On the eigensolution of elastically connected column

Andrew Watson (1251807)

W.P. Howson (7120709)

English

Loughborough University Institutional Repository

ISVCS10, 10th International Symposium on Vibrations of Continuous SystemsStanley Hotel, Estes Park, Colorado, USA, July 26 - 31, 2015On the eigensolution of elastically connected columnsW. Paul Howson∗ and Andrew Watson#∗ Independent ConsultantGwanwyn, Craig Penlline, CF717RT, UKw.p.howson@gmail.com# Dept. of Aero. and Auto. EngineeringLoughborough UniversityLoughborough, LE11 3TU, UKa.watson@lboro.ac.ukIntroductionConsideration is given to determining the eigensolution of a class of structures comprising n suit-ably related parallel columns that are connected to each other, and possibly also to foundations,by uniformly distributed Winkler interfaces of unequal stiffness. For conciseness, the body of thework is written in the context of elastic critical buckling. However, since the duality of the eigen-problem posed by buckling and vibration is well known, the extension to vibration is straightfor-ward.Initially the coupled fourth order differential equations that define the system are developed fromfirst principles and arranged in the form of a generalized symmetric linear eigenvalue problem.Exact solution of these equations leads to n, uncoupled substitute systems, each of which yieldsan infinite number of critical buckling loads that, when arranged in ascending order, comprise thecomplete spectrum of critical buckling loads of the original problem. Thus, if only the fundamentalcritical buckling load is required, then only one substitute system needs to be solved.Each substitute system is relatively simple and describes the buckling of a single unified mem-ber, but supported on a Winkler foundation of different magnitude in each case. However, theexact solutions required from each substitute system necessitate the closed form solution of a tran-scendental eigenvalue problem. This is achieved in the present case by utilizing an exact elasticstiffness matrix in conjunction with the Wittrick-Williams algorithm, which guarantees that anydesired critical buckling load can be calculated to any desired accuracy with the certain knowledgethat none have been missed. The corresponding modes of vibration are then recovered by backsubstitution for each substitute system and subsequently related back to the individual membersof the original structure. This approach also enables some of the powerful features of the stiffnessmethod to be utilized to model more complex structures. A simple example is given to clarify theapproach.TheoryFigure 1 defines the structural configuration and the positive directions of the member forces anddisplacements, from which the equations of vertical and moment equilibrium for typical memberi can be deduced straightforwardly. These are presented with the appropriate constitutive relation-ship in Equations (1)-(2)−Qi+(Qi+ dQidx dx)+ kidxVi−1− (ki+ ki−1)dxVi+ ki+1dxVi+1 = 0 (1)Qidx+Mi− (Mi+ dMidx dx)+PidVidxdx= 0 Mi =−EiIi d2Vidx2(2)Lk11 1 2k21 2 21 i−1 2ki1 i 2ki+11 i+1 21 n−1 2kn1 n 2kn+1(a)xy, V , QΨ, MdxVi+1Qi+1Pi+1Mi+1Qi+1+∂Qi+1∂xdxPi+1Mi+1+∂Mi+1∂xdx∂Vi+1∂xdxki+1ViQiPiMiQi+∂Qi∂xdxPiMi+∂Mi∂xdx∂Vi∂xdxkiVi−1Qi−1Pi−1Mi−1Qi−1+∂Qi−1∂xdxPi−1Mi−1+∂Mi−1∂xdx∂Vi−1∂xdx(b)Figure 1: (a) General structure orientation. ki is a typical Winkler stiffness per unit length.(b) Positive member forces and displacements relating to an elemental length of typical member i.Eliminating Qi and Mi and introducing the non-dimensional length parameter ξ = x/L yields−κi[D4+ pi2D2]Vi+ kiVi−1− (ki+ ki+1)Vi+ ki+1Vi+1 = 0 (3)whereD=ddξ, κi =EiIiL4, pi2 =PiL2EiIi(4)The current approach now requires that pi2 is a constant for each member. i.e. thatpi2 = p2 (5)Equation (3) can then be written for a general member as−kiVi−1+(ki+ ki+1)Vi− ki+1Vi+1−κiλVi = 0 (6)whereλ =−[D4+ p2D2] (7)Hence the corresponding equations for the first (i= 1) and last (i= n) members are, respectively(k1+ k2)V1− k2V2−κ1λV1 = 0 (8)and−knVn−1+(kn+ kn+1)Vn−κnλVn = 0 (9)A complete set of equations for an n level system can now be assembled from Equations (6), (8)and (9)k1+ k2 −k2. . .