Equations of Motion for an Inextensible Beam Undergoing Large Deflections

2016 ◽  
Vol 83 (5) ◽  
Author(s):  
Earl Dowell ◽  
Kevin McHugh
Keyword(s):  
Boundary Conditions ◽  
Lagrange Multiplier ◽  
Equations Of Motion ◽  
Equation Of Motion ◽  
Large Deflections ◽  
Lagrange Equations ◽  
Cantilevered Beam ◽  
Alternative Approaches ◽  
Transverse Deflection ◽  
Free Beam

The Euler–Lagrange equations and the associated boundary conditions have been derived for an inextensible beam undergoing large deflections. The inextensibility constraint between axial and transverse deflection is considered via two alternative approaches based upon Hamilton's principle, which have been proved to yield equivalent results. In one approach, the constraint has been appended to the system Lagrangian via a Lagrange multiplier, while in the other approach the axial deflection has been expressed in terms of the transverse deflection, and the equation of motion for the transverse deflection has been determined directly. Boundary conditions for a cantilevered beam and a free–free beam have been considered and allow for explicit results for each system's equations of motion. Finally, the Lagrange multiplier approach has been extended to equations of motion of cantilevered and free–free plates.

Nonlinear Responses of Inextensible Cantilever and Free-Free Beams Undergoing Large Deflections

2017 ◽  
Author(s):  
Kevin McHugh ◽  
Earl Dowell
Keyword(s):  
Equations Of Motion ◽  
Ritz Method ◽  
Large Deflections ◽  
Constraint Force ◽  
Nonlinear Responses ◽  
Nonlinear Motion ◽  
Resonant Modes ◽  
Cantilevered Beam ◽  
Point Forces ◽  
Free Beam

A theoretical and computational model has been developed for the nonlinear motion of an inextensible beam undergoing large deflections for cantilevered and free-free boundary conditions. The inextensibility condition was enforced through a Lagrange multiplier which acted as a constraint force. The Rayleigh-Ritz method was used by expanding the deflections and the constraint force in modal series. Lagrange’s Equations were used to derive the equations of motion of the system, and a 4th order Runge-Kutta solver was used to solve them. Comparisons for the cantilevered beam were drawn to experimental and computational results previously published and show good agreement for responses to both static and dynamic point forces. Some physical insights into the cantilevered beam response at the 1st and 2nd resonant modes were obtained. The free-free beam condition was investigated at the 1st and 3rd resonant modes and the nonlinearity (primarily inertia) was shown to shift the resonant frequency significantly from the linear natural frequency and lead to hysteresis in both modes.

scholarly journals Nonlinear Responses of Inextensible Cantilever and Free–Free Beams Undergoing Large Deflections

2018 ◽  
Vol 85 (5) ◽  
Author(s):  
Kevin McHugh ◽  
Earl Dowell
Keyword(s):  
Equations Of Motion ◽  
Ritz Method ◽  
Large Deflections ◽  
Constraint Force ◽  
Nonlinear Responses ◽  
Nonlinear Motion ◽  
Resonant Modes ◽  
Cantilevered Beam ◽  
Point Forces ◽  
Free Beam

A theoretical and computational model has been developed for the nonlinear motion of an inextensible beam undergoing large deflections for cantilevered and free–free boundary conditions. The inextensibility condition was enforced through a Lagrange multiplier which acted as a constraint force. The Rayleigh–Ritz method was used by expanding the deflections and the constraint force in modal series. Lagrange's equations were used to derive the equations of motion of the system, and a fourth-order Runge–Kutta solver was used to solve them. Comparisons for the cantilevered beam were drawn to experimental and computational results previously published and show good agreement for responses to both static and dynamic point forces. Some physical insights into the cantilevered beam response at the first and second resonant modes were obtained. The free–free beam condition was investigated at the first and third resonant modes, and the nonlinearity (primarily inertia) was shown to shift the resonant frequency significantly from the linear natural frequency and lead to hysteresis in both modes.

Approximate Formulas for Cylindrical Shell Free Vibration Based on Vlasov’s and Enhanced Vlasov’s Semi-Momentless Theory

2018 ◽  
Author(s):  
Igor Orynyak ◽  
Yaroslav Dubyk
Keyword(s):  
Cylindrical Shell ◽  
Boundary Conditions ◽  
Circular Cylindrical Shell ◽  
Equations Of Motion ◽  
Middle Surface ◽  
Axial Direction ◽  
Mode Shapes ◽  
Natural Mode ◽  
Transverse Deflection ◽  
Approximate Formulas

