Research On Free Vibration And Parametric Instability Of Thin-walled Shell Structures

Though the thickness of the shell is very thin,it has has higher strength and stiffness,and is widely used in aerospace,civil,mechanical,marine and other engineering industries.The free vibration analysis based on the linear elastic theory is the basis to do further study of the shell subjected to various loadings.Though many methods have been proposed to analyze the free vibration of thin-walled shells,it is still of need and great significance to develop a simple and efficient method to investigate this type of structures.In recent years,the Haar wavelet method has been applied to the free vibration analysis of static thin-walled shells.However,the method has not been employed to analyze the parametric instability of shell structures.When the shell elements are subjected to periodic in-plane loads,it may lead to parametric resonance/parametric instability due to certain combinations of the values of load parameters and natural frequency of transverse vibration.The shell structures may be destroyed when the phenomenon of parametric resonance occurs.Most of the previous work on parametric instability of static shells only deal with the first-order approximations of principal instability regions using Bolotin's method.To obtain more accurate results of principal instability regions is necessary to some extent.The assumption of Floquet multipliers in Bolotin's method cannot be satisfied for the gyroscopic system.Furthermore,the multiple scales method is essentially a small parameter method and is not capable of analyzing the parametric instability of rotating cylindrical shells subjected to various periodic axial loads.Thus,it is of great technical importance to propose a new method to solve such problems.The thesis initially investgated the free vibration characteristics of thin-walled shells,and then carried out research on the parametric instability of shells subjected to periodic axial loads.What's more,the thesis expanded the application of the Haar wavelet method in the field of structural dynamics and propsed the new method to analyze the parametric stability behaviours of rotating thin-walled shells.The detailed research contents and achievements can be summarized as follows:The solution process of the Haar wavelet method for static shells is simplified and extended to the vibration analysis of rotating truncated conical shells.The governing differential equations are established based on the Love first-approximation theory.The displacement field is expressed as the Haar wavelet series in the axial direction and trigonometric functions in the circumferential direction.Considering boundary conditions,the eigenvalue equation is obtained to determine the vibration behaviors of rotating conical shells.The effects of geometrical parameters,rotating speed and boundary conditions on the vibration characteristics are studied.It is shown that the effect of semi-vertex angle on the forward and backward frequencies is noticeably different for different cases.When When the length of the shell in axial direction changes,For one case,the forward and backward frequencies decrease monotonically with the increase of semi-vertex angle.For the other case,the frequencies initially increase with the increase of semi-vertex angle and upon reaching a peak,and then decrease as semi-vertex angle is further increased.When the length of the shell in axial direction not changes,the forward and backward frequencies decrease monotonically with the increase of semi-vertex angle.The critical rotating speed only occurs for mode of 1.With rotation,the natural frequencies decrease at this mode,and eventually approach zero when their critical rotating speeds are reached.At this certain rotating speed,the rotating conical shell may become dynamically unstable.Axial mode shapes of the rotating truncated conical shells under four different boundary conditions are very similar.Using the Haar wavelet method combined with Bolotin's method,parametric instability of truncated conical shells under periodic axial loads has been studied.The difference of first-order and second-order approximations of principal instability regions of under compressive and tensile loads is compared.It is shown that the difference for compressive loading is more obvious than that for tensile loading when the relative amplitude of parametric excitation is large.The width of principal instability regions significantly increases with the increase of the static component of the applied loading.The decrease of length-to-radius ratio and increase of thickness-to-radius ratio would both result in the enhancement of the instability regions.The effects of semi-vertex angle for different different cases on on the principal instability regions are discussed.It is found that the moving directions and width of the principal instability region are significantly distinct for all cases with the increase of semi-vertex angle.Based on the assumed mode method and Floquet exponent method,parametric instability behaviours of simply-supported rotating cylindrical shells under periodic axial loads has been studied.The effects of rotation speed,static load factors,viscous damping and shell geometrical characteristics on the location and width of the instability regions of rotating cylindrical shells are investigated.The results reveal that for rotating cylindrical shells under periodic axial loads there are only combination instability regions.It is found that the lower of the starting points of excitation frequency for circumferential wave number,the wider is the combination instability region.The viscous damping has different influence on the parametric instability behaviors of non-rotating and rotating cylindrical shells.For non-rotating cylindrical shells,the increase of damping continuously results in the reduction instability regions.However,for rotating cylindrical shells,boundaries of parametric instability regions with viscous damping intersect with those no damping.This phenomenon means that the instability of rotating cylindrical shells may be enhanced under some cases due to the existence of viscous damping.The Haar wavelet method combined with Floquet exponent method have been used to investigate the parametric instability of rotating truncated conical shells.The effects of different parameters on the the instability regions under four different boundary conditions are analyzed.Similar with the results for rotating cylindrical shells,for rotating conical shells there are only combination instability regions,of which the width decreases with the increase of the boundary constraint.It is found that for circumferential wave number of 1 the variation trends of the instability regions for different boundary conditions are noticeably distinct with the increase of rotating speed.However,for circumferential wave number of 3,the instability regions for different boundary conditions shift to higher frequency range with the increase of rotating speed.And the width of the instability regions also decreases in the speed growth process.The thesis studied the free vibration characteristics of cylindrical and conical shells and the parametric instability of such structures subjected to periodic axial loads.The research contents and results of the thesis enrich the dynamics of thin shell structures and may be served as the guidance in the design of enhancing the stability of the shells subjected to dynamical loads.