91Ó°ÊÓ

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 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: $$ \ddot{x}+ x +\varepsilon \frac{dx}{dt}=M $$

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: $$ \ddot{x} +\varepsilon \frac{dx}{dt} + x = 0 $$

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: $$ x_p = M $$

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.