Bending-Bending Mode of a Rotating Tapered-Twisted Turbomachine Blade Including Rotatory Inertia and Shear Deformation

1969 ◽  
Vol 91 (4) ◽  
pp. 1017-1024 ◽  
Author(s):  
R. M. Krupka ◽  
A. M. Baumanis
Keyword(s):  
Shear Deformation ◽  
Equations Of Motion ◽  
Torsional Vibrations ◽  
Parameter System ◽  
Field Equations ◽  
Lagrange Equations ◽  
Rotatory Inertia ◽  
Lumped Parameter ◽  
Bending Mode ◽  
Definition Of

This paper presents the effect on natural frequency and mode shape of the inclusion of terms that are present in the general equations of motion to describe phenomena associated with Rotatory Inertia and Shear Deformation. The coupling that exists between the flexural and torsional vibrations is not considered. Carnegie’s formulation of the Lagrange Equations of motion is used and the set of field equations solved using Myklestad’s adaptation of the Holzer method. The definition of the lumped parameter system used and the derivation of the associated discrete “difference equations,” which are utilized in the computer approach to the boundary value problem considered, constitute an extension of the Carnegie work.

Lie Symmetry and Approximate Hojman Conserved Quantity of Lagrange Equations for a Weakly Nonholonomic System

2013 ◽  
Vol 30 (1) ◽  
pp. 21-27 ◽  
Author(s):  
Y.-L. Han ◽  
X.-X. Wang ◽  
M.-L. Zhang ◽  
L.-Q. Jia
Keyword(s):  
Nonholonomic System ◽  
Equations Of Motion ◽  
Lie Symmetry ◽  
Conserved Quantity ◽  
Holonomic System ◽  
Lagrange Equations ◽  
Definition Of ◽  
Weakly Nonholonomic System ◽  
Infinitesimal Transformations ◽  
Hojman Conserved Quantity

ABSTRACTThe Lie symmetry and Hojman conserved quantity of Lagrange equations for a weakly nonholonomic system and its first-degree approximate holonomic system are studied. The differential equations of motion for the system are established. Under the special infinitesimal transformations of group in which the time is invariable, the definition of the Lie symmetry for the weakly nonholonomic system and its first-degree approximate holonomic system are given, and the exact and approximate Hojman conserved quantities deduced directly from the Lie symmetry are obtained. Finally, an example is given to study the exact and approximate Hojman conserved quantity for the system.

scholarly journals Nonlinear interactions in deformable container cranes

2015 ◽  
Vol 230 (1) ◽  
pp. 5-20 ◽  
Author(s):  
Andrea Arena ◽  
Walter Lacarbonara ◽  
Matthew P Cartmell
Keyword(s):  
Rigid Body ◽  
Internal Resonance ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Higher Order ◽  
Mode Shapes ◽  
Lagrange Equations ◽  
Internal Resonances ◽  
Container Cranes ◽  
Bending Mode

Nonlinear dynamic interactions in harbour quayside cranes due to a two-to-one internal resonance between the lowest bending mode of the deformable boom and the in-plane pendular mode of the container are investigated. To this end, a three-dimensional model of container cranes accounting for the elastic interaction between the crane boom and the container dynamics is proposed. The container is modelled as a three-dimensional rigid body elastically suspended through hoisting cables from the trolley moving along the crane boom modelled as an Euler-Bernoulli beam. The reduced governing equations of motion are obtained through the Euler-Lagrange equations employing the boom kinetic and stored energies, derived via a Galerkin discretisation based on the mode shapes of the two-span crane boom used as trial functions, and the kinetic and stored energies of the rigid body container and the elastic hoisting cables. First, conditions for the onset of internal resonances between the boom and the container are found. A higher order perturbation treatment of the Taylor expanded equations of motion in the neighbourhood of a two-to-one internal resonance between the lowest boom bending mode and the lowest pendular mode of the container is carried out. Continuation of the fixed points of the modulation equations together with stability analysis yields a rich bifurcation behaviour, which features Hopf bifurcations. It is shown that consideration of higher order terms (cubic nonlinearities) beyond the quadratic geometric and inertia nonlinearities breaks the symmetry of the bifurcation equations, shifts the bifurcation points and the stability ranges, and leads to bifurcations not predicted by the low order analysis.

THE UNIVERSAL EQUATIONS OF MOTION FOR NONLINEAR FIELDS IN DIFFERENT BASES DESCRIBING STRONGLY INTERACTING MANY-BODY SYSTEMS WITH TWO-BODY INTERACTIONS

1995 ◽  
Vol 09 (13n14) ◽  
pp. 1611-1637 ◽  
Author(s):  
J.M. DIXON ◽  
J.A. TUSZYŃSKI
Keyword(s):  
Plane Wave ◽  
Coherent Structures ◽  
Equations Of Motion ◽  
Quantum Field ◽  
Many Body ◽  
Field Equations ◽  
Strongly Interacting ◽  
Highly Nonlinear ◽  
Many Body Systems ◽  
Definition Of

A brief account of the Method of Coherent Structures (MCS) is presented using a plane-wave basis to define a quantum field. It is also demonstrated that the form of the quantum field equations, obtained by MCS, although highly nonlinear for many-body systems with two-body interactions, is independent of the basis of states used for the definition of the field.

