-Mechanical Vibration-Lect-08-Two-Degree-of-Freedom Systems

Lecture 8

Two-Degree-of-Freedom Systems
Contents
Equations of Motion for Forced Vibration
Forced Vibration Analysis

Equations of Motion for Forced Vibration
Consider a viscously damped two-degree-of-freedom spring-mass system, shown in Fig. 1(a). The motion of the system is completely described by the coordinates x_1 (t) and x_2 (t) , which define the positions of the masses m_1 and m_2 at any time t from the respective equilibrium positions. The external forces F_1 (t) and F_2 (t) act on the masses m_1 and m_2 respectively. The free-body diagrams of the masses m_1 and m_2 are shown in Fig. 1(b).

The application of Newton s second law of motion to each of the masses gives the equations of motion:
m_1 x ?_1+(c_1+c_2 ) x ?_1-c_2 x ?_2+(k_1+k_2 ) x_1-k_2 x_2=f_1…(1)
m_2 x ?_2-c_2 x ?_1+(c_2+c_3 ) x ?_2-k_2 x_1+(k_2+k_3 ) x_2=f_2…(2)
It can be seen that Eq. (1) contains terms involving (namely, -c_2 x ?_2 and -k_2 x_2), whereas Eq. (2) contains terms involving (namely, -c_2 x ?_1 and -k_2 x_1). Hence they represent a system of two coupled second-order differential equations. We can therefore expect that the motion of the mass m_1 will influence the motion of the mass m_2 and vice versa.
[m] x ? ?(t)+[c] x ? ?(t)+[k] x ?(t)=f ?(t) …(3)
where [m], [c], and [k] are called the mass, damping, and stiffness matrices, respectively, and are Equations (1) and (2) can be written in matrix form as follow;
[m]=[?(m_1&0@0&m_2 )]
[c]=[?(c_1+c_2&-c_2@-c_2&c_2+c_3 )]
[k]=[?(k_1+k_2&-k_2@-k_2&k_2+k_3 )]
and x ?(t) and f ?(t) are called the displacement and force vectors, respectively, and are given by
x ?(t)={?(x_1 (t)@x_2 (t))}
and
f ?(t)={?(f_1 (t)@f_2 (t))}
It can be seen that [m], [c], and [k] are all 2x2 matrices whose elements are the known masses, damping coefficients, and stiffnesses of the system, respectively. Further, these matrices can be seen to be symmetric, so that
[m]^T=[m] , [c]^T=[c] ,? [k]?^T=[k]
where the superscript T denotes the transpose of the matrix.
Notice that the equations of motion (1) and (2) become uncoupled (independent of one another) only when c_2=k_2=0 , which implies that the two masses m_1 and m_2 are not physically connected. In such a case, the matrices [m], [c], and [k] become diagonal.
The solution of the equations of motion (1) and (2) for any arbitrary forces f_1 (t) and f_2 (t) is difficult to obtain, mainly due to the coupling of the variables x_1 (t) and x_2 (t) . The solution of Eqs. (1) and (2) involves four constants of integration (two for each equation).