Eigenvalues of Trusses and Beams Using the Accurate Element Method
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- Sophia Wiggins
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1 Eigenvalues of russes and Beams Using the Accurate Element Method Maty Blumenfeld Department of Strength of Materials Universitatea Politehnica Bucharest, Romania Paul Cizmas Department of Aerospace Engineering eas A&M University College Station, eas Abstract he accurate element method (AEM) is a method developed for the numerical integration of the ordinary differential equations. he differential equations are discretized by dividing the computational domain in elements and reducing the solution to nodal values, similar to the finite element method. he solution over the elements is approimated using high-degree interpolation functions. A high-degree interpolation function would require a large number of unknowns per element. A prominent attribute of the AEM is the methodology developed for eliminating unknowns inside the element. As a result, the salient feature of the AEM is the decoupling between the solution accuracy and the number of unknowns. Consequently, high accuracy solutions are obtained using a reduced computational cost compared to traditional methods, such as the finite element method. he AEM uses the same approach to solve initial value problems, boundary value problems or eigenvalue problems. his paper is focused on computing the eigenvalues and eigenvectors for the aial vibration of trusses and transverse vibration of beams. he results obtained using the AEM were compared against finite element results obtained using ANSYS. For both trusses and beams, the accuracy of the eigenvalues computed using the AEM was several orders of magnitude higher than that of the finite element analysis, while the computational time was approimately the same. Introduction he various numerical methods recently developed in engineering may be considered as serving a common goal: solving in the best possible way the ordinary or partial differential equations describing the physical phenomena. his paper, restricted at this initial stage to solving ordinary differential equations (ODEs), presents a new method that allows generating accurate results with a small computational effort. his method, which we call the accurate element method (AEM), will be used herein to compute the eigenvalues of straight trusses and beams with constant and stepwise variable cross-section. he AEM can be compared with the displacement method (DM) and the finite element method (FEM). he strategy of these methods can be summarized by the following three steps: () discretization of the domain D on which the governing equations must be integrated; () local approimation of the solution of the governing equations, in which the information given by each element is transferred to the nodes of the element; and (3) reconstruction of the domain D, obtained by writing nodal equations that bring together the information given by the elements adjacent to each node. he solution of the system of equations obtained in this way represents the nodal unknowns. he first and third steps of the AEM and the DM/FEM are similar. Differences between the AEM and the DM/FEM eist in the second step. he second step is crucial in obtaining an accurate solution of the governing equations. As will be shown herein, the solution accuracy obtained with the AEM using one or a few elements can be reached by the DM only if a large number of elements is used. wo important concepts introduced by the AEM will be presented in the net section: the complete transfer relation (CR) and the concordant functions (CF). hese two concepts will then be applied to calculate eigenvalues and eigenvectors for the aial vibration of trusses. he AEM results will be
2 compared against the DM results. he section preceding the conclusions will present the computation of eigenvalues for the transverse vibration of straight beams. he Accurate Element Method he AEM has been developed and implemented to integrate linear and nonlinear ODEs with constant and variable coefficients []. An important aspect of the AEM is the complete transfer relation. o introduce the complete transfer relation, let us consider the ODE that describes the beam deflection [5], p. dw d p =. () EI his equation links the fourth derivative of the displacement to the transverse distributed load p, the elasticity modulus E and the moment of inertia I. o accurately solve the ODE it is necessary to integrate it eactly four times. he ODE integration is usually done numerically, either using Runge- Kutta or similar type methods [], or by accepting from the beginning an approimation, such as in the DM where the variation of the distributed load is neglected []. Instead, the AEM integrates the ODE without any approimation, leading to one or more integral equations. For the solution of equation (), the AEM generates four integral equations, each one including a term f ( ) ( ) d, () where f ( ) is a known function and, are the limits of the integration domain D. he ensemble of the four equations previously mentioned will be referred as the complete transfer relation (CR), because it transfers the information to the nodes, and because this transfer is complete, i.e., without any approimation. he problem is not yet solved because equation () cannot be integrated since () is unknown. In order to make the integration possible, () is usually replaced by an interpolation function, which is a low-degree polynomial leading therefore to approimate results. Instead, the AEM uses high-degree interpolation functions, called herein concordant functions. o eliminate part of the unknowns, the governing equation and its derivatives are used, as showed in the net section. he eigenvalues of trusses he governing equation he behavior of a truss is described by two differential equations: () the equilibrium equation, and () the deformation equation. he equilibrium equation is dn q ( ) =, (3) d where q ( ) is the distributed aial load and N is the internal aial force. he deformation equation, based on Bernoulli hypothesis, is du N =, () d EA where the aial displacement is u = u( ) and A is the constant transverse area of the truss. If equation (3) is substituted into the differentiated form of equation (), the following second-order ODE is obtained q ( ) + =. (5) d EA du
3 he eigenvalue problem results by replacing q ( ) with an inertia load. If ρ is the density of the material and ω is the circular frequency, the inertia force df is given by the product between the mass ( dm = ρ Ad ) and the acceleration ( uω ). As a result, df = uω ρa d and therefore q ( ) dfd uω ρ A If q ( ) is substituted in (5) and u ( ) is replaced by (), then where β ρ ω = ( / E) and k ( k ) d = k. d = / =. (6) () d d + β( ) = or + β =, (7) he numerical eamples analyzed below will calculate the eigenvalues of equation (7), with the following boundary conditions, corresponding to = and = ( = = ) = = (8) = = = =. (9) () ( ) he eact analytic solution of (7) is ( ) = Asin β + Bcos β. he boundary condition (8) yields B =. he boundary condition (9), applied to the derivative ( ) = A β cos β, π gives cos ( β ) =. Consequently β = (k ), k =,, 3,... and { π } β = (k ) /, k =,, 3,... k he Complete ransfer Relation Let us suppose that the integration of equation (7) must be performed between and, the current abscissa being. After the first integration, equation (7) becomes () + ( ) d+ K =, () β where K is an integration constant. he integration constant K can be eliminated by evaluating equation () at both ends of the integration interval. At =, it results and at = K = () () () K = β ( d ). (3) If K is eliminated between equations () and (3), the first integral of equation (7) is obtained () () ( d ) =. () β he procedure continues with the integration of equation () ( ) + ( ) d d+ K + K = (5) β ξ ξ
4 o eliminate the integration constant K, equation (5) is evaluated at both ends of the integration interval. At =, one obtains and at = ( ) (6) d d K K + β ( ξ) ξ + + =, ( ) (7) d d K K + β ( ξ) ξ + + = he integration constant K is replaced in (6) by () and in (7) by (3). aking into account that ( d ) ( ξ) ξ d=, the difference between (6) and (7) leads to { ( ) } (8) + + β ( ξ) dξ d+ d = () () Using integration by parts, the terms from the parenthesis in equation (8) reduce to ( ( ) ) ξ d ξ d d d (9) + = Consequently, equation (7) integrated twice becomes () () + + β ( ) d =. he ensemble of the two relations () and, which is the result of transferring the two successive integrals of (7) to the ends of the computational domain, represents the complete transfer relations (CR). he Concordant Functions he third-degree concordant function CF As it was stated previously the challenge is how to replace function ( ) in order to integrate it. he function that approimates ( ) must be based on the four end unknowns which have been used in () () the in equations () and, namely,, and. Based on these unknowns, one obtains a unique description of a third-degree polynomial whose first derivative is ( ) = C + C + C + C () 3 3 ( ) = d / d= C + C + 3C () () 3 he relation () will be further referred to as the CF concordant (or interpolation) function. Its coefficients can be obtained by imposing the following end conditions: ( = ) = C + C + C + C = (3) 3 3 ( = ) = C + C + C + C = () 3 3 ( = ) = C + C + 3C = (5) () () 3
