Integral Method for Static, Dynamic, Stability and Aeroelastic Analysis of Beam-like Structure Configurations

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1 Integral Method for Static, Dynamic, Stability and Aeroelastic Analysis of Beam-like Structure Configurations Viorel ANGHEL* *Corresponding author POLITEHNICA University of Bucharest, Strength of Materials Department, Splaiul Independenţei 33, 06004, Bucharest, Romania, DOI: 0.3/ Received: 3 October 07/ Accepted: 3 October 07/ Published: December 07 Copyright 07. Published by INCAS. This is an open access article under the CC BY-NC-ND license ( Abstract: This work presents a synthesis of the use of an integral approximate method based on structural influence functions (Green s functions) concerning the behavior of beam-like structures. This integral method is used in the areas of static, dynamic, aeroelasticity and stability analysis. The method starts from the differential equations governing the bending or/and torsional behavior of a beam. These equations are put in integral form by using appropriate Green s functions, according to the boundary conditions. Choosing a number of n collocation points on the beam axis, each integral are then computed by a summation using weighting numbers. This approach is suitable for conventional Euler-Bernoulli beams and also for the thin-walled open or closed cross-section beams which can have bending-torsion coupling. Generally, for a static analysis this approach leads to a linear system of equations (the case of the lift aeroelastic distribution analysis) or to an eigenvalues and eigenvectors problem in the case of dynamic, stability or divergence analysis. Key Words: Integral Method, Green Functions, Collocation, Static, Dynamic, Stability, Aeroelasticity. INTRODUCTION The use of the structural influence functions (Green s functions) in the structural and aeroelastic analysis are presented in []. In Romania this approach is widely used by Professor A. Petre in his works on aeroelasticity in fixed wing aircraft [, 3]. In the case of the rotating beams and blades the method using Green s functions was presented for simple configurations in order to obtain the natural frequencies for the bending and bending-torsion vibration analysis [4, 5]. The coupled bending vibration analysis in the case of pretwisted blades was presented in [6]. Then, a general case of the coupled bending-bending-torsion vibration analysis of straight beams and blades was described in [7]. The papers [8, 9] concern the dynamic analysis of rotating beams with tip mass. Other works related to the dynamic analysis of rotating beams and blades are [0, ]. The method of Green s functions was then applied for composite beams both for dynamic analysis [] and static analysis [3]. Other applications are then presented in [4, 5]. In [5] a short application concerning the buckling analysis of a straight uniform beam was also performed. Aspects concerning the aeroelastic analysis of wings are presented in work [6]. New developments of the methods based on Green s functions in dynamic analysis of beams and blades are recently reported in [7, 8]. INCAS BULLETIN, Volume 9, Issue 4/ 07, pp. 3 0 (P) ISSN , (E) ISSN

2 Viorel ANGHEL 4 In the case of the dynamic, stability or aeroelastic analysis this method leads to an eigenvalues and eigenvectors problem. For the static analysis, the use of Green s functions leads to a linear system. This paper presents the general formulation of the method for the case of the static, dynamic, aeroelastic and stability analysis of beam-like structures which can be used for the study of some wings or rotor blade configurations. As examples, some simple numerical applications are also discussed in comparison with analytical results.. STATIC ANALYSIS The bending behavior of a straight beam, having the length L and loaded transversally by the distributed force p(x), can be described by a differential equation: It can take the integral form, []: ( x) w' ' '' p( x) EI () w( x) L Gw( x, ) p( ) d () 0 The previous equation is based on the Green s function G w (x,ξ) representing the bending deflection w(x,ξ) at distance x due to a unit force applied at ξ (Fig. ). The differential equation governing the Saint Venant torsional behavior of a beam, having the length L and loaded by the distributed torsion moment m t (x), is: ( x) ' ' m ( x) 0 It can be written in the integral form as follows: GJ t (3) L ( x) G ( x, ) m ( ) d (4) 0 t using the Green s function G t (x,ξ) that represents the twist deflection angle ϕ(x,ξ) at distance x due to a unit torsion moment applied at location ξ (Fig. ). t Fig. Physical significance of Green s functions The material of the beam is considered metallic, isotropic having the longitudinal elastic modulus E and the shear modulus G. The notations I(x) and J(x) are for the moment of inertia of the beam cross section and for the torsional stiffness constant, respectively. The equations for the torsional and bending behavior of a thin walled-beam can be also put in an integral form, [3]. The integrals involved in such type of approach can be approximated by a summation using n collocation points ξ i with f i = f(ξ i ): 0 L n f ( ) d fi Wi (5) i INCAS BULLETIN, Volume 9, Issue 4/ 07

