Periodic Axial Load Research Articles

Parametric instability of a cantilever beam with magnetic field and periodic axial load

The present work deals with the parametric instability regions of a cantilever beam with tip mass subjected to time-varying magnetic field and axial force. The nonlinear temporal differential equation of motion having two frequency parametric excitations is solved using second-order method of multiple scales. The closed-form expressions for the parametric instability regions for three different resonance conditions are determined. The influence of magnetic filed, axial load, damping constant and mass ratio on the parametric instability regions are investigated. These results obtained from perturbation analysis are verified by solving the temporal equation of motion using fourth-order Runge–Kutta method. The instability regions obtained using this method is found to be in good agreement with the experimental result.

Parametric instability regions of three-layered soft-cored sandwich beam using higher-order theory

The purpose of the present work is to study the parametric instability of a three-layered, soft-cored, symmetric sandwich beam subjected to a periodic axial load. Due to soft core, the displacements of the top and bottom skins are different and hence, instead of using the classical theory a higher-order theory is used. Using extended Hamilton's principle and taking beam theory for the skins and a two-dimensional theory for the core, the governing equations of motion and boundary conditions are derived. A generalized Galerkin's method is used to reduce the equations of motion to a set of non-dimensional coupled Mathieu–Hill's equations with complex coefficients. The parametric instability regions for simple and combination resonances are investigated for simply supported, clamped–guided, clamped–free riveted and clamped–free end conditions by modified Hsu's method. The influences of shear parameter; the core loss-factor and the ratio of core thickness to skin thickness on the zones of instability have been studied. This general analysis can be applied to sandwich beams with a flexible core and any type of construction. The results are compared with those reported for classical theories.

Non-linear analysis of dynamic stability for functionally graded cylindrical shells under periodic axial loading

Abstract This paper presents a simple solution of the dynamic stability of functionally graded shells under periodic axial loading based on large deflection theory. Material properties are assumed to be temperature-dependent, and graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of the constituents. The equations of motion are solved by Galerkin procedure. Bolotin’s method is then employed to obtain the steady-state vibrations for non-linear Mathieu equations. The effect of the volume fraction of the material constituents and their distribution on the parametric resonance, in particular steady-state vibrations amplitude and also the effect of the length-to-radius and thickness-to-radius ratios of the cylinder are examined and compared. A good agreement is obtained by comparing the present analysis with other available literature.

Dynamic stability of rotating cylindrical shells subjected to periodic axial loads

In this paper, the dynamic stability of rotating cylindrical shells under static and periodic axial forces is investigated using a combination of the Ritz method and Bolotin’s first approximation. The kernel particle estimate is employed in hybridized form with harmonic functions, to approximate the 2-D transverse displacement field. A system of Mathieu–Hill equations is obtained through the application of the Ritz energy minimization procedure. The principal instability regions are then obtained via Bolotin’s first approximation. In this formulation, both the hoop tension and Coriolis effects due to the rotation are accounted for. Various boundary conditions are considered, and the present results represent the first instance in which, the effects of boundary conditions for this class of problems, have been reported in open literature. Effects of rotational speeds on the instability regions for different modes are also examined in detail.

Dynamic instability and chaos of empty and fluid-filled circular cylindrical shells under periodic axial loads

In the present paper the dynamic stability of circular cylindrical shells subjected to static and dynamic axial loads is investigated. Both Donnell's nonlinear shallow shell and Sanders–Koiter shell theories have been applied to model finite-amplitude static and dynamic deformations. Results are compared in order to evaluate the accuracy of these theories in predicting instability onset and post-critical nonlinear response. The effect of a contained fluid on the stability and the post-critical behaviour is analyzed in detail. Geometric imperfections are considered and their influence on the dynamic instability and post-critical behaviour is investigated. Chaotic dynamics of pre-compressed shells is investigated by means of nonlinear time-series techniques, extracting correlation dimension and Lyapunov exponents.

Stability analysis of a cantilever composite beam on elastic supports

In this study, static and dynamic stabilities of a laminated composite cantilever beam having linear translation spring and torsional spring as elastic supports subjected to periodic axial loading are examined. The beam is assumed to be a Euler beam and modeled by using the finite element method. The model is considered to have symmetric and asymmetric lay-ups. The effective flexural modulus of the beam is used in the analysis. Using energy expressions in conjunction with Bolotin’s approach, the study is carried out employing various disturbing frequency ranges in which the beam is to be unstable. The effects of the variation of cross-section in one direction, the ratio of length to thickness, orientation angle, static and dynamic load parameters, stiffness of elastic supports having linear translation spring and torsional spring, and position of the elastic support on stability are examined. The results obtained are given in tables and graphics. In addition, the obtained results of the fundamental natural frequency and critical buckling load parameters are compared with the results of other investigators in existing literature.

