An Analytical Model for the Vibration of Laminated Beams Including the Effects of Both Shear and Thickness Deformation in the Adhesive Layer

1986 ◽  
Vol 108 (1) ◽  
pp. 56-64 ◽  
Author(s):  
R. N. Miles ◽  
P. G. Reinhall
Keyword(s):  
Shear Deformation ◽  
Analytical Model ◽  
Equations Of Motion ◽  
Ritz Method ◽  
Adhesive Layer ◽  
Order Theory ◽  
Sixth Order ◽  
Viscoelastic Damping ◽  
Laminated Beams ◽  
Metal Layers

The equations of motion governing the vibration of a beam consisting of two metal layers bonded together with a soft viscoelastic damping adhesive are derived and solved. The adhesive is assumed to undergo both shear and thickness deformations during the vibration of the beam. In previous investigations the thickness deformation has been assumed to have negligible effect on the total damping. However, if the adhesive is very soft, and if at least one of the metal layers is stiff in bending, the thickness deformation in the adhesive can become the dominant damping mechanism. The analysis presented here comprises an extension of the well-known sixth order theory of DiTaranto, Mead, and Markus to include thickness deformation. The equations of motion are derived using Hamilton’s Principle and solutions are obtained by the Ritz method. It is shown that the use of a lightweight constraining layer which is stiff in bending will result in a design which is considerably more damped than a conventional configuration in which the adhesive is undergoing predominant shear deformation.

An analytical model of a rotating radial cantilever beam considering the coupling between bending, stretching and torsion

2021 ◽  
pp. 1-30
Author(s):  
Xianglin Wu ◽  
Yinghou Jiao ◽  
Zhao-Bo Chen
Keyword(s):  
Analytical Model ◽  
Cantilever Beam ◽  
Rotational Speed ◽  
Mode Coupling ◽  
Twist Angle ◽  
Equations Of Motion ◽  
Ritz Method ◽  
Arbitrary Cross Section ◽  
Softening Effect ◽  
Setting Angle

Abstract In this paper, the mode coupling between bending, stretching and torsional deformations is mainly studied by presenting an analytical model of a rotating cantilever beam with pre-twist angle and arbitrary cross section. Equations of motion of the beam are derived using Hamilton's principle. The Coriolis effect due to the coupling of the bending deformation and stretching deformation, the eccentricity caused by inconsistency between elastic center and centroid, spin softening effect, stress stiffening effect, shear deformation, and rotary inertia are included in the model. Equations of motion are solved by the Rayleigh-Ritz method. The natural frequencies obtained by the proposed analytical modal are in good agreement with those obtained by Finite Element Method (FEM) which proved the accuracy of the analytical model. Finally, the coupling between different mode components is studied in detail based on a quantitative method. The transformation/ conversion between different mode components is revealed, the influence of rotational speed, setting angle and pre-twist angle on this conversion mode is studied. Results show that a specific mode shape is usually composed of multiple mode components. The essence of mode coupling is the coupling between different mode components. The influence of rotational speed, setting angle and pre-twist angle on the mode coupling is that they cause the transformation/ conversion between different mode components.

Analysis of Electromechanical Performance of Energy Harvesting Skin Based on the Kirchhoff Plate Theory

2014 ◽  
Author(s):  
Heonjun Yoon ◽  
Byeng D. Youn ◽  
Heung S. Kim
Keyword(s):  
Energy Harvesting ◽  
Differential Equations ◽  
Analytical Model ◽  
Plate Theory ◽  
Equations Of Motion ◽  
Ritz Method ◽  
Electrical Circuit ◽  
Kirchhoff Plate ◽  
Mode Shapes ◽  
Kirchhoff Plate Theory

As a compact and durable design concept, energy harvesting skin (EH skin), which consists of piezoelectric patches directly attached onto the surface of a vibrating structure as one embodiment, has been recently proposed. This study aims at developing an electromechanically-coupled analytical model of the EH skin so as to understand its electromechanical behavior and get physical insights about important design considerations. Based on the Kirchhoff plate theory, the Hamilton’s principle is used to derive the differential equations of motion. The Rayleigh-Ritz method is implemented to calculate the natural frequency and the corresponding mode shapes of the EH skin. The electrical circuit equation is derived by substituting the piezoelectric constitutive relation into Gauss’s law. Finally, the steady-state output voltage is obtained by solving the differential equations of motion and electrical circuit equation simultaneously. The results of the analytical model are verified by comparing those of the finite element analysis (FEA) in a hierarchical manner.

