Eq. (170):
e(0) = u (1, 0) =
ρ gl 2 mc 1 ( + ) EA ρ l 2
(171)
From Eq. (75), we also have
n
u (ξ , 0) = ∑ φk (ξ )qk (0) + ξ e(0)
(172)
k =1
Combining Eq. (171), (172) yields
65
ρ gl 2 mc 1 2 ρ gl 2 mc 1 sin kπξ qk (0) = [( + 1)ξ − ξ ) − ( + )ξ ∑ 2 EA ρ l EA ρ l 2 k =1 n
=
ρ gl 2 ξ − ξ 2 EA
(173)
2
Multiplying sin nπξ and integrating on both sides of Eq. (173) yields
qn (0) =
=
ρ gl 2 EA
ρ gl 2
1
∫ (ξ − ξ 0
2
EA (nπ )
3
2
) sin nπξ dξ
[1 − (−1)n ]
(174)
Apparently, uc (0) = e(0) + mc g / kc
(175)
Eqs. (171), (174), (175) are the initial conditions for method 2.
5.2
Equilibrium
In the longitudinal vibration of the elevator system, the equilibrium position of the cable can not be obtained as simple as that in conventional way because the translation and the vibration are coupled. In this section, the equilibrium position is defined as the deflection caused solely by the translating motion. Thus, the vibratory displacement can be refined as the displacement defined in chapter 2 minus the displacement resulted from the translation. 66
For the same infinitesimal segment in Figure 3 in Lagrangian reference frame ( Χ , t ) , only the deformation caused by the translation is considered instead of taking into account the vibration deformation. Applying Newton’s 2nd law yields:
ρΔXv = ρΔXg + T ( X , t ) + ΔT − T ( X , t )
(176)
where T ( X ) is the tension at position X .
Rearranging Eq. (176) and integrating yields T ( X , t ) = T0 − ρ ( g − v) X
(177)
where T0 is the tension of the cable before the translation. Obviously, at X = L (the original length of the cable), the tension is T ( L, t ) = mc ( g − v)
(178)
Substituting Eq. (178) into Eq. (177) yields T0 = (mc + ρ L)( g − v)
(179)
Consequently, T ( X , t ) = [mc + ρ ( L − X )]( g − v)
(180)
From mechanics of materials, we have U X ( X , t ) = T ( X , t ) / EA
(181)
67
Integrating Eq. (181) yields U ( X , t) =
g −v 1 ( mc X + ρ LX − ρ X 2 ) + U 0 EA 2
(182) t
Applying the boundary condition of the upper end U (− ∫ vdτ , t ) = 0 yields 0
U ( X , t) =
t t t g −v 1 {( mc ( X + ∫ vdτ ) + ρ L( X + ∫ vdτ ) − ρ [ X 2 − ( ∫ vdτ ) 2 )} (183) 0 0 0 EA 2
t
Transforming Eq. (183) into the Eulerian reference frame by using x = X + ∫ vdτ yields 0
u ( x, t ) =
t g −v 1 [mc x + ρ ( L + ∫ vdτ ) x − ρ x 2 ] 0 EA 2
(184)
Let u ( x, t ) = u ( x , t ) + u ( x , t )
(185)
Without considering the distributed damping, the governing equation of the cable and the boundary conditions obtained in chapter 2 can be rewritten as
ρ
D 2u D 2u EAu − = − ρ xx Dt 2 Dt 2
(186)
u (0, t ) = 0
(187)
Du (l , t ) ⎤ ⎡ EAu x (l , t ) − kc [uc (t ) − u (l , t ) ] − cc ⎢uc (t ) − Dt ⎥⎦ ⎣ = − mc ( g − v) − kc u (l , t ) − cc
Du (l , t ) Dt
(188)
Based on the new vibratory displacement defined above, the dynamic response from method 2 can be achieved similarly following the procedures in chapter 3. 68
5.3
Example and Discussion
A robust program (Matlab) follows the procedures described in Chapter 3 and gives the numerical response of the translating elevator cable-car system with an arbitrary moving profile, initial conditions. The typical parameters for a hoist cable in a high-rise, high-speed elevator are ρ = 0.6 kg/m, mc = 1000 kg and kc = 4 × 105 N/m.
The axial stiffness of the hoist cable is
EA = 6 × 106 N. Ignoring the bending stiffness of the cable and tension change due to
speed variation, the hoist cable is modeled as a vertically translating string with a massspring attached at its lower end, governed by Eq. (7). The damping and applied force are neglected; f (t ) = c = cc = 0 . The flight time for a travel distance of 150 m (50 stories) is 38 sec. The upward movement profile, as shown in Figure 7, is divided into seven regions. In the region i ( i = 1, 2,
,7 ), the function L(T ) is given by a polynomial,
L(T ) = L(0i ) + L1(i ) (T − T( i −1) ) + L(2i ) (T − T(i −1) ) 2 + L(3i ) (T − T(i −1) )3 + L(4i ) (T − T(i −1) ) 4 + L(5i ) (T − T(i −1) )5 , where Ti −1 ≤ T ≤ Ti and L(mi ) ( m = 0,1,
,
(189)
,5 ) are given in Table 1. The initial and final
lengths of the cable are 171 m and 21 m respectively.
The maximum velocity,
acceleration and jerk are 5 m/s, 0.75 m/s2 and 0.84 m/ s3, respectively. To improve the accuracy of the solution all the integrals in the discretized equations are evaluated analytically and the expressions are given in Appendix C for method 1. Unless stated otherwise, n = 25 . 69
For simplicity purpose, method 1 refers to the discretization scheme in 3.3, n
u ( x, t ) = ∑ψ j (ξ )q j (t ) j =1
n
Method 2 refers to the scheme in 3.4.1, u ( x, t ) = ∑ φk (ξ ) qk (τ ) + ξ e(τ ) k =1
n
Method 3 refers to that in 3.4.2, u (ξ , t ) = ∑ψ (ξ ) qk (t ) + ϑ (ξ ) w(t ) . k =1
Firstly, the results from method 1 and method 2 are compared. Secondly, method 2 and method 3 are compared to verify the correctness of both improved discretized schemes. Lastly, the dynamic response from the defined equilibrium position is demonstrated.
