scholarly journals Modeling and Dynamical Behavior of Rotating Composite Shafts with SMA Wires

2014 ◽  
Vol 2014 ◽  
pp. 1-16 ◽  
Author(s):  
Yongsheng Ren ◽  
Qiyi Dai ◽  
Ruijun An ◽  
Youfeng Zhu
Keyword(s):  
Differential Equations ◽  
Natural Frequencies ◽  
Equations Of Motion ◽  
Dynamical Behavior ◽  
Circular Cross Section ◽  
Longitudinal Axis ◽  
Initial Strain ◽  
Marginal Effect ◽  
Rotating Shaft ◽  
Speed Increase

A dynamical model is developed for the rotating composite shaft with shape-memory alloy (SMA) wires embedded in. The rotating shaft is represented as a thin-walled composite of circular cross-section with SMA wires embedded parallel to shaft’s longitudinal axis. A thermomechanical constitutive equation of SMA proposed by Brinson is employed and the recovery stress of the constrained SMA wires is derived. The equations of motion are derived based on the variational-asymptotical method (VAM) and Hamilton’s principle. The partial differential equations of motion are reduced to the ordinary differential equations of motion by using the Galerkin method. The model incorporates the transverse shear, rotary inertia, and anisotropy of composite material. Numerical results of natural frequencies and critical speeds are obtained. It is shown that the natural frequencies of the nonrotating shaft and the critical rotating speed increase as SMA wire fraction and initial strain increase and the increase in natural frequencies becomes more significant as SMA wire fraction increases. The initial strain of SMA wires appears to have marginal effect on dynamical behaviors of the shaft. The actuation performance of SMA wires is found to be closely related to the ply-angle.

Effect of Slenderness Ratio on the Natural Frequencies of Pre-Twisted Cantilever Beams of Uniform Rectangular Cross-Section

1971 ◽  
Vol 13 (1) ◽  
pp. 51-59 ◽  
Author(s):  
B. Dawson ◽  
N. G. Ghosh ◽  
W. Carnegie
Keyword(s):  
Differential Equations ◽  
Shear Deformation ◽  
Cross Section ◽  
Natural Frequencies ◽  
Rectangular Cross Section ◽  
Slenderness Ratio ◽  
Twist Angle ◽  
Equations Of Motion ◽  
Rotary Inertia ◽  
Cantilever Beams

This paper is concerned with the vibrational characteristics of pre-twisted cantilever beams of uniform rectangular cross-section allowing for shear deformation and rotary inertia. A method of solution of the differential equations of motion allowing for shear deformation and rotary inertia is presented which is an extension of the method introduced by Dawson (1)§ for the solution of the differential equations of motion of pre-twisted beams neglecting shear and rotary inertia effects. The natural frequencies for the first five modes of vibration are obtained for beams of various breadth to depth ratios and lengths ranging from 3 to 20 in and pre-twist angle in the range 0–90°. The results are compared with those obtained by an alternative method (2), where available, and also to experimental results.

Free Vibration Analysis of an Inflated Toroidal Shell

2002 ◽  
Vol 124 (3) ◽  
pp. 387-396 ◽  
Author(s):  
Akhilesh K. Jha ◽  
Daniel J. Inman ◽  
Raymond H. Plaut
Keyword(s):  
Differential Equations ◽  
Free Vibration ◽  
Vibration Analysis ◽  
Natural Frequencies ◽  
Circular Cross Section ◽  
Free Vibration Analysis ◽  
Longitudinal Direction ◽  
Variable Coefficients ◽  
Mode Shapes ◽  
Toroidal Shell

Free vibration analysis of a free inflated torus of circular cross-section is presented. The shell theory of Sanders, including the effect of pressure, is used in formulating the governing equations. These partial differential equations are reduced to ordinary differential equations with variable coefficients using complete waves in the form of trigonometric functions in the longitudinal direction. The assumed mode shapes are divided into symmetric and antisymmetric groups, each given by a Fourier series in the meridional coordinate. The solutions (natural frequencies and mode shapes) are obtained using Galerkin’s method and verified with published results. The natural frequencies are also obtained for a circular cylinder with shear diaphragm boundary condition as a special case of the toroidal shell. Finally, the effects of aspect ratio, pressure, and thickness on the natural frequencies of the inflated torus are studied.

