Showing papers on "Equations of motion published in 1981"
Book•
01 Sep 1981
TL;DR: In this article, the authors discuss the development of Vibration Analysis of Continuous Structural Elements (SSA) and their application in the field of deep shell physics, including the following:
Abstract: Preface to the Third Edition Preface to the Second Edition Preface to the First Edition Historical Development of Vibration Analysis of Continuous Structural Elements References Deep Shell Equations Shell Coordinates and Infinitesimal Distances in Shell Layers Stress-Strain Relationships Strain-Displacement Relationships Love Simplifications Membrane Forces and Bending Moments Energy Expressions Love's Equations by Way of Hamilton's Principle Boundary Conditions Hamilton's Principle Other Deep Shell Theories Shells of Nonuniform Thickness References Radii of Curvature References Equations of Motion for Commonly Occurring Geometries Shells of Revolution Circular Conical Shell Circular Cylindrical Shell Spherical Shell Other Geometries References Nonshell Structures Arch Beam and Rod Circular Ring Plate Torsional Vibration of Circular Cylindrical Shell and Reduction to a Torsion Bar References Natural Frequencies and Modes General Approach Transversely Vibrating Beams Circular Ring Rectangular Plates That are Simply Supported Along Two Opposing Edges Circular Cylindrical Shell Simply Supported Circular Plates Vibrating Transversely Examples: Plate Clamped at Boundary Orthogonality Property of Natural Modes Superposition Modes Orthogonal Modes from Nonorthogonal Superposition Modes Distortion of Experimental Modes Because of Damping Separating Time Formally Uncoupling of Equations of Motion In-Plane Vibrations of Rectangular Plates In-Plane Vibration of Circular Plates Deep Circular Cylindrical Panel Simply Supported at All Edges Natural Mode Solutions by Power Series On Regularities Concerning Nodelines References Simplified Shell Equations Membrane Approximations Axisymmetric Eigenvalues of a Spherical Shell Bending Approximation Circular Cylindrical Shell Zero In-Plane Deflection Approximation Example: Curved Fan Blade Donnell-Mushtari-Vlasov Equations Natural Frequencies and Modes Circular Cylindrical Shell Circular Duct Clamped at Both Ends Vibrations of a Freestanding Smokestack Special Cases of the Simply Supported Closed Shell and Curved Panel Barrel-Shaped Shell Spherical Cap Inextensional Approximation: Ring Toroidal Shell The Barrel-Shaped Shell Using Modified Love Equations Doubly Curved Rectangular Plate References Approximate Solution Techniques Approximate Solutions by Way of the Variational Integral Use of Beam Functions Galerkin's Method Applied to Shell Equations Rayleigh-Ritz Method Southwell's Principle Dunkerley's Principle Strain Energy Expressions References Forced Vibrations of Shells by Modal Expansion Model Participation Factor Initial Conditions Solution of the Modal Participation Factor Equation Reduced Systems Steady-State Harmonic Response Step and Impulse Response Influence of Load Distribution Point Loads Line Loads Point Impact Impulsive Forces and Point Forces Described by Dirac Delta Functions Definitions and Integration Property of the Dirac Delta Function Selection of Mode Phase Angles for Shells of Revolution Steady-State Circular Cylindrical Shell Response to Harmonic Point Load with All Mode Components Considered Initial Velocity Excitation of a Simply Supported Cylindrical Shell Static Deflections Rectangular Plate Response to Initial Displacement Caused by Static Sag The Concept of Modal Mass, Stiffness Damping, and Forcing Steady State Response of Shells to Periodic Forcing Plate Response to a Periodic Square Wave Forcing Beating Response to Steady State Harmonic Forcing References Dynamic Influence (Green's) Function Formulation of the Influence Function Solution to General Forcing Using the Dynamic Influence Function Reduced Systems Dynamic Influence Function for the Simply Supported Shell Dynamic Influence Function for the Closed Circular Ring Traveling Point Load on a Simply Supported Cylindrical Shell Point Load Traveling Around a Closed Circular Cylindrical Shell in Circumferential Direction Steady-State Harmonic Green's Function Rectangular Plate Examples Floating Ring Impacted by a Point Mass References Moment Loading Formulation of Shell Equations That Include Moment Loading Modal Expansion Solution Rotating Point Moment on a Plate Rotating Point Moment on