scholarly journals Vibration Analysis of Beams Using Alternative Admissible Functions With Penalties

2021 ◽  
Author(s):  
Srividyadhare Kateel ◽  
Natalie Baddour
Keyword(s):  
Boundary Conditions ◽  
Vibration Analysis ◽  
Natural Frequencies ◽  
Mathematical Formulation ◽  
Mode Shapes ◽  
Geometrical Parameters ◽  
Bernoulli Beam ◽  
Euler Bernoulli Beam ◽  
Assumed Mode ◽  
Admissible Functions

Abstract Assumed mode methods are often used in vibrations analysis, where the choice of assumed mode affects the stability and useability of the method. System eigenfunctions are often used for these expansions, however a change in the boundary conditions usually results in a change in eigenfunction. This paper investigates the use of Alternative Admissible Functions (AAF) with penalties for the vibration analysis of an Euler-Bernoulli beam for different boundary conditions. A key advantage of the proposed approach is that the choice of AAF does not depend on the boundary conditions since the boundary conditions are modelled via penalty functions. The mathematical formulation of the system matrices, and the effect of beam geometry changes on the computed natural frequencies and modeshapes are presented. The computed natural frequencies and mode shapes show an excellent agreement when compared with closed-form Euler-Bernoulli beam values. The study reveals that with an increase in the stiffness of the beam, the values of the penalties need to be increased. The results of this study suggest that boundary conditions, as well as beam geometrical parameters should be considered when selecting appropriate values of the penalties.

scholarly journals The Effect of Axial Force on the Free Vibration of an Euler-Bernoulli Beam Carrying a Number of Various Concentrated Elements

2013 ◽  
Vol 20 (3) ◽  
pp. 357-367 ◽  
Author(s):  
Gürkan Şcedilakar
Keyword(s):  
Free Vibration ◽  
Vibration Analysis ◽  
Axial Load ◽  
Natural Frequencies ◽  
Free Vibration Analysis ◽  
Mode Shapes ◽  
The Finite Element Method ◽  
Bernoulli Beam ◽  
Euler Beam ◽  
Euler Bernoulli Beam

In this study, free vibration analysis of beams carrying a number of various concentrated elements including point masses, rotary inertias, linear springs, rotational springs and spring-mass systems subjected to the axial load was performed. All analyses were performed using an Euler beam assumption and the Finite Element Method. The beam used in the analyses is accepted as pinned-pinned. The axial load applied to the beam from the free ends is either compressive or tensile. The effects of parameters such as the number of spring-mass systems on the beam, their locations and the axial load on the natural frequencies were investigated. The mode shapes of beams under axial load were also obtained.

Calculation of the natural frequencies and mode shapes of a Euler–Bernoulli beam which has any combination of linear boundary conditions

2019 ◽  
Vol 25 (18) ◽  
pp. 2473-2479 ◽  
Author(s):  
Paulo J. Paupitz Gonçalves ◽  
Michael J. Brennan ◽  
Andrew Peplow ◽  
Bin Tang
Keyword(s):  
Boundary Conditions ◽  
Natural Frequencies ◽  
Dynamic Stiffness ◽  
New Method ◽  
Mode Shapes ◽  
Linear Boundary ◽  
Bernoulli Beam ◽  
Classical Boundary ◽  
Euler Bernoulli Beam

There are well-known expressions for natural frequencies and mode shapes of a Euler-Bernoulli beam which has classical boundary conditions, such as free, fixed, and pinned. There are also expressions for particular boundary conditions, such as attached springs and masses. Surprisingly, however, there is not a method to calculate the natural frequencies and mode shapes for a Euler–Bernoulli beam which has any combination of linear boundary conditions. This paper describes a new method to achieve this, by writing the boundary conditions in terms of dynamic stiffness of attached elements. The method is valid for any boundaries provided they are linear, including dissipative boundaries. Ways to overcome numerical issues that can occur when computing higher natural frequencies and mode shapes are also discussed. Some examples are given to illustrate the applicability of the proposed method.

An Exact Solution for the Natural Frequencies of Flexible Beams Undergoing Overall Motions

2003 ◽  
Vol 9 (11) ◽  
pp. 1221-1229 ◽  
Author(s):  
Ali H Nayfeh ◽  
S.A. Emam ◽  
Sergio Preidikman ◽  
D.T. Mook
Keyword(s):  
Exact Solution ◽  
Natural Frequencies ◽  
Three Dimensional ◽  
Beam Theory ◽  
Free Vibrations ◽  
Mode Shapes ◽  
Flexible Beam ◽  
Bernoulli Beam ◽  
Flexible Beams ◽  
Euler Bernoulli Beam

We investigate the free vibrations of a flexible beam undergoing an overall two-dimensional motion. The beam is modeled using the Euler-Bernoulli beam theory. An exact solution for the natural frequencies and corresponding mode shapes of the beam is obtained. The model can be extended to beams undergoing three-dimensional motions.

