On the Vibrations of a Laminated Body

1968 ◽  
Vol 35 (4) ◽  
pp. 689-696 ◽  
Author(s):  
J. D. Achenbach ◽  
C. T. Sun ◽  
G. Herrmann
Keyword(s):  
Boundary Conditions ◽  
Equations Of Motion ◽  
Sufficient Conditions ◽  
Low Frequency ◽  
Continuum Theory ◽  
Acoustic Mode ◽  
Governing Equations ◽  
Natural Boundary ◽  
Natural Boundary Conditions ◽  
Laminated Layer

A continuum theory for a laminated medium is further developed in this paper. Constitutive equations, stress equations of motion, and natural boundary conditions are presented, and sufficient conditions for a unique solution are discussed. The governing equations and boundary conditions are employed to study the thickness-twist motion of a laminated layer. For every nodal number there is a low frequency acoustic mode and a high frequency optical mode. The frequencies of the acoustic modes are compared with the corresponding frequencies predicted by the effective modulus theory, and the relative magnitudes of the material parameters for which these frequencies are substantially at variance are indicated.

Meshless Integral Method for Analysis of Elastoplastic Geotechnical Materials

2010 ◽  
Author(s):  
Jianfeng Ma ◽  
Joshua David Summers ◽  
Paul F. Joseph
Keyword(s):  
Boundary Conditions ◽  
Function Approximation ◽  
Integral Method ◽  
Weak Form ◽  
Constitutive Law ◽  
Governing Equations ◽  
Natural Boundary ◽  
Pressure Sensitive ◽  
Support Domain ◽  
Natural Boundary Conditions

The meshless integral method based on regularized boundary equation [1][2] is extended to analyze elastoplastic geotechnical materials. In this formulation, the problem domain is clouded with a node set using automatic node generation. The sub-domain and the support domain related to each node are also generated automatically using algorithms developed for this purpose. The governing integral equation is obtained from the weak form of elastoplasticity over a local sub-domain and the moving least-squares approximation is employed for meshless function approximation. The geotechnical materials are described by pressure-sensitive multi-surface Drucker-Prager/Cap plasticity constitutive law with hardening. A generalized collocation method is used to impose the essential boundary conditions and natural boundary conditions are incorporated in the system governing equations. A comparison of the meshless results with the FEM results shows that the meshless integral method is accurate and robust enough to solve geotechnical materials.

Governing Equations for Vibrating Constrained-Layer Damping Sandwich Plates and Beams

1972 ◽  
Vol 39 (4) ◽  
pp. 1041-1046 ◽  
Author(s):  
M.-J. Yan ◽  
E. H. Dowell
Keyword(s):  
Differential Equation ◽  
Boundary Conditions ◽  
Numerical Results ◽  
Sandwich Plates ◽  
Constrained Layer Damping ◽  
Governing Equations ◽  
Natural Boundary ◽  
Comparison With Experiment ◽  
Preliminary Comparison ◽  
Natural Boundary Conditions

For constrained-layer damping a simple differential equation for nonsymmetric sandwich plates or beams made of isotropic and homogeneous layers is deduced. The natural boundary conditions associated with this equation are also derived. Typical numerical results are presented including a preliminary comparison with experiment.

Dynamic Variational Principles for Discontinuous Elastic Fields

1979 ◽  
Vol 23 (02) ◽  
pp. 115-122 ◽  
Author(s):  
M. Cengiz Dökmeci
Keyword(s):  
Variational Principles ◽  
Initial Conditions ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Governing Equations ◽  
Natural Boundary ◽  
Dimensional Theory ◽  
Elastic Fields ◽  
Complete Set ◽  
Natural Boundary Conditions

Various forms of variational principles are derived for the three-dimensional theory of elastodynamics. The continuity requirements on the fields of stresses or strains and/or displacements are relaxed through Friedrichs's transformation. Thus, the generalized forms of certain types of earlier variational principles' are systematically constructed using a basic principle of physics. The variational principles derived herein are shown to generate, as the appropriate Euler equations, the complete set of the governing equations of linear elastodynamics, that is, the stress equations of motion, the strain displacement relations, the mixed natural boundary conditions, the constitutive equations, the natural initial conditions, and the jump conditions. Similarly, generalized variational principles are established for the nonlinear theory of elastodynamics, for the incremental motions in linear elasticity, and for an elastic Cosserat continuum, as well.

Simulation of Engine Vibration on Nonlinear Hydraulic Engine Mounts

2005 ◽  
Author(s):  
A. R. Ohadi ◽  
G. Maghsoodi
Keyword(s):  
Equations Of Motion ◽  
Low Frequency ◽  
Governing Equations ◽  
Engine Mounts ◽  
Engine Vibration ◽  
Engine Mount ◽  
Rotational Motions ◽  
The Time Domain ◽  
Nonlinear Equations Of Motion ◽  
Four Cylinders

In this paper, vibration behavior of engine on nonlinear hydraulic engine mount including inertia track and decoupler is studied. In this regard, after introducing the nonlinear factors of this mount (i.e. inertia and decoupler resistances in turbulent region), the vibration governing equations of engine on one hydraulic engine mount are solved and the effect of nonlinearity is investigated. In order to have a comparison between rubber and hydraulic engine mounts, a 6 degree of freedom four cylinders V-shaped engine under inertia and balancing masses forces and torques is considered. By solving the time domain nonlinear equations of motion of engine on three inclined mounts, translational and rotational motions of engines body are obtained for different engine speeds. Transmitted base forces are also determined for both types of engine mount. Comparison of rubber and hydraulic mounts indicates the efficiency of hydraulic one in low frequency region.

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.

scholarly journals Elements of Bi-cubic Polynomial Natural Spline Interpolation for Scattered Data: Boundary Conditions Meet Partition of Unity Technique

2020 ◽  
Vol 8 (4) ◽  
pp. 994-1010
Author(s):  
Weizhi Xu
Keyword(s):  
Boundary Conditions ◽  
Spline Interpolation ◽  
Partition Of Unity ◽  
Scattered Data ◽  
Rectangular Domain ◽  
Natural Boundary ◽  
Natural Spline ◽  
Cubic Polynomial ◽  
Natural Boundary Conditions ◽  
Cubic Polynomials

This paper investigates one kind of interpolation for scattered data by bi-cubic polynomial natural spline, in which the integral of square of partial derivative of two orders to x and to y for the interpolating function is minimal (with natural boundary conditions). Firstly, bi-cubic polynomial natural spline interpolations with four kinds of boundary conditions are studied. By the spline function methods of Hilbert space, their solutions are constructed as the sum of bi-linear polynomials and piecewise bi-cubic polynomials. Some properties of the solutions are also studied. In fact, bi-cubic natural spline interpolation on a rectangular domain is a generalization of the cubic natural spline interpolation on an interval. Secondly, based on bi-cubic polynomial natural spline interpolations of four kinds of boundary conditions, and using partition of unity technique, a Partition of Unity Interpolation Element Method (PUIEM) for fitting scattered data is proposed. Numerical experiments show that the PUIEM is adaptive and outperforms state-of-the-art competitions, such as the thin plate spline interpolation and the bi-cubic polynomial natural spline interpolations for scattered data.

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