scholarly journals Free Vibrations of Beam System Structures with Elastic Boundary Conditions and an Internal Elastic Hinge

2013 ◽  
Vol 2013 ◽  
pp. 1-10
Author(s):  
Alejandro R. Ratazzi ◽  
Diana V. Bambill ◽  
Carlos A. Rossit
Keyword(s):  
Boundary Conditions ◽  
Dynamic Properties ◽  
Separation Of Variables ◽  
Equations Of Motion ◽  
Free Vibrations ◽  
Exact Expression ◽  
The Finite Element Method ◽  
Bending Vibration ◽  
Elastic Boundary ◽  
Beam System

The study of the dynamic properties of beam structures is extremely important for proper structural design. This present paper deals with the free in-plane vibrations of a system of two orthogonal beam members with an internal elastic hinge. The system is clamped at one end and is elastically connected at the other. Vibrations are analyzed for different boundary conditions at the elastically connected end, including classical conditions such as clamped, simply supported, and free. The beam system is assumed to behave according to the Bernoulli-Euler theory. The governing equations of motion of the structural system in free bending vibration are derived using Hamilton's principle. The exact expression for natural frequencies is obtained using the calculus of variations technique and the method of separation of variables. In the frequency analysis, special attention is paid to the influence of the flexibility and location of the elastic hinge. Results are very similar with those obtained using the finite element method, with values of particular cases of the model available in the literature, and with measurements in an experimental device.

scholarly journals Analytical Solution for Free Vibrations of a Moderately Thick Rectangular Plate

2013 ◽  
Vol 2013 ◽  
pp. 1-13 ◽  
Author(s):  
Ivo Senjanović ◽  
Marko Tomić ◽  
Nikola Vladimir ◽  
Dae Seung Cho
Keyword(s):  
Analytical Solution ◽  
Boundary Conditions ◽  
Thick Plate ◽  
Mixed Boundary ◽  
Separation Of Variables ◽  
Equations Of Motion ◽  
Mixed Boundary Conditions ◽  
Free Vibrations ◽  
Force Displacement ◽  
Good Agreement

In the present thick plate vibration theory, governing equations of force-displacement relations and equilibrium of forces are reduced to the system of three partial differential equations of motion with total deflection, which consists of bending and shear contribution, and angles of rotation as the basic unknown functions. The system is starting one for the application of any analytical or numerical method. Most of the analytical methods deal with those three equations, some of them with two (total and bending deflection), and recently a solution based on one equation related to total deflection has been proposed. In this paper, a system of three equations is reduced to one equation with bending deflection acting as a potential function. Method of separation of variables is applied and analytical solution of differential equation is obtained in closed form. Any combination of boundary conditions can be considered. However, the exact solution of boundary value problem is achieved for a plate with two opposite simply supported edges, while for mixed boundary conditions, an approximate solution is derived. Numerical results of illustrative examples are compared with those known in the literature, and very good agreement is achieved.

A Nonlinear Shear Deformable Nanoplate Model Including Surface Effects for Large Amplitude Vibrations of Rectangular Nanoplates with Various Boundary Conditions

2015 ◽  
Vol 07 (05) ◽  
pp. 1550076 ◽  
Author(s):  
Reza Ansari ◽  
Mostafa Faghih Shojaei ◽  
Vahid Mohammadi ◽  
Raheb Gholami ◽  
Mohammad Ali Darabi
Keyword(s):  
Boundary Conditions ◽  
Surface Stress ◽  
Equations Of Motion ◽  
Continuation Method ◽  
Free Vibrations ◽  
Model Parameters ◽  
Geometrically Nonlinear ◽  
Residual Surface Stress ◽  
Various Boundary Conditions ◽  
Shear Deformable

In this paper, a geometrically nonlinear first-order shear deformable nanoplate model is developed to investigate the size-dependent geometrically nonlinear free vibrations of rectangular nanoplates considering surface stress effects. For this purpose, according to the Gurtin–Murdoch elasticity theory and Hamilton's principle, the governing equations of motion and associated boundary conditions of nanoplates are derived first. Afterwards, the set of obtained nonlinear equations is discretized using the generalized differential quadrature (GDQ) method and then solved by a numerical Galerkin scheme and pseudo arc-length continuation method. Finally, the effects of important model parameters including surface elastic modulus, residual surface stress, surface density, thickness and boundary conditions on the vibration characteristics of rectangular nanoplates are thoroughly investigated. It is found that with the increase of the thickness, nanoplates can experience different vibrational behavior depending on the type of boundary conditions.

