Mass–Spring–Damper Calculator: Step-by-Step Solutions

Free and forced vibrations: damping type, solution, amplitude–phase form, steady state, graph. Every step of the solution is shown, as you would write it on paper, using exact fractions.

What it solves

Solve $m{x}^{\prime \prime }+c{x}^{\prime }+kx=F_0\cos \omega t$ with $x(0)=x_0$, $x^{\prime }(0)=v_0$. Use $F_0=0$ for free motion; $\omega =0$ means a constant force $F_0$.

How to use it

1. Enter your own numbers or functions in the boxes (an example is already filled in, or press Random example).
2. Check the equation preview, then press Solve step by step.
3. Reveal the solution one step at a time with Next step, or show everything at once.

Need the theory first? Read Section 5.3: Free, Damped Motion, which explains the method in small steps with worked examples and exercises, then come back and test yourself with the Chapter 5 quiz.

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