Dynamic Analysis Of Axially Moving Cantilever Beam Featuring Time-Varying Velocity

In engineering practice,certain five-axis linkage machine tool systems,band saws,satellite antennas,etc.can be modeled as cantilever beams with axial variable speed motion.At the same time,it is also a time-varying parameter continuum system,and its motion differential equations and boundary conditions are time-varying,which brings many technical problems to dynamic analysis.Therefore,studying the dynamic modeling,dynamic characteristics,and stability of a cantilever beam with time-varying parameters in axially variable motion has great theoretical significance and engineering application value.The main research contents of this article include:(1)The dynamic equation of a cantilever beam with axially variable motion with time-varying parameters is established,and the transient dynamic behavior of the beam under low-speed uniform motion is studied.The telescopic deformation structure is simplified into a cantilever beam with variable length and variable mass with axial motion based on the Euler-Bernoulli beam theory,and the differential equation of beam lateral motion is established by using the D’Alembert principle.Based on the separation of variables method and the multidimensional method,the equation of motion of the number of fluctuations when the trigger oscillates freely at low velocity is obtained.The modulus functions of axially moving cantilever and non axially moving cantilever are analyzed.The modulus functions and natural frequency characteristics of axially moving cantilever are different from those of ordinary cantilever.(2)The differential equation of transverse motion of axially variable-speed cantilever beam with self-excited force at the end is established,and the Galerkin method is used to solve the differential equation.The vibration displacement solutions of the system under the first,second,third and fourth order truncation approximation are obtained.The approximate solutions obtained under different truncations are compared,and the influence of self-excited force on the structural mode of the system is discussed.The feasibility of Galerkin method for solving partial differential equations is also discussed.(3)The vibration of the axially moving cantilever beam during the retracting movement is studied,and the vibration of the system is studied by simulating the actual operating state of an axially moving structure.Propose an axial velocity distribution curve to simulate the actual system working trajectory,simulate the actual extension and retraction trajectory of the beam,so that the beam movement starts from a static state,and returns to a static state after reaching the final required length.The study expands/The influence of the retraction time on the dynamic response of the beam provides a theoretical basis for selecting the best trajectory execution time when the system is deployed and retracted in actual engineering applications.(4)Based on the system dynamics equation,study the influence of the basic parameters of the axially moving cantilever with time-varying parameters on its dynamic performance.Through numerical examples,the influence of the initial length,motion acceleration,excitation amplitude and other parameters of the beam are analyzed for the dynamic performance and stability of beams,which provides an effective reference for determining the optimal system design parameters.Through the research of this article,it is concluded that the vibration form of the axially moving cantilever beam changes with time.The natural frequency of the beam is a function of time.The beam expansion is inherently unstable,while the beam retracts are stable.These research results have important practical significance for studying the dynamic characteristics of time-varying structural systems.