70
0
Velocity (m/s)
(a)
120 80 40 0
0.4
(b)
-2 -4
1.0 (c)
3
0.8
Jerk (m/s )
2
Acceleration (m/s )
Position (m)
160
0.0 -0.4 -0.8 0 5 10 15 20 25 30 35
(d)
0.5
3
5
2
0.0
6 4
-0.5 1 -1.0
7
0 5 10 15 20 25 30 35
t (s)
t (s)
Figure 5 The prescribed moving profile: (a) position l (t ) , (b) velocity v ( t ) , (c) acceleration v ( t ) , and (d) jerk v (t ) .
71
Table 1 The moving profile regions and polynomial coefficients
Region
tip
L(0i )
L(1i )
L(2i )
L(3i )
L(4i )
L(5i )
(sec)
(m)
(m/s)
(m/s2)
(m/s3)
(m/s4)
(m/s5)
1
1.33
171.0
0
0
0
-0.106
0.0316
2
6.67
170.8
-0.5
-0.375
0
0
0
3
8
157.5
-4.5
-0.375
0
0.106
-0.0316
4
30
151.0
-5
0
0
0
0
5
31.33
41.0
-5
0
0
0.106
-0.0316
6
36.67
34.5
-4.5
0.375
0
0
0
7
38
21.2
-0.5
0.375
0
-0.106
0.0316
i
72
5.3.1 Method 1 vs. Method 2 Theoretically, the dynamic responses from method 1 will eventually converge if the included modes are large enough, which, however, is not numerically efficient.
0.4 Method 1 Method 2
0.35
Disp. of the Car (m)
0.3
0.25
0.2
0.15
0.1
0.05
0
5
10
15
20 t (s)
73
25
30
35
0.35 Method 1 Method 2
0.34
Method 1 Method 2
0.26
0.33
0.24 Disp. of the Car (m)
Disp. of the Car (m)
0.32 0.31 0.3 0.29 0.28
0.22
0.2
0.18
0.27
0.16
0.26 1
2
3
4
5 t (s)
6
7
8
11
9
13
14
15 t (s)
16
17
18
0.1
Method 1 Method 2
0.19
12
19
Method 1 Method 2
0.095
0.18
0.09 Disp. of the Car (m)
Disp. of the Car (m)
0.17 0.16 0.15 0.14 0.13
0.085 0.08 0.075 0.07 0.065
0.12
0.06
0.11
0.055 0.05 21
22
23
24
25 t (s)
26
27
28
29
31
30
32
33
34 t (s)
Figure 6 Vibration of the car with details (M1, 2)
74
35
36
37
20
Disp. of the end of the cable at x=l(t) (m)
0.35 Method 1 Method 2
0.3
0.25
0.2
0.15
0.1
0.05
0
0
5
10
15
25
30
35
0.25
0.32
Method 1 Method 2
Method 1 Method 2
0.24
0.31
Disp. of the end of the cable at x=l(t) (m)
Disp. of the end of the cable at x=l(t) (m)
20 t (s)
0.3 0.29 0.28 0.27 0.26 0.25
0.23 0.22 0.21 0.2 0.19 0.18 0.17 0.16
0.24 0.15 0
1
2
3
4
5 t (s)
6
7
8
10
11
12
13
14
15 t (s)
16
17
18
0.08
Method 1 Method 2 Disp. of the end of the cable at x=l(t) (m)
0.15 Disp. of the end of the cable at x=l(t) (m)
9
0.14 0.13 0.12 0.11 0.1 0.09 0.08 0.07
19
Method 1 Method 2
0.07
0.06
0.05
0.04
0.03
0.06 21
22
23
24
25 t (s)
26
27
28
0.02 30
29
31
32
33
34 t (s)
35
Figure 7 Vibration of the lower end of the cable with details (M1, 2)
75
36
37
20
0.35 Method 1 Method 2
0.3
u(lt-10,t) (m)
0.25
0.2
0.15
0.1
0.05
0
0
5
10
15
0.3
20 t (s)
Method 1 Method 2
25
30
35
0.23
Method 1 Method 2
0.22
0.29
0.21
0.28
u(lt-10,t) (m)
u(lt-10,t) (m)
0.2
0.27 0.26 0.25
0.19 0.18 0.17 0.16
0.24
0.15
0.23 0.14
0.22
0.13
1
2
3
4
5 t (s)
6
7
8
9
11
Method 1 Method 2
0.14
12
13
14
15 t (s)
16
17
18
19
20
Method 1 Method 2
0.06 0.055
0.13
0.05 0.12 0.11
u(lt-10,t) (m)
u(lt-10,t) (m)
0.045
0.1 0.09
0.04 0.035 0.03 0.025
0.08
0.02
0.07
0.015 0.06 0.01 21
22
23
24
25 t (s)
26
27
28
29
30
31
32
33
34 t (s)
35
36
37
Figure 8 Vibration of physical particle at 10 m above the lower end with details (M1, 2)
76
38
4
1.15
x 10
Method 1 Method 2
Axial force T(l(t)-10,t) (N)
1.1
1.05
1
0.95
0.9
0.85
0
5
10
15
20 t (s)
25
30
35
10800 Method 1 Method 2
10600 10400
Spring force (N)
10200 10000 9800 9600 9400 9200 9000 8800
0
5
10
15
20 t (s)
25
30
Figure 9 Axial force and spring force (M1, 2)
77
35
6
-0.2
x 10
Method 1 Method 2
-0.4 -0.6
Energy (J)
-0.8 -1 -1.2 -1.4 -1.6 -1.8
0
5
10
15
20 t (s)
25
30
Figure 10 Total mechanical energy profile (M1, 2)
78
35
4
6
x 10
Method 1 Method 2
Rate of change of energy (J/s)
5
4
3
2
1
0
0
5
10
15
20 t (s)
25
4
30
35
4
x 10
x 10
5.5 Method 1 Method 2
5
5 Method 1 Method 2
4.5 Rate of change of energy (J/s)
4 3.5 3 2.5 2 1.5
4 3.5 3 2.5 2 1.5
1
1
0.5
0.5
1
2
3
4 t (s)
5
6
0
7
30
31
32
33
34 t (s)
4
x 10 5.6
Method 1 Method 2
5.5 Rate of change of energy (J/s)
Rate of change of energy (J/s)
4.5
5.4 5.3 5.2 5.1 5 4.9 4.8
10
15
20
25
30
t (s)
Figure 11 Rate of change of energy with details (M1, 2)
79
35
36
37
5.3.2 Method 2 vs. Method 3 The numerical output demonstrates the effectiveness of these two discretization scheme. The results are in fairly good agreement with each other.
