ASE 365 - Lecture 31 - Free vibration of continuous systems...

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Unformatted text preview: Free vibration of continuous systems approach. Newtonian a by vibration, transverse in beams and vibration, torsional in shafts vibration, axial in rods vibration, transverse in strings for ICs) BCs, (PDEs, problems value boundary derived have We length. unit per torque or force nt; displaceme angular or axial , transverse ; or , ; or where : problems order Second = = = = = - f u GJ EA T s J A m f u s u m ) ( f v EI v A = + ) ( : problems order Fourth L x t x f t x v T t x v x A < < = = - , ) , ( ) ) , ( ( ) , ( ) ( : string a of vibration free for PDE visualize. to easiest is vibration transverse problems, order second the Of ) variables. of n (Separatio ? which in one I.e., solution? motion" s synchronou " a there Is : question Key ) ( ) ( ) , ( t F x V t x v = ) ! (separable : or : is which : becomes PDE so, If ) ( ) ( ) ) ( ( ) ( ) ( ) ( ) ) ( ( ) ( ) ( ) ( , ) ) ( ) ( ( ) ( ) ( ) ( x V x A x V T t F t F t F x V T t F x V x A L x t F x V T t F x V x A = = < < = - . : time in ODE an gives This =- CF F C x V x A x V T t F t F x x V x A x V T t F t F = = = ) ( ) ( ) ) ( ( ) ( ) ( ) ( ) ( ) ) ( ( ) ( ) ( : constant a to equal be must they Therefore, only. of function a is RHS the and only, time of function a is LHS the , Since system. ve conservati a being string the with consistent not is This lly. exponentia decay and grow solutions two s ' ) ( t F . is equation stic characteri the ODE, this For 2 =- C s negative. one and positive one : roots real two are there , If C ). if (linear solution harmonic a has which , becomes ODE time the Then . or , So 2 2 2 = = +- = F F C C ). ( ) ( ) ) ( ( ) ( ) ( ) ) ( ( ) ( ) ( 2 2 x V x A x V T x x V x A x V T t F t F = -- = = : in ODE an obtain we which from , have we Then vibration. of modes and s frequencie natural obtain to solved are types EVP Both . EVP the to analogous (EVP), problem eigenvalue a is This u u M K 2 = algebraic al differenti problems. eigenvalue al differenti analogous at arrive to taken be can steps same the problems, vibration...
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ASE 365 - Lecture 31 - Free vibration of continuous systems...

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