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Dynamic contact analysis of a tensioned beam with a moving mass–spring system
Journal of Sound and Vibration 331 (2012) 2520–2531
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Dynamic contact analysis of a tensioned beam with a moving mass–spring system Kyuho Lee a, Yonghyeon Cho b, Jintai Chung a,n a b
Department of Mechanical Engineering, Hanyang University, 1271 Sa-3-dong, Sangnok-gu, Ansan, Kyeonggi-do 425-791, Republic of Korea Korea Railroad Research Institute, Woram-dong, Uiwang City, Kyeonggi-do 437-757, Republic of Korea
a r t i c l e in f o
abstract
Article history: Received 11 June 2011 Received in revised form 12 January 2012 Accepted 12 January 2012 Handling Editor: H. Ouyang Available online 6 February 2012
The dynamic contact problem of a tensioned beam with clamped-pinned ends is analyzed when the beam contacts a moving mass–spring system. The contact and contact loss conditions are expressed in terms of constraint equations after considering the dynamic contact between the beam and the moving mass. Using these constraints and equations of motion for the beam and moving mass, dynamic contact equations are derived and then discretized using the finite element method, which is based on the Lagrange multiplier method. The time responses for the contact forces are computed from these discretized equations. The contact force variations and contact loss are investigated for the variations of the moving mass velocity, the beam tension, the moving mass, and the stiffness of the moving mass–spring system. In addition, the possibility of contact loss and safe contact conditions between the moving mass and the tensioned beam are also studied. & 2012 Elsevier Ltd. All rights reserved.
1. Introduction The dynamic analyses of flexible tensioned beams with moving mass or load are very important in engineering because such structures are widely used in various applications, e.g., contact wires in high-speed railways, suspension bridges, overhead cranes, and tethered satellite systems. Many recent studies of beams with moving mass focus on railway systems related to the development of high-speed railway systems. In these applications, system properties determine the dynamic behavior and stability of high-speed railways. Dynamic stability is crucial in contact conditions because it concerns the collection of electric energy in a high-speed railway. If loss of contact or sudden change of the contact force occurs during railway operations, serious problems such as suspension of service or derailment may occur due to the unstable collection of electric energy. Therefore, stable contact conditions should be derived to ensure safe operation of high-speed railways. Many researchers have studied the dynamic behaviors and contact of flexible tensioned beams subjected to moving masses or loads at high speed. Rieker et al. [1] and Rieker and Trethewey [2] performed finite element analyses of beams subjected to moving masses and they assumed that the moving mass is always attached to the beam. However, this assumption does not represent contact loss or contact force between the beam and the moving mass. Lee [3] and Stancioiu et al. [4] considered the separation of the beam from the moving-mass system. It was found in [4] that the response was closer to the experimental results when loss of contact due to separation was considered. On the other hand, Lee [5]
n
Corresponding author. Tel.: þ 82 31 400 5287; fax: þ82 31 406 5550. E-mail address: [email protected] (J. Chung).
0022-460X/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.jsv.2012.01.014
K. Lee et al. / Journal of Sound and Vibration 331 (2012) 2520–2531
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analyzed the influence of an axial force on the onset of separation between a moving mass and beam. He found that separation of the mass from the beam could be avoided by the application of suitable tensile axial forces. As tensioned beams with moving mass are often examined in the context of high-speed railway systems, governing equations for the tensioned beam have been introduced in numerous studies [6–11]. Park et al. [9], Wu and Brennan [12], and Lee [13] performed finite-element analyses of tensioned beams with moving mass while considering the contact between the beam and the mass. Many researchers used the penalty method in order to solve these contact problems, such as wire-pantographs used for high-speed railways [14–16]. However, in the penalty method, the contact constraints for preventing penetration were not imposed precisely because the penalty method uses an assumed parameter that may change both the contact conditions and contact force. Therefore, Teichelmann et al. [10], Lee [13], and Seo et al. [17] carried out contact analyses using the Lagrange multiplier method, which introduces the unknown contact force as the Lagrange multiplier. This method is able to obtain more precise results than the penalty method. Although many analyses of dynamic contact between beam and moving mass have been conducted, there have been few studies of the stable contact conditions between a simple tensioned beam and a moving mass because the majority of previous dynamic contact analyses have focused on the complex wire/pantograph in high-speed railways. Aboshi et al. [18] studied the causes of contact loss and confirmed that the contact force fluctuation is influenced by the wave reflection. Specifically, they found that rubber damping hangers or friction damping hangers reduce contact loss of the pantograph and wear of the contact wire. Park et al. [19] investigated the optimal design parameters of pantographs in order to minimize contact loss. However, they did not investigate contact loss or force variation for various design parameters, such as tension, velocity, moving mass, and stiffness. In this study, we analyze the dynamic contact problem of a tensioned beam with clamped-pinned ends using the finite element method when the beam is in contact with a moving mass–spring system. When the tension is considered in the beam and the moving mass–spring system is assumed to have a single degree of freedom, the equations for the beam motion are derived from the Euler beam theory. Furthermore, the dynamic contact conditions are expressed in terms of constraint equations. With these constraints and equations of motion for the beam and moving mass, dynamic-contact equations are derived and discretized through the finite element method using the Lagrange multiplier method. From these equations, the time responses are computed for the contact forces. Moreover, the contact force variations are simulated for various design parameters, such as the moving speed, tension, the mass and stiffness of the moving mass system. Based on these simulations, safe contact conditions between tensioned beam and the moving mass are obtained. These conditions must satisfy the small magnitude variation of the contact force and minimize the contact loss. The possibility of contact loss is also investigated in this study. 2. Modeling and equation of motion Consider an extensible straight beam with uniform tension and a moving mass–spring system, as shown in Fig. 1, where Figs. 1(a) and (b) represent the beam before and after deformation, respectively. The straight beam, fixed at the left end and supported by a roller at the right end, has length L, cross-sectional area A, Young’s modulus E, density r, and distributed force per unit length p(x, t). The beam with tension T applied at the right end is excited by the moving mass–spring system with a time-variant velocity V. The moving mass–spring system has mass m and spring constant k. Due to this excitation, which is caused by the moving mass, contact force and displacements occur between the tensioned
y
L p ( x, t ) x
T
m
k
V
y
p
v
ξ
u
T
x
V Fig. 1. Schematics of a tensioned beam with a moving mass–spring system: (a) before deformation and (b) after deformation.
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beam and the moving mass. The displacement of the moving mass is represented as x(x, t). There is a contact force fc between the beam and moving mass when the mass contacts the beam. It is assumed in this study that the contact force has only normal directional force because the tangential contact force is negligible. This study derives the equations of motion for the beam using the Euler beam theory because the beam length L is much longer than the cross-sectional dimensions. In order to consider large deflection, the von Karman strain theory is used along with the Euler beam theory. According to the Euler beam theory, the deflections in the x and y directions, ux(x, y, t) and uy(x, y, t), are expressed by ux ðx,y,tÞ ¼ uðx,tÞy
@vðx,tÞ , @x
uy ðx,y,tÞ ¼ vðx,tÞ
(1)
where u(x, t) and v(x, t) are the longitudinal and transverse displacements of an arbitrary point on the centerline of the beam. In order to consider nonlinearity due to large deflection of the beam, the longitudinal normal strain ex of the von Karman strain theory is used: @u 1 @uy 2 ex ¼ x þ (2) 2 @x @x Next, consider the stress–strain relationship of the beam. It is assumed that the material of the beam is homogeneous, isotropic, and elastic. Since the tensioned beam is slender and the cross-sectional area is very small when compared to the other dimensions, the linearized stress is adopted in this study. The linearized axial stress sLx is represented by
sLx ¼ EeLx
(3)
where eLx is given by the linear term of Eq. (2). The equations of motion for the tensioned beam and the moving mass–spring system are obtained from the Hamilton principle, which is written as Z t2 ðdKdU þ dWÞ dt ¼ 0 (4) t1
where d is the variation operator, t1 and t2 are any instances, K is the kinetic energy, U is the potential energy, and W is the work done by non-conservative forces. The rotary inertia effect of the beam is ignored because the beam of this study is thin. For this case, the kinetic energy for the beam and moving mass–spring system is expressed as Z L " 2 2 # 2 1 @u @v 1 K ¼ rA þ (5) dx þ mðx_ þ V 2 Þ 2 @t @t 2 0 where the superposed dot stands for differentiation with respect time. Since the geometric nonlinearity is a source of the nonlinear beam behavior, the nonlinear strain and the linearized stress are used to derive the equations of motion. This type of modeling can be found in Refs. [20–22]. Using this modeling, the variation of the potential energy for the tensioned beam with the moving mass–spring system is given by Z LZ dU ¼ sLx dex dA dx þkxdx (6) 0
A
When the distributed force p is applied to the beam and the moving mass is located at x¼xm, the virtual work done by the non-conservative forces is represented by Z L dW ¼ ½p þf c dðxxm Þdv dx þ f c dx (7) 0
where dðxxm Þ represents the Dirac delta function at x¼xm. By introducing Eqs. (5)–(7) into Eq. (4), the nonlinear equations of motion for the beam and moving mass can be derived as follows: @2 u @2 u EA 2 ¼ 0 2 @x @t
