Chapter 2
sheetsheet

\begin{align*} \int _{a}^{b}e^{c\tau }\sin \left ( t-\tau \right ) d\tau &=\frac {c\left [ e^{c\tau }\sin \left ( t-\tau \right ) \right ] _{a}^{b}+\left [ e^{c\tau } \cos \left ( t-\tau \right ) \right ] _{a}^{b}}{\left ( 1+c^{2}\right ) } \\ \int _{a}^{b}e^{c\tau }\cos \left ( t-\tau \right ) d\tau &=\frac {c\left [ e^{c\tau }\cos \left ( t-\tau \right ) \right ] _{a}^{b}-\left [ e^{c\tau } \sin \left ( t-\tau \right ) \right ] _{a}^{b}}{1+c^{2}} \\ \int \cos at&=\frac {\sin at}{a} \\ \int \sin at&=\frac {-\cos at}{a} \\ \end{align*}

F(t)	Guess
\(ke^{bt}\)	\(Ae^{bt}\)
\(kt^{n}\)	\(A_{n}t^{n}+\cdots +A_{0}\)
\(\cos \omega t\) or \(\sin \omega \)	\(c_{1}\cos \omega t+c_{2}\sin \omega t\)
\(ke^{at}\cos \omega t\)	\(e^{at}\left ( c_{1}\cos \omega t+c_{2}\sin \omega t\right ) \)
\(x^{\prime \prime }+x^{\prime }+x=f\left ( t\right ) \)	\(\left [ s^{2}X-sx\left ( 0\right ) -x^{\prime }\left ( 0\right ) \right ] \)
\(L\left ( t\right ) =\int _{0}^{\infty }x\left ( t\right ) e^{-st}dt\)	\(+\left [ sX-x\left ( 0\right ) \right ] +X=F\)

roots  \(\int udv=uv-\int vdu\)	\(x\left ( t\right ) \)
real and distinct	\(Ae^{\lambda _{1}t}+Be^{\lambda _{2}t}\)
double real	\(Ae^{\lambda t}+Bte^{\lambda t}\)
complex \(\alpha \pm j\beta \)	\(e^{\alpha t}\left ( A\cos \beta t+B\sin \beta t\right ) \)
\(x\left ( t\right ) =A\cos \omega _{n}t+B\sin \omega _{n}t\)	\(x\left ( t\right ) =C\sin \left ( \omega _{n}t+\theta \right ) \)
\(A=u\left ( 0\right ) B=\frac {v\left ( 0\right ) }{\omega _{n}},\theta =\arctan \left ( \frac {A}{B}\right ) \)	\(C=\sqrt {A^{2}+B^{2}}\ P=\left [ v_{1}v_{2}\right ] \)

\(\begin {pmatrix} \frac {\partial g}{\partial x_{1}} & \frac {\partial g}{\partial x_{2}}\\ \frac {\partial f}{\partial x_{1}} & \frac {\partial f}{\partial x_{2}}\end {pmatrix} \)

modal: \(x=M^{-\frac {1}{2}}q\)

Find eigenvalus of

\(\tilde {k}=M^{-\frac {1}{2}}kM^{-\frac {1}{2}}\)

\(r^{\prime \prime }=\Lambda r,q=P\ r,\)

\(P^{T}=P^{-1}(orthon.)\)

\(r\left ( 0\right ) =P^{T}M^{\frac {1}{2}}x\left ( 0\right ) \)

\(x\left ( t\right ) =M^{-\frac {1}{2}}P\ r\left ( t\right ) \)

\(\omega _{n}^{2}=\frac {k}{m}\)

\(r=\frac {\omega }{\omega _{n}}\)

\(\zeta =\frac {c}{c_{cr}}\)

\(c_{cr}=2\sqrt {km}=2\omega _{n}m\)

\(\omega _{d}=\omega _{n}\sqrt {1-\zeta ^{2}}\)  \(\zeta <1\)

