Solve displacement, velocity, acceleration, and damping parameters. Compare regimes with clean inputs and calculated results. Export summaries, study formulas, and use sample values confidently.
Use consistent SI units for reliable outputs.
Example values for a typical underdamped spring mass system.
| Mass (kg) | k (N/m) | c (N·s/m) | x0 (m) | v0 (m/s) | t (s) | Regime | ω_n (rad/s) | ζ | x(t) (m) |
|---|---|---|---|---|---|---|---|---|---|
| 1.00 | 25.00 | 2.00 | 0.08 | 0.00 | 2.00 | Underdamped | 5.000000 | 0.200000 | -0.010887 |
The governing differential equation is:
m x″ + c x′ + k x = 0
The main derived parameters are:
If ζ < 1, then ω_d = √(ω_n² − α²).
x(t) = e^(−αt)[A cos(ω_d t) + B sin(ω_d t)]
If ζ = 1, then x(t) = e^(−αt)(A + Bt).
If ζ > 1, then x(t) = C1 e^(r1 t) + C2 e^(r2 t).
Here, r1 = −α + √(α² − ω_n²) and r2 = −α − √(α² − ω_n²).
The calculator differentiates the displacement model to get velocity.
Acceleration is also checked with: a(t) = −(c/m)v(t) − (k/m)x(t).
Kinetic energy = 0.5mv². Potential energy = 0.5kx². Total energy is their sum.
Damped harmonic motion describes a vibrating system that loses energy over time. A spring mass system is the classic example. The spring pulls the mass back toward equilibrium. The damper resists motion. That resistance reduces amplitude during every cycle.
Damping changes both the shape and speed of the response. Light damping allows oscillation to continue for many cycles. Critical damping removes oscillation and returns the system quickly. Heavy damping also avoids oscillation, but the return becomes slower. Engineers use these differences to tune comfort, safety, and control.
The natural angular frequency shows how fast the system would oscillate without damping. The damping ratio compares actual damping to the critical value. The decay rate shows how quickly amplitude shrinks. The damped frequency tells you the real oscillation speed when damping is present. Energy values reveal how much motion remains at a chosen time.
This calculator is useful in vibration analysis, suspension design, instrument response, packaging tests, and machine isolation. It helps students verify textbook problems. It also helps engineers inspect transient behavior before building a full simulation. The tool gives displacement, velocity, acceleration, force, period, and energy in one place.
Start with the regime label. That instantly tells you the type of motion. Next, review the damping ratio and frequencies. Then inspect displacement and velocity at the selected time. If the energy ratio is small, the system has already dissipated most of its initial motion. The time history preview makes the decay trend easier to see.
Keep all units consistent. Use kilograms, meters, seconds, and newtons. Small unit mistakes can distort damping ratio and energy. When comparing designs, keep mass and stiffness fixed and vary only damping. That makes the effect of the damper clear and measurable.
It is oscillatory motion with energy loss. A resistive force, often proportional to velocity, reduces amplitude as time increases.
The damping ratio compares actual damping with critical damping. It tells you whether the system is underdamped, critically damped, or overdamped.
Negative displacement means the mass moved to the opposite side of equilibrium. It is normal in oscillatory motion.
ω_n is the undamped natural frequency. ω_d is the reduced oscillation frequency after damping is included.
The damper converts mechanical energy into heat or other losses. That is why total mechanical energy drops over time.
Yes. If the damping coefficient is zero, the tool treats the system as undamped and removes exponential decay.
Use SI units for best consistency. Enter mass in kilograms, stiffness in newtons per meter, damping in newton second per meter, and time in seconds.
Critical damping is useful when fast settling is needed without oscillation. Common examples include instruments, door closers, and control mechanisms.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.