0.4 Method 2 Method 3
0.35
Disp. of the Car (m)
0.3
0.25
0.2
0.15
0.1
0.05
0
5
10
15
20 t (s)
0.35
30
35
0.15 Method 2 Method 3
0.34
Method 2 Method 3
0.14 0.13
0.33
0.12
0.32
Disp. of the Car (m)
Disp. of the Car (m)
25
0.31 0.3 0.29
0.11 0.1 0.09 0.08
0.28
0.07
0.27
0.06
0.26 1
2
3
4
5 t (s)
6
7
8
9
26
28
30
Figure 12 Vibration of the car with details (M2, 3)
80
32 t (s)
34
36
38
0.35 Method 2 Method 3
Disp. of the lower end of the cable(m)
0.3
0.25
0.2
0.15
0.1
0.05
0
0
5
10
15
25
30
35
0.045
Method 2 Method 3 Disp. of the lower end of the cable(m)
0.32 Disp. of the lower end of the cable(m)
20 t (s)
0.315
0.31
0.305
0.3
Method 2 Method 3
0.04
0.035
0.03
0.025
0.295 0
0.5
1
1.5
2
2.5 t (s)
3
3.5
4
0.02 35
4.5
35.5
36
36.5 t (s)
37
Figure 13 Vibration of the lower end of the cable with details (M2, 3)
81
37.5
38
0.35 Method 2 Method 3
0.3
u(l(t) -10,t) (m)
0.25
0.2
0.15
0.1
0.05
0
0
5
10
15
20 t (s)
Method 2 Method 3
0.3
30
35
Method 2 Method 3
0.03
0.295
0.025 u(l(t) -10,t) (m)
u(l(t) -10,t) (m)
25
0.29
0.285
0.02
0.015
0.28 0.01
0.275 0
0.5
1
1.5
2
2.5 t (s)
3
3.5
4
4.5
5
34
34.5
35
35.5
36
36.5
37
t (s)
Figure 14 Vibration of physical particle at 10 m above the lower end with details (M2, 3)
82
37.5
10800 Method 2 Method 3
10600
Axial force F(l(t) -10,t) (N)
10400 10200 10000 9800 9600 9400 9200 9000 8800
0
5
10
15
20 t (s)
25
30
35
10800 Method 2 Method 3
10600
Axial forces F (lower end) (N)
10400 10200 10000 9800 9600 9400 9200 9000 8800
0
5
10
15
20 t (s)
25
30
Figure 15 Axial force at different location of the cable (M2, 3)
83
35
6
-0.2
x 10
Method 2 Method 3
-0.4 -0.6
Energy (J)
-0.8 -1 -1.2 -1.4 -1.6 -1.8
0
5
10
15
20 t (s)
25
Figure 16 Total mechanical energy profile (M2, 3)
84
30
35
4
6
x 10
Method 2 Method 3 Rate of change of total energy (J/s)
5
4
3
2
1
0
0
5
10
15
20 t (s)
25
4
30
35
4
x 10
x 10
5.5 Method 2 Method 3
5
Rate of change of total energy (J/s)
4 3.5 3 2.5 2 1.5
4 3.5 3 2.5 2 1.5
1
1
0.5
0.5
1
2
3
4 t (s)
Method 2 Method 3
4.5
4.5
5
6
7
30
31
32
33
34 t (s)
4
x 10
Method 2 Method 3
5.6 Rate of change of total energy (J/s)
Rate of change of total energy (J/s)
5
5.5 5.4 5.3 5.2 5.1 5 4.9 10
15
20
25
30
t (s)
Figure 17 Rate of change of total energy with details (M2, 3)
85
35
36
37
38
5.3.3 Dynamic Responses Based on the Equilibrium Position (Method 2)
Disp. of the lower end from translation (m)
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
0
5
10
15
20 t (s)
25
30
Figure 18 Displacement resulted from the translating motion
86
35
-3
5
x 10
Disp. of the lower end from equilibrium(m)
4 3 2 1 0 -1 -2 -3 -4
0
5
10
15
20 t (s)
25
30
35
Figure 19 Displacement of the lower end of the cable from the newly defined equilibrium position
87
Chapter 6: Conclusions and Future Work
In the present work, the governing equation of a vertically translating elevator cable-car system with an arbitrarily varying length is obtained by using Newton’s Law and Hamilton’s Principle. Both approaches yield the same results. The assumed modes method and a new spatial discretization method are used to discretize either the governing partial differential equation of the moving cable or the Lagrangian of the system before the application of Lagrange’s equations. Both yield the same discretized governing equations for all three methods. Compared to the assumed modes discretization method, the other two methods presented here are more accurate and provide more physical insight. The rate of change in the total mechanical energy from the system viewpoint and control volume viewpoint is derived, the former of which can establish an instantaneous work and energy relation. The expression of energy flux at a fixed spatial point of the elevator cable is given, and the relations between energy flux and total energy by are presented in differential form, integral form. While the assumed modes method can relatively accurately estimate the energy of the cable, it cannot accurately predict the displacement of the cable due to slow convergence, the displacement of the car, and the axial force in the cable. The new discretization 88
methods ensures that all the boundary conditions are satisfied and can be used to accurately determine the displacements of the cable and the car, and the axial force in the cable. The equilibrium position resulted from the translation is introduced.
The dynamic
response with respect to this equilibrium is numerically achieved. In future work, a mathematical model for more complicated elevator systems, which reveals more physical meanings, could be introduced.
The coupling of transverse
vibration and longitudinal vibration need to be investigated, as well as the nonlinear vibration for the translating media.