Dynamic Behavior of Vehicles Travelling on Bridges

1993 ◽  
Author(s):  
Ebrahim Esmailzadeh ◽  
Mehrdaad Ghorashi
Keyword(s):  
Boundary Conditions ◽  
Dynamic Behavior ◽  
Numerical Solutions ◽  
Equations Of Motion ◽  
Beam Theory ◽  
Parameter System ◽  
Moving Vehicle ◽  
Lumped Parameter ◽  
Euler Bernoulli Beam ◽  
Two Degree Of Freedom

Abstract An investigation into the dynamic behavior of a bridge with simply supported boundary conditions, carrying a moving vehicle, is performed. The vehicle has been modelled as a two degree of freedom lumped-parameter system travelling at a uniform speed. Furthermore, the bridge is assumed to obey the Euler-Bernoulli beam theory of vibration. This analysis may well be applied to a beam with different boundary conditions, but the computer simulation results given in this paper are set for only the case of freely hinged ends. Numerical solutions for the derived differential equations of motion are obtained and their close agreement, in some extreme cases, with those reported earlier by the authors are observed. Finally, the effect of speed on the maximum dynamic deflection of bridge is shown to be of much importance and hence an estimation for the critical speed of the vehicle is presented.

Equations of motion in a generalized theory of gravitation

1981 ◽  
Vol 59 (11) ◽  
pp. 1723-1729 ◽  
Author(s):  
R. B. Mann ◽  
J. W. Moffat
Keyword(s):  
World Line ◽  
Equations Of Motion ◽  
Static Solution ◽  
Coupling Constant ◽  
Field Equations ◽  
Complete Space ◽  
Test Particles ◽  
Theory Of Gravitation ◽  
Definition Of ◽  
Universal Coupling

The problem of the motion of test particles is studied in a theory of gravitation based on a nonsymmetric gμν. According to the conservation laws the test particles can follow two kinds of geodesies, depending on the definition of a local inertial frame in the theory. One of these geodesies is nonmaximal and leads to a timelike and null world line complete space when a new parameter l, that occurs as a constant of integration in the spherically symmetric, static solution of the field equations, satisfies [Formula: see text]. In the theory, the parameter [Formula: see text] where N is the number of fermions in a system and a is a new universal coupling constant that satisfies [Formula: see text]. The physical implications of l and the associated conservation law of fermion number is discussed in detail.

scholarly journals I. Identifications

1938 ◽  
Vol 165 (922) ◽  
pp. 313-332 ◽  
Keyword(s):  
Classical Theory ◽  
Equations Of Motion ◽  
Laws Of Nature ◽  
Mechanical Force ◽  
Magnetic Intensity ◽  
Field Equations ◽  
Physical Assumption ◽  
General Object ◽  
Definition Of ◽  
Magnetic Poles

1. The general object of the following papers is to ascertain what form the equations of electromagnetism take when derived on a purely kinematic basis. Maxwell’s theory is not assumed. The only physical assumption made, namely, that a system of moving charges conserves its energy (defined kinematically) when the accelerations of the charges vanish, is a very slight one and is certainly satisfied in classical electromagnetism, but the resulting equations and laws, whilst coinciding with the classical theory to a considerable extent, differ in certain essential particulars. This arises from the avoidance of the empirical laws and hypothetical assumptions from which Maxwell’s theory starts. In particular we avoid the formal inconsistency in the classical theory by which a magnetic intensity H is defined via the mechanical force on an isolated magnetic pole, yet isolated magnetic poles do not occur in the classical “theory of electrons”. In the present treatment a magnetic intensity is defined via the mechanical force on a moving “charged” particle, as an element entering into the calculation of such force. The general method is, adopting the dynamics constructed in previous papers on a purely kinematic basis (Milne 1936, 1937), to formulate equations of motion containing the next most general type of “external” force arising after “gravitational” forces have been dealt with. Such forces arise from the double differentiation of scalar “superpotentials”, but we do not lay down what form these scalars are to take. Instead we allow them to determine themselves, by imposing the single physical assumption above-mentioned, after the equation of energy has been derived. Once the scalar superpotentials have been so determined, their double differentiation yields symbols E, H, which are then compared with the empirical laws governing the interaction of “charges”; this allows us to identify the adopted definition of charge and the symbols E, H with the similar quantities occurring in the experimental formulation. Lastly, we derive the identities satisfied by the resulting E, H; these partly coincide with, and partly differ from, the “field equations” with which the classical theory starts, and thus we end with theorems which play the part of the “laws of nature” assumed at the outset in the classical theory.

