Effective mass (spring-mass system)

In a spring-mass system the mass of the spring m, as well as the suspended mass M, have an influence on the motion. However, since not all of the spring moves at the same velocity as the suspended mass, the mass of the spring cannot be simply added to the suspended mass. The effective mass of the spring is the mass which must be added to the suspended mass to correctly predict the behavior of the system.

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Ideal spring

The effective mass of the spring in a spring-mass system when using an ideal spring is independent whether the direction of the spring-mass system is horizontal, vertical or oblique, is also 1/3 of the mass of the spring, i.e., m/3. This can be shown by integration:

Its length is dy, mass is dm, velocity is u and so

dm=\left(\frac{dy}{L}\right)m

where L is the length of the spring.

Hence the total kinetic energy of the spring and mass is

E=\int_m\frac{1}{2}u^2\,dm
=\int_0^L\frac{1}{2}u^2\left(\frac{dy}{L}\right)m\!
=\frac{1}{2}\frac{m}{L}\int_0^L u^2\,dy

But the velocity of each position of the spring is directly proportional to its length and so

u:v=y:L
u=\frac{vy}{L}

Hence

\frac{1}{2}\frac{m}{L}\int_0^L u^2\,dy
=\frac{1}{2}\frac{m}{L}\int_0^L\left(\frac{vy}{L}\right)^2\,dy
=\frac{1}{2}\frac{m}{L^3}v^2\int_0^L y^2\,dy
=\frac{1}{2}\frac{m}{L^3}v^2\left[\frac{y^3}{3}\right]_0^L
=\frac{1}{2}\frac{m}{3}v^2

With comparison of the original kinetic energy formula \frac{1}{2}mv^2, we can conclude that effective mass of spring in this case is m/3, i.e., 1/3 of the mass of the spring.

Real spring

However, the above calculations are only suitable for small values of \frac{M}{m}. Since Jun-ichi Ueda and Yoshiro Sadamoto have done the experiment. It is found that, as the ratio \frac{M}{m} increases beyond 7, the effective mass of a spring in a vertical spring-mass system becomes smaller than Rayleigh's value \frac{m}{3} and eventually reaches negative values. This unexpected behavior of the effective mass can be explainable in terms of the elastic after-effect.

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