Formula & Calculator
Natural Frequency of a Spring-Mass System
Calculates the natural (undamped) angular frequency at which a spring-mass system oscillates when disturbed from equilibrium.
Interpretation
The natural frequency of a spring‑mass system is the frequency at which it oscillates freely without external forces. It is ω_n = √(k/m). This is a critical parameter in vibration analysis and resonance avoidance.
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| omega_n | Natural angular frequency | rad/s |
| k | Spring stiffness | N/m |
| m | Mass attached to the spring | kg |
What it means
The natural frequency is the fundamental frequency at which a system tends to oscillate in the absence of any driving or damping forces. For a simple spring‑mass system with stiffness k and mass m, the undamped natural angular frequency is ω_n = √(k/m) (rad/s). The cyclic natural frequency is f_n = (1/2π)√(k/m) (Hz). This frequency is determined solely by the system's physical properties. If an external excitation matches this frequency, resonance occurs, leading to large amplitude oscillations that can cause failure. Therefore, engineers must design systems to avoid resonance. The natural frequency concept extends to multi‑degree‑of‑freedom systems, where there are multiple modes. It is used in designing machine foundations, suspension systems, and vibration isolators. Damping reduces the response at resonance. Understanding natural frequencies is also crucial in acoustics, aerospace, and civil engineering for ensuring structural integrity and comfort.
Worked example
Natural Frequency – Two Examples
Real‑World| Parameter | Value |
|---|---|
| k | 1000 N/m |
| m | 10 kg |
| Parameter | Value |
|---|---|
| k | 5000 N/m |
| m | 20 kg |
Common mistakes
- Mass m: Use mass in kg, not weight.
- Stiffness k: Spring constant in N/m.
- Units: k in N/m, m in kg → ω in rad/s.
- Natural frequency vs. damped frequency: This is the undamped natural frequency; damping reduces it.
- Linear assumption: Assumes spring is linear (Hooke’s law) and mass is concentrated.
Applications
The natural frequency of a spring‑mass system is the frequency at which it oscillates freely, determined by the square root of stiffness divided by mass. This parameter is critical in vibration analysis and control, as operating near natural frequencies can lead to resonance, causing excessive displacements and potential failure. Engineers use it to design vibration isolators, tuned mass dampers, and suspension systems. In automotive engineering, it helps set suspension stiffness to improve ride comfort. In aerospace, it is essential for avoiding flutter in wings and control surfaces. The concept also appears in civil engineering for earthquake‑resistant design, where building frequencies are tuned away from seismic frequencies. Understanding natural frequency enables safer and more comfortable designs.
- Vibration isolation and damping system design
- Automotive suspension tuning
- Aircraft flutter avoidance and control
- Earthquake‑resistant building design
- Precision instrument and MEMS design
Frequently Asked Questions
The natural frequency is the frequency at which a system oscillates when disturbed and then allowed to move freely (without external forcing). For a simple mass‑spring system, the angular natural frequency is ω_n = √(k/m), where k is the spring stiffness and m is the mass. The cyclic frequency in Hz is f_n = ω_n/(2π).
ω is in radians per second and is the rate of change of the phase angle. f is in Hertz (cycles per second). They are related by ω = 2π·f. The formula ω_n = √(k/m) gives the angular natural frequency. To get the frequency in Hz, divide by 2π: f_n = (1/(2π))·√(k/m).
- Using f instead of ω without dividing by 2π – a very common error, leading to a value that is 6.28 times too large.
- Using the wrong units for k and m – ensure k is in N/m and m in kg to get ω in rad/s.
- Ignoring the mass of the spring – for a heavy spring, the effective mass is m + m_spring/3.
- Applying it to non‑linear systems – for large deflections, the spring may not be linear, and the frequency depends on amplitude.
The natural frequency increases with √k and decreases with √m. To increase the natural frequency, you can make the spring stiffer or reduce the mass. This is used in design to avoid resonance: you either increase f_n (stiffen the system) or decrease it (add mass) to move it away from the forcing frequency.
Resonance occurs when the frequency of an external forcing matches the system's natural frequency. At resonance, the amplitude of vibration grows very large (theoretically infinite in an undamped system). In practice, damping limits the response, but large vibrations can cause fatigue failure. Engineers design systems so that the natural frequency is at least 20‑30% away from expected forcing frequencies.
Damping reduces the natural frequency slightly. For a damped system, the damped natural frequency is ω_d = ω_n · √(1 − ζ²), where ζ is the damping ratio. For light damping (ζ < 0.2), the difference is negligible. For heavy damping (ζ > 1), the system is overdamped and no longer oscillates.
Gravity shifts the equilibrium position but does not affect the natural frequency. The equation of motion for a vertical spring‑mass is m·x¨ + k·x = 0 (where x is measured from the static equilibrium). Therefore, the natural frequency is still √(k/m).
For springs in series, the equivalent stiffness is 1/k_eq = 1/k₁ + 1/k₂. For springs in parallel, k_eq = k₁ + k₂. For multiple masses, you need to derive the equations of motion (using Lagrange's equations or Newton's laws) and find the eigenvalues of the system matrix. For a two‑mass system, there are two natural frequencies.
For a disk (moment of inertia J) attached to a torsional spring (stiffness k_t), the angular natural frequency is ω_n = √(k_t / J). This applies to shafts in torsion and rotating machinery.
Vibration isolators (e.g., rubber mounts) are designed to have a natural frequency much lower than the forcing frequency. The transmissibility (ratio of transmitted force to input force) decreases when the forcing frequency is > √2 times the natural frequency. So, by choosing a soft spring (low k) and/or a large mass, you lower the natural frequency and achieve good isolation.