89
Appendix A: Spatial discretization results for assumed modes method by using Lagrange’s Equation For assumed modes method, the solution form is assumed as n
u (ξ ,τ ) = ∑ψ k (ξ )qk (τ ) k =1
The total kinetic energy and its derivatives are
T=
ρl 2
1
∫ [ψ 0
k
v 1 qk + ϑ e + (1 − ξ )ψ k′ qk + v]2 dξ + mc (uc + v) 2 l 2
1 1 1 1 ∂T = l ∫ ψ kψ h d ξ qh + l ∫ ψ k dξ v + v ∫ (1 − ξ )ψ kψ h′ dξ qh 0 0 0 ρ ∂qk
1 1 1 ∂T v2 = v ∫ (1 − ξ )ψ k′ψ h d ξ qh + v 2 ∫ (1 − ξ )ψ k′ d ξ + 0 0 ρ ∂qk l
1
(A-1)
∫ (1 − ξ ) ψ ′ψ ′ dξ q 2
0
k
h
h
(A-2)
The total potential energy from Eq. (13) and Eq. (15) becomes
V=
=
1 EA 1 ∂u 2 1 ( ) d ξ + k[uc − u (1, t )]2 − ρ gl ∫ (u + lξ )dξ − mc g (uc + l ) ∫ 0 2l 0 ∂ξ 2
1 EA 1 1 (ψ k′ qk ) 2 dξ + kc [uc −ψ k (1)qk ]2 − ρ gl ∫ (ψ k qk + lξ )dξ − mc g (uc + l ) ∫ 0 0 2l 2
Taking partial derivative of potential energy V yields
90
1 ∂V EA 1 ψ k′ψ h′ d ξ qh − ρ gl ∫ ψ k dξ + kcψ k (1)ψ k (1)qh − kcψ k (1)uc = ∫ 0 ∂qk l 0
(A-3)
∂V = kc uc − kcψ k (1)qk − mc g ∂uc
(A-4)
Substituting (A-1-4) into Lagrange Equation with non-conservative force d ∂T ∂T ∂V − + = Qi dt ∂qi ∂qi ∂qi where Qi is non-conservative generalized force. Since virtual work done by non-conservative force equals virtual work done by generalized force, Qi can be achieved by similarly as in chapter 3. Rearranging Lagrange Equation yields 1
1
1
0
0
0
' l ∫ ψ iψ j d ξ q j + v{∫ ψ iψ j dξ + ∫ (1 − ξ )(ψ iψ 'j −ψ iψ j )dξ +
+[
1 EA 1 v2 ′ ′ ′ d v d + − − ψ ψ ξ (1 ξ ) ψ ψ ξ i j ∫0 l ρ l ∫0 i j
−
mc
ρ
uc +
cc
ρ
cc
ρ
ψ i (1)uc −
uc −
cc
ρ
kc
ρ
1
i
0
j
ρ
kc
ρ
ψ i (1)qi =
Putting in matrix form
91
mc
ρ
i
j
(1)] q j
(A-5)
0
uc −
ψ i (1)ψ j (1)}q j
kc
2
1
kc
ρ
∫ (1 − ξ ) ψ ′ψ ′ dξ + ρ ψ (1)ψ
ψ i (1)uc = l ( g − v) ∫ ψ i dξ
ψ i (1)qi +
cc
( g − v)
(A-6)
[ M ]x + [C ]x + [ K ]x = f where
(
x = ( q uc ) , f = f q T
f uc
)
T
⎛ W q W quc ⎞ [W ] = ⎜ u q (W = M , C , K ) , uc ⎟ c W W ⎝ ⎠
,
In which 1
M ijq = l ∫ ψ iψ j d ξ ,
M iuc q = M iquc = 0 ,
0
1
1
0
0
M uc =
' Cijq = v[ ∫ ψ iψ j d ξ + ∫ (1 − ξ )(ψ iψ 'j −ψ iψ j )d ξ +
Ciquc = Ciuc q = −
K ijq =
cc
ψ i (1) ,
ρ
C uc =
cc
ρ
kc
ρ
ψ i (1) ,
K uc =
kc
ρ
1 Since ψ k (ξ ) = sin(k − )πξ , k = 1, 2, 2 1
∫ ψψ 0
i
j
1 dξ = δ ij , 2
1
1
∫ (1 − ξ )ψ ′dξ = ∫ ψ 0
i
0
1
1
k
dξ =
i
ψ i (1)ψ j (1)] ,
1
kc
∫ (1 − ξ ) ψ ′ψ ′ dξ + ρ ψ (1)ψ 2
i
0
j
i
1
fi q = l ( g − v) ∫ ψ i dξ ,
,
0
, and
1
1
∫ ψ ′ dξ = ψ
∫ ψ ′ψ ′ dξ = 2 (k − 2 ) δ 0
ρ
,
ρ
,
1 EA 1 v2 ′ + − − ψ ψ ξ (1 ξ ) ψ ψ ξ d v d i j ∫0 ρ l ∫0 i j l
K iquc = K iuc q = −
cc
mc
2
j
1 , (k − 1/ 2)π 92
k
0
ij
,
k
j
(1) ,
f uc =
(1) = (−1) k −1
mc
ρ
( g − v)
If i = j ,
∫
1
0
(1 − ξ )ψ iψ i′dξ =
1 , 4
∫
1
0
(1 − ξ ) 2ψ i′ψ i′dξ =
1 π2 1 (k − )2 , + 4 6 2
If i ≠ j ,
1
∫ (1 − ξ )ψ ψ ′ dξ = i
0
1
j
(i − 1/ 2)( j − 1/ 2) , (i − j )(i + j − 1) 1
∫ (1 − ξ ) ψ ′ψ ′ dξ = (i − 1/ 2)( j − 1/ 2)[ (i − j ) 0
2
i
j
93
2
+
1 ] (i + j − 1) 2
