The system shown in Fig. 1.11(a) consists of the two masses m1, and m2 which move in the horizontal direction on a friction-free surface. The two masses are connected to each other and to the surface by springs and dampers, as shown in the figure. The external forces F1 and F2 act, respectively, on

Question

The system shown in Fig. 1.11(a) consists of the two masses [latex]m_{1}[/latex], and [latex]m_{2}[/latex] which move in the horizontal direction on a friction-free surface. The two masses are connected to each other and to the surface by springs and dampers, as shown in the figure. The external forces [latex]F_{1}[/latex] and [latex]F_{2}[/latex] act, respectively, on the masses [latex]m_{1}[/latex] and [latex]m_{2}[/latex]. Obtain the differential equations of motion of this system.

Accepted Answer

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

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

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

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