91Ó°ÊÓ

In mechanical systems, the natural frequency refers to the frequency at which a system tends to oscillate when disturbed and then allowed to move without any further external forces. For the setup described in the problem, the natural frequency \( \omega_n \) is a key parameter. It is determined by the properties of the motor and the supporting springs. Mathematically, the natural frequency is given by the formula: \[ \omega_n = \sqrt{\frac{k}{M}} \] Here, \(k\) is the spring constant, which measures the stiffness of the springs, and \(M\) is the mass of the motor. This expression tells us that the natural frequency depends on how easily the system springs back to equilibrium when displaced. When oscillating at this frequency, a system experiences its maximum amplitude of vibration if not damped. Understanding the natural frequency is crucial for evaluating how the system will respond to external forces such as those induced by an unbalanced motor.

Centrifugal force comes into play in systems involving rotational motion. In the scenario given, the motor's rotor is slightly unbalanced, meaning there is a mass \(m\) at a perpendicular distance \(r\) from the axis of rotation. This unbalance causes what is known as centrifugal force, calculated with the formula: \[ F = m r \omega_{f}^{2} \] Where \( \omega_f \) is the angular velocity of the motor. This force acts outward from the axis of rotation, as if trying to "flee" from the center. It can cause parts of the machine, like the motor housing, to vibrate unnaturally. Understanding centrifugal force is pivotal in predicting these kinds of forced vibrations and determining their effects on system motion.

Steady-state vibrations occur when a system oscillates at a constant amplitude over time, under continued external excitation. In the context of the motor problem, after the initial transient responses have died out, the system reaches a steady state, dictated by the balance of centrifugal force and spring restoring forces. The amplitude \( x_m \) then becomes a function of both the exciting frequency \( \omega_f \) and the natural frequency \( \omega_n \): \[ x_{m} = \frac{r (m / M) (\omega_{f} / \omega_{n})^{2}}{1 - (\omega_{f} / \omega_{n})^{2}} \] This equation demonstrates how the amplitude of vibrations is influenced by the relationship between the forcing frequency and the natural frequency. If \( \omega_f \) is close to \( \omega_n \), the system can reach very large amplitudes. This scenario is referred to as resonance and can potentially lead to mechanical failure if not controlled or damped. Thus, understanding steady-state vibrations helps in designing systems to avoid detrimental resonance effects.