Question

Study the self-oscillating modes of the motion of a pendulum with small friction under the action of a constant torque \(M\) : $$ \tilde{x}+\sin x+\varepsilon \dot{x}=M $$

Short answer

Answer: The self-oscillating modes of a pendulum with small friction under the action of constant torque depend on the friction coefficient, ε. For ε=0, the pendulum exhibits simple harmonic motion without damping. For ε>0, the pendulum exhibits damped oscillatory motion. The general solution for the case of small oscillations with damping is given by x(t)= e^(-εt/2)(C_1*cos(ω_1 t) + C_2*sin(ω_1 t)) + M, where C_1 and C_2 depend on the initial conditions.

Step-by-step solution

Rewrite the equation in the form of a second-order ODE

Rewrite the given equation as a second-order ODE, by noting that \(\tilde{x}\) refers to the second derivative of \(x\) with respect to time (\(\ddot{x}\)), and \(\dot{x}\) refers to the first derivative of \(x\) with respect to time (\(\frac{d}{dt}x\) or \(\frac{dx}{dt}\)): $$ \ddot{x}+\sin x+\varepsilon \frac{dx}{dt}=M $$

Linearize the equation

Linearize the equation by approximating \(sin x \approx x\) for small oscillations. This approximation works well when the angle \(x\) is sufficiently small, as the pendulum's motion closely follows a sinusoidal pattern and can be analyzed using simple harmonic motion: $$ \ddot{x}+ x +\varepsilon \frac{dx}{dt}=M $$

Find the homogeneous equation and the particular solution

The equation we derived in step 2 is a non-homogeneous, linear second-order ODE with a constant forcing term. It can be rewritten as: $$ \ddot{x} +\varepsilon \frac{dx}{dt} + x = M $$ To find the solution, we need to find both the homogeneous equation and the particular solution. The homogeneous equation looks like: $$ \ddot{x} +\varepsilon \frac{dx}{dt} + x = 0 $$

Analyze the effect of friction on the motion

The equation can be further rewritten into a standard form of the second-order ODE: $$ \frac{d^2x}{dt^2} +\varepsilon \frac{dx}{dt} + x = 0 $$ Now, let's analyze the equation by considering the cases of different values of the friction coefficient, \(\varepsilon\). 1. If \(\varepsilon=0\), the equation represents a simple harmonic oscillator without damping given by: $$ \frac{d^2x}{dt^2} + x = 0 $$ This equation has solutions in the form of sinusoidal functions, that is \(x(t) = A\cos(\omega_0 t) + B\sin(\omega_0 t),\) where \(A\) and \(B\) are constants determined by initial conditions, and \(\omega_0^2=1\). 2. If \(\varepsilon>0\), we have a damped oscillatory motion, and the motion is determined by the roots of the characteristic equation, which are complex numbers. The general solution takes the form of: $$ x(t)= e^{-\frac{\varepsilon}{2}t}(C_1\cos(\omega_1 t) + C_2\sin(\omega_1 t)) $$ The values of \(C_1\) and \(C_2\) depend on the initial conditions. Now, we will find the particular solution by assuming that the pendulum eventually reaches a steady oscillation with constant amplitude. This final steady-state solution can be assumed as a constant, say \(x_p\). So, we have: $$ \ddot{x_p} +\varepsilon \dot{x_p} + x_p = M $$ Since the steady-state solution is a constant, its derivatives are zero. The equation simplifies to: $$ x_p = M $$

Combine the homogeneous and particular solution

The general solution of the given equation can be obtained by combining the homogeneous solution from step 4 with the particular solution. For the case of small oscillations with damping (\(\varepsilon>0\)), the general solution is: $$ x(t)= e^{-\frac{\varepsilon}{2}t}(C_1\cos(\omega_1 t) + C_2\sin(\omega_1 t)) + M $$ In conclusion, we have studied the self-oscillating modes of a pendulum with small friction under the action of constant torque \(M\). The general solution of the motion is determined by the friction coefficient \(\varepsilon\), and the initial conditions.

Key concepts

Pendulum motion

A pendulum is an excellent system for studying oscillatory motion. It consists of a weight suspended from a pivot that swings back and forth under the influence of gravity. In ideal conditions, a pendulum's motion can be described as simple harmonic motion, repeating at regular intervals. This can be modeled with a differential equation:

• If there are no external forces or friction, the motion is purely sinusoidal.
• A constant external torque (\(M\)) can modify this, leading to different oscillation frequencies.
• The displacement of the pendulum (\(x\)) will typically be a function of time.

Understanding pendulum motion essential for delving into complex oscillatory systems.

Linearization

Linearization is a mathematical process used to simplify the models of systems that are typically non-linear. For small oscillations, the sine function (\(\sin x\)) can be approximated as its argument (\(x\)). This assumption simplifies the differential equation of the pendulum:

• This approximation holds true when the angle of displacement is small.
• The resulting linear equation is significantly easier to solve.
• It allows for analysis using linear techniques like those for harmonic oscillators.

By using linearization, it's possible to examine a pendulum's behavior without complex computations.

Damping effects

Damping refers to the effects of forces that reduce the amplitude of oscillations. In a pendulum, damping causes the motion to gradually cease over time. This is represented by a frictional term in the differential equation:

• The damping coefficient (\(\varepsilon\)) affects how quickly the motion loses energy.
• In a damped system, the motion is explained by an exponential decay.
• A larger (\(\varepsilon\)) suggests stronger damping, leading to faster cessation of motion.

Understanding damping helps in predicting and controlling oscillatory systems in real-world applications like engineering and physics.

Harmonic oscillation

Harmonic oscillation describes a type of motion where the restoring force is directly proportional to the displacement. For a pendulum, this leads to sinusoidal motion, described by a second-order differential equation:

• In the absence of damping (\(\varepsilon=0\)), the motion is undamped simple harmonic motion.
• The presence of damping leads to a damped harmonic oscillator.
• In both cases, the solution involves sinusoidal components, influenced by initial conditions and external forces.

Harmonic oscillation forms the basis for understanding many dynamic systems, providing insights for designing clocks, suspension systems, and other oscillatory devices.