Spring basic equations

General spring data	Spring free motion	Spring damped free motion
Forced vibration no damping	Damped forced vibration	Helical compression spring design

Spring Equations Summary

Work done on an elastic spring during compression or extension:
Work done on an elastic spring during compression or extension from rest, is known as the elastic potential energy.
The force exerted by the spring on the body which deforms it:
The equivalent spring constant K of n springs connected in series.
The equivalent spring constant K of n springs connected in parallel.

Spring free motion equations

when there is no external force and no damping (Second order homogeneous system).

The motion of a spring with initial conditions x_(t=0) = 2 and x'_(t=0) = 0

From Newton second law:

The solution of this equation is an oscillatory motion of the form

The solution can be expressed in more general form as:

Where A and Φ are constants that depends on the initial displacement and the initial velocity at t = 0.

The oscillation frequency is:
The time of one cycle is:
Angular velocity:

Spring damped free motion equations

Motion equation of damped free motion spring is:

This is a second order homogenous differential equation with constant coefficients, we assume an exponential solution of the form x(t) = A est (all values of m, c and k are > 0). After substituting this solution to the motion equation we have:

This is the characteristic equation whose solutions are:

Solutions of the characteristic equation

Because the values of m, c and k are all positive the value of the square root is less then c/2m and therefore the values of s1 and s2 are always negative.

We have to distinguish between three cases which depends on the value under the root:

Overdamped motion	r1 and r2 < 0
critically damped motion	The critical damping is: c_c is the boundary of changing the motion from vibratory mode to non vibratory motion.
underdamped motion	Where If we replace A and B by the values: We obtain:

Forced vibration with no damping

The motion equation of forced spring with periodic external force is:

The general solution to this differential equation is:

A and B are constants to be determined by the initial conditions.

Where ω₀ is the natural period:

This motion is the sum of two periodic functions of different periods and the same amplitude (variation of the amplitude with time is called amplitude modulation in electronics).

Assume a case when the initial conditions are:

The solution after evaluating constants A and B is:

After some geometric substitutions we get another form of the solution:

Sketched at right
the values:

ω = 5 and ω₀ = 6

It is interesting to analyse the case when the forcing period ω is very close or equal to the natural period ω₀ in this case the solution of the motion equation turns to be:

Because of the term t sin (ω₀t) this answer is diverging when t is becoming very large values no matter of the values of A and B, this phenomenon is known as resonance. and is the reason why in real mechanical systems

the excitation frequency should be different as much as possible from the natural frequency.
A real spring can not elongate more then its designed length otherwise it will break, in this range the spring constant k is assumed to be linear.

If the spring data are F₀ = 36 N,
m = 2 kgm, ω₀ = ω = 3
and the initial conditions are
x(t=0) = 0 and dx/dt(t=0) = 0.

A = 0, B = 0 the motion x(t) will be:

x(t) = 5t Sin(3t)

Forced and damped vibration

The more accurate motion equation of spring with damping and external periodic force is:

Spring with damping and periodic external force equation

The general solution of this differential equation of motion consists of two parts, the first is the solution of the

homogeneous equation which is the damped free vibration and is equal to:

And the steady state solution
X is the amplitude and is given by:
And phase angle δ is given by:

s₁ and s₂ are the solutions of the characteristic equation (see spring free motion solution) hence the term

approach zero when t gets large values, that is because s₁ and s₂ are both negative numbers,
this term has impact only on the beginning of the motion and is known as the transient state, as t increases the

solution of the equation becomes the steady state form.

Because the value under the root of the amplitude is never zero even when ω = ω₀ , it is obvious that the motion is a constant amplitude sinusoid motion and the amplitude is depending only on the values of m, c, k and the

frequencies ω and ω₀ .

The values of X and δ can be expressed as nondimensional values by dividing the numerator and denominator

by k to obtain:

and

We can also define the following relationship

natural frequency
critical damping
damping factor
and

then the nondimensional expressions for the amplitude X and phase δ becomes:

and

The equations reveal that the amplitude and the phase are a function of the frequency ratio ω/ω₀ and the damping factor ζ. These equations can be plotted as follows.