Question 2.3.3: As an example of using Laplace transforms to solve a homogen......
As an example of using Laplace transforms to solve a homogeneous differential equation, consider the undamped single-degree-of-freedom system described by
\ddot{x}(t)+\omega_n^2 x(t)=0, \quad x(0)=x_0, \quad \dot{x}(0)=v_0
Taking the Laplace transform of \ddot{x}+\omega_n^2 x=0 for these nonzero initial conditions results in
s^2 X(s)-s x_0-v_0+\omega_n^2 X(s)=0
by direct application of the definition given in Appendix B and the linear nature of the Laplace transform. Algebraically solving this last expression for X(s) yields
X(s)=\frac{x_0+s v_0}{s^2+\omega_n^2}
Using L^{-1}[X(s)]=x(t) and entries (6) and (5) of Table B.1 yields that the solution is
x(t)=x_0 \cos \omega_n t+\frac{v_0}{\omega_n} \sin \omega_n t
This is, of course, in total agreement with the solution obtained in Chapter 1 .
| Table B.1 Partial List of Functions and Their Laplace Transforms with Zero Initial Conditions and t > 0 | ||
| F(s) | f(t) | |
| (1) | 1 | \delta\left(t_0\right) \text { unit impulse at } t_0 |
| (2) | \frac1s | 1, unit step |
| (3) | \frac{1}{s+a}\left(\frac{1}{s-a}\right) | e^{-a t} \quad\left(e^{a t}\right) |
| (4) | \frac{1}{(s+a)(s+b)} | \frac{1}{b-a}\left(e^{-a t}-e^{-b t}\right) |
| (5) | \frac{\omega}{s^2+\omega^2} | \sin \omega t |
| (6) | \frac{s}{s^2+\omega^2} | \cos \omega t |
| (7) | \frac{1}{s\left(s^2+\omega^2\right)} | \frac{1}{\omega^2}(1-\cos \omega t) |
| (8) | \frac{1}{s^2+2 \zeta \omega s+\omega^2} | \frac{1}{\omega_d} e^{-\zeta \omega t} \sin \omega_d t, \zeta<1, \omega_d=\omega \sqrt{1-\zeta^2} |
| (9) | \frac{\omega^2}{s\left(s^2+2 \zeta \omega s+\omega^2\right)} | 1-\frac{\omega}{\omega_d} e^{-\zeta \omega t} \sin \left(\omega_d t+\phi\right), \phi=\cos ^{-1} \zeta, \zeta<1 |
| (10) | \frac{1}{s^n} | \frac{t^{n-1}}{(n-1) !}, n=1,2 \ldots |
| (11) | \frac{n !}{(s-\omega)^{n+1}} | t^n e^{\omega t}, n=1,2 \ldots |
| (12) | \frac{1}{s(s+\omega)} | \frac{1}{\omega}\left(1-e^{-\omega t}\right) |
| (13) | \frac{1}{s^2(s+\omega)} | \frac{1}{\omega^2}\left(e^{-\omega t}+\omega t-1\right) |
| (14) | \frac{\omega}{s^2-\omega^2} | \sinh \omega t |
| (15) | \frac{1}{s^2\left(s^2+\omega^2\right)} | \cosh \omega t |
| (16) | \frac{1}{s^2\left(s^2+\omega^2\right)} | \frac{1}{\omega^3}(\omega t-\sin \omega t) |
| (17) | \frac{1}{\left(s^2+\omega^2\right)^2} | \frac{1}{2 \omega^3}(\sin \omega t-\omega t \cos \omega t) |
| (18) | \frac{s}{\left(s^2+\omega^2\right)^2} | \frac{t}{2 \omega} \sin \omega t |
| (19) | \frac{s^2-\omega^2}{\left(s^2+\omega^2\right)^2} | t \cos \omega t |