Problem: "A rod with a length of $l$, mass $m$, is attached by a thread of length $l/2$ according to figure. The rod may perform small, planar swings. Determine its eigen-frequencies."


enter image description here

Solution attempt: So I started out describing the kinetic energies $T$ and potential energy $V$, expressed as cartesian coordinates, then expressed in terms of $\theta_1$ and $\theta_2$ with their respective time derivatives.

Cartesians: $T = \frac{1}{2}mu^2 + \frac{1}{2}I_{x'}\dot{\theta}_2^2$, $V = mgy$, where $x'$ is referring to the axis passing through the center of mass, CM, of the (assumed homogeneous) thin rod; and where u is the speed of CM, (x, y).

Expressing $x$ and $y$ (CM) in terms of the angles:

$$x=\frac{l}{2}(\sin(\theta_1) + \sin(\theta_2)),$$ $$y=-\frac{l}{2}(\cos(\theta_1) + \cos(\theta_2)).$$

Taking time-derivatives:

$$\dot{x} = \frac{l\dot{\theta}_1}{2}\cos(\theta_1) + \frac{l\dot{\theta}_2}{2}\cos(\theta_2),$$ $$\dot{y} = \frac{l\dot{\theta}_1}{2}\sin(\theta_1) + \frac{l\dot{\theta}_2}{2}\sin(\theta_2)$$

Inserting into $u$:

$$u^2 = \dot{x}^2 + \dot{y}^2$$

Using inertia for thin rod through CM: $I = \frac{1}{12}ml^2$, we end up with:

Same as before: sorry...

$$ L = T - V = \frac{ml^2}{24}[3\dot{\theta}_1^2 + 4\dot{\theta}_2^2 + 6\dot{\theta}_1\dot{\theta}_2\cos(\theta_1 - \theta_2)] + mg\frac{l}{2}[\cos(\theta_1) + \cos(\theta_2)] $$

Using the Euler-Lagrange equation: $$ \frac{\delta}{\delta t}(\frac{\delta L}{\delta \dot{\theta}_i}) - \frac{\delta L}{\delta \theta_i} = 0 $$

After some work, we end up with two differential equations:

$$\ddot{\theta}_1 + \ddot{\theta}_2\cos(\theta_1 - \theta_2) + \dot{\theta}_2^2\sin(\theta_1 - \theta_2) + 2\frac{g}{l}\sin(\theta_1) = 0$$ $$\ddot{\theta}_2 + \frac{3}{4}\ddot{\theta}_1\cos(\theta_1 - \theta_2) - \frac{3}{4}\dot{\theta}_1^2\sin(\theta_1 - \theta_2) + \frac{3g}{2l}\sin(\theta_2) = 0 $$

Comments: This is as far as I've gotten without making assumptions regarding the "smallness" of the oscillations hinted at in the problem statement. I realise now that this problem of mine might be better published in a mathematics forum, that is, if my problem indeed is limited by my mathematical ability (or lack of).

Sorry for this expansive post, and thanks in advance!

EDIT: Regarding lack of algebraic and mathematical steps taken, this is due to the restriction of links allowed when below 10 reputation on this site; sorry for this.

  • $\begingroup$ se.u.94y.info may help $\endgroup$ – user854 Apr 12 '16 at 19:55