Free vibration analysis of a rectangular plate carrying multiple various concentrated elements

2003 ◽  
Vol 217 (2) ◽  
pp. 171-183 ◽  
Author(s):  
J-S Wu ◽  
H-M Chou ◽  
D-W Chen
Keyword(s):  
Finite Element ◽  
Free Vibration ◽  
Boundary Conditions ◽  
Rectangular Plate ◽  
Normal Mode ◽  
Vibration Analysis ◽  
Natural Frequencies ◽  
Free Vibration Analysis ◽  
Equation Of Motion ◽  
Mode Shapes

The dynamic characteristic of a uniform rectangular plate with four boundary conditions and carrying three kinds of multiple concentrated element (rigidly attached point masses, linear springs and elastically mounted point masses) was investigated. Firstly, the closed-form solutions for the natural frequencies and the corresponding normal mode shapes of a rectangular ‘bare’ (or ‘unconstrained’) plate (without any attachments) with the specified boundary conditions were determined analytically. Next, by using these natural frequencies and normal mode shapes incorporated with the expansion theory, the equation of motion of the ‘constrained’ plate (carrying the three kinds of multiple concentrated element) were derived. Finally, numerical methods were used to solve this equation of motion to give the natural frequencies and mode shapes of the ‘constrained’ plate. To confirm the reliability of previous free vibration analysis results, a finite element analysis was also conducted. It was found that the results obtained from the above-mentioned two approaches were in good agreement. Compared with the conventional finite element method (FEM), the approach employed in this paper has the advantages of saving computing time and achieving better accuracy, as can be seen from the existing literature.

scholarly journals Natural Frequencies and Mode Shapes of Statically Deformed Inclined Risers

2016 ◽  
Author(s):  
Feras K. Alfosail ◽  
Ali H. Nayfeh ◽  
Mohammad I. Younis
Keyword(s):  
Natural Frequencies ◽  
Equilibrium Configuration ◽  
Mode Shapes ◽  
Bernoulli Beam ◽  
Galerkin Approach ◽  
Inclination Angles ◽  
Linear Vibrations ◽  
Euler Bernoulli Beam ◽  
Beam Mode ◽  
Good Agreement

In this work, we investigate numerically the linear vibrations of inclined risers using the Galerkin approach. The riser is modeled as an Euler-Bernoulli beam accounting for the nonlinear mid-plane stretching and self-weight. After solving for the initial deflection of the riser due to self-weight, a Galerkin expansion of fifteen axially loaded beam mode shapes are used to solve the eigenvalue problem of the riser around the static equilibrium configuration. This yields the riser natural frequencies and exact mode shapes for various values of inclination angles and applied tension. The obtained results are validated against a boundary-layer analytical solution and are found in good agreement. This constructs a basis to study the nonlinear forced vibrations of inclined risers.

Vibration Analysis of a Stepped Beam by Using Adomian Decomposition Method

2012 ◽  
Vol 160 ◽  
pp. 292-296
Author(s):  
Qi Bo Mao ◽  
Yan Ping Nie ◽  
Wei Zhang
Keyword(s):  
Decomposition Method ◽  
Natural Frequencies ◽  
Adomian Decomposition Method ◽  
Free Vibrations ◽  
Continuity Condition ◽  
Mode Shapes ◽  
Bernoulli Beam ◽  
Stepped Beam ◽  
Adomian Decomposition ◽  
Euler Bernoulli Beam

The free vibrations of a stepped Euler-Bernoulli beam are investigated by using the Adomian decomposition method (ADM). The stepped beam consists two uniform sections and each section is considered a substructure which can be modeled using ADM. By using boundary condition and continuity condition equations, the dimensionless natural frequencies and corresponding mode shapes can be easily obtained simultaneously. The computed results for different boundary conditions are presented. Comparing the results using ADM to those given in the literature, excellent agreement is achieved.

Vibration and flutter analyses of cantilever trapezoidal honeycomb sandwich plates

2017 ◽  
Vol 21 (8) ◽  
pp. 2887-2920 ◽  
Author(s):  
Keivan Torabi ◽  
Hasan Afshari ◽  
Farhad Haji Aboutalebi
Keyword(s):  
Boundary Conditions ◽  
Critical Speed ◽  
Natural Frequencies ◽  
Shear Deformation Theory ◽  
Mode Shapes ◽  
Geometrical Parameters ◽  
Honeycomb Core ◽  
Governing Equations ◽  
Flow Angle ◽  
Aerodynamic Pressure

