Spring Surge & Natural Frequency – Helical Spring Dynamics

A spring subjected to rapidly varying loading can experience dynamic oscillation in addition to its normal static deflection. Of particular importance in helical springs is spring surge, in which longitudinal disturbance waves travel through the spring.

If the excitation frequency approaches a natural frequency of the spring system or one of the spring's own vibration modes, resonance can produce large dynamic deflections and stresses. Natural-frequency and surge behaviour should therefore be considered in springs subjected to rapid cyclic operation.

Need help choosing or understanding a spring arrangement?
See the Spring Selection and Design Guide before using the detailed spring design pages.

Spring Rate and Deflection
Natural Frequency of a Spring-Mass System
Spring Surge
Spring Element Analysis
Spring Wave Equation
Boundary Conditions
Natural Surge Frequencies
Fundamental Surge Frequency
Fixed / Driven End Conditions
Practical Design Considerations

Nomenclature

Symbol	Description	Unit
A, B	Arbitrary constants in the vibration solution	—
c	Frequency parameter used in the spring vibration equation	Dimensionless
C	Spring index, D/d	Dimensionless
d	Spring wire diameter	m
D	Mean coil diameter	m
F	Axial spring force	N
dF_i	Inertia force on an elemental portion of the spring	N
dF_s	Change in spring force across an elemental portion of the spring	N
f_n	Natural frequency	Hz (cycles/s)
G	Shear modulus of spring material	Pa
k	Linear spring rate	N/m
M	Supported mass	kg
m_s	Mass of active part of spring	kg
dm	Mass of elemental portion of spring	kg
n	Coil coordinate measured along the active spring	Dimensionless
dn	Elemental number of coils	Dimensionless
n_a	Number of active coils	Dimensionless
u	Axial vibration displacement amplitude	m
δ	Axial spring deflection	m
ρ	Spring material density	kg/m³
ω	Circular frequency	rad/s
ω_n	Natural circular frequency	rad/s

Spring Rate and Deflection

For a linear spring, the deflection is proportional to the applied axial load over the working range. The spring rate is the force per unit elastic deflection:

k = \frac{F}{δ}

When force is expressed in newtons and deflection in metres, k is expressed in N/m. N/mm may also be used provided the units are kept consistent throughout the calculation.

Natural Frequency of a Spring-Mass System

Consider a mass M supported by an idealised weightless spring having spring rate k.

Mass supported by a spring for natural frequency analysis

For an undamped single-degree-of-freedom spring-mass system, the natural circular frequency is:

ω_{n} = \sqrt{\frac{k}{M}}

in radians per second.

The corresponding natural frequency in cycles per second (Hz) is:

f_{n} = \frac{1}{2 π} \sqrt{\frac{k}{M}}

This relationship describes the fundamental natural frequency of the idealised spring-supported mass. A real spring, however, has distributed mass and therefore possesses vibration characteristics and natural frequencies of its own.

Spring Surge

When a helical spring is subjected to rapidly varying axial motion, elastic disturbances can propagate along the active coils. Reflections of these disturbances can establish standing-wave patterns within the spring. This phenomenon is commonly known as spring surge.

If the operating excitation approaches one of these natural surge frequencies, the dynamic displacement and stress may become significantly greater than predicted by a static spring calculation.

The spring's own vibration frequencies can be investigated by considering a small elemental portion of the active spring in harmonic motion.

Spring Element Analysis

Consider a small element of a helical spring comprising dn coils. The length of wire in the element is approximately:

π D dn

The cross-sectional area of the spring wire is:

\frac{π d^{2}}{4}

For spring-material density ρ, the mass of the spring element is therefore:

dm = \frac{π^{2} d^{2} ρ D dn}{4}

For vibration displacement amplitude u at circular frequency ω, the inertia-force amplitude on the element is:

{dF}_{i} = dm ω^{2} u = \frac{π^{2} d^{2} ρ ω^{2} u D dn}{4}

Element of helical spring showing spring forces and inertia force

Helical Spring Rate

For a close-coiled helical compression spring, the spring-rate expression used in the original RoyMech derivation is:

k = \frac{F}{δ} = \frac{G d^{3}}{8 C^{3} n_{a}} (\frac{C^{2}}{C^{2} + 0.5})

where:

C = \frac{D}{d}

For normal spring indices, the factor C²/(C² + 0.5) is close to unity. Neglecting this small correction gives:

k \approx \frac{G d^{4}}{8 n_{a} D^{3}}

For the vibration analysis, the variation of spring force through an element is related to the axial displacement gradient. The original RoyMech derivation gives:

{dF}_{s} = \frac{G d^{4}}{8 n_{a} D^{3}} \cdot \frac{d^{2}}{d n^{2}} u dn

Spring Wave Equation

Equating the forces acting on the spring element gives the differential equation:

\frac{d^{2} u}{d n^{2}} + c^{2} u = 0

where:

c = \sqrt{\frac{2 π^{2} ρ D^{4} ω^{2}}{G d^{2}}}

The differential equation has the harmonic solution:

u = A \sin (c n) + B \cos (c n)

where A and B are arbitrary constants determined by the spring boundary conditions.

Boundary Conditions – One End Fixed, One End Free

For a spring fixed at one end and free at the other, take n = 0 at the fixed end and n = n_a at the free end.

At the fixed end:

u = 0 at n = 0

At the free end, the force condition is:

\frac{du}{dn} = 0 at n = n_{a}

The first condition gives B = 0. Applying the second condition gives:

\cos (c n_{a}) = 0

and therefore:

c n_{a} = \frac{a π}{2} where a = 1, 3, 5, \dots

Natural Surge Frequencies

Applying the boundary conditions to the spring wave equation gives the following series of natural circular frequencies:

ω_{n} = \frac{a d}{2 n_{a} D^{2}} \sqrt{\frac{G}{2 ρ}}

in radians per second, where:

a = 1, 3, 5, \dots

The corresponding frequencies in cycles per second are:

f_{n} = \frac{a d}{4 π n_{a} D^{2}} \sqrt{\frac{G}{2 ρ}}

Fundamental Surge Frequency

For the fundamental mode of the spring with one end fixed and the other free, a = 1. The frequency can also be expressed using the spring rate and the mass of the active portion of the spring:

f_{n} = \frac{1}{4} \sqrt{\frac{k}{m_{s}}}

where the mass of the active part of the spring is:

m_{s} = \frac{π^{2} d^{2} ρ D n_{a}}{4}

and, using the simplified helical spring-rate expression:

k = \frac{G d^{4}}{8 n_{a} D^{3}}

Fixed / Driven End Conditions

The preceding derivation applies to a spring with one end fixed and the other end free.

For a spring restrained between two plates, or for the related case where one end is fixed and the other is driven with sinusoidal axial motion, the boundary conditions produce the even sequence:

a = 2, 4, 6, \dots

The fundamental natural circular frequency for this condition is:

ω_{n} = \frac{d}{n_{a} D^{2}} \sqrt{\frac{G}{2 ρ}}

in radians per second.

The corresponding frequency in cycles per second is:

f_{n} = \frac{d}{2 π n_{a} D^{2}} \sqrt{\frac{G}{2 ρ}}

Boundary conditions matter: The natural surge frequencies depend on the way the ends of the spring are restrained or driven. The appropriate frequency relationship should therefore be selected for the actual installation rather than treating a single surge-frequency equation as universal.

Practical Spring Surge Considerations

The possibility of surge becomes increasingly important as the operating speed or cyclic loading frequency increases. Important considerations include:

Excitation frequency: avoid operation at or sufficiently near significant natural frequencies of the spring and spring-mass system.
Harmonic excitation: a periodic input can contain harmonics above its fundamental operating frequency, and these may also excite spring vibration modes.
Spring mass: the distributed mass of the spring is fundamental to its surge behaviour and cannot be ignored in high-speed applications.
Active coils: the number of active coils directly affects both spring rate and surge frequency.
Spring geometry: wire diameter and mean coil diameter have a strong influence on the natural surge frequencies.
Material: shear modulus and material density both influence the propagation of vibration through the spring.
End conditions: the natural modes depend on whether the spring ends are free, restrained or subjected to imposed motion.
Fatigue: resonant dynamic stresses may substantially increase fatigue damage even where the static spring stresses appear satisfactory.

Frequency separation: A spring intended for rapid cyclic operation should be designed with adequate separation between operating excitation frequencies and significant natural frequencies. The required separation should be established from the application, excitation spectrum, damping, permitted dynamic response and the applicable design requirements rather than relying on a universal frequency ratio.

Engineering reference: The equations on this page describe idealised longitudinal vibration of a helical spring. Actual dynamic behaviour may also be influenced by attached masses, damping, friction, coil contact, support flexibility and other features of the complete mechanism.

Spring Surge and Natural Frequency

Contents