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7.3: Simple IC Response and Step Response of Undamped Second Order Systems

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    7665
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    We find in this section two important special case solutions of Equations 7.2.4 and 7.2.5:

    1. pure initial condition response for zero input, \(u(t) = 0\); and
    2. pure step response with both initial conditions being zero.

    For the case of pure initial condition response (also called free vibration), with \(u(t) = 0\), Equations 7.2.4 and 7.2.5 become

    \[x(t)=x_{0} \cos \omega_{n} t+\frac{\dot{x}_{0}}{\omega_{n}} \sin \omega_{n} t, 0 \leq t\label{eqn:7.10} \]

    Let us use a trigonometric identity [see Equations 4.3.2, 4.3.4, and 4.3.5] to combine the two sinusoids of Equations \(\ref{eqn:7.10}\) into a single term. The following definition will reduce the writing required and also will turn out to be physically meaningful:

    \[x_{\max } \equiv \sqrt{x_{0}^{2}+\left(\dot{x}_{0} / \omega_{n}\right)^{2}}\label{eqn:7.11} \]

    \[\Rightarrow \quad x(t)=x_{\max }\left[\cos \omega_{n} t \times\left(\frac{x_{0}}{x_{\max }}\right)-\sin \omega_{n} t \times\left(\frac{-\dot{x}_{0} / \omega_{n}}{x_{\max }}\right)\right] \nonumber \]

    \[\Rightarrow \quad x(t)=x_{\max } \cos \left(\omega_{n} t+\phi\right), 0 \leq t, \quad \phi=\tan ^{-1}\left(\frac{-\dot{x}_{0} / \omega_{n}}{x_{0}}\right)\label{eqn:7.12} \]

    Figure \(\PageIndex{1}\) is an annotated graph of response Equation \(\ref{eqn:7.11}\)-Equation \(\ref{eqn:7.12}\) for positive values of the ICs, \(x_{0}>0\) and \(\dot{x}_{0}>0\). Clearly, the output is a pure sinusoid of amplitude \(x_{\max }\) and phase angle \(\phi\). This form of response is called free vibration, because it occurs without any forcing input, \(u(t)\) = 0. The circular frequency of vibration is \(\omega_n\) rad/s, the natural frequency defined in Equation 7.1.3. The cyclic natural frequency is \(f_{n}=\omega_{n} / 2 \pi\) cycles/s (Hz), and the natural period annotated on Figure \(\PageIndex{1}\) is \(T_{n}=1 / f_{n}=2 \pi / \omega_{n}\) s/cycle. Free vibration at the natural frequency is one of the most important characteristics of undamped systems and, more realistically, of lightly damped systems. Damped 2nd order systems are discussed in Chapter 9.

    clipboard_ee3c936a62e79a1d6cc11a474e395f54c.png
    Figure \(\PageIndex{1}\): IC response of an undamped 2nd order system

    For the case of pure 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 \(U\):

    \[u(t)=U H(t)\label{eqn:7.13} \]

    The appropriate form of the general solution to use in this case is Equation 7.2.4,

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

    Here we used the property of the unit-step function that \(H(t-\tau)=1\) for \(t-\tau>0\) (from Equation 2.4.2 and Figure 2.4.2); this inequality obviously is satisfied for \(\tau\) over the limits of the definite integral. For most applications of the convolution integrals in Equations 7.2.4 and 7.2.5, the form in Equation 7.2.5 is preferable because the integrand term \(u(t-\tau)\) in Equation 7.2.4 is usually difficult to interpret and/or awkward to handle in the integration. This case, however, is an exception since \(u(t-\tau)\) is easy to interpret and is extremely simple. Completing the integration gives

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

    \[\Rightarrow \quad x(t)=U\left(1-\cos \omega_{n} t\right), 0 \leq t\label{eqn:7.14} \]

    Equation \(\ref{eqn:7.14}\) is graphed for a few cycles of response in Figure \(\PageIndex{2}\). The response is sinusoidal and periodic with the system natural period \(T_{n}\), as is the free-vibration IC response. However, Equation \(\ref{eqn:7.14}\) is forced response, and the input \(u(t)\) is non-zero and constant for \(t > 0\), so this step response vibrates around a non-zero mean value (the pseudo-static output value \(x_{p s}=U\)), oscillating between 0 and 2\(U\). In contrast, the free-vibration IC response of Equation \(\ref{eqn:7.12}\) and Figure \(\PageIndex{1}\) vibrates around the zero mean value, oscillating between \(-x_{\max }\) and \(+x_{\max }\).

    clipboard_e51a1506bcd080229df837cf157a74b4f.png
    Figure \(\PageIndex{2}\): Step response of an undamped 2nd order system

    This page titled 7.3: Simple IC Response and Step Response of Undamped 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.