A new method for free vibration of beams using theory of elasticity

J R Banerjee, Huijuan Su

Research output: Contribution to Book/Report typesConference contribution

Abstract

A refined theory which accounts for both longitudinal and transverse displacements as well as for the Poisson’s effect is developed for free vibration analysis of beams. This is achieved by using the theory of elasticity and symbolic computation. The formulation is based on a two dimensional (time dependent) stress field in which both normal and shear stresses along the length and depth/thickness of the beam are considered but the stresses in the lateral/width direction are ignored. Both Newton’s law and Hamilton’s principle are used to derive the governing differential equations of motion in free vibration. The derivation led to two coupled partial differential equations from which the time dependent terms are eliminated by assuming harmonic oscillation. In order to secure an approximate, but accurate formulation, the ordinary differential equations obtained in this way (still coupled in the space variables) are then solved by expanding the solution in series and then selecting appropriate terms from the series. An important feature of this work is the application of symbolic computation when obtaining the differential equations and their solution in explicit analytical form. The theory developed provides general solution for longitudinal and transverse displacements of the beam in free vibration in terms of arbitrary constants. Results for specific cases such as cantilever, simply-supported and built-in beams can be obtained by imposing the boundary conditions and then eliminating the arbitrary constants. Numerical evaluation of the theory has not been possible at present because the volume of the work needed has proved to be enormous and would take the paper further than it is intended. However, a numerical study will be undertaken at a later stage which will constitute a separate investigation of the problem. The theory presented can be extended further to develop the frequency dependent dynamic stiffness matrix of the beam for its free vibration analysis using the Wittrick-Williams algorithm.
Original languageEnglish
Title of host publication49th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, 16th AIAA/ASME/AHS Adaptive Structures Conference,10th AIAA Non-Deterministic Approaches Conference, 9th AIAA Gossamer Spacecraft Forum, 4th AIAA Multidisciplinary Design Optimization Specialists Conference
PublisherAmerican Institute of Aeronautics and Astronautics
ISBN (Electronic)9781600869938
ISBN (Print)9781563479380
DOIs
Publication statusE-pub ahead of print - 14 Jun 2012
Event49th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, 16th AIAA/ASME/AHS Adaptive Structures Conference,10th AIAA Non-Deterministic Approaches Conference, 9th AIAA Gossamer Spacecraft Forum, 4th AIAA Multidisciplinary Design Optimization Specialists Conference - Schaumburg, United States
Duration: 7 Apr 200810 Apr 2008

Conference

Conference49th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, 16th AIAA/ASME/AHS Adaptive Structures Conference,10th AIAA Non-Deterministic Approaches Conference, 9th AIAA Gossamer Spacecraft Forum, 4th AIAA Multidisciplinary Design Optimization Specialists Conference
CountryUnited States
CitySchaumburg
Period7/04/0810/04/08

Fingerprint

free vibration
elastic properties
differential equations
harmonic oscillation
formulations
stiffness matrix
partial differential equations
newton
shear stress
stress distribution
equations of motion
derivation
boundary conditions
evaluation

Keywords

  • Vibration
  • Vibration analysis
  • Beams
  • Theory of elasticity

Cite this

Banerjee, J. R., & Su, H. (2012). A new method for free vibration of beams using theory of elasticity. In 49th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, 16th AIAA/ASME/AHS Adaptive Structures Conference,10th AIAA Non-Deterministic Approaches Conference, 9th AIAA Gossamer Spacecraft Forum, 4th AIAA Multidisciplinary Design Optimization Specialists Conference American Institute of Aeronautics and Astronautics. https://doi.org/10.2514/6.2008-2083
Banerjee, J R ; Su, Huijuan. / A new method for free vibration of beams using theory of elasticity. 49th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, 16th AIAA/ASME/AHS Adaptive Structures Conference,10th AIAA Non-Deterministic Approaches Conference, 9th AIAA Gossamer Spacecraft Forum, 4th AIAA Multidisciplinary Design Optimization Specialists Conference. American Institute of Aeronautics and Astronautics, 2012.
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abstract = "A refined theory which accounts for both longitudinal and transverse displacements as well as for the Poisson’s effect is developed for free vibration analysis of beams. This is achieved by using the theory of elasticity and symbolic computation. The formulation is based on a two dimensional (time dependent) stress field in which both normal and shear stresses along the length and depth/thickness of the beam are considered but the stresses in the lateral/width direction are ignored. Both Newton’s law and Hamilton’s principle are used to derive the governing differential equations of motion in free vibration. The derivation led to two coupled partial differential equations from which the time dependent terms are eliminated by assuming harmonic oscillation. In order to secure an approximate, but accurate formulation, the ordinary differential equations obtained in this way (still coupled in the space variables) are then solved by expanding the solution in series and then selecting appropriate terms from the series. An important feature of this work is the application of symbolic computation when obtaining the differential equations and their solution in explicit analytical form. The theory developed provides general solution for longitudinal and transverse displacements of the beam in free vibration in terms of arbitrary constants. Results for specific cases such as cantilever, simply-supported and built-in beams can be obtained by imposing the boundary conditions and then eliminating the arbitrary constants. Numerical evaluation of the theory has not been possible at present because the volume of the work needed has proved to be enormous and would take the paper further than it is intended. However, a numerical study will be undertaken at a later stage which will constitute a separate investigation of the problem. The theory presented can be extended further to develop the frequency dependent dynamic stiffness matrix of the beam for its free vibration analysis using the Wittrick-Williams algorithm.",
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Banerjee, JR & Su, H 2012, A new method for free vibration of beams using theory of elasticity. in 49th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, 16th AIAA/ASME/AHS Adaptive Structures Conference,10th AIAA Non-Deterministic Approaches Conference, 9th AIAA Gossamer Spacecraft Forum, 4th AIAA Multidisciplinary Design Optimization Specialists Conference. American Institute of Aeronautics and Astronautics, 49th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, 16th AIAA/ASME/AHS Adaptive Structures Conference,10th AIAA Non-Deterministic Approaches Conference, 9th AIAA Gossamer Spacecraft Forum, 4th AIAA Multidisciplinary Design Optimization Specialists Conference, Schaumburg, United States, 7/04/08. https://doi.org/10.2514/6.2008-2083

