Understanding the Frequency of Damped Free Vibration in Vibrating Systems

The concept of damped free vibration is fundamental in the study of mechanical and structural engineering. In this type of vibration, a system oscillates with energy loss over time due to damping forces such as friction and air resistance. This article will explore the relationship between the damped natural frequency and the natural frequency of a system, and how damping affects the frequency of vibration.

Defining Damped Free Vibration

Damped free vibration refers to the oscillation of a system in the absence of external forces, but with energy dissipation due to damping. Unlike undamped free vibration, which maintains a constant amplitude, damped free vibration experiences a gradual decrease in amplitude over time. This phenomenon is widely observed in mechanical systems and is crucial for understanding the dynamics of machinery and structures.

The Frequency of Damped Free Vibration

The damped natural frequency, denoted as (f_d), is the frequency at which the system vibrates when damping is present. This frequency is always less than the natural frequency, (f_n), of the system when it is undamped. This relationship can be mathematically expressed as:

where (zeta) is the damping ratio. The damping ratio (zeta) is defined as the ratio of the actual damping to the critical damping. When (zeta > 0), the frequency of damped free vibration is always less than the natural frequency of the system, indicating that damping reduces the oscillation frequency.

The Equation of Motion and the Impact of Damping

The equation of motion for a damped harmonic oscillator is given by:

[mfrac{d^2x}{dt^2} bfrac{dx}{dt} kx 0]

where (m) is the mass, (b) is the damping coefficient, and (k) is the spring constant. In the absence of damping, the natural frequency (w) is given by:

[w sqrt{frac{k}{m}}]

When damping is present, the equation can be rewritten as:

[mfrac{d^2x}{dt^2} bfrac{dx}{dt} kx 0]

Substituting (omega frac{b}{2m}) and (omega_n sqrt{frac{k}{m}}), the equation becomes:

[mfrac{d^2x}{dt^2} 2mfrac{b}{2m}frac{dx}{dt} kx 0]

[omega'^2x 2zetaomega_nomega'x omega_n^2x 0]

This simplifies to:

[x'' 2zetaomega_nx' omega_n^2x 0]

The solution to this equation is:

[x(t) e^{-zetaomega_nt}(Acos(omega_dt) Bsin(omega_dt))]

where:

[omega_d omega_nsqrt{1 - zeta^2}]

Here, (omega_d) is the damped natural frequency, and it is always less than (omega_n) when (zeta > 0).

Conclusion

In conclusion, the frequency of damped free vibration is always smaller than the natural frequency of the system due to the effects of damping, which reduces the oscillation frequency. This phenomenon is vital in the design and analysis of mechanical systems, as it affects the stability and performance of the system over time.