Mechanical engineering 2Question 2.A Fourier series approximation to the force show a sequence of approximations.Given P=23−1−3−40561Let A= (v1, v2, v3)By the given conditionsPu1= v1Pu2 = v2Pu3 = v3That is matrix equation PA= V = (v1 v2 v3) PA=VP-1PA = P-1VA= P-1VNow |P| =-1 and adjoint (p) = −432−973−431 ^ TThen P-1 =494−3−2−3−2−1−3Therefore A = P-1V =494−3−2−3−2−1−3−3−6−8462545 =44466−3436−5−11−105
Mechanical engineering 3U1=44−34−11U2=4636−10U3=6−55Question 3.The amplitude is given by:A=Fo/s√(1¿−(wwn)n+(23wwn)2)¿Wn=√s/m =√3000/6 =10√5A=80/3000√¿¿¿ =80/3000√¿¿¿ =0.023668mTherefore A= 2.3668cmWhere by FO=80N f=8cps rs =3000N/m w=2πf
Mechanical engineering 4 m= 6kgBut ᵶ =0Then y =Asin(wt+∅)Note: ∅ is the phase difference (not given)QUESTION 4 By Duhamel's method the response of the system is given by the convolution of the unit impulse response function and the forcing function. For the undamped case, the unit impulse response function is:The unit impulse response function for the undamped motion is:The forcing function is given by:The overall response is thus given by:
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