Determine the effect of the mass of the spring on the natural frequency of the system shown in figure

 

Solution:

Let x be the displacement of mass m and so the velocity will be 𝑥̇. The velocity of spring element at a

distance y from the fixed end maybe written as,

where l is the total length of the spring.

The kinetic energy of spring element dy per unit area is written as,

And, the kinetic energy of the mass m,

Hence, for the interval y = 0 to y = 𝑙, Total kinetic energy of the spring mass system,

The potential energy of spring element dy is written as,

The potential energy of the mass,

(𝑃. 𝐸)𝑚 = 𝑚𝑔 × 0 = 0

Therefore, Total Potential Energy of the system written as,

Hence, Total Energy = Total Potential Energy + Total Kinetic Energy

The total energy of the system is constant. Therefore,

This is the differential equation of motion of the spring mass system.

If the motion is simple harmonic, then let, the solution of this second order differential equation is,

𝑥 = 𝐴 sin 𝜔𝑛𝑡

Where, 𝜔𝑛 = natural frequency

A = Maximum displacement of the mass

Therefore,

Hence, we can conclude if the mass m is increased the natural frequency can be decreased.

Tags: determine the effect of the mass of the spring on the natural frequency of the system,
The Effect of Mass on Frequency, Investigate the effect of varying the mass of an object on the acceleration, The effect of the mass of a spring on the natural frequency of harmonic oscillation