Simple approximate formulas for the natural frequencies of circular cylindrical shells are presented for modes in which transverse deflection dominates. Based on the Donnell-Mushtari thin shell theory the equations of motion of the circular cylindrical shell are introduced, using Vlasov assumptions and Fourier series for the circumferential direction, an exact solution in the axial direction is obtained. To improve the results assumptions of Vlasov’s semimomentless theory are enhanced, i.e. we have used only the hypothesis of middle surface inextensibility to obtain a solution in axial direction. Nonlinear characteristic equations and natural mode shapes, are derived for all type of boundary conditions. Good agreement with experimental data and FEM is shown and advantage over the existing formulas for a variety of boundary conditions is presented.

scholarly journals A NOTE ON THE (ANTI-)BRST INVARIANT LAGRANGIAN DENSITIES FOR THE FREE ABELIAN 2-FORM GAUGE THEORY

2010 ◽  
Vol 25 (28) ◽  
pp. 2457-2467
Author(s):  
SAURABH GUPTA ◽  
R. P. MALIK
Keyword(s):  
Gauge Theory ◽  
Vector Field ◽  
Lagrange Multiplier ◽  
Field Condition ◽  
Equations Of Motion ◽  
Present Theory ◽  
Lagrange Equations ◽  
Symmetry Transformations ◽  
The Absolute ◽  
Lagrangian Densities

We show that the previously known off-shell nilpotent [Formula: see text] and absolutely anticommuting (sb sab + sab sb = 0) Becchi–Rouet–Stora–Tyutin (BRST) transformations (sb) and anti-BRST transformations (sab) are the symmetry transformations of the appropriate Lagrangian densities of a four (3+1)-dimensional (4D) free Abelian 2-form gauge theory which do not explicitly incorporate a very specific constrained field condition through a Lagrange multiplier 4D vector field. The above condition, which is the analogue of the Curci–Ferrari restriction of the non-Abelian 1-form gauge theory, emerges from the Euler–Lagrange equations of motion of our present theory and ensures the absolute anticommutativity of the transformations s(a)b. Thus, the coupled Lagrangian densities, proposed in our present investigation, are aesthetically more appealing and more economical.

PROPAGATION AND RESONANCE OF COMPOSITE WAVES IN PRISMATIC RODS

1936 ◽  
Vol 14a (3) ◽  
pp. 66-70
Author(s):  
R. Ruedy
Keyword(s):  
Boundary Conditions ◽  
Equations Of Motion ◽  
Equation Of Motion ◽  
Experimental Results ◽  
Cubic Equation ◽  
Resonance Frequencies ◽  
Good Agreement

By taking into account the three main terms of the equation of motion of the prismatic rod, there is obtained for the frequency a cubic equation which is in good agreement with the experimental results when the thickness of the rod is not negligible compared with its length but does not exceed about one-fifth of the length. It corresponds to the equation obtained for a system with three degrees of freedom.For a composite vibration consisting of a wave of dilatation and a wave of distortion in the direction of the smallest dimension of the rod, and waves of dilatation in the two other directions, the equations of motion combined with some of the boundary conditions yield another cubic equation for the resonance frequencies.

A Generalized Formulation of the Vectorial Equations of Motion for Nonprismatic Thin Space Beams

1971 ◽  
Vol 38 (4) ◽  
pp. 955-960 ◽  
Author(s):  
M. F. Massoud
Keyword(s):  
Boundary Conditions ◽  
Cross Section ◽  
Displacement Vector ◽  
Equations Of Motion ◽  
Equation Of Motion ◽  
Linear Displacement ◽  
Characteristic Dimension ◽  
Center Line ◽  
Thin Beam ◽  
Arbitrary Space

The formulation of a generalized vectorial equation of motion for small vibrations of any nonprismatic thin beam, which center line is an arbitrary space curve, is presented. The thin beam is such that any characteristic dimension of any cross section is assumed to be small compared to the local radii of curvature and geometric torsion. The equations of motion are given in terms of two independent vectors; a linear displacement vector of the centroid of the cross section and a rotation displacement vector about the centroid. A brief discussion of the boundary conditions in terms of these two vectors is given. The effects of rotary inertia and the shear deformation upon the general derived expressions are discussed.

Vibration Control of Beams with Negative Capacitive Shunting of Interdigital Electrode Piezoceramics

2005 ◽  
Vol 11 (3) ◽  
pp. 331-346 ◽  
Author(s):  
Chul H. Park ◽  
Amr Baz
Keyword(s):  
Mathematical Model ◽  
Boundary Conditions ◽  
Vibration Control ◽  
Equations Of Motion ◽  
Vibration Modes ◽  
Frequency Range ◽  
Governing Equations ◽  
Interdigital Electrode ◽  
Beam System ◽  
Cantilevered Beam

A pair of interdigital electrode (IDE) piezoceramics is used to simultaneously suppress multimode vibrations of a cantilevered beam. This is achieved by connecting the IDE piezoceramics in parallel to a negative capacitive shunt circuit. The governing equations of motion of an IDE piezo/beam system and associated boundary conditions are derived using the Hamilton principle. The obtained mathematical model is validated experimentally Attenuations ranging between 5 and 20 dB are obtained for all the vibration modes over the frequency range of 0-3000 Hz. The presented theoretical and experimental techniques provide invaluable tools for designing simple and effective passive vibration dampers for structures with closely packed modes.

