In a real spring–mass system, the spring has a non-negligible mass
Contents
Ideal uniform spring
The effective mass of the spring in a spring-mass system when using an ideal spring of uniform linear density is 1/3 of the mass of the spring and is independent of the direction of the spring-mass system (i.e., horizontal, vertical, and oblique systems all have the same effective mass). This is because external acceleration does not affect the period of motion around the equilibrium point.
We can find the effective mass of the spring by finding its kinetic energy. This requires adding all the mass elements' kinetic energy, and requires the following integral, where
Since the spring is uniform,
The velocity of each mass element of the spring is directly proportional to its length, i.e.
Comparing to the expected original kinetic energy formula
Note that
We can find the equilibrium point
Defining
This is the equation for a simple harmonic oscillator with period:
So we can see that the effective mass of the spring added to the mass of the load gives us the "effective total mass" of the system that must be used in the standard formula
General case
As seen above, the effective mass of a spring does not depend upon "external" factors such as the acceleration of gravity along it. In fact, for a non-uniform spring, the effective mass solely depends on its linear density
So the effective mass of a spring is:
This result also shows that
Real spring
The above calculations assume that the stiffness coefficient of the spring does not depend on its length. However, this is not the case for real springs. For small values of