(8)
@2 v @4 v @ @u @v ¼ p þ f c dðxxm Þ þ EI EA @x @x @x @x4 @t 2
(9)
rA rA
mx€ þkx ¼ f c
(10)
where I is the area moment of inertia about the centerline of the beam. The corresponding boundary conditions are given by u¼v¼
@v ¼ 0 at x ¼ 0 @x
(11)
K. Lee et al. / Journal of Sound and Vibration 331 (2012) 2520–2531
v¼
@2 v ¼ 0, @x2
EA
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@u ¼ T at x ¼ L @x
(12)
The equations of motion, given in Eqs. (8)–(10), can be simplified under the assumption that the longitudinal displacement u is much smaller than the transverse displacement v. With this assumption, we can consider that the axial normal strain qu/qx is uniform along the beam and independent of time. This means that there is no need to solve Eq. (8) because the time response of the longitudinal displacement is always constant. Since EAqu/qx in Eq. (9) is replaced by tension T, the equations of motion can be rewritten as
rA
@2 v @4 v @2 v þ EI 4 T 2 ¼ p þ f c dðxxm Þ 2 @x @x @t
(13)
mx€ þ kx ¼ f c
(14)
The relative gap between the tensioned beam and the moving mass is important because it has an influence on the dynamic behaviors of the beam and mass. The moving mass is not always attached to the tensioned beam, which means that separation is possible due to the relative displacements between the beam and mass. Therefore, the equations of motion should be considered for both cases involving contact and non-contact. First, in order to define the conditions of contact and non-contact, the relative displacement between the moving mass and the beam should be considered. The gap between the beam and the mass is defined by gðx,tÞ ¼ vðx,tÞxðx,tÞ at x ¼ xm
(15)
When g(x, t)40, contact does not occur because the position of the beam at x¼xm is higher than the position of the moving mass. Conversely, when g(x, t)¼ 0, contact is generated because the relative displacement between the beam and the moving mass is zero. When the gap is equal to zero, the Lagrange multiplier method should be used in order to prevent penetration between the beam and mass. Next, the equations of motion considering the loss of contact are investigated. As shown in Eqs. (13) and (14), they are divided by the gap condition. When g ¼0, a contact force fc is applied to the equations; however, when g40, it is not applied. It is ineffectual to solve these equations independently, because the moving mass is not always attached to the tensioned beam. When the moving mass contacts the beam, the contact force has a nonzero value and the equations of motion are expressed as
rA
@2 v @4 v @2 v þ EI 4 T 2 ¼ p þf c dðxxm Þ, mx€ þ kx ¼ f c , when g ¼ 0 @x @x @t 2
(16)
On the other hand, when the mass is separated from the beam, the contact force becomes zero and the equations of motion are given by
rA
@2 v @4 v @2 v þ EI 4 T 2 ¼ p, 2 @x @x @t
mx€ þ kx ¼ 0, when g 40
(17)
After the mass and beam are separated, the reattachment between them is judged by the value of gap g. If the value of g changes from a positive value to zero, the moving mass becomes reattached to the beam and the motions of the mass and beam should be described by Eq. (16). If the moving mass is quite less than the beam mass, the impact between the mass and beam can be neglected during reattachment. Otherwise, the impact should be considered because the momentum of the mass is not negligible. In fact, Cheng et al. [23] and Stancioiu et al. [24] showed that the impact during reattachment has an influence on the dynamic responses of the beam with moving mass. However, in this study, the impact at reattachment is not considered. 3. Finite element formulation with the Lagrange multiplier method In order to obtain approximate solutions from the derived equations of motion and boundary conditions, the finite element method is used in this study. The equations of motion and the boundary conditions are transformed into their weak forms. Next, the weak forms are spatially discretized with two-node beam elements, as shown in Fig. 2. In this figure, the numbers above the beam elements represent the element numbers, and the numbers below the beam elements represent the node numbers. The discretized equations for the beam motion can be expressed as Mb d€ b þKb db ¼ f b þ f c rb
1
e
2
1
2
(18)
3
e
N
e +1
N
N +1
Fig. 2. Element and node numbers of the finite element model for the tensioned beam.
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e
e +1
e m
a he Fig. 3. Model for the calculation of vector rb when the mass, which has distance a from node e, is located on the eth element of the finite element model for the tensioned beam.
where db is the global displacement vector, Mb and Kb are the global mass and stiffness matrices of the beam, and fb and rb are the global distributed force and contact coefficient vector, respectively. The global matrices and vectors are represented by T
db ¼ fv2 , y2 ,v3 , y3 ,. . .,vN , yN , yN þ 1 g
(19)
As v1, y1, and vN þ 1 are known degrees of freedom or non-active degrees of freedom in the established model, they are not considered in Eq. (19). When the global contact coefficient vector rb is assembled, the position of the moving mass should be carefully considered because rb changes according to the position of the moving mass. If the moving mass is on the eth element and the distance from node e to the moving mass is a, as shown in Fig. 3, the coefficient vector rb is given by T
rb ¼ f0,0,. . .,N e1 ,Ne2 ,Ne3 ,N e4 ,. . .,0,0g In Eq. (20), expressed as
Ne1 ,
N e2 ,
N e3
and
Ne4
(20)
are the eth, (eþ1)th, (eþ2)th and (eþ3)th elements of the rb vector and they are 3
Ne1 ¼ ðaxe þ 1 Þ2 ð2a3xe þxe þ 1 Þ=he , 3 Ne3 ¼ ðaxe Þ2 ð2a þ xe 3xe þ 1 Þ=he ,
2
N e2 ¼ ðaxe Þðaxe þ 1 Þ2 =he , 2
N e4 ¼ ðaxe Þ2 ðaxe þ 1 Þ=he
(21)
where xe is the nodal coordinate of node e and he represents the element size given by he ¼ xe þ 1 xe . A contact finite element analysis model should be established to consider the motion of moving mass and the constraints given in Eqs. (16) and (17). For this purpose, this study uses the contact finite element method presented by Zhong [25]. The gap g between the beam and the moving mass, defined by Eq. (15), can be rewritten as g ¼ rT d
(22)
where d is the system displacement vector including the beam and moving mass displacements and r is the system contact coefficient vector: ( ) db rb d¼ (23) , r¼ x 1 Considering whether the moving mass contacts the beam or not, the equations of motion for the beam–moving mass system are dependent on the sign of g. Therefore, the discretized equations corresponding to Eqs. (16) and (17) are represented by Md€ þKd ¼ f þ f c r, when g ¼ 0
(24)
Md€ þ Kd ¼ f, when g 40
(25)
where M and K are the system mass and stiffness matrices, respectively, and f is the system load vector: Kb 0 Mb 0 fb , K¼ , f¼ M¼ 0 m 0 k 0
(26)
When the gap between the beam and moving mass is equal to zero, Eq. (24) should be applied to the analysis because contact force is generated. Contrarily, if the gap is larger than zero, Eq. (25) is adopted during the analysis. In order to obtain the time response and contact force between the tensioned beam and the moving mass, the generalized-a time integration method is used [26]. Before applying the generalized-a method, it is necessary to define the _ and d€ at time t ¼t as d , v , and a , respectively. In the generalized-a method, Eqs. (22)–(25) approximate values of d, d, n n n n are transformed to g ¼ rTn þ 1af dn þ 1af
(27)
Man þ 1am þ Kdn þ 1af ¼ f n þ 1af þ f c rn þ 1af , when g ¼ 0
(28)
K. Lee et al. / Journal of Sound and Vibration 331 (2012) 2520–2531
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Man þ 1am þ Kdn þ 1af ¼ f n þ 1af , when g 4 0
(29)
where f c , f n þ 1af and rn þ 1af are computed at time t n þ 1af ¼ ð1af Þt n þ 1 þ af t n and dn þ 1af ¼ ð1af Þdn þ 1 þ af dn ,
an þ 1am ¼ ð1am Þan þ 1 þ am an
(30)
The updated equations for the displacement and velocity vectors are expressed as dn þ 1 ¼ d~ n þ bDt 2 an þ 1 ,
vn þ 1 ¼ v~ n þ g Dtan þ 1
(31)
where d~ n and v~ n are the predictors of the displacement and velocity given by d~ n ¼ dn þ Dtvn þð1=2bÞ Dt 2 an ,
v~ n ¼ vn þð1gÞ Dtan
(32)
In the above equations, Dt is the time step size between t n and t n þ 1 (i.e., Dt ¼ t n þ 1 t n ), and af, am, b, and g represent the algorithmic parameters of the generalized-a method. In this study, the following values are used: af ¼ am ¼1/2, b ¼1/4 and g ¼1/2. See Ref. [26] for more details about the generalized-a method. The Lagrange multiplier method is used to solve the contact problem between the moving mass and tensioned beam. When there is no contact between the mass and beam or when g 40, the Lagrange multiplier method is not required to obtain the dynamic response because Eq. (25) can be solved easily. However, when the mass and beam have contact or when g ¼ 0, the algebraic equation, Eq. (22), and the differential equation, Eq. (24), need to be solved simultaneously. In order to solve the algebraic-differential equations, the Lagrange multiplier method should be used. Consider the solution procedure using the Lagrange multiplier method. To obtain the time responses dn and vn, the acceleration an þ 1 needs to be computed. Substituting Eqs. (30) and (31) into Eq. (28), the following equation was obtained: Aan þ 1 f c rn þ 1af ¼ F
(33)
where A ¼ ð1am ÞM þð1af ÞbDt 2 K,
F ¼ f n þ 1af am Man ð1af ÞKd~ n af Kdn
(34)
As shown in Eq. (33), this equation cannot be solved because it has two unknowns of an þ 1 and fc. One more equation, which describes the kinematic constraint, is required. This equation is given by g ¼ rTn þ 1af dn þ 1af ¼ 0
(35)
Introduction of Eqs. (30) and (31) to Eq. (35) leads to ð1af ÞbDt 2 rTn þ 1af an þ 1 ¼ G
(36)
G ¼ ð1af ÞrTn þ 1af d~ n af dn
(37)
where
Therefore, combining Eqs. (33) and (36), the Lagrange multiplier method to determine an þ 1 and fc can be expressed as the following matrix–vector equation: 2 3( ) A rn þ 1af an þ 1 F 4 5 ¼ (38) ð1af ÞbDt 2 rTn þ 1af 0 fc G After solving Eq. (38) for the updated acceleration an þ 1 and contact force fc, the updated displacement dn þ 1 and updated velocity vn þ 1 can be determined by using Eqs. (31) and (32). 4. Analysis and discussion The proposed finite element formulation is verified by comparing the result obtained in this study with the result of Ref. [27]. The untensioned cantilever beam with a moving mass is shown in Fig. 4, where the mass m travels along the beam with a constant speed V. The gravitational effect of only the mass is considered in the computation for comparison. This means that the gravity effect of the beam is neglected. It is assumed that the initial position of the mass is the left end of the beam and the initial beam deflection is zero. Furthermore, it is also assumed that there is no separation between the mass and beam. The following physical parameters are used for computation: m¼2629 kg, V¼50.8 m/s, L¼7.62 m,
y m
V
x L Fig. 4. Untensioned cantilever beam with a moving mass.
