**1. Introduction**

Transmission mechanisms are frequently used in machines for power transmission, varia‐ tion of speed and/or working direction and conversion of rotary motion into reciprocating motion. At high speeds, the vibration of mechanisms causes wear, noise and transmission errors. The vibration problem of transmission mechanisms has been investigated for a long time, both theoretically and experimentally. In dynamic modelling, a transmission mecha‐ nism is usually modelled as a multibody system. The differential equations of motion of a multibody system that undergo large displacements and rotations are fully nonlinear in *n* generalized coordinates in vector of variable q [1–4].

$$\mathbf{M}(\mathbf{q},\ t)\stackrel{\bullet}{\dot{\mathbf{q}}} + \mathbf{k}(\dot{\mathbf{q}},\ q,\ t) = \mathbf{h}\left(\dot{\mathbf{q}},\ q,\ t\right) \tag{1}$$

It is very difficult or impossible to find the solution of Eq. (1) with the analytical way. Never‐ theless, the numerical methods are efficient to solve the problem [5-9].

Besides, many technical systems work mostly on the proximity of an equilibrium position or, especially, in the neighbourhood of a desired motion which is usually called "program‐ med motion", "desired motion", "fundamental motion", "input–output motion" and etc. according to specific problems. In this chapter, the term "desired fundamental motion" is used for this object. The desired fundamental motion of a robotic system, for instance, is usually described through state variables determined by prescribed motions of the endeffector. For a mechanical transmission system, the desired fundamental motion can be the motion of working components of the system, in which the driver output rotates uniform‐ ly and all components are assumed to be rigid. It is very convenient to linearize the equa‐

© 2012 Khang and Dien; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 Khang and Dien; licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

tions of motion about this configuration to take advantage of the linear analysis tools [10-18]. In other words, linearization makes it possible to use tools for studying linear systems to analyze the behavior of multibody systems in the vicinity of a desired fundamental mo‐ tion. For this reason, the linearization of the equations of motion is most useful in the study of control [12-13], machinery vibrations [14-19] and the stability of motion [20-21]. Mathe‐ matically, the linearized equations of motion of a multibody system form usually a set of linear differential equations with time-varying coefficients. Considering steady-state mo‐ tions of the multibody system only, one obtains a set of linear differential equations hav‐ ing time-periodic coefficients.

$$\mathbf{M}(t)\ddot{\tilde{q}}(t) + \mathbf{C}(t)\dot{q}(t) + \mathbf{K}(t)q(t) = \mathbf{d}(t) \tag{2}$$

Note that Eq. (2) can be expressed in the compact form as

$$
\dot{\mathbf{x}} = \mathbf{P}(t)\mathbf{x} + f(t) \tag{3}
$$

alized techniques such as the harmonic balance method, the method of conventional oscilla‐

Parametric Vibration Analysis of Transmission Mechanisms Using Numerical Methods

http://dx.doi.org/10.5772/51157

303

Following the above introduction, an overview of the numerical calculation of dynamic stability conditions of linear dynamic systems with time-periodic coefficients is presented in Section 2. Sections 3 presents numerical procedures based on Runge-Kutta method and Newmark method to find periodic solutions of linear systems with time-periodic coeffi‐ cients. In Section 4, the proposed approach is demonstrated and validated by dynamic models of transmission mechanisms and measurements on real objects. The improvement in the computational efficiency of Newmark method comparing with Runge-Kutta method for

**2. Numerical calculation of dynamic stability conditions of linear dynamic systems with time-periodic coefficients: An overview**

where *P*(*t*) is a continuous *T*-periodic *n* ×*n* matrix. According to Floquet theory [17, 18, 20, 21], the characteristic equation of Eq. (6) is independent of the chosen fundamental set of solutions. Therefore, the characteristic equation can be formulated by the following way.

> ( ) 1 when (0) 0 otherwhile

*s i <sup>x</sup>* <sup>ì</sup> <sup>=</sup> <sup>=</sup> <sup>í</sup>

and *x*1(0), *x*2(0), ..., *xn*(0) = *I*. By implementing numerical integration of Eq. (6) within inter‐

is called the monodromy matrix of Eq. (6) [20]. The characteristic equation of Eq. (6) can then

*s i*

val 0, *T* for *n* given initial conditions respectively, we obtain *n* vectors *x<sup>i</sup>*

*x***˙** =*P*(*t*)*x* (6)

(0) for *i* =1, ..., *n* , their elements

*Φ*(*T* )= *x*1(*T* ), *x*2(*T* ), ..., *xn*(*T* ) (8)

<sup>î</sup> (7)

(*T* ), *i* =1, ..., *n* .

We shall consider a system of homogeneous differential equations

Firstly, we specify a set of *n* initial conditions *x<sup>i</sup>*

The matrix *Φ*(*t*) defined by

be written in the form

tor, the WKB method [14-16, 23, 24].

linear systems is also discussed.

where we use the state variable *x*

$$\mathbf{x} = \begin{bmatrix} q \\ \dot{q} \end{bmatrix}, \ \dot{\mathbf{x}} = \begin{bmatrix} \dot{q} \\ \ddot{q} \end{bmatrix} \tag{4}$$

and the matrix of coefficients *P*(*t*), vector *f*(*t*) are defined by

$$P(t) = \begin{bmatrix} 0 & I \\ -\mathbf{M}^{-1}\mathbf{K} & -\mathbf{M}^{-1}\mathbf{C} \end{bmatrix} f(t) = \begin{bmatrix} 0 \\ \mathbf{M}^{-1}\mathbf{d} \end{bmatrix} \tag{5}$$

where *I* denotes the *n* ×*n* identity matrix.

In the steady state of a machine, the working components perform stationary motions [14-18], matrices *M* (*t*), *C*(*t*), *K*(*t*) and vector *d*(*t*) in Eq. (2) are time-periodic with the least period *T*. Hence, Eq. (2) represents a parametrically excited system. For calculating the steady-state periodic vibrations of systems described by differential equations (1) or (2) the harmonic balance method, the shooting method and the finite difference method are usually used [8,11,14]. In addition, the numerical integration methods as Newmark method and Runge-Kutta method can also be applied to calculate the periodic vibration of parametric vi‐ bration systems governed by Eq. (2) [5-9].

Since periodic vibrations are a commonly observed phenomenon of transmission mecha‐ nisms in the steady-state motion, a number of methods and algorithms were developed to find a *T*-periodic solution of the system described by Eq. (2). A common approach is by im‐ posing an arbitrary set of initial conditions, and solving Eq. (2) in time using numerical methods until the transient term of the solution vanishes and only the periodic steady-state solution remains [14,22]. Besides, the periodic solution can be found directly by other speci‐ alized techniques such as the harmonic balance method, the method of conventional oscilla‐ tor, the WKB method [14-16, 23, 24].

Following the above introduction, an overview of the numerical calculation of dynamic stability conditions of linear dynamic systems with time-periodic coefficients is presented in Section 2. Sections 3 presents numerical procedures based on Runge-Kutta method and Newmark method to find periodic solutions of linear systems with time-periodic coeffi‐ cients. In Section 4, the proposed approach is demonstrated and validated by dynamic models of transmission mechanisms and measurements on real objects. The improvement in the computational efficiency of Newmark method comparing with Runge-Kutta method for linear systems is also discussed.