\(mx^{\prime \prime }\left ( t\right ) +kx\left ( t\right ) =0\rightarrow x\left ( t\right ) =e^{-\zeta \omega _{n}t}\left ( A\cos \omega _{d}t+B\sin \omega _{d}t\right ) \) or \(x\left ( t\right ) =Ce^{-\zeta \omega _{n}t}\sin \left ( \omega _{d}t-\theta \right ) \)

\(A=u\left ( 0\right ) ,B=\frac {v\left ( 0\right ) +x\left ( 0\right ) \zeta \omega _{n}}{\omega _{d}},C=\sqrt {A^{2}+B^{2}},\theta =\tan ^{-1}\left ( \frac {B}{A}\right ) \)

\(L=T-U\), \(\frac {d}{dt}\frac {\partial L}{\partial q_{i}^{\prime }}-\frac {\partial L}{\partial q_{i}}=Q_{i},\ \)_____________________Rayleigh \(\frac {d}{dt}\frac {\partial L}{\partial q^{\prime }}-\frac {\partial L}{\partial q}+\frac {\partial R}{\partial q^{\prime }}=0\ \)where \(R=\frac {1}{2}c\left ( q^{\prime }\right ) ^{2}\) \(\ \)\(\delta W=Q_{i}\delta q_{i}\)

\(mx^{\prime \prime }\left ( t\right ) +kx\left ( t\right ) =F_{0}\sin \omega t\rightarrow x\left ( t\right ) =A\cos \omega _{n}t+B\sin \omega _{n}t+\frac {F_{0}}{k}\frac {1}{1-r^{2}}\sin \omega t,A=x\left ( 0\right ) ,\ B=\frac {v\left ( 0\right ) }{\omega _{n}}-\frac {r}{1-r^{2}}\)

\(mx^{\prime \prime }\left ( t\right ) +cx^{\prime }\left ( t\right ) +kx\left ( t\right ) =F_{0}\sin \omega t\rightarrow x\left ( t\right ) =e^{-\zeta \omega _{n}t}\left ( A\cos \omega _{d}t+B\sin \omega _{d}t\right ) +\frac {F_{0}}{k}\frac {1}{\sqrt {\left ( 1-r^{2}\right ) ^{2}+\left ( 2\zeta r\right ) ^{2}}}\sin \left ( \omega t-\theta \right ) \)

\(\theta =\arctan \left ( \frac {2\zeta r}{1-r^{2}}\right ) ,\ \lambda _{1,2}=\frac {-b}{2a}\pm \frac {\sqrt {b^{2}-4ac}}{2a}\) where \(ax^{2}+bx+c=0,\ \)

\(\sin 2A\)	\(2\sin A\cos A\)
\(\cos 2A\)	\(2\cos ^{2}A-1\)
\(\sin A\sin B\)	\(\frac {1}{2}\left ( \cos \left ( A-B\right ) -\cos \left ( A+B\right ) \right ) \)
\(\cos A\cos B\)	\(\frac {1}{2}\left ( \cos \left ( A-B\right ) +\cos \left ( A+B\right ) \right ) \)
\(\sin A\cos B\)	\(\frac {1}{2}\left ( \sin \left ( A-B\right ) +\sin \left ( A+B\right ) \right ) \)
\(h=v_{i}t+\frac {1}{2}gt^{2}\)	\(h=\frac {v_{i}+v_{f}}{2}t\)
\(v_{f}^{2}=v_{i}^{2}+2gh\)	\(v_{f}=v_{i}+gt\)
speed is \(\sqrt {2gh}\)	\(h_{u}\left ( t\right ) =\frac {\hat {F}}{m\omega _{n}}\sin \omega _{n}t\)
	\(\hat {F}=F\Delta t=mv\)

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 betwen\(\frac {x_{2}^{2}}{2}-\frac {g}{l}\cos x_{1}=c\)

\(\frac {dx_{1}}{dt}=\pm \sqrt {c_{1}+\frac {2g}{l}\cos x_{1}}\)