5 hese four conditions yield the system of equations or using the notation ( = ) = C + C + 3C = (6) () () 3 3 C 3 C = () 3 C () 3 C 3 where [ A ] is the square matri and [ A] C = (7) 3 C = C C C C (8) he vector of coefficients [ ] () () =. (9) [ A ] C can be obtained using the inverse of the matri [ ] C = A. (3) he integrals remaining in equations () and can now be evaluated by using equation () Int = d = C + C + C + C d = C + C + C + C ( ) ( ) Int = d = ( C + C + C + C ) d = C + C + C + C If one denotes Int = [ ] (3) then [ Int ] = 3 5 [ ] [ ] [ ] [ ] [ ] [ ][ ] () () Int = Int C = Int A = P = P + P + P + P (33) 3 (3) [ ] [ ] [ ] [ ] [ ] [ ][ ] () () Int = Int C = Int A = Q = Q + Q + Q + Q (3) 3
6 where P and Q are two matrices with four terms, which can be computed for each pair of abscissas and. If = and =, the terms of P and Q are those given in the CF rows of the ables and, respectively. It is now possible to obtain a final form the CR by replacing (33) and (3) in both equations () and ( P P P P ) β = (35) () () () () 3 ( ) + + β = (36) () () Q Q Q () Q () 3 able. he coefficients P i used in equation (35) for different CFs Coefficients P P P P 3 P P 5 P 6 P 7 P 8 CF / / / -/ CF6 / / / -/ / / CF8 / / 3/8-3/8 /8 /8 /68 -/68 able. he coefficients Q i used in equation (36) for different CFs Coefficients Q Q Q Q 3 Q Q 5 Q 6 Q 7 Q 8 CF 3/ 7/ /3 -/ CF6 /7 5/ /5-3/ /8 / CF8 5/36 3/36 5/6-7/5 5/8 / /378 -/3 he CR includes two equations, (35) and (36), which have four unknowns. he problem can be solved, however, because the boundary conditions (8) and (9) are also available. he four equations thus obtained form a homogeneous system that can be written as ( = ) ( = ) Q Q Q Q + = P P P P 3 3 β () () () () In the system of equations (37), the first row is equation (35), the second row is equation (36), the third row is equation (8) and the last row is equation (9). he system of equations can be written in compact form as [ ] [ ] A β A + =. Equation 38 represents a classical eigenvalue problem. he eigenvalues corresponding to the CF interpolation functions, obtained by solving equation (37), are shown in able 3. able 3 also shows the eact solution and the relative errors (37) (38)
7 error = ( eigenvalue eigenvalue )/ eigenvalue (39) CF eact eact computed with respect to the eact solution. able 3. he eact and the AEM solutions based on low-degree polynomials Errors (referred to the eact analytic solution) NE IF CF ω ω ω 3 ω ω ω 3 * * * E- * * E- * * 3.7 E- E- E- 3.6 E-.6E- * 7.9 E- E- E-.6 E- E-.9 E-.6 E- 3.7 E-3.6 E- 5.6 E- 6 E-. E- E-.6 E-3 7. E-3 E-3.6 E- 3.7 E- 6.7 E-6. E- 5. E- E-3 E-3 9 E-3. E E-6 3. E E-.8 E-3. E E-8.3 E E-6.6 E- E-3.3 E-3 * * * 5.6 E- 6.6 E-.5 E-3 * * * Note that equation () represents a Hermite type polynomial. he methodology presented above will be used in the net sections to obtain higher-degree CFs. he relative error (39) allows to evaluate how many digits of the computed result coincide with eact solution. he result for CF8 corresponding to the first eigenfrequency has an error of. E-6. he eponent of the error, 6 in this case, indicates a coincidence of 6 digits. his is not always true, but a coincidence of 6± digits is to be epected for the CF8 interpolation function. he fifth-degree concordant function CF6 Let us suppose that the Concordant Function is a fifth-degree polynomial ( ) = C + C + C + C + C + C where C, C, C, C3, C and C 5 are si unknown constants. hese si constants will be obtained by using si conditions. Four of these conditions are identical to equations (5-8) presented in the previous section. he other two necessary conditions will be obtained by using the second derivative () at the ends of the domain: ( = ) = () () () ( = ) = () () () Based on these si conditions a system of equations similar to (7) can be written [ A] [ C] [ ] =, (3) where [ A ] 6 is a si-by-si square matri and [ ] = C C C C C C 3 C 6 5
