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Transient Vibrations: Response of Spring–Mass System to a Ramp Function

Ramp Function

The equation of motion in this case is

    \begin{equation*} m \ddot{x} + k x = \beta t. \end{equation*}

It is easy to verify by direct substitution that the particular solution is

    \begin{equation*} x_P(t) = \frac{\beta}{k} t \end{equation*}

so the total solution is

(7.5)   \begin{equation*} \label{eq:RampResponseGeneral} x(t) = A \sin{pt} + B \cos{pt} + \frac{\beta}{k} t. \end{equation*}

If the initial conditions are

    \begin{equation*} x(0) = 0, \qquad \dot{x}(0) = 0, \end{equation*}

then A = \dfrac{-\beta}{k p} and B=0 so we get

(7.6)   \begin{equation*} \label{eq:RampResponseIC} \boxed{x(t) = \frac{\beta}{k p} \Bigl(p t - \sin{pt} \Bigr)} \end{equation*}

which is illustrated below.

Response to Ramp Function with x(0) = ˙x(0) = 0

We can interpret this response as an oscillation with amplitude \dfrac{\beta}{k p} about an equilibrium position that is increasing linearly with time.

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