By Carl L. DeVito

Many branches of arithmetic come jointly in harmonic research, every one including richness to the topic and every giving insights into this attention-grabbing box. Devito's Harmonic research provides a complete creation to Fourier research and Harmonic research and offers a number of examples and versions in order that scholars depart with a transparent knowing of the idea.

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**Example text**

2n) vanishing. As a result the terms on the Bi-periodic Mass-Spring Systems 51 right side of Eq. 4) are equal to zero. Corresponding frequency equation can be obtained as where rn denotes the half wave number of the mode for the original system. Taking a specific example as shown in Fig. 2. 2 Bi-periodic mass spring system with fixed ends, p=3 and n=2 The parameters are leading to Substituting Eq. 20)into Eq. 21) can be rewritten, in term of Ro as The roots of Eq. 23b) Noting Eq. 19) becomes That leads to The two roots are R, = l ( m = 2 ) , 3(m=4) There are five natural frequencies altogether.

N) denote the real loads acting on the original system. The term AMw2xj acts as a kind of loading. The expressions on the left sides of Eqs. lb) possess cyclic periodicity. We can now apply the U-transformation to Eq. 1). Let 7T where y = -, i = f i and 2N denotes the total number of subsystems. N 1 ZN Prernultiplying Eq. =I Introducing Eq. 2) into Eq. 4) 48 Exact Analysis of Bi-periodic Structures As a result f; = O and fiN= O From Eq. 6~) W ~- 2k cos m y ) Substituting Eqs. ,2N) represents the solution of the perfect periodic system with AM vanishing.

1 may be expressed as where w denotes the natural frequency, x j denotes the amplitude of j-th subsystem and the term - (AK - A M w 2 ) x j may be formally treated as the load. 2) to Eq. 31) where and then Substituting Eqs. 33) into Eq. 2a) yields x. =- (AK - MU') N Introducing the notation X , in Eq. 34) we have ei(j-l)mve-i(u-l)mpy 2U=I2 K m =+~2 k - M u 2 -2kcosmy = x,+(,-,,, and inserting j XI+(~-~)~ = 1+ (s - l)p (s=1,2,. ,n) where p,' denotes the harmonic influence coefficient for the considered system with AK=AM=o.