Simulation of a MEMS RF Filter

2005 ◽  
Author(s):  
Eihab M. Abdel-Rahman ◽  
Bashar K. Hammad ◽  
Ali H. Nayfeh
Keyword(s):  
Equations Of Motion ◽  
Distributed Parameter ◽  
Parameter System ◽  
Transmission Characteristics ◽  
Coupled Resonators ◽  
Lagrange Equations ◽  
Filter Model ◽  
Rf Filter ◽  
Two Degree Of Freedom ◽  
Filter Response

We simulate the motions in a MEMS bandpass Radio-Frequency (RF) filter. The filter model is obtained by discretizing the Lagrangian of the distributed-parameter system using a Galerkin procedure. The Euler-Lagrange equations are then used to obtain a two-degree-of-freedom model consisting of two non-linearly coupled ordinary-differential equations of motion. We use the model to study the transmission characteristics of a bandpass filter made up of two coupled resonators. Three distinct response regimes, separated by two critical amplification levels Vcr1 and Vcr2, are identified in the filter response. For amplification levels up to Vcr1, the pass signal is artifact free. Two types of artifacts due to the filter dynamics appear and distort the signal for amplification levels beyond Vcr1.

A Multiple-Mode Approach to Large-Amplitude Vibration of Moderately Thick Elliptical Plates

1984 ◽  
Vol 51 (1) ◽  
pp. 153-158 ◽  
Author(s):  
M. Sathyamoorthy ◽  
M. E. Prasad
Keyword(s):  
Shear Deformation ◽  
Stress Function ◽  
Lateral Displacement ◽  
Equations Of Motion ◽  
Mode Analysis ◽  
Rotatory Inertia ◽  
Multiple Mode ◽  
Large Amplitude Vibration ◽  
Nonlinear Equations Of Motion ◽  
Plate Geometry

Based on a multiple-mode analysis, solutions to the nonlinear equations of motion are presented for elliptical plates in terms of variations of nonlinear periods with amplitudes of vibration. The governing equations are written in terms of lateral displacement w and stress function F and the effects of transverse shear deformation and rotatory inertia are incorporated into these equations. For the multiple-mode approach considered in this paper, an exact solution to the stress function is determined. Effects of geometric nonlinearity, shear deformation, rotatory inertia, plate geometry, and modal interaction on the vibration behaviors of elliptical plates are investigated in detail.

scholarly journals Four-dimensional gravity as an almost-Poisson system

2015 ◽  
Vol 24 (07) ◽  
pp. 1550047
Author(s):  
Eyo Eyo Ita
Keyword(s):  
Phase Space ◽  
Poisson Bracket ◽  
Jacobi Identity ◽  
Equations Of Motion ◽  
Space Structure ◽  
Lagrange Equations ◽  
Phase Space Structure ◽  
Full Phase Space ◽  
Dimensional Gravity ◽  
Definition Of

In this paper, we examine the phase space structure of a noncanonical formulation of four-dimensional gravity referred to as the Instanton representation of Plebanski gravity (IRPG). The typical Hamiltonian (symplectic) approach leads to an obstruction to the definition of a symplectic structure on the full phase space of the IRPG. We circumvent this obstruction, using the Lagrange equations of motion, to find the appropriate generalization of the Poisson bracket. It is shown that the IRPG does not support a Poisson bracket except on the vector constraint surface. Yet there exists a fundamental bilinear operation on its phase space which produces the correct equations of motion and induces the correct transformation properties of the basic fields. This bilinear operation is known as the almost-Poisson bracket, which fails to satisfy the Jacobi identity and in this case also the condition of antisymmetry. We place these results into the overall context of nonsymplectic systems.

Numerical Study of One Dimensional Pipe Flow Under Pump Cavitation Surge

2017 ◽  
Author(s):  
Motohiko Nohmi ◽  
Satoshi Yamazaki ◽  
Shusaku Kagawa ◽  
Byungjin An ◽  
Donghyuk Kang ◽  
...  
Keyword(s):  
Pipe Flow ◽  
Numerical Study ◽  
Equations Of Motion ◽  
Gain Factor ◽  
Distributed Parameter ◽  
Piping System ◽  
Distributed Parameter System ◽  
Parameter System ◽  
Lumped Parameter ◽  
System Calculation

Pump cavitation surge is highly coupled phenomenon with unsteady cavitation inside a pump and system dynamics of the pipe flow surrounding the pump. The piping system flow dynamics can be calculated under two kinds of assumptions; lumped parameter system (LPS) and distributed parameter system (DPS). In the lumped parameter system, the equations of motion of water columns inside pipes are calculated upstream and downstream of the pump. In the distributed parameter system, wave propagations along the pipes are calculated. In this study a simple system that consists of an upstream tank, an upstream pipe, a pump with cavitation, a downstream pipe and a downstream tank is analyzed by using two methods. Cavitation inside the pump is featured in the lumped parameters of cavitation compliance and mass flow gain factor. In the lumped parameter system case, equations of motion are calculated numerically by Runge-Kutta methods. In the distributed parameter system case, wave propagations are calculated by Method of Characteristics. From the comparison of two method results, appropriate criterion for practical piping system calculation is discussed.

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