J. Cent. South Univ. Technol. (2011) 18: 108-115

DOI: 10.1007/s11771-011-0667-9

Modal analysis on transverse vibration of axially moving roller chain coupled with lumped mass

XU Li-xin(许立新)¹, YANG Yu-hu(杨玉虎)¹, CHANG Zong-yu(常宗瑜)², LIU Jian-ping(刘建平)¹

1. School of Mechanical Engineering, Tianjin University, Tianjin 300072, China;

2. College of Engineering, Ocean University of China, Qingdao 266100, China

? Central South University Press and Springer-Verlag Berlin Heidelberg 2011

Abstract: The modal characteristics of the transverse vibration of an axially moving roller chain coupled with lumped mass were analyzed. The chain system was modeled by using the multi-body dynamics theory and the governing equations were derived by means of Lagrange’s equations. The effects of the parameters, such as the axially moving velocity of the chain, the tension force, the weight of lumped mass and its time-variable assign position in chain span, on the modal characteristics of transverse vibration for roller chain were investigated. The numerical examples were given. It is found that the natural frequencies and the corresponding mode shapes of the transverse vibration for roller chain coupled with lumped mass change significantly when the variations of above parameters are considered. With the movement of the chain strand, the natural frequencies present a fluctuating phenomenon, which is different from the uniform chain. The higher the order of mode is, the greater the fluctuating magnitude and frequency are.

Key words: roller chain; modal analysis; transverse vibration; lumped mass

1 Introduction

Modal analysis method, as an engineering tool, is used to calculate the natural frequencies and mode shapes of a structure or a mechanical system. In general, an important objective of application of modal analysis is the identification and evaluation of vibration phenomena [1-3].

Transverse vibration of an axially moving chain has been investigated in the past 40 years because of a large number of applications to mechanical systems [4-12]. The calculation of natural frequencies and mode shapes of transverse vibration is an important research content.

Study on the modal characteristics of a chain system can be dated back to the study of MAHALINGAM [4]. ARIARATNAM and ASOKANTHAN [5] investigated the instability of the chain span due to polygonal action using MAHALINGAM’s previous results. Based on the theory of string model, WANG [6-7] proposed a complete roller chain dynamic model with the tight and slack spans and identified the parametric resonances and the instability zones of the chain system. In later work, CHOI [8-9] researched the mode shapes of chain transverse vibrations. The studies mentioned above consider the chain span to be a one dimensional continuous string. CHEN [13] proposed a comprehensive review on the transverse vibration and modal characteristics of a continuous string. However, a chain is composed of a series of links connected by joints. Therefore, a continuous string model cannot represent a chain accurately due to the discrete nature of the Chain.

When the emphasis is put on the analysis of the detailed link-to-link dynamic behavior of a chain system, LIN et al [10] proposed a model of an automotive timing chain with complete standard geometry of sprockets and all components of chain links. The natural frequencies of the system and the associated mode shapes are analyzed. More recently, ZHANG [11] presented a research on transverse vibration of an axially moving silent chain using the multi-body dynamics theory and the effects of the axially moving velocity and the link moment of inertia on the natural characteristic of the transverse vibration were investigated. WANG et al [12] established the finite element model of the silent chain by ANSYS software and the modal characteristic of the model was simulated and verified.

In these studies, only the modal characteristics of a uniform chain were investigated. None of them has studied the modal characteristics of an inhomogeneous chain coupled with lumped mass. Practically, the inhomogeneous chains are widely used in the field of light and automatic industry such as the die-cutting machine, medicine and food automatic filling machine and packaging machinery. They have a same characteristic that one or more lumped masses, such as gripper bars with a function to make materials processed to the expected position, are distributed on the chain spans [14-15]. Due to the coupling effect of the lumped masses, the modal characteristics of an axially moving roller chain in transverse vibration direction are different from those of uniform chains. Obviously, the natural frequencies and the mode shapes of the inhomogeneous chain coupled with lumped masses cannot be accurately predicted by a uniform chain analysis.

The purpose of the present study is to analyze the modal characteristics of an axially moving roller chain coupled with the lumped mass. The chain span was modeled by using the multi-body dynamics theory and the governing equation of transverse vibration was derived by means of Lagrange’s equations as used in Ref.[11]. The effects of the parameters such as the axially moving velocity, the tension force and the weight of the lumped mass and its time-variable assign position in chain span on the modal characteristics of the chain were investigated.