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E¼ 2.07 1011 N/m2, I¼ 4.58 10 5 m4 and rA ¼46 kg/m. The vertical displacements at the free end of the cantilever beam are plotted in Fig. 5, where the solid line represents the displacement computed by the proposed finite element formulation while the triangular symbol represents the displacement computed by Bowe and Mullarkey [27]. As shown in Fig. 5, the result obtained in this study is in good agreement with the result of Bowe and Mullarkey. Before discussing contact analysis, it is necessary to analyze the wave propagation speed in the tensioned beam because the contact force is influenced by the encounter between the moving mass and the reflected wave. The encounter position is also related to the speeds of both the moving mass and the reflected wave. In general, the wave propagation speed in the tensioned beam varies, as does the wave speed in the string. This means that wave speed varies with the wavelength. The wave speed in the tensioned beam, c, is given by vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u u4p2 EI þ l2 T c¼t (39) rAl2 flexible rigidity EI is negligible when compared where l is the wavelength. Since the beam of this study is very slender, the ffi pffiffiffiffiffiffiffiffiffiffiffi to the tension T. In this case, the wave speed may be expressed by c ¼ T=rA. Therefore, the wave propagation speed of the tensioned slender beam can be assumed as the speed of the string case [9]. In order to verify this assumption, the contact force is analyzed when the moving mass velocity and gravity force are zero. The stationary mass is located 1 m from the left end of the beam, and the initial compression of the spring is given by 0.05 m. The mechanical properties and dimensions used in the computations in this paper, if there is no other specific remark, are given by m¼ 1 kg, k¼9.87 102 N/m, L¼50 m, A ¼1.50 10 4 m2, E¼1.34 1011 N/m2, I ¼9.64 10 10 m4, r ¼ 8900 kg/m3, and T ¼3.0 104 N. The beam mass M is computed by M ¼ rAL. To evaluate the effect of flexible rigidity on the wave speed, the force variation when EI¼0 is compared with the variation when EIa0. The time histories of the contact forces for the both cases of EI¼ 0 and a0 are plotted in Fig. 6, when the mass is stationary. The contact force variations are caused by the encounters between the mass and the reflected wave. It is shown in Fig. 6 that the periodic variations of the contact forces similar for the both cases, regardless of the existence of flexible rigidity. The period of contact force wave is about 0.657 s and the wave travel distance during this period is 98 m. If this distance is divided by 0.01
Vertical Displacement (m)
0.00 -0.01 -0.02 -0.03 -0.04 -0.05
Present study Ref. [27]
-0.06 -0.07 0.00
0.03
0.06
0.09
0.12
0.15
Time (s) Fig. 5. Vertical displacement at the free end of the cantilever beam when the mass moves to the right with a constant speed of 50.8 m/s.
61 60
fc (N)
59
EI = 0
58 57 EI ≠ 0
56 55
0
1
2
3
4
5
t (s) Fig. 6. Time histories of the contact forces for EI¼ 0 and a0 when the stationary mass is located at x¼ 1 m and the initial compression of spring is 0.05 m.
K. Lee et al. / Journal of Sound and Vibration 331 (2012) 2520–2531
2527
the wave period, the wave speed is computed as 149.2 m/s. This computed wave is nearly the same as the theoretical wave speed of 149.9 m/s. It may be concluded that tension is a dominant parameter in determining the wave speed in the tensioned beam. 4.1. Effect of the moving mass velocity This study uses a velocity profile with continuous acceleration, which can avoid a large contact force due to abrupt change of acceleration. The velocity profile with continuous acceleration may be expressed as 8
< V 0 t 1 sin 2pt if 0 rt r t 0 t0 2p t0 VðtÞ ¼ (40) : V0 if t 0 rt r t f where tf and t0 represent the time for the moving mass to travel the lengths L and L/5, respectively, and V0 is a constant moving velocity after time t0. Since integration of the velocity given in Eq. (40) from 0 to t0 is equal to L/5, the relation between t0 and V0 can be expressed as t0 ¼2L/(5V0). The velocity profile corresponding to Eq. (40) is plotted in Fig. 7. For the change of the moving mass velocity, the contact force and the loss of contact are investigated. The ffi pffiffiffiffiffiffiffiffiffiffiffi dimensionless velocity V/c, where c is the wave speed given by c ¼ T=rA, changes from 0.4 to 1.3. Using the discretized equations of Eqs. (27)–(32) and (38), the contact force between the moving mass and the tensioned beam is calculated. Fig. 8 shows the contact analysis results for the travel distance of the moving mass versus the velocity of the moving mass. The contact forces for the various sets of x/L and V/c are plotted in Fig. 8(a), and the contact loss is presented in Fig. 8(b). In general, the wave with a uniform speed propagates through the material and is then reflected at the boundary. When the position of the moving mass changes by the velocity given in Eq. (40), the reflected wave and moving mass encounter each other and cause the variation of contact force. As the dimensionless velocity of the moving mass increases from 0.4 to 1.3 in this study while the wave speed is uniform at 149.9 m/s, the first encounter of the moving mass and the wave occurs and the encounter
V V0
t0
tf
t
Fig. 7. Profile of the moving mass velocity without sudden change of acceleration.
Fig. 8. Contact analysis results for the variations of the moving mass velocity V and the travel distance of the moving mass x: (a) the contact force and (b) the loss of contact.
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position moves to the right. For this reason, the contact force variation line due to the first encounter can be clearly seen in Fig. 8(a), and the contact force variation line of the second encounter can be seen near the boundary. This figure also shows that the contact force corresponding to the first encounter becomes maximized when the dimensionless velocity V/c is near to the unity or when the moving velocity approaches the wave speed. It is interesting to observe that the contact force becomes zero near the boundary when the dimensionless velocity is about 0.7–0.9. This means that the fluctuation of contact force due to the encounter of the moving mass and the reflected wave may cause the separation between the moving mass and beam. The loss of contact occurs when the contact force becomes zero. In order to conveniently discuss this phenomenon, a two-dimensional plot is presented in Fig. 8(b), which shows the contact loss. The black area represents separation region while the white area represents the contact region. It is shown that the contact condition near the boundary of the beam is maintained when V/c is less than 0.7 or larger than 0.9, while contact loss occurs when V/c is about 0.7–0.9. Therefore, in order to obtain safe contact conditions, the dimensionless velocity should avoid the range of 0.7–0.9. 4.2. Effect of the tension The contact force variations are observed when the dimensionless tension T/EA changes. The mechanical properties for computation are the same as mentioned before, but the constant moving mass velocity V0 is given by 100 m/s. The contact analysis results for the travel distance versus the applied tension are illustrated in Fig. 9, where T/EA varies from 0.001 to 0.0018. It is shown in this figure that the overall contact force when the dimensionless tension of 0.001 is applied is much larger than the contact force when the dimensionless tension of 0.002 is applied. The reason for this is that the contact force is influenced to a larger extent by the beam flexibility at the tension of 0.001 than at the tension of 0.002. As the transverse flexibility of the beam decreases with the applied tension, the beam deflections due to the flexibility are reduced with the increase of tension. Thus, the overall contact force decreases as the tension increases. Moreover, in Fig. 9(a), the force fluctuation due to the encounter of the moving mass and the reflected wave is shown to become much larger when low tension, rather than high tension, is applied to the beam. The wave speed at the dimensionless tension of 0.001 is determined to be 122.7 m/s. When this wave speed is close to the moving mass speed of 100 m/s, rapid contact force variation caused by the encounter of the moving mass and the reflected wave is observed near the right end of the beam, as shown in Fig. 9(a). However, rapid contact force variation is not observed at the dimensionless tension of 0.002 because the wave speed at this tension is much faster than the moving mass velocity. It is also shown that the first encounter position of the moving mass and the reflected wave moves to the left when the tension increases. The reason for this is that the wave speed increases according to the increase of tension and the wave for higher tension moves faster when compared to the wave for lower tension. Therefore, the first encounter positions shifts to the left and the contact force magnitude decreases with the tension. The two-dimensional plot of the contact loss is also investigated, as illustrated in Fig. 9(b), where the black area is the region of contact loss and the white area is the contact region. It can be seen that the contact loss occurs near the boundary when the dimensionless tension changed from 0.0010 to 0.0012. Therefore, in order to obtain safe contact conditions and to minimize the variation in contact force, a large value of tension is desirable because a large tension makes the wave speed much faster than the speed of the moving mass.
