J.R. Banerjee

Dynamic stiffness formulation for structural elements: A general approach

Abstract A general theory to develop the dynamic stiffness matrix of a structural element is outlined. Substantial saving in computer time can be achieved if explicit analytical expressions for the elements of the dynamic stiffness matrix are used instead of numerical methods. Such expressions can be derived with the help of symbolic computation. The application of the dynamic stiffness matrix to calculate the natural frequencies of a structure is discussed with particular reference to the Wittrick-Williams algorithm. The method presented is fairly general.

Free vibration of composite beams - An exact method using symbolic computation

An exact dynamic stiffness matrix method has been developed to predict the free vibration characteristics of composite beams (or simple structures assembled from them) for which the bending and torsional displacements are (materially) coupled. To achieve this, an explicit expression is presented for each of the elements of the dynamic stiffness matrix of a bending-torsion coupled composite beam. This was made possible by performing symbolic computing with the help of the package Reduce. Programming the stiffness expressions in Fortran on a SUN SPARC station indicates about 75% savings in computer time when compared with the matrix inversion method normally adopted in the absence of such expressions. The derived dynamic stiffness matrix is then used in conjunction with the Wittrick-Williams algorithm to compute the natural frequencies and mode shapes of composite beams with substantial coupling between bending and torsional displacements. The results obtained from the present theory are compared with those available in the literature and discussed.

Free vibration of axially loaded composite Timoshenko beams using the dynamic stiffness matrix method

Abstract The free vibration analysis of axially loaded composite Timoshenko beams is carried out by using the dynamic stiffness matrix method. This is accomplished by developing an exact dynamic stiffness matrix of a composite beam with the effects of axial force, shear deformation and rotatory inertia taken into account, i.e. it is for an axially loaded composite Timoshenko beam. The theory includes the (material) coupling between the bending and torsional modes of deformations which is usually present in laminated composite beams due to ply orientation. An analytical expression for each of the elements of the dynamic stiffness matrix is derived by rigorous application of the symbolic computing package reduce . Use of such expressions leads to substantial savings in computer time when compared with numerical methods usually adopted in the absence of such expressions. The application of the dynamic stiffness matrix is demonstrated by investigating the free vibration characteristics of an example composite beam for which some comparative results are available. The solution technique used to yield the natural frequencies is that of the Wittrick–Williams algorithm. The effects of axial force, shear deformation and rotatory inertia on the natural frequencies are demonstrated. The theory developed has applications to composite wings and helicopter blades.

Development of an exact dynamic stiffness matrix for free vibration analysis of a twisted Timoshenko beam

Abstract An exact dynamic stiffness matrix for a twisted Timoshenko beam is developed in this paper in order to investigate its free vibration characteristics. First the governing differential equations of motion and the associated natural boundary conditions of a twisted Timoshenko beam undergoing free natural vibration are derived using Hamiltons principle. The inclusion of a given pretwist together with the effects of shear deformation and rotatory inertia, gives rise in free vibration to four coupled second order partial differential equations of motion involving bending displacements and bending rotations in two planes. For harmonic oscillation these four partial differential equations are combined into an eighth order ordinary differential equation, which is identically satisfied by all components of bending displacements and bending rotations. This difficult task has become possible only with the help of symbolic computation. Next the exact solution of the differential equation is obtained in completely general form in terms of eight arbitrary constants. This is followed by application of boundary conditions for displacements and forces. The procedure leads to the formation of the dynamics stiffness matrix of the twisted Timoshenko beam relating harmonically varying forces with harmonically varying displacements at its ends. The resulting dynamic stiffness matrix is used with particular reference to the Wittrick–Williams algorithm to compute the natural frequencies and mode shapes of a twisted Timoshenko beam with cantilever end condition. The exact results from the present theory are compared with numerically simulated results using simpler theories, and some conclusions are drawn.

Coupled bending-torsional dynamic stiffness matrix for timoshenko beam elements

Abstract Analytical expressions for the coupled bending-torsional dynamic stiffness matrix elements of a uniform Timoshenko beam element are derived in an exact sense by solving the governing differential equations of motion of the element. Application of the developed theory in the context of wings, blades and grillages is discussed with particular reference to an established algorithm. Programming the derived stiffness expressions on a VAX computer indicates about 87% savings in computer time when compared with the matrix inversion method normally adopted in the absence of such expressions. The correctness of the stiffness expressions is numerically checked up to machine accuracy against the corresponding stiffnesses from the inversion method. The stiffnesses are also checked up to nine figure accuracy against those obtained from a comparable approximate method.

