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Newbie

## Example of Using Laplace Transforms to Solve an ODE

By Philip J. Pritchard, Ph.D.

This worksheet will:

• Demonstrate how to find the motion of a mass attached to a spring and dashpot due to a known applied force using Laplace transforms
• Apply to dynamics, mechanical engineering, etc.
• Perform using Laplace transforms, spring-mass-dashpot system, equation of motion, plots, etc.

This worksheet illustrates PTC Mathcad's ability to symbolically solve an ordinary differential equation using Laplace transforms. In this example, from dynamics, the worksheet demonstrates how to find the motion x(t) of a mass m attached to a spring (strength k) and dashpot (coefficient c) due to a known applied force F(t).

To begin, this worksheet provides you with an example of a forced vibration of a spring-mass- dashpot system. All data surrounding the problem is laid out along with the relevant equations. To use the Laplace transformation you are shown how to convert each differential of motion x(t) into a corresponding expression in s space, how to find the Laplace transform of applied force F(t), assemble the Laplace transform of the equation of motion, solve that algebraic equation, and transform the expression of space X(s) back into the solution motion x(t).

All notation, formulas, data, computations, plots, and solutions are included in this worksheet to aid you in your own calculations.