Fig. 9. Contact analysis results for the variations of the beam tension T and the travel distance of the moving mass x: (a) the contact force and (b) the loss of contact.
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4.3. Effects of the mass and spring of the moving system The contact force and the contact loss between the beam and moving mass are investigated when the mass of the moving system changes with other mechanical properties specified before. The contact force for the variation of the moving mass and the travel distance is presented in Fig. 10(a), where the dimensionless mass m/M changes from 0.001 to 0.040. As shown in this figure, the overall contact force as well as the force fluctuation becomes large as the mass increases. The loss of contact between the beam and moving mass is illustrated in Fig. 10(b), from which it is observed that the contact loss occurs at a relatively large value of the mass. Therefore, it may be concluded that a large mass of the moving system has a tendency to increase the contact force fluctuation and contact loss. The contact force variation and contact loss are also analyzed when the stiffness of the moving system changes. The dimensionless stiffness of the spring, which is defined by k/(T/L), is changed from 0.1 to 4, while the other parameter values for computations are the same as before. The contact analysis results for the variation of the spring stiffness and the travel distance are presented in Fig. 11. It is observed in Fig. 11(a) that the overall contact force at high stiffness is much larger than the force at low stiffness. Moreover, rapid contact force variations are shown at high stiffness near the boundary. Despite the rapid contact force variation, contact loss does not occur at high stiffness shown in Fig. 11(b). Hence, the rapid contact force variation is not a sufficient condition for generating contact loss. However, regardless of the contact maintenance, excessively high contact force is undesirable because it may increase the wear of the beam and moving mass. Therefore, a low contact force is recommended if contact is guaranteed. On the other hand, as shown in Fig. 11(b), contact loss is observed at a low dimensionless stiffness region of 0.1–1.7. In order to obtain safe contact conditions, an extremely low stiffness of the moving system should be avoided.
4.4. Possibility of contact loss The Possibility of contact loss for the variations of the tension and velocity are also investigated. The Possibility of contact loss, which is denoted by Pl, is defined as the contact loss time divided by the total analysis time. For example, if contact loss is not observed during the analysis time, the Possibility of contact loss becomes zero. The Possibility of contact loss is plotted in Fig. 12 when the tension and the velocity change. Fig. 12(a) shows the three-dimensional mesh plot of the contact loss Possibility while Fig. 12(b) shows the contour plot corresponding to Fig. 12(a). These figures demonstrate that loss of contact occurs only in a certain region, such as region B. Regions A and C correspond to stable contact conditions because these regions have no contact loss. In these regions, any parameter combination of the tension and velocity is useful to obtain a stable contact condition. The border lines dividing regions A–C are closely related to the ratio of the moving pffiffiffiffiffiffiffiffiffiffiffiffimass velocity to the wave speed. As mentioned above, the wave speed c for a given tension T is computed by c ¼ T=rA. Consider the border line between regions A and B. This border line can be described approximately by the curve defined by V/c ¼0.64. Similarly, the border line between regions B and C can be approximated by the curve of V/c¼0.96. In Fig. 12(b), the curves of V/c ¼0.64 and 0.96
Fig. 10. Contact analysis results for the variations of the mass of moving mass system m and the travel distance of the moving mass x: (a) the contact force and (b) the loss of contact.
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Fig. 11. Contact analysis results for the variations of the stiffness of the moving mass system k and the travel distance of the moving mass x: (a) the contact force and (b) the loss of contact.
Fig. 12. Possibility of contact loss for the variations of the moving mass velocity V and the beam tension T: (a) the three-dimensional mesh plot and (b) the contour plot.
are plotted with dashed lines. Even though a theoretical background for these border lines cannot be found in this study, it is clear that the Possibility of contact loss becomes high for a specific range of V/c. For the case of this study, the specific range of V/c is from 0.64 to 0.96. Therefore, in order to minimize contact loss, the moving velocity should be selected out of the rage of V/c corresponding to contact loss.
5. Conclusions Dynamic contact analysis between the tensioned beam and the moving mass–spring system is performed when the beam with clamped-pinned ends contacts the moving mass–spring system. This study models the tensioned beam as an Euler–Bernoulli beam, and the moving-mass system as a single-degree-of-freedom system. The dynamic contact equations are derived based on this model, which describe the transverse displacements of the beam and moving mass. These equations are discretized using the finite-element method, and the contact analysis is performed using the Lagrange
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multiplier method. The dynamic behaviors of the beam and moving mass are investigated and the safe contact conditions are discussed. The results of the contact analysis can be summarized as follows: (1) The magnitude of the contact force between the moving mass and the beam becomes large when the moving mass velocity approaches the wave speed. (2) The fluctuation of contact force due to the encounter of the moving mass and the reflected wave may cause the separation between the moving mass and beam. (3) The contact loss occurs near the boundary of the beam when the dimensionless moving mass speed V/c is in a specific range. (4) In order to minimize the variation in contact force, a large beam tension, low moving mass, and low moving system stiffness are desirable for given values of the speed of the moving system. (5) In order to obtain safe contact conditions, a large beam tension, high moving system stiffness, and low moving mass are desirable.