Appendix B: spatial discretization results for method 3 If ϑ (ξ ) = ξ 3 − ξ 2 is given in section 3.4.2 for method 3 and the same set of trial functions used for method 1 are applied, the discretized equations can be obtained by following the same procedures as 1
1
1
1
0
0
0
0
l ∫ ψ iψ j d ξ q j + l ∫ ϑ (ξ )ψ i d ξ e + v{∫ ψ iϑ (ξ )dξ + ∫ (1 − ξ )(ψ iϑ ′(ξ ) −ψ i'ϑ (ξ ))dξ }e
1
1
0
0
cc
' + v{∫ ψ iψ j d ξ + ∫ (1 − ξ )(ψ iψ 'j −ψ iψ j )dξ +
+[
1 EA 1 v2 ′ ′ ′ d v d + − − ψ ψ ξ (1 ξ ) ψ ψ ξ i j ∫0 l ρ l ∫0 i j
1
+{∫ [ 0
ρ
ψ i (1)ψ j (1)}q j
1
kc
∫ (1 − ξ ) ψ ′ψ ′ dξ + ρ ψ (1)ψ 2
i
0
j
i
j
(1)] q j
1 EA v 2 − (1 − ξ ) 2 ]ϑ ′(ξ )ψ i′d ξ + v ∫ (1 − ξ )ϑ ′(ξ )ψ i′dξ }e 0 ρl l
−
cc
ρ
ψ i (1)uc −
kc
ρ
1
ψ i (1)uc = l ( g − v) ∫ ψ i dξ
(B-1)
0
1
1
1
1
0
0
0
0
l ∫ ϑ (ξ )ψ j d ξ q j + l ∫ ϑ 2 (ξ )d ξ e + v{∫ ϑ (ξ )ψ j d ξ + ∫ (1 − ξ )[ϑ (ξ )ψ ′j − ϑ ′(ξ )ψ j ]d ξ }q j
1
1
0
0
+ v ∫ ϑ 2 (ξ )d ξ e + {∫ [
1
+{∫ [ 0
1 EA v 2 − (1 − ξ ) 2 ]ϑ ′(ξ )ψ ′j dξ + v ∫ (1 − ξ )ϑ (ξ )ψ ′p dξ }q j 0 ρl l
1 1 EA v 2 − (1 − ξ ) 2 ]ϑ ′(ξ )ϑ ′(ξ )d ξ + v ∫ (1 − ξ )ϑ (ξ )ϑ ′(ξ )d ξ }e = l ( g − v) ∫ ϑ (ξ )dξ 0 0 ρl l
(B-2)
94
mc
ρ
uc +
cc
ρ
uc −
cc
ρ
ψ i (1)qi +
kc
ρ
uc −
kc
ρ
ψ i (1)qi =
mc
ρ
( g − v)
(B-3)
The discretized equations of motion in matrix form is [ M ]x + [C ]x + [ K ]x = f where
(
x = ( q e uc ) ; f = f q T
f uc
fe
)
T
⎛ W q W qe W quc ⎞ ⎜ ⎟ , [W ] = ⎜ W eq W e W euc ⎟ (W = M , C , K ) ⎜ W uc q W uc e W uc ⎟ ⎝ ⎠
In which 1
1
M ijq = l ∫ ψ iψ j d ξ ,
M ieq = M iqe = l ∫ ϑ (ξ )ψ i d ξ , M iuc q = M iquc = M iuce = M ieuc = 0 ,
0
0
1
M e = l ∫ ϑ 2 (ξ )d ξ ,
M uc =
0
1
1
0
0
mc
ρ
,
' Cijq = v[ ∫ ψ iψ j d ξ + ∫ (1 − ξ )(ψ iψ 'j −ψ iψ j )d ξ +
1
1
0
0
cc
ρ
ψ i (1)ψ j (1)] ,
Ciqe = v{∫ ϑ (ξ )ψ j d ξ + ∫ (1 − ξ )[ϑ ′(ξ )ψ j − ϑ (ξ )ψ ′j ]dξ } , 1
1
0
0
′ ′ C eq j = v{∫ ϑ (ξ )ψ j d ξ + ∫ (1 − ξ )[ϑ (ξ )ψ j − ϑ (ξ )ψ j ]d ξ } ,
1
C e = v ∫ ϑ 2 (ξ )d ξ , 0
Ciquc = Ciuc q = −
cc
ρ
ψ i (1) ,
95
C uc =
cc
ρ
,
K ijq =
1 EA 1 v2 ′ d v d + − − ψ ψ ξ (1 ξ ) ψ ψ ξ i j ∫0 l ρ l ∫0 i j
1
K iqe = ∫ [ 0
1
K eq j = ∫ [ 0
1
K ee = ∫ [ 0
1
2
i
0
1 EA v 2 − (1 − ξ ) 2 ]ϑ ′(ξ )ϑ ′(ξ )d ξ + v ∫ (1 − ξ )ϑ (ξ )ϑ ′(ξ )d ξ , 0 ρl l
1
1
0
0
1
1
∫ ψ ϑ dξ = R
2 k
k
[
1
1
2 k
k
0
2
1
∫ (1 − ξ )ψ ϑ ′dξ = R
2 k
k
2
∫ ψ ′ϑ ′dξ = R 0
∫
1
0
(1) ,
j
k
[
[(−1) k (1 −
k
(1 − ξ ) 2ψ k′ϑ ′dξ =
6 ), Rk2
ρ
,
( g − v)
ρ
1
1
∫ ϑ dξ = 105 , 2
0
18 10 ) − ], Rk2 Rk
6 12 + (−1) k (1 − 2 )] , Rk Rk
+ (−1) k −1 (1 −
mc
kc
K uc =
1 Denote Rk = (k − )π , then 2
2 6 + (−1) k −1 (1 − 2 )] , Rk Rk
∫ (1 − ξ )ϑ ′ψ ϑ dξ = R
1
i
1 EA v 2 − (1 − ξ ) 2 ]ϑ ′(ξ )ψ ′j d ξ + v ∫ (1 − ξ )ϑ (ξ )ψ i′d ξ , 0 ρl l
1 Note that ψ k (ξ ) = sin(k − )πξ , k = 1, 2, 2
0
j
1 k EA v 2 − (1 − ξ ) 2 ]ϑ ′(ξ )ψ i′d ξ + v ∫ (1 − ξ )ϑ ′(ξ )ψ i d ξ , K iquc = K iuc q = − c ψ i (1) 0 ρ ρl l
fi q = l ( g − v) ∫ ψ i d ξ , fi e = l ( g − v) ∫ ψ i d ξ , f uc =
0
kc
∫ (1 − ξ ) ψ ′ψ ′ dξ + ρ ψ (1)ψ
1
2
∫ ϑ ′ dξ = 15 , 2
0
1
2
∫ (1 − ξ ) ϑ ′ dξ = 105 , 2
2
0
1
1
∫ (1 − ξ )ϑϑ ′dξ = 210 , 0
2 (−1) k 24 3 [1 + ] − 3 [2 + (−1) k ] , Rk Rk Rk Rk 96
