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 2^nd 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 2^nd order ODE, Equation 7.1.5, with ICs \(x(0) \equiv x_{0}\) and \(\dot{x}(0) \equiv \dot{x}_{0}\):

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

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