Response of a Damped System Under Rotating Unbalance

Apuntes

# Response of a Damped System Under Rotating Unbalance  (a)  (b) Unbalance in rotating machinery is one of the main causes of vibration. A simplified model of such a machine is shown in Fig. 3.19. The total mass of the machine is $M$, and there are two eccentric masses $m/2$ rotating in opposite directions with a constant angular velocity $\omega$. The centrifugal force $(me\omega^2)/2$ due to each mass will cause excitation of the mass $M$. We consider two equal masses $m/2$ rotating in opposite directions in order to have the horizontal components of excitation of the two masses cancel each other. However, the vertical components of excitation add together and act along the axis of symmetry $A - A$ in $$ F_0 \sin \omega t $$ Fig. 3.19. If the angular position of the masses is measured from a horizontal position, the total vertical component of the excitation is always given by $F(t) = me\omega^2 \sin \omega t$. The equation of motion can be derived by the usual procedure: $$ M \ddot{x} + c \dot{x} + k x = \underbrace{me\omega^2}_{(me\omega^2)} \sin \omega t \tag{3.78} $$ The solution of this equation will be identical to Eq. (3.60...