Research On The Nonlinear Mode Characteristics And Vibration Suppression Of Beam Structures With Dry Friction

Dry friction is a kind of nonlinear damping,which widely exists in the nature.The damping is usually used in mechanical structures,such as blades of aero-engines and freight car bogies,the aim of which is to restrain vibration and control vibration characteristics.In recent years,the modal characteristic of structures with dry friction is barely focused in the literature,compared with the response research of this kind of structures.The action mechanism of the dry friction on the modes of beam-like structures should be studied intensively.Furthermore,the transition of nonlinear normal modes and the damping characteristics under the action of dry friction need to be studied.These researches are important for design of beam structures with dry friction.The modal characteristics of beam-like structures under dry friction and the mechanism of vibration suppression are studied by using experimental,analytical and numerical methods.The experimental research mainly answers the question that whether the structural modes is influenced by dry friction or not.The analytical and numerical researches present what are the modal characteristics of beam-like structures under dry friction,and how to design structures and absorbers by utilizing the characteristics.The main work of the thesis is listed as follows:(1)Based on the established double beams model with multi-points friction,the LMS Polymax method is used to obtain the first 16 orders modes,including elastic modes and rigid modes.Results show that the frequencies of rigid modes decrease and the frequencies of elastic modes increase slightly as the non-dimensional excitation force increases.The non-dimensional excitation force is the ratio of the amplitude excitation force and dry friction.The middle-order modes are greatly influenced by friction more than low-or high-order modes.The rational range of the amplitude of excitation force is determined to fully identify the modes of structures.(2)An operating modal experiment method is obtained to apply in large engineering structures.The modal experiments of horizontal and vertical direction excitation are respectively carried out under the friction produced by 20 N normal pressure.The obtained modal parameters are superposed together,the result of which is consistent with that of by a modal experiment hammer.The proposed method is regularly influenced by some parameters,such as reference direction and sampling time.(3)An energy equivalent method is proposed.The mode frequencies and mode shapes of a cantilever beam under dry friction are equivalent to that of a single cantilever beam.In order to consider the boundary condition of dry friction,the Lagrange's equation of the first kind is used to establish the motion equations of the cantilever beam under friction.The motion equations of the equivalent model(a single cantilever beam)are obtained using the Lagrange's equation of the second kind.The analytical expression between frequency and excitation energy of the two models is obtained.Then based on the principle of energy equivalence,the natural frequency of the cantilever beam with friction is deduced.The mode shape of the frictional beam is also obtained.The natural frequencies of the first 1-5 orders are obtained by using numerical method to verify the effectiveness of the proposed method.The two kinds of results are well consistent.(4)There exist some features in the influence of friction magnitude,frictional location and energy on the beam's natural frequency.Understanding these features are helpful for the design of vibration suppression.For example,the different friction location can greatly influence the efficiency of damping.If the frictional location is located in the mode node of certain mode,this mode does not be influenced by friction damping but other mode motions will be restrained.This feature is the mode filter of the dry friction.Furthermore,the frequency energy plot of the frictional cantilever beam shows being with a upper critical value,which is different from the lower critical value of the frequency energy plot of nonlinear stiffness systems.(5)The configuration of dry friction determines the modes transition.A 2-DOF model is established.The grounded friction and ungrounded friction are respectively considered in the model.The Hamilton frequency energy plot(FEP)and the transient FEP are employed to study the mode transition of the 1:1 inphase and outphase modes of the two models.The grounded friction model and the ungrounded friction model possess the same Hamilton FEP,but the features of their mode transition are completely different.For the mode transition of the outphase mode,the model of the grounded friction keeps the single mode motion,however,the model of the ungrounded friction shows the two different kinds of mode motions.It is more complicated for the mode transition of inphase mode.The model of the grounded friction goes through inphase mode motion,3:1 mode motion and outphase mode motion.The model of the ungrounded friction experiences the inphase and 3:1 mode motions.(6)The mechanism of nonlinear absorber is derived from the characteristics of modes transition.The characteristics are different for the various mode motions.The configuration of dry friction determines the mode transition and the result of the transition.The modes of nonlinear systems control the damping characteristics of nonlinear absorber.In the aspect of damping effect,the model of ungrounded friction is superior to that of grounded friction.The damping effect of 3:1 mode is better than that of outphase mode.The work in the thesis will be helpful to understand the nonlinear normal modes of beam structures with dry friction.The result of the thesis will be an important supplement to the theory of the nonlinear normal modes and be a meaningful instruction to design nonlinear absorbers based on dry friction.