Acknowledgments This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) grant funded by the Ministry of Education, Science, and Technology (2010-0016354). References [1] J.R. Rieker, Y.H. Lin, M.W. Trethewey, Discretization considerations in moving load finite element beam models, Finite Element in Analysis and Design 21 (1996) 129–144. [2] J.R. Rieker, M.W. Trethewey, Finite element analysis of an elastic beam structure subjected to a moving distributed mass train, Mechanical Systems and Signal Processing 13 (1999) 31–51. [3] U. Lee, Separation between the flexible structure and the moving mass sliding on it, Journal of Sound and Vibration 209 (1998) 867–877. [4] D. Stancioiu, H. Ouyang, J.E. Mottershead, Dynamics of a beam and a moving two-axle system with separation, Proceeding of the Institution of Mechanical Engineering Part C-Journal of Mechanical Science 222 (2008) 1947–1956. [5] H.P. Lee, On the separation of a mass travelling on a beam with axial forces, Mechanics Research Communications 22 (1995) 371–376. [6] J.R. Ockendon, A.B. Tayler, The dynamics of a current collection system for an electric locomotive, Proceedings of the Royal Society of London. Series A, Mathematical and Physical Science 322 (1971) 447–468. [7] G. Poetsch, J. Evans, R. Meisinger, W. Kortum, W. Baldauf, A. Veitl, J. Wallaschek, Pantograph/catenary dynamics and control, Vehicle System Dynamics 28 (1997) 159–195. [8] M. Arnold, B. Simeon, Pantograph and catenary dynamics: a benchmark problem and its numerical solution, Applied Numerical Mathematics 34 (2000) 345–362. [9] T.J. Park, B.J. Kim, Y.Y. Wang, C.H. Han, A catenary analysis for studying the dynamic characteristic of a high speed rail pantograph, KSME International Journal 16 (2002) 436–447. [10] G. Teichelmann, M. Schaub, B. Simeon, Modelling and simulation of railway cable systems, Zamm-Zeitschrift fur Angeeandte Mathematik und Mechanik 85 (2005) 864–877. [11] Y.H. Cho, K. Lee, Y. Park, B. Kang, K. Kim, Influence of contact wire pre-sag on the dynamics of pantograph-railway catenary, International Journal of Mechanical Science 52 (2010) 1471–1490. [12] T.X. Wu, M.J. Brennan, Basic analytical study of pantograph-catenary system dynamics, Vehicle System Dynamics 30 (1998) 443–456. [13] K. Lee, Analysis of dynamic contact between overhead wire and pantograph of a high-speed electric train, Proceeding of the Institution of Mechanical Engineers Part F-Journal of Rail and Rapid Transit 221 (2007) 157–166. [14] A. Collina, S. Bruni, Numerical simulation of pantograph–overhead equipment interaction, Vehicle System Dynamics 38 (2002) 261–291. [15] P. Harell, L. Drugge, M. Reijm, Study of critical sections in catenary systems during multiple pantograph operation, Proceeding of the Institution of Mechanical Engineers Part F-Journal of Rail and Rapid Transit 219 (2005) 203–211. [16] Y.H. Cho, Numerical simulation of the dynamic responses of railway overhead contact lines to a moving pantograph, considering a nonlinear dropper, Journal of Sound and Vibration 315 (2008) 433–454. [17] J.H. Seo, H. Sugiyama, A.A. Shabana, Three-dimensional large deformation analysis of the multibody pantograph/catenary systems, Nonlinear Dynamics 42 (2005) 199–215. [18] M. Aboshi, I. Nakai, H. Knoshita, Current collection characteristics and improvement methods of high-tension overhead catenary systems, Electric Engineering in Japan 123 (1998) 67–76. [19] T.J. Park, C.S. Han, J.H. Jang, Dynamic sensitivity analysis for the pantograph of a high-speed rail vehicle, Journal of Sound and Vibration 266 (2003) 235–260. [20] J. Chung, J.-E. Oh, H.H. Yoo, Non-linear vibration of a flexible spinning disc with angular acceleration, Journal of Sound and Vibration 231 (2000) 375–391. [21] J.W. Heo, J. Chung, Vibration analysis of a flexible rotating disk with angular misalignment, Journal of Sound and Vibration 231 (2004) 821–841. [22] D. Jung, J. Chung, In-plane and out-of-plane motions of an extensible semi-circular pipe conveying fluid, Journal of Sound and Vibration 311 (2008) 408–420. [23] Y.S. Cheng, F.T.K. Au, Y.K. Cheung, D.Y. Zheng, On the separation between moving vehicles and Bridge, Journal of Sound and Vibration 222 (1999) 781–801. [24] D. Stancioiu, H. Ouyang, J.E. Mottershead, Vibration of a beam excited by a moving oscillator considering separation and reattachment, Journal of Sound and Vibration 310 (2008) 1128–1140. [25] Z.-H. Zhong, Finite Element Procedures for Contact-impact Problems, Oxford University Press, Oxford, 1993. [26] J. Chung, G.M. Hulbert, A time integration algorithm for structural dynamics with improved numerical dissipation: the generalized-a method, American Society of Mechanical Engineers Journal of Applied Mechanics 60 (1993) 371–375. [27] C.J. Bowe, T.P. Mullarkey, Unsprung wheel–beam interactions using modal finite element models, Advances in Engineering Software 39 (2008) 911–922.
Contents lists available at SciVerse ScienceDirect
Journal of Sound and Vibration journal homepage: www.elsevier.com/locate/jsvi
Dynamic contact analysis of a tensioned beam with a moving mass–spring system Kyuho Lee a, Yonghyeon Cho b, Jintai Chung a,n a b
Department of Mechanical Engineering, Hanyang University, 1271 Sa-3-dong, Sangnok-gu, Ansan, Kyeonggi-do 425-791, Republic of Korea Korea Railroad Research Institute, Woram-dong, Uiwang City, Kyeonggi-do 437-757, Republic of Korea
a r t i c l e in f o
abstract
Article history: Received 11 June 2011 Received in revised form 12 January 2012 Accepted 12 January 2012 Handling Editor: H. Ouyang Available online 6 February 2012
The dynamic contact problem of a tensioned beam with clamped-pinned ends is analyzed when the beam contacts a moving mass–spring system. The contact and contact loss conditions are expressed in terms of constraint equations after considering the dynamic contact between the beam and the moving mass. Using these constraints and equations of motion for the beam and moving mass, dynamic contact equations are derived and then discretized using the finite element method, which is based on the Lagrange multiplier method. The time responses for the contact forces are computed from these discretized equations. The contact force variations and contact loss are investigated for the variations of the moving mass velocity, the beam tension, the moving mass, and the stiffness of the moving mass–spring system. In addition, the possibility of contact loss and safe contact conditions between the moving mass and the tensioned beam are also studied. & 2012 Elsevier Ltd. All rights reserved.
1. Introduction The dynamic analyses of flexible tensioned beams with moving mass or load are very important in engineering because such structures are widely used in various applications, e.g., contact wires in high-speed railways, suspension bridges, overhead cranes, and tethered satellite systems. Many recent studies of beams with moving mass focus on railway systems related to the development of high-speed railway systems. In these applications, system properties determine the dynamic behavior and stability of high-speed railways. Dynamic stability is crucial in contact conditions because it concerns the collection of electric energy in a high-speed railway. If loss of contact or sudden change of the contact force occurs during railway operations, serious problems such as suspension of service or derailment may occur due to the unstable collection of electric energy. Therefore, stable contact conditions should be derived to ensure safe operation of high-speed railways. Many researchers have studied the dynamic behaviors and contact of flexible tensioned beams subjected to moving masses or loads at high speed. Rieker et al. [1] and Rieker and Trethewey [2] performed finite element analyses of beams subjected to moving masses and they assumed that the moving mass is always attached to the beam. However, this assumption does not represent contact loss or contact force between the beam and the moving mass. Lee [3] and Stancioiu et al. [4] considered the separation of the beam from the moving-mass system. It was found in [4] that the response was closer to the experimental results when loss of contact due to separation was considered. On the other hand, Lee [5]
n
Corresponding author. Tel.: þ 82 31 400 5287; fax: þ82 31 406 5550. E-mail address: [email protected] (J. Chung).
0022-460X/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.jsv.2012.01.014
K. Lee et al. / Journal of Sound and Vibration 331 (2012) 2520–2531
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analyzed the influence of an axial force on the onset of separation between a moving mass and beam. He found that separation of the mass from the beam could be avoided by the application of suitable tensile axial forces. As tensioned beams with moving mass are often examined in the context of high-speed railway systems, governing equations for the tensioned beam have been introduced in numerous studies [6–11]. Park et al. [9], Wu and Brennan [12], and Lee [13] performed finite-element analyses of tensioned beams with moving mass while considering the contact between the beam and the mass. Many researchers used the penalty method in order to solve these contact problems, such as wire-pantographs used for high-speed railways [14–16]. However, in the penalty method, the contact constraints for preventing penetration were not imposed precisely because the penalty method uses an assumed parameter that may change both the contact conditions and contact force. Therefore, Teichelmann et al. [10], Lee [13], and Seo et al. [17] carried out contact analyses using the Lagrange multiplier method, which introduces the unknown contact force as the Lagrange multiplier. This method is able to obtain more precise results than the penalty method. Although many analyses of dynamic contact between beam and moving mass have been conducted, there have been few studies of the stable contact conditions between a simple tensioned beam and a moving mass because the majority of previous dynamic contact analyses have focused on the complex wire/pantograph in high-speed railways. Aboshi et al. [18] studied the causes of contact loss and confirmed that the contact force fluctuation is influenced by the wave reflection. Specifically, they found that rubber damping hangers or friction damping hangers reduce contact loss of the pantograph and wear of the contact wire. Park et al. [19] investigated the optimal design parameters of pantographs in order to minimize contact loss. However, they did not investigate contact loss or force variation for various design parameters, such as tension, velocity, moving mass, and stiffness. In this study, we analyze the dynamic contact problem of a tensioned beam with clamped-pinned ends using the finite element method when the beam is in contact with a moving mass–spring system. When the tension is considered in the beam and the moving mass–spring system is assumed to have a single degree of freedom, the equations for the beam motion are derived from the Euler beam theory. Furthermore, the dynamic contact conditions are expressed in terms of constraint equations. With these constraints and equations of motion for the beam and moving mass, dynamic-contact equations are derived and discretized through the finite element method using the Lagrange multiplier method. From these equations, the time responses are computed for the contact forces. Moreover, the contact force variations are simulated for various design parameters, such as the moving speed, tension, the mass and stiffness of the moving mass system. Based on these simulations, safe contact conditions between tensioned beam and the moving mass are obtained. These conditions must satisfy the small magnitude variation of the contact force and minimize the contact loss. The possibility of contact loss is also investigated in this study. 2. Modeling and equation of motion Consider an extensible straight beam with uniform tension and a moving mass–spring system, as shown in Fig. 1, where Figs. 1(a) and (b) represent the beam before and after deformation, respectively. The straight beam, fixed at the left end and supported by a roller at the right end, has length L, cross-sectional area A, Young’s modulus E, density r, and distributed force per unit length p(x, t). The beam with tension T applied at the right end is excited by the moving mass–spring system with a time-variant velocity V. The moving mass–spring system has mass m and spring constant k. Due to this excitation, which is caused by the moving mass, contact force and displacements occur between the tensioned
y
L p ( x, t ) x
T
m
k
V
y
p
v
ξ
u
T
x
V Fig. 1. Schematics of a tensioned beam with a moving mass–spring system: (a) before deformation and (b) after deformation.
