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Fourier Transform
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DERIVATION AND Integration
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Consider ma maon rightsystem ci cathe beof motion incan plannedequation s sfa the lawof Newtonanumber secondexampleways using È Xif ha xi theGmi xitca Xa XiInCatiafa XiXi hamathe same equations È cadaXi fitaxiKi theGtmi citcari da hai trattoXe fakecit G tmi fr fpmi pCz0 ti i1 I 1E tee the EonFreeio synchronousvibration unbarredanof
SYSTEM MoranLets fecewiththeconsider andEsystemprevious hai triabarkithmithxI taxKi tratto omi aKarKaiLet's ofconsidera hand motiona special followscalled definedofvibrationmotion ModesSynchronous asa editX2Xxv pare I011 Lt Egenesimare inIv thetimeanddiwhere Ct isconstant historyamplituderepresents thethe tuothat ratio betweenthe displacementsImposing constantmotionssynchronous means isSo havenow we datheir theirfatiri Olavvimi viin o D miothendai dithe da oma thirdHaithe theofwritewe cannow synchronousequationgoverning lorinmoto iiii feativ roIwhereIF trii dide theirhtmco owe assume tiro dabrr cima oFanmatrixthein problemeigenvalueHi haCrimi It.IE iootsn m.oFanThe theof problemeigenvaluegeneral is leci MettaThe theoftheof I are synchronous motionamplitudecomponentsthis the of firstthe matrixif todeterminantSolutionhassystem zeroequalisSo the ofturns Waissolution in haid than 4mmtema t.kzthanmadettami II o µ 2in maThe tuo ofthenahalthecui circularroots systema
Frequencies represent the non-eigenvectors of the system. We obtain the modes of vibrations by combining the phase and frequency of the hand motion. What's interesting is that any combination of frequencies can result in synchronous motion. Note that this system displays general properties of natural frequencies and vibrating degrees of freedom. The response of the system to a forcing function is harmonic. Let's consider a symmetric system that is excited by a general forcing function. The complexity of the system is represented by the different frequencies acting on it. Also, we're interested in the steady-state solution of the ordinary differential equations that govern the system. The presence of damping guarantees that the transient solution will eventually expire. The solution is given by the matrix edit.