Reflections on the Theory of Elastic Plates

1985 ◽  
Vol 38 (11) ◽  
pp. 1453-1464 ◽  
Author(s):  
Eric Reissner
Keyword(s):  
Shear Deformation ◽  
Fourth Order ◽  
Transverse Shear ◽  
Three Dimensional ◽  
Order Theory ◽  
Sixth Order ◽  
Interior Solution ◽  
Transverse Shear Deformation ◽  
Two Dimensional ◽  
Elastic Plates

We depart from a three-dimensional statement of the problem of small bending of elastic plates, for a survey of approximate two-dimensional theories, beginning with Kirchhoff’s fourth-order formulation. After discussing various variational statements of the three-dimensional problem, we describe the development of two-dimensional sixth-order theories by Bolle´, Hencky, Mindlin, and Reissner which take account of the effect of transverse shear deformation. Additionally, we report on an early analysis by Le´vy, on a direct two-dimensional formulation of sixth-order theory, on constitutive coupling of bending and stretching of laminated plates, on higher than sixth-order theories, and on an asymptotic analysis of sixth-order theory which leads to a fourth-order interior solution contribution with first-order transverse shear deformation effects included, as well as to a sequentially determined second-order edge zone solution contribution.

Ion acceleration by multiple laser pulses and their spontaneous electromagnetic fields in plasma

2008 ◽  
Vol 74 (1) ◽  
pp. 111-118
Author(s):  
FEN-CE CHEN
Keyword(s):  
Magnetic Fields ◽  
Electric Field ◽  
Analytical Model ◽  
Electromagnetic Fields ◽  
Equations Of Motion ◽  
Charged Particles ◽  
Laser Pulses ◽  
The Self ◽  
Electric And Magnetic Fields ◽  
Relativistic Equations

AbstractThe acceleration of ions by multiple laser pulses and their spontaneously generated electric and magnetic fields is investigated by using an analytical model for the latter. The relativistic equations of motion of test charged particles are solved numerically. It is found that the self-generated axial electric field plays an important role in the acceleration, and the energy of heavy test ions can reach several gigaelectronvolts.

scholarly journals Parametric structural modelling of fish bone active camber morphing aerofoils

2018 ◽  
Vol 29 (9) ◽  
pp. 2008-2026 ◽  
Author(s):  
Andres E Rivero ◽  
Paul M Weaver ◽  
Jonathan E Cooper ◽  
Benjamin KS Woods
Keyword(s):  
Analytical Model ◽  
Composite Laminates ◽  
Plate Theory ◽  
Ritz Method ◽  
Linear Equations ◽  
Variable Stiffness ◽  
Automated Analysis ◽  
Fish Bone ◽  
Two Dimensional ◽  
Wide Range

Camber morphing aerofoils have the potential to significantly improve the efficiency of fixed and rotary wing aircraft by providing significant lift control authority to a wing, at a lower drag penalty than traditional plain flaps. A rapid, mesh-independent and two-dimensional analytical model of the fish bone active camber concept is presented. Existing structural models of this concept are one-dimensional and isotropic and therefore unable to capture either material anisotropy or spanwise variations in loading/deformation. The proposed model addresses these shortcomings by being able to analyse composite laminates and solve for static two-dimensional displacement fields. Kirchhoff–Love plate theory, along with the Rayleigh–Ritz method, are used to capture the complex and variable stiffness nature of the fish bone active camber concept in a single system of linear equations. Results show errors between 0.5% and 8% for static deflections under representative uniform pressure loadings and applied actuation moments (except when transverse shear exists), compared to finite element method. The robustness, mesh-independence and analytical nature of this model, combined with a modular, parameter-driven geometry definition, facilitate a fast and automated analysis of a wide range of fish bone active camber concept configurations. This analytical model is therefore a powerful tool for use in trade studies, fluid–structure interaction and design optimisation.

scholarly journals A Semianalytical Approach for Free Vibration Characteristics of Functionally Graded Spherical Shell Based on First-Order Shear Deformation Theory

2019 ◽  
Vol 2019 ◽  
pp. 1-18 ◽  
Author(s):  
Fuzhen Pang ◽  
Cong Gao ◽  
Jie Cui ◽  
Yi Ren ◽  
Haichao Li ◽  
...  
Keyword(s):  
Free Vibration ◽  
Spherical Shell ◽  
Shear Deformation ◽  
Deformation Theory ◽  
Ritz Method ◽  
Shear Deformation Theory ◽  
Vibration Characteristics ◽  
Functionally Graded ◽  
Numerical Verification ◽  
First Order

This paper describes a unified solution to investigate free vibration solutions of functionally graded (FG) spherical shell with general boundary restraints. The analytical model is established based on the first-order shear deformation theory, and the material varies uniformly along the thickness of FG spherical shell which is divided into several sections along the meridian direction. The displacement functions along circumferential and axial direction are, respectively, composed by Fourier series and Jacobi polynomial regardless of boundary restraints. The boundary restraints of FG spherical shell can be easily simulated according to penalty method of spring stiffness technique, and the vibration solutions are obtained by Rayleigh–Ritz method. To verify the reliability and accuracy of the present solutions, the convergence and numerical verification have been conducted about different boundary parameters, Jacobi parameter, etc. The results obtained by the present method closely agree with those obtained from the published literatures, experiments, and finite element method (FEM). The impacts of geometric dimensions and boundary conditions on the vibration characteristics of FG spherical shell structure are also presented.

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