2024-03-29 Coupled Oscillations

HW Problem 3 Review

$$I=\frac{2}{5}M{R}^2\cdot \mathbf{I}$$ $$=\frac{2}{5}M{R}^2\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$$ $$I_{ij}=J_{ij}-M({\delta }_{ij}{a}^2-a_i{a}_j)$$ $$I_{ij}=\int \rho \left({\delta }_{ij}\sum _k{x}_{\alpha k}^2-x_{\alpha i}{x}_{\alpha j}\right)dx$$ $$T=\frac{1}{2}\vec{\omega }\{\mathbf{I}\}\vec{\omega }$$ $$({\omega }_1,{\omega }_2,{\omega }_3)\left(\begin{array}{ccc}{I}_{11}& I_{12}& I_{13}\\ I_{21}& I_{22}& I_{23}\\ I_{31}& I_{32}& I_{33}\end{array}\right)\cdot \left(\begin{array}{c}{\omega }_1\\ {\omega }_2\\ {\omega }_3\end{array}\right)$$

This simplified is

$$\frac{1}{2}[I_{11}{\omega }_1^2+I_{12}{\omega }_1{\omega }_2+I_{12}{\omega }_1{\omega }_3+I_{21}{\omega }_2{\omega }_1+I_{22}{\omega }_2^2+I_{23}{\omega }_2{\omega }_3+I_{31}{\omega }_3{\omega }_1+I_{32}{\omega }_3{\omega }_2+I_{33}{\omega }_3^2]$$

NB: Need to memorize the parallel axis theorem, discrete and continuous inertia tensor thing.

E X A M topics

Homeworks 5-9:

• HW 5: Lagrange Multiplier Method ***
• HW 6: Hamiltonian Mechanics
• HW 7: Central Force Problems
• ~~HW 8: Scattering & Non-Inertial frames ~~(not on exam)
• HW 9: Inertia Tensors

4 problems on exam

• One on Lagrange multipliers
• One on Hamiltonian
• One on Central forces
• One on inertia tensor

Hamiltonian Review:

$$H=\sum {\dot{q}}_j\frac{\partial L}{\partial {\dot{q}}_j}-L$$

Also might be

$$T+U$$

Central forces Review centrifugal potential energy

$$U_{cf}=\frac{l^2}{2\mu r^2}$$

Inertia Tensors Discrete, continuous, how to diagonal matrices using eigenvalues

Dont memorize how to solve individual problems, review homework.

Non-lambda method?

How to generate Hamiltonian Generalized Momentum Central Forces Oscillating (Taylor series)

$$\tilde{L}=L+\lambda f$$

Review gravity and spring potentials

Force of constraint F is equal to “something”… Central forces, know the oscillations, taylor series. Inertia tensors should be about 8 equations

The two topics that will be on the final exam are coupled oscillations and vibrations

Coupled Oscillations

In general, this is a complex motion that can always be described using a system of normal coordinates (each oscillates with a single, well defined frequency)

Normal Modes- Making use of initial conditions, motion can be constrained so only 1 normal coordinate varies with time.

For a system of “n” oscillators, there will be “n” normal modes, which may be degenerate.

figure 1

$$T=\frac{1}{2}{m}_1{\dot{x}}_1^2+\frac{1}{2}{m}_2{\dot{x}}_2^2$$ $$U=\frac{1}{2}\kappa x_1^2+\frac{1}{2}\kappa x_2^2+\frac{1}{2}{\kappa }_{12}(x_1-x_2{)}^2$$ $$L=T-U$$ $$L=\frac{1}{2}{m}_1{\dot{x}}_1^2+\frac{1}{2}{m}_2{\dot{x}}_2^2-\frac{1}{2}\kappa x_1^2+\frac{1}{2}\kappa x_2^2+\frac{1}{2}{\kappa }_{12}(x_1-x_2{)}^2$$

First Euler-Lagrange Equation

$$\frac{d}{dt}\frac{\partial L}{\partial {\dot{x}}_1}-\frac{\partial L}{\partial x_1}=0$$ $$M{\ddot{x}}_1+\kappa x_1+{\kappa }_{12}(x_1-x_2)=0$$ $$\frac{d}{dt}\frac{\partial L}{\partial {\dot{x}}_2}-\frac{\partial L}{\partial x_2}=0$$ $$M{\ddot{x}}_1+(\kappa +{\kappa }_{12})x_1-{\kappa }_{12}{x}_2=0$$ $$M{\ddot{x}}_2+(\kappa +{\kappa }_{12})x_2-{\kappa }_{12}{x}_1=0$$

