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7.2: General Solution for Output of Undamped Second Order Systems

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  • Let us solve Equation 7.1.5 for output \(x(t)\), given any physically realistic input \(u(t)\), for time \(t\) > 0, and given appropriate initial conditions at \(t\) = 0. We use Laplace transformation with application of the inverse convolution transform from Chapter 6. To simplify the notation, we denote \(X(s) \equiv L[x(t)]\). Transforming Equation 7.1.5 with use of Equation 2.2.11 gives

    \[s^{2} X(s)-s x(0)-\dot{x}(0)+\omega_{n}^{2} X(s)=\omega_{n}^{2} L[u(t)]\label{eqn:7.6}\]

    Equation \(\ref{eqn:7.6}\) tells us that we need two initial conditions, one on the output and one on the derivative of the output, for this 2nd order ODE. Accordingly, we simplify the writing with the definitions

    \[\text{ICs for }2^{\text {nd }}\text{ order ODE: }x_{0} \equiv x(0),\text{ initial "position"; }\dot{x}_{0} \equiv \dot{x}(0),\text{ initial "velocity"}\label{eqn:7.7}\]

    Solving Equation \(\ref{eqn:7.6}\) for \(X(s)\) with the use of notation Equation \(\ref{eqn:7.7}\) gives

    \[X(s)=x_{0} \frac{s}{s^{2}+\omega_{n}^{2}}+\frac{\dot{x}_{0}}{\omega_{n}} \frac{\omega_{n}}{s^{2}+\omega_{n}^{2}}+\omega_{n} \overbrace{\frac{\omega_{n}}{s^{2}+\omega_{n}^{2}}}^{F_{1}(s)} \overbrace{L[u(t)]}^{F_{2}(s)}\label{eqn:7.8}\]

    We invert Equation \(\ref{eqn:7.8}\) using transforms Equation 2.4.7 and Equation 2.4.8, and the inverse convolution transform Equation 6.1.5, to find the two equivalent general solution equations for the standard undamped 2nd order ODE, Equation 7.1.5, with ICs \(x(0) \equiv x_{0}\) and \(\dot{x}(0) \equiv \dot{x}_{0}\):

    \[x(t)=\overbrace{x_{0} \cos \omega_{n} t+\frac{\dot{x}_{0}}{\omega_{n}} \sin \omega_{n} t}^{\text{IC response}}+\overbrace{\omega_{n} \int_{\tau=0}^{\tau=t} \sin \omega_{n} \tau \times u(t-\tau) d \tau}^{\text{forced response}}\label{eqn:7.9a}\]

    \[x(t)=\overbrace{x_{0} \cos \omega_{n} t+\frac{\dot{x}_{0}}{\omega_{n}} \sin \omega_{n} t}^{\text{IC response}} + \overbrace{\omega_{n} \int_{\tau=0}^{\tau=t} \sin \omega_{n}(t-\tau) \times u(\tau) d \tau}^{\text {forced response}}\label{eqn:7.9b}\]