Vibration Equation Solver Calculator

Calculator inputs

Use consistent units for displacement, equilibrium, and velocity. Frequency is in hertz, time is in seconds, and mass is in kilograms.

Reset

Initial displacement x(0)

Set the starting position.

Initial velocity v(0)

Positive or negative values are allowed.

Natural frequency fn

Enter cycles per second.

Damping ratio ζ

0 is undamped. 1 is critical.

Equilibrium offset c

Reference position of equilibrium.

Mass m

Used for stiffness and energy.

Evaluation time t

Main solution point shown above.

Series start time

First row in the response table.

Series end time

Must be greater than start time.

Series points

Choose between 2 and 200 points.

Formula used

Base model

The calculator solves the second-order vibration equation y″ + 2ζω_ny′ + ω_n²y = 0, where y = x − c and ω_n = 2πf_n.

Underdamped case

For ζ < 1, y(t) = e^−ζω_nt[y₀cos(ω_dt) + Bsin(ω_dt)], with ω_d = ω_n√(1 − ζ²) and B = (v₀ + ζω_ny₀) / ω_d.

Critical and overdamped cases

For ζ = 1, y(t) = [y₀ + (v₀ + ω_ny₀)t]e^−ω_nt. For ζ > 1, y(t) = C₁e^r₁t + C₂e^r₂t, where r_1,2 = −ω_n(ζ ∓ √(ζ² − 1)).

Derived values

Acceleration comes from a = −2ζω_nv − ω_n²y. Stiffness is k = mω_n². Total energy equals ½mv² + ½ky².

How to use this calculator

Enter the initial displacement and initial velocity at time zero.
Provide the natural frequency and damping ratio for the system.
Set the equilibrium offset if your motion oscillates around a shifted center.
Add mass to compute stiffness, kinetic energy, potential energy, and total energy.
Choose the evaluation time and series range, then submit the form.
Review the solved response, characteristic roots, and generated response table above the form.
Export the computed summary and time series to CSV or PDF.

Example data table

Example inputs: x(0) = 0.05, v(0) = −0.02, fn = 2.00 Hz, ζ = 0.12, c = 0.00, and m = 1.50.

Time	Displacement	Velocity	Acceleration	Total energy
0.00	0.050000	-0.020000	-7.835365	0.296388
0.40	0.005166	0.328683	-1.807107	0.084186
0.80	-0.013414	0.104622	1.802675	0.029519
1.20	-0.005572	-0.066972	1.081806	0.007040
1.60	0.002342	-0.051409	-0.214863	0.002632
2.00	0.002370	0.004616	-0.388243	0.000681

Frequently asked questions

1. What does this solver calculate?

It solves the free vibration response of a second-order system and returns displacement, velocity, acceleration, stiffness, energies, damping behavior, and characteristic roots.

2. Can I use any displacement unit?

Yes. Keep displacement and equilibrium in the same unit. Velocity and acceleration should follow the matching time-based versions of that same unit.

3. What happens when the damping ratio equals one?

The system becomes critically damped. It returns to equilibrium without oscillating and usually settles faster than an overdamped response.

4. Why is the damped frequency sometimes zero?

For critical or overdamped motion, oscillation disappears. In those cases, a damped angular frequency is not meaningful, so the value is shown as zero.

5. Why do I need mass?

Mass lets the calculator derive stiffness from the natural frequency and compute kinetic, potential, and total energy consistently.

6. What is the equilibrium offset?

It is the center position around which motion occurs. The solver converts your displacement into relative motion before applying the vibration equations.

7. Does this handle forced vibration?

No. This version solves the homogeneous response only. It is best for free vibration and decay analysis from known initial conditions.

8. What do the exported files contain?

The exports include the solved summary metrics and the generated response series table, making it easy to review or share your results.