1
1
∫ ϑ dξ = − 12 0
Appendix C: energy discretization results for method 3 The kinetic energy
T=
ρl 2
v v 1 qk + ϑ e + (1 − ξ )ψ k′ qk + (1 − ξ )ϑ ′e + v]2 dξ + mc (uc + v) 2 l l 2
1
∫ [ψ 0
k
Putting Eq. (C-1) in matrix form yields
T=
1 T 1 v ⎡⎣ M ⎤⎦ v + mc (uc + v) 2 2 2
where ⎛q⎞ ⎜ ⎟ ⎜e⎟ ν = ⎜q⎟ , ⎜ ⎟ ⎜e⎟ ⎜v⎟ ⎝ ⎠
⎡ [ M qq ] ⎢ qe T ⎢ (m ) ⎡⎣ M ⎤⎦ = ⎢[ M qq ]T ⎢ qe T ⎢ (m ) ⎢ (m qv )T ⎣
m qe mee
[ M qq ] m eq
m qe mee
m qe mee
[ M qq ] (m qe )T
m qe mee
mev
(m qv )T
mev
m qv ⎤ ⎥ mev ⎥ m qv ⎥ ⎥ mev ⎥ mvv ⎥⎦
In which 1
1
1
0
0
0
M ijqq = ρ l ∫ ψ iψ j dξ , miqe = ρ l ∫ ϑψ i d ξ , M ijqq = ρ v ∫ (1 − ξ )ψ iψ 'j dξ , 1
miqe = ρ v ∫ (1 − ξ )ϑ ′ψ i dξ , 0
1
mieq = ρ v ∫ ξ (1 − ξ )ϑψ i' dξ , 0
M ijqq =
ρv2 l
∫
1
0
(1 − ξ ) 2ψ i′ψ ′j dξ ,
1
1
miqv = ρ l ∫ ψ i d ξ ,
mee = ρ l ∫ ϑ 2 d ξ
0
0
1
1
mee = ρ v ∫ (1 − ξ )ϑϑ ′d ξ ,
mev = ρ l ∫ ϑ d ξ
0
miqe =
ρv2
97
l
1
0
∫ (1 − ξ ) ϑ ′ψ ′dξ , 0
2
i
(C-1)
1
m = ρ v ∫ (1 − ξ )ψ i′dξ , qv i
0
m = ee
ρ v2 l
∫
1
0
1
(1 − ξ ) 2 ϑ ′ϑ ′d ξ , mev = ρ v ∫ (1 − ξ )ϑ ′d ξ , 0
mvv = ρ l The potential energy is
V=
1 EA 1 1 (ψ k′ qk + ϑ ′e) 2 dξ + k[uc −ψ k (1)qk ]2 − ρ gl ∫ (ψ k qk + ϑ e + lξ )dξ − mc g (uc + l ) ∫ 0 0 2l 2 (C-2)
Putting in matrix form yields
V=
1 T l w ⎡⎣ K ⎤⎦ w − ρ gl (kT q+κ e+ ) − mc g (uc + l ) 2 2
where ⎛q⎞ ⎜ ⎟ w = ⎜e ⎟, ⎜u ⎟ ⎝ c⎠
⎛ K qq ⎜ ⎡⎣ K ⎤⎦ = ⎜ K eq ⎜ K uc q ⎝
⎞ 1 ⎟ κ = , ⎟ ∫0 ϑ dξ , uc uc ⎟ K ⎠ K quc K euc
K qe K ee K uc e
1
ki = ∫ ψ i d ξ 0
In which
K ijqq =
EA 1 ψ i′ψ ′j dξ + kψ i (1)ψ j (1) , l ∫0
K iquc = − kψ i (1) ,
K ee =
K iqe = K ieq =
EA 1 2 ϑ ′ dξ , l ∫0
Also, the rate of change of kinetic energy is
98
EA 1 ϑ ′ψ i′dξ , l ∫0
K euc = K uc e = 0 ,
K ucuc = k
dT 1 = vT ⎡⎣ M ⎤⎦ v+ vT ⎡ M ⎤ v+m(uc + v)(uc + v) 2 ⎣ ⎦ dt
(C-3)
where ⎛q⎞ ⎜ ⎟ ⎜e⎟ ν = ⎜q⎟ , ⎜ ⎟ ⎜e⎟ ⎜v⎟ ⎝ ⎠
⎡ [ M qq ] ⎢ qe T ⎢ (m ) ⎡ Mˆ ⎤ = ⎢[ M qq ]T ⎢⎣ ⎥⎦ ⎢ qe T ⎢ (m ) ⎢ (m qv )T ⎣
m qe
[ M qq ]
m qe
mee m qe
m eq [ M qq ]
mee m qe
mee
(m qe )T
mee
mev
(m qv )T
mev
m qv ⎤ ⎥ mev ⎥ m qv ⎥ ⎥ mev ⎥ mvv ⎥⎦
In which 1
1
0
0
1
M ijqq = ρ v ∫ ψ iψ j d ξ , miqe = ρ v ∫ ϑψ i d ξ , 1
miqe = ρ v ∫ (1 − ξ )ϑ ′ψ i d ξ , 0
1
mieq = ρ v ∫ ξ (1 − ξ )ϑψ i′d ξ , 0
M ijqq =
ρv l
(2v −
1
0
1
mev = ρ v ∫ (1 − ξ )ϑ ′d ξ , 0
0
1
miqv = ρ v ∫ ψ i dξ , 0
1
mee = ρ v ∫ ϑ 2 d ξ , 0
1
mee = ρ v ∫ (1 − ξ )ϑϑ ′d ξ , 0
v2 1 ) (1 − ξ ) 2ψ i′ψ ′j dξ , l ∫0
m = ρ v ∫ (1 − ξ )ψ i′d ξ , qv i
M ijqq = ρ v ∫ (1 − ξ )ψ iψ ′j dξ ,
miqe =
ρv
ρv l
(2v −
1
mev = ρ v ∫ ϑ d ξ , 0
v2 1 ) (1 − ξ ) 2 ϑ ′ψ i′d ξ l ∫0
v2 1 m = (2v − ) ∫ (1 − ξ ) 2 ϑ ′ϑ ′dξ , l l 0 ee
mvv = ρ v
Similarly, the rate of change of potential energy is
99
dV 1 = qT ⎡⎣ K ⎤⎦ q + qT ⎡ K ⎤ q − ρ gv(k T q+κ e) − ρ gv(k T q+κ e) − mc guc − ( ρ l + mc ) gv dt 2 ⎣ ⎦ where
K ijqq = −
EAv 1 ψ i′ψ ′j dξ , l 2 ∫0
K iquc = 0 ,
K iee = −
K iqe = K iqe = −
EAv 1 ϑ ′ψ i′d ξ , l 2 ∫0
EAv 1 2 ϑ ′d ξ , K euc = K uc e = 0 , K ucuc = 0 2 ∫0 i l