2 Derivation of equations of motion

Fig.1 shows the model of an axially moving roller chain, where y_i and y_i-1 represent the transverse displacements of joints; θ_i represents the rotation angle;  represents the tension force; v represents the axially moving velocity; I_i (iN) represents the moment of inertia; m_i represents the mass of the i-th chain link. It is supposed that the center of gravity of chain link i locates at its geometrical center o_i. Then, the transverse displacement of the link i from its equilibrium position can be expressed as

                               (1)

It is assumed that the rotation angle θ_i of each link around its center of gravity is small. Thus, it can be expressed by using the displacement of joints:

                                (2)

Fig.1 Model of roller chain span

where h is the length of the chain pitch. The chain link is approximated as the bar with length h when calculating its moment of inertia I_i. There is

                                   (3)

When some lumped masses are distributed in the chain span, it is assumed that the weight of the lumped mass is equivalent to the corresponding chain links. Therefore, the weight of the link will increase, as well as the moment of inertia.

Fig.2 shows the boundary conditions of the chain at the start point and end point. When the study on response of forced vibration of chain is required, the excitations such as the meshing incentive can be imposed on the boundary y₀ and y_N. Since the gravitational force is sufficiently small compared with the tension force  the equilibrium configuration of the chain is a straight line.

Fig.2 Boundary conditions of model of roller chain

The velocity in y direction of the center of gravity for link i is given as

                     (4)

The second item in the above equation represents the effect of the axially moving velocity on the velocity in y direction [11]. The chain tension force considering the effect of centrifugal force can be expressed as

                                (5)

where F₀ represents the initial tension force in the chain and  represents the average mass per length of the chain.

The kinetic and potential energy of link i are given as follows, respectively:

(6)

                           (7)

Then, the total kinetic and potential energy for the chain with N links can be obtained from Eqs.(6) and (7):

   (8)

                        (9)

Substituting Eqs.(8) and (9) into Lagrange’s equations, there is

                     (10)

where q_i represents the generalized coordinate;  represents the generalized velocity and Q_i represents the generalized force. Then, the application of Lagrange equations leads to N-1 equations of motion as follows:

                (11a)

, i=2, 3, …, N-2     (11b)

  i=N                   (11c)

Eq.(11) can be simply expressed in the form of matrix as

                         (12)

The detailed description of matrix M, K, C, and vector Y and F are shown as follows.

,

.

For modal analysis, damping in the chain is not needed to be considered. Additionally, to solve the eigenvalue problem of Eq.(12), zero was put the right side of the equation, and the equation of free vibration of the moving roller chain is obtained as

                              (13)

The solution of the above differential equations for the transverse vibration of roller chain can be expressed as

 i=1, 2, …, N-1               (14)

or in vector notation

              (15)

where φ_i represents the amplitude of motion of the i-th coordinate; N-1 represents the number of degree of freedom; φ represents constant vector and ω is a constant. Substituting Eq.(15) into Eq.(13) derives the following equation:

             (16)

or rearranging terms:

                            (17)

where for the general case it is a set of N-1 homogeneous  algebraic system of linear equations with N-1 unknown displacements φ_i and an unknown parameter ω². The formulation of Eq.(17) is an important mathematical problem known as an eigenvalue problem. Its nontrivial solution requires that the determinant of the matrix factor of φ is equal to zero. In this case,

                           (18)

The root ω² of this equation provides the natural frequency ω_i. It is then possible to solve the unknown φ_i in terms of relative values. The displacement φ_i corresponding to the root  is the modal shape (eigenvector) of the chain system. For the chain system with multi-degree of freedom, the eigenvalue problem can be solved by using the computational software such as MATLAB.

3 Numerical example

The parameters of roller chain used in numerical calculations are listed in Table 1. A chain is normally made by two kinds of link pair: one outer pair and one inner pair (seen in Fig.1). The outer pair is normally smaller than the inner pair. In this study, it is assumed that the outer pair and the inner pair have the same mass and moment of inertia.

Table 1 Parameters of roller chain used in example

In numerical example, it is assumed that only one lumped mass exists in the chain span, as shown in Fig.3. With the movement of the chain, the lumped mass moves from the first link to the last link position. Then, in a whole moving period, the lumped mass has N positions to select.

Fig.3 Axially moving roller chain coupled with lumped mass

In simulation, for the chain system with multi-degree of freedom, the analysis of the first five order modes is necessary and enough.