a Shell Rectangular Plate Excited by a Line Moment Response of a Ring on an Elastic Foundation to a Harmonic Point Moment Moment Green's Function References Vibration of Shells and Membranes Under the Influence of Initial Stresses Strain-Displacement Relationships Equations of Motion Pure Membranes Example: The Circular Membrane Spinning Saw Blade Donnell-Mushtari-Vlasov Equations Extended to Include Initial Stresses References Shell Equations with Shear Deformation and Rotary Inertia Equations of Motion Beams with Shear Deflection and Rotary Inertia Plates with Transverse Shear Deflection and Rotary Inertia Circular Cylindrical Shells with Transverse Shear Deflection and Rotary Inertia References Combinations of Structures Receptance Method Mass Attached to Cylindrical Panel Spring Attached to Shallow Cylindrical Panel Harmonic Response of a System in Terms of Its Component Receptances Dynamic Absorber Harmonic Force Applied Through a Spring Steady-State Response to Harmonic Displacement Excitation Complex Receptances Stiffening of Shells Two Systems Joined by Two or More Displacement Suspension of an Instrument Package in a Shell Subtracting Structural Subsystems Three and More Systems Connected Examples of Three Systems Connected to Each Other References Hysteresis Damping Equivalent Viscous Damping Coefficient Hysteresis Damping Direct Utilization of Hysteresis Model in Analysis Hysteretically Damped Plate Excited by Shaker Steady State Response to Periodic Forcing References Shells Made of Composite Material Nature of Composites Lamina-Constitutive Relationship Laminated Composite Equation of Motion Orthotropic Plate Circular Cylindrical Shell Orthotropic Nets or Textiles Under Tension Hanging Net or Curtain Shells Made of Homogeneous and Isotropic Lamina Simply Supported Sandwich Plates and Beams Composed of Three Homogeneous and Isotropic Lamina References Rotating Structures String Parallel to Axis of Rotation Beam Parallel to Axis of Rotation Rotating Ring Rotating Ring Using Inextensional Approximation Cylindrical Shell Rotating with Constant Spin About Its Axis General Rotations of Elastic Systems Shells of Revolution with Constant Spin About Their Axes of Rotation Spinning Disk References Thermal Effects Stress Resultants Equations of Motion Plate Arch, Ring, Beam, and Rod Limitations Elastic Foundations Equations of Motion for Shells on Elastic Foundations Natural Frequencies and Modes Plates on Elastic Foundations Ring on Elastic Foundation Donnell-Mushtari-Vlasov Equations with Transverse Elastic Foundation Forces Transmitted Into the Base of the Elastic Foundation Vertical Force Transmission Through the Elastic Foundation of a Ring on a Rigid Wheel Response of a Shell on an Elastic Foundation to Base Excitation Plate Examples of Base Excitation and Force Transmission Natural Frequencies and Modes of a Ring on an Elastic Foundation in Ground Contact at a Point Response of a Ring on an Elastic Foundation to a Harmonic Point Displacement References Similitude General Similitude Derivation of Exact Similitude Relationships for Natural Frequencies of Thin Shells Plates Shallow Spherical Panels of Arbitrary Contours (Influence of Curvature) Forced Response Approximate Scaling of Shells Controlled by Membrane Stiffness Approximate Scaling of Shells Controlled by Bending Stiffness References Interactions with Liquids and Gases Fundamental Form in Three-Dimensional Curvilinear Coordinates Stress-Strain-Displacement Relationships Energy Expressions Equations of Motion of Vibroelasticity with Shear Example: Cylindrical Coordinates Example: Cartesian Coordinates One-Dimensional Wave Equations for Solids Three-Dimensional Wave Equations for Solids Three-Dimensional Wave Equations for Inviscid Compressible Liquids and Gases (Acoustics) Interface Boundary Conditions Example: Acoustic Radiation Incompressible Liquids Example: Liquid on a Plate Orthogonality of Natural Modes for Three-Dimensional Solids, Liquids, and Gases References Discretizing Approaches Finite Differences Finite Elements Free and Forced Vibration Solutions References Index
1,166 citations
CERN1
TL;DR: In this paper, the authors study the properties of interacting field theories which are invariant under duality rotations which transform a vector field strength into its dual, and show that the largest group for n interacting field strengths is the non-compact Sp(2 n,R), which has U( n ) as its maximal compact subgroup.
