Find the solution of a spring-mass-damper system governed by the equation \(m \ddot{x}+c \dot{x}+k x=F(t)=\delta F . t\) with \(m=c=k=1\) and \(\delta F=1\). Assume the initial values of \(x\) and \(\dot{x}\) to be zero and \(\Delta t=0.5\). Compare the central difference solution with the exact solution given in Example 4.9.

Data From Example 4.9:-

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A compacting machine, modeled as a single-degree-of-freedom system, is shown in Fig. 4.10(a). The force acting on the mass m (m includes the masses of the piston, the platform, and the material being compacted) due to a sudden application of the pressure can be idealized as a step force, as shown in Fig. 4.10(b). Determine the response of the system.