Investigation of Damping Potential of Strip Damper on a Real Turbine Blade

2016 ◽  
Author(s):  
M. Afzal ◽  
I. Lopez Arteaga ◽  
L. Kari ◽  
V. Kharyton
Keyword(s):  
Boundary Conditions ◽  
Dynamic Properties ◽  
Equations Of Motion ◽  
Iterative Solution ◽  
Element Model ◽  
Forced Response ◽  
Contact Forces ◽  
Response Analysis ◽  
Bladed Disk ◽  
Different Types

This paper investigates the damping potential of strip dampers on a real turbine bladed disk. A 3D numerical friction contact model is used to compute the contact forces by means of the Alternate Frequency Time domain method. The Jacobian matrix required during the iterative solution is computed in parallel with the contact forces, by a quasi-analytical method. A finite element model of the strip dampers, that allows for an accurate description of their dynamic properties, is included in the steady-state forced response analysis of the bladed disk. Cyclic symmetry boundary conditions and the multiharmonic balance method are applied in the formulation of the equations of motion in the frequency domain. The nonlinear forced response analysis is performed with two different types of boundary conditions on the strip: (a) free-free and (b) elastic, and their influence is analyzed. The effect of the strip mass, thickness and the excitation levels on the forced response curve is investigated in detail.

An Asymptotic Method to Analyze the Vibrations of an Elastic Layer

1969 ◽  
Vol 36 (1) ◽  
pp. 65-72 ◽  
Author(s):  
J. D. Achenbach
Keyword(s):  
Power Series ◽  
Boundary Conditions ◽  
Equations Of Motion ◽  
Free Vibrations ◽  
Forced Vibrations ◽  
Wave Number ◽  
Dimensionless Wave Number ◽  
Analogous Manner ◽  
Independent Variable ◽  
Thickness Variable

The displacement components for both free and forced vibrations are sought as power series of the dimensionless wave number ε, where ε = 2π × layer thickness/wavelength. For the free vibration problem the object is to determine the frequencies, which are also sought as power series of the dimensionless wave number. The displacement and frequency expansions are substituted in the displacement equations of motion and in the boundary conditions. By collecting terms of the same order εn, a system of second-order, inhomogeneous, ordinary differential equations of the Helmholtz type is obtained, with the thickness variable as independent variable, and with associated boundary conditions. For free vibrations, subsequent integration yields the coefficients of εn for the displacements and the frequencies for all modes, and in the whole range of frequencies, but in a range of dimensionless wave numbers 0 < ε < ε* < 1, where ε* increases as more terms are retained in the expansions. For forced vibrations, the amplitudes are determined in an analogous manner if the external surface tractions are of sinusoidal dependence on the in-plane coordinates and on time. The response to surface tractions of more general spatial dependence is obtained by Fourier superposition.

Forced Vibrations of Viscoelastic Timoshenko Beams

1976 ◽  
Vol 98 (3) ◽  
pp. 820-826 ◽  
Author(s):  
C. C. Huang ◽  
T. C. Huang
Keyword(s):  
Boundary Conditions ◽  
Laplace Transform ◽  
Attenuation Factor ◽  
Equations Of Motion ◽  
Bending Moment ◽  
Expansion Method ◽  
Free Vibrations ◽  
Forced Vibrations ◽  
Mode Shapes ◽  
Timoshenko Beams