Mechanical Behavior of an Electrostatically Actuated Microplate

2003 ◽  
Author(s):  
Xiaopeng Zhao ◽  
Eihab M. Abdel-Rahman ◽  
Ali H. Nayfeh
Keyword(s):  
Differential Equations ◽  
Natural Frequencies ◽  
Finite Dimension ◽  
Equations Of Motion ◽  
Galerkin Approximation ◽  
Mode Shapes ◽  
Design Parameters ◽  
Electrostatically Actuated ◽  
Linear Eigenvalue Problem ◽  
Electrically Actuated

We present a nonlinear model of electrically actuated microplates. The model accounts for the nonlinearity in the electric forcing as well as mid-plane stretching of the plate. We use a Galerkin approximation to reduce the partial-differential equations of motion to a finite-dimension system of nonlinearly coupled second-order ordinary-differential equations. We find the deflection of the microplate under DC voltage and study the pull-in phenomenon. The natural frequencies and mode shapes are then obtained around the deflected position of the microplate by solving the linear eigenvalue problem. The effect of various design parameters on both the static response and the dynamic characteristics are studied.

scholarly journals Dynamic Analysis of a Tapered Composite Thin-Walled Rotating Shaft Using the Generalized Differential Quadrature Method

2020 ◽  
Vol 2020 ◽  
pp. 1-14
Author(s):  
Weiyan Zhong ◽  
Feng Gao ◽  
Yongsheng Ren ◽  
Xiaoxiao Wu ◽  
Hongcan Ma
Keyword(s):  
Differential Equations ◽  
Dynamic Model ◽  
Differential Quadrature Method ◽  
Equations Of Motion ◽  
Differential Quadrature ◽  
Quadrature Method ◽  
Rotating Shaft ◽  
Generalized Differential Quadrature Method ◽  
Thin Walled ◽  
Generalized Differential

A dynamic model of a tapered composite thin-walled rotating shaft is presented. In this model, the transverse shear deformation, rotary inertia, and gyroscopic effects have been incorporated. The equations of motion are derived based on a refined variational asymptotic method (VAM) and Hamilton’s principle. The partial differential equations of motion are reduced to the ordinary differential equations of motion by using the generalized differential quadrature method (GDQM). The validity of the dynamic model is proved by comparing the numerical results with those obtained in the literature and by using ANSYS. The effects of taper ratio, boundary conditions, ply angle, length to mean radius ratios, and mean radius to thickness ratios on the natural frequencies and critical rotating speeds are investigated.

Dynamic Behavior of Ground-Effect Machines in Motion Over Waves

1963 ◽  
Vol 7 (02) ◽  
pp. 1-10
Author(s):  
J. D. Lin
Keyword(s):  
Differential Equations ◽  
Dynamic Response ◽  
Mass Flow ◽  
Natural Frequencies ◽  
Equations Of Motion ◽  
Ground Effect ◽  
Small Disturbance ◽  
Third Order ◽  
Dynamic Motion ◽  
Theoretical Predictions

Dynamic motion in heaving and pitching of a ground-effect machine traveling over a train of sinusoidal waves is formulated on the basis of the small-disturbance theory. The equations of motion are derived as two third-order ordinary differential equations with coefficients which depend on the characteristics of the machine and the mass flow into or out of the cavity under the machine. The coefficienis are linearized at the equilibrium height; then, the solutions are obtained for the natural frequencies, the damping, and the steady periodic responses to motion over waves. It is found that a ground-effect machine with peripheral and central jet curtains is generally stable for heaving and pitching, but the damping is rather small. The effect of coupling due to the discontinuity of mass-flow coefficients is also shown to be very weak and thus may be neglected in this linearized analysis. A program, prepared for the IBM 1 620 digital computers at Hydronautics, Incorporated, allows study of the dynamic stability of a machine through variation of pertinent parameters, and also investigations of its dynamic response to various wave-encounter conditions. Theoretical predictions for practical machines, which have in model tests experienced dynamic motions over waves, are presented here and compared with experimental results. Satisfactory agreements are indicated in the natural frequency and response both for heaving and pitching.

