Parametric Resonance Instability Of Elastically Supported Circular Arches In-Plane Under Vertical Uniform Periodic Loads

Arch has a beautiful shape and excellent mechanical properties and is widely used in bridges,buildings,machinery,water conservancy and other fields.Working environment of the arch is often complex and changeable,and load on it is diverse.The dynamic stability of the arch has become one of the key factors affecting its bearing capacity.The occurrence of dynamic instability of the arch depends on its natural dynamic characteristics and the characteristics of external dynamic loads.Even if the dynamic load is far smaller than its static critical buckling load,the arch may still experience dynamic instability.Therefore,the mechanism of dynamic instability of the arch is more complicated than that of static instability and its occurrence is unexpected.Existing research focuses on the dynamic stability of arches under ideal boundary conditions.However,in practical engineering,arches are often constrained by elastic boundaries.Therefore,it is of great significance and valuable to study the dynamic stability of elastically supported circular arches under periodic loads.In order to reveal the dynamic instability mechanism of elastically supported circular arches,this paper is therefore focuses on the parameter resonance dynamic instability of elastically supported circular arches under vertical uniform periodic loads.It is follows that:(1)An analytical solution for the dynamic instability domain of an elastically supported circular arch under a vertically uniform harmonic load was derived.Based on the energy method,the energy equation of the dynamic system of an elastically supported circular arch under a vertically uniform harmonic load was derived.The Hamilton principle was used to obtain the coupled axial and radial motion control differential equations of the elastically supported circular arch.By introducing the assumption of incompressibility of the arch axis,the motion control differential equations of the elastically supported circular arch were decoupled and the analytical results of the free vibration frequency corresponding to the axial and radial modal functions of the elastically supported circular arch were solved.The Galerkin method was used to establish the second-order ordinary differential dynamic stability equation of an elastically supported circular arch under a vertically uniform harmonic load.The Bolotin method was adopted to derive an analytical solution for the dynamic instability domain of an elastically supported circular arch under a vertically uniform harmonic load.The effects such as elastic support flexibility,arch rise-to-span ratio,and damping ratio on the dynamic instability domain were analyzed.The correctness of the theoretical results was verified by finite element numerical simulation analysis,revealing the mechanism of dynamic instability of an elastically supported circular arch.(2)Analytical solutions for the dynamic instability domain of an elastically supported circular arch under vertically uniform square wave,triangular wave,and sawtooth wave loads were derived.Taking three common periodic loads as examples,by representing them in the form of Fourier series,the periodic loads were transformed into a linear superposition of a series of harmonic loads.The Bolotin method was used to derive analytical solutions for the dynamic instability domain of an elastically supported circular arch under vertically uniform square wave,triangular wave,and sawtooth wave loads.The effects of the aforementioned periodic loads on the dynamic instability domain of the elastically supported circular arch under the same load amplitude were analyzed.The correctness of the theoretical results was verified by finite element numerical simulation analysis,revealing the variation trend of the dynamic instability domain of the elastically supported circular arch with periodic loads.(3)The nonlinear dynamic response of an elastically supported circular arch under vertically uniform square wave,triangular wave,sawtooth wave,and harmonic loads was solved.Considering the geometric nonlinearity of the elastically supported circular arch,the Duffing equation for the in-plane vibration of an elastically supported circular arch under a vertically uniform periodic load was derived.The Runge-Kutta method was used to iteratively solve the nonlinear dynamic response of an elastically supported circular arch under vertically uniform square wave,triangular wave,sawtooth wave,and harmonic loads.The variation trend of the dynamic response amplitude and parametric resonance of an elastically supported circular arch under different periodic loads was analyzed.The correctness of the theoretical results was verified by finite element numerical simulation analysis of frequency sweeping.