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beam and the moving mass. The displacement of the moving mass is represented as x(x, t). There is a contact force fc between the beam and moving mass when the mass contacts the beam. It is assumed in this study that the contact force has only normal directional force because the tangential contact force is negligible. This study derives the equations of motion for the beam using the Euler beam theory because the beam length L is much longer than the cross-sectional dimensions. In order to consider large deflection, the von Karman strain theory is used along with the Euler beam theory. According to the Euler beam theory, the deflections in the x and y directions, ux(x, y, t) and uy(x, y, t), are expressed by ux ðx,y,tÞ ¼ uðx,tÞy
@vðx,tÞ , @x
uy ðx,y,tÞ ¼ vðx,tÞ
(1)
where u(x, t) and v(x, t) are the longitudinal and transverse displacements of an arbitrary point on the centerline of the beam. In order to consider nonlinearity due to large deflection of the beam, the longitudinal normal strain ex of the von Karman strain theory is used: @u 1 @uy 2 ex ¼ x þ (2) 2 @x @x Next, consider the stress–strain relationship of the beam. It is assumed that the material of the beam is homogeneous, isotropic, and elastic. Since the tensioned beam is slender and the cross-sectional area is very small when compared to the other dimensions, the linearized stress is adopted in this study. The linearized axial stress sLx is represented by
sLx ¼ EeLx
(3)
where eLx is given by the linear term of Eq. (2). The equations of motion for the tensioned beam and the moving mass–spring system are obtained from the Hamilton principle, which is written as Z t2 ðdKdU þ dWÞ dt ¼ 0 (4) t1
where d is the variation operator, t1 and t2 are any instances, K is the kinetic energy, U is the potential energy, and W is the work done by non-conservative forces. The rotary inertia effect of the beam is ignored because the beam of this study is thin. For this case, the kinetic energy for the beam and moving mass–spring system is expressed as Z L " 2 2 # 2 1 @u @v 1 K ¼ rA þ (5) dx þ mðx_ þ V 2 Þ 2 @t @t 2 0 where the superposed dot stands for differentiation with respect time. Since the geometric nonlinearity is a source of the nonlinear beam behavior, the nonlinear strain and the linearized stress are used to derive the equations of motion. This type of modeling can be found in Refs. [20–22]. Using this modeling, the variation of the potential energy for the tensioned beam with the moving mass–spring system is given by Z LZ dU ¼ sLx dex dA dx þkxdx (6) 0
A
When the distributed force p is applied to the beam and the moving mass is located at x¼xm, the virtual work done by the non-conservative forces is represented by Z L dW ¼ ½p þf c dðxxm Þdv dx þ f c dx (7) 0
where dðxxm Þ represents the Dirac delta function at x¼xm. By introducing Eqs. (5)–(7) into Eq. (4), the nonlinear equations of motion for the beam and moving mass can be derived as follows: @2 u @2 u EA 2 ¼ 0 2 @x @t
(8)
@2 v @4 v @ @u @v ¼ p þ f c dðxxm Þ þ EI EA @x @x @x @x4 @t 2
(9)
rA rA
mx€ þkx ¼ f c
(10)
where I is the area moment of inertia about the centerline of the beam. The corresponding boundary conditions are given by u¼v¼
@v ¼ 0 at x ¼ 0 @x
(11)
K. Lee et al. / Journal of Sound and Vibration 331 (2012) 2520–2531
v¼
@2 v ¼ 0, @x2
EA
2523
@u ¼ T at x ¼ L @x
(12)
The equations of motion, given in Eqs. (8)–(10), can be simplified under the assumption that the longitudinal displacement u is much smaller than the transverse displacement v. With this assumption, we can consider that the axial normal strain qu/qx is uniform along the beam and independent of time. This means that there is no need to solve Eq. (8) because the time response of the longitudinal displacement is always constant. Since EAqu/qx in Eq. (9) is replaced by tension T, the equations of motion can be rewritten as
rA
@2 v @4 v @2 v þ EI 4 T 2 ¼ p þ f c dðxxm Þ 2 @x @x @t
(13)
mx€ þ kx ¼ f c
(14)
The relative gap between the tensioned beam and the moving mass is important because it has an influence on the dynamic behaviors of the beam and mass. The moving mass is not always attached to the tensioned beam, which means that separation is possible due to the relative displacements between the beam and mass. Therefore, the equations of motion should be considered for both cases involving contact and non-contact. First, in order to define the conditions of contact and non-contact, the relative displacement between the moving mass and the beam should be considered. The gap between the beam and the mass is defined by gðx,tÞ ¼ vðx,tÞxðx,tÞ at x ¼ xm
(15)
When g(x, t)40, contact does not occur because the position of the beam at x¼xm is higher than the position of the moving mass. Conversely, when g(x, t)¼ 0, contact is generated because the relative displacement between the beam and the moving mass is zero. When the gap is equal to zero, the Lagrange multiplier method should be used in order to prevent penetration between the beam and mass. Next, the equations of motion considering the loss of contact are investigated. As shown in Eqs. (13) and (14), they are divided by the gap condition. When g ¼0, a contact force fc is applied to the equations; however, when g40, it is not applied. It is ineffectual to solve these equations independently, because the moving mass is not always attached to the tensioned beam. When the moving mass contacts the beam, the contact force has a nonzero value and the equations of motion are expressed as
rA
@2 v @4 v @2 v þ EI 4 T 2 ¼ p þf c dðxxm Þ, mx€ þ kx ¼ f c , when g ¼ 0 @x @x @t 2
(16)
On the other hand, when the mass is separated from the beam, the contact force becomes zero and the equations of motion are given by
rA
@2 v @4 v @2 v þ EI 4 T 2 ¼ p, 2 @x @x @t
mx€ þ kx ¼ 0, when g 40
(17)
After the mass and beam are separated, the reattachment between them is judged by the value of gap g. If the value of g changes from a positive value to zero, the moving mass becomes reattached to the beam and the motions of the mass and beam should be described by Eq. (16). If the moving mass is quite less than the beam mass, the impact between the mass and beam can be neglected during reattachment. Otherwise, the impact should be considered because the momentum of the mass is not negligible. In fact, Cheng et al. [23] and Stancioiu et al. [24] showed that the impact during reattachment has an influence on the dynamic responses of the beam with moving mass. However, in this study, the impact at reattachment is not considered. 3. Finite element formulation with the Lagrange multiplier method In order to obtain approximate solutions from the derived equations of motion and boundary conditions, the finite element method is used in this study. The equations of motion and the boundary conditions are transformed into their weak forms. Next, the weak forms are spatially discretized with two-node beam elements, as shown in Fig. 2. In this figure, the numbers above the beam elements represent the element numbers, and the numbers below the beam elements represent the node numbers. The discretized equations for the beam motion can be expressed as Mb d€ b þKb db ¼ f b þ f c rb
1
e
2
1
2
(18)
3
e
N
e +1
N
N +1
Fig. 2. Element and node numbers of the finite element model for the tensioned beam.
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e
e +1
e m
a he Fig. 3. Model for the calculation of vector rb when the mass, which has distance a from node e, is located on the eth element of the finite element model for the tensioned beam.
