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9.6: Step Response of Underdamped Second Order Systems

  • Page ID
    7682
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    For step response, we set the ICs to zero, and we define the input to be a step function at time \(t = 0\), with step magnitude \(U\): \(u(t)=U H(t)\). The appropriate form of the general solution to use is Equation 9.3.8, which becomes [with \(H(t-\tau)=1\) for \(\tau<t\)]

    \[x(t)=\frac{\omega_{n}^{2}}{\omega_{d}} \int_{\tau=0}^{\tau=t} e^{-\zeta \omega_{n} \tau} \sin \omega_{d} \tau \times u(t-\tau) d \tau=\frac{\omega_{n}^{2}}{\omega_{d}} \int_{\tau=0}^{\tau=t} e^{-\zeta \omega_{n} \tau} \sin \omega_{d} \tau \times U H(t-\tau) d \tau \nonumber \]

    \[=U \frac{\omega_{n}^{2}}{\omega_{d}} \int_{\tau=0}^{\tau=t} e^{-\zeta \omega_{n} \tau} \sin \omega_{d} \tau d \tau \nonumber \]

    This integral can be evaluated by hand (e.g., using integration by parts), but the process is tedious. The following evaluation was completed with use of a table of integrals:

    \[x(t)=U \frac{\omega_{n}^{2}}{\omega_{d}} \frac{1}{\left(-\zeta \omega_{n}\right)^{2}+\omega_{d}^{2}}\left[e^{-\zeta \omega_{n} \tau}\left(-\zeta \omega_{n} \sin \omega_{d} \tau-\omega_{d} \cos \omega_{d} \tau\right)\right]_{\tau=0}^{\tau=t} \nonumber \]

    \[=U \frac{1}{\omega_{d}}\left[e^{-\zeta \omega_{n} t}\left(-\zeta \omega_{n} \sin \omega_{d} t-\omega_{d} \cos \omega_{d} t\right)-1\left(-\omega_{d}\right)\right] \nonumber \]

    \[x(t)=U\left[1-e^{-\zeta \omega_{n} t}\left(\cos \omega_{d} t+\frac{\zeta \omega_{n}}{\omega_{d}} \sin \omega_{d} t\right)\right], \text { for } 0 \leq t \text { and } 0 \leq \zeta<1\label{eqn:9.29} \]

    Note that the coefficient of \(\sin \omega_{d} t\) in Equation \(\ref{eqn:9.29}\) is dependent only on the damping ratio:

    \[\frac{\zeta \omega_{n}}{\omega_{d}}=\frac{\zeta \omega_{n}}{\omega_{n} \sqrt{1-\zeta^{2}}}=\frac{\zeta}{\sqrt{1-\zeta^{2}}}=\zeta_{s} \nonumber \]

    Step response Equation \(\ref{eqn:9.29}\) for small damping ratio \(\zeta=0.11\) is plotted over a few cycles of response on Figure \(\PageIndex{1}\). Relative to the pseudo-static response, \(x_{p s}=U\), the actual step response of a damped system initially overshoots, then undershoots, then overshoots again, then undershoots again, etc., etc. But damping dissipates the energy of vibration, causing the response eventually (not shown on Figure \(\PageIndex{1}\)) to settle statically at \(\lim _{t \rightarrow \infty} x(t)=x_{p s}=U\). Several step-response characteristics (called specifications, or specs in engineering jargon) of a system can be quantified and often are of great interest in practice. For example, the rise time is the time required for the response first to reach \(U\), which on Figure \(\PageIndex{1}\) is just a bit longer than \(\frac{1}{2} \pi / \omega_{n}\). However, before studying those characteristics in more detail, it is appropriate that we first consider impulse response.

    clipboard_e66661e7fce1da0fd820ae73a0f4d1a43.png
    Figure \(\PageIndex{1}\): Step response of a damped 2nd order system, \(\zeta=0.11\)

    This page titled 9.6: Step 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.