Forced Vibration of a Mass Spring System without Damping

Engineeering Projects

In the case forced vibration of a mass spring system without damping the mass spring system is acted by a periodic oscillation force and the damping is absent. Unbalanced rotating machinery such as compressor piston produces such type of motion. This is shown in fig.1 (a). For this case the equation of motion can be written as:

W-(W+kz)+Q_oSin\omega t = m\dfrac{d^2z}{dt^2}

m\dfrac{d^2z}{dt^2}+kz=Q_oSin\omega t ———-(1)

The solution of equation includes the solution for free vibration and solution satisfying the right hand side. Concept of rotating vector is used to solve the equation (1). s per the vector diagram (fig.1 c) we can write,

force vibration of mass and spring

kA-m\omega^2A-Q_0 = 0 ———-(2)

A(k-m\omega^2)=Q_0 A=\dfrac{Q_0}{k-m\omega^2} A=\dfrac{\dfrac{Q_0}{k}}{1-\dfrac{m\omega^2}{k}}

A=\dfrac{\dfrac{Q_0}{k}}{1-\left(\dfrac{\omega}{\omega_n}\right)^2} ———-(3)

Hence the complete solution is:

A=\dfrac{\dfrac{Q_0}{k}}{1-\left(\dfrac{\omega}{\omega_n}\right)^2}Sin\omega t+C_1Sin\omega_n t+C_2Cos\omega_n t ———-(4)

The last two terms will eventually vanish because of the damping in real system. Therefore, the steady solution is:

A=\dfrac{\dfrac{Q_0}{k}}{1-\left(\dfrac{\omega}{\omega_n}\right)^2}Sin\omega t ———-(5)

z_{max}= A=\dfrac{\dfrac{Q_0}{k}}{1-\left(\dfrac{\omega}{\omega_n}\right)^2}Sin\omega t ———-(6)

M=\dfrac{A}{\dfrac{Q_0}{k}}=\dfrac{1}{1-\left(\dfrac{\omega}{\omega_n}\right)^2} ———-(7)

Where, M = Magnification Factor and \dfrac{Q_0}{k} = Static deflection under Q0. A graph of the above equation is shown in fig. 1(d).

Leave a Comment