where db is the global displacement vector, Mb and Kb are the global mass and stiffness matrices of the beam, and fb and rb are the global distributed force and contact coefficient vector, respectively. The global matrices and vectors are represented by T
db ¼ fv2 , y2 ,v3 , y3 ,. . .,vN , yN , yN þ 1 g
(19)
As v1, y1, and vN þ 1 are known degrees of freedom or non-active degrees of freedom in the established model, they are not considered in Eq. (19). When the global contact coefficient vector rb is assembled, the position of the moving mass should be carefully considered because rb changes according to the position of the moving mass. If the moving mass is on the eth element and the distance from node e to the moving mass is a, as shown in Fig. 3, the coefficient vector rb is given by T
rb ¼ f0,0,. . .,N e1 ,Ne2 ,Ne3 ,N e4 ,. . .,0,0g In Eq. (20), expressed as
Ne1 ,
N e2 ,
N e3
and
Ne4
(20)
are the eth, (eþ1)th, (eþ2)th and (eþ3)th elements of the rb vector and they are 3
Ne1 ¼ ðaxe þ 1 Þ2 ð2a3xe þxe þ 1 Þ=he , 3 Ne3 ¼ ðaxe Þ2 ð2a þ xe 3xe þ 1 Þ=he ,
2
N e2 ¼ ðaxe Þðaxe þ 1 Þ2 =he , 2
N e4 ¼ ðaxe Þ2 ðaxe þ 1 Þ=he
(21)
where xe is the nodal coordinate of node e and he represents the element size given by he ¼ xe þ 1 xe . A contact finite element analysis model should be established to consider the motion of moving mass and the constraints given in Eqs. (16) and (17). For this purpose, this study uses the contact finite element method presented by Zhong [25]. The gap g between the beam and the moving mass, defined by Eq. (15), can be rewritten as g ¼ rT d
(22)
where d is the system displacement vector including the beam and moving mass displacements and r is the system contact coefficient vector: ( ) db rb d¼ (23) , r¼ x 1 Considering whether the moving mass contacts the beam or not, the equations of motion for the beam–moving mass system are dependent on the sign of g. Therefore, the discretized equations corresponding to Eqs. (16) and (17) are represented by Md€ þKd ¼ f þ f c r, when g ¼ 0
(24)
Md€ þ Kd ¼ f, when g 40
(25)
where M and K are the system mass and stiffness matrices, respectively, and f is the system load vector: Kb 0 Mb 0 fb , K¼ , f¼ M¼ 0 m 0 k 0
(26)
When the gap between the beam and moving mass is equal to zero, Eq. (24) should be applied to the analysis because contact force is generated. Contrarily, if the gap is larger than zero, Eq. (25) is adopted during the analysis. In order to obtain the time response and contact force between the tensioned beam and the moving mass, the generalized-a time integration method is used [26]. Before applying the generalized-a method, it is necessary to define the _ and d€ at time t ¼t as d , v , and a , respectively. In the generalized-a method, Eqs. (22)–(25) approximate values of d, d, n n n n are transformed to g ¼ rTn þ 1af dn þ 1af
(27)
Man þ 1am þ Kdn þ 1af ¼ f n þ 1af þ f c rn þ 1af , when g ¼ 0
(28)
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Man þ 1am þ Kdn þ 1af ¼ f n þ 1af , when g 4 0
(29)
where f c , f n þ 1af and rn þ 1af are computed at time t n þ 1af ¼ ð1af Þt n þ 1 þ af t n and dn þ 1af ¼ ð1af Þdn þ 1 þ af dn ,
an þ 1am ¼ ð1am Þan þ 1 þ am an
(30)
The updated equations for the displacement and velocity vectors are expressed as dn þ 1 ¼ d~ n þ bDt 2 an þ 1 ,
vn þ 1 ¼ v~ n þ g Dtan þ 1
(31)
where d~ n and v~ n are the predictors of the displacement and velocity given by d~ n ¼ dn þ Dtvn þð1=2bÞ Dt 2 an ,
v~ n ¼ vn þð1gÞ Dtan
(32)
In the above equations, Dt is the time step size between t n and t n þ 1 (i.e., Dt ¼ t n þ 1 t n ), and af, am, b, and g represent the algorithmic parameters of the generalized-a method. In this study, the following values are used: af ¼ am ¼1/2, b ¼1/4 and g ¼1/2. See Ref. [26] for more details about the generalized-a method. The Lagrange multiplier method is used to solve the contact problem between the moving mass and tensioned beam. When there is no contact between the mass and beam or when g 40, the Lagrange multiplier method is not required to obtain the dynamic response because Eq. (25) can be solved easily. However, when the mass and beam have contact or when g ¼ 0, the algebraic equation, Eq. (22), and the differential equation, Eq. (24), need to be solved simultaneously. In order to solve the algebraic-differential equations, the Lagrange multiplier method should be used. Consider the solution procedure using the Lagrange multiplier method. To obtain the time responses dn and vn, the acceleration an þ 1 needs to be computed. Substituting Eqs. (30) and (31) into Eq. (28), the following equation was obtained: Aan þ 1 f c rn þ 1af ¼ F
(33)
where A ¼ ð1am ÞM þð1af ÞbDt 2 K,
F ¼ f n þ 1af am Man ð1af ÞKd~ n af Kdn
(34)
As shown in Eq. (33), this equation cannot be solved because it has two unknowns of an þ 1 and fc. One more equation, which describes the kinematic constraint, is required. This equation is given by g ¼ rTn þ 1af dn þ 1af ¼ 0
(35)
Introduction of Eqs. (30) and (31) to Eq. (35) leads to ð1af ÞbDt 2 rTn þ 1af an þ 1 ¼ G
(36)
G ¼ ð1af ÞrTn þ 1af d~ n af dn
(37)
where
Therefore, combining Eqs. (33) and (36), the Lagrange multiplier method to determine an þ 1 and fc can be expressed as the following matrix–vector equation: 2 3( ) A rn þ 1af an þ 1 F 4 5 ¼ (38) ð1af ÞbDt 2 rTn þ 1af 0 fc G After solving Eq. (38) for the updated acceleration an þ 1 and contact force fc, the updated displacement dn þ 1 and updated velocity vn þ 1 can be determined by using Eqs. (31) and (32). 4. Analysis and discussion The proposed finite element formulation is verified by comparing the result obtained in this study with the result of Ref. [27]. The untensioned cantilever beam with a moving mass is shown in Fig. 4, where the mass m travels along the beam with a constant speed V. The gravitational effect of only the mass is considered in the computation for comparison. This means that the gravity effect of the beam is neglected. It is assumed that the initial position of the mass is the left end of the beam and the initial beam deflection is zero. Furthermore, it is also assumed that there is no separation between the mass and beam. The following physical parameters are used for computation: m¼2629 kg, V¼50.8 m/s, L¼7.62 m,
y m
V
x L Fig. 4. Untensioned cantilever beam with a moving mass.
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E¼ 2.07 1011 N/m2, I¼ 4.58 10 5 m4 and rA ¼46 kg/m. The vertical displacements at the free end of the cantilever beam are plotted in Fig. 5, where the solid line represents the displacement computed by the proposed finite element formulation while the triangular symbol represents the displacement computed by Bowe and Mullarkey [27]. As shown in Fig. 5, the result obtained in this study is in good agreement with the result of Bowe and Mullarkey. Before discussing contact analysis, it is necessary to analyze the wave propagation speed in the tensioned beam because the contact force is influenced by the encounter between the moving mass and the reflected wave. The encounter position is also related to the speeds of both the moving mass and the reflected wave. In general, the wave propagation speed in the tensioned beam varies, as does the wave speed in the string. This means that wave speed varies with the wavelength. The wave speed in the tensioned beam, c, is given by vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u u4p2 EI þ l2 T c¼t (39) rAl2 flexible rigidity EI is negligible when compared where l is the wavelength. Since the beam of this study is very slender, the ffi pffiffiffiffiffiffiffiffiffiffiffi to the tension T. In this case, the wave speed may be expressed by c ¼ T=rA. Therefore, the wave propagation speed of the tensioned slender beam can be assumed as the speed of the string case [9]. In order to verify this assumption, the contact force is analyzed when the moving mass velocity and gravity force are zero. The stationary mass is located 1 m from the left end of the beam, and the initial compression of the spring is given by 0.05 m. The mechanical properties and dimensions used in the computations in this paper, if there is no other specific remark, are given by m¼ 1 kg, k¼9.87 102 N/m, L¼50 m, A ¼1.50 10 4 m2, E¼1.34 1011 N/m2, I ¼9.64 10 10 m4, r ¼ 8900 kg/m3, and T ¼3.0 104 N. The beam mass M is computed by M ¼ rAL. To evaluate the effect of flexible rigidity on the wave speed, the force variation when EI¼0 is compared with the variation when EIa0. The time histories of the contact forces for the both cases of EI¼ 0 and a0 are plotted in Fig. 6, when the mass is stationary. The contact force variations are caused by the encounters between the mass and the reflected wave. It is shown in Fig. 6 that the periodic variations of the contact forces similar for the both cases, regardless of the existence of flexible rigidity. The period of contact force wave is about 0.657 s and the wave travel distance during this period is 98 m. If this distance is divided by 0.01
Vertical Displacement (m)
0.00 -0.01 -0.02 -0.03 -0.04 -0.05
Present study Ref. [27]
-0.06 -0.07 0.00
0.03
0.06
0.09
0.12
0.15
Time (s) Fig. 5. Vertical displacement at the free end of the cantilever beam when the mass moves to the right with a constant speed of 50.8 m/s.
61 60
fc (N)
59
EI = 0
58 57 EI ≠ 0
56 55
0
1
2
3
4
5
t (s) Fig. 6. Time histories of the contact forces for EI¼ 0 and a0 when the stationary mass is located at x¼ 1 m and the initial compression of spring is 0.05 m.
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the wave period, the wave speed is computed as 149.2 m/s. This computed wave is nearly the same as the theoretical wave speed of 149.9 m/s. It may be concluded that tension is a dominant parameter in determining the wave speed in the tensioned beam. 4.1. Effect of the moving mass velocity This study uses a velocity profile with continuous acceleration, which can avoid a large contact force due to abrupt change of acceleration. The velocity profile with continuous acceleration may be expressed as 8
< V 0 t 1 sin 2pt if 0 rt r t 0 t0 2p t0 VðtÞ ¼ (40) : V0 if t 0 rt r t f where tf and t0 represent the time for the moving mass to travel the lengths L and L/5, respectively, and V0 is a constant moving velocity after time t0. Since integration of the velocity given in Eq. (40) from 0 to t0 is equal to L/5, the relation between t0 and V0 can be expressed as t0 ¼2L/(5V0). The velocity profile corresponding to Eq. (40) is plotted in Fig. 7. For the change of the moving mass velocity, the contact force and the loss of contact are investigated. The ffi pffiffiffiffiffiffiffiffiffiffiffi dimensionless velocity V/c, where c is the wave speed given by c ¼ T=rA, changes from 0.4 to 1.3. Using the discretized equations of Eqs. (27)–(32) and (38), the contact force between the moving mass and the tensioned beam is calculated. Fig. 8 shows the contact analysis results for the travel distance of the moving mass versus the velocity of the moving mass. The contact forces for the various sets of x/L and V/c are plotted in Fig. 8(a), and the contact loss is presented in Fig. 8(b). In general, the wave with a uniform speed propagates through the material and is then reflected at the boundary. When the position of the moving mass changes by the velocity given in Eq. (40), the reflected wave and moving mass encounter each other and cause the variation of contact force. As the dimensionless velocity of the moving mass increases from 0.4 to 1.3 in this study while the wave speed is uniform at 149.9 m/s, the first encounter of the moving mass and the wave occurs and the encounter
V V0
t0
tf
t
Fig. 7. Profile of the moving mass velocity without sudden change of acceleration.
