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Fourier Series Representation Of An Arbitrary Periodic Function (Part - 1) - Mechanical Engineering PDF Download

Fourier Series Representation of an Arbitrary Periodic Function Consider now an arbitrary periodic function f (t) with period T, i.e., f (t) satisfies the property

f (t + nT ) = f (t), ∀n=I                                                                                   (2–352)

where I is the set of integers. Examples of arbitrary periodic functions include a square-wave (see Fig. 2–16) and a sawtooth (see Fig. 2–17). It is known that any arbitrary periodic function

Fourier Series Representation Of An Arbitrary Periodic Function (Part - 1) - Mechanical Engineering
Figure 2–16 Square-Wave Function

can be expressed as an infinite series of sines and cosines. This infinite series is called a Fourier series. Suppose now that we consider a function f (t) that is periodic with period T on the interval from zero to T. Then, in terms of a Fourier series expansion, the periodic function f (t) can be written as

Fourier Series Representation Of An Arbitrary Periodic Function (Part - 1) - Mechanical Engineering

Fourier Series Representation Of An Arbitrary Periodic Function (Part - 1) - Mechanical Engineering

where Ω = 2π/T is the fundamental frequency. It is known that the functions e ikΩt , (k = 0, ±1, ±2, . . .) are orthogonal over the time interval t ∈ [0, T ], i.e.,

 

Fourier Series Representation Of An Arbitrary Periodic Function (Part - 1) - Mechanical Engineering

The coefficients ck, (k = 0, ±1, ±2, . . .) are obtained as follows. Suppose we multiply both sides of Eq. (2–353) by e −ilΩt (where l ∈ I) and integrate over the period of the function (i.e., from zero to T). We then obtain

 

Fourier Series Representation Of An Arbitrary Periodic Function (Part - 1) - Mechanical Engineering

 

where  Fourier Series Representation Of An Arbitrary Periodic Function (Part - 1) - Mechanical Engineering

Noting that Ω = 2π/T, we have

Fourier Series Representation Of An Arbitrary Periodic Function (Part - 1) - Mechanical Engineering

Suppose now that we let m = k − l (we note that, because k and l are integers, m is also an integer). Then when m ≠ 0 we have

Fourier Series Representation Of An Arbitrary Periodic Function (Part - 1) - Mechanical Engineering

Furthermore, for the case that m = 0, we need to take the limit as m → 0 as

Fourier Series Representation Of An Arbitrary Periodic Function (Part - 1) - Mechanical Engineering

Because both the numerator and denominator approach zero as m → 0, we can use L’Hopital’s rule to obtain

Fourier Series Representation Of An Arbitrary Periodic Function (Part - 1) - Mechanical Engineering

Noting that the condition m = 0 is equivalent to the condition that k = l, we have

Fourier Series Representation Of An Arbitrary Periodic Function (Part - 1) - Mechanical Engineering

which implies

Fourier Series Representation Of An Arbitrary Periodic Function (Part - 1) - Mechanical Engineering

The expression for ck from Eq. (2–362) can then be used in Eq. (2–353) to obtain the Fourier series expansion of the periodic function f (t). It is noted that a Fourier series can be written in real form as follows. First, we note that

Fourier Series Representation Of An Arbitrary Periodic Function (Part - 1) - Mechanical Engineering

Then, we have

Fourier Series Representation Of An Arbitrary Periodic Function (Part - 1) - Mechanical Engineering

Now we have

Fourier Series Representation Of An Arbitrary Periodic Function (Part - 1) - Mechanical Engineering

from which we obtain

Fourier Series Representation Of An Arbitrary Periodic Function (Part - 1) - Mechanical Engineering

Similarly,

Fourier Series Representation Of An Arbitrary Periodic Function (Part - 1) - Mechanical Engineering

from which we obtain

Fourier Series Representation Of An Arbitrary Periodic Function (Part - 1) - Mechanical Engineering

Then we can define

Fourier Series Representation Of An Arbitrary Periodic Function (Part - 1) - Mechanical Engineering

Solving for ck and c−k in terms of ak and bk, we obtain

Fourier Series Representation Of An Arbitrary Periodic Function (Part - 1) - Mechanical Engineering

We can then write

 Fourier Series Representation Of An Arbitrary Periodic Function (Part - 1) - Mechanical Engineering

 

Substituting the expressions for cand c−k into this last equation, we obtain

Fourier Series Representation Of An Arbitrary Periodic Function (Part - 1) - Mechanical Engineering

Rearranging, we have

Fourier Series Representation Of An Arbitrary Periodic Function (Part - 1) - Mechanical Engineering

Then, using Eqs. (2–365) and (2–368), we obtain

Fourier Series Representation Of An Arbitrary Periodic Function (Part - 1) - Mechanical Engineering
Eq. (2–377) is a real form of a Fourier series for an arbitrary periodic function f (t).

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FAQs on Fourier Series Representation Of An Arbitrary Periodic Function (Part - 1) - Mechanical Engineering

1. What is a Fourier series representation?
Ans. A Fourier series representation is a mathematical technique used to express an arbitrary periodic function as a sum of sinusoidal functions. It allows us to decompose a complex waveform into simpler harmonic components, which can be analyzed and manipulated more easily.
2. How does a Fourier series represent an arbitrary periodic function?
Ans. A Fourier series represents an arbitrary periodic function by using a combination of sine and cosine functions with different amplitudes and frequencies. By adjusting these amplitudes and frequencies appropriately, the Fourier series can closely approximate the original function over its entire period.
3. Why is a Fourier series representation useful in mechanical engineering?
Ans. Fourier series representations are useful in mechanical engineering for various reasons. They allow engineers to analyze and understand the harmonic content of periodic mechanical signals, such as vibrations or oscillations. This knowledge is crucial in designing efficient and reliable mechanical systems, as it helps in predicting and mitigating potential issues related to resonances or unwanted vibrations.
4. Can any periodic function be accurately represented by a Fourier series?
Ans. Yes, any periodic function that satisfies certain mathematical conditions, such as having a finite number of discontinuities within its period, can be accurately represented by a Fourier series. The accuracy of the representation depends on the number of harmonic terms used in the series and the complexity of the original function.
5. How can Fourier series be applied to practical mechanical engineering problems?
Ans. Fourier series can be applied to practical mechanical engineering problems in various ways. For example, they can be used to analyze the vibration characteristics of mechanical structures, determine the natural frequencies of systems, and design filters to remove unwanted harmonic content from signals. Additionally, Fourier series can be employed in signal processing techniques, such as noise reduction or signal compression, which are relevant in areas like condition monitoring or control systems design.
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