Calculate the natural frequency of mechanical systems and structures
This calculator helps engineers determine the natural frequencies of various mechanical systems, which is crucial for avoiding resonance and designing vibration-resistant structures.
Natural frequency is the frequency at which a system tends to oscillate when disturbed and left to vibrate freely. Every mechanical system has one or more natural frequencies, and when a system is excited at one of its natural frequencies, it will vibrate with increased amplitude, a phenomenon known as resonance.
Understanding natural frequencies is crucial in engineering design to:
The simplest vibrating system consists of a mass attached to a spring. The natural frequency (in Hz) is given by:
f = (1/2π) × √(k/m)
Where:
For beams, the natural frequency depends on the support conditions, material properties, and geometry:
For a uniform cantilever beam, the natural frequency of the nth mode is:
fn = (λn²/2π) × √(EI/mL⁴)
Where:
For a uniform simply supported beam:
fn = (n²π²/2L²) × √(EI/m)
Where n is the mode number (1, 2, 3, ...)
For a uniform beam with both ends fixed:
fn = (λn²/2π) × √(EI/mL⁴)
Where:
For a simple pendulum, the natural frequency is:
f = (1/2π) × √(g/L)
Where:
For a torsional system (rotational mass-spring):
f = (1/2π) × √(kt/J)
Where:
In structural engineering, natural frequency analysis is used to:
In mechanical engineering, natural frequency analysis helps:
In aerospace engineering, natural frequency analysis is critical for:
In acoustics and music, natural frequency principles are used to:
Resonance occurs when a system is excited at or near its natural frequency, causing amplified vibrations that can lead to structural damage or failure. Famous examples include:
To avoid resonance problems:
This calculator provides estimates based on simplified models and does not account for all real-world factors. For critical applications: