Study On The Dynamic Behavior Of The Flexible Beam System With The Consideration Of Cubic Stiffness Nonlinearity

As one of the basic structure elements,beam structures are widely employed in various engineering fields.In engineering,beam structures are inevitably subjected to the vibration excitation introduced by the energy equipment,working environment,and others.Working in the vibration environment for a long time,vibration problems present in beam structures,including fatigue damage,structural instability,and others.Vibration suppression and isolation devices are introduced into the beam structure vibration system to suppress its unwanted vibration.The existing vibration suppression and isolation devices are mainly designed based on the linear theory,limiting their comprehensive performance,including working frequency band,vibration suppression effectiveness,and others,which is hard to satisfy the increasing demand for engineering vibration suppression.To improve the comprehensive performance of traditional vibration suppression and isolation devices,cubic stiffness nonlinearity is attempted to design the nonlinear vibration suppression and isolation devices.Against this background,researchers employ nonlinear vibration reduction and isolation devices to improve the vibration control effectiveness of beam structures.Considering limitations in the current research,this study employs the cubic stiffness nonlinearity in the nonlinear vibration-absorbing element,the nonlinear support element,and the nonlinear coupling element.The dynamic behavior of the flexible beam system with different application forms of cubic stiffness nonlinearity under arbitrary flexible boundary conditions is studied in the following.Based on the modal information of the beam structure with arbitrary flexible boundaries,a unified prediction model for the dynamic behavior of flexible beam structure with different application forms of cubic nonlinear stiffness(nonlinear vibration-absorbing element,nonlinear supporting element,and nonlinear coupling element)is established.The Galerkin truncation method is utilized to predict the dynamic behavior of the beam structure with cubic stiffness nonlinearities under arbitrary flexible boundary conditions.The numerical results indicate that the Galerkin truncation method based on the modal information can accurately predict the dynamic behavior of the flexible beam structure with different application forms of cubic stiffness nonlinearities,where the flexible beam has arbitrary flexible boundary conditions.As the boundary conditions change,dynamic responses of the vibration system can be predicted without modifying the model.Then,the cubic stiffness nonlinearity is employed as the vibration-absorbing element.A dynamic behavior analysis model of a flexible beam system with multiple nonlinear energy sinks and different types of nonlinear vibration-absorbing elements is established.On this basis,the influence of nonlinear energy sinks and the form of nonlinear absorbers on the dynamic behavior of beam structures is studied.Numerical results suggest that suitable parameters of nonlinear energy sinks can simultaneously suppress the vibration of multiple primary resonance regions in amplitude-frequency responses of the beam.The dynamic behavior of the beam structure is different in sensitivity to the parameter variation of different nonlinear vibrationabsorbing elements.Under a certain parameter range,the targeted energy transfer occurs between the nonlinear energy sinks and flexible beam structures.Secondly,the cubic stiffness nonlinearity is employed as the nonlinear supporting element.The dynamic behavior analysis model of the flexible beam system with nonlinear point support and the local uniform nonlinear foundation is established.Then,the influence of the nonlinear point support and local uniform nonlinear foundation on the dynamic behavior of beam structures is studied.The numerical result suggests that the dynamic behavior of flexible beams with nonlinear supporting elements is sensitive to their initial values.The complex dynamic behavior of the flexible beam system can be motivated by the nonlinear point support and local uniform nonlinear foundation,including amplitude frequency response peak jumping,amplitude unstable,multi-period vibration,quasi-periodic vibration,and other complex dynamic behavior.Additionally,a suitable simplified form should be selected according to the actual width of the local nonlinear uniform foundation.Thirdly,the cubic stiffness nonlinearity is employed as the nonlinear coupling element.According to whether additional degrees of freedom are introduced or not,the nonlinear coupling element is divided into a nonlinear stiffness coupling system and a nonlinear springmass coupling system.Based on this,the dynamic behavior analysis model of coupling beam structures connected through the nonlinear stiffness coupling system and nonlinear spring-mass coupling system is established.Then,the influence of nonlinear coupling elements on the dynamic behavior of coupling beam structure is studied.Numerical results suggest that the influence of the nonlinear stiffness coupling system and nonlinear spring-mass coupling system on the dynamic behavior of the coupling beam structures has vibration state converted values under a single-frequency excitation.Under a certain parameter range,the targeted energy transfer occurs between flexible beam structures.Moreover,appropriate nonlinear coupling element parameters can suppress the vibration level of coupling beam structures.Fourthly,the adjustable flexible boundary,passive nonlinear stiffness mechanism,and adjustable nonlinear stiffness mechanism are designed by utilizing the design principle of adjustable mechanism and nonlinear mechanism.Then,the mechanical properties of the corresponding mechanism are tested.Results suggest that the equivalent stiffness of the adjustable boundary presents linear characteristics,where its control unit can accurately control its equivalent stiffness.For the adjustable nonlinear stiffness mechanism,the experimental results of the nonlinear restoring force of the adjustable nonlinear stiffness mechanism match well with its theoretical results.The equivalent nonlinear stiffness of the adjustable nonlinear stiffness mechanism can be effectively controlled by adjusting the initial length of the horizontal springs.Furthermore,passive nonlinear stiffness mechanisms exhibit cubic stiffness characteristics.The passive nonlinear stiffness mechanisms as well as the adjustable nonlinear stiffness mechanism can be applied to control the vibration of beams as nonlinear vibrationabsorbing elements,nonlinear supporting elements,and nonlinear coupling elements.Eventually,the dynamic response test system of the beam structure with adjustable flexible boundary and cubic stiffness nonlinearities is established based on the Lab View programming platform.The influence of adjustable flexible boundary and cubic stiffness nonlinearities on the dynamic behavior of beam structures is studied by experiments.Results suggest that the adjustable flexible boundary can suppress the vibration of beam structures by changing the position of the primary resonance region.The different application forms of cubic stiffness nonlinearities can effectively suppress the vibration at the observation point of flexible beam structures under appropriate parameters.Meanwhile,the complex dynamic behavior of the flexible beam system can be motivated by different application forms of cubic stiffness nonlinearity.According to the experiment results,the correctness of the theoretical research on the dynamic behavior of flexible beams with cubic stiffness nonlinearity is qualitatively verified.