4 Results and discussion

Fig.4 shows the relationship between natural frequencies of transverse vibration of roller chain and its axially moving velocity under the condition that the lumped mass is 0.2 kg and the initial tension force is    1 000 N. The velocity of the chain increases from 0 to  8 m/s. It is seen that, with the increase of the velocity, the natural frequencies decrease correspondingly. Under the high speed condition, the natural frequencies, especially the high orders, decrease significantly. The simulation results are in agreement with the research conclusion on the uniform chain made by ZHANG [11]. More important is that the natural frequencies of transverse vibration of roller chain coupled with lumped mass have a time-variable characteristic that is different from the uniform chain. This is caused by the fact that the position of the lumped mass changes with the movement of roller chain. The natural frequencies are different when the lumped mass locates at different positions in chain span. From Fig.4, it is interesting to observe that the natural frequencies present fluctuating phenomenon with the movement of chain. The higher the order number is, the greater the fluctuating magnitude and the frequency of the transverse vibration are. Additionally, in a whole moving period, all order natural frequencies appear symmetrically at the middle point of the chain.

Fig.5 shows the relationship between natural frequencies of transverse vibration of roller chain and the tension force (F) under the condition that the lumped mass is 0.2 kg and the axially moving velocity is 4 m/s. The initial tension force increases from 400 to 1 200 N. It is observed that the values of natural frequencies increase with the increase of tension force, but for the low order modes, the increase of natural frequencies is smaller than that of the high order modes. Additionally, the natural frequencies also present fluctuating phenomenon with the movement of chain.

Fig.6 shows the relationship between natural frequencies of transverse vibration of roller chain and the mass of lumped mass under the condition that the axially moving velocity is 4 m/s and the tension force is 1 000 N. The mass of the lumped mass increases from 0.1 to 0.5 kg. It is observed that the values of natural frequencies decrease with the mass of the lumped mass increasing. But, with the increase of the mass of lumped mass, the fluctuating magnitude of natural frequencies increases.

Fig.7 presents the first five order mode shapes of transverse vibration of roller chain. For comparison, the lumped mass is supposed to locate at five different positions: at the first chain link position (case 1); at the tenth chain link position (case 2); at the twentieth chain link position (case 3); at the thirtieth chain link position

Fig.4 Natural frequencies of transverse vibration of roller chain versus axially moving velocity: (a) First order mode; (b) Second order mode; (c) Third order mode; (d) Fourth order mode; (e) Fifth order mode

Fig.5 Natural frequencies of transverse vibration of roller chain versus tension force: (a) First order mode; (b) Second order mode;  (c) Third order mode; (d) Fourth order mode; (e) Fifth order mode

Fig.6 Natural frequencies of transverse vibration of roller chain versus mass of lumped mass: (a) First order mode; (b) Second order mode; (c) Third order mode; (d) Fourth order mode; (e) Fifth order mode

(case 4); at the thirty-ninth chain link position (case 5). The simulation condition is that the axially moving velocity is 4 m/s, the tension force is 1 000 N and the mass of lumped mass is 0.2 kg. It can be seen that the mode shapes are different when the lumped mass locates at different positions in chain span.

Fig.7 Mode shapes of transverse vibration of roller chain with lumped mass locating at different positions in chain span: (a) Mode shape 1; (b) Mode shape 2; (c) Mode shape 3; (d) Mode shape 4; (e) Mode shape 5

5 Conclusions

1) For the roller chain system coupled with the lumped mass, the natural frequencies of transverse vibration present a fluctuating phenomenon with the movement of the chain strand. The higher the order of mode is, the greater the fluctuating magnitude and the frequency are.

2) With the increase of the tension force, the natural frequencies of transverse vibration of roller chain increase, but for the low order modes the increase of natural frequencies is smaller than that of the high order modes.

3) With the increase of the mass of lumped mass, the natural frequencies of transverse vibration of roller chain decreases, but the fluctuating magnitude increases.

4) The mode shapes of transverse vibration of roller chain are different when the lumped mass locates at different positions in chain span.

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(Edited by LIU Hua-sen)

Foundation item: Project(50605060) supported by the National Natural Science Foundation of China; Project(20050056058) supported by the Research Fund for the Doctoral Program of Higher Education of China; Project(06YFJMJC03300) supported by the National Science Foundation of Tianjin, China

Received date: 2010-07-20; Accepted date: 2010-10-12

Corresponding author: YANG Yu-hu, Professor, PhD; Tel: +86-22-27404071; E-mail: yangyuhu@tju.edu.cn