A new method for free vibration of beams using theory of elasticity. / Banerjee, J R; Su, Huijuan.

49th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, 16th AIAA/ASME/AHS Adaptive Structures Conference,10th AIAA Non-Deterministic Approaches Conference, 9th AIAA Gossamer Spacecraft Forum, 4th AIAA Multidisciplinary Design Optimization Specialists Conference. American Institute of Aeronautics and Astronautics, 2012.

Research output: Contribution to Book/Report typesConference contribution

TY - GEN

T1 - A new method for free vibration of beams using theory of elasticity

AU - Banerjee, J R

AU - Su, Huijuan

PY - 2012/6/14

Y1 - 2012/6/14

N2 - A refined theory which accounts for both longitudinal and transverse displacements as well as for the Poisson’s effect is developed for free vibration analysis of beams. This is achieved by using the theory of elasticity and symbolic computation. The formulation is based on a two dimensional (time dependent) stress field in which both normal and shear stresses along the length and depth/thickness of the beam are considered but the stresses in the lateral/width direction are ignored. Both Newton’s law and Hamilton’s principle are used to derive the governing differential equations of motion in free vibration. The derivation led to two coupled partial differential equations from which the time dependent terms are eliminated by assuming harmonic oscillation. In order to secure an approximate, but accurate formulation, the ordinary differential equations obtained in this way (still coupled in the space variables) are then solved by expanding the solution in series and then selecting appropriate terms from the series. An important feature of this work is the application of symbolic computation when obtaining the differential equations and their solution in explicit analytical form. The theory developed provides general solution for longitudinal and transverse displacements of the beam in free vibration in terms of arbitrary constants. Results for specific cases such as cantilever, simply-supported and built-in beams can be obtained by imposing the boundary conditions and then eliminating the arbitrary constants. Numerical evaluation of the theory has not been possible at present because the volume of the work needed has proved to be enormous and would take the paper further than it is intended. However, a numerical study will be undertaken at a later stage which will constitute a separate investigation of the problem. The theory presented can be extended further to develop the frequency dependent dynamic stiffness matrix of the beam for its free vibration analysis using the Wittrick-Williams algorithm.

AB - A refined theory which accounts for both longitudinal and transverse displacements as well as for the Poisson’s effect is developed for free vibration analysis of beams. This is achieved by using the theory of elasticity and symbolic computation. The formulation is based on a two dimensional (time dependent) stress field in which both normal and shear stresses along the length and depth/thickness of the beam are considered but the stresses in the lateral/width direction are ignored. Both Newton’s law and Hamilton’s principle are used to derive the governing differential equations of motion in free vibration. The derivation led to two coupled partial differential equations from which the time dependent terms are eliminated by assuming harmonic oscillation. In order to secure an approximate, but accurate formulation, the ordinary differential equations obtained in this way (still coupled in the space variables) are then solved by expanding the solution in series and then selecting appropriate terms from the series. An important feature of this work is the application of symbolic computation when obtaining the differential equations and their solution in explicit analytical form. The theory developed provides general solution for longitudinal and transverse displacements of the beam in free vibration in terms of arbitrary constants. Results for specific cases such as cantilever, simply-supported and built-in beams can be obtained by imposing the boundary conditions and then eliminating the arbitrary constants. Numerical evaluation of the theory has not been possible at present because the volume of the work needed has proved to be enormous and would take the paper further than it is intended. However, a numerical study will be undertaken at a later stage which will constitute a separate investigation of the problem. The theory presented can be extended further to develop the frequency dependent dynamic stiffness matrix of the beam for its free vibration analysis using the Wittrick-Williams algorithm.

KW - Vibration

KW - Vibration analysis

KW - Beams

KW - Theory of elasticity

U2 - 10.2514/6.2008-2083

DO - 10.2514/6.2008-2083

M3 - Conference contribution

SN - 9781563479380

BT - 49th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, 16th AIAA/ASME/AHS Adaptive Structures Conference,10th AIAA Non-Deterministic Approaches Conference, 9th AIAA Gossamer Spacecraft Forum, 4th AIAA Multidisciplinary Design Optimization Specialists Conference

PB - American Institute of Aeronautics and Astronautics

ER -

Banerjee JR, Su H. A new method for free vibration of beams using theory of elasticity. In 49th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, 16th AIAA/ASME/AHS Adaptive Structures Conference,10th AIAA Non-Deterministic Approaches Conference, 9th AIAA Gossamer Spacecraft Forum, 4th AIAA Multidisciplinary Design Optimization Specialists Conference. American Institute of Aeronautics and Astronautics. 2012 https://doi.org/10.2514/6.2008-2083