On lateral-torsional instability of arches subjected to motion-dependent loading

The problem of dynamic lateral-torsional instability of circular arches acted upon by generally non-conservative forces is treated herein. The influence of motion-dependency of loading on the stability characteristics is determined in certain cases. It is also found that a semi-circular arch which is only capable of rigid-mode-buckling according to static analysis may lose its stability through non-rigid modes if the dynamic criterion is utilized.     

Nonlinear dynamic buckling of shallow circular arches under a sudden uniform radial load

The structural behavior of a shallow arch is highly nonlinear, and so when the amplitude of the oscillation of the arch produced by a suddenly applied load is sufficiently large, the oscillation of the arch may reach a position at its primary unstable equilibrium path or secondary bifurcation unstable equilibrium path, leading the arch to buckle dynamically. This paper presents an analytical study of the nonlinear dynamic in-plane buckling of a shallow circular arch under a uniform radial load that is applied suddenly and with an infinite duration. The principle of conservation of energy is used to establish the criterion for dynamic buckling of the arch, and the analytical solution for the dynamic buckling load is derived. It is shown that under a suddenly applied uniform radial load, a shallow pinned–fixed arch has a unique possible dynamic buckling load, while shallow pinned–pinned and fixed–fixed arches may have two possible dynamic buckling loads: a lower dynamic buckling load and an upper dynamic buckling load. The dynamic buckling loads of a shallow arch under a suddenly applied uniform radial load with infinite duration are found to be lower than their static counterparts, and to increase with an increase of the arch included angle and slenderness. The effect of static preloading on the dynamic buckling of an arch is also investigated. It is found that the pre-applied static load decreases the dynamic buckling load of the arch, but increases the sum of the pre-applied load and the dynamic buckling load.     

Transverse Vibration and Stability Analysis of Circular Plate Subjected to Follower Force and Thermal Load

Transverse vibration and stability analysis of circular plate subjected to follower force and thermal load are analyzed . B ased on the thin plate theory in involving the variable temperature, the differential equation of transverse vibration for the axisymmetric circular plate subjected to follower force and thermal load is established. Then, the differential equation of vibration and corresponding boundary conditions are discretized by the differential quadrature method. Meanwhile, the generalized eigenvalue under three different boundary conditions are calculated. In this case, the change curve of the first order dimensionless complex frequency of the circular plate subjected to the follower force in the different conditions with the variable temperature coefficient and temperature load is analyzed. The stability and corresponding critical loads of the circular plate subjected to follower force and thermal load with simply supported edge, clamped edge and free edge are discussed. The results provide theoretical basis for improving the dynamic stability of the circular plate.     

Dynamic analysis of ring-stiffened circular cylindrical shells

A numerical method is developed for the dynamic analysis of ring-stiffened circular cylindrical thin elastic shells. Only circular symmetric vibrations of the shell segments and radial and torsional vibrations of the rings are considered. The geometric and material properties of the shell segments and the rings may vary from segment to segment. Free vibrations or forced vibrations due to harmonic pressure loading are treated with the aid of dynamic stiffness influence coefficients for shell segments and rings. Forced vibrations due to transient pressure loading are treated with the aid of dynamic stiffness influence coefficients for shell segments and rings defined in the Laplace transform domain. The time domain response is then obtained by a numerical inversion of the transformed solution. The effect of external viscous or internal viscoelastic damping is also investigated by the proposed method. In all the cases, the dynamic problem is reduced to a static-like form and the “exact” solution of the problem is numerically obtained.     

Identification and updating of loading in frameworks using dynamic measurements

This paper is concerned with the effect of structural loading on dynamic performance. This topic is recognised as being of importance when validating finite element (FE) models with experimental data. A strategy for including axial load effects in a model updating procedure is developed. The method can be used to identify loading in structural frameworks using measured dynamic data.The effectiveness of the new method is demonstrated by means of case studies involving both simulated and experimental data. The theoretical study allows aspects of the sensitivity of the method to realistic levels of experimental noise to be studied as well as the way in which dynamic load identification can be enhanced with static measurements. The experimental case study proves the practical success of the technique. Updated axial load parameters are compared with static measurements of the same quantities.     

Axially symmetric stability of a completely free circular plate subjected to a non-conservative edge load

This paper presents a study on the behaviour of the vibration and stability of a two dimensional structure: i.e., a completely free circular plate subjected to non-conservative radial loading. The eigencurves and mode shapes of the circular plate are presented for various values of the non-conservativeness parameter. Some interesting conclusions concerning the behaviour of a completely free plate are drawn from the analytical investigation of the solution of the problem and from the numerical calculations as well.     

Fracture of metallic rings during magnetic-pulse shock loading

Metallic rings made of aluminum and copper foils are studied after the action of a distributed radial magnetic-pulse load. Two loading approach modifications allowed us to substantially decrease the period of an applied sinusoidal load and to determine the time from load application to sample failure. A method is proposed to estimate the radial force on a metallic ring from coil turns. The profiles of radial pressure on the inner ring surface are measured, and the circumferential tensile stresses in ring fracture are determined. Microstructural studies of failed ring samples show that they underwent dynamic recrystallization. It is found that, as the operating load period shortens, the fraction of the ductile component in a fracture surface decreases and the samples undergo more brittle fracture.     