8 [ ] =. () 6 () () () () he vector of coefficients C can be obtained using the inverse of the [ ] 6 6 [ C] [ A] [ ] A matri =. (5) he integrals of equations () and will be denoted as Int 6 and Int 6. hese integrals can be evaluated similarly to the integrals Int and Int, if equation is used instead of equation (). If one denotes Int = 6 [ ] then [ Int ] = 6 [ ][ ] [ ][ ] [ ] [ ][ ] () () () () Int 6 = Int C = Int A = P = P + P + P + P + P + P (6) [ ][ ] [ ][ ] [ ] [ ][ ] () () () () Int 6 = Int C = Int A = Q = Q + Q + Q + Q + Q + Q where P and Q 6 are one-row matrices with si terms, which can be computed for each pair of 6 abscissas and. If, for instance, = and =, the terms of P and Q 6 are those given 6 in the CF6 rows of the ables and, respectively. It is now possible to obtain the final form of the CR by replacing equations (6) and (7) in equations () and (7) ( P P P P P P ) ( ) β = (8) () () () () () () β = () () Q Q Q () Q () Q () Q () A new problem arises now, whose solution is the essence of the accurate element method presented herein. he conditions of the problem have not changed, therefore only four equations are available, () while there are si unknowns involved, corresponding to (). he unknowns,, and were used to determine the CF interpolation function. We will consider these unknowns to be the () () basic unknowns, while the unknowns and will be considered the apparent unknowns, and they will be eliminated. o eliminate the apparent unknowns () and () additional information is needed. Such information is usually obtained from some relations with adjacent elements or by accepting a reasonable approimation. Neither of these options is used by the AEM. Instead, the necessary information is obtained from the governing equations. he second derivative is obtained from the governing equation (7), that is ()
9 = β. (9) () As a result, the second derivatives at the ends of the integration domain are given by two eact additional relations = β (5) () If the derivatives () () = β (5) (), are replaced in equation (8) it results ( P P P P ) ( P P ) ( ) ( ) () () () () β β + = () () () () + + β Q + Q + Q + Q + β Q + Q = (5) Equations (5), (5) and (5) can be written in matri form as P P P P 3 P P ( P) ( P) Q Q Q Q Q Q ( Q ) ( Q ) β + β =. () () () () () () (53) Note that equation (53) is identical to equation (37) ecept for the additional term that includes β. For the interpolation functions CF6, the terms P7, P8, Q7 and Q 8 are null. he system of equations (53) can be written in compact form as [ ] [ ] A β A β A + + =. (5) he equation (5) represents a polynomial eigenvalue problem. he eigenvalues β were obtained herein using the routine POLYEIG of MALAB. his function solves the polynomial eigenvalue problem of degree p. If either one but not both [A ] and [A p ] are singular, the problem is well posed, but some of the eigenvalues may be zero or infinite. From equation (5) three eigenvalues result, which are given in the CF6 row of able 3, together with the relative errors. Note that the first four digits of the first eigenvalue are eact although only one AEM element with CF6 interpolation functions has been used. he seventh-degree concordant function CF8 As shown in the previous section, the results improve if the degree of the CF increases. he methodology presented for the elimination of the apparent unknowns can be applied for any higherdegree CF. For instance if a CF8 with eight terms is considered ( ) = C + C + C + C + C + C + C + C (55) two additional conditions compared to the CF6 case are necessary to obtain all the constants. hese (3) additional conditions will be based on the third derivative,, at ends of the domain ( = ) = (56) (3) (3) (3) (3) ( = ) =. (57)
10 A system of equations similar to (7) can be written using the eight available conditions (5-8), (), (), (56) and (57) [ A] [ C] [ ] =, (58) where [ A ] 8 is an eight-by-eight square matri and [ C ] = C C C C C C C C (59) [ ] =. (6) 8 () () () () (3) (3) he vector of coefficients C can be obtained using the inverse of the [ ] 8 8 [ C] [ A] [ ] A =. (6) he integrals of equations () and will be denoted as Int 8 and Int 8. hese integrals can be evaluated in the same manner as the integrals Int 6 and Int 6, if equation (55) is used instead of equation. Following the same procedure as in the previous section yields Int8 = P + P + P + P + P + P + P + P (6) () () () () (3) (3) Int 8 = Q + Q + Q + Q + Q + Q + Q + Q, (63) () () () () (3) (3) where the eight terms P i, Q i, are given for = and = in the CF8 rows of the ables and. he final form of the CR results by replacing (6) and (63) in equations () and ( P P P P P P P P ) β = (6) () () () () () () (3) (3) ( ) + + β = () () Q Q Q () Q () Q () Q () Q (3) Q (3) Because the basic unknowns remain () () (3) now four, namely,, and (),, and (3) (65) (), the number of apparent unknowns is. he apparent unknowns must be eliminated using the governing equation. Besides the two relations (5) and (5) established previously, two more relations can be obtained by differentiating equation (9) (3) () =. (66) As a result, two eact additional relations specify the third derivatives at the ends of the integration domain β β = (67) (3) () = β. (68) (3) () If equations (5), (5), (67) and (68) are replaced in equation (6), this yields
