Coupling of In-Plane Flexural, Tangential, and Shear Wave Modes of a Curved Beam

2011 ◽  
Vol 134 (1) ◽  
Author(s):  
B. Kang ◽  
C. H. Riedel
Keyword(s):  
Shear Wave ◽  
Equations Of Motion ◽  
High Order ◽  
Curved Beam ◽  
Beam Model ◽  
Elastic Coupling ◽  
Dynamic Coupling ◽  
Frequency Spectra ◽  
Coupling Terms ◽  
Wave Modes

In this paper, the coupling effects among three elastic wave modes, flexural, tangential, and radial shear, on the dynamics of a planar curved beam are assessed. Two sets of equations of motion governing the in-plane motion of a curved beam are derived, in a consistent manner, based on the theory of elasticity and Hamilton’s principle. The first set of equations retains all resulting linear coupling terms that includes both static and dynamic coupling among the three wave modes. In the derivation of the second set of equations, the effects of Coriolis acceleration and high-order elastic coupling terms are neglected, which leads to a set of equations without dynamic coupling terms between the tangential and shear wave modes. This second set of equations of motion is the one most commonly used in studies on thick curved beams that include the effects of centerline extensibility, rotary inertia, and shear deformation. The assessment is carried out by comparing the dynamic behavior predicted by each curved beam model in terms of the dispersion relations, frequency spectra, cutoff frequencies, natural frequencies and mode shapes, and frequency responses. In order to ensure the comparison is based on accurate results, the wave propagation technique is applied to obtain exact wave solutions. The results suggest that the contributions of the dynamic and high-order elastic coupling terms to the response of a thick curved beam are quite significant and that these coupling terms should not be neglected for an accurate analysis of a thick curved beam with a large curvature parameter.

Effect of an Original Gear Mesh Modeling on the Gearbox Dynamic Behavior

2007 ◽  
Author(s):  
Emmanuel Rigaud ◽  
Joe¨l Perret-Liaudet ◽  
Mohamed-Salah Mecibah
Keyword(s):  
Load Distribution ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Elastic Coupling ◽  
Gear Mesh ◽  
Meshing Stiffness ◽  
Face Width ◽  
Building Process ◽  
Coupling Terms

Prediction of the vibratory and acoustical behavior of gearboxes is generally based on characterization of the excitation sources, overall modeling of the gearbox, modal analysis and solving of the parametric equations of motion generated by these models. On the building process of such large degrees of freedom models, the elastic coupling induced by the gear mesh is generally described by the parametric meshing stiffness k(t) along the line of action. This kind of model is not able to take into account the load distribution along the tooth face width, even though the resulting moment can constrain rotational angles associated to wheels tilting and flexural deformation of shafts. The scope of this study is to introduce the coupling terms between wheels associated to these phenomena. Some examples show how they can influence the gear modal characteristics and dynamic response and, consequently, the vibratory and acoustical response of the gearbox.

Linear Dynamic Coupling in Geared Rotor Systems

1986 ◽  
Vol 108 (2) ◽  
pp. 171-176 ◽  
Author(s):  
J. W. David ◽  
L. D. Mitchell
Keyword(s):  
Gear Tooth ◽  
Equations Of Motion ◽  
Dynamic Coupling ◽  
Rotating Shaft ◽  
Rotor Systems ◽  
Linear Dynamic ◽  
Heavy Lift ◽  
Reaction Forces ◽  
Coupling Terms ◽  
Nonlinear Equations Of Motion

The ability to analyze accurately the torsional-axial-lateral coupled response of geared systems is the key to the prediction of dynamic gear forces, shaft moments and torques, dynamic reaction forces, and moments at all bearing points. These predictions can, in turn, be used to estimate gear-tooth lives, shaft lives, housing vibrational response, and noise generation. Typical applications would be the design and analysis of gear drives in heavy-lift helicopters, industrial speed reducers, Naval propulsion systems, and heavy, off-road equipment. In this paper, the importance of certain linear dynamic coupling terms on the predicted response of geared rotor systems is addressed. The coupling terms investigated are associated with those components of a geared system that can be modeled as rigid disks. First, the coupled, nonlinear equations of motion for a disk attached to a rotating shaft are presented. The conventional argument for ignoring these dynamic coupling terms is presented and the error in this argument is revealed. It is shown that in a geared system containing gears with more than about 50 teeth, the magnitude of some of the dynamic-coupling terms is potentially as large as the magnitude of the linear terms that are included in most rotor analyses. In addition, it is shown that the dynamic coupling terms produce the multi-frequency responses seen in geared systems. To quantitatively determine the effects of the linear dynamic-coupling terms on the predicted response of geared rotor systems, a trial problem is formulated in which these effects are included. The results of this trial problem shows that the inclusion of the linear dynamic-coupling terms changed the predicted response up to eight orders of magnitude, depending on the response frequency. In addition, these terms are shown to produce sideband responses greater than the unbalanced response of the system.