On using the dynamic snap-through motion of MEMS initially curved microbeams for filtering applications

Numerical and experimental investigations of the dynamics of micromachined shallow arches (initially curved microbeams) and the possibility of using their dynamic snap-through motion for filtering purposes are presented. The considered MEMS arches are actuated by a DC electrostatic load along with an AC harmonic load. Their dynamics is examined numerically using a Galerkin-based reduced-order model when excited near both their first and third natural frequencies. Several simulation results are presented demonstrating interesting jumps and dynamic snap-through behavior of the MEMS arches and their attractive features for uses as band-pass filters, such as their sharp roll-off from pass-bands to stop-bands and their flat response. Experimental work is conducted to test arches realized of curved polysilicon microbeams when excited by DC and AC loads. Experimental data of the micromachined curved beams are shown for the softening and hardening behavior near the first and third natural frequencies, respectively, as well as dynamic snap-through motion.     

TRANSIENT DYNAMIC RESPONSE ANALYSIS OF ORTHOTROPIC CIRCULAR CYLINDRICAL SHELL UNDER EXTERNAL HYDROSTATIC PRESSURE

Following Flügge's exact derivation for the buckling of cylindrical shells, the equations of motion for transient dynamic loading of orthotropic circular cylindrical shells under external hydrostatic pressure have been formulated. The normal mode theory is used to provide transient dynamic response for the equations of motion. The effect of shell's parameters, external hydrostatic pressure and material properties on the shell response has been studied in detail. A part of tables and figures are given in this paper.     

Numerical study of friction-induced instability and acoustic radiation – Effect of ramp loading on the squeal propensity for a simplified brake model

This paper presents a numerical study of the influence of loading conditions on the vibrational and acoustic responses of a disc brake system subjected to squeal. A simplified model composed of a circular disc and a pad is proposed. Nonlinear effects of contact and friction over the frictional interface are modelled with a cubic law and a classical Coulomb?s law with a constant friction coefficient. The stability analysis of this system shows the presence of two instabilities with one and two unstable modes that lead to friction-induced nonlinear vibrations and squeal noise. Nonlinear time analysis by temporal integration is conducted for two cases of loadings and initial conditions: a static load near the associated sliding equilibrium and a slow and a fast ramp loading. The analysis of the time responses shows that a sufficiently fast ramp loading can destabilize a stable configuration and generate nonlinear vibrations. Moreover, the fast ramp loading applied for the two unstable cases generates higher amplitudes of velocity than for the static load cases. The frequency analysis shows that the fast ramp loading generates a more complex spectrum than for the static load with the appearance of new resonance peaks. The acoustic responses for these cases are estimated by applying the multi-frequency acoustic calculation method based on the Fourier series decomposition of the velocity and the Boundary Element Method. Squeal noise emissions for the fast ramp loading present lower or higher levels than for the static load due to the different amplitudes of velocities. Moreover, the directivity is more complex for the fast ramp loading due to the appearance of new harmonic components in the velocity spectrum. Finally, the sound pressure convergence study shows that only the first harmonic components are sufficient to well describe the acoustic response.     

Postbuckling and vibration of end-supported elastica pipes conveying fluid and columns under follower loads

Fluid-conveying pipes with supported ends buckle when the fluid velocity reaches a critical value. For higher velocities, the postbuckled equilibrium shape can be directly related to that for a column under a follower end load. However, the corresponding vibration frequencies are different due to the Coriolis force associated with the fluid flow. Clamped–clamped, pinned–pinned, and clamped–pinned pipes are considered first. Axial sliding is permitted at the downstream end. The pipe is modeled as an inextensible elastica. The equilibrium shape may have large displacements, and small motions about that shape are analyzed. The behavior is conservative in the prebuckling range and nonconservative in the postbuckling range (during which the Coriolis force does work and the motions decay). Next, related columns are studied, first with a concentrated follower load at the axially sliding end, and then with a distributed follower load. In all cases, a shooting method is used to solve the nonlinear boundary-value problem for the equilibrium configuration, and to solve the linear boundary-value problem for the first four vibration frequencies. The results for the three different types of loading are compared.     

The influence of the behaviour of the load on the frequencies and critical loads of arches with flexibly supported ends

In this work, exact and approximate formulae for determining the lowest natural frequencies and critical loads for arches with flexibly supported ends have been obtained in three cases of load behaviour during the deformation process. The influence of the manner of loading on the frequencies and on the critical load has been shown as a function of the opening angle of the arch and the rigidity of the end supports.     