In this article, free flexural vibration and supersonic flutter analyses are studied for cantilevered trapezoidal plates composed of two homogeneous isotropic face sheets and an orthotropic honeycomb core. The plate is modeled based on the first-order shear deformation theory, and aerodynamic pressure of external flow with desired flow angle is estimated via the piston theory. For this goal, first applying the Hamilton's principle, the set of governing equations and boundary conditions are derived. Then, using a transformation of coordinates, the governing equations and boundary conditions are converted from the original coordinates into new computational ones. Finally, the differential quadrature method is employed and natural frequencies, corresponding mode shapes, and critical speed are numerically achieved. Accuracy of the proposed solution is confirmed by the finite element simulations and published experimental results. After the validation, effect of various parameters on the vibration and flutter characteristics of the plate are investigated. It is concluded that geometry of hexagonal cells in the honeycomb core has a weak effect on the natural frequencies and critical speed of the sandwich plate, whereas thickness of the honeycomb core has main influence on the natural frequencies and the critical speed. Besides, it is shown that the honeycomb core thickness has optimum values that lead to the most growth in the natural frequencies or critical speed. These optimum magnitudes can be taken into account by designers to increase the natural frequencies or expand flutter boundaries and make aircrafts safer in supersonic flights. It is also concluded that geometrical parameters of the hexagonal cells and thickness of the honeycomb core have no significant effect on the value of the critical flow angle.

Dynamic Responses of Two Beams Connected by a Spring-Mass Device

2012 ◽  
Vol 29 (1) ◽  
pp. 143-155 ◽  
Author(s):  
H.- P. Lin ◽  
D. Yang
Keyword(s):  
Natural Frequencies ◽  
Beam Theory ◽  
Free Vibrations ◽  
Dynamic Responses ◽  
Mode Shapes ◽  
Entire System ◽  
Bernoulli Beam ◽  
Continuous Beams ◽  
Experimental Validations ◽  
Euler Bernoulli Beam

AbstractThis paper deals with the transverse free vibrations of a system in which two beams are coupled with a spring-mass device. The dynamics of this system are coupled through the motion of the mass. The entire system is modeled as two two-span beams and each span of the continuous beams is assumed to obey the Euler-Bernoulli beam theory. Considering the compatibility requirements across each spring con-nection position, the eigensolutions (natural frequencies and mode shapes) of this system can be obtained for different boundary conditions. Some numerical results and experimental validations are presented to demonstrate the method proposed in this article.

Vibration analysis of a uniform pre-twisted rotating Euler–Bernoulli beam using the modified Adomian decomposition method

2017 ◽  
Vol 23 (9) ◽  
pp. 1345-1363 ◽  
Author(s):  
Desmond Adair ◽  
Martin Jaeger
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Vibration Analysis ◽  
Decomposition Method ◽  
Adomian Decomposition Method ◽  
Free Vibration Analysis ◽  
Published Data ◽  
Bernoulli Beam ◽  
Adomian Decomposition ◽  
Euler Bernoulli Beam

The governing equations for a pre-twisted rotating cantilever beam are derived and used for free vibration analysis of a pre-twisted rotating beam whose flexural displacements are coupled in two planes. First differential equations of motion of a rotating twisted beam, including terms due to centrifugal stiffening, are derived for an Euler–Bernoulli beam undergoing free natural vibrations. The general solutions of these equations are obtained on applying the Adomian modified decomposition method (AMDM). The AMDM allows the governing differential equations to become recursive algebraic equations and the boundary conditions to become simple algebraic frequency equations suitable for symbolic computation. With additional simple mathematical operations on the model, the natural frequencies and corresponding closed-form series solution of the mode shape can be obtained simultaneously. Two main advantages of the application of the AMDM are, for the cases considered here, its fast convergence rate to the solution with the high degree of accuracy. As the AMDM technique is systematic, it is found straight-forward to modify boundary conditions from one case to the next. Comparison of results with published data showed the present calculations to be in reasonable agreement.

Dynamics of a beam with multiple spring/mass attachments using binary asymptotic admissible functions

2020 ◽  
pp. 095440622094156
Author(s):  
S Vijay Kumar ◽  
Abhijit Sarkar ◽  
Chandramouli Padmanabhan
Keyword(s):  
Boundary Conditions ◽  
Natural Frequencies ◽  
Computational Procedure ◽  
Mode Shapes ◽  
Spring Stiffness ◽  
Lumped Mass ◽  
Element Simulation ◽  
Extremal Values ◽  
Parameter Ranges ◽  
Admissible Functions

The dynamics of an Euler–Bernoulli beam, with multiple spring/mass attachments, is investigated in this paper. A novel analytical approach for determining the natural frequencies and mode shapes of the system is presented. The formulation is applicable for any parameter value of the attached mass or spring stiffness. The calculation is performed using a minimal binary set of admissible functions for each spring/mass connection, with extremal values (zero or infinity) of the parameter. For example, to study a beam with n intermediate spring/mass connections, only 2 n admissible functions are required. These functions are used in a Rayleigh–Ritz-based energy formulation. The small number of functions used in the formulation leads to an efficient computational procedure. The results from the proposed formulation are compared with those from a finite element simulation, for different boundary conditions. The results obtained by the two methods are in excellent agreement for all boundary conditions as well as for different parameter ranges. Detailed design studies, on the effect of spring stiffness and lumped mass values, as well as their locations, on the natural frequencies and mode shapes, have been carried out. Design charts have also been generated to aid the designer of such structures.

Sign in / Sign up

Export Citation Format

Share Document