ASE 365 - Lecture 29

# ASE 365 - Lecture 29 - Continuous systems visualized easily...

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Unformatted text preview: Continuous systems . visualized easily and ally mathematic simple : vibration transverse in string a analyzing by begin will We media. structural the throughout d distribute are properties elastic and inertial springs, and masses discrete of instead systems, parameter)- ed (distribut continuous In ones. discrete than rather freedom of degrees are there so , e.g., time, as well as position spatial of functions continuous become nts Displaceme n t x u ∞ ) , ( String analysis y(x, t) x f(x,t)=force/le ngth L dx t x y x T t dx x y dx x T t x f t x y x A dx dx dx ) , ( ) ( ) , ( ) ( lim ) , ( ) , ( ) ( lim ′- + ′ + + = → → ρ : take , by Divide x A T vary with could h mass/lengt tension = = ρ : dx length of element Consider T(x+d x) d x f(x,t) dx T( x) ) , ( ) ( ) , ( ) ( ) , ( ) ( ) , ( t x y dx x A t dx x y dx x T t x y x T dx t x f F y ρ = + ′ + + ′- = ∑ ) constant. is ( ) ( x T F x → = ∑ ( 29 L x t x y x T x t x f t x y x A < < ′ ∂ ∂ + = , ) , ( ) ( ) , ( ) , ( ) ( ρ : (PDE) equation al differenti partial a gives This L x y T f y A < < ′ ′ + = , ) ( ρ : compactly More ) (like : form standard to Rearranged F kx x m L x f y T y A = + < < = ′ ′- , ) ( ρ ) in order (2nd or : ICs and ) in order (2nd : BCs need we (BVP), problem value boundary For t x v x y x y x y x y x t L y t y ) ( ) ( ) , ( ), ( ) , ( ) , ( , ) , ( = = = = An analogous system f(x,t)=force/le ngth u(x, t) x x x A x EA vary with h mass/lengt rigidity axial = = ) ( ) ( ρ : dx length of element Consider f(x,t) dx P(x+dx ,t) P(x, t) ) , ( ) ( ) , ( ) , ( ) , ( t x u dx x A t dx x P t x P dx t x f F x ρ = + +- = ∑ x u x EA x u u x EA A E A t x P x x x ∂ ∂ = ∆- = = = ∆ + ) ( ) ( ) ( ) , ( ε σ dx t x u x EA t dx x u dx x EA t x f t x u x A dx dx dx ) , ( ) ( ) , ( ) ( lim ) , ( ) , ( ) ( lim ′- + ′ + + = → → ρ : take , by Divide : vibration axial in Rod ( 29 L x t x u x EA x t x f t x u x A < < ′ ∂ ∂ + = , ) , ( ) ( ) , ( ) , ( ) ( ρ : PDE L x u EA f u A < < ′ ′ + = , ) ( ρ : compactly More ) (like : form standard to Rearranged F kx x m L x f u EA u A = + < < = ′ ′-...
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ASE 365 - Lecture 29 - Continuous systems visualized easily...

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