# The Damped Harmonic Oscillator

*Pieter P*

#### Table of Contents *list*

On the previous page, we derived the model of the moving-coil galvanometer, and we concluded that it's
identical to the damped harmonic oscillator with an external driving force:

On this page, we'll solve the equation of the damped harmonic oscillator analytically, discussing the different solution regimes, and calculating the key features of the step response, such as overshoot and rise time.

## Solution of the step response of the damped harmonic oscillator

To make solving the equation easier, we'll define two constants:

Equation

Equation

In this section, we'll derive the solution of the step response, that is, the solution for

### Homogeneous solution

The homogeneous solution

If

### Particular solution for the step response

The particular solution

The Heaviside step function

A guess for the particular solution could be

### General solution for the step response

The general solution is the sum of the homogeneous and the particular solution:

### Determining the constants of integration

To find

#### Case 1: ${\lambda}_{1}\ne {\lambda}_{2}$

In order to be able to evaluate the second boundary condition, we need an expression for

#### Case 2: ${\lambda}_{1}={\lambda}_{2}=\lambda $

In this case, the derivative is a bit more complicated:

## Solution regimes of the damped harmonic oscillator

Depending on the parameters **overdamped regime**, and if the roots have a nonzero imaginary part, we say
that the oscillator is in the **underdamped regime**. On the boundary between these two regimes, the
discriminant of **critically damped regime**.

Looking at equation

Regime | Damping factor | Physical constants |
---|---|---|

Underdamped | ||

Critically damped | ||

Overdamped |

The figure below shows the qualitative differences between the different regimes.

Image source code### Underdamped regime

We know that in this case,

Substituting this into the solution (equation

#### Peak time

Now that we have a nice sinusoidal expression for the solution, we can determine the position of the first peak, the global maximum of the step response (see the previous figure).

The factor

#### Overshoot

By plugging in this value

#### Rise time

The rise time

Since the rise time is inversely proportional to the natural frequency of the system, it's common to eliminate it from
the equation:

The lower the damping ratio, the faster the output rises, so the smaller the rise time. As the damping ration
increases, the system reacts more slowly, and the rise time increases. For

#### Settling time

The settling time

In equation

The amplitude of these extrema can be used to approximate the settling time:

The following image visualizes the exponential decay of the extrema, as derived in equation

#### Damped frequency

The period of the sinusoidal factor of the step response (equation

The corresponding frequency

#### Natural frequency

Looking at equation

### Overdamped regime

In this case,

At

#### Rise time

Unlike in the underdamped case, the step response of an overdamped oscillator never reaches its final value

To compute the rise time, we need to find

Note that the coefficients

Unfortunately, this type of equation doesn't have a general analytical solution, so we'll have to settle for an approximated or numerical solution.

A first observation is that as the damping factor

As a consequence of

This means that if

We can simplify the denominator of equation

To solve equation

The following image shows the numerical solutions for the true rise time, the approximation by ignoring the second
exponential, and the linear approximation, i.e. the asymptote for

As expected, the approximation that ignores the second exponential gives poor results for

To get a better look at the differences between the true solution and the approximation, we subtract both of them from their asymptote. That results in the following figure.

Image source code
Both curves have a similar shape. A possible approach to refine our approximation for the rise time could be to
start from

The constraint completely determines the value of

For example, using

#### Settling time

In the overdamped case, the rise time and the settling time are tightly coupled.
The settling time gives requires solving the same kind of transcendental equation of the same form as

The same kind of approximation is used as well, by ignoring the second exponential term.

As expected, the approximation is more accurate for the settling time than for the rise time, because the second exponential has more time to decay.

### Critically damped regime

The critically damped regime is the regime with the fastest rise time while not having any overshoot or oscillations.

Image source code#### Rise time

Like the overdamped regime, the

#### Settling time

The formulation of the settling time is of same the form of equation