Proposed Solution Methodology for the Dynamically Coupled Nonlinear Geared Rotor Mechanics Equations

1985 ◽  
Vol 107 (1) ◽  
pp. 112-116 ◽  
Author(s):  
L. D. Mitchell ◽  
J. W. David
Keyword(s):  
Steady State ◽  
Numerical Simulations ◽  
Equations Of Motion ◽  
Linear Equations ◽  
Three Dimensional ◽  
Dynamic Coupling ◽  
State Response ◽  
Shaft System ◽  
Coupling Terms ◽  
Solution Methodology

The equations which describe the three-dimensional motion of an unbalanced rigid disk in a shaft system are nonlinear and contain dynamic-coupling terms. Traditionally, investigators have used an order analysis to justify ignoring the nonlinear terms in the equations of motion, producing a set of linear equations. This paper will show that, when gears are included in such a rotor system, the nonlinear dynamic-coupling terms are potentially as large as the linear terms. Because of this, one must attempt to solve the nonlinear rotor mechanics equations. A solution methodology is investigated to obtain approximate steady-state solutions to these equations. As an example of the use of the technique, a simpler set of equations is solved and the results compared to numerical simulations. These equations represent the forced, steady-state response of a spring-supported pendulum. These equations were chosen because they contain the type of nonlinear terms found in the dynamically-coupled nonlinear rotor equations. The numerical simulations indicate this method is reasonably accurate even when the nonlinearities are large.

Dynamic Analysis of Mechanical Systems With Elastic Telescopic Members

1994 ◽  
Author(s):  
B. O. Al-Bedoor ◽  
Y. A. Khulief
Keyword(s):  
Dynamic Model ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Elastic Beam ◽  
Vibrational Motion ◽  
Dynamic Coupling ◽  
Gravitational Effects ◽  
Coupling Terms ◽  
Tip Mass ◽  
Stiffening Effect

Abstract A dynamic model for the vibrational motion of an elastic beam-like telescopic member is presented. In addition to translation, the elastic member is allowed to execute large reference rotation. The Lagrangian approach in conjunction with the assumed modes technique are employed in deriving the equations of motion. The developed model accounts for all the dynamic coupling terms, as well as the stiffening effect due to the beam reference rotation. The tip mass dynamics is included together with the associated dynamic coupling between the modal degrees of freedom. In addition, the devised dynamic model takes into account the gravitational effects, thus permitting motions in either vertical or horizontal planes. Numerical simulation of a mechanical system with an elastic telescopic member is presented.

Validation of a Belt Model for Prediction of Hub Forces from a Rolling Tire4

2009 ◽  
Vol 37 (2) ◽  
pp. 62-102 ◽  
Author(s):  
C. Lecomte ◽  
W. R. Graham ◽  
D. J. O’Boy
Keyword(s):  
Internal Pressure ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Prediction Method ◽  
Analytical Models ◽  
Beam Model ◽  
Impedance Boundary ◽  
Bending Waves ◽  
Tire Rotation ◽  
Three Dimensional Models

Abstract An integrated model is under development which will be able to predict the interior noise due to the vibrations of a rolling tire structurally transmitted to the hub of a vehicle. Here, the tire belt model used as part of this prediction method is first briefly presented and discussed, and it is then compared to other models available in the literature. This component will be linked to the tread blocks through normal and tangential forces and to the sidewalls through impedance boundary conditions. The tire belt is modeled as an orthotropic cylindrical ring of negligible thickness with rotational effects, internal pressure, and prestresses included. The associated equations of motion are derived by a variational approach and are investigated for both unforced and forced motions. The model supports extensional and bending waves, which are believed to be the important features to correctly predict the hub forces in the midfrequency (50–500 Hz) range of interest. The predicted waves and forced responses of a benchmark structure are compared to the predictions of several alternative analytical models: two three dimensional models that can support multiple isotropic layers, one of these models include curvature and the other one is flat; a one-dimensional beam model which does not consider axial variations; and several shell models. Finally, the effects of internal pressure, prestress, curvature, and tire rotation on free waves are discussed.