Modal Analysis of a Rotating Shaft-Disk-Blades System With a Cracked Blade

1997 ◽  
Author(s):  
Ming-Chuan Wu ◽  
Shyh-Chin Huang
Keyword(s):  
Natural Frequencies ◽  
Equations Of Motion ◽  
Rotating Shaft ◽  
Coupled Modes ◽  
Energy Principle ◽  
Assumed Mode Method ◽  
Discrete Equations ◽  
Mode Method ◽  
Released Energy ◽  
Assumed Mode

Abstract The dynamic behavior of a rotating shaft-disk-blades system containing a cracked blade is investigated. With the crack released energy, the flexibility due to crack is evaluated. An energy principle in conjunction with the assumed-mode method is applied to yield the discrete equations of motion. Numerical examples are given for cases with between two and five symmetrically arrayed blades. The results show that there exist both torsion-bending coupled modes and blade-coupling modes, which occur at repeated frequencies. When there is a cracked blade, the frequencies of torsion-bending coupled modes decrease due to the crack, and blade-coupling modes have the phenomena of frequency bifurcation. Finally, the effects of shaft speed on the natural frequencies are illustrated.

scholarly journals Nonlinear Dynamical Behavior of Axially Accelerating Beams: Three-Dimensional Analysis

2016 ◽  
Vol 11 (1) ◽  
Author(s):  
Mergen H. Ghayesh ◽  
Hamed Farokhi
Keyword(s):  
Differential Equations ◽  
Fourier Transforms ◽  
Equations Of Motion ◽  
Nonlinear Partial Differential Equations ◽  
Dynamical Behavior ◽  
Three Dimensional ◽  
Global Dynamics ◽  
Nonlinear Dynamical ◽  
Time Histories ◽  
Lateral Displacements

The three-dimensional (3D) nonlinear dynamics of an axially accelerating beam is examined numerically taking into account all of the longitudinal, transverse, and lateral displacements and inertia. Hamilton’s principle is employed in order to derive the nonlinear partial differential equations governing the longitudinal, transverse, and lateral motions. These equations are transformed into a set of nonlinear ordinary differential equations by means of the Galerkin discretization technique. The nonlinear global dynamics of the system is then examined by time-integrating the discretized equations of motion. The results are presented in the form of bifurcation diagrams of Poincaré maps, time histories, phase-plane portraits, Poincaré sections, and fast Fourier transforms (FFTs).

Elastodynamic Modeling and Analysis for an Exechon Parallel Kinematic Machine

2015 ◽  
Vol 138 (3) ◽  
Author(s):  
Jun Zhang ◽  
Yan Q. Zhao ◽  
Yan Jin
Keyword(s):  
Differential Equations ◽  
Dynamic Characteristics ◽  
Natural Frequencies ◽  
Equations Of Motion ◽  
Structural Features ◽  
Parallel Kinematic Machine ◽  
Dynamic Behaviors ◽  
Elastodynamic Modeling ◽  
Parallel Kinematic ◽  
Moving Platform

