Differential Equations: Mechanical and Electrical Vibrations

Using the systematic approach, find the initial and final values of the current through the inductor and the voltage across the capacitor for a second-order circuit before and after the switch is closed or a source is turned on.

Modeling forced mechanical vibrations using second-order non-homogeneous linear differential equations.

Derive the particular solution $x_p(t)$ for a forced damped oscillation given the differential equation $m x'' + c x' + k x = F_0 \cos(\omega t)$ and express it in terms of a single cosine term using the phase angle.

Derive the equation of motion for a mass-spring-damper system using Newton's second law.

A mass $m$ is attached to a spring on a frictionless surface. Initially at rest at an equilibrium position. When displaced and released, determine the differential equation describing its motion using Newton's Second Law and Hooke's Law. Solve the differential equation for the position of the mass over time.

A typical vibrating mechanical system consisting of mass, spring and a viscous damper. This is an over-damped case where the roots of the characteristic equation are real and not repeated.

Given a mechanical system with a mass, spring, and damper, disturbed by initial displacement with no initial velocity, derive and solve the differential equation: $m x'' + c x' + k x = 0$ with given values: $m = 1 \text{ kg}$, $c = 3 \text{ Ns/m}$, $k = 2 \text{ N/m}$, and initial conditions $x(0) = 0.01 \text{ m}$, $x'(0) = 0$.

Given a mass oscillating on a spring with friction, derive the second order constant coefficient differential equation governing its motion.

Solve the differential equation $mx'' + cx' + kx = 0$ for the three cases: underdamped, overdamped, and critically damped.