Question 1.2: The system shown in Fig. 1.11(a) consists of the two masses ...

The system shown in Fig. 1.11 has two degrees of freedom which can be represented by two independent coordinates x_{1} and x_{2}. Without any loss of generality, we assume that x_{2} is greater than x_{1}, and \dot{x_{2} } is greater than \dot{x_{1} }. By using the free body diagram shown in the figure, the differential equation of motion for the mass m_{1} is given by

m \ddot{x_{1} } = Fx_{1}
= F_{1}(t) − k_{1}x_{1} − c_{1}\dot{x_{1} } + k_{2}(x_{2} − x_{1}) + c_{2}(\dot{x_{2} } − \dot{x_{1} })
which can be written as
m_{1}\ddot{x_{1} } + (c_{1}+ c_{2})\dot{x_{1} } − c_{2}\dot{x_{2} } + (k_{1} + k_{2})x_{1} − k_{2}x_{2} = F_{1}(t)
Similarly, for the second mass, we have
m_{2}\ddot{x_{2} } = F_{x_{2}}
= F_{2}(t) − k_{2}(x_{2} − x_{1}) + c_{2}(\dot{x_{2} } − \dot{x_{1} })
or
m_{2}\ddot{x_{2} }  + c_{2}\dot{x_{2} } − c_{2}\dot{x_{1} } + k_{2}x_{2} − k_{2}x_{1} = F_{2}(t)
Note that there are two second-order differential equations of motion since the system has two degrees of freedom.