Damped Harmonic Oscillator

Closed-form solution of all four regimes · m(d²x/dt²) + c(dx/dt) + kx = 0

Input parameters

Mass

Block mass (m > 0)

0.120

Spring constant

Spring stiffness (k > 0)

N/m

1500

Initial position

Displacement from equilibrium at t = 0 (negatives allowed)

-2020

Initial velocity

Velocity at t = 0 (negatives allowed)

-55

Damping

Damping coefficient

Viscous drag c ≥ 0, in kg/s (= N·s/m)

kg/s

050

Speed

Results

Detected regime

Underdamped

0 < ζ < 1: oscillates with an amplitude that decays exponentially under the envelope.

Natural frequency ω₀

6.325

rad/s

Frequency f₀

1.007

Natural period T₀

0.9935

Damping ratio ζ

0.1186

Secondary Parameters

Critical damping c_c

12.65 kg/s

Initial energy E₀

0.05 J

Settling time t_s

5.087 s

Damped frequency ω_d

6.28 rad/s

Pseudo-period T_d

1.001 s

Quality factor Q

4.216

Log decrement δ

0.7504

Fundamentals & Explanation

The Oscillator Equation and its Two Universal Parameters

A mass m on a spring of stiffness k, subject to a viscous drag proportional to velocity (coefficient c), obeys Newton's second law as a linear, homogeneous, second-order differential equation:

m\,\ddot{x} + c\,\dot{x} + k\,x = 0

Dividing by m gives the canonical form, in which all of the physics is captured by just two parameters: the natural frequency ω₀ and the dimensionless damping ratio ζ.

\ddot{x} + 2\zeta\omega_0\,\dot{x} + \omega_0^2\,x = 0,\qquad \omega_0=\sqrt{\tfrac{k}{m}},\quad \zeta=\frac{c}{2\sqrt{mk}}

The denominator of ζ is the critical damping coefficient c_c = 2√(mk) = 2mω₀: the boundary between oscillating and not oscillating. That is why ζ = c/c_c is the natural, unit-independent measure of damping.

The Four Regimes

The roots of the characteristic equation r² + 2ζω₀r + ω₀² = 0, namely r = ω₀(−ζ ± √(ζ²−1)), decide the nature of the response. The sign of the discriminant ζ²−1 separates the regimes:

Regime	Condition	x(t)
Ideal (S.H.M.)	ζ = 0	A cos(ω₀t − φ)
Underdamped	0 < ζ < 1	e^−ζω₀t(C₁cos ω_d t + C₂sin ω_d t)
Critical	ζ = 1	(C₁ + C₂t) e^−ω₀t
Overdamped	ζ > 1	C₁e^r₁t + C₂e^r₂t

The ideal case is the ζ → 0 limit of the underdamped one: with ζ = 0 we have ω_d = ω₀ and the exponential envelope equals 1, so the oscillation never decays and energy is conserved. The constants C₁, C₂ are fixed by the initial conditions x₀ and v₀; in the underdamped case C₁ = x₀ and C₂ = (v₀ + ζω₀x₀)/ω_d.

Damped Frequency and Pseudo-period

When the system oscillates (ζ < 1), it does not vibrate at its natural frequency but at the always-lower damped frequency: friction slows the swing.

\omega_d = \omega_0\sqrt{1-\zeta^2}\,,\qquad T_d = \frac{2\pi}{\omega_d}

The amplitude is bounded by the envelope ±A·e^−ζω₀t: each peak is smaller than the previous one by a fixed factor. Velocity and acceleration follow by differentiation; here the acceleration is taken straight from the equation itself, a = −2ζω₀v − ω₀²x, which keeps it exact across every regime.

Energy and Dissipation

Kinetic

E_k = \tfrac{1}{2} m v^2

Potential

E_p = \tfrac{1}{2} k x^2

Total

E = E_k + E_p

With no damping, total energy is constant: kinetic and potential trade back and forth without loss (a closed ellipse in the phase portrait). With damping, friction dissipates energy at the rate dE/dt = −c·v² ≤ 0, never positive: the energy decreases monotonically until the system comes to rest.

Underdamped Metrics

Three indicators summarise how damped an oscillating system is:

Q = \frac{1}{2\zeta}\qquad \delta = \frac{2\pi\zeta}{\sqrt{1-\zeta^2}}\qquad t_s \approx \frac{4}{\zeta\omega_0}

Quality factor Q: how many oscillations survive before dying out; high Q = light damping.
Logarithmic decrement δ: the log of the ratio of two successive peaks; the classic experimental way to measure ζ.
Settling time t_s: when the envelope drops to 2 % (the standard control-engineering criterion).

The Phase Portrait

Plotting velocity v against position x reveals the dynamics at a glance, with time eliminated. The ideal system traces a closed ellipse (constant energy); the underdamped one a spiral winding into the origin; the critical and overdamped ones a direct curve to the origin without coiling. The origin is the attractor: the rest state every dissipative system eventually reaches.

Critical vs. Overdamped

Both return to equilibrium without oscillating, but the critical case (ζ = 1) does it in the shortest possible time: the optimal tuning of a car shock absorber or a door closer. The overdamped case (ζ > 1) features two time constants (τ₁ and τ₂). The response decays as a sum of two exponentials, but the overall dynamics are slower and governed almost entirely by its slow time constant τ₁ = −1/r₁; the heavier the friction, the more sluggish the return. In the results panel, we compute both so you can compare them.

Model Limits and Assumptions

Linearity

The spring obeys Hooke's law (F = −kx) and the friction is viscous, proportional to velocity. Large amplitudes (a non-linear spring) or dry (Coulomb, constant) friction fall outside the model.

Free oscillation

There is no external force: the equation is homogeneous. Adding a periodic drive introduces resonance and a steady-state regime — the subject of a separate forced-oscillations calculator.

Constant parameters

m, k, c are assumed time-invariant. The same model describes second-order RLC circuits (with L↔m, 1/C↔k, R↔c), a direct bridge to electronics.

Quantity	Symbol	SI Unit
Mass	m	kilogram (kg)
Spring constant	k	newton/metre (N/m)
Damping coefficient	c	kg/s (= N·s/m)
Natural frequency	ω₀	rad/s
Damping ratio	ζ	dimensionless
Energy	E	joule (J)

References and Resources

[1] French, A. P. Vibrations and Waves, Ch. 3–4: "Free vibrations of physical systems" and "The forced vibrations of damped systems". MIT Introductory Physics Series.
[2] Taylor, J. R. Classical Mechanics, Ch. 5: "Oscillations". University Science Books.
[3] Thornton, S. & Marion, J. Classical Dynamics of Particles and Systems (5th ed.), Ch. 3: "Oscillations". Brooks/Cole.

Damped Harmonic Oscillator

Input parameters

Results

Secondary Parameters

Fundamentals & Explanation

The Oscillator Equation and its Two Universal Parameters

The Four Regimes

Damped Frequency and Pseudo-period

Energy and Dissipation

Underdamped Metrics

The Phase Portrait

Critical vs. Overdamped

Model Limits and Assumptions

Linearity

Free oscillation

Constant parameters

References and Resources

You may also like:

The Oscillator Equation and its Two Universal Parameters

The Four Regimes

Damped Frequency and Pseudo-period

Energy and Dissipation

Underdamped Metrics

The Phase Portrait

Critical vs. Overdamped

Model Limits and Assumptions

Linearity

Free oscillation

Constant parameters

References and Resources