As a newly invented parallel kinematic machine (PKM), Exechon has attracted intensive attention from both academic and industrial fields due to its conceptual high performance. Nevertheless, the dynamic behaviors of Exechon PKM have not been thoroughly investigated because of its structural and kinematic complexities. To identify the dynamic characteristics of Exechon PKM, an elastodynamic model is proposed with the substructure synthesis technique in this paper. The Exechon PKM is divided into a moving platform subsystem, a fixed base subsystem and three limb subsystems according to its structural features. Differential equations of motion for the limb subsystem are derived through finite element (FE) formulations by modeling the complex limb structure as a spatial beam with corresponding geometric cross sections. Meanwhile, revolute, universal, and spherical joints are simplified into virtual lumped springs associated with equivalent stiffnesses and mass at their geometric centers. Differential equations of motion for the moving platform are derived with Newton's second law after treating the platform as a rigid body due to its comparatively high rigidity. After introducing the deformation compatibility conditions between the platform and the limbs, governing differential equations of motion for Exechon PKM are derived. The solution to characteristic equations leads to natural frequencies and corresponding modal shapes of the PKM at any typical configuration. In order to predict the dynamic behaviors in a quick manner, an algorithm is proposed to numerically compute the distributions of natural frequencies throughout the workspace. Simulation results reveal that the lower natural frequencies are strongly position-dependent and distributed axial-symmetrically due to the structure symmetry of the limbs. At the last stage, a parametric analysis is carried out to identify the effects of structural, dimensional, and stiffness parameters on the system's dynamic characteristics with the purpose of providing useful information for optimal design and performance improvement of the Exechon PKM. The elastodynamic modeling methodology and dynamic analysis procedure can be well extended to other overconstrained PKMs with minor modifications.

Nonlinear Equations of Motion for the Maneuvering Flexible Aircraft Wings

2006 ◽  
Author(s):  
S. Ahmad Fazelzadeh ◽  
Abbas Mazidi
Keyword(s):  
Differential Equations ◽  
Natural Frequencies ◽  
Equations Of Motion ◽  
Body Motion ◽  
Dynamical Equations ◽  
Flexible Wings ◽  
Aircraft Wings ◽  
Angular Velocities ◽  
Nonlinear Equations Of Motion ◽  
Structural Motion

In this paper, the complete dynamical equations for the general maneuvering flexible wings with sweep and dihedral angles are formulated. These equations are valid for an isotropic non-uniform wing; include transverse shear and warping effects. The equations of motion and boundary conditions are derived using Hamilton’s variational principle. Interaction between rigid-body motion caused by the angular velocities of the general maneuver, and elastic deformations of the wing, results in nonlinear terms, form an important contribution to the final equations. For model validation, the simplified partial differential equations of pull-up maneuver are transformed into a set of differential equations through a Galerkin approach and finally the results of numerical simulation are presented. The combination of flexible structural motion and maneuver parameters are very effective on natural frequencies and instability boundaries.

scholarly journals Influence of rotatory inertia on stochastic stability of a viscoelastic rotating shaft

2008 ◽  
Vol 35 (4) ◽  
pp. 363-379
Author(s):  
Ratko Pavlovic ◽  
P. Kozic ◽  
G. Janevski
Keyword(s):  
Cross Section ◽  
Stochastic Stability ◽  
Circular Cross Section ◽  
Longitudinal Axis ◽  
Physical Parameters ◽  
Rotatory Inertia ◽  
Rotating Shaft ◽  
Axial Forces ◽  
Constant Rate ◽  
Mean And Variance

The stochastic stability problem of a viscoelastic Voigt-Kelvin balanced rotating shaft subjected to action of axial forces at the ends is studied. The shaft is of circular cross-section, it rotates at a constant rate about its longitudinal axis of symmetry. The effect of rotatory inertia of the shaft cross-section and external viscous damping are included into account. The force consists of a constant part and a time-dependent stochastic function. Closed form analytical solutions are obtained for simply supported boundary conditions. By using the direct Liapunov method almost sure asymptotic stability conditions are obtained as the function of stochastic process variance, external damping coefficient, retardation time, angular velocity, and geometric and physical parameters of the shaft. Numerical calculations are performed for the Gaussian process with a zero mean and variance ?2 as well as for harmonic process with amplitude H.

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