11 ( P P P P ) ( P P P P ) ( ) ( ) () () () () () () β β = () () () () () () + + β Q + Q + Q + Q + β Q + Q + Q + Q = he final system of equations in matri form is given by equation (53). Consequently, equations (5) remain the same, leading to the four eigenvalues that are given in the CF8 row of able 3. Note that the first si digits of the first eigenvalue are eact although only one AEM element with CF8 interpolation functions has been used. A numerical comparison between the DM and the AEM he DM solution is based on the two-by-two stiffness matri [ K ], leading to the well-known relation [], p. [ F] = [ K][ δ ], (69) where [ F ] = F F is the nodal forces vector and [ δ ] u u displacements vector. Because δ used to replace u ( ) = ( ) has only a linear variation given by ( ) ( ) ( ) = is the nodal aial is based on only two end unknowns, the interpolation function u = u+ u u / L= + / L (7) where L =. he interpolation function (7) is referred herein as IF. By using IF, the CR becomes [], p. 78 () + L = β ( L / 3+ L / 6) () + L = β ( L / 6+ L / 3) (7) By applying the boundary conditions (8) and (9) and taking into account that L = =, the second equation (7) yields β = 3. IF he relative error with respect to the analytical value of the eigenvalue is error = ( eigenvalue eigenvalue )/ eigenvalue = 6.. IF IF eact eact Note that this error is at least two-to-three orders of magnitude larger than the AEM errors shown in able 3 for the first eigenvalues. he rest of this section presents a comparison between the eigenvalues obtained using the IF and CF interpolation functions. he comparison will be done for a truss with constant area, using a different number of elements, NE. he truss is clamped at both ends such as the boundary conditions are ( = = ) = = u = (7) = = π = = =. (73) ( ) u he length of the truss is L = π. If ρ/ E =, the governing equation (7) can be written as d + ω ( ) =. (7) d he eact eigenfrequencies of equation (7) with the boundary conditions (7) and (73) are given by ω =, ω =, ω 3 =3, ω =,, ω =k. k
12 able. Errors for the IF and CF interpolation functions Errors (referred to the eact analytic solution) NE IF CF ω ω ω 3 ω ω ω 3 * * * E- * * E- * * 3.7 E- E- E- 3.6 E-.6E- * 7.9 E- E- E-.6 E- E-.9 E-.6 E- 3.7 E-3.6 E- 5.6 E- 6 E-. E- E-.6 E-3 7. E-3 E-3.6 E- 3.7 E- 6.7 E-6. E- 5. E- E-3 E-3 9 E-3. E E-6 3. E E-.8 E-3. E E-8.3 E E-6.6 E- E-3.3 E-3 * * * 5.6 E- 6.6 E-.5 E-3 * * * As shown in able, the results obtained using the IF interpolation function could be considered rather poor. Even the results obtained using 5 elements with the IF interpolation function are quite far from the eact analytic solution. he eigenvalues were obtained by using a program written by the authors and by using ANSYS, the results being the same. Note that commercial codes are not able to use a higher-degree interpolation function, because they are based on a stiffness matri that allows only the use of low-degree interpolation functions. able 5. Errors for the CF8, CF and CF6 interpolation functions Errors (referred to the eact analytic solution) ω CF8 CF CF6 NE= NE= NE=3 NE= NE= NE=3 NE= NE= NE=3 ω.8 E- 6.8 E-7.8 E-8.3 E-8.9 E-. E-3.7 E- E-6 8 E-6 ω E-3.8 E- 6.5 E-6.7 E-7.3 E E-.3 E-8.7 E-.3 E- ω 3 *.6 E-3.8 E- E- 6.9 E-6.3 E-7. E-.7 E-9.7 E- ω * 3.6 E-3 E-3 3. E-.7 E-5.8 E-6. E-3.3 E E- ω 5 * 6 E- 5 E-3 *. E-3. E E- 9.3 E-6.3 E-8 ω 6 * * 3 E-3 * E-.7 E-5 9. E-. E-.3 E-7 ω 7 * * E- *.6 E- 7. E- * 8 E- 3.5 E-6 ω 8 * * 8 E- * 3 E-.6 E-3 *. E-3.3 E-5 ω 9 * * * * *.3 E- *.3 E-. E- ω * * * * *.8 E- * 5.7 E-. E-
13 able 5 shows the variation of the eigenvalue errors for interpolation functions CF8, CF and CF6. he number of elements was less or equal to three. he first ten eigenvalues, if available, were listed in the table. he eigenvalues were calculated using the routine POLYEIG of MALAB. Recall that the CF interpolation function did not require the elimination of any apparent unknowns, therefore the only difference between IF and CF is the use of a different number of unknowns per element. he result obtained for ω using CF with three elements, that is twelve unknowns, is similar to that obtained using IF with thirty elements, that is sity unknowns. Consequently, for the same accuracy, the computational time of the AEM using the CF interpolation function is smaller than that of the FEM using the IF interpolation function. One can also compare the errors between the CF8, CF and CF6 interpolation functions with the same number of elements. Let us consider the results corresponding to the first eigenvalue for three elements. In this case, the number of unknowns after eliminating the