Fig. 8. Contact analysis results for the variations of the moving mass velocity V and the travel distance of the moving mass x: (a) the contact force and (b) the loss of contact.
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position moves to the right. For this reason, the contact force variation line due to the first encounter can be clearly seen in Fig. 8(a), and the contact force variation line of the second encounter can be seen near the boundary. This figure also shows that the contact force corresponding to the first encounter becomes maximized when the dimensionless velocity V/c is near to the unity or when the moving velocity approaches the wave speed. It is interesting to observe that the contact force becomes zero near the boundary when the dimensionless velocity is about 0.7–0.9. This means that the fluctuation of contact force due to the encounter of the moving mass and the reflected wave may cause the separation between the moving mass and beam. The loss of contact occurs when the contact force becomes zero. In order to conveniently discuss this phenomenon, a two-dimensional plot is presented in Fig. 8(b), which shows the contact loss. The black area represents separation region while the white area represents the contact region. It is shown that the contact condition near the boundary of the beam is maintained when V/c is less than 0.7 or larger than 0.9, while contact loss occurs when V/c is about 0.7–0.9. Therefore, in order to obtain safe contact conditions, the dimensionless velocity should avoid the range of 0.7–0.9. 4.2. Effect of the tension The contact force variations are observed when the dimensionless tension T/EA changes. The mechanical properties for computation are the same as mentioned before, but the constant moving mass velocity V0 is given by 100 m/s. The contact analysis results for the travel distance versus the applied tension are illustrated in Fig. 9, where T/EA varies from 0.001 to 0.0018. It is shown in this figure that the overall contact force when the dimensionless tension of 0.001 is applied is much larger than the contact force when the dimensionless tension of 0.002 is applied. The reason for this is that the contact force is influenced to a larger extent by the beam flexibility at the tension of 0.001 than at the tension of 0.002. As the transverse flexibility of the beam decreases with the applied tension, the beam deflections due to the flexibility are reduced with the increase of tension. Thus, the overall contact force decreases as the tension increases. Moreover, in Fig. 9(a), the force fluctuation due to the encounter of the moving mass and the reflected wave is shown to become much larger when low tension, rather than high tension, is applied to the beam. The wave speed at the dimensionless tension of 0.001 is determined to be 122.7 m/s. When this wave speed is close to the moving mass speed of 100 m/s, rapid contact force variation caused by the encounter of the moving mass and the reflected wave is observed near the right end of the beam, as shown in Fig. 9(a). However, rapid contact force variation is not observed at the dimensionless tension of 0.002 because the wave speed at this tension is much faster than the moving mass velocity. It is also shown that the first encounter position of the moving mass and the reflected wave moves to the left when the tension increases. The reason for this is that the wave speed increases according to the increase of tension and the wave for higher tension moves faster when compared to the wave for lower tension. Therefore, the first encounter positions shifts to the left and the contact force magnitude decreases with the tension. The two-dimensional plot of the contact loss is also investigated, as illustrated in Fig. 9(b), where the black area is the region of contact loss and the white area is the contact region. It can be seen that the contact loss occurs near the boundary when the dimensionless tension changed from 0.0010 to 0.0012. Therefore, in order to obtain safe contact conditions and to minimize the variation in contact force, a large value of tension is desirable because a large tension makes the wave speed much faster than the speed of the moving mass.
Fig. 9. Contact analysis results for the variations of the beam tension T and the travel distance of the moving mass x: (a) the contact force and (b) the loss of contact.
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4.3. Effects of the mass and spring of the moving system The contact force and the contact loss between the beam and moving mass are investigated when the mass of the moving system changes with other mechanical properties specified before. The contact force for the variation of the moving mass and the travel distance is presented in Fig. 10(a), where the dimensionless mass m/M changes from 0.001 to 0.040. As shown in this figure, the overall contact force as well as the force fluctuation becomes large as the mass increases. The loss of contact between the beam and moving mass is illustrated in Fig. 10(b), from which it is observed that the contact loss occurs at a relatively large value of the mass. Therefore, it may be concluded that a large mass of the moving system has a tendency to increase the contact force fluctuation and contact loss. The contact force variation and contact loss are also analyzed when the stiffness of the moving system changes. The dimensionless stiffness of the spring, which is defined by k/(T/L), is changed from 0.1 to 4, while the other parameter values for computations are the same as before. The contact analysis results for the variation of the spring stiffness and the travel distance are presented in Fig. 11. It is observed in Fig. 11(a) that the overall contact force at high stiffness is much larger than the force at low stiffness. Moreover, rapid contact force variations are shown at high stiffness near the boundary. Despite the rapid contact force variation, contact loss does not occur at high stiffness shown in Fig. 11(b). Hence, the rapid contact force variation is not a sufficient condition for generating contact loss. However, regardless of the contact maintenance, excessively high contact force is undesirable because it may increase the wear of the beam and moving mass. Therefore, a low contact force is recommended if contact is guaranteed. On the other hand, as shown in Fig. 11(b), contact loss is observed at a low dimensionless stiffness region of 0.1–1.7. In order to obtain safe contact conditions, an extremely low stiffness of the moving system should be avoided.
4.4. Possibility of contact loss The Possibility of contact loss for the variations of the tension and velocity are also investigated. The Possibility of contact loss, which is denoted by Pl, is defined as the contact loss time divided by the total analysis time. For example, if contact loss is not observed during the analysis time, the Possibility of contact loss becomes zero. The Possibility of contact loss is plotted in Fig. 12 when the tension and the velocity change. Fig. 12(a) shows the three-dimensional mesh plot of the contact loss Possibility while Fig. 12(b) shows the contour plot corresponding to Fig. 12(a). These figures demonstrate that loss of contact occurs only in a certain region, such as region B. Regions A and C correspond to stable contact conditions because these regions have no contact loss. In these regions, any parameter combination of the tension and velocity is useful to obtain a stable contact condition. The border lines dividing regions A–C are closely related to the ratio of the moving pffiffiffiffiffiffiffiffiffiffiffiffimass velocity to the wave speed. As mentioned above, the wave speed c for a given tension T is computed by c ¼ T=rA. Consider the border line between regions A and B. This border line can be described approximately by the curve defined by V/c ¼0.64. Similarly, the border line between regions B and C can be approximated by the curve of V/c¼0.96. In Fig. 12(b), the curves of V/c ¼0.64 and 0.96
Fig. 10. Contact analysis results for the variations of the mass of moving mass system m and the travel distance of the moving mass x: (a) the contact force and (b) the loss of contact.
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Fig. 11. Contact analysis results for the variations of the stiffness of the moving mass system k and the travel distance of the moving mass x: (a) the contact force and (b) the loss of contact.
Fig. 12. Possibility of contact loss for the variations of the moving mass velocity V and the beam tension T: (a) the three-dimensional mesh plot and (b) the contour plot.
are plotted with dashed lines. Even though a theoretical background for these border lines cannot be found in this study, it is clear that the Possibility of contact loss becomes high for a specific range of V/c. For the case of this study, the specific range of V/c is from 0.64 to 0.96. Therefore, in order to minimize contact loss, the moving velocity should be selected out of the rage of V/c corresponding to contact loss.
5. Conclusions Dynamic contact analysis between the tensioned beam and the moving mass–spring system is performed when the beam with clamped-pinned ends contacts the moving mass–spring system. This study models the tensioned beam as an Euler–Bernoulli beam, and the moving-mass system as a single-degree-of-freedom system. The dynamic contact equations are derived based on this model, which describe the transverse displacements of the beam and moving mass. These equations are discretized using the finite-element method, and the contact analysis is performed using the Lagrange
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multiplier method. The dynamic behaviors of the beam and moving mass are investigated and the safe contact conditions are discussed. The results of the contact analysis can be summarized as follows: (1) The magnitude of the contact force between the moving mass and the beam becomes large when the moving mass velocity approaches the wave speed. (2) The fluctuation of contact force due to the encounter of the moving mass and the reflected wave may cause the separation between the moving mass and beam. (3) The contact loss occurs near the boundary of the beam when the dimensionless moving mass speed V/c is in a specific range. (4) In order to minimize the variation in contact force, a large beam tension, low moving mass, and low moving system stiffness are desirable for given values of the speed of the moving system. (5) In order to obtain safe contact conditions, a large beam tension, high moving system stiffness, and low moving mass are desirable.
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