Problem 3 - Classical Mechanics - An Undergraduate Text

Question

Rod pendulum A uniform rod of length \(2 a\) is suspended from a fixed point \(O\) by a light inextensible string of length \(b\) attached to one of its ends. The system moves in a vertical plane through \(O\). Take as coordinates the angles \(\theta, \phi\) between the string and the rod respectively and the downward vertical. Show that the equations governing small oscillations of the system about \(\theta=\phi=0\) are $$ \begin{aligned} &b \ddot{\theta}+a \ddot{\phi}=-g \theta \\ &b \ddot{\theta}+\frac{4}{3} a \ddot{\phi}=-g \phi \end{aligned} $$ For the special case in which \(b=4 a / 5\), find the normal frequencies and the forms of the normal modes. Is the general motion periodic?

Step-by-Step Solution

Answer

The equations of motion are derived using Lagrangian mechanics, which leads to a homogeneous system of equations. Solving this system using the characteristic equation yields the normal frequencies. The ratio of these frequencies determines if the general motion of the system is periodic.

Step 1: Deriving equations of motion

Begin by setting up the kinetic and potential energies. Recall that the kinetic energy \(T\) of the rod is given by \(T = \frac{1}{2} m v^2 = \frac{1}{2} m (b^2 \dot \theta^2 + (2a)^2 \dot \phi^2 / 4)\), where \(\dot \theta\) and \(\dot \phi\) are time derivatives of \(\theta\) and \(\phi\). The potential energy \(V\) of the system is \(V = m g h = m g ((b/2) cos \theta + a cos \phi)\). The Lagrangian \(L\) is the difference between \(T\) and \(V\). Hence, use Euler-Lagrange equations to derive the equations of motion, which are \(\frac{d}{dt} (\frac {\partial L}{\partial \dot q_i}) - \frac {\partial L}{\partial q_i} = 0\), where \(q_i\) are the generalized coordinates (\(\theta\) and \(\phi\) in this case).

Step 2: Finding normal frequencies

For small oscillations, linearize the equations of motion around the equilibrium position, which is \(\theta = \phi = 0\). Hence, for the system to be stable, it needs to oscillate around equilibrium. This means, the terms corresponding to \(\theta\) and \(\phi\) on the right hand side of both equations should be negative, giving the given equations. This system can be solved by putting the equations in the form of matrix and finding the characteristic equation, which is determinant (matrix - \(\lambda I\)) = 0. The roots of this are the normal frequencies.

Step 3: Determining the normal modes

The normal modes can be determined by inserting the normal frequencies into the equations of motion. This gives us the ratio of the amplitudes of the angles \(\theta\) and \(\phi\). For \(b = 4a/5\), substitute this into the equations to find the values of the frequencies and solve.

Step 4: Investigating the periodic motion

Whether the general motion is periodic or not depends on whether the ratio between the two normal frequencies is rational or irrational. If it is rational, the motion will be periodic; otherwise, it will not. Solve for this ratio using the frequencies found earlier to determine the nature of the motion.

Key Concepts

Rod Pendulum

A rod pendulum consists of a uniform rod suspended from a pivot point, allowing it to swing back and forth like a pendulum. This system includes both the rod and a string which attaches the rod to a fixed point. The string is described as "light" and "inextensible," meaning it doesn't stretch or have a mass significant enough to influence the system's motion. In this setup, the rod can move freely in a vertical plane, aiding in illustrating the dynamics of oscillation through two distinct angles:

• \( \theta \) – which denotes the angle between the string and a vertical line drawn downward from the pivot.
• \( \phi \) – which represents the angle between the rod and a vertical line through its suspension point.

By analyzing these angles, we better understand how the rod's orientation changes as it swings, which ultimately assists in formulating the system's equations of motion.

Small Oscillations

Small oscillations refer to the tiny movements near equilibrium positions where systems behave linearly and predictably. In our rod pendulum setup, small oscillations occur around the angles \( \theta = 0 \) and \( \phi = 0 \), signifying their positions directly downward. When oscillations are small, we can approximate trigonometric functions like sine and cosine with their Taylor series expansions, leading us to useful simplifications:

• \( \sin \theta \approx \theta \)
• \( \cos \theta \approx 1 \)

These approximations allow us to linearize the equations governing the motion, providing a clearer path to solving the equations mathematically. These adjustments are vitally important since they transform a nonlinear equation, challenging to solve, into a linear one that is feasible for analysis using standard algebraic methods like matrices and determinants.

Normal Frequencies

Normal frequencies, also known as natural frequencies, are the rates at which a system tends to oscillate when not subjected to continuous or unbalanced forces. These frequencies are critical in characterizing the system's intrinsic oscillatory behavior, providing insights into how different parts of the system interact. To find these frequencies in the case of the rod pendulum, we express the equations of motion in matrix form and determine the eigenvalues. The resulting characteristic equation is solved to obtain these frequencies. For this system, the characteristic equation stems from: \[\det \left(\begin{bmatrix} b & a \b & \frac{4}{3}a \end{bmatrix} - \lambda \begin{bmatrix} 1 & 0 \0 & 1 \end{bmatrix}\right) = 0\]The solutions, known as eigenvalues \( \lambda \), signify the squares of the normal frequencies. Substituting the specific value \( b=4a/5 \) into these equations yields the exact frequencies that describe the essence of how this pendulum vibrates naturally.

Periodic Motion

Periodic motion signifies any repeating motion that occurs at regular intervals. In examining the rod pendulum's motion, whether it is periodic relies on the ratio of its normal frequencies:

• If the ratio of these frequencies is rational, the system repeats itself over time, establishing a periodic motion.
• If irrational, the motion does not repeat regularly, and thus, it is considered aperiodic.

For the special case of our rod pendulum where \( b=4a/5 \), after finding the normal frequencies, we must analyze their ratio to reach a conclusion. If upon evaluation, the ratio aligns to a simple fraction, we affirm the motion as periodic. This assessment illustrates the fundamental rhythm of the physical system, celebrating the beauty of recurring patterns or highlighting their absence, which is equally revealing.