Study On The Dynamics And Control Of Axially Moving Beams

Axially moving beams are very important mechanism. Mechanisms, such as rigid in-flightrefuelling pipe, bridge of the bridge laying truck, rapid raising military radar antenna, flying rocketand missile, can all be modeled as axially moving beams. The studies on the dynamics and vibrationcontrol of axially moving beams are very valuable.Two dynamic configurations of axially moving beams are investigated in the article. One is anaxially moving cantilever beam, where the dynamic modeling and vibration control of the transverseoscillation of the beam are studied. The other is an axially moving free-free beam, where the dynamicmodeling and the stability of the beam are analyzed, the thermal effect on the dynamics of the beamand the dynamics of the metallic thermal protection system (MTPS) are studied as well. The mainpoints of the concrete content are as follows.(1) The influence of the end mass on the modes of the axially moving cantilever beam isinvestigated. Firstly, the transverse vibration equation of an axially moving cantilever beam with tipmass is formulated by the D’Alembert principle. The instant linearized equations are set up based on伽辽金法’s method subsequently. At last, two modal shapes that satisfy different orthogonallyconditions are studied in the vbrational calculation of the axially moving cantilever beam. It is foundthat the modal shapes of the uniform beam can substitute the ones of the beam with light lumpedmass in the伽辽金法’method.(2) Based on the theoretical analysis, the control of the transverse vibration of an axially movingcantilever beam is studied, and the control configurations for the active vibrator and the active forceare presented. Firstly, the transverse vibration equation and boundary conditions are formulated by theHamilton principle. Then, the instant linearized equations are set up based on伽辽金法’s method forthe approximation solution. The controllers for the active vibrator and the active force are designedbased on linear quadratic regulator (LQR) method subsequently, where the optimal Q and R matricesare selected by the weighting coefficient method. At last, the two control methods are simulated bynumerical examples. The results show that the two control methods can all suppress the transversevibration effectively. The control result of the active force method is better than that of the activevibrator method, but the active vibrator is more realizable than the active force in practice.(3) An experimental platform is designed and presented for the axially moving cantilever beam,and the damping and boundary condition of the axially moving beam are updated by experiments.The beam which slides through a prismatic joint is considered as an axially moving cantilever beam, the transverse vibration equation is given, and the discretized equations of motion of the translatingbeam with time-dependent coefficients is derived by mode superposition method. Firstly, the fixedboundary condition is modified by the adjustment of the cantilever length of the theoretical beammodel with the measured results of its first order natural frequencies under various cantilever lengths.It is found that the correction lengths to the model beam have no explicit relationship with thecantilever lengths. Secondly, the first order decay coefficients are identified by LogarithmicDecrement method and it is find that the decay coefficients of the beam decrease with the increase ofthe cantilever length. At last, the calculated responses by the modified model are fit well with theexperimental results. It verifies the effectiveness of the proposed model modification method.(4) A noncontact exciting equipment is designed and put forward. Based on the experimentalplatform and the exciting equipment, vibration control of the axially moving cantilever beam isstudied, where the velocity feedback control method, the linear quadratic Gaussian (LQG) controlmethod and the H∞control method are studied, respectively. According to the control methods,numerical simulation and actual experiments are implemented for the beam with constant length andvarying length, respectively. And the results show that the control methods are effective and thesimulations fit well with the experiments.(5) Frequency characteristics and stability of the axially moving free-free beam with high velocitywhich simulates a filying missile is investigated. Firstly, the transverse vibration equation of theaxially moving free-free beam is derived by Hamilton’s principle, where the lumped mass attached atarbitrary position of the beam is taken into account, and the cross section is consecutive variation. Theinstant linearized equations are set up based on伽辽金法’s method for the approximation solutionsubsequently. Secondly, the influence of the axially movment, the changing of the mass density, andthe temperature on the natural frequencies of the beam are investigated, respectively. At last, thesufficient conditions for uniform stability and uniform exponential stability of the beam areestablished via Lyapunov stability criteria. It is found that the damping can be increased if the lumpedmass is placed suitably, the slope of the mass per unit length of the beam influences system stabilitysignificantly, and the axially movment and temperature can decrease the stiffness of the beam whichinduces the vibration frequencies descending.(6) The response and control of the transverse vibration of an axially moving free-free beam withhigh velocity subjected to the thermal shock is studied. Considered the thermal shock and the controlforce, the transverse vibration equation and boundary conditions are formulated by the Hamiltonprinciple. The displacement responses are subsequently simulated, where the axially velocity is2Ma.Then the controller is designed based on LQR method, and the displacement response for the controlmethod is simulated by a numerical example. The result shows that the control method can suppress the transverse vibration effectively.(7) Dynamic modeling of the beam with MTPS is investigated, and the simplified method forcalculation of thermal modes is studied. Firstly, the effective thermal conductivities, specific heatcapacity, density and elastic coefficient of metallic honeycombs are presented. Then a completemodel including outer honeycombs, middle thermal protection layer, inner honeycombs, isolation andload-carrying structure are modeling, and the thermodynamical response and the bending modes aresubsequently studied. At last, the simplified model for bending modes calculation is investigated, andthe results show that it is effective in the fast calculation of the modes of the structure with MTPS.Also it’s verified that the influence of the high temperature on structure’s modes is notable.