apparent unknowns was twelve, the same for the CF8, CF and CF6 interpolation functions. he ratios between the errors corresponding to the CF and the IF interpolation functions, shown in able 6, indicate the net advantage of the AEM with high-degree interpolation functions. able 6. Errors for the CF8, CF and CF6 interpolation functions Error IF /Error CF8 Error IF /Error CF Error IF /Error CF6.6E 3.3E9 5.7E he eigenvectors his section briefly describes how to calculate the eigenvectors once the eigenvalues have been computed. First, the eigenvalue β is substituted in either equation (37) or (53). Because equations (37) and (53) are homogeneous systems, the first CR equation must be replaced by an equation that gives an arbitrary nonzero value to one of the basic end unknowns (obviously not to those involved in the ( k ) boundary conditions). he solution of this system of equations provides the end unknowns, and ( k ). he apparent unknowns can be calculated using equations (5) and (5) for CF6 or equations (5), (5), (67) and (68) for CF8. he vector of coefficients C results from equations (5) or (6) and the eigenvector is obtained from or (55). If the solution obtained with the CF8 interpolation function is considered, and = is imposed arbitrarily, the eigenvector corresponding to the first eigenvalue is given by 3 ( ) = he value of the eigenvector at = 5. is given by ( =. 5) =. 77, while the value corresponding to the eact solution (73) is he first five digits of the AEM eigenvector coincide to the eact solution, although only one element was used. Methods for improving the accuracy of the AEM he results obtained using one element with the CF8 interpolation function have satisfactory accuracy. It is useful, however, to eamine ways of improving them. wo options will be discussed herein: () increasing the degree of the interpolation polynomial, and () increasing the number of elements. Increasing the degree of the polynomial, that is, increasing the number of terms of in the CF interpolation function can be done without special difficulties by following the procedure already described. his requires higher derivatives of the governing equation (7) in order to eliminate the new apparent unknowns. his strategy has already been used for CF6 (fifteenth-degree polynomial) and CF (nineteenth-degree polynomial) [], pp For the case of CF, the error corresponding to
14 the first eigenvalue obtained with one element dropped to , that is, the first 5 digits coincide with the eact result. Increasing the number of elements can be done together or independently of varying the degree of the interpolation function. Let us suppose the domain of integration is divided in two elements noted and 3, respectively. he lengths of these elements can be equal or not. he number of nodal unknowns is double the number of unknowns per element, that is, two times four. he unknowns of the element - are (),, and () (), and the unknowns of the element 3- are,, and () 3 3. o solve the problem, eight equations are necessary. hese equations can be divided in three groups: () equations based on the CR, () equations based on the connective relations between the elements, and (3) boundary conditions. he equations based on the CR will rely on the equations () and and will generate four equations. he equations based on the connective relations will impose the continuity of the function and its first derivative at the adjacent nodes = 3 = () () 3 In addition, two boundary conditions, such as the relations (8) and (9) can be imposed at the ends of the integration domain. Consequently, a total of eight equations will be available from the CR, connective relations and boundary conditions, such that the constants of the interpolation function CF8 can be determined. he eigenvalues of straight beams he governing equation he deformation analysis of a straight beam, written in the z coordinate system, is based on relation (). Following the procedure used in section, p ( ) is given by the inertia load (6), where u will be replaced by w : p( ) wω ρ A Substituting p( ) from equation (75) into () yields [5], p. =. (75) where w β w =, (76) () β ρ A ω EI =. (77) z Let us replace the displacement w by, such that the fourth-order ODE (76) becomes β = (78) () he Complete ransfer Relation he ODE (78) must be integrated four times. hese successive integrations lead to the following four equations that represent the CR