Model of vortex flow of charge in a vessel with turbine impeller

1987 ◽  
Vol 52 (8) ◽  
pp. 1888-1904
Author(s):  
Miloslav Hošťálek ◽  
Ivan Fořt
Keyword(s):  
Boundary Conditions ◽  
Equations Of Motion ◽  
Theoretical Data ◽  
Eddy Diffusion ◽  
Quantitative Agreement ◽  
Creeping Flow ◽  
Model Parameters ◽  
Mean Flow ◽  
Two Dimensional Flow ◽  
Vorticity Transport

A theoretical model is described of the mean two-dimensional flow of homogeneous charge in a flat-bottomed cylindrical tank with radial baffles and six-blade turbine disc impeller. The model starts from the concept of vorticity transport in the bulk of vortex liquid flow through the mechanism of eddy diffusion characterized by a constant value of turbulent (eddy) viscosity. The result of solution of the equation which is analogous to the Stokes simplification of equations of motion for creeping flow is the description of field of the stream function and of the axial and radial velocity components of mean flow in the whole charge. The results of modelling are compared with the experimental and theoretical data published by different authors, a good qualitative and quantitative agreement being stated. Advantage of the model proposed is a very simple schematization of the system volume necessary to introduce the boundary conditions (only the parts above the impeller plane of symmetry and below it are distinguished), the explicit character of the model with respect to the model parameters (model lucidity, low demands on the capacity of computer), and, in the end, the possibility to modify the given model by changing boundary conditions even for another agitating set-up with radially-axial character of flow.

scholarly journals A least action principle for interceptive walking

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Soon Ho Kim ◽  
Jong Won Kim ◽  
Hyun Chae Chung ◽  
MooYoung Choi
Keyword(s):  
Social Systems ◽  
Equations Of Motion ◽  
Classical Mechanics ◽  
Action Principle ◽  
Lagrange Equations ◽  
Least Action Principle ◽  
Least Action ◽  
The Least Action Principle ◽  
Principle Of Least Action ◽  
Human Participants

AbstractThe principle of least effort has been widely used to explain phenomena related to human behavior ranging from topics in language to those in social systems. It has precedence in the principle of least action from the Lagrangian formulation of classical mechanics. In this study, we present a model for interceptive human walking based on the least action principle. Taking inspiration from Lagrangian mechanics, a Lagrangian is defined as effort minus security, with two different specific mathematical forms. The resulting Euler–Lagrange equations are then solved to obtain the equations of motion. The model is validated using experimental data from a virtual reality crossing simulation with human participants. We thus conclude that the least action principle provides a useful tool in the study of interceptive walking.

scholarly journals Pole-skipping and hydrodynamic analysis in Lifshitz, AdS2 and Rindler geometries

2021 ◽  
Vol 2021 (6) ◽  
Author(s):  
Haiming Yuan ◽  
Xian-Hui Ge
Keyword(s):  
Boundary Condition ◽  
Green’S Function ◽  
Green's Function ◽  
Equations Of Motion ◽  
Equation Of Motion ◽  
Boundary Theory ◽  
Hydrodynamic Analysis ◽  
Dimensionless Variables ◽  
Pass Through ◽  
Special Points

Abstract The “pole-skipping” phenomenon reflects that the retarded Green’s function is not unique at a pole-skipping point in momentum space (ω, k). We explore the universality of pole-skipping in different geometries. In holography, near horizon analysis of the bulk equation of motion is a more straightforward way to derive a pole-skipping point. We use this method in Lifshitz, AdS2 and Rindler geometries. We also study the complex hydrodynamic analyses and find that the dispersion relations in terms of dimensionless variables $$ \frac{\omega }{2\pi T} $$ ω 2 πT and $$ \frac{\left|k\right|}{2\pi T} $$ k 2 πT pass through pole-skipping points $$ \left(\frac{\omega_n}{2\pi T},\frac{\left|{k}_n\right|}{2\pi T}\right) $$ ω n 2 πT k n 2 πT at small ω and k in the Lifshitz background. We verify that the position of the pole-skipping points does not depend on the standard quantization or alternative quantization of the boundary theory in AdS2× ℝd−1 geometry. In the Rindler geometry, we cannot find the corresponding Green’s function to calculate pole-skipping points because it is difficult to impose the boundary condition. However, we can still obtain “special points” near the horizon where bulk equations of motion have two incoming solutions. These “special points” correspond to the nonuniqueness of the Green’s function in physical meaning from the perspective of holography.

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