Spring Oscillation Frequency Calculator

Definition of the oscillation frequency of a mass–spring system

The oscillation frequency of a mass–spring system says how many full cycles of harmonic motion occur in one second. If a spring of constant k is attached to a mass m and the mass is displaced from equilibrium, the restoring force pulls it back. The mass passes the equilibrium position and swings out the other way. The time of one such round trip is the period T, and the frequency is its reciprocal, f = 1/T. When the force is proportional to the displacement, F = −k·x, the motion is harmonic and the frequency does not depend on the amplitude.

The formula this calculator evaluates is f = (1/(2π))√(k/m). The unit is the hertz: 1 Hz is one cycle per second. The constant k is the stiffness of the spring in newtons per meter, and m is the oscillating mass in kilograms. A stiffer spring raises the frequency. A larger mass lowers it, as a square root, not in proportion. For k = 200 N/m and m = 0.5 kg the quotient k/m is 400, its square root is 20 rad/s, and f = 20/(2π) = 3.183 Hz. The period is then 0.3142 s. That pair of numbers is the reference value for the whole page.

History: Hooke, Huygens, and the hertz

Robert Hooke announced that force is proportional to extension. In 1676, he deposited the anagram “ceiiinosssttuv” with the Royal Society, and in 1678, in the treatise De Potentia Restitutiva, he solved it as “ut tensio, sic vis”: as the extension, so the force. Without that linear restoring force there is no harmonic motion of a spring. Hooke’s experiments covered springs, wires, and beams; the constant k is exactly the coefficient of that proportion.

Independence of the period from the amplitude, isochronism, had been studied earlier by Christiaan Huygens for the pendulum. In Horologium oscillatorium (1673) he derived T = 2π√(l/g) and showed that the isochronism of a pendulum is only an approximation for small angles, while motion along a cycloid is exactly isochronous. For a spring, the linearity of Hooke’s law gives strict isochronism over the whole range in which F = −k·x remains true. The equation m·x″ + k·x = 0 and the solution with angular frequency ω = √(k/m) entered analytical mechanics in the 18th century. They are in every textbook on oscillations today.

The name of the unit came later. Heinrich Hertz measured electromagnetic waves in the late 19th century, and the 11th General Conference on Weights and Measures in 1960 adopted the hertz as the SI unit of frequency in place of the “cycle per second.” Resolution 12 of that conference established the SI and listed the hertz among the derived units. The BIPM SI Brochure defines 1 Hz = 1 s⁻¹. The radian per second, in which ω is reported, has the same dimension, because the radian is dimensionless; for clarity one writes rad/s.

Derivation of f = (1/2π)√(k/m)

Step 1. Newton’s second law for a mass on a spring, with friction neglected, reads m·a = −k·x, or m·x″ + k·x = 0. This is the equation of the harmonic oscillator. Step 2. Look for a periodic solution x = A·cos(ωt + φ). Differentiating twice gives x″ = −ω²·x, so −m·ω² + k = 0 and ω² = k/m. The angular frequency is ω = √(k/m), and its unit is the radian per second. Step 3. One full cycle corresponds to a phase advance of 2π, so the period is T = 2π/ω = 2π√(m/k), and the frequency is f = ω/(2π) = (1/(2π))√(k/m).

Step 4. For k = 200 N/m and m = 0.5 kg: k/m = 400, ω = 20 rad/s, f = 3.183 Hz, T = 0.3142 s. Doubling the mass gives ω = √200 ≈ 14.14 rad/s and f = 2.251 Hz. The same result follows from energy. The sum ½mv² + ½kx² is constant: at the extreme displacement all the energy is elastic, and at equilibrium all of it is kinetic. The time between those states is a quarter of the period, and the calculation again gives T = 2π√(m/k).

Assumptions: an ideal spring, no damping, a point mass

The calculator assumes an ideal spring: massless, without preload, linear without limit, together with a point mass and no friction. There is no amplitude field, because in this model the amplitude does not enter the formula. A real spring has mass. For a uniform spring fixed at one end, one third of the spring’s mass is added to the oscillating mass, because the speed of the coils grows linearly from zero at the support to the speed of the free end. The calculator does not add that correction. If you know the spring’s mass, enter m increased by one third of that mass.

