Longitudinal Vibration and Damping Analysis of Adhesively Bonded Double-Strap Joints

1992 ◽  
Vol 114 (3) ◽  
pp. 330-337 ◽  
Author(s):  
S. He ◽  
M. D. Rao
Keyword(s):  
Boundary Conditions ◽  
Complex Modulus ◽  
Longitudinal Vibration ◽  
Structural Parameters ◽  
Equations Of Motion ◽  
Adhesive Layer ◽  
Adhesively Bonded ◽  
Searching Strategy ◽  
Resonance Frequencies ◽  
Loss Factors

A mathematical model to study the longitudinal vibration of an adhesively bonded double-strap joint is presented in this paper. Energy method and Hamilton’s principle are used to derive the governing equations of motion and natural boundary conditions of the joint system. The adhesive is modeled as a viscoelastic material using complex modulus approach. Both the shear and longitudinal deformation in the adhesive layer are included in the analysis. The equations to predict the system resonance frequencies and loss factors are derived from the system natural and forced boundary conditions for the case of simply supported boundary conditions. A special searching strategy for finding the zeros of a complex determinant has been utilized to obtain the numerical results. The effects of the adhesive shear modulus and structural parameters such as lap ratio, adhesive and strap thickness on the system resonance frequencies and loss factors are also studied.

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.

Coupled Flexural and Torsional Vibration Analysis of Composite Beams

2019 ◽  
Author(s):  
Ehsan Sarfaraz ◽  
Jeremiah O. Afolabi ◽  
Hamid R. Hamidzadeh
Keyword(s):  
Boundary Conditions ◽  
Torsional Vibration ◽  
Analytical Method ◽  
Composite Beam ◽  
Equations Of Motion ◽  
Closed Form Solution ◽  
Composite Beams ◽  
Form Solution ◽  
Mode Shapes ◽  
Loss Factors

Abstract An analytical method is presented to determine the dynamic response of a coupled flexural and torsional vibration of composite beams. The general governing equations of motion are presented and a closed form solution for free vibration of the composite beam that demonstrates both geometric and material coupling is developed. The proposed solution is used to compute the natural frequencies, modal loss factors, and mode shapes of the composite beam for several modes of the coupled bending and torsional vibrations for any boundary conditions. In this analysis, hysteretic damping for the composite beam is considered and its effects on mode shapes and modal loss factors are determined. In addition, the variation of modal parameters such as bending displacement, bending slope, torsional rotation, shear force, bending moment, and torque along the beam for the first four mode shapes are presented for the clamped-free boundary conditions. Moreover, to verify the validity of the presented analytical method, results for the cases with no damping are compared with the previously established results and good agreement is achieved. The presented results can provide a guideline for selecting appropriate geometries and materials to design composite beams.

PROPAGATION AND RESONANCE OF LONGITUDINAL WAVES IN PRISMATIC RODS

1936 ◽  
Vol 14a (2) ◽  
pp. 48-55
Author(s):  
R. Ruedy
Keyword(s):  
Boundary Conditions ◽  
Poisson’S Ratio ◽  
Equations Of Motion ◽  
Poisson's Ratio ◽  
Longitudinal Waves ◽  
Fundamental Frequencies ◽  
Cubic Equation ◽  
Resonance Frequencies ◽  
Fourth Series ◽  
Shearing Motion

For vibrations involving shearing and rotation, and for those involving both distortion and dilatation, the equations of motion combined with the boundary conditions yield in the simplest case a cubic equation for the resonance frequencies; its solution depends on Poisson's ratio and on the resonance frequencies fx, fy, fz, which the rod possesses when in pure shearing motion in the direction of its three axes. Three series of resonance frequencies are obtained when fy and fz are constant and the frequencies of the overtones are inserted for fx. A fourth series of resonance frequencies begins above the highest of the fundamental frequencies fx, fy, fz.

Frequency and Loss Factors of Sandwich Beams under Various Boundary Conditions

1978 ◽  
Vol 20 (5) ◽  
pp. 271-282 ◽  
Author(s):  
D. K. Rao
Keyword(s):  
Exact Solution ◽  
Boundary Conditions ◽  
Equations Of Motion ◽  
Energy Approach ◽  
Sandwich Beams ◽  
Exact Results ◽  
Various Boundary Conditions ◽  
Complete Set ◽  
Loss Factors

A complete set of equations of motion and boundary conditions governing the vibration of sandwich beams are derived by using the energy approach. They are solved exactly for important boundary conditions. The computational difficulties that were encountered in previous attempts at the exact solution of these equations have been overcome by careful programming. These exact results are presented in the form of design graphs and formulae, and their usage is illustrated by examples.