3 5 Integral Method for Static, Dynamic, Stability and Aeroelastic Analysis of Beam like Structure Configurations where W i are weighting numbers corresponding to Simpson s method of integration adopted here. The equations () and (4) give the possibility to obtain the static bending and torsion deflections for known distributed force p(x) and distributed torsion moment m t (x). Aerodynamic loads are distributed loads, other effects like concentrated forces, concentrated moments or discrete attachments can be also introduced [9,5]. 3. DYNAMIC ANALYSIS For example, the differential equation in the case of the transverse free vibrations of a rotating beam has the following form: EI( x) w' ' ' ' m( x) w T ( x) w' ' (6) where the axial force (tension due to the angular velocity of rotation Ω) is: L m( ) x T ( x) d (7) The term containing T(x) gives the stiffening effect due to the rotation. In equation (6), m(x) is the mass of unit length of the beam and ω is its natural circular frequency. This equation can be considered as having the form (). It takes a matrix form using n collocation points ξ i and relations (), (5): w G W M w G W M D w G W M D w w w in w x (8) In the previous equation: [G w ] is a matrix containing the measured or calculated influence coefficients G w (ξ i,ξ j ), [W] is a weighting matrix depending on the integration method (Simpson, here), [D ], [D ] are differentiating matrices based on central difference operator, [M ], [M in ] and [M x ] are diagonal matrices with the values m(x), m ( ) d and m(x) x respectively, along the main diagonal. The equation takes the form: which is a standard eigenvalue problem: with: G w G G w w 3 (9) I w 0 L x A (0) A G I G G3 a matrix of n n dimension and [ I ] an unity matrix having also the dimension n n. The dimension of the eigenvalue problem can be reduced by the use of collocation functions corresponding to the boundary conditions. The displacement w is written as: p k x () w( x) C f () k k INCAS BULLETIN, Volume 9, Issue 4/ 07

4 Viorel ANGHEL 6 where f k (x) are a number of p known functions and C k are constant coefficients. For the n collocation points: '' F C; w ' F C ; w F C w (3) ' '' where [F], [F ], [F ] are matrices of dimension (n,p) containing the values f k, f k, f k in the collocation points. Using these relations the differentiating matrices are no more necessary and (8) becomes: F C G w W M F C G W M F C G W M F C w in Multiplying with transpose of the matrix [F] the above relation takes the form: C B C E C A One obtains a standard eigenvalue problem: w x (4) (5) I C 0 B (6) where [A ], [B ], [E ] and [B] are now p p matrices (p < n). The results are the natural circular frequencies ω k = πf k, with f k the natural frequencies [Hz]. As example, one considers the free vibrations analysis of a clamped-free uniform beam having the constant bending rigidity EI =, the mass per unit length m = ρa = and L =. For such type of beam, the natural circular frequencies are obtained analytically as: i i L EI A (7) Here ρ is the material density and A is the cross-section area. In our case i i. The first three analytical values β i are: β =.875, β = 4.694, β 3 = 5π/, [9]. Using the relation (9) or (0) with Ω = 0, the first three natural frequencies obtained using n collocation points are given in the table below: Table Results for the first three natural circular frequencies (collocation points) n = 0 n = 0 n = 40 n = 60 n = 00 Exact ω ω ω These results are in close agreement with the analytical ones. 4. STATIC AEROELASTICITY ANALYSIS Work [6] presents some examples of the use of this integral formulation in the wing divergence analysis. The wing is considered like a straight clamped-free beam. In the torsion divergence analysis one can also obtain a standard eigenvalue problem: ( q) I 0 3 A (8) INCAS BULLETIN, Volume 9, Issue 4/ 07

