MATH
MATH
MATH
MATH

F(t) Guess
$ke^{bt}$ $Ae^{bt}$
$kt^{n}$ MATH
$\cos\omega t$ or $\sin\omega$ MATH
MATH MATH
MATH MATH
MATH MATH
roots
MATH
$x\left( t\right) $
real and distinct MATH
double real MATH
complex $\alpha\pm j\beta$ MATH
MATH MATH
MATH MATH
MATH
modal: MATH
Find eigenvalus of
MATH
MATH
MATH
MATH
MATH
MATH
MATH
MATH
MATH
MATH $\zeta<1$
MATH or MATH
MATH
$L=T-U$, MATH
Rayleigh MATHwhere MATH
$\ $
MATH
MATH
MATH
MATH where $ax^{2}+bx+c=0,\ $

$\sin2A$ $2\sin A\cos A$
$\cos2A$ $2\cos^{2}A-1$
$\sin A\sin B$ MATH
$\cos A\cos B$ MATH
$\sin A\cos B$ MATH
MATH MATH
MATH $v_{f}=v_{i}+gt$
speed is $\sqrt{2gh}$ MATH
MATH
phase roots $\lambda_{1}$ and $\lambda_{2}>0$ Unstable, repelling
phase roots $\lambda_{1}$ and $\lambda_{2}<0$ stable, attracting
both real, one >0 and one <0 unstable saddle point
equal roots and >0 unstable, degenrate
equal roots and <0 stable, degenrate
complex, real part>0 unstabe, spiral out
complex, real part<0 stable, spiral in
pure complex conjugrates marginaly stable, cirlce
time betwenMATH
MATH
MATH
convert: MATH
MATH
MATH
MATH
MATH
MATH MATH
MATH MATH
$\sin^{2}a$ MATH
MATH MATH
MATH MATH
MATH MATH
laws_of_cosine.png MATH

$\ $
$M\ddot{x}+kx=0$, assume MATHPlug in, rewrite as MATH, find eigens of sys,each $\omega_{i},$find MATH
MATH, use MATH MATH
MATH

MATH
MATH
MATH
MATH
MATH
$f\left( t\right) =$impulseMATH

solid disk, around center $I=\frac{mr^{2}}{2}$
t
hin loop, around center $I=mr^{3}$
solid sphere MATH
rod, axis at center of rod MATH

rod, axis at end of rod $I=\frac{ML^{2}}{3}$
series: MATHpar $k=k_{1}+k_{2}$
MATH

d1.png d3.png d4.png d2.png feedback.png

2 equations of motions for unbalanced: MATH and MATH, where MATH eq for $M$ is

MATH, guess MATH, we obtain MATH perturbation: MATHhence MATH. Sub in ODE, generate 2 ODE's and solve for $x_{0}$ and use result to find $x_{1}.$watch for IC and resonanse. For system ID, set up MATH and from the spectrum, find $m,c,k$