Damping, for example from a shock absorber, lowers the frequency of a decaying oscillation to √(ω² − (b/(2m))²), provided the expression under the square root is positive, and it makes the amplitude decrease with time. Outside the elastic range the force is no longer equal to k·x and the period begins to depend on the amplitude. Hanging the spring vertically does not change the frequency: the weight shifts the equilibrium by mg/k, and the oscillation is counted from that new zero with the same ω. The acceleration g does not enter the formula for f. A simple pendulum is different: there T = 2π√(l/g) and the bob’s mass does not appear.

Springs combined in parallel add their constants, and springs in series add the reciprocals: 1/k = 1/k₁ + 1/k₂. This calculator takes the already combined k. It does not build a spring network, and it does not distinguish extension from compression, because the frequency does not depend on the sign of the displacement.

Frequency, period, angular frequency, and energy

The panel reports f in hertz, the period T = 1/f, the angular frequency ω = 2πf, the number of cycles per minute 60·f, and the frequency at double the mass. For the reference values that is 3.183 Hz, 0.3142 s, 20 rad/s, 191 cycles per minute, and 2.251 Hz after the mass is doubled. Quadrupling the mass halves f, to 1.592 Hz, because the square root of four is two. The frequency at half the mass would be higher by the same factor √2.

The elastic energy ½kx² and the force k·x depend on the amplitude, which this calculator does not accept. The greatest speed of the mass is the product of the amplitude and ω, but the calculator does not compute that speed, because the formula has no amplitude. The force at a known x is covered by the Hooke’s law calculator, the energy by the elastic energy calculator, and the plain conversion of hertz into seconds by the period calculator T = 1/f. The angular velocity calculator also gives the angular frequency from the period alone, by ω = 2π/T. That is the same ω, with a different input.

Units, orders of magnitude, and oscillations around us

The hertz is the SI unit of frequency. Engineering also uses cycles per minute, that is 60·f; the panel reports both numbers. The constant k stays in N/m after the header is switched to US units. The mass then changes to pounds, and the panel shows it in pounds and in kilograms, and k in lbf/ft. A catalog value in N/mm is multiplied by 1,000 before it is entered as k. Grams convert to kilograms: 200 g is 0.2 kg.

The orders of magnitude below illustrate the formula for the stated k and m. They are not a measurement from a manufacturer’s catalog. A soft demonstration spring and a mass of a few hundred grams give about 1–3 Hz; the reference value on this page is 3.183 Hz. Car-body bounce is often designed near 1–1.5 Hz: a spring of 30,000 N/m and 400 kg on one wheel give 1.378 Hz in this model, and the shock absorber damps that motion. A trampoline mat with an effective stiffness of about 2,800 N/m under a 70 kg jumper gives 1.007 Hz, almost one second per cycle. A bungee cord is soft and works only in tension: 60 N/m and 80 kg give 0.1378 Hz, but a real jump is not the full symmetric period of this formula. A stiff spring of 1,000 N/m and a mass of 0.1 kg give 15.92 Hz.

The same ratio k/m gives the same frequency. A spring of 20 N/m with a mass of 0.2 kg and a combined stiffness of 8,000 N/m under a mass of 80 kg both have k/m = 100 and both give 1.592 Hz. That is why k and m enter the formula together, not each number on its own. A motorcycle fork of about k = 10,000 N/m and a sprung mass of about 80 kg gives 1.779 Hz in this model. A mattress zone with a combined stiffness of about 8,000 N/m under an 80 kg person falls at the same 1.592 Hz as a school spring of 20 N/m with a 200 g mass.

Sources

The definition, the derivation, and the historical note come from physics textbooks and historical sources: R. Hooke, Lectures de Potentia Restitutiva, or of Spring, London 1678; C. Huygens, Horologium oscillatorium, Paris 1673; D. Halliday, R. Resnick, J. Walker, Fundamentals of Physics, the chapter on oscillations (a mass on a spring); H. D. Young, R. A. Freedman, University Physics, the chapter “Periodic Motion,” the section on a mass on a spring; R. P. Feynman, R. B. Leighton, M. Sands, The Feynman Lectures on Physics, vol. I, chapter 21 (The Harmonic Oscillator); L. D. Landau, E. M. Lifshitz, Mechanics, the chapter on small oscillations; BIPM, SI Brochure, 9th edition (the hertz); Resolution 12 of the 11th CGPM, 1960, which established the SI and listed the hertz. This page is for educational purposes.