Consider a SDOF system shown in Figure 2.1. It undergoes undamped free vibrations. The mass = 40 kNs²/m, stiffness k = 3500 kN/m. Determine the natural frequency and natural period of vibration. The initial conditions are given as: x(0) = 0.01 m, x(0) = 0.1 m/s. Hence, plot a graph of

Question

Consider a SDOF system shown in Figure 2.1. It undergoes undamped free vibrations. The mass = 40 kNs²/m, stiffness k = 3500 kN/m. Determine the natural frequency and natural period of vibration. The initial conditions are given as: x(0) = 0.01 m, [latex]\dot{x}[/latex] (0) = 0.1 m/s. Hence, plot a graph of displacement–time history, and inertia and elastic forces.

Accepted Answer

The complete solution is given by Equations (2.6) and (2.9). These equations can be plotted either using MS EXCEL or MATLAB.

[latex]\omega =\sqrt{\frac{k}{m} }[/latex] (2.6)

[latex]x(t)=\frac{\dot{x}(0) }{\omega } \sin \omega t+\cos \omega t[/latex] (2.9a)

[latex]\dot{x} (t)=\dot{x} (0)\cos \omega t -x(0)\omega \sin \omega t[/latex] (2.9b)

[latex]\ddot{x} (t)=-\omega ^{2}x(t)[/latex] (2.9c)

The frequency of vibration is given by [latex]\omega =\sqrt{\frac{k}{m} }=\sqrt{\frac{3500}{40} } [/latex] = 9.3541 rad/sec

Natural period is given by [latex]T =\frac{2\pi }{\omega } [/latex] = 0.6717 sec

The plots of displacement, velocity and acceleration are shown in Figure 2.6. Knowing the displacement and acceleration, elastic force (= kx) and inertia forces (= m[latex]\ddot{x} [/latex] ) can be estimated. At any point of time, the sum of elastic and inertia forces is zero to keep the system in equilibrium.

Question 2.1: Consider a SDOF system shown in Figure 2.1. It undergoes und...

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