A Study On Dynamic Properties Of Rotating Beams And Plates Using Chebyshev-Ritz Method

Rotating beam/plate structures are the typical rigid-flexible coupling systems,which are widely used in modern engineering fields,such as spacescraft,robots and high-speed rotary mechanism.For these rigid-flexible coupling systems,the coupling dynamic charactericstics of the large overall rigidbody motion and the elastic deformation are shown.The traditional hybrid-coordinate method is essentially a zero-order approximated coupling modeling method,in which the small elastic deformation assumption of the structure dynamics is adopted and the high-order coupling terms between the large overall rigidbody motion and the elastic deformation of the flexible body are ignored.When the large overall motion of the system is at high speed,the result of the zero-order approximated model will be divergent.In 1987,Kane et al.studied the dynamic characteristics of a rotating cantilevered beam in non-inertial reference frame and proposed the "dynamic stiffening" effect.Many different approximated methods have been employed to "capture" the dynamic stiffness terms.However,some of these methods were not convincing.In recent two decades,the first-order approximated model including the second-order coupling deformation was proposed based on the basic principles of mechanics.The first-order approximated model was validated by experiments.The studies on the dynamic characteristics of rotating beam/plate structures are divided into two aspects of the dynamic responses and modal characteristics.In this dissertation,based on the superiority of the Chebyshev polynomials in numerical calculation,the dynamic characteristics of rotating beam/plate structures are studied in detail as follows:(1)The three-dimensional dynamic characteristics of rotating hub-Euler-Bernoulli beams are studied by using the Chebyshev-Ritz method.The three-dimensional rigid-flexible dynamic model of the system is established by using the second-kind Lagrange's equation.The non-linear coupling deformations are taken into account.The deformations of the beam are discretized by the Ritz method,and the modal functions are composed of the Chebyshev polynomials multiplied by the boundary functions in such a way that the geometric boundary conditions of the beam are implicitly satisfied.The first-order approximated model of the out-of-plane lateral bending motion and the first-order approximated coupling model of thein-plane transverse bending and longitudinal stretching motions are given.The corresponding zero-order approximated models are given by ignoring the non-linear coupling deformations.Firstly,the out-of-plane bending free vibration characteristics of rotating Euler-Bernoulli beams are studied.The influences of the rotational speed,hub radius ratio,taper ratios andboundary conditions on the mode characteristics are discussed.Then,the coupling effects between the in-plane transverse bending and longitudinal stretching motions are studied.The differences among the zero-order approximated model,the first-order approximated simplied model and the first-order approximated coupling model are presented.It is revealed that for dealing with the problem of the high-frequency excitation response,using Chebyshev polynomials is more efficient in computing precision than using other polynomials.The relations between the tuned speed and the resonance of the flexible beam are discussed.The frequency veering and mode shape conversions in the in-plane coupling vibration are studied in detail.(2)The out-of-plane bending free vibration analysis of rotating axially functionally graded Timoshenko beams with different combinations of boundary conditions is investigated by using the Chebyshev-Ritz method.The effects of the hub radius ratio,rotational speed ratio,taper ratios,rotary inertia and material gradient index on the modal characteristics of the Timoshenko beams with six different sets of boundary conditions are studied.It is shown that the influence of the gradient of the material composition on the natural frequency is related to the boundary condition.(3)The dynamic properties of rotating rectangular thin plates by using the Chebyshev-Ritz method.The rigid-flexible dynamic equations of rotating thin plates are derived by the second-kind Lagrange's equation.The difference between the zero-order approximated model and the first-order approximated model is studied.The great influence of different boundary conditions on dynamic responses of rotating thin plates is revealed.(4)The free vibration analysis of rotating tapered rectangular Mindlin plates by using the Chebyshev-Ritz method.Based on the Mindlin plate theory,the governing eigenvalue equations of rotating tapered moderately thick plates with different combinations of boundary conditions are derived.The effects of hub radius ratio,rotational speed ratio,taper ratio,aspect ratio and thickness ratio on the modal characteristics of the Mindlin plates.The effects of rotational speed on the frequency veering and mode shape conversion are focused on.