From the control volume viewpoint without considering damping and external impact dEcv (t ) Du 1 Du (0, t ) + v] + v{EAu x2 (0, t ) + ρ[ (0, t ) + v]2 } = − EAu x (0, t )[ dt Dt 2 Dt Applying the approximation method 3 in above equation yields 1 u x (0, t ) = ψ k′ (0)qk , l
Du v Du (0, t ) = ψ k′ (0)qk , (l , t ) = ψ k (1)qk Dt l Dt
So we have dEcv (t ) 1 ρ v 2 − EA [ψ k′ (0)qk + l ]2 } = v{EA + 2 dt 2 l
100
Bibliography [1]
C. D. Mote, Jr., "Dynamic Stability of Axially Moving Materials," The Shock
and Vibration Digest, vol. 4, pp. 2-11, 1971. [2]
J. A. Wickert and C. D. Mote, Jr., "Current Research on the Vibration and
Stability of Axially-Moving Materials," The Shock and Vibration Digest, vol. 20, pp. 3-13, 1988. [3]
W. D. Zhu, "Vibration and Stability of Time-Dependent Translating Media,"
The Shock and Vibration Digest, vol. 32, pp. 369-397, 2000. [4]
L. Q. Chen, "Analysis and Control of Transverse Vibrations of Axially
Moving Strings," Applied Mechanics Reviews, vol. 58, pp. 91-116, 2005. [5]
G. F. Carrier, "The Spaghetti Problem," American Mathematical Monthly, vol.
56, pp. 669-672, 1949. [6]
W. L. Miranker, "The Wave Motion in a Medium in Motion," IBM Journal of
Research and Development, vol. 4, pp. 36-42, 1960. [7]
W. J. Schaffers, "The Vibration of Shaft Ropes with Time-Variable Length,
Treated by Means of Riemann’s Method," ASME Journal of Engineering for Industry, Series B, vol. 83, pp. 68-72, 1961. [8]
R. D. Swope and W. F. Ames, "Vibrations of a Moving Threadline," Journal
of Franklin Institute, vol. 275, pp. 36-55, 1963.
101
[9]
W. D. Zhu and J. Ni, "Energetics and Stability of Translating Media with an
Arbitrarily Varying Length," ASME Journal of Vibration and Acoustics, vol. 122, pp. 295-304, 2000. [10]
W. F. Ames, S. Y. Lee, and J. N. Zaiser, "Non-linear Vibration of a Traveling
Threadline," International Journal of Non-linear Mechanics, vol. 3, pp. 449-469, 1968. [11]
L. Mansfield and J. G. Simmonds, "The Reverse Spaghetti Problem: Drooping
Motion of an Elastica Issuing from a Horizontal Guide," ASME Journal of Applied Mechanics, vol. 54, pp. 147-150, 1987. [12]
J. Stolte and R. C. Benson, "Dynamic Deflection of Paper Emerging from a
Channel," ASME Journal of Vibration and Acoustics, vol. 114, pp. 187-193, 1992. [13]
J. A. Wickert, "Non-linear Vibration of a Traveling Tensioned Beam,"
International Journal of Non-linear Mechanics, vol. 27, pp. 503-517, 1992. [14]
J. Moon and J. A. Wickert, "Non-linear Vibration of Power Transmission
Belts," Journal of Sound and Vibration, vol. 200, pp. 419-431, 1997. [15]
S.-Y. Lee and C. D. Mote, "Wave Characteristics and Vibration Control of
Translating Beams by Optimal Boundary Damping," ASME Journal of Vibration and Acoustics, vol. 121, pp. 18-25, January 1999.
102
[16]
S.-Y. Lee and C. D. Mote, Jr., "Vibration Control of an Axially Moving String
by Boundary Control," ASME Journal of Dynamic System, Measurement and Control, vol. 118, pp. 66-74, 1996. [17]
