Problem 4 - Classical Mechanics

Question

Draw the phase portrait of the damped linear oscillator, whose displacement \(x(t)\) satisfies \(\ddot{x}+\mu \dot{x}+\omega_{0}^{2} x=0\), in the phase plane \((x, y)\), where \(y=\dot{x} .\) Distinguish the cases (a) under- (or light) damping \(0<\mu<2 \omega_{0}\), (b) over-damping \(\quad \mu>2 \omega_{0}\) (c) critical damping \(\quad \mu=2 \omega_{0}\)

Step-by-Step Solution

Answer

For the damped linear oscillator described by the equation \(\ddot{x}+\mu \dot{x}+\omega_{0}^{2} x=0\), we have analyzed the motion for different damping coefficients and distinguished three cases: 1. Under-damping (0 < \(\mu\) < \(2\omega_{0}\)): The phase portrait shows spirals converging to the origin, representing oscillatory behavior with decreasing amplitude. 2. Over-damping (\(\mu\) > \(2\omega_{0}\)): The phase portrait shows two families of curves converging to the origin, corresponding to different quadrants where the trajectories start, representing non-oscillatory, decaying motion. 3. Critical damping (\(\mu\) = \(2\omega_{0}\)): The phase portrait displays unique trajectories for each initial condition, all converging to the origin, representing the fastest possible decay without oscillation.

Step 1: Find the general solution of the given equation for each case

(a) Under-damping (0 < \(\mu\) < \(2\omega_{0}\)): In this case, the roots of the characteristic equation are complex: \(\lambda_{1,2} = \frac{-\mu \pm \sqrt{\mu^2 - 4\omega_0^2}}{2}\) The general solution is: \(x(t) = e^{-\frac{\mu}{2}t}(A \cos(\sqrt{\omega_0^2 - \frac{\mu^2}{4}}t)+B \sin(\sqrt{\omega_0^2 - \frac{\mu^2}{4}}t))\) (b) Over-damping (\(\mu\) > \(2\omega_{0}\)): In this case, the roots of the characteristic equation are real and distinct: \(\lambda_{1,2} = \frac{-\mu \pm \sqrt{\mu^2 - 4\omega_0^2}}{2}\) The general solution is: \(x(t) = C e^{\lambda_1 t} + D e^{\lambda_2 t}\) (c) Critical damping (\(\mu\) = \(2\omega_{0}\)): In this case, the roots of the characteristic equation are real and equal: \(\lambda_{1,2} = \frac{-\mu}{2}\) The general solution is: \(x(t) = (E + Ft)e^{-\frac{\mu}{2}t}\)

Step 2: Find the velocity function for each case

(a) Under-damping: \(y(t) = \dot{x}(t) = e^{-\frac{\mu}{2}t}(-\frac{\mu}{2}(A \cos(\sqrt{\omega_0^2 - \frac{\mu^2}{4}}t)+B \sin(\sqrt{\omega_0^2 - \frac{\mu^2}{4}}t)) - \sqrt{\omega_0^2 - \frac{\mu^2}{4}}(- A \sin(\sqrt{\omega_0^2 - \frac{\mu^2}{4}}t) + B \cos(\sqrt{\omega_0^2 - \frac{\mu^2}{4}}t)))\) (b) Over-damping: \(y(t) = \dot{x}(t) = C \lambda_1 e^{\lambda_1 t} + D \lambda_2 e^{\lambda_2 t}\) (c) Critical damping: \(y(t) = \dot{x}(t) = -\frac{\mu}{2}(E + Ft)e^{-\frac{\mu}{2}t} + Fe^{-\frac{\mu}{2}t}\)

Step 3: Draw phase portraits for each case

The phase portraits represent various possible trajectories for the given system depending on the initial conditions. (a) Under-damping: In this case, the phase portrait of the motion has the shape of spirals converging to the origin. (b) Over-damping: In this case, the phase portrait shows two families of curves converging to the origin, one corresponding to trajectories that start in the first/second quadrant and the other corresponding to trajectories starting in the third/fourth quadrant. (c) Critical damping: In this case, the phase portrait shows a unique trajectory for each initial condition, converging to the origin.

Key Concepts

phase portrait

A phase portrait is a graphical representation that shows all possible trajectories of a dynamical system in a phase plane. For a damped harmonic oscillator, this usually involves the displacement, denoted as \(x(t)\), and the velocity \(y(t) = \dot{x}(t)\). In the context of a damped harmonic oscillator, the phase portrait provides insight into how different types of damping affect the behavior of the system over time. By plotting \(x(t)\) against \(y(t)\), one can observe the nature of the motion and the system's return to equilibrium.

For each type of damping—underdamping, overdamping, and critical damping—the phase portrait will have distinct characteristics. Understanding these differences can help in forecasting the system's response to certain initial conditions.

underdamping

In the case of underdamping, where the damping coefficient \(\mu\) satisfies \(0 < \mu < 2\omega_{0}\), the system exhibits an oscillatory motion. This situation arises when the damping is not strong enough to halt the oscillations immediately. Instead, the system gradually loses energy over each cycle until it eventually comes to rest.

The phase portrait for an underdamped system typically shows spirals that converge to the origin. These spirals represent how the amplitude of oscillations decreases over time. The system tends to oscillate back and forth, crossing the equilibrium point multiple times before stopping.

• Oscillates back and forth
• Gradual energy loss
• Spiral-shaped phase portrait

overdamping

Overdamping occurs when the damping coefficient \(\mu\) is larger than \(2\omega_{0}\). In this scenario, the system returns to equilibrium more sluggishly than in critical damping, but without oscillating. It takes a longer time to reach the equilibrium state because the system is excessively damped.

The phase portrait for overdamping features two distinct families of curves, each converging to the origin. These curves suggest a simple, monotonic return to the equilibrium position. Unlike underdamped systems, there is no overshooting, simply a slow, steady decline.

• No oscillations
• Sluggish return to equilibrium
• Distinct curves in phase portrait

critical damping

Critical damping represents the borderline condition, where \(\mu = 2\omega_{0}\). In this case, the system returns to equilibrium as fast as possible without oscillating. It provides the quickest return to the rest position, making it highly efficient for systems where a rapid response is desired.

The phase portrait for a critically damped system displays a unique trajectory for each initial condition, all converging smoothly and quickly to the origin. This characteristic makes critical damping an essential concept in engineering, ensuring that systems such as doors, car suspensions, and measuring instruments operate efficiently.

• No oscillations
• Fastest return to equilibrium
• Unique trajectory in phase portrait