559 citations
TL;DR: In this paper, a new theory for beams of rectangular cross-section which includes warping of the cross-sections is presented, and results for two typical static examples are given for both the new theory and Timoshenko beam theory.
471 citations
Abstract: We present a new direct method of estimating the three-dimensional motion parameters of a rigid planar patch from two time-sequential perspective views (image frames). First, a set of eight pure parameters are defined. These parameters can be determined uniquely from the two given image frames by solving a set of linear equations. Then, the actual motion parameters are determined from these pure parameters by a method which requires the solution of a sixth-order polynomial of one variable only, and there exists a certain efficient algorithm for solving a sixth-order polynomial. Aside from a scale factor for the translation parameters, the number of real solutions never exceeds two. In the special case of three-dimensional translation, the motion parameters can be expressed directly as some simple functions of the eight pure parameters. Thus, only a few arithmetic operations are needed.
391 citations
TL;DR: In this article, the authors present a similar method, but with the exception that the problem is solved in the physical plane and finite depth is introduced, whereas in this paper, the same problem is stated in the same way, except that certain other effects can be included without much modification of the program.
216 citations
TL;DR: In this article, the authors reformulated QCD in terms of white composite fields, the traces of the loop products, and developed a new gauge-invariant perturbation theory in the loop space, reproducing asymptotic freedom.
208 citations
TL;DR: In this paper, an exact solution to the Navier-Stokes equations for the flow in a channel or tube with an accelerating surface velocity is presented, where the equations of motion are reduced to a single ordinary differential equation for the similarity function which is solved numerically.
Abstract: An exact solution to the Navier–Stokes equations for the flow in a channel or tube with an accelerating surface velocity is presented. By means of a similarity transformation the equations of motion are reduced to a single ordinary differential equation for the similarity function which is solved numerically. For the two-dimensional flow in a channel, a single solution is found to exist when the Reynolds number R is less than 310. When R exceeds 310, two additional solutions appear and form a closed branch connecting two different asymptotic states at infinite R. The large R structure of the solutions consists of an inviscid fluid core plus an O(R−1) thin boundary layer adjacent to the moving wall. Matched-asymptotic-expansion techniques are used to construct asymptotic series that are consistent with each of the numerical solutions.For the axisymmetric non-swirling flow in a tube, however, the situation is quite different. For R [Lt ] 10[sdot ]25, two solutions exist which form a closed branch. Beyond 10[sdot ]25, no similarity solutions exist within the range 10[sdot ]25 0. These solutions, however, do not evolve from the R = 0 state nor do they bifurcate from the non-swirling solutions at any finite value of R.
191 citations
TL;DR: In this article, the second-order gravitational field and equations of motion for two point-like objects in algebraically closed form were obtained using a fast-motion approximation method and a regularization procedure was used to guarantee the consistency of the approximation scheme.
Abstract: Using a fast-motion approximation method we obtain the second-order gravitational field and equations of motion for two pointlike objects in algebraically closed form. A regularization procedure is used which is shown to guarantee the consistency of the approximation scheme. The equations of motion are then transformed within the framework of relativistic predictive mechanics into a system of ordinary differential equations.
169 citations
TL;DR: In this article, the critical velocities of cylinders supported by elastic spars were calculated by using the measured unsteady fluid dynamic forces, which are induced by the vibrating cylinders.
166 citations
Journal Article•
136 citations
TL;DR: In this article, the Poincare invariant post-linear approximation of general relativity was used to obtain newtonian-like equations of motion for two slowly moving point masses.
TL;DR: In this article, the temporal behavior of a parametrically excited mathematical pendulum is examined by means of the numerical solution of its equation of motion, and it is shown that for certain parameter intervals the pendulum behaves in an apparently chaotic way.
TL;DR: The functional integral method for the statistical solution of stochastic differential equations is extended to a broad class of nonlinear dynamical equations with random coefficients and initial conditions as mentioned in this paper, which have applications in the calculation of particle motion in stochastically magnetic fields, in the solution of Stochastic wave equations, and in the description of electromagnetic plasma turbulence.