Size Effects in Dynamics of Dipolar Planar Nanosystems

2004 ◽  
Vol 99-100 ◽  
pp. 223-226
Author(s):  
H. Puszkarski ◽  
J.-C.S. Lévy ◽  
M. Krawczyk
Keyword(s):  
Size Effects ◽  
Equations Of Motion ◽  
Sample Surface ◽  
Dipolar Field ◽  
Dipolar Interactions ◽  
Planar System ◽  
Frequency Spectra ◽  
Magnetostatic Waves ◽  
Dynamic Components ◽  
Field Static

The equations of motion are derived for a magnetic planar system with dipolar interactions taken into account. Magnetostatic waves propagating perpendicularly to the sample surface and dipolar field static and dynamic components are calculated for the case when saturating field is applied perpendicularly to the sample surface. The corresponding frequency spectra and mode profiles are computed numerically with emphasis laid on size effects. It is established that two lowest-frequency modes are surface-localized modes. These modes preserve their surface-localized character with growing sample dimensions.

scholarly journals Vibration Characteristics of Piezoelectric Microbeams Based on the Modified Couple Stress Theory

2014 ◽  
Vol 2014 ◽  
pp. 1-12 ◽  
Author(s):  
R. Ansari ◽  
M. A. Ashrafi ◽  
S. Hosseinzadeh
Keyword(s):  
Boundary Conditions ◽  
Couple Stress ◽  
Natural Frequencies ◽  
Couple Stress Theory ◽  
Equations Of Motion ◽  
Modified Couple Stress Theory ◽  
Beam Model ◽  
Governing Equations ◽  
Stress Theory ◽  
Modified Couple Stress

The vibration behavior of piezoelectric microbeams is studied on the basis of the modified couple stress theory. The governing equations of motion and boundary conditions for the Euler-Bernoulli and Timoshenko beam models are derived using Hamilton’s principle. By the exact solution of the governing equations, an expression for natural frequencies of microbeams with simply supported boundary conditions is obtained. Numerical results for both beam models are presented and the effects of piezoelectricity and length scale parameter are illustrated. It is found that the influences of piezoelectricity and size effects are more prominent when the length of microbeams decreases. A comparison between two beam models also reveals that the Euler-Bernoulli beam model tends to overestimate the natural frequencies of microbeams as compared to its Timoshenko counterpart.

Nonlinear Dynamic Study on Effects of Rub-Impact Caused by Oil-Rupture in a Multi-Shafts Turbine With a Squeeze Film Damper

2014 ◽  
Author(s):  
T. N. Shiau ◽  
C. R. Wang ◽  
D. S. Liu ◽  
W. C. Hsu ◽  
T. H. Young
Keyword(s):  
Dynamic Behavior ◽  
Nonlinear Dynamic ◽  
Equations Of Motion ◽  
Rotor System ◽  
Squeeze Film ◽  
Speed Ratio ◽  
Frequency Spectra ◽  
Squeeze Film Damper ◽  
Assumed Mode Method ◽  
Nonlinear Dynamic Behavior

An investigation is carried out the analysis of nonlinear dynamic behavior on effects of rub-impact caused by oil-rupture in a multi-shafts turbine system with a squeeze film damper. Main components of a multi-shafts turbine system includes an outer shaft, an inner shaft, an impeller shaft, ball bearings and a squeeze film damper. In the squeeze film damper, oil forces can be derived from the short bearing approximation and cavitated film assumption. The system equations of motion are formulated by the global assumed mode method (GAMM) and Lagrange’s approach. The nonlinear behavior of a multi-shafts turbine system which includes the trajectories in time domain, frequency spectra, Poincaré maps, and bifurcation diagrams are investigated. Numerical results show that large vibration amplitude is observed in steady state at rotating speed ratio adjacent to the first natural frequency when there is no squeeze film damper. The nonlinear dynamic behavior of a multi-shafts turbine system goes in its way into aperiodic motion due to oil-rupture and it is unlike the usual way (1T = >2T = >4T = >8T etc) as compared to one shaft rotor system. The typical routes of bifurcation to aperiodic motion are observed in a multi-shafts turbine rotor system and they suddenly turn into aperiodic motion from the periodic motion without any transition. Consequently, the increasing of geometric or oil parameters such as clearance or lubricant viscosity will improve the performance of SFD bearing.