In a previous paper, the correspondence principle has been applied to derive the differential equations of motion of viscoelastic Timoshenko beams with or without external viscous damping. To study free vibrations these equations are solved by Laplace transform and boundary conditions are applied to obtain the attenuation factor and the frequency of the damped free vibrations and mode shapes. The present paper continues to analyze this subject and deals with the responses in deflection, bending slope, bending moment and shear for forced vibrations. Laplace transform and appropriate boundary conditions have been applied. Examples are given and results are plotted. The solution of forced vibrations of elastic Timoshenko beams obtained as a result of reduction from viscoelastic case and by eigenfunction expansion method concludes the paper.

scholarly journals Vibration analysis of multi-stepped and multi-damaged parabolic arches using GDQ

2014 ◽  
Vol 2 (1) ◽  
Author(s):  
Erasmo Viola ◽  
Marco Miniaci ◽  
Nicholas Fantuzzi ◽  
Alessandro Marzani
Keyword(s):  
Boundary Conditions ◽  
Cross Sections ◽  
Natural Frequencies ◽  
Equations Of Motion ◽  
Free Vibrations ◽  
Mode Shapes ◽  
Internal Boundary ◽  
Radius Of Curvature ◽  
Inertia Effects ◽  
Circular Arch

AbstractThis paper investigates the in-plane free vibrations of multi-stepped and multi-damaged parabolic arches, for various boundary conditions. The axial extension, transverse shear deformation and rotatory inertia effects are taken into account. The constitutive equations relating the stress resultants to the corresponding deformation components refer to an isotropic and linear elastic material. Starting from the kinematic hypothesis for the in-plane displacement of the shear-deformable arch, the equations of motion are deduced by using Hamilton’s principle. Natural frequencies and mode shapes are computed using the Generalized Differential Quadrature (GDQ) method. The variable radius of curvature along the axis of the parabolic arch requires, compared to the circular arch, a more complex formulation and numerical implementation of the motion equations as well as the external and internal boundary conditions. Each damage is modelled as a combination of one rotational and two translational elastic springs. A parametric study is performed to illustrate the influence of the damage parameters on the natural frequencies of parabolic arches for different boundary conditions and cross-sections with localizeddamage.Results for the circular arch, derived from the proposed parabolic model with the derivatives of some parameters set to zero, agree well with those published over the past years.

Theoretical Analysis of Free Vibrations Based on a New High Order Shell Theory for Cylindrical Shells Conveying Fluid

2017 ◽  
Author(s):  
Ming Ji ◽  
Kazuaki Inaba
Keyword(s):  
Boundary Conditions ◽  
Shell Theory ◽  
Natural Frequencies ◽  
Equations Of Motion ◽  
Cylindrical Shells ◽  
Fluid Pressure ◽  
Free Vibrations ◽  
High Order ◽  
Out Of Plane ◽  
Conveying Fluid

The natural frequencies of free vibrations for thick cylindrical shells with clamped-clamped ends conveying fluid are investigated. Equations of motion and boundary conditions are derived by Hamilton’s principle based on the new high order shell theory. The hydrodynamic force is derived from the linearized potential flow theory. Besides, fluid pressure acting on the shell wall is gotten by the assumption of non-penetration condition. The out-of-plane and in-plane vibrations are coupled together due to the existence of fluid-solid-interaction (FSI). Under the assumption of harmonic motion, the dispersion relationships are presented. Using the method of frequency sweeping, the natural frequencies of symmetric modes and asymmetric modes corresponding to each flow velocity are found by satisfying the dispersion relationship equations and boundary conditions. Several numerical examples with different flow velocities and thickness are presented compared with previous thin shell theory and FEM results and show reasonable agreement. The effects of thickness are discussed.