Abstract: The functional integral method for the statistical solution of stochastic differential equations is extended to a broad, new class of nonlinear dynamical equations with random coefficients and initial conditions. This work encompasses previous results for classical systems with random forces and initial conditions with arbitrary statistics and provides new results for systems with nonlinear interactions which are nonlocal in time. Closed equations of motion for the correlation and response functions are derived which have applications in the calculation of particle motion in stochastic magnetic fields, in the solution of stochastic wave equations, and in the description of electromagnetic plasma turbulence. As an illustration of the new results for nonlocal interactions, the electromagnetic dispersion tensor is calculated to first order in renormalized theory.
TL;DR: In this article, a class of exact solutions to the equations of motion of a second grade fluid were established wherein the nonlinearities which occur in the equations are self-cancelling though individually non-vanishing.
TL;DR: In this paper, the Euler parameters, Lagrange's form of d'Alembert's principle, generalized speeds, quasi-coordinates, relative coordinates, and structural analysis techniques are used for solving the equations of motion of multi-body mechanical systems with flexible links.
TL;DR: In this article, it is shown that the equations of motion in the realm of the model are obtainable from those of the real world by replacing any linear or angular acceleration term by corresponding linear and angular velocity term.
TL;DR: In this article, it is shown that geometrodynamics does not have any conditional symmetry: such a symmetry should be generated by a dynamical variable K[gab, pab] which is linear and homogeneous in the gravitational momentum pab and which has a weakly vanishing Poisson bracket with the super-Hamiltonian and supermomentum.
Abstract: The concept of conditional symmetry is introduced for a parametrized relativistic particle model and generalized to geometrodynamics. Its role in maintaining a one‐system interpretation of the quantized theory is emphasized. It is shown that geometrodynamics does not have any conditional symmetry: Such a symmetry should be generated by a dynamical variable K[gab, pab] which is linear and homogeneous in the gravitational momentum pab and which has a weakly vanishing Poisson bracket with the super‐Hamiltonian and supermomentum. The generators K fall into equivalence classes modulo the supermomentum constraint. It is shown that each equivalence class can be represented by a member which is a spatial invariant. The remaining weak equations are turned into strong equations by the method of Lagrange multipliers. The local structure of the super‐Hamiltonian and supermomentum imposes locality restrictions on the multipliers. These restrictions imply that the generator must be weakly equivalent to a local generator. A recursive argument then shows that the local generator must actually be weakly ultralocal. This uniquely determines the generator as the conformal Killing (super)vector of the local supermetric. However, the curvature scalar in the super‐Hamiltonian breaks the conditional symmetry of the supermetric term and turns geometrodynamics into a theory without any symmetry. This result is generalized to inhomogeneous generators.
TL;DR: A method of formulating and automatically integrating the equations of motion of quite general constrained dynamic systems is presented and design sensitivity analysis is carried out using a state space adjoint variable method that has been employed extensively in optimal control and structural design optimization.
Abstract: A method of formulating and automatically integrating the equations of motion of quite general constrained dynamic systems is presented Design sensitivity analysis is carried out using a state space adjoint variable method that has been employed extensively in optimal control and structural design optimization Both dynamic analysis and design sensitivity analysis formulations are automated and numerical solution of state and adjoint differential equations are carried out using a stiff numerical integration method that treats mixed systems of differential and algebraic equations A computer code that implements the method is applied to two numerical examples The first example concerns a relatively simple slider-crank mechanism The second example treats a more complex agricultural trip plow that undergoes intermittent motion
TL;DR: In this article, a unified variational formulation of field equations and conservation laws in general mechanical continua is presented, which is applicable to dynamic processes in any medium which admits a Lagrangian.
TL;DR: In this article, an analytical calculation of the torque generated by orthogonal waves is reported, which is a result of a viscous effect, rather than the Bernoulli effect as in Rayleigh's torque.
Abstract: An analytical calculation of the torque generated by orthogonal waves is reported. This torque is a result of a viscous effect, rather than the Bernoulli effect as in Rayleigh's torque. The agreement between the reported experimental values and this calculation is excellent.
TL;DR: In this article, a new analytical approach to the investigation of the regions of instability of multiple-degree-of-freedom parametric dynamic systems is presented, based on the harmonic balance method.
TL;DR: In this article, a complete correspondence between the quantum mechanical equations of motion and the classical equations for extended isolated bodies or pole-dipole particles is found based on a WKB scheme and a Gordon decomposition of the Dirac probability four-current.