\(t=t_{0}+\int _{x_{1}\left ( t_{0}\right ) }^{x_{1}\left ( t\right ) }\frac {dx_{1}}{\sqrt {c_{1}+\frac {2g}{l}\cos x_{1}}}\)

convert: \(x^{\prime \prime }+kx=0\)

\(\frac {dx_{2}}{dt}+kx_{1}=0,\frac {dx_{2}}{dx_{1}}\frac {dx_{1}}{dt}=-kx_{1}\)

\(\frac {dx_{2}}{dx_{1}}x_{2}=-kx_{1}\)

\(\frac {x_{2}^{2}}{2}=-k\frac {x_{1}^{2}}{2}+C\)

\(h_{d}\left ( t\right ) =\frac {1}{m\omega _{d}}e^{-\zeta \omega _{n}t}\sin \omega _{d}t\)

\(\sin \left ( a\pm b\right ) \)	\(\sin a\cos b\pm \cos a\sin b\)
\(\cos \left ( a\pm b\right ) \)	\(\cos a\cos b\mp \sin a\sin b\)
\(\sin ^{2}a\)	\(\frac {1}{2}\left ( 1-\cos 2a\right ) \)

\(\cos ^{2}a=\frac {1}{2}\left ( 1+\cos 2a\right ) \)	\(sin\left ( A\pm 90\right ) =\cos A\)
\(\cos \left ( a\pm 90^{0}\right ) =\mp \sin a\)	\(sin\left ( A\pm 180\right ) =\mp \sin A\)
\(\cos \left ( a\pm 90^{0}\right ) =\cos a\)	\(\cos \left ( A\pm 180\right ) =-\cos A\)

Figure 2.1: laws of cosine

\(c^{2}=a^{2}+b^{2}-2ab\cos \left ( \lambda \right ) \)

\(\underset {t->0^{+}}{\lim }f\left ( t\right ) =\underset {s->\infty }{\lim }sF\left ( s\right ) \)

\(\underset {s->0}{\lim }sF\left ( s\right ) =\underset {t->\infty }{\lim }f\left ( t\right ) \)

\(M\ddot {x}+kx=0\), assume \(x_{i}=A_{i}\cos \left ( \omega t+\phi _{i}\right ) ,\)Plug in, rewrite as \(\left [ sys\right ] \left [ A\right ] =0\), find eigens of sys,each \(\omega _{i},\)find \(r_{1}=\left ( \frac {A_{2}^{\left ( 1\right ) }}{A_{1}^{\left ( 1\right ) }}\right ) ,r_{2}=\left ( \frac {A_{2}^{\left ( 2\right ) }}{A_{1}^{\left ( 2\right ) }}\right ) \)

\(x_{1}=A_{1}^{\left ( 1\right ) }\cos \left ( \omega _{1}t+\phi _{1}\right ) +A_{1}^{\left ( 2\right ) }\cos \left ( \omega _{2}t+\phi _{2}\right ) ,\ \ \ \ x_{2}=A_{2}^{\left ( 1\right ) }\cos \left ( \omega _{1}t+\phi _{1}\right ) +A_{2}^{\left ( 2\right ) }\cos \left ( \omega _{2}t+\phi _{2}\right ) \), use \(A_{2}^{\left ( 1\right ) }=r_{1}A_{1}^{\left ( 1\right ) },\) \(A_{2}^{\left ( 2\right ) }=r_{2}A_{1}^{\left ( 2\right ) }\)

\(x_{1}=A_{1}^{\left ( 1\right ) }\cos \left ( \omega _{1}t+\phi _{1}\right ) +A_{1}^{\left ( 2\right ) }\cos \left ( \omega _{2}t+\phi _{2}\right ) ,\ \ \ \ x_{2}=r_{1}A_{1}^{\left ( 1\right ) }\cos \left ( \omega _{1}t+\phi _{1}\right ) +r_{2}A_{1}^{\left ( 2\right ) }\cos \left ( \omega _{2}t+\phi _{2}\right ) \)