Impact Response of an Adhesively Bonded Lap Joint

2014 ◽  
Author(s):  
Jack Chiu ◽  
Feridun Delale ◽  
Niell Elvin
Keyword(s):  
Numerical Solutions ◽  
Crack Nucleation ◽  
Equations Of Motion ◽  
Adhesive Layer ◽  
Numerical Approach ◽  
Lap Joints ◽  
Lap Joint ◽  
Orthotropic Plates ◽  
Adhesively Bonded ◽  
Special Cases

Failure of adhesively bonded joints is often dictated by the stresses developed within the adhesive layer, which are difficult to measure experimentally. While solutions, including closed-form solutions, exist for static cases, even numerical solutions are not easily obtainable for dynamic cases where the bonded layers are dissimilar in material and/or geometry. In this paper, we present a method to determine the dynamic stresses in the adhesive and adherends of adhesively bonded lap joints subjected to arbitrary dynamic end loads. In the formulation the adherends are treated as orthotropic plates while the adhesive layer is approximated as a tension-shear spring. The equations of motion result in a complex system of fourteen (14) partial differential equations in time and space. The equations are solved numerically using the Finite Difference Method (FDM). First, special cases where known solutions exist are solved to verify both the formulation and the numerical approach. Next, the problem of a lap joint subject to a remote, transmitted impact is considered and results are obtained. The planar distribution of stresses within the adhesive layer shows areas of dynamic stress concentration which may act as crack nucleation sites.

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 Actions, topological terms and boundaries in first-order gravity: A review

2016 ◽  
Vol 25 (04) ◽  
pp. 1630011 ◽  
Author(s):  
Alejandro Corichi ◽  
Irais Rubalcava-García ◽  
Tatjana Vukašinac
Keyword(s):  
Boundary Conditions ◽  
Symplectic Structure ◽  
Equations Of Motion ◽  
Hamiltonian Formalism ◽  
Action Principle ◽  
Internal Boundary ◽  
Four Dimensions ◽  
First Order ◽  
Conserved Charges ◽  
Boundary Terms

In this review, we consider first-order gravity in four dimensions. In particular, we focus our attention in formulations where the fundamental variables are a tetrad [Formula: see text] and a [Formula: see text] connection [Formula: see text]. We study the most general action principle compatible with diffeomorphism invariance. This implies, in particular, considering besides the standard Einstein–Hilbert–Palatini term, other terms that either do not change the equations of motion, or are topological in nature. Having a well defined action principle sometimes involves the need for additional boundary terms, whose detailed form may depend on the particular boundary conditions at hand. In this work, we consider spacetimes that include a boundary at infinity, satisfying asymptotically flat boundary conditions and/or an internal boundary satisfying isolated horizons boundary conditions. We focus on the covariant Hamiltonian formalism where the phase space [Formula: see text] is given by solutions to the equations of motion. For each of the possible terms contributing to the action, we consider the well-posedness of the action, its finiteness, the contribution to the symplectic structure, and the Hamiltonian and Noether charges. For the chosen boundary conditions, standard boundary terms warrant a well posed theory. Furthermore, the boundary and topological terms do not contribute to the symplectic structure, nor the Hamiltonian conserved charges. The Noether conserved charges, on the other hand, do depend on such additional terms. The aim of this manuscript is to present a comprehensive and self-contained treatment of the subject, so the style is somewhat pedagogical. Furthermore, along the way, we point out and clarify some issues that have not been clearly understood in the literature.

Experimental and Theoretical Nonlinear Dynamic Response of Intact and Cracked Bone-Like Specimens With Various Boundary Conditions

2007 ◽  
Vol 129 (5) ◽  
pp. 541-549 ◽  
Author(s):  
Erick Ogam ◽  
Armand Wirgin ◽  
Z. E. A. Fellah ◽  
Yongzhi Xu
Keyword(s):  
Boundary Conditions ◽  
Duffing Oscillator ◽  
Contact Friction ◽  
Nonlinear Dynamic Response ◽  
Various Boundary Conditions ◽  
Vibration Data ◽  
Element Simulation ◽  
Resonance Frequencies ◽  
Vibration Experiment ◽  
Freely Suspended

The potentiality of employing nonlinear vibrations as a method for the detection of osteoporosis in human bones is assessed. We show that if the boundary conditions (BC), relative to the connection of the specimen to its surroundings, are not taken into account, the method is apparently unable to differentiate between defects (whose detection is the purpose of the method) and nonrelevant features (related to the boundary conditions). A simple nonlinear vibration experiment is described which employs piezoelectric transducers (PZT) and two idealized long bones in the form of nominally-identical drinking glasses, one intact, but in friction contact with a support, and the second cracked, but freely-suspended in air. The nonlinear dynamics of these specimens is described by the Duffing oscillator model. The nonlinear parameters recovered from vibration data coupled to the linear phenomena of mode splitting and shifting of resonance frequencies, show that, despite the similar soft spring behavior of the two dynamic systems, a crack is distinguishable from a contact friction BC. The frequency response of the intact glass with contact friction BC is modeled using a direct steady state finite element simulation with contact friction.

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