When a system is subjected to forced vibrations, then under steady-state conditions
it vibrates at the imposed frequency
Understanding Steady-State Forced Vibrations
Forced vibration occurs when a system is acted upon by an external periodic force. This force drives the system to oscillate.
Every system has a natural frequency (${\omega_n}$), which is the frequency at which it would vibrate if disturbed and then left alone, assuming no damping.
In forced vibrations, an external force applies a specific imposed frequency (${\omega}$). Initially, the system's response includes components related to its natural frequency (transient response).
Steady-State Response in Forced Vibrations
However, as time progresses, the effect of damping causes the transient vibrations to decay. Under steady-state conditions, the only vibrations that remain are those directly driven by the external force.
Therefore, when a system is subjected to forced vibrations and reaches a steady-state condition, it vibrates at the frequency of the external applied force, which is the imposed frequency (${\omega}$). The system's response is dictated by the driving frequency, not its own natural frequency (${\omega_n}$), after the initial transient effects have disappeared.
Important Questions from Forced Vibration
Which method is used to find the natural frequencies of a system with multi degrees of freedom?
The ratio of the amplitude of the steady-state response of forced vibrations to the static deflection under the action of a static force is known as
When the frequency of the forced vibration is equal to the frequency of the free vibration, this condition is called ______.
The amplitude of the steady-state reaction divided by the static deflection under force application is known as _______.
Consider a forced single degree-of-freedom system governed by \(\rm \ddot x(t) + 2 ζ ω_n \dot x (t) + ω_n^2 x(t) = ω_n^2 \cos (ω t)\), where ζ and ωn are the damping ratio and undamped natural frequency of the system, respectively, while ω is the forcing frequency. The amplitude of the forced steady state response of this system is given by [(1 − r2)2 + (2ζr)2]-1/2, where 𝑟 = ω/ω n. The peak amplitude of this response occurs at a frequency ω = ωp. If ωd denotes the damped natural frequency of this system, which one of the following options is true?
