Spring Mass Frequency Calculator

Calculator

Calculation mode Find frequency from k and m Find stiffness from target frequency and m Find mass from target frequency and k

Use SI or imperial units, mixed safely.

Spring stiffness (k)

N/mN/mmkN/mlbf/inlbf/ft

Higher stiffness increases frequency.

Mass (m)

kgglbmslug

Larger mass lowers frequency.

Target frequency

Hz

Used to design k or m for a target.

Damping coefficient (optional)

N·s/mN·s/mmlbf·s/inlbf·s/ft

Adds ζ, Q, and damped frequency.

Calculate Reset

Formula used

How to use this calculator

1. Select a mode: compute frequency, required stiffness, or required mass.
2. Enter known values and choose units for each input.
3. Optionally enter damping to estimate ζ, Q, and f_d.
4. Press Calculate to view results above the form.
5. Use Download CSV or Download PDF to save results.

Spring–mass frequency notes

1) What the natural frequency means

Natural frequency is the rate a mass oscillates on a linear spring after a small disturbance. It is the baseline for vibration design, isolation, and resonance checks. When external forcing approaches this value, response can grow rapidly unless damping or control limits the motion.

2) Typical input ranges in practice

For lab rigs and product mechanisms, stiffness often falls between 100 and 10,000 N/m, while effective mass may range from 0.1 to 50 kg. These ranges commonly produce frequencies from below 1 Hz (slow sway) up to tens of hertz (tight mechanisms).

3) Units and safe conversions

This tool converts stiffness and mass to consistent base units before calculating. For example, 1 N/mm equals 1000 N/m, and 1 lbm equals 0.453592 kg. Converting internally avoids hidden errors when comparing designs that mix metric and imperial specifications.

4) Period and angular speed for timing

Two outputs help timing and control: the angular natural frequency ω_n (rad/s) and the period T (s). As a reference, k = 1200 N/m with m = 3.5 kg gives ω_n ≈ 18.52 rad/s, f_n ≈ 2.95 Hz, and T ≈ 0.339 s.

5) Damping ratio, bandwidth, and Q

If a viscous damping coefficient is provided, the calculator estimates the damping ratio ζ and the quality factor Q = 1/(2ζ). Light damping (ζ ≈ 0.01 to 0.05) yields sharp resonance and higher Q, while moderate damping (ζ ≈ 0.1 to 0.3) broadens response and reduces peak vibration.

6) Designing for a target frequency

Use the “required stiffness” or “required mass” mode when you have a frequency goal. The ideal relationship is k = (2πf)² m and m = k/(2πf)². This is useful for tuning mounts, selecting springs, or estimating added mass needed to shift resonance away from a forcing source.

7) Sensitivity and tolerance awareness

Frequency scales with the square root of stiffness and inverse square root of mass. That means a 10% stiffness increase raises f by about 4.9%, while a 10% mass increase lowers f by about 4.9%. This insight helps prioritize manufacturing tolerances and measurement accuracy.

8) Practical modeling tips

Use the effective moving mass, not the total assembly mass, and include any attached hardware. Springs in parallel add stiffness, while series springs reduce it. Keep deflections within the linear spring region, and treat large-motion systems separately. For safety, avoid operating near resonance without sufficient damping margin.

FAQs

1) Is natural frequency the same as resonance frequency?

They are effectively the same for a lightly damped, linear system. With damping, the peak response occurs slightly below the undamped natural frequency, and the shift grows as damping increases.

2) What mass should I enter if parts rotate or flex?

Enter the effective mass at the spring location. Convert rotational inertia using an equivalent mass model, or estimate using energy methods. If unsure, test the assembled system and back-calculate from the measured frequency.

3) How do I combine multiple springs?

Springs in parallel add: k_eq = k₁ + k₂ + …. Springs in series reduce: 1/k_eq = 1/k₁ + 1/k₂ + ….

4) Why does added mass lower the frequency?

Frequency follows f ∝ 1/√m. Increasing mass makes the system harder to accelerate, so it oscillates more slowly for the same restoring force.

5) When should I enter damping?

Use damping when you want a more realistic response estimate, such as damped frequency, Q, or decay behavior. If damping is unknown, leave it blank to focus on the ideal natural frequency and period.

6) What does “overdamped” mean in the results?

Overdamped means ζ > 1, so the system returns to equilibrium without oscillating. It settles smoothly but more slowly than the critically damped case, depending on the damping level.

7) Can this calculator handle large deflections or nonlinear springs?

It assumes a linear spring and small oscillations. For nonlinear stiffness, large travel, or contact effects, the effective frequency changes with amplitude, so a nonlinear analysis or measured data is recommended.