Use of computer algebra in Hamiltonian calculations

Banerjee, J.R.; Sobey, Adam; Su, Huijuan; Fitch, John

doi:10.1016/j.advengsoft.2007.03.013

Cited by 15 publications

(21 citation statements)

References 12 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Because of the harmonic oscillation hypothesis adopted for the freely vibrating Timoshenko beam as indicated by Eqs. (31) and (32) and also by the introduction of the non-dimensional length  = x/L, the expressions for the amplitudes of the shear force (V) and bending moment (M) arising from Eqs. (29), (30) and (48) will take the following form.…”

Section: Dynamic Stiffness Matrix Of a Timoshenko Beammentioning

confidence: 99%

“…(3), using the  operator, integrating by parts and then collecting terms. In an earlier publication, the entire procedure to generate the governing differential equations of motion and natural boundary conditions for bar or beam type structures was automated by Banerjee et al[31] by applying symbolic computation. In this way, the governing differential equation of motion of the Rayleigh-Love bar is obtained as[7,21] -product of the Hamiltonian formulation, the expression for the axial force f(x, t) follows from the natural boundary condition to give[7,21]…”

mentioning

confidence: 99%

See 1 more Smart Citation

An exact dynamic stiffness matrix for a beam incorporating Rayleigh–Love and Timoshenko theories

Banerjee

Ananthapuvirajah

2019

International Journal of Mechanical Sciences

Self Cite

View full text Add to dashboard Cite

An exact dynamic stiffness matrix for a beam is developed by integrating the Rayleigh-Love theory for longitudinal vibration into the Timoshenko theory for bending vibration. In the formulation, the Rayleigh-Love theory accounted for the transverse inertia in longitudinal vibration whereas the Timoshenko beam theory accounted for the effects of shear deformation and rotating inertia in bending vibration. The dynamic stiffness matrix is developed by solving the governing differential equations of motion in free vibration of a Rayleigh-Love bar and a Timoshenko beam and then imposing the boundary conditions for displacements and forces.Next the two dynamic stiffness theories are combined using a unified notation. The ensuing dynamic stiffness matrix is subsequently used for free vibration analysis of uniform and stepped bars as well as frameworks through the application of the Wittrick-Williams algorithm as solution technique. Illustrative examples are given to demonstrate the usefulness of the theory and some of the computed results are compared with published ones. The paper closes with some concluding remarks.

show abstract

Section: Dynamic Stiffness Matrix Of a Timoshenko Beammentioning

confidence: 99%

mentioning

confidence: 99%

An exact dynamic stiffness matrix for a beam incorporating Rayleigh–Love and Timoshenko theories

Banerjee

Ananthapuvirajah

2019

International Journal of Mechanical Sciences

Self Cite

View full text Add to dashboard Cite

show abstract

“…The entire formulation using DSM in this paper is accomplished in the real domain as opposed to previous formulations which used complex arithmetic when developing the element dynamic stiffness matrices [31,32]. Another important further development reported in this paper is the derivation of explicit algebraic expressions for the dynamic stiffness elements using symbolic computation [33][34][35]. The explicit expressions for the dynamic stiffness elements are particularly useful in optimisation studies and also when some, but not all of the stiffnesses are needed.…”

Section: Introductionmentioning

confidence: 99%

Free vibration of functionally graded beams and frameworks using the dynamic stiffness method

Banerjee

Ananthapuvirajah

2018

Journal of Sound and Vibration

Self Cite

View full text Add to dashboard Cite

The free vibration analysis of functionally graded beams (FGBs) and frameworks containing FGBs is carried out by applying the dynamic stiffness method and deriving the elements of the dynamic stiffness matrix in explicit algebraic form. The usually adopted rule that the material properties of the FGB vary continuously through the thickness according to a power law forms the fundamental basis of the governing differential equations of motion in free vibration. The differential equations are solved in closed analytical form when the free vibratory motion is harmonic. The dynamic stiffness matrix is then formulated by relating the amplitudes of forces to those of the displacements at the two ends of the beam. Next, the explicit algebraic expressions for the dynamic stiffness elements are derived with the help of symbolic computation. Finally the Wittrick-Williams algorithm is applied as solution technique to solve the free vibration problems of FGBs with uniform cross-section, stepped FGBs and frameworks consisting of FGBs. Some numerical results are validated against published results, but in the absence of published results for frameworks containing FGBs, consistency checks on the reliability of results are performed. The paper closes with discussion of results and conclusions.

show abstract

“…Substituting T and U from Equations (8) and 9into Equation 11, using the δ operator, integrating each term by parts, and then collecting terms yield the governing differential equations and natural boundary conditions in free vibration of the FGB. The entire procedure has been processed through the application of symbolic computation [12]. The following governing differential equations are eventually obtained as,…”

Section: Property Parametersmentioning

confidence: 99%

“…The kinetic and potential energies of the FGB are then formulated by using the Timoshenko beam theory which accounts for the effects of shear deformation and rotary inertia. Next, the governing differential equations of motion in free vibration are derived using Hamilton's principle and making use of symbolic computation [12]. The expressions for axial force, shear force and bending moment at any crosssection of the FGB are obtained as a by-product of the Hamiltonian formulation.…”

Section: Introductionmentioning

confidence: 99%

Free Vibration of a Functionally Graded Timoshenko Beam using the Dynamic Stiffness Method

Su¹

Civil-Comp Proceedings

Self Cite

View full text Add to dashboard Cite

The dynamic stiffness method is used to investigate the free vibration behaviour of a functionally graded beam (FGB). The material properties of the FGB are assumed to vary in the thickness direction based on a power-law. The kinetic and potential energies of the beam are formulated using the Timoshenko beam theory. The governing differential equations of motion in free vibration for the FGB are derived using Hamilton's principle. The analytical expressions for axial force, shear force and bending moments at any cross-section of the FGB are obtained as a by-product of the Hamiltonian formulation. The differential equations are solved in closed analytical form for harmonic oscillation. The dynamic stiffness matrix of the FGB is then formulated by relating the amplitudes of forces and displacements at the ends of the beam. The Wittrick-Williams algorithm is used as solution technique to yield natural frequencies and mode shapes of the FGB. A parametric study is carried out by varying significant beam parameters and boundary conditions. The investigation required a substantial amount of validation exercise to confirm the predictable accuracy of the dynamic stiffness method. The results are discussed and some concluding remarks are made.

show abstract

Use of computer algebra in Hamiltonian calculations

Cited by 15 publications

References 12 publications

An exact dynamic stiffness matrix for a beam incorporating Rayleigh–Love and Timoshenko theories

An exact dynamic stiffness matrix for a beam incorporating Rayleigh–Love and Timoshenko theories

Free vibration of functionally graded beams and frameworks using the dynamic stiffness method

Free Vibration of a Functionally Graded Timoshenko Beam using the Dynamic Stiffness Method

Contact Info

Product

Resources

About