\(g\left ( t\right ) =\frac {a_{0}}{2}+\sum _{n=1}^{\infty }a_{n}\cos n\left ( 2\pi f\right ) t+b_{n}\sin n\left ( 2\pi f\right ) t\)

\(a_{0}=\frac {1}{T/2}\int _{0}^{T}f\left ( t\right ) \ \ \ \ b_{n}=\frac {1}{T/2}\int _{0}^{T}f\left ( t\right ) \sin n\left ( 2\pi f\right ) tdt\)

\(a_{n}=\frac {1}{T/2}\int _{0}^{T}f\left ( t\right ) \cos n\left ( 2\pi f\right ) tdt\ \ \ \ \ \ T=period\ of\ f\left ( t\right ) \)

\(h_{over}\left ( t\right ) =\frac {1}{2m\omega _{n}\sqrt {\xi ^{2}-1}}e^{-\xi \omega _{n}t}\left ( e^{\omega _{n}\sqrt {\xi ^{2}-1}t}-e^{-\omega _{n}\sqrt {\xi ^{2}-1}t}\right ) \)

\(h_{critical}\left ( t\right ) =\frac {1}{m}te^{-\xi \omega _{n}t}\)

\(f\left ( t\right ) =\)impulse\(=F\Delta t=\left [ mv\left ( 0^{-}\right ) -mv\left ( 0^{+}\right ) \right ] \delta \left ( t\right ) \)

solid disk, around center \(I=\frac {mr^{2}}{2}\)

thin loop, around center \(I=mr^{3}\)

solid sphere \(I=\frac {2}{5}mr^{2}\)

rod, axis at center of rod \(I=\frac {ML^{2}}{12}\)

rod, axis at end of rod \(I=\frac {ML^{2}}{3}\)

series: \(\frac {1}{k}=\frac {1}{k_{1}}+\frac {1}{k_{2}}\)par \(k=k_{1}+k_{2}\)

\( {\textstyle \int \limits _{a}^{b}} \tau \sin \omega \left ( t-\tau \right ) d\tau =\frac {-\sin \left ( \omega \left ( t-a\right ) \right ) -a\omega \cos \left ( \omega \left ( t-a\right ) \right ) +\sin \left ( \omega \left ( t-b\right ) \right ) +b\omega \cos \left ( \omega \left ( t-b\right ) \right ) }{\omega }\)

2 equations of motions for unbalanced: \(\left ( M-m\right ) \ddot {x}+c\dot {x}+kx=F_{r}\) and \(m\left ( \ddot {x}+\ddot {x}_{r}\right ) =-F_{r}\), where \(x_{r}=e\sin \omega t,\) eq for \(M\) is

\(M\ddot {x}+c\dot {x}+kx=me\omega ^{2}\sin \omega t\), guess \(X_{p}=X\sin \left ( \omega t-\theta \right ) \), we obtain \(X=\frac {Me}{m}\frac {r^{2}}{\sqrt {\left ( 1-r^{2}\right ) ^{2}+\left ( 2\xi r\right ) ^{2}}},\theta =\tan ^{-1}\frac {2\xi r}{1-r^{2}}\) perturbation: \(x^{\prime \prime }+\omega _{0}^{2}x+\alpha x^{3}=0\rightarrow x=x_{0}+\alpha x_{1}+\alpha ^{2}x_{2}+\cdots ,\omega ^{2}=\omega _{0}^{2}+\alpha \omega _{1}^{2}\left ( A\right ) +\alpha \omega _{2}^{2}\left ( A\right ) +\cdots ,\)hence \(\omega _{0}^{2}=\omega ^{2}-\alpha \omega _{1}^{2}\left ( A\right ) \). 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 \(\left \vert G\left ( j\omega \right ) \right \vert =\frac {1}{\sqrt {\left ( c\omega \right ) ^{2}+\left ( k-m\omega ^{2}\right ) ^{2}}}\) and from the spectrum, find \(m,c,k\)