5 7 Integral Method for Static, Dynamic, Stability and Aeroelastic Analysis of Beam like Structure Configurations The eigenvalues λ depend on the dynamic pressure q = ρv /, with ρ and v representing the air density and air velocity, respectively. The first minimum eigenvalue λ gives the divergence velocity v D. For the bending divergence analysis of the same type of wing one can also obtain a standard eigenvalue problem: qi w 0 B (9) Here, the minimum eigenvalue q = ρv / gives the divergence velocity v D. 5. STABILITY ANALYSIS One considers here the standard problem of the buckling of a pin-ended straight beam subjected to an axial compression force P. In this boundary conditions case the Green s functions G w (x,) = w(x,), see figure below. Fig. Buckling fundamental study case and the corresponding used Green function The equation governing the bending displacements is: EI x EIxw' ' ' ' Pw' ' w' ' Pw; or (0) Using the same steps, the matrix form of the last equations is: w PG w W D w PG w () in the case of the use of collocation points. This represents an eigenvalue problem: PI w 0 A () where [A ]=inv[g]. The eigenvalues of the matrix [A ] give the critical buckling loads ( = - P c ). When collocation functions are also used this matrix form becomes: C PG W F C F w (3) After the left multiplication with transpose of the matrix [F], the above relation takes the form: or: A C B A C This represents an eigenvalue problem: C P B (4) P C PI C 0 3 (5) A (6) Now, the eigenvalues of the matrix [A 3 ] give the critical buckling loads ( = - P c ). For example, it is considered a beam having the constant bending rigidity EI = and L =. INCAS BULLETIN, Volume 9, Issue 4/ 07

6 Viorel ANGHEL 8 The analytical results concerning the first three critical buckling loads are the followings: EI P c ; P 4 ; 3 9 ; c Pc (7) L The first critical buckling loads when using n collocation points are shown in the next table. Table Results for the first three buckling loads (collocation points) n = 0 n = 0 n = 40 n = 60 n = 00 Exact P c =9.869 P c = P c = The results are better when one increases the collocation points number n. A source of errors is the differentiating matrix [D ]. In order to avoid the use of this matrix, one can consider the collocation functions approach. For example one can take a family of maximum p = 3 polynomial functions in order to describe the bending deflection w(x). These functions are compatible with the boundary conditions: x x x x x x x x x w 3 L L L L L L L 3 L 3 L x, w x, w x. Using these collocation functions for p =,, 3, the results are the following: Table 3 Results for the first three buckling loads (collocation points and collocation functions) n = 0 n = 0 n = 40 n = 60 n = 00 Exact P c (p=) =9.869 P c (p=) =9.869 P c (p=3) =9.869 P c (p=) = P c (p=3) = P c3 (p=3) = These results are better, especially for the first buckling critical load P c. In order to increase the precision for higher buckling critical loads (P c and P c3 ) one can take a greater number of collocation functions p > CONCLUSIONS This work presents the static, dynamic and stability analysis of several simple beam configurations using an integral formulation (I.F.) based on the use of structural influence functions (Green s functions). These functions are computed by specific methods of Strength of materials as they can be interpreted as displacement in a point of a beam due to a unit force applied in another point. The presented integral approach needs the use of a number of collocation points. For the numerical integration, integration matrices based on Simpson s method of integration were here employed. Differentiating matrices are also necessary in the case of rotating beams and INCAS BULLETIN, Volume 9, Issue 4/ 07

7 9 Integral Method for Static, Dynamic, Stability and Aeroelastic Analysis of Beam like Structure Configurations buckling analysis. In these cases, the use of collocation functions can lead to a better accuracy of the calculations. The simple examples given for the dynamic and buckling analysis show a good agreement with the analytical results. The formulation presented here can be also used in the case of non-uniform beams or for