-Mechanical Vibration-Lect-07-Equation of motion for Forced vibration

1. 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 ??1(??) and ??2(??) , which define the
positions of the masses ??1 and ??2 at any time t from the
respective equilibrium positions. The external forces ??1(??) and
??2(??) act on the masses ??1 and ??2 respectively. The free-body
diagrams of the masses ??1 and ??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:
??1???1+(??1+??2)???1???2???2+(??1+??2)??1???2??2=??1…(1)
??2???2???2???1+(??2+??3)???2???2??1+(??2+??3)??2=??2…(2)
It can be seen that Eq. (1) contains terms involving (namely, ???2???2 and ???2??2), whereas Eq. (2) contains terms involving (namely, ???2???1 and ???2??1). Hence they represent a system of two coupled second-order differential equations. We can therefore expect that the motion of the mass ??1 will influence the motion of the mass ??2 and vice versa. [??] ????(??)+[??]????(??)+[??]???(??)=???(??) …(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; [??]=[??1??????2] [??]=[??1+??2???2???2??2+??3] [??]=[??1+??2???2???2??2+??3]
and ???(??) and ???(??) are called the displacement and force vectors, respectively, and are given by
???(??)={??1(??)??2(??)}
???(??)={??1(??)??2(??)}
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
[??]??=[??] , [??]??=[??] , [??]??=[??]
where the superscript ?? denotes the transpose of the matrix.
Notice that the equations of motion (1) and (2) become uncoupled (independent of one another) only when ??2=??2=0 , which implies that the two masses ??1 and ??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 ??1(??) and ??2(??) is difficult to obtain, mainly due to the coupling of the variables ??1(??) and ??2(??) . The solution of Eqs. (1) and (2) involves four constants of integration (two for each equation).