Dynamic stability of spinning pre-twisted sandwich beams with a constrained damping layer subjected to periodic axial loads

The dynamic stability problems of spinning pre-twisted sandwich beams with a constrained damping layer subjected to periodic axial loads are studied by using the finite element method. The complex modulus representation is used for the viscoelastic material. The gyroscopic effects for small perturbations are linearized as a skew symmetric damping matrix. The effects of pre-twisted angles and spinning speeds on unstable regions are studied. The influences of the core thickness, shear parameter, core loss factor and stiffness of constraining layer on the unstable regions are also discussed.

Dynamic stability of rotating pre-twisted blades with a constrained damping layer

The stability problems of a rotating pre-twisted blade with a viscoelastic core constrained by a laminated face layer subjected to a periodic axial load are studied by using the finite element method. The complex modulus representation is used for the viscoelastic material. A set of equations of motion governing the bending and extensional displacements are derived by Hamilton’s principle. The regions of instability are determined by using Bolotin’s procedure that modified for the complex case. The effects of rotating speed, pre-twisted angle, setting angle and static axial load on the first static buckling load under different shear parameter and core loss factor are presented. The influences of core loss factor, core thickness ratio and stiffness parameter on the unstable regions are also studied. The shear parameter exists an optimal range in which the width of unstable region is narrowest.

Erratum to “Parametric resonance of a rotating cylindrical shell subjected to periodic axial loads” [Journal of Sound and Vibration 214 (1998) 513–529

Erratum to “Parametric resonance of a rotating cylindrical shell subjected to periodic axial loads” [Journal of Sound and Vibration 214 (1998) 513–529

Stability and vibration of empty and fluid-filled circular cylindrical shells under static and periodic axial loads

In the present study, the dynamic stability of simply supported, circular cylindrical shells subjected to dynamic axial loads is analysed. Geometric nonlinearities due to finite-amplitude shell motion are considered by using the Donnell’s nonlinear shallow-shell theory. The effect of structural damping is taken into account. A discretization method based on a series expansion involving a relatively large number of linear modes, including axisymmetric and asymmetric modes, and on the Galerkin procedure is developed. Axisymmetric modes are included; indeed, they are essential in simulating the inward deflection of the mean oscillation with respect to the equilibrium position and in describing the axisymmetric deflection due to axial loads. A finite length, simply supported shell is considered; the boundary conditions are satisfied, including the contribution of external axial loads acting at the shell edges. The effect of a contained liquid is investigated. The linear dynamic stability and nonlinear response are analysed by using continuation techniques and direct simulations.

Dynamic stability analysis of functionally graded cylindrical shells under periodic axial loading

Abstract In this paper, a formulation for the dynamic stability analysis of functionally graded shells under harmonic axial loading is presented. A profile for the volume fraction is assumed and a normal-mode expansion of the equations of motion yields a system of Mathieu–Hill equations the stability of which is analyzed by the Bolotin’s method. The present study examines the effects of the volume fraction of the material constituents and their distribution on the parametric response, in particular the positions and sizes of the instability regions.

Dynamic stability analysis of laminated composite cylindrical shells subjected to conservative periodic axial loads

Abstract The dynamic stability of thin, laminated cylindrical shells under combined static and periodic axial forces is studied here using Love's theory for thin shells. A system of Mathieu–Hill equations is obtained by a normal-mode expansion of the equations of motion, the stability of which is examined by Bolotin's method. The dynamic instability regions are investigated for different lamination schemes. The effects of the length-to-radius and thickness-to-radius ratios of the cylinder on the instability regions are also examined.

PARAMETRIC RESONANCE OF A ROTATING CYLINDRICAL SHELL SUBJECTED TO PERIODIC AXIAL LOADS

The parametric resonance of rotating cylindrical shells under periodic axial loading is investigated. The formulation is based on the dynamic version of Donnell's equation for thin rotating cylindrical shells. A modified assumed-mode method is used to reduce the partial differential equations of motion to a system of coupled second order differential equations with periodic coefficients of the Mathieu–Hill type. The instability regions are determined based on the principle of Bolotin's method. Of special interest here are the effects of the centrifugal and Coriolis forces on the instability regions which were examined in detail.

DYNAMIC STABILITY OF ROTATING COMPOSITE SHAFTS UNDER PERIODIC AXIAL COMPRESSIVE LOADS

This paper describes the dynamic behavior of a rotating composite shaft subjected to axial periodic forces using the finite element method. Laminated composite shafts are modelled using Timoshenko beams. The numerical results show good agreement with the reported beams models. Effects of static and time dependent components of axial loads on the stability of the composite shaft are studied. This paper also investigates the effect of the rotational speeds and the disk on the unstable regions of the shaft. The numerical results show that for the same geometric parameters, a steel shaft has a lower frequency than that of the composite shafts; however, the steel shaft is more stable than composite shafts because the shaft-disk system is subjected to axial periodic forces at lower rotational speeds. Also, the effect of the gyroscopic moments makes the steel shaft more sensitive to the periodic axial load than the composite one.

