Forced Vibrations Phase Lag

When a mechanical system is subjected to a continuous external periodic force, it undergoes what is known as forced vibrations. If these vibrations continue for a long enough time such that any transient effects die out, the system reaches a stable condition called steady-state forced vibrations. In this state, the system vibrates at the frequency of the applied external force.

Understanding Phase Lag

In forced vibrations, there's often a time delay between the application of the exciting force and the resulting displacement or response of the system. This delay is represented by a phase lag, which is an angular difference between the force and the displacement. It indicates how much the response lags behind the excitation.

Resonance in Vibrations

Resonance is a critical phenomenon in forced vibrations. It occurs when the frequency of the exciting force ($\omega$) becomes equal to or very close to the natural frequency ($\omega_n$) of the system. At resonance, if damping is very small, the amplitude of vibration can theoretically become very large, leading to potential system failure.

Phase Lag at Resonance

For a single-degree-of-freedom system with viscous damping, the phase angle ($\phi$) between the exciting force and the resulting displacement is given by the formula:

$$\tan \phi = \frac{2 \zeta (\omega/\omega_n)}{1 - (\omega/\omega_n)^2}$$

where:

• $\phi$ is the phase lag
• $\zeta$ (zeta) is the damping ratio
• $\omega$ is the frequency of the exciting force
• $\omega_n$ is the natural frequency of the system

At resonance, the exciting frequency equals the natural frequency, meaning $\omega = \omega_n$. Therefore, the frequency ratio $(\omega/\omega_n)$ becomes 1.

Let's substitute $\omega/\omega_n = 1$ into the phase angle formula:

$$\tan \phi = \frac{2 \zeta (1)}{1 - (1)^2}$$

$$\tan \phi = \frac{2 \zeta}{1 - 1}$$

$$\tan \phi = \frac{2 \zeta}{0}$$

As the denominator approaches zero, $\tan \phi$ approaches infinity. For $\tan \phi$ to be infinite, the angle $\phi$ must be $90^\circ$ (or $\pi/2$ radians).

Therefore, for steady-state forced vibrations, the phase lag at resonance is $90^\circ$. This means that at resonance, the displacement of the system lags the exciting force by exactly a quarter of a cycle. This $90^\circ$ phase shift is a characteristic feature of resonant behavior in damped systems, regardless of the amount of damping (as long as damping is present).