Expressing a Periodic Excitation Force Acting on a System in Terms of the Fourier Series

May 15, 2017 Abdul Siraj, P.E.

In this post, I will cover show how to express a periodic excitation force acting on a system in terms of the fourier series.

F(t) represents the periodic excitation force acting on a system. In order for a function to be considered periodic, it must satisfy the following equality

 

 

In which To is the period or in other words, the minimum time required for the periodic excitation force to repeat itself. If a function is periodic, then it can be expressed by the Fourier series, which is a convergent infinite series of sine and cosine terms. The following expression is the fourier series for a periodic exciting force F(t) with a period of To:

 

 

Where omega is the frequency of the forcing function and is equal to two times pi divided by the period and “n” is the set of positive integers . The values of the coefficients ao, an, bn are found by computing the following integrals:

 

 

If the forcing function can be identified as being odd or even, the the computation of the Fourier coefficients can be greatly expedited. It may be a good idea to refresh the concepts of odd and even functions. An odd forcing function is antisymmetric with respect to the time origin, that is, F(t) = -F(-t). If the forcing function is odd, then the Fourier coefficients ao = an = 0. An even forcing function is symmetric with respect to the time origin, that is, F(t) = F(-t). If the forcing function is even, then the Fourier coefficient bn = 0. Therefore, an odd function can be expressed through the Fourier sine series while an even function can be represented by the Fourier cosine series. In the case where the forcing function is neither odd nor even, then the full Fourier series must be deployed.

 

From a theoretical point of view, the preceding equations suggest that the convergence of a Fourier series requires an infinite number of terms. Practically speaking, a relatively small number of terms will typically yield an accurate approximation of the periodic excitation force.

 

Let us now consider a viscously damped single degree of freedom system that is subjected to a periodic, non-harmonic excitation of period To. The differential equation of motion for the steady-state response is given by the following equation:

 

 

In which the full Fourier series expansion has been substituted for the periodic exciting forcing function. Since the transient response will decay with time, we will only focus on the particular solution for the steady-state response. The principle of superposition is valid in this instance because the differential equation is linear. Therefore, the steady-state displacement can simply be expressed as the summation of the individual particular solutions for all harmonic terms that represent the forcing function, F(t). The particular solution is shown below:

 

 

 

In which psi is the phase angle of the the steady-state response and “r” is equal to the excitation frequency divided by the natural frequency.