M. Otsuki, K. Yoshida, S. Nakagaki, T. Nakagawa, S. Fujimoto, and H.
Kimura, "Experimental Examination of non-stationary robust vibration control for an elevator rope," Proceedings of the Institution of Mechanical Engineers. Part I: Journal of Systems and Control Engineering, vol. 218, pp. 531-544, November 2004. [18]
M. Otsuki, Y. Ushijima, K. Yoshida, H. Kimura, and T. Nakagawa,
"Nonstationary sliding mode control using time-varying switching hyperplane for transverse vibration of elevator rope," in Proceedings of the ASME Interantional Design Engineering Technical Conferences and Computers and Information in Engineering Conference - DETC2005, 2005, pp. 925-934. [19]
M. Otsuki, Y. Ushijima, K. Yoshida, H. Kimura, and T. Nakagawa,
"Application of nonstationary sliding mode control to suppression of transverse vibration of elevator rope using input device with gaps," JSME International Journal. Series C, Dynamics, Control, Robotics, Design and Manufacturing, vol. 49, pp. 385394, Dec 2006. [20]
M. Otsuki, K. Yoshida, K. Nagata, S. Fujimoto, and T. Nakagawa,
"Experimental Study on vibration control for rope-sway of elevator of high-rise building," in Proceedings of the 2002 American Control Conference, 2002, pp. 238243
103
[21]
Y. Yoshiaki, T. Masao, O. Kouji, and S. Yoshiki, "Vibration control of super-
high-speed elevators : Car vibration control method based on computer simulation and experiment using a full-size car model," JSME international journal. Series C, dynamics, control, robotics, design and manufacturing vol. 40, pp. 74-81, 1997. [22]
Y. Yoshiaki, T. Masao, O. Kouji, and S. Yoshiki, "Vibration control of super-
high-speed elevators: 1st report control method of car horizontal vibration based on computer simulation," Nippon Kikai Gakkai Ronbunshu, C Hen/Transactions of the Japan Society of Mechanical Engineers, Part C, vol. 60, pp. 3776-3781, Nov 1994. [23]
O. Kouji, S. Yoshiki, Y. Yoshiaki, and T. Masao, "Vibration control of super-
high-speed elevators: 2nd reprt, study on vibration suppression method using an actual system," Transactions of the Japan Society of Mechanical Engineers, Part C, vol. 60, pp. 3782-3788, Nov 1994. [24]
Y. Yoshiaki, T. Masao, O. Kouji, S. Yoshiki, K. Kazushi, and S. Kazuki,
"Vibration control of super-high-speed elevators: 3rd reprt, A field test of car traveling," Transaction of the Japan Society of Mechanical Engineers, Part C, vol. 61, pp. 15-21, Jan 1995. [25]
A. Arakawa and K. Miyata, "A Variable-structure Control Method for the
Suppression of Elevator-Cage Vibration," in Proceeding of IEEE 2002 28th Annual Conference of the Industrial Electronics Society(IECON 02), 2002, pp. 1830-1835
104
[26]
H. Kimura and T. Nakagawa, "Vibration analysis of elevator rope with
vibration suppressor," Nippon Kikai Gakkai Ronbunshu, C Hen/Transactions of the Japan Society of Mechanical Engineers, Part C, vol. 71, pp. 442-447, February 2005. [27]
H. Kimura, H. Ito, and T. Nakagawa, "Vibration analysis of elevator rop
(forced vibration of rope with time-varying length)," Nippon Kikai Gakkai Ronbunshu, C Hen/Transactions of the Japan Society of Mechanical Engineers, Part C, vol. 71, pp. 1871-1876, June 2005. [28]
H. Kimura, H. Ito, and T. Nakagawa, "Vibration analysis of elevator rope
(2nd reprt, forced vibration of rope with damping)," Nippon Kikai Gakkai Ronbunshu, C Hen/Transactions of the Japan Society of Mechanical Engineers, Part C, vol. 72, pp. 1440-1445, 2006. [29]
H. Kimura, H. Ito, Y. Fujita, and T. Nakagawa, "Forced bivration analysis of
an elevator rope with both ends moving," Journal of Vibration and Acoustics, Transactions of the ASME, vol. 129, pp. 471-477, 2007. [30]
H. Kimura, T. Iijima, S. Matsuo, and Y. Fujita, "Vibration analysis of elevator
rope (comparison between experimental results and calculated results)," Nippon Kikai Gakkai Ronbunshu, C Hen/Transactions of the Japan Society of Mechanical Engineers, Part C, vol. 74, pp. 31-36, January 2008. [31]
W. D. Zhu and B. Z. Guo, "Free and Forced vibration of an axially moving
string with an arbitary velocity profile," ASME Journal of Vibration and Acoustics, vol. 65, pp. 901-907, 1998.
105
[32]
W. D. Zhu and G. Y. Xu, "Vibration of Elevator Cables with Small Bending
Stiffness," Journal of Sound and Vibration, vol. 263, pp. 679-699, Jun 2003. [33]
W. D. Zhu, "Control volume and system formulations for translating media
and stationary media with moving boudaries," Journal of Sound and Vibration, vol. 254, pp. 189-201, Jun 2002. [34]
W. D. Zhu, J. Ni, and J. Huang, "Active Control of Translating Media with
Arbitrarily Varying Length," ASME Journal of Vibration and Acoustics, vol. 123, pp. 347-358, 2001. [35]
W. D. Zhu and Y. Chen, "Forced response of translating media with variable
length and tension: application to high-speed elevators," Institute of Mechanical Engineers Journal of Multibody Dynamics, vol. 219, pp. 35-53, 2005. [36]
W. D. Zhu and Y. Chen, "Theoretical and Experimental Investigation of
Elevator Cable Dynamics and Control," ASME Journal of Vibration and Acoustics, vol. 128, pp. 66-78 February 2006. [37]
W. D. Zhu and N. A. Zheng, "Exact Response of a Translating String With
Arbitrarily Varying Length Under General Excitation," Journal of Applied Mechanics, vol. 75, pp. 031003-1 - 031003-14, May 2008. [38]
R. M. Chi and H. T. Shu, "Longitudinal Vibration of a Hoist Rope Coupled
with the Vertical Vibration of an Elevator Car," Journal of Sound and Vibration, vol. 148, pp. 154-159, 1991.
106
[39]
Y. H. Zhang, "Longitudinal Vibration Modeling and Control of a Flexible
Transporter System with Arbitrarily Varying Cable Lengths," Journal of Vibration and Control, vol. 11, pp. 431-456, 2005. [40]
Y. Terumichi, M. Yoshizawa, Y. Fukawa, and Y. Tsujioka, "Vibrations of
Moving Elevator Rope and Cab in a High-Rise Building (Vertical Vibration of a Cab in Consideration of Longitudinal Vibration of a Hoist Rope)," Nippon Kikai Gakkai Ronbunshu, C Hen/Transactions of the Japan Society of Mechanical Engineers, Part C, vol. 60, pp. 1487-1494, May 1994. [41]
S. Kaczmarczyk, J. P. Andrew, and J. P. Adams, "The Modelling and
Prediction of the Influence of Building Vibration on the Dynamic Response of Elevator Ropes," MATERIALS SCIENCE FORUM, vol. 440-441, pp. 489-496, 2003. [42]
H. Ito, M. Morishita, and H. Kimura, "The modeling of longitudinal and
lateral vibrations on the elevator rope," Nippon Kikai Gakkai Ronbunshu, C Hen/Transactions of the Japan Society of Mechanical Engineers, Part C, vol. 72, pp. 1435-1439, 2006.
107
108