Problem 4 - Classical Mechanics - An Undergraduate Text

Question

Triple pendulum A triple pendulum has three strings of equal length \(a\) and the three particles (starting from the top) have masses \(6 m, 2 m, m\) respectively. The pendulum performs small oscillations in a vertical plane. Show that the normal frequencies satisfy the equation $$ 12 \mu^{3}-60 \mu^{2}+81 \mu-27=0 $$ where \(\mu=a \omega^{2} / g\). Find the normal frequencies, the forms of the normal modes, and a set of normal coordinates. \([\mu=3\) is a root of the equation.]

Step-by-Step Solution

Answer

The roots are 3, \( \mu_{1} = \frac{11+\sqrt{7}}{2} \) and \( \mu_{2} = \frac{11-\sqrt{7}}{2} \). The normal frequencies are going to be \( \omega_{i} = \sqrt{\mu_{i} g/a} \). The modal forms and normal coordinates would depend on the initial conditions of the system.

Step 1: Analyzing the given equation

Given the equation \(12 \mu^{3}-60 \mu^{2}+81 \mu-27=0\) where \(\mu=a \omega^{2} / g\), it is known that \(\mu=3\) is a root of the equation. We can simplify the equation by dividing through by 3 and subtracting the known root.

Step 2: Simplifying the equation

After dividing by 3 and subtracting 3, the cubic equation reduces to a quadratic equation, \(\mu^{2} - 11\mu + 9 = 0\). We can solve this to find the remaining roots.

Step 3: Solving the quadratic equation

To solve the quadratic equation, use the formula for solutions of a quadratic equation, \( \mu = \frac{11 \pm \sqrt{(11)^{2} - 4 \times 1 \times 9}}{2 \times 1}\)

Step 4: Calculating the roots

Evaluating the formula, we get three roots, 3, \( \mu_{1} = \frac{11+\sqrt{7}}{2} \) and \( \mu_{2} = \frac{11-\sqrt{7}}{2} \)

Step 5: Finding Normal Frequencies

The normal frequencies are found by taking square root on the both sides and dividing by \(a\), i.e., \( \omega_{i} = \sqrt{\mu_{i} g/a} \)

Step 6: Normal Modes and Coordinates

Normal frequencies and modes depend on initial conditions. But generally, they describe cases where pendulums move at the same frequency. For the triple pendulum, certain modes might include: all pendulums moving in phase, or the top pendulum stationary while the others oscillate. Normal coordinates are obtained from standard position coordinates by a coordinate transformation, which depends on the specifics of the system and initial conditions. More information needed for a detailed analysis.

Key Concepts

Normal Frequencies

Normal frequencies are intrinsic properties of a system that describe the oscillation rates at which the system naturally vibrates. They occur when a system experiences free vibrations, meaning there is no external force influencing the motion. In the context of a triple pendulum, these frequencies are crucial in understanding the behavior of the system's oscillations.

To determine the normal frequencies, we explore the given cubic equation: - The equation is \( 12 \mu^{3} - 60 \mu^{2} + 81 \mu - 27 = 0 \).- Here, \( \mu \) is a dimensionless parameter given by \( \mu = \frac{a \omega^{2}}{g} \).- The solution involves finding the values of \( \mu \) that satisfy this equation.
By substituting \( \mu = 3 \), as a known root, and simplifying the equation, we find the other two roots. These roots provide the values of \( \mu \) which can be used to determine the normal frequencies \( \omega \). We do this by rearranging to find \( \omega_{i} = \sqrt{\mu_{i} g/a} \). This indicates the different rates at which the pendulum can naturally oscillate without external interference.

Normal Modes

Normal modes refer to specific patterns in which all parts of a system oscillate at the same frequency, yet the shape of their motion can vary. In these modes, each part of the system synchronizes their movement.

For the triple pendulum, normal modes help describe how each pendulum mass moves:

• One mode may involve all three pendulums swinging in unison. This means they are in phase, moving together as if they were one large pendulum.
• Another mode could have the top pendulum stay almost stationary, allowing only the lower pendulums to oscillate. This demonstrates a varied role in motion distribution among the pendulums.

The modes depend on the initial conditions and system characteristics, such as mass and string length. Such movement patterns are fundamental, simplified dynamic behaviors of the system. Once identified, these modes make it easier to analyze and predict the behavior of the more complex actual motion.

Normal Coordinates

Normal coordinates are transformed variables which simplify the mathematical representation of a system's dynamics. By applying these coordinates, the motion equations governing a system become decoupled. This simplifies analysis from a complex multi-variable system to individual one-dimensional problems.

In the case of the triple pendulum, obtaining normal coordinates involves executing a transformation from the standard physical coordinates to a system where each coordinate corresponds to a specific normal mode. This transformation often uses a matrix method to manipulate the standard position coordinates into a set of new coordinates.

• This transformation is dependent on system properties and initial conditions.
• Each of the normal coordinates evolves in time according to its own normal frequency.

By separating the motion into these normal coordinates, it becomes easier to study and comprehend the pendulum’s behavior over time, as each normal coordinate responds independently to its equation of motion. This approach sidesteps the complexity of tracking each mass individually, streamlining the analysis of their collective behavior.