Analysis Of Vibration Characteristics Of Rotating Viscoelastic Rectangular Plate With Fractional Derivative

Rotating rectangular plate is widely used in aerospace,mechanical engineering and other fields,such as engines,turbines and etc.The vibration of rectangular plate is affected by the centrifugal force and the working environment under the fixed axis rotation state,and the structure is prone to be damaged.Therefore,it is of great significance to study the vibration characteristics of the rotating rectangular plate.The vibration characteristics of the rotating viscoelastic rectangular plate with fractional derivative are analyzed in this paper.The main work is as follows:(1)The differential equation of motion and boundary conditions of the rotating viscoelastic rectangular plate with fractional derivative are derived.The mechanical model of the rotating viscoelastic rectangular plate with fractional derivative is established with three coordinate systems,which called inertial coordinate system,rotating coordinate system and local coordinate system respectively.Based on the two-dimensional fractional Kelvin-Voigt viscoelastic material constitutive equation,the kinetic energy,strain energy,centrifugal potential energy and dissipated energy of the system are obtained.By means of Hamilton's principle and Riemann-Liouville fractional derivative properties,the differential equations of transverse motion and boundary conditions of the rotating viscoelastic rectangular plate with fractional derivative are derived.(2)The vibration characteristics of the rotating viscoelastic rectangular plate are analyzed.When the viscous coefficient is 0,the differential equation of motion of the rotating viscoelastic rectangular plate with fractional derivative degenerates to the differential equation of motion of the rotating elastic rectangular plate.The differential equation is solved by differential quadrature method.Comparing the obtained results with the literature results,the validity of the differential quadrature method is verified.When the fractional order is 1,the fractional derivative transforms into the integer order derivative,and the differential equation of motion of the rotating viscoelastic rectangular plate with fractional derivative transforms into the differential equation of motion of the rotating viscoelastic rectangular plate.The influence of parameters on the imaginary part of dimensionless complex frequency is discussed.The results show that,at a constant angular velocity,as the thickness to length ratio,the setting angle and the viscous coefficient increase,the imaginary part of dimensionless complex frequency decreases.And as the width to length ratio and the radius to length ratio increase,the imaginary part of dimensionless complex frequency increases.(3)The vibration characteristics of the rotating viscoelastic rectangular plate with fractional derivative are analyzed.The differential quadrature method is used to solve the differential equation of motion,the characteristic equation of the system with fractional order is obtained,and the influence of the fractional order,viscous coefficient and other parameters on the imaginary part of the dimensionless complex frequency is discussed.The results show that,at a constant angular velocity,the imaginary part of dimensionless complex frequency decreases with the increase of the fractional order.The change trend of imaginary part of dimensionless complex frequency with the thickness to length ratio,setting angle,viscous coefficient,width to length ratio and radius to length ratio is consistent with the conclusion in(2).The change of parameters has a larger impact on the third-order imaginary part of dimensionless complex frequency than the first-order imaginary part of dimensionless complex frequency.Under the same parameters,the imaginary part of dimensionless complex frequency described by the integer order derivative is lower than the imaginary part of dimensionless complex frequency described by the fractional derivative.