Analytical and numerical results for vibration analysis of multi-stepped and multi-damaged circular arches

This paper investigates the in-plane linear dynamic behaviour of multi-stepped and multi-damaged circular arches under different boundary conditions. Cracked cross-sections are modelled as massless elastic rotational hinges. In damaged configuration, cracks can be located both at the interface between two adjacent portions as well as inside the portion itself. For each arch portion bounded by two cracks, the differential equations of motion have been established considering axial extension, transverse shear effects and rotatory inertia. The equilibrium equations of arch portions are combined in the coupled fundamental system in terms of radial displacement, tangential displacement and rotation. Analytical and numerical solutions for multi-stepped arches, in undamaged as well as in damaged configurations, are proposed. The analytical solution is based on the Euler characteristic exponent procedure involving the roots of characteristic polynomials, while the numerical method is focused on the Generalized Differential Quadrature (GDQ) method and the Generalized Differential Quadrature Element (GDQE) technique. Numerical results are shown in terms of the first 10 analytical and numerical frequencies of multi-stepped and multi-damaged arches with different boundary conditions. Finally, convergence and stability characteristics of the GDQE procedure are investigated. The convergence rate of the natural frequencies is shown to be very fast and the stability of the numerical procedure is very good.     

Influence of a viscoelastic foundation on the stability of Beck's column: An exact analysis

The stability of Beck's column supported by three different viscoelastic foundations, viz., the standard linear solid, the Maxwell and the Kelvin-Voigt, is investigated. Closed form stability criteria are obtained for the entire range of system parameters through an exact dynamic analysis for each foundation model. The results for the Kelvin-Voigt model show that for a given stiffness parameter of the foundation the critical load increases with an increase in damping and reaches a limiting value for large damping. Unlike the case of conservative loading, the Maxwell foundation is shown to have a positive influence on the stability of this non-conservative problem. Furthermore, for this model, an optimum combination of foundation parameters exists to yield the maximum flutter load. The standard linear solid foundation combines the characteristics of Maxwell and Kelvin-Voigt models, as expected.     

Vibration and stability of a free circular plate subjected to non-conservative loading

This paper is concerned with the stability and vibration of a completely free circular plate subjected to a non-conservative edge loading. The eigencurves and mode shapes of the plate are obtained for various values of the non-conservativeness parameter. Numerical results are presented for the asymmetrical mode shapes of the plate. Interesting conclusions are drawn from these results some of which are verified analytically.     

圆薄板大撓度问题

Equations for the large deflection of thin plate established by Th. von Karman has been well known for many years. But so far there have been only a few iproblems studied with numerical certainty. S. Way was the first to apply these equations to solve the problem of a clamped plate under uniform pressure by the method of power series. After this, S. Levy found the solution of the simply supported rectangular plate under uniform load by the method of double trigonometric series. Both methods are too labourious to be applicable to other more important cases. Lately, Chien Wei-zang treated Way's problem again by means of the perturbation method and obtained excellent results. By the method as given by Chien Wei-zang, Yeh Kai-yuan worked out the problem of circular plate with a central hole under central concentrated load.In this paper, more results are given for various circular plates under various edge conditions. These include uniformly loaded circular plate under various edge conditions (section 2) and central concentrated loaded circular plate under various edge conditions (section 3). Such edge conditions are: (1) simply supported, (2) simply hinged, (3) rigidly clamped, (4) clamped but free to slip, (5) edge clamped but with possible slipping in horizontal direction, (6) edge simply supported but elastically fastened, and (7) edge clamped in elastic wall.All these results are presented in such a form that direct application in design problem is possible. In particular cases, under edge conditions (1) to (4), as σ=0.3, design formulae and curves for central deflection, radial tensile stress and radial bending stress are presented.     

热力联合作用弹性薄圆板的弯曲与屈曲

 导出了热力联合作用下弹性薄圆板的弯曲动力响应控制方程,讨论了其弯曲变形特点及影响失效的因素。分析表明在短时热能沉积作用下,热屈曲是弹性薄板失效的主要方式之一;反鼓包或反冲塞是热屈曲的后继行为;增加外载和热能沉积功率水平都将加速热屈曲的发生;材料的温度相关性与热能沉积的时空分布对薄板的力学行为都有重要影响,同为产生和影响热剪切失效(反冲塞)的重要因素。     

Stability of an axially accelerating string subjected to frictional guiding forces

The dynamic response of an axially translating continuum subjected to the combined effects of a pair of spring supported frictional guides and axial acceleration is investigated; such systems are both non-conservative and gyroscopic. The continuum is modeled as a tensioned string translating between two rigid supports with a time-dependent velocity profile. The equations of motion are derived with the extended Hamilton's principle and discretized in the space domain with the finite element method. The stability of the system is analyzed with the Floquet theory for cases where the transport velocity is a periodic function of time. Direct time integration using an adaptive step Runge-Kutta algorithm is used to verify the results of the Floquet theory. This approach can also be employed in the general case of arbitrary time-varying velocity. Results are given in the form of time history diagrams and instability point grids for different sets of parameters such as the location of the stationary load, the stiffness of the elastic support, and the values of initial tension. This work showed that presence of friction adversely affects stability, but using non-zero spring stiffness on the guiding force has a stabilizing effect. This work also showed that the use of the finite element method and Floquet theory is an effective combination to analyze stability in gyroscopic systems with stationary friction loads.