# Ordinary Differential Equations/Motion with a Damping Force

Applications of Second-Order Differential Equations **>** **Motion with a Damping Force**

Simple Harmonic Motion with a Damping Force can be used to describe the motion of a mass at the end of a spring under the influence of friction.

## Contents

## Laws of Motion[edit]

The friction force is considered to obey a linear law, that to say, it is given by the following expression:

- where

- is a positive constant and represents the coefficient of friction,
- represents the friction force and
- is the velocity.

Note that the minus sign indicates that the friction force always opposes the movement.

## The Differential Equation of the Motion[edit]

The differential equation of the motion with a damping force will be given by:

In order to obtain the leading coefficient equal to **1**, we divide this equation by the mass:

### Non-conservation of energy[edit]

We may multiply the equation of motion by the velocity in order to get an integrable form:

Now we integrate this equation from **0** to **t** to obtain an expression for the energy:

Denoting the mechanical energy by

the variation of energy is given by:

That is to say, if the velocity does not vanish, the system is losing energy. Physically speaking, friction converts mechanical energy into thermal energy.

### Initial condition[edit]

With the free motion equation, there are generally two bits of information one must have to appropriately describe the mass's motion.

- The starting position of the mass.
- The starting direction and magnitude of motion.

Generally, one isn't present without the other. For simplicity, we will consider all displacement below the equilibrium point as and above as .

For upward motion , and for downward motion .

### Solution[edit]

We look for a general solution in the following form:

substituting this solution into the equation, we find the quadratic equation:

the solution of this equation is given by: