Laplace Transform of Mass-Spring-Damper System

Published on May 29, 2025

Calculus

Find the Laplace Transform of the solution for a mass-spring-damper system with an applied force, given initial conditions. The solution involves setting up the differential equation, applying the Laplace Transform, and solving for X(s).

Question

Answer

Here’s a step-by-step solution to the problem:

The problem describes a mass-spring-damper system with an external force. We need to set up the differential equation for the system and then find the Laplace Transform of its solution.

1. Identify the given parameters and forces:

Mass,
Spring constant,
Damping force: The damping constant is “twice the velocity”. This means the damping force is proportional to the velocity, . The problem states “damping constant of twice the velocity”, which is a bit ambiguous. However, in the context of a damping force , if the damping constant itself is “twice the velocity”, it would imply , making the damping force , which is non-linear. A more standard interpretation for “damping constant of twice the velocity” in a linear system is that the damping coefficient is 2. Let’s assume the damping force is .
Applied force, (positive force)