Sensitivity of the Shear Wave Speed-Stress Relationship to Soft Tissue Material Properties and Fiber Alignment

2021 ◽  
Author(s):  
Jonathon Blank ◽  
Darryl Thelen ◽  
Matthew S. Allen ◽  
Joshua Roth
Keyword(s):  
Wave Propagation ◽  
Shear Modulus ◽  
Shear Wave ◽  
Wave Speed ◽  
Axial Stress ◽  
Beam Model ◽  
Fiber Alignment ◽  
Wave Speeds ◽  
Shear Wave Speed ◽  
Constitutive Properties

The use of shear wave propagation to noninvasively gauge material properties and loading in tendons and ligaments is a growing area of interest in biomechanics. Prior models and experiments suggest that shear wave speed primarily depends on the apparent shear modulus (i.e., shear modulus accounting for contributions from all constituents) at low loads, and then increases with axial stress when axially loaded. However, differences in the magnitudes of shear wave speeds between ligaments and tendons, which have different substructures, suggest that the tissue’s composition and fiber alignment may also affect shear wave propagation. Accordingly, the objectives of this study were to (1) characterize changes in the apparent shear modulus induced by variations in constitutive properties and fiber alignment, and (2) determine the sensitivity of the shear wave speed-stress relationship to variations in constitutive properties and fiber alignment. To enable systematic variations of both constitutive properties and fiber alignment, we developed a finite element model that represented an isotropic ground matrix with an embedded fiber distribution. Using this model, we performed dynamic simulations of shear wave propagation at axial strains from 0% to 10%. We characterized the shear wave speed-stress relationship using a simple linear regression between shear wave speed squared and axial stress, which is based on an analytical relationship derived from a tensioned beam model. We found that predicted shear wave speeds were both in-range with shear wave speeds in previous in vivo and ex vivo studies, and strongly correlated with the axial stress (R2 = 0.99). The slope of the squared shear wave speed-axial stress relationship was highly sensitive to changes in tissue density. Both the intercept of this relationship and the apparent shear modulus were sensitive to both the shear modulus of the ground matrix and the stiffness of the fibers’ toe-region when the fibers were less well-aligned to the loading direction. We also determined that the tensioned beam model overpredicted the axial tissue stress with increasing load when the model had less well-aligned fibers. This indicates that the shear wave speed increases likely in response to a load-dependent increase in the apparent shear modulus. Our findings suggest that researchers may need to consider both the material and structural properties (i.e., fiber alignment) of tendon and ligament when measuring shear wave speeds in pathological tissues or tissues with less well-aligned fibers.

The Effects of Harmonic Drive Gears on Robot Dynamics

1991 ◽  
Author(s):  
Tsung-Chieh Lin ◽  
K. Harold Yae
Keyword(s):  
Mathematical Models ◽  
Nonlinear Dynamic ◽  
Equations Of Motion ◽  
Robot Dynamics ◽  
Harmonic Drive ◽  
Dynamic Coupling ◽  
Manipulator Dynamics ◽  
Dynamic Formulation ◽  
System Equations ◽  
Modelling Assumptions

Abstract Mathematical models of the harmonic drive have been developed, and their effects on manipulator dynamics have been examined. The harmonic drive is modelled as a flexible gear with a high gear reduction ratio. The recursive Newton-Euler dynamic formulation is applied to deriving the system equations of motion that include the effects of the geared actuation. The equations include not only the nonlinear dynamic coupling between rotors and links but the gyroscopic effect due to the spinning rotors. Different modelling assumptions creates four models and their time responses are compared. As an example, a seven degree of freedom robot was chosen to make comparisons in time responses.

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