Abstract: From covariant Dirac theory in curved space-time, dynamical equations for the motion of the spin and the spin-induced non-geodesic behaviour of the particle trajectories are deduced. This is done for arbitrary space-times in a generally covariant and observer- independent way. The procedure is thereby based on a WKB scheme and a Gordon decomposition of the Dirac probability four-current. A complete, correspondence between the quantum mechanical equations of motion and the classical equations for extended isolated bodies or pole-dipole particles is found. This can as well be taken as a confirmation that to the first order of a WKB approximation the gyro-gravitational factors of the classical angular nromentum and of the intrinsic quantum mechanical spin agree.
TL;DR: In this article, the authors extended the quantal description of a cavity from the case of constant mass to that of variable mass, corresponding to a decaying or driven cavity, in such a way that both Maxwell's equations and quantal equations of motion are invariant.
Abstract: The quantal description of a cavity is extended from the case of constant mass to that of variable mass, corresponding to a decaying or driven cavity, in such a way that both Maxwell's equations and the quantal equations of motion are invariant. The method is illustrated for the case of a decaying cavity.
TL;DR: In this paper, the authors studied the special case where one surface is fixed and the other is rapidly accelerated from a state of rest to a uniform motion, based in part on linear theory and in substance on a finite difference analysis of the full nonlinear equations of motion.
Abstract: We study the motion which results when a fluid film is squeezed between two parallel plane surfaces in relative motion. Particular attention is given to the special case where one surface is fixed and the other is rapidly accelerated from a state of rest to a state of uniform motion. The analysis is based in part on linear theory and in substance on a finite-difference analysis of the full nonlinear equations of motion.
16 Jun 1981
TL;DR: In this paper, a Gaussian elimination algorithm with full pivoting decomposes the constraint Jacobian matrix, identifies dependent variables, and constructs an influence coefficient matrix relating variations in dependent and independent variables.
Abstract: : A computer-based method for formulation and efficient solution of nonlinear, constrained differential equations of motion is developed for planar mechanical systems. Nonlinear holonomic constraint equations and differential equations of motion are written in terms of a maximal set of Cartesian generalized coordinates, to facilitate the general formulation of constraints and forcing functions. A Gaussian elimination algorithm with full pivoting decomposes the constraint Jacobian matrix, identifies dependent variables, and constructs an influence coefficient matrix relating variations in dependent and independent variables. This information is employed to numerically construct a reduced system of differential equations whose solution yields the total system dynamic response. A numerical integration algorithm with positive-error control, employing a predictor-corrector algorithm with variable order and step-size, is developed that integrates for only the independent variables, yet effectively determines dependent variables. A general method is developed for dynamic analysis of systems with impulsive forces, impact, discontinuous constraints, and discontinuous velocities. This class of systems includes discontinuous kinematic and geometric constraints that characterize backlash and impact within systems.
01 Mar 1981
TL;DR: A priori estimates for solutions of the quasilinear hyperbolic-parabolic equations governing the initial value problem describing the motion of compressible, viscous and heat-conductive, Newtonian fluids are derived by means of a new energy method as mentioned in this paper.
Abstract: : A priori estimates for solutions of the quasilinear hyperbolic-parabolic equations governing the initial value problem describing the motion of compressible, viscous and heat-conductive, Newtonian fluids are derived by means of a new energy method. This technique enables us to simplify and unify our previous results on the global existence in time and uniqueness of smooth solutions of these equations for sufficiently smooth and 'small' initial data and to obtain their rate of decay. (Author)
TL;DR: In this paper, a general method for constructing path intergrals for the nuclear many-body problem using continuous and overcomplete sets of vectors in the Hilbert space is presented. But this method is restricted to the case where the state labels play the role of classical coordinates which are quantized as bosons.
Abstract: We present a general method for constructing path intergrals for the nuclear many-body problem. This method uses continuous and overcomplete sets of vectors in the Hilbert space. The state labels play the role of classical coordinates which are quantized as bosons. The equations of motion for the classical coordinates are obtained by calculating the functional integral in the saddle point approximation. In the particular case where the over-complete set considered is the set of all Slater determinants, the classical equations of motion are the time-dependent Hartree-Fock equations. The functional integral provides a way of requantizing these classical equations. This quantization involves boson degrees of freedom and is in some cases very similar to the method of boson expansion. It is shown that the functional integral formalism provides a unifying framework to describe various approaches to the nuclear many-body problem.