15 (3) (3) β d = () () (3) (3) + + β d = () () () () (3) (3) + + β d = () () () () 3 (3) 3 (3) 3 β d = (79) he Concordant Functions he seventh-degree concordant function CF8 Because equation (78) has been integrated four times, the CR (79) is based on eight end unknowns (3) (3) () () () (),,,,,, and. hese eight unknowns are the basic unknowns. In this case the basic concordant function will be a CF8 interpolation function given by equation (55) for which there are no apparent unknowns. Consequently, the relations (55), (6), (63), can be used directly without any elimination procedure. For a fourth-order ODE, the major modification compared to second-order ODEs is that two additional integrals must be considered in addition to equations (6) and (63). hese additional integrals result from the third and fourth equations of (79) () () () () (3) (3) 8 = = Int d R R R R R R R R (8) 3 3 () () () () (3) (3) Int = d = S + S + S + S + S + S + S + S. (8) Following a procedure similar to that presented in section 3, a homogeneous system of equations similar to equation (7) results [ ] [ ] A β A 8 + =. Note that in this case [ A ] and [ A ] are eight-by-eight matrices and is an eight-component 8 vector given by relation (6). Four equations of the system of equations (8) are provided by the CR (79). he other four necessary equations are represented by the boundary conditions. If the beam is simply supported, then both ends displacements and bending moments are zero. By taking into account () () the equation w = M( ) / EI = the boundary conditions are = w = ; = w = ; = w = ; = w =. (83) () () () () (8) he fifteenth-degree concordant function CF6 As shown in section 3, if a concordant function with more terms than those corresponding to the basic unknowns is used, the result is a polynomial eigenvalue problem described by (5). Recall that for the
16 second-order ODE (7), the coefficients of the CF interpolation function were calculated using only basic unknowns, without apparent unknowns. In this case, the relation (5) was the same for the CF6 and CF8 interpolation functions. his had a favorable influence on the accuracy of the results. For the fourth-order ODE (78) the coefficients of the CF8 interpolation function can be calculated using only basic unknowns, without apparent unknowns. In this case, equation (5) is the same whether the CF, CF, CF or CF6 interpolation functions are used. o achieve higher accuracy it is recommended therefore to use the CF6 interpolation function, which is a fifteenth-degree polynomial. Using the same procedure presented in section 3, the integral (6) will now depend on 6 terms = () () () () (3) = (8) (3) () () (5) (5) (6) (6) (7) (7) ) β Int6 β d β ( P P P P P P P + P + P + P + P + P + P + P + P + P. In this case, eight apparent unknowns must be eliminated. his can be done using the governing equation (78) and its first three derivatives β = () β = (5) () β = (6) () β =. (7) (3) (85) By applying them to both ends, the following eight relations result Replaced in equation (8), these relations lead to β = β = () () β = β = (5) () (5) () β = β = (6) () (6) () β = (7) (3) β =. (7) (3) (86) () () () () (3) (3) 6 ( ) β Int = β d = β P + P + P + P + P + P + P + P + (87) () () () () (3) (3) + β ( P + P + P + P + P + P + P + P ) A similar procedure applied to Q i, R i and S i, (i =,,,6) will lead to equation (5). A numerical comparison between the DM and the AEM he DM is based on equation (69), where for the straight beam [ ] () () () () w w w w δ = =. (88) Because only four end unknowns are involved, the DM can only use a third-degree polynomial (Hermite type) to obtain the eigenvalues. his limitation leads to quite poor results, compared to using the CF8 and especially CF6 interpolation functions. he third-degree polynomial will be denoted IF. he eigenvalues of a simply supported beam with constant cross-area are given by the eact solution
17 π ω = n n L E I. ρ A (89) If A =, I =, ρ /E= and L = π, the eigenvalues are given by ω =, ω =, 3 ω =3, ω =,..., ω n = n. he results obtained by using the DM with the IF interpolation function are shown in able 6. he relative errors were calculated according to (39). As known by the users of structural analysis programs, even for the simple case analyzed herein, the beam must be divided in a large number of elements in order to generate accurate results, especially if higher eigenvalues must be computed. able 7. Results obtained with IF (Hermite interpolation function) Elements Relative errors NE = ω ω ω 3 No solution No solution No solution -7.3 E-3 No solution No solution 3 -. E E- No solution -3. E E E E- -.5 E E E-5 -. E E E-5-3. E- -.8 E E-6 -. E E- -.3 E E E-5 -. E E E-5 he results obtained by using the CF8 and CF6 interpolation functions are shown in able 8 [], pp he number of eigenfrequencies was increased up to ω 5, though the analysis was based only on one, two and three elements. he comparison of the results shown in ables 7 and 8 illustrates that the accuracy of the AEM results is much better than that of the DM results. Let us underline few aspects. he result obtained for ω using IF with twenty elements ( = 8 unknowns) is similar to that obtained using CF8 with two elements ( 8 = 6 unknowns). If compared to CF6 with two elements, which has also 6 unknowns, the ratio between the relative errors is error (elements, 8 unknowns). 