A mechanical structure such as a beam attached at one end will oscillate at a particular frequency if it is knocked or set into motion; this is known as its natural frequency. Systems can have single or multiple degrees of freedom, depending on the number of coordinates required to describe the oscillation.
The natural frequency of a system can be determined using the mass of the system, m, and its stiffness, k. A system with a single degree of freedom requires only a single coordinate to describe its motion and/or oscillations; this is the simplest type of system and its natural frequency can be derived using this equation:wn2 = k / m
The terms used in the equation are described as follows:
wn2 = natural frequency, with units of radians per second (rad/s)
k = stiffness of the system, with units of newtons per meter (N/m)
m = mass of the system, with units of kilograms (kg)
The natural frequency can be converted into units of hertz (Hz) using the following equation:
fn = wn / (2π)
Where fn is in units of hertz.
Note that systems with more than a single degree of freedom require several equations to determine the system’s natural frequency. The natural frequency is determined using empirical methods in many applications due to the complexity of deriving the value using numerical and analytical techniques.