| Beam structures are integral components of various engineering systems,and their vibration behavior directly impacts the stability and reliability of the systems.Due to factors such as external loads,material nonlinearity,geometric nonlinearity,and smallscale effects,beam structures often exhibit nonlinear or even strongly nonlinear vibration states.In many cases,this nonlinear vibration may lead to structural damage,performance degradation,or even failure,posing a significant threat to the normal operation of engineering structures.Therefore,conducting nonlinear dynamic studies on various beam structures and proposing reliable vibration control strategies are of paramount importance for enhancing the safety and stability of engineering structures.This thesis applies time delay feedback control methods to several common types of beam structures,focusing on the investigation of displacement and velocity time delay feedback control for their free vibration,axial motion,and forced vibration.The main findings and contributions are summarized as follows:(1)The strong nonlinear vibration characteristics of nanobeams under displacement and velocity time delay feedback control are studied.The homotopy analysis method is employed in the nano-scale research domain to study the periodic motion induced by Hopf bifurcation.The effects of nonlocal parameters and time delay control parameters on the system’s hardening behavior,Hopf bifurcation,and periodic solutions are discussed.It is found that both the feedback gain coefficient and time delay can effectively control the Hopf bifurcation in the system and alter the number and amplitude of periodic solutions.Nonlocal parameters not only increase the response range and peak amplitude of the system but also affect its hardening behavior.(2)The parametric resonance of axially moving nanobeams under time delay feedback control is investigated.The frequency-response equation for the system is obtained using the multiple scales method,and stable intervals for zero and non-zero solutions are determined,along with the sufficient and necessary conditions for Hopf bifurcation.Optimal time delay ranges for vibration reduction are determined by plotting stable solution regions associated with displacement and velocity time delays.The effects of nonlocal parameters,displacement feedback gain coefficients,velocity feedback gain coefficients,displacement time delay,and velocity time delay on system dynamics are discussed.It is observed that nonlocal parameters not only reduce the system’s amplitude but also alter the instability regions of non-zero solutions.The time delay parameters can affect the number,amplitude,and stability of non-zero solutions.(3)The primary and secondary resonance of cantilever beam systems with concentrated mass under displacement and velocity time delay feedback control are investigated.Using the homotopy analysis method,the amplitude-frequency response characteristics of the control system under primary and secondary resonance conditions are discussed.The optimal time delay for suppressing system amplitude is determined with damping ratio as the objective function.It is found that velocity feedback control is not always superior to displacement feedback control,and the relative magnitude of the natural frequency to the excitation frequency affects the effectiveness of displacement and velocity time delay feedback control.Reasonable selection of displacement and velocity time delay parameters can effectively control the Hopf bifurcation,suppress the amplitude,adjust the response region and jump frequency.This theoretical research provides a foundation for the application of time delay feedback controllers in both macro and microstructures and advances the use of homotopy analysis methods in time delay dynamic systems. |
