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9.4: IC Transient Response of Underdamped Second Order Systems

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    7680
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    One form of transient, free vibration is response from non-zero initial conditions, with input \(u(t)=0\) for all time. With \(u(t)=0\), response Equations 9.3.8 and 9.3.9 becomes

    \[x(t)=e^{-\zeta \omega_{n} t}\left[x_{0} \cos \omega_{d} t+\left(\frac{\dot{x}_{0}+\zeta \omega_{n} x_{0}}{\omega_{d}}\right) \sin \omega_{d} t\right], \text { for } 0 \leq t \text { and } 0 \leq \zeta<1\label{eqn:9.21a} \]

    By following the procedure of Equations 7.3.2-7.3.4 for combining the two sinusoids of Equation \(\ref{eqn:9.21a}\), we can also express the IC, free-vibration response as

    \[x(t)=x_{\max } e^{-\zeta \omega_{n} t} \cos \left(\omega_{d} t+\phi\right), \text { for } 0 \leq t \text { and } 0 \leq \zeta<1\label{eqn:9.21b} \]

    where \(x_{\max }=\sqrt{x_{0}^{2}+\left(\frac{\dot{x}_{0}+\zeta \omega_{n} x_{0}}{\omega_{d}}\right)^{2}}\) and \(\phi=\tan ^{-1}\left[-\left(\frac{\dot{x}_{0}+\zeta \omega_{n} x_{0}}{\omega_{d}}\right) / x_{0}\right]\)

    Note that for \(\zeta=0\), response Equations \(\ref{eqn:9.21a}\) and \(\ref{eqn:9.21b}\) reduce to the results Equation 7.3.1 and Equation 7.3.4 derived for an undamped system in Chapter 7. In this regard, it is useful to keep in mind the definition of the damped natural frequency, \(\omega_{d} \equiv \omega_{n} \sqrt{1-\zeta^{2}}\).

    The equation \(\omega_{d} \equiv \omega_{n} \sqrt{1-\zeta^{2}}\) also shows that damping reduces the frequency of free vibration, and increases the period, \(T_{d} \equiv 2 \pi / \omega_{d}\). However, the words “frequency” and “period” are used loosely in this case, because the damped response is not truly periodic. More correctly, \(T_{d}\) is defined as the time between successive local crests or troughs of the response, and between successive positive-going or negative-going zeros.

    Figure \(\PageIndex{2}\) on the next page is an annotated sketch of response Equations \(\ref{eqn:9.21a}\) and \(\ref{eqn:9.21b}\) for positive values of the ICs, \(x_{0}>0\) and \(\dot{x}_{0}>0\), and for the small damping ratio \(\zeta=0.11\). The output is a sinusoid, \(\cos \left(\omega_{d} t+\phi\right)\), modulated by a decaying exponential envelope, \(\pm x_{\max } e^{-\zeta \omega_{n} t} \equiv \pm x_{\max } e^{-t / \tau_{2}}\). In this equation for the exponential envelope, we define the time constant \(\tau_{2}\) appropriate for underdamped 2nd order systems as

    \[\tau_{2} \equiv \frac{1}{\zeta \omega_{n}}\label{eqn:9.22} \]

    clipboard_ea37482f8d222da81e340cf97d0469295.png
    Figure \(\PageIndex{1}\): IC response of a damped 2nd order system, \(\zeta = 0.11\)

    It is often necessary to measure experimentally the dynamic response of mechanical systems, which requires sensors (transducers) that sense a response quantity and convert it into an electrical voltage. The sensors most abundant and relatively inexpensive for mechanical systems are translational accelerometers (see homework Problem 10.12). Velocity sensors and translation (displacement) sensors are also used, but less commonly. It is appropriate, therefore, that we find from “displacement” Equations \(\ref{eqn:9.21a}\) and \(\ref{eqn:9.21b}\) the corresponding equations for free-vibration “velocity” and “acceleration.” One differentiation of Equation \(\ref{eqn:9.21b}\) gives

    \[\dot{x}(t)=x_{\max } e^{-\zeta \omega_{n} t}\left[-\zeta \omega_{n} \cos \left(\omega_{d} t+\phi\right)-\omega_{d} \sin \left(\omega_{d} t+\phi\right)\right] \nonumber \]

    Next, we combine the two sinusoidal terms with use of the trigonometric identity \(\sin A \cos B+\cos A \sin B=\sin (A+B)\) and the equation \(\omega_{d} \equiv \omega_{n} \sqrt{1-\zeta^{2}}\), leading to

    \[\dot{x}(t)=-\omega_{n} x_{\max } e^{-\zeta \omega_{n} t} \sin \left(\omega_{d} t+\phi+\sin ^{-1} \zeta\right), \text { for } 0 \leq t \text { and } 0 \leq \zeta<1\label{eqn:9.23} \]

    One more differentiation of Equation \(\ref{eqn:9.23}\) followed by a similar combination procedure gives

    \[\ddot{x}(t)=-\omega_{n}^{2} x_{\max } e^{-\zeta \omega_{n} t} \cos \left(\omega_{d} t+\phi+2 \sin ^{-1} \zeta\right), \text { for } 0 \leq t \text { and } 0 \leq \zeta<1\label{eqn:9.24} \]

    We see that displacement, velocity, and acceleration all have the same damped sinusoidal form. Therefore, experimental measurements of any of the three can be used for identification of system parameters such as \(\zeta\) and \(\omega_d\), a subject addressed in subsequent sections.


    This page titled 9.4: IC Transient Response of Underdamped Second Order Systems is shared under a CC BY-NC 4.0 license and was authored, remixed, and/or curated by William L. Hallauer Jr. (Virginia Tech Libraries' Open Education Initiative) via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.