振动理论与分析8.pptVIP

Orthognnallity of vibration mode Longitudinal vibration of rod U1 U2 U3 Ui ┴ Uj Review: Set up partial differential equation of motion Set solution form u(x,t)=U(x)ejωt or u(x,t)=U(x)est get mode solution and (3) Boundary condition (4) Determine coefficient A or B by one boundary condition (5) Determine natural frequency by another boundary condition (6) Mode equation and shape (7) Solution of equation General method for continuous system Vibrating String Lagrange’s equation x y o (1)Kinetic energy Potential energy Lagrange integration Vibrating String (2)From Lagrange’ formula Get equation of motion Vibrating String Let (3)Solution of equation for One of methods Vibration mode depend space x Vibrating String Solution form substitute in motion equation obtain Solution is Vibrating String (4) Boundary condition y(0,t)=0; y( l, t)=0 x y o Vibrating String y(0, t)=0; B=0 Because of From y( l, t )=0 or n=1,2,3… (5)Natural frequency (6) Mode function while A=1 Vibrating String Vibration mode shape Vibrating String (7) Solution of vibration Vibrating String Vibration equation for moving string x y o Kinetic energy Potential energy Lagrange integration y(x, t) is a function of x and t v Vibrating String let Equation of motion Vibrating String Let solution form： Root of characteristic equation: Then： 0 Vibrating String Boundary condition: ; Form condition , Let Y(x) can be changed: Vibrating String Form condition , From previous formula: ; Natural frequency of moving string Vibrating String Then the models function: -complex model Natural frequency of moving object is different to that o