These are coupled equations

Because we expect oscillatory behavior

Solutions should be of an oscillatory form…

$$x_1(t)=B_1{e}^{i\omega t}$$ $$x_2(t)=B_2{e}^{i\omega t}$$

Aside:

$$\ddot{x}=-{\omega }^{2}x$$ $$x=A{e}^{i\omega t}$$ $$\dot{x}=i\omega A{e}^{i\omega t}$$ $$\ddot{x}=-A{\omega }^2{e}^{i\omega t}$$ $$={\omega }^{2}x$$

Plugging this back into coupled equation to cancel $e^{i\omega t}$

$$-M{B}_1{\omega }^2+(\kappa +{\kappa }_{12})B_1-{\kappa }_{12}{B}_2=0$$ $$-M{B}_2{\omega }^2+(\kappa +{\kappa }_{12})B_2-{\kappa }_{12}{B}_1=0$$

Reorganize

$$(\kappa +{\kappa }_{12}-M{\omega }^2)B_1-{\kappa }_{12}{B}_2=0$$ $$-{\kappa }_{12}{B}_1+(\kappa +{\kappa }_{12}-M{\omega }^2)B_2=0$$

Finding solutions: Trivial solution $B_1$ and $B_2$ are zero. No movement.

For a nontrivial solution determinant of $B_1$ and $B_2$ coefficient matrix must be 0

$$\det \left[\begin{array}{cc}\kappa +{\kappa }_{12}-M{\omega }^2& -{\kappa }_{12}\\ -{\kappa }_{12}& \kappa +{\kappa }_{12}-M{\omega }^2\end{array}\right]$$ $$(\kappa +{\kappa }_{12}-M{\omega }^2{)}^2-{\kappa }_{12}^2=0$$ $$\kappa +{\kappa }_{12}-M{\omega }^2=\pm {\kappa }_{12}$$ $$\omega =\pm \sqrt{\frac{\kappa +{\kappa }_{12}\pm {\kappa }_{12}}{M}}$$ $${\omega }_1=\pm \sqrt{\frac{\kappa +2{\kappa }_{12}}{M}}$$ $${\omega }_2=\pm \sqrt{\frac{\kappa }{M}}$$

General Solutions

$$x_1(t)=B_1^{+}{e}^{i{\omega }_{1}t}+B_1^{-}{e}^{-i\omega t}+B_2^{+}{e}^{i{\omega }_{2}t}+B_2^{-}{e}^{-i{\omega }_{2}t}$$ $$x_2(t)=-B_1^{+}{e}^{i{\omega }_{1}t}-B_1^{-}{e}^{-i\omega t}+B_2^{+}{e}^{i{\omega }_{2}t}+B_2^{-}{e}^{-i{\omega }_{2}t}$$

define coordinates

$${\eta }_1=x_1-x_2$$ $${\eta }_2=x_1+x_2$$ $$x_1=\frac{1}{2}({\eta }_2+{\eta }_1)$$ $$x_2=\frac{1}{2}({\eta }_2-{\eta }_1)$$

Our EL equations

$$M({\ddot{\eta }}_2+{\ddot{\eta }}_1)+(\kappa +{\kappa }_{12})({\eta }_2+{\eta }_1)-{\kappa }_{12}({\eta }_2-{\eta }_1)=0$$ $$M({\ddot{\eta }}_2-{\ddot{\eta }}_1)+(\kappa +{\kappa }_{12})({\eta }_2-{\eta }_1)-{\kappa }_{12}({\eta }_2+{\eta }_1)=0$$

Rewrite

$$M({\ddot{\eta }}_2+{\ddot{\eta }}_1)+(\kappa +2{\kappa }_{12}){\eta }_1+\kappa {\eta }_2=0$$ $$M({\ddot{\eta }}_2-{\ddot{\eta }}_1)-(\kappa +2{\kappa }_{12}){\eta }_1+\kappa {\eta }_2=0$$

Add/subtract equations

Subtract:

$$M{\ddot{\eta }}_1+(\kappa +2{\kappa }_{12}){\eta }_1=0$$

Add:

$$M{\ddot{\eta }}_2+2\kappa {\eta }_1=0$$