3E 7 IF = = 8. error (elements, 6 unknowns). E 9 CF6 If the same number of unknowns is considered, namely 6, then error (elements, 6 unknowns) 3. E IF = = 3. error (elements, 6 unknowns). E 9 CF6 Note also that the error.9 E- obtained for ω using three elements with CF6 corresponds to ω =.9. he analytical solution is ω eact =, such that the AEM result has ten eact digits. 5
18 able 8. Relative errors for the first five eigenfrequencies obtained with CF8 and CF6 ω NE = element NE = elements NE = 3 elements CF8 CF6 CF8 CF6 CF8 CF6 ω 3 E- 3. E E-7. E-9.8 E-8.9 E- ω 5. E E-7 3 E- 3. E E-6 7 E-9 ω E-. E- 3 E E-8 3 E- 3. E-8 ω 7.5 E-.3 E-3. E E-7.3 E E-8 ω 5 No solution 8. E- 8 E- 7. E-6 6. E E-7 Conclusions he AEM, a new method for the integration of ordinary differential equations, has been developed and implemented to integrate linear and nonlinear ODEs with constant and variable coefficients [, 3]. he AEM has been applied herein to compute accurately and efficiently the aial vibration of trusses and the transverse vibration of straight beams. he AEM can model two- and three-dimensional systems of beams and trusses with constant and variable cross section, as well as beam deformation due to shear forces []. he results presented herein were limited to two-dimensional trusses and straight beams. he truss element solved a second-order ODE, for which four basic unknowns must be used. he basic concordant function CF was therefore a four-term, third-degree polynomial. he straight-beam element solved a fourth-order ODE, for which eight basic unknowns must be used. he basic concordant (or interpolation) function CF8 is an eight-term, seventh-degree polynomial. As shown herein, the concordant functions have an essential role in the accurate element method, because they allow the use of high-degree polynomials without increasing the number of the end unknowns. An important step in obtaining the concordant functions is the calculation of the inverse matri [ A]. his step may become time consuming, especially for the very high-degree concordant functions. he methodology presented here is a simplified version intended to facilitate understanding the AEM. he methodology implemented in the code, however, uses concordant functions written for a natural (non-dimensional) coordinate system. In this last case the inverse matri [ A] is always the same, being calculated only once. he inverse matrices [ ] []. A are given several concordant functions in his paper was focused on the solution of beam and trusses free vibration. he AEM allowed us to obtain a large number of high frequencies using a small number of elements, thereby reducing the computational time. he results obtained using the AEM were compared against finite element results obtained using ANSYS. For both trusses and beams, the accuracy of the eigenvalues computed using the AEM was several orders of magnitude higher than that of the finite element analysis, while the computational time was approimately the same. Because the strategies of AEM and DM are similar, the AEM elements can be included without special problems in a structural analysis program, such as ANSYS. he use of high-degree concordant functions can be handled by a subprogram that will automatically eliminate the apparent unknowns, returning to the main program the Complete ransfer Relation including only the small number of the basic unknowns. References []. M. Blumenfeld, A new method for accurate solving of ordinary differential equations, Editura ehnica, Bucharest,.
19 []. M. Blumenfeld and P. G. A. Cizmas, he accurate element method: A new paradigm for numerical solution of ordinary differential equations. Proceedings of the Romanian Academy, (3), 3. [3]. M. Blumenfeld and P. G. A. Cizmas, he Accurate Element Method: A novel method for integrating ordinary differential equations. In John Junkins Astrodynamics Symposium, College Station, eas, May 3. []. C. S. Desai and J. F. Abel. Introduction to the Finite Element Method. Van Nostrand Reinhold Company, 97. [5]. L. Meirovitch. Elements of Vibration Analysis. Mc Graw-Hill, New York, second edition, 986.
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