Dynamic Analysis Of Rotating Tapered Centilever Beam And Deploying Centilever Pipe Conveying Fluid

Rotating cantilever beams,especially tapered and variable length ones,are found in several practical engineering examples such as the rotating machinery,helicopter blades,wind turbine blades,the swinging arm of LED chip sorting machine,and the rotating cantilever beams with internal flow,such as the liquid injection tube of high speed filling machine,etc.Considering the complex coupling effect of rotary,deployment and elastic vibration,and the effect of the internal flow,the research of transverse vibration and stability of these cantilever system has important practical significance.Therefore,the vibration and stability of the rotating,tapered cantilever beam structure with complex installation conditions and variable length pipe with internal flow are analyzed in this paper,respectively.The main research work is as follows.(1)The dynamic behaviour of rotating cantilever Timoshenko beams are investigated.Based on the nonlinear von Kármán strain and the associated linear stress,the nonlinear dynamic equations of a rotating,tapered,cantilever Timoshenko beam are derived by using the Hamilton principle.The nonlinear responses of the Timoshenko beam are investigated via the numerical calculation;After the dynamic equations are linearized in the neighborhood of the equilibrium position when the rotation beam is in a steady state,the effect of angular speed,hub radius,slenderness ratio,and the height and width taper ratios on the frequencies of the rotating Timoshenko beam is investigated,in which the extensional deformation of the beam is considered.(2)The transverse vibration and stability of the tapered cantilever beam with oscillating rotating velocity are analyzed.For oscillating rotating or swing tapered cantilever beam,the oscillating change of centrifugal inertia force may cause the parameter resonance of the beam.The method of multiple scales is applied to investigate the parametric resonances of the rotating and swing tapered cantilever beam,then the effect of the oscillation amplitude,frequency and mean rotating velocity on the parametric resonance regions and the tip responses are discussed.(3)The transverse vibration and stability of the axially deploying cantilever pipe without fluid with varying length are analyzed.Based on the longitudinal and transverse coupling motion differential equation,the system dynamic response is obtained using the Runge-Kutta method.For stability analysis of the pipe with constant-speed deployment,the eigenvalues of motion equation and variation rate of transverse vibration energy are used to investigate the stability of the axially deploying cantilever pipe.(4)The parametric resonances and transverse vibration of an axially oscillating deploying cantilever pipe without fluid are analyzed to investigate the dynamical stability of the pipe.The differential equation of motion with time-dependent coefficients is discretized by the Galerkin method,and the method of multiple scales is applied to investigate the parametric resonances of the oscillating deploying cantilever beam.The effect of the oscillation amplitude,frequency and mean deploying velocity on the parametric resonance regions and the tip responses are discussed.In addition,the ‘jump' phenomenon on the tip response amplitude of the axially oscillating deploying cantilever pipe is discussed also.(5)The differential equation of a vertical cantilever pipe conveying fluid is derived when the pipe has deploying and rotating motion based on Euler-Bernoulli beam theory and Hamilton's principle.Then,the dynamic responses of the vertical cantilever pipe conveying fluid are analyzed,and the beat phenomenon,tip trajectory and effects of the centrifugal,Coriolis and tangential inertia forces on the dynamical responses are discussed also.Furthermore,the stability of the vertical cantilever pipe conveying fluid are investigated based the eigenvalue and vibrational energy,the effects of deploying or retracting speed,mass ratio,and fluid velocity on the transverse vibration and stability of the vertically deploying cantilever pipe conveying fluid are discussed.