Free and Forced Vibration of Coupled Beam Systems Resting on Variable Viscoelastic Foundations

2020 ◽  
Vol 20 (12) ◽  
pp. 2050141
Author(s):  
Jinpeng Su ◽  
Kun Zhang ◽  
Qiang Zhang ◽  
Ying Tian
Keyword(s):  
Boundary Conditions ◽  
Forced Vibration ◽  
Dynamic Properties ◽  
Least Square ◽  
Weighted Residual Method ◽  
Transient Vibration ◽  
Various Boundary Conditions ◽  
Beam System ◽  
Interface Potential ◽  
Forced Vibration Analysis

This paper presents a modified variational method for free and forced vibration analysis of coupled beam systems resting on various viscoelastic foundations. Non-uniform as well as uniform curved and straight Timoshenko beam components are considered in the coupled beam system. Using proper coordinate transformations, interactions among the beam components of the coupled beam system are accommodated by combining Lagrange multiplier method and least-square weighted residual method. Interface potential energy for various boundary conditions including the elastic ones is simultaneously formulated. Thus, the proposed method allows flexible choice of the admissible functions, regardless of the boundary conditions. Based on the proposed energy method, Winkler, Pasternak or even variable foundations distributed in a parabolic or sinusoidal manner can be easily introduced into the coupled beam systems. Two kinds of damping, namely the proportional and viscous damping, are also employed to model the energy dissipation of the viscoelastic foundations. Corresponding finite element (FE) simulations are performed where possible and good agreement is observed. Thus, great efficiency and accuracy of the present approach are demonstrated for free, steady-state and transient vibration of the coupled beam systems. The influences of the parameters of the variable viscoelastic foundations on the dynamic properties of the coupled beam system are also examined.

Formulation of Problem of Free Vibration Analysis of Isotropic Homogeneous Rectangular Mindlin Plates Having Variable Thickness With General Elastic Boundary Conditions

2017 ◽  
Author(s):  
Sangle Sourabh ◽  
Verma Shesha ◽  
Mali Kiran
Keyword(s):  
Shear Stress ◽  
Boundary Conditions ◽  
Variable Thickness ◽  
Ritz Method ◽  
Free Vibration Analysis ◽  
Free Vibrations ◽  
Element Analysis ◽  
Elastic Boundary ◽  
Constant Shear Stress ◽  
Transverse Deflection

This work presents a formulation for the free vibrations of isotropic homogeneous rectangular Mindlin plates with variable thickness. These plates are subjected to general boundary supports in present study. To obtain arbitrarily supported boundary conditions, new form of trigonometric series expansion functions is used as the admissible functions for transverse deflection and rotation due to bending. In order to account the constant shear stress assumption, a shear stress correction factor is taken into consideration. The Rayleigh-Ritz Method is employed in this formulation. The boundaries are assumed to have three set of springs to achieve required boundary condition. Thus the changes in boundary conditions can be easily obtained by varying the stiffness of these springs, without actually making any changes in the shape functions. In this study, FEA (Finite Element Analysis) has been carried out for the Mindlin plates, for simply supported and constrained on two opposite sides.

scholarly journals Free vibrations of planar serial frame structures in the case of axially functionally graded materials

2020 ◽  
pp. 17-17
Author(s):  
Aleksandar Obradovic ◽  
Slavisa Salinic ◽  
Aleksandar Tomovic
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Functionally Graded Materials ◽  
Separation Of Variables ◽  
Closed Form Solution ◽  
Free Vibrations ◽  
Functionally Graded ◽  
Frame Structures ◽  
Variable Cross ◽  
Graded Materials

This paper considers the problem of modal analysis and finding the closed-form solution to free vibrations of planar serial frame structures composed of Euler-Bernoulli beams of variable cross-sectional geometric characteristics in the case of axially functionally graded materials. Each of these beams is performing coupled axial and bending vibrations, where coupling occurs due to the boundary conditions at their joints. The numerical procedure for solving the system of partial differential equations, after the separation of variables, is reduced to solving the two-point boundary value problem of ordinary linear differential equations with nonlinear coefficients and linear boundary conditions. In this case, it is possible to transfer the boundary conditions and reduce the problem to the Cauchy initial value problem. Also, it is possible to analyze the influence of different parameters on the structure dynamic behavior. The method is applicable in the case of different boundary conditions at the right and left ends of such structures, as illustrated by an appropriate numerical example.

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