−ki ki+ ki+1 −ki+1. . .−kn kn+ kn+1−λκ1. . .κi. . .κnV1...Vi...Vn=0...0...0 (10)where zeros have been omitted for clarity. Equation (10) can therefore be written for any appro-priate formulation as(k−λκ)V= 0 (11)The form of Equation (11) is that of a generalized symmetric linear eigenvalue problem, for whicha number of standard routines are available for calculating the eigenvalues, λ , and correspondingeigenvectors, V.Substitute systemsThe n values of λ that satisfy the linear eigenvalue problem define a family of second order differ-ential operators that satisfy the original problem and which are given by Equation (7) asD4+ p2D2 =−λi i= 1,2, . . . ,n (12)Equation (12) can be assigned a physical context by noting that it is a property of such differentialoperators that they can be written as[D4+ p2D2]V =−λiV i= 1,2, . . . ,n (13)and hence that[D4+ p2D2+λi]V = 0 i= 1,2, . . . ,n (14)and V is a typical lateral displacement.Each of these equations now describe the elastic critical buckling of a single unified member, butsupported on a Winkler foundation of different magnitude in each case. Equation (14) thereforerepresent n substitute systems, each of which yields an infinite number of critical buckling loadsthat, when arranged in ascending order, comprise the complete spectrum of critical buckling loadsof the original problem. It therefore follows that when only the lowest critical buckling load is re-quired, it is only necessary to solve the substitute system that contains the lowest linear eigenvalueobtained from Equations (10) and (11).In the current context, each substitute system is solved by transforming to a stiffness formulationin conjunction with the Wittrick-Williams algorithm, the boundary conditions being the single,identical set imposed on each of the original members.ExampleThe simple illustrative problem selected can be envisaged by considering Figure 1(a) as comprisingfive simply supported members (n= 5) with ki = κi = k (i= 1,2, . . . ,5) and k6 = 0. The five lineareigenvalues and their corresponding eigenvectors that then stem from Equations (10) and (11) aregiven in Table 1.Table 1: Linear eigenvalues and their corresponding eigenvectors.Vi, ji λi j = 1 j = 2 j = 3 j = 4 j = 51 0.0810 0.2817 0.5406 0.7557 0.9096 0.98982 0.6903 0.7557 0.9898 0.5406 −0.2817 −0.90963 1.715 0.9898 0.2817 −0.9096 −0.5406 0.75574 2.831 0.9096 −0.7557 −0.2817 0.9898 −0.54065 3.683 0.5406 −0.9096 0.9898 −0.7557 0.2817The first three of a possible infinite number of critical buckling load parameters, p2, is then givenin Table 2. Hence, the critical buckling loads in each member can be deduced from Equations (4)- (5).Table 2: The first three critical buckling load parameters for each linear eigenvalue. The modalnumbers are given in brackets.p2 for each λim i= 1 i= 2 i= 3 i= 4 i= 51 9.8778 (1) 9.9395 (2) 10.043 (3) 10.156 (4) 10.243 (5)2 39.481 (6) 39.496 (7) 39.522 (8) 39.550 (9) 39.572 (10)3 88.827 (11) 88.834 (12) 88.846 (13) 88.858 (14) 88.868 (15)Mode shape recoveryThe buckling mode shape of the original members can be recovered by multiplying the substitutesystem mode shape by the appropriate element of the appropriate linear eigenvector. Now it isclear that the substitute system mode shapes for a simply supported member are defined byV = Asin(mpiξ ) m= 1,2, . . . ,∞ (15)where A is an arbitrary constant.As an example, consider the eigenvector corresponding to the fourteenth critical load of the orig-inal structure. It can be seen from Table 2 that this corresponds to p2 = 88.858, with i = 4 andm= 3. Thus the eigenvector defining the mode shapes of the five original members, top to bottom,isV˜= AV4, j sin(3piξ ) j = 1,2, . . . ,5 or V˜= A0.9096−0.7557−0.28170.9898−0.5406sin(3piξ ) (16)

On the eigensolution of elastically connected columns

https://repository.lboro.ac.uk/articles/On_the_eigensolution_of_elastically_connected_columns/9221612/files/16799069.pdf

On the eigensolution of elastically connected columns

Abstract

Similar works

Full text

Available Versions

Loughborough University Institutional Repository