Axial Inertia Research Articles

Critical states of thin-wall conical gyroscopes elastically attached to a rotating platform

The analysis of effects of emergence of critical states of rigid and elastic thin-wall conical shell frustums hinged or elastically attached by their smaller edges to a rotating platform is performed on the basis of hybrid differential equations. The cases of simple and compound rotations of the carrying platform are considered. It is established that in the case of simple rotation the peculiarities of the critical state onset for the rigid and elastic shells are determined by relationships between their axial inertia moments, whereas in compound rotation the first frequencies of precession resonant vibrations of elastic shells do not depend on elastic pliability of the ties between the shell and the platform.

Analytical Solution for a Multi-layer Thick Cylindrical Shell Subjected to Axial Inertia Applicable for Slump Estimations of Solid Propellant Rocket Motor Grains

Analytical Solution for a Multi-layer Thick Cylindrical Shell Subjected to Axial Inertia Applicable for Slump Estimations of Solid Propellant Rocket Motor Grains

112 端部質量を有するはりの非線形連成振動における軸方向慣性を考慮した解析

112 端部質量を有するはりの非線形連成振動における軸方向慣性を考慮した解析

Free and Forced Wave Vibration Analysis of Axially Loaded Materially Coupled Composite Timoshenko Beam Structures

In this paper, wave vibration analysis of axially loaded bending-torsion coupled composite beam structures is presented. It includes the effects of axial force, shear deformation, and rotary inertia; namely, it is for an axially loaded composite Timoshenko beam. The study also includes the material coupling between the bending and torsional modes of deformations that is usually present in laminated composite beam due to ply orientation. From a wave standpoint, vibrations propagate, reflect, and transmit in a structure. The transmission and reflection matrices for various discontinuities on an axially loaded materially coupled composite Timoshenko beam are derived. Such discontinuities include general point supports, boundaries, and changes in section. The matrix relations between the injected waves and externally applied forces and moments are also derived. These matrices can be combined to provide a concise and systematic approach to vibration analysis of axially loaded materially coupled composite Timoshenko beams or complex structures consisting of such beam components. The systematic approach is illustrated through numerical examples for which comparative results are available in the literature.

C03 1リンクフレキシブルマニピュレータの残留振動抑制のためのモデリングと軌道生成法(OS13-1 一般-1)

This paper presents the optimal trajectory planning for a one-link flexible manipulator to suppress the residual vibration. In the trajectory planning, we express the joint angle by cubic spline function, and then use the tabu search method to find an optimal trajectory. The optimal trajectory thus obtained has minimum vibration requirement. The manipulator is modeled by considering axial inertia based on inextensibility condition. The Lagrangian approach in conjunction with the assumed modes method is applied to obtain the equation of motion of the manipulator that is used in the trajectory planning. The effectiveness of the proposed method is verified by experiments.

632 端部質量ばねを有するはりの非線形曲げ振動における軸方向慣性力の影響

In this research, effects of axial inertia force on nonlinear bending vibrations of a beam with a tip mass spring are analyzed. Although the added mass at the end of the beam decreases the natural frequency of the longitudinal vibration, vibration of the beams have been analyzed disregarding the inertia force in the axial direction. First, analytical procedure of the vibration of the beam is derived including both effects of the axial inertia force and the geometrical nonlinearity of the beam. Next, results of natural frequency of the beam are presented and effects of the axial inertia force are inspected. Finally, frequency response curves of the beam are obtained by the harmonic balance method and influence of the axial inertia force is examined.

On the non-linear vibrations of an inextensible rotating arm with setting angle and flexible hub

This work is concerned with the development of a dynamic model for a slender flexible arm undergoing relatively large planar flexural deformations, and clamped with a setting angle to a compliant rotating hub. The model is derived assuming an inextensible Euler–Bernoulli beam, but takes into account the axial inertia and nonlinear curvature. Two coordinate transformations were used to obtain the inertial position vector of a typical material point along the beam span. The inextensibility constraint is used to relate the axial and transverse displacements as well as velocities of the material point in the beam body coordinate system. The displacement and velocity vectors obtained are used in the system kinetic energy and exact curvature to eliminate dependence of the system Lagrangian on the beam axial deflection and velocity. The assumed mode method is used to discretize the system Lagrangian and to derive, directly, the nonlinear equivalent temporal solutions to the problem. The resulting dynamic model is a set of four strongly coupled nonlinear ordinary differential equations, which, in the present work, is solved only numerically. Examples of the results of the numerical solutions for the effects of setting angle and defined physical parameter ratios on the system dynamic response characteristics for a sinusoidal hub torque profile are presented and discussed.

Several phenomena in structural impact and structural crashworthiness

Abstract This article discusses the dynamic plastic instability of various basic structural members, subjected to large axial impact loads, which is relevant to the fields of structural impact and structural crashworthiness. In particular, studies on the dynamic axial response of idealised elastic–plastic models, rods, cylindrical shells and square tubes, is discussed in some detail. The analyses for some of the rods, shells and tubes retain the simultaneous influence of elastic and plastic stress waves (axial inertia) and the structural response (lateral inertia). The predictions reveal the profound influence of stress wave propagation phenomena which explain the characteristics of many of the experimental results obtained in laboratories over the years. These more complete analyses are vital for the higher velocity impact scenarios encountered increasingly in designs and for comparing properly the relative merits of different ductile materials in potential energy absorbing systems.