Free vibration of sandwich beams using the dynamic stiffness method

The free vibration analysis of symmetric sandwich beams is carried out in this paper by using the dynamic stiffness method. First the governing partial differential equations of motion of a three-layered symmetric sandwich beam undergoing free natural vibration are derived using Hamiltons principle. The formulation led to two partial differential equations that are both coupled in axial and bending deformations. While seeking solution for harmonic oscillation, the two equations are combined into one sixth-order ordinary differential equation, which applies to both axial and bending displacements. This procedure was facilitated by the use of symbolic computation with the package REDUCE. Closed form analytical solution of the sixth order differential equation is then obtained in its most general form in terms of six arbitrary constants. Expressions for axial force, shear force and bending moment are also obtained in terms of the six arbitrary constants. Next the boundary conditions for displacements and forces at the ends of the sandwich beam are applied to eliminate the constants. This essentially casts the equations in the form of element dynamic stiffness matrix of the sandwich beam relating harmonically varying forces with harmonically varying displacements. The resulting dynamic stiffness matrix is then applied in conjunction with the Wittrick-Williams algorithm to compute the natural frequencies and mode shapes of an example sandwich beam. Numerical results are discussed and compared with those available in the literature. This is followed by conclusions.

Exact dynamic stiffness matrix of a bending-torsion coupled beam including warping

Abstract It is known that an allowance for warping stiffness can change the natural frequencies of thin-walled open section beams substantially. The purpose of this paper is to investigate the magnitude of such changes by using an exact member theory. This is achieved by formulating an exact dynamic stiffness matrix for a typical beam member from established theory and linking this to a new and convenient procedure which extends the well-known Wittrick-Williams algorithm to ensure convergence upon any desired natural frequency. Numerical results are given for both single and continuous beams of the channel section for which some comparative results are available in the literature. The effect of warping stiffness on the natural frequencies is discussed and it is concluded that substantial error can be incurred if the effect is ignored.

Frequency equation and mode shape formulae for composite Timoshenko beams

Exact expressions for the frequency equation and mode shapes of composite Timoshenko beams with cantilever end conditions are derived in explicit analytical form by using symbolic computation. The effect of material coupling between the bending and torsional modes of deformation together with the effects of shear deformation and rotatory inertia is taken into account when formulating the theory (and thus it applies to a composite Timoshenko beam). The governing differential equations for the composite Timoshenko beam in free vibration are solved analytically for bending displacements, bending rotation and torsional rotations. The application of boundary conditions for displacement and forces for cantilever end condition of the beam yields the frequency equation in determinantal form. The determinant is expanded algebraically, and simplified in an explicit form by extensive use of symbolic computation. The expressions for the mode shapes are also derived in explicit form using symbolic computation. The method is demonstrated by an illustrative example of a composite Timoshenko beam for which some published results are available.

Coupled bending-torsional dynamic stiffness matrix of an axially loaded Timoshenko beam element

Abstract Analytical expressions for the coupled bending-torsional dynamic stiffness matrix terms of an axially loaded uniform Timoshenko beam element are derived in an exact sense by solving the governing differential equations of motion of the element. The symbolic computing package REDUCE has been used to generate an analytical expression for each of the dynamic stiffness terms in a concise form. For check purposes, numerical values of the dynamic stiffness matrix terms were obtained using the derived explicit expressions as well as by an alternative nonanalytical method based on matrix inversions and matrix multiplications. Stiffnesses obtained from both methods agreed with each other to machine accuracy. Application of the developed theory is discussed with particular reference to an established algorithm. The influence of axial force, shear deformation and rotatory inertia on the natural frequencies of a bending-torsion coupled beam with cantilever end-conditions is demonstrated by numerical results. Such results are not generally available in the literature. Therefore, results obtained by partially restricting the present theory are compared with the existing literature wherever possible. The results indicate that the method is accurate and efficient.

Free vibration analysis of a twisted beam using the dynamic stiffness method

Abstract An exact dynamics stiffness matrix is developed and subsequently used for free vibration analysis of a twisted beam whose flexural displacements are coupled in two planes. First the governing differential equations of motion of the twisted beam undergoing free natural vibration are derived using Hamiltons principle. Next the general solutions of these equations are obtained when the oscillatory motion of the beam is harmonic. This is followed by application of boundary conditions for displacements and forces, which essentially leads to the formation of the dynamics stiffness matrix of the twisted beam relating harmonically varying forces with harmonically varying displacements at its ends. The resulting dynamic stiffness matrix is used in connection with the Wittrick–Williams algorithm to compute natural frequencies and mode shapes of a twisted beam with cantilever end condition. These are compared with previously published results to confirm the accuracy of the method, and some conclusions are drawn.