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

We shall consider a system of homogeneous differential equations

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‐

*M* (*t*)*q***¨**(*t*) + *C*(*t*)*q***˙**(*t*) + *K*(*t*)*q*(*t*)=*d*(*t*) (2)

*x***˙** =*P*(*t*)*x* + *f* (*t*) (3)

*<sup>q</sup>***¨** (4)

*<sup>d</sup>* , (5)

ing time-periodic coefficients.

302 Advances in Vibration Engineering and Structural Dynamics

where we use the state variable *x*

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

bration systems governed by Eq. (2) [5-9].

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

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

*<sup>P</sup>*(*t*)= **<sup>0</sup>** *<sup>I</sup>* −*M* <sup>−</sup><sup>1</sup>

*<sup>x</sup>* <sup>=</sup> *<sup>q</sup>*

*K* −*M* <sup>−</sup><sup>1</sup>

*<sup>q</sup>***˙** , *<sup>x</sup>***˙** <sup>=</sup> *<sup>q</sup>***˙**

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‐

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‐

*<sup>C</sup>* , *<sup>f</sup>* (*t*)= **<sup>0</sup>**

*M* <sup>−</sup><sup>1</sup>

$$
\dot{\mathbf{x}} = \mathbf{P}(t)\mathbf{x} \tag{6}
$$

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. Firstly, we specify a set of *n* initial conditions *x<sup>i</sup>* (0) for *i* =1, ..., *n* , their elements

$$\mathbf{x}\_i^{(v)}(\mathbf{0}) = \begin{cases} 1 & \text{when } s = i \\ 0 & \text{otherwise} \end{cases} \tag{7}$$

and *x*1(0), *x*2(0), ..., *xn*(0) = *I*. By implementing numerical integration of Eq. (6) within inter‐ val 0, *T* for *n* given initial conditions respectively, we obtain *n* vectors *x<sup>i</sup>* (*T* ), *i* =1, ..., *n* . The matrix *Φ*(*t*) defined by

$$\mathbf{OP}(T) = \begin{bmatrix} \mathbf{x}\_1(T), \; \mathbf{x}\_2(T), \; \dots, \; \mathbf{x}\_n(T) \end{bmatrix} \tag{8}$$

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

$$\left|\Phi(T) - \rho\mathbb{I}\right| = \begin{vmatrix} \mathbf{x}\_1^{(0)}(T) - \rho & \mathbf{x}\_2^{(0)}(T) & \dots & \mathbf{x}\_n^{(0)}(T) \\ \mathbf{x}\_1^{(2)}(T) & \mathbf{x}\_2^{(2)}(T) - \rho & \dots & \mathbf{x}\_n^{(2)}(T) \\ \dots & \dots & \dots & \dots \\ \mathbf{x}\_1^{(n)}(T) & \mathbf{x}\_2^{(n)}(T) & \dots & \mathbf{x}\_n^{(n)}(T) - \rho \end{vmatrix} = \mathbf{0} \tag{9}$$

Expansion of Eq. (9) yields a *n*-order algebraic equation

$$a\_1 \rho^n + a\_1 \rho^{n-1} + a\_2 \rho^{n-2} + \dots + a\_{n-1} \rho + a\_n = 0 \tag{10}$$

*x***˙** =*P*(*t*)*x* + *f* (*t*) (12)

Parametric Vibration Analysis of Transmission Mechanisms Using Numerical Methods

*x***˙** =*P*(*t*)*x* (13)

*x*(0)= *x*(*T* ) (14)

) and *xi*+1 = *x*(*ti*+1) represent the

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

305

(*i*−1) (15)

(16)

where *x* is the vector of state variables, matrix *P*(*t*) and vector *f* (*t*) are periodic in time with period *T*. The system of homogeneous differential equations corresponding to Eq. (12) is

As well known from the theory of differential equations, if Eq. (13) has only non-periodic solutions except the trivial solution, then Eq. (12) has an unique *T*-periodic solution. This pe‐ riodic solution can be obtained by choosing the appropriate initial condition for the vector of variables x and then implementing numerical integration of Eq. (12) within interval 0, *T* . An algorithm is developed to find the initial value for the periodic solution [18, 19]. Firstly,

The interval 0, *T* is now divided into *m* equal subintervals with the step-size

states of the system, respectively. Using the fourth-order Runge-Kutta method, we get a nu‐

(*i*−1) <sup>+</sup> <sup>2</sup>*k*<sup>3</sup>

1 <sup>2</sup> *<sup>k</sup>*<sup>1</sup> (*i*−1)

1 <sup>2</sup> *<sup>k</sup>*<sup>2</sup> (*i*−1)

) + *f* (*ti* ) .

(*i*−1)

(*i*−1) <sup>+</sup> *<sup>k</sup>*<sup>4</sup>

) + *f* (*ti*−<sup>1</sup> +

) + *f* (*ti*−<sup>1</sup> +

*h* <sup>2</sup> ) ,

*h* <sup>2</sup> ) ,

*x<sup>i</sup>* = *Ai*−1*xi*−<sup>1</sup> + *bi*−<sup>1</sup> (17)

and *ti*+1, *x<sup>i</sup>* = *x*(*ti*

the *T*-periodic solution must satisfy the following condition

*h* =*ti* −*ti*−<sup>1</sup> =*T* / *m*. At the discrete times *ti*

*k*1 (*i*−1)

*k*2 (*i*−1)

*k*3 (*i*−1)

*k*4 (*i*−1)

where matrix *Ai*−1 is given by

Substituting Eq. (16) into Eq. (15), we obtain

*x<sup>i</sup>* = *xi*−<sup>1</sup> +

=*h P*(*ti*−<sup>1</sup> +

=*h P*(*ti*−<sup>1</sup> +

=*h P*(*ti*

1 <sup>6</sup> *<sup>k</sup>*<sup>1</sup>

=*h P*(*ti*−1)*xi*−<sup>1</sup> + *f* (*ti*−1) ,

*h* <sup>2</sup> )(*xi*−<sup>1</sup> <sup>+</sup>

*h* <sup>2</sup> )(*xi*−<sup>1</sup> <sup>+</sup>

)(*xi*−<sup>1</sup> + *k*<sup>3</sup>

(*i*−1) <sup>+</sup> <sup>2</sup>*k*<sup>2</sup>

merical solution [5]

where

where unknowns *ρ<sup>k</sup>* (*k* =1, ..., *n*), called Floquet multipliers, can be determined from Eq. (10). Floquet exponents are given by

$$
\lambda\_k = \frac{1}{T} \text{Im}\rho\_{k'} \text{ ( $k=1, \dots, n$ )}\tag{11}
$$

When the Floquet multipliers or Floquet exponents are known, the stability conditions of solutions of the system of linear differential equations with periodic coefficients can be easi‐ ly determined according to the Floquet theorem [17–20]. The concept of stability according to Floquet multipliers can be expressed as follows.

If |*ρ<sup>k</sup>* |1, the trivial solution *x* =0 of Eq. (6) will be asymptotically stable. Conversely, the solution *x* =0 of Eq. (6) becomes unstable if at least one Floquet multiplier has modulus be‐ ing larger than 1.

If |*ρ<sup>k</sup>* | ≤1 and Floquet multipliers with modulus 1 are single roots of the characteristic equation, the solution *x* =0 of Eq. (6) is stable.

If |*ρ<sup>k</sup>* | ≤1 and Floquet multipliers with modulus 1 are multiple roots of the characteristic equation, and the algebraic multiplicity is equal to their geometric multiplicity, then the sol‐ ution *x* =0 of Eq. (6) is also stable.