325 1リンクフレキシブルマニピュレータの最適軌道に関する一考察

This paper presents the optimal trajectory planning for a one-link flexible manipulator to suppress the residual vibration. In the trajectory planning, we express the joint angle by cubic spline function, and then use genetic algorithm to find an optimal trajectory. The optimal trajectory thus obtained has minimum vibration requirement. The manipulator is modeled by considering axial inertia, nonlinear curvature and inextensibility condition. The Lagrangian approach in conjunction with the assumed modes method is applied to obtain the equation of motion of the manipulator that is used in the trajectory planning. In the numerical calculation, we verify the effectiveness of the trajectory planning and investigate the effect of nonlinear curvature on the dynamics.

Dynamic analysis of a spatial coupled timoshenko rotating shaft with large displacements

The dynamic simulation is presented for an axial moving flexible rotating shafts, which have large rigid motions and small elastic deformation. The effects of the axial inertia, shear deformation, rotating inertia, gyroscopic moment, and dynamic unbalance are considered based on the Timoshenko rotating shaft theory. The equations of motion and boundary conditions are derived by Hamilton principle, and the solution is obtained by using the perturbation approach and assuming mode method. This study confirms that the influence of the axial rigid motion, shear deformation, slenderness ratio and rotating speed on the dynamic behavior of Timoshenko rotating shaft is evident, especially to a high-angular velocity rotor.

BIFURCATIONS AND CHAOS OF AN IMMERSED CANTILEVER BEAM IN A FLUID AND CARRYING AN INTERMEDIATE MASS

The concern of this work is the local stability and period-doubling bifurcations of the response to a transverse harmonic excitation of a slender cantilever beam partially immersed in a fluid and carrying an intermediate lumped mass. The unimodal form of the non-linear dynamic model describing the beam–mass in-plane large-amplitude flexural vibration, which accounts for axial inertia, non-linear curvature and inextensibility condition, developed in Al-Qaisia et al. (2000Shock and Vibration7 , 179–194), is analyzed and studied for the resonance responses of the first three modes of vibration, using two-term harmonic balance method. Then a consistent second order stability analysis of the associated linearized variational equation is carried out using approximate methods to predict the zones of symmetry breaking leading to period-doubling bifurcation and chaos on the resonance response curves. The results of the present work are verified for selected physical system parameters by numerical simulations using methods of the qualitative theory, and good agreement was obtained between the analytical and numerical results. Also, analytical prediction of the period-doubling bifurcation and chaos boundaries obtained using a period-doubling bifurcation criterion proposed in Al-Qaisia and Hamdan (2001 Journal of Sound and Vibration244, 453–479) are compared with those of computer simulations. In addition, results of the effect of fluid density, fluid depth, mass ratio, mass position and damping on the period-doubling bifurcation diagrams are studies and presented.

NON-LINEAR FREE VIBRATIONS OF A ROTATING FLEXIBLE ARM

The non-linear, moderately large amplitude flexural free vibrations of an arm clamped with a setting angle to a rigid rotating hub are studied. The shear deformation and rotary inertia effects are assumed to be negligible, but account is taken of axial inertia, non-linear curvature and the inextensibility condition. The Lagrangian approach in conjunction with the assumed modes method, assuming constant hub rotation speed, is used in a consistent manner to obtain the third order non-linear uni-modal temporal problem. Because of the strength of the non-linearities in the temporal problem, which includes elastic and inertial geometric stiffening as well as inertial softening terms, a time transformation method is employed to obtain an approximate solution to the frequency–amplitude relation of arm free oscillation. Results in non-dimensional form are presented graphically, for the effect of hub rotation speed, blade setting angle, and hub radius on the variation of the natural frequency with vibration amplitude.

Enhancing performances of SHPB for determination of flow curves

A compression test on a Split Hopkinson Pressure Bar (SHPB) is the most common testing method adopted to determine material flow curves in a quite wide range of high strain rates. However, in most ordinary SHPB applications material rheology can only be investigated up to a rather low strain. Furthermore, several phenomena, including friction at interfaces and radial and axial inertia, are potential sources of inaccuracy for flow stress data. The paper presents the main design-guidelines followed when developing a high-performance SHPB system for the determination of accurate flow curves. The results from hot compression tests carried out on Aluminium- and Nickel-alloys prove that with the developed system most of the limitations of ordinary-SHPBs are overcome and system performances significantly improve.

