Damped Simple Harmonic Motion Calculator

Calculator inputs

This calculator uses the underdamped trigonometric motion form. Enter either physical system parameters or direct damping parameters.

Input mode

Initial amplitude A

Phase angle φ (degrees)

Evaluation time t

Graph duration

Graph points

Sample rows

Time unit label

Displacement unit label

Physical parameter inputs

Mass m

Damping coefficient c

Spring constant k

Direct motion inputs

Equivalent mass m

Damping factor β

Damped angular frequency ωd

Reset

Example data table

This example uses A = 5, β = 0.15, ωd = 2.8 rad/s, and φ = 20°.

Time (s)	Displacement x(t) (m)
0	4.698463
0.5	-0.822569
1	-4.30342
1.5	-0.649186
2	3.499253
2.5	1.662325

Formula used

Underdamped displacement:
x(t) = A e^-βt cos(ω_dt + φ)

Velocity:
v(t) = -A e^-βt [β cos(ω_dt + φ) + ω_d sin(ω_dt + φ)]

Acceleration:
a(t) = A e^-βt [(β² - ω_d²) cos(ω_dt + φ) + 2βω_d sin(ω_dt + φ)]

From a mass-spring-damper system:
β = c / 2m, ω_n = √(k/m), ζ = c / 2√(km), ω_d = ω_n√(1 - ζ²)

Mechanical energy:
E(t) = ½kx² + ½mv²

The trigonometric damped form applies only to the underdamped case, where ζ < 1.

How to use this calculator

Choose the input mode. Use physical parameters if you know mass, damping, and spring stiffness.
Enter amplitude, phase angle, evaluation time, graph duration, and number of plotted points.
Fill either the physical parameter fields or the direct damping parameter fields.
Set time and displacement unit labels to match your application.
Press Calculate motion to show the result above the form.
Review displacement, velocity, acceleration, energy, and derived constants.
Use the graph buttons to switch among displacement, velocity, acceleration, and energy plots.
Export the sampled data with the CSV or PDF buttons.

How to graph a trigonometric damped motion function on the calculator

Enter amplitude, phase, damping factor, and damped angular frequency, or derive them from mass, damping, and spring values. Set the graph duration and point count, then calculate. The chart first shows displacement with exponential envelopes. Use the graph buttons to switch to velocity, acceleration, or energy and inspect the motion shape clearly.

FAQs

1. What does this calculator solve?

It solves underdamped simple harmonic motion using the trigonometric form. It returns displacement, velocity, acceleration, energy, damping ratio, frequencies, period, decay time, half-life, sampled tables, and graphs.

2. When is the trigonometric damped form valid?

Use it only for underdamped motion, where the damping ratio is below one. In physical terms, that means the damping coefficient must stay below the critical damping limit.

3. What is the meaning of β?

β is the exponential decay rate. Larger β values shrink the oscillation amplitude faster. It controls how quickly the motion envelope drops over time.

4. Why are there two frequency values?

The natural frequency ωn belongs to the undamped system. The damped frequency ωd is the actual oscillation rate after damping reduces the motion frequency slightly.

5. What does the phase angle change?

The phase angle shifts the waveform left or right in time. It changes the initial position and initial direction without changing the decay rate or oscillation frequency.

6. What does the energy result represent?

The energy value is the sum of spring potential energy and kinetic energy at each time. In a damped oscillator, this mechanical energy decreases as damping removes energy from the system.

7. Can I use custom units?

Yes. The calculator lets you set time and displacement unit labels. These labels are used in results, tables, and graph axes for clearer reporting.

8. What should I do if the calculator shows an underdamped error?

Reduce damping, increase stiffness, or increase mass so the damping ratio becomes less than one. Then the system will follow the trigonometric damped motion form used here.