周期的軸圧縮負荷を受けるアングルプライ積層円筒殻の動的安定性

This paper deals with the problem of dynamic stability of angle-ply laminated cylindrical shells subjected to static and periodic axial compressive loading. First, the axially symmetric motion of the shell under loading is determined. Subsequently, certain perturbations are superimposed on this motion, and their behavior in time is investigated. The symmetric state of motion of the shell is called stable if the perturbations remain bounded. The solutions for the prebuckling motion and the perturbated motion are obtained by the use of Galerkin's method. Stability regions are examined by utilizing Mathieu's equation. The inevitability of dynamically unstable behavior is proved analytically and the effects of various factors, such as fundamental natural frequency, vibrated amplitude and dynamic unstable mode, are clarified.

PARAMETRIC INSTABILITY OF MULTI-LAYERED SANDWICH BEAMS

The present work deals with the parametric instability of multi-layered symmetric sandwich beams with alternate elastic and viscoelastic layers, subjected to periodic axial load. The equations of motion and boundary conditions are derived by applying Hamilton's principle, and the general Galerkin method is utilized to convert the equations of motion into a set of coupled Hill's equations with complex coefficients in the time domain. Numerical results are obtained for beams having, three, five and seven layers. The effects of the shear and core-thickness parameters as well as the core loss factor upon the regions of parametric instability are considered. Zones of instability are obtained for the cases in which beams with various number of layers have the same weight or size or flexural rigidity. Finally, for the criteria of constant size and flexural rigidity as well as constant weight and flexural rigidity, the effects of various parameters on the stability of the system are studied.

Parametric instability of a sandwich beam under various boundary conditions

Parametric instability of a three-layered symmetric sandwich beam subjected to a periodic axial load is considered under nine different boundary conditions. The influence of static load parameter and core-thickness parameter on the system loss factor is investigated. The effect of shear parameter on the static buckling loads is also considered, besides, the effects of shear parameter, core thickness parameter and core loss factor on the regions of parametric instability are studied.

Dynamic stability of a pre-twisted, three layered, symmetric sandwich beam

Abstract This paper deals with the parametric instability of a pre-twisted, cantilevered, three-layered symmetric sandwich beam subjected to a periodic axial load at the free end. The non-dimensional governing equations of motion and the associated boundary conditions are derived by Hamilton’s principle. These equations are reduced to the time domain by the use of the generalized Galerkin method. This gives rise to a set of coupled Hill’s equations with complex coefficients. The regions of instability are determined by using Hsu’s method, modified for the complex case. The static buckling loads are determined and the influences of the core loss factor and the static load parameter on the system loss factor are investigated. In addition, the effects of pre-twist angle and geometric, shear and static load parameters on the regions of parametric instability are studied.

Vibration analysis of elastic coupler in a planar mechanism

This paper presents an analysis of the transverse vibration and stability of an elastic beam simultaneously subject to a periodic axial load, a distributed transverse load, and time-dependent displacement excitations at both ends. The equation of motion derived from Bernoulli–Euler beam theory is a fourth-order partial differential equation with periodic coefficients. To obtain approximate solutions, the assumed-modes method is used. The unknown time-dependent function in the assumed-modes method is determined by a generalized inhomogeneous Hill’s equation. The instability regions possessed by this generalized Hill’s equation are obtained by both the perturbation technique up to the second order and the harmonic balance method. The dynamic response and the corresponding spectrum of the transversely oscillating elastic beam are calculated by the weighted-residual method.

周期的軸圧縮荷重を受ける積層円筒殻の動的応答

Advanced fiber-reinforced composite materials have been used for structural members, because of their high specific strength and stiffness. This paper deals with the problem of dynamic stability of angle-ply laminated cylindrical shells subjected to static and periodic axial compressive loads. First, the axially symmetric motion of the shell under the loading is determined. Subsequently, certain perturbations are superimposed on this motion, and their effect on the behavior of the shell is investigated. The symmetric state of motion of the shell is assumed to be stable if the perturbations remain to the bounds. The solutions for the prebuckling motion and the perturbated motion are obtained by the use of Galerkin's method. Stable regions are examined by utilizing Mathieu equation. The inevitability of dynamically unstable behaviors and the effects of various factors, such as compressive load ratio, lamination angle, fundamental natural frequency, driving amplitude of the vibration and dynamic unstable mode are analytically clarified.