Nonlinear vibration of Timoshenko beam due to moving loads including the effects of weight and longitudinal inertia of beam

The effects of weight and axial inertia of a beam are taken into account for studying the nonlinear vibration of the Timoshenko beam due to external loads. The combination of Galerkins method and Runge-Kutta method are employed to obtain the dynamic responses of the beam. A concentrated force and a two-axle vehicle traversing on the beam are taken as two examples to investigate the response characteristics of the beam. Results show that the effect of axial inertia of the beam increases the fundamental period of the beam. Further, both the dynamic deflection and the dynamic moment of the beam obtained with including the effect of axial inertia of the beam are greater than those of the beam without including that effect of the beam.

The stability of rapidly deforming rods and jets

When a uniform rod is deformed in tension under conditions of pure axial stress it will undergo indefinite uniform deformation. In practice these conditions are not achievable because neither perfect uniformity nor perfect uniaxial stress can be attained. In slow to moderately fast tension, instability (necking) starts at relatively small strains and is governed by the boundary conditions (uniformity and end effects) and the deformation model. If the strain rate is very high, however, then strains of several thousand per cent can be achieved before instability develops as is observed in shaped charge jets. It is shown in the present paper that, at very high strain rates, very large stable strains can be achieved either through certain deformation models (this is uncommon) or, more generally, through the effects of inertia on stability. A quantitative theory is developed which shows the important role of both radial and axial inertia. These inertial effects are intimately linked to the deformation model in such a way that, for any perturbation or other effect to initiate instability, a complex set of conditions involving the axial and radial inertia and the deformation model must be satisfied. These define the geometrical, inertial and material characteristics required for large stable extensions.

On dynamic buckling of elastic–plastic beams

A theoretical study of dynamic buckling of elastic–plastic beams under compressive axial forces is presented. It is assumed that the loading process is so slow that axial inertia effects may be neglected. Four types of axial loading are considered. It is supposed that the beam's material has linear strain hardening. Equations of motion are derived from Hamilton's principle. A numerical method of integrating these equations is presented. The main aim of the paper is to shed light on the problem how the deflection shapes develop in the post-critical stage. For this purpose six examples are presented.

Non-uniform and dynamic torsion of elastic beams Part 1: Governing equations and particular solutions

The double torsion fracture test has been widely used in the past for measuring resistance to crack growth under static loading and, more recently, under high-loading-rate conditions. Specimen deformation in this test is conventionally analysed on the basis of a one-dimensional torsion equation. Under dynamic conditions, it is imperative to establish an accurate torsion equation which can be used to model the double torsion test. The present paper describes the development, in this context, of a dynamic fourth-order partial differential equation for one-dimensional torsion, based on Barr's equation. The equation obtained in this paper accounts for the axial inertia associated with cross-sectional warping and the axial stresses which arise when warping is constrained. The equation can also accommodate non-linear elasticity. The equation is validated, using Barr's own data, for the torsional resonance problem which he studied and, using finite-element analysis, for a problem of constrained static torsion analysed by Timoshenko.

VIBRATION OF MULTI-SPAN TIMOSHENKO FRAMES DUE TO MOVING LOADS

An analytical method is presented in this paper to study the vibration of multi-span Timoshenko frames. The combined effects of axial inertia, rotatory inertia and shear deformation of each branch of those frames are simultaneously considered. Any two distinct sets of the mode shape functions are shown to be orthogonal. The method of modal analysis is then adopted to investigate the forced vibration of the frames. A concentrated load and a uniformly distributed load moving on these frames are treated as two examples. Results show that as the span number gets larger, the first modal frequency gets smaller. Furthermore, the longer column implies a smaller first modal frequency. The absolute maximum deflection of the Timoshenko frame is larger than that of the Bernoulli–Euler frame.

Vibration of T-type Timoshenko frames subjected to moving loads

In this study, a theoretical method to analyze the vibration of a T-type Timoshenko frame is proposed. The effects of axial inertia, rotatory inertia and shear deformation of each branch are considered. The orthogonality of any two distinct sets of mode shape functions is also demonstrated. Vibration of the frame due to moving loads is studied by the method and the response characteristics of the frame are investigated. Furthermore, the effect of column length on the response of the frame is also studied.

SIMPLIFIED EQUATIONS OF MOTION FOR THE RADIAL–AXIAL VIBRATIONS OF FLUID FILLED PIPES

The equations of motion for straight fluid filled pipes are greatly simplified. It is found, for frequencies below a third of the ring frequency, that the radial–axial waves in cylinders are as if the circumferential motion were inextensional. This is the fundamental assumption for the analysis. The derivation is also based on the assumption of long axial wavelengths, resulting in the axial inertia of the fluid and the axial flexural stiffness of the pipe wall being negligible. The formulation is restricted to frequencies well below the cut-on of higher order fluid modes. For such frequencies, the compressibility of the fluid is neglected and the internal fluid loading, on the pipe, is approximated as an increase in the radial inertia. Upon this basis, the equations of motion, for each circumferential mode, are similar to those for a Timoshenko beam on a Winkler foundation. Numerical experiments are made, comparing the approximate theory with results from calculations from the Helmholtz equation for the fluid and accurate thin-walled cylinder theory. Criteria of the application of the simplified theory are formulated.