Explore the fascinating physics of damped harmonic motion with our free online Damped Oscillation Graph Calculator. Input initial amplitude, natural frequency, and damping ratio to instantly compute the damped frequency, time constant, and system classification, revealing how oscillatory systems lose energy over time and visualize their behavior.
Formula:
The displacement x(t) of a damped oscillating system over time t can be described by the general formula:
x(t) = A₀ * e-ζω₀t * cos(ωdt + φ)
Where:
- A₀: Initial Amplitude (maximum displacement at t=0, often assumed φ=0 for basic cases)
- e: Euler's number (base of the natural logarithm, approx. 2.71828)
- ζ (zeta): Damping Ratio (a dimensionless value)
- ω₀ (omega-naught): Natural Frequency (undamped, in rad/s)
- ωd (omega-d): Damped Natural Frequency =
ω₀ * √(1 - ζ²)(in rad/s, for underdamped systems) - t: Time (in seconds)
- φ (phi): Phase Angle (initial phase, often assumed 0 for simplicity)
Additionally, key characteristics to understand damped oscillation behavior include:
- Decay Rate:
ζω₀(determines how fast the amplitude decreases, in 1/s) - Time Constant (τ):
1 / (ζω₀)(the time it takes for the amplitude to decay to 1/e of its initial value, in seconds)