Use the convolution integral to find the response of the single-DOF system represented by Eq. (3.2), assuming that the damping is zero, when a step force of magnitude $\overline{F}$ is applied at t = 0, if the initial displacement, z, and the initial velocity, $\dot{z}$ , are both zero at t = 0.

$\ddot{z} + 2\gamma \omega _{n}\dot{z} + \omega ^{2}_{n}z = F/m$ (3.2)

Question

Use the convolution integral to find the response of the single-DOF system represented by Eq. (3.2), assuming that the damping is zero, when a step force of magnitude $\overline{F}$ is applied at t = 0, if the initial displacement, z, and the initial velocity, $\dot{z}$ , are both zero at t = 0.

$\ddot{z} + 2\gamma \omega _{n}\dot{z} + \omega ^{2}_{n}z = F/m$ (3.2)

[latex]\ddot{z} + 2\gamma \omega _{n}\dot{z} + \omega ^{2}_{n}z = F/m[/latex] (3.2)

Accepted Answer

From Eq. (C) of Example 3.2, the unit impulse response of the system is

[latex]h(t)= \frac{1}{m \omega _{n}}\sin \omega _{n}t[/latex] (A)

The applied force, [latex]F( au)[/latex], is

[latex]\begin{aligned}&\begin{array}{lll}F( au ) = 0 &&& au < 0 \F( au ) = \overline{F} &&& 0 < au <\infty &&&&&& (B) \end{array} \end{aligned} [/latex]

From Eq. (3.21) and Eq. (A) (bearing in mind that [latex]\sin \left(- heta ight) = - \sin heta \mathrm{, and} \cos \left(- heta ight) = \cos heta[/latex] ):

[latex]z(t) = \int_{0}^{t}{F( au)} \cdot h(t- au ) \;d au = \frac{\overline{F} }{m\omega _{n}} \int_{0}^{t}{\sin \omega _{n}(t - au )} \;d au = \frac{\overline{F} }{m\omega _{n}}\int_{0}^{t}{\sin (-\omega _{n} au + \omega _{n}t)} \;d au = \frac{\overline{F} }{m\omega _{n}}\int_{0}^{t}{-\sin (\omega _{n} au - \omega _{n}t)} \;d au = \frac{\overline{F} }{m\omega _{n}^{2}} {{}_{0}^{t}}\left[\cos (\omega _{n} au - \omega _{n}t) ight][/latex]

[latex]\begin{aligned}&\begin{array}{lll} z(t) = \frac{\overline{F} }{m\omega _{n}^{2}} \left(1 - \cos \omega _{n}t\ ight) = \frac{\overline{F} }{k} \left(1 - \cos \omega _{n}t\ ight) \ (\mathrm{since} k = m\omega _{n}^{2}). \end{array} (C) \end{aligned}[/latex]

Question 3.3: Use the convolution integral to find the response of the sin......

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