Math 2A Cheat Sheet

What to Put on It

Derivative rules for sinh and cosh:

\frac{d}{dx}\sinh{x} = \cosh{x} \hspace{2em} \frac{d}{dx}\cosh{x} = \sinh{x} \tag{1}

Governing equation for a damped oscillator:

\frac{d^2x}{dt^2} + 2 \xi \omega \frac{dx}{dt} + \omega^2x = 0

Roots:

s = - \xi \omega \pm \sqrt{\xi^2 - 1}

Exponential response to forced damping:

Ay'' + By' + Cy = e^{st} \hspace{0.5em} \textnormal{is solved by} \hspace{0.5em} y = Ye^{st} = \frac{1}{As^2 + Bs + C}e^{st}

That Y is the transfer function. s may be any complex number a + i\omega. For resonance introduce an extra t: y_p = Yte^{st}.

Undamped forced oscillation with f(t) = \sin(n\pi t) or \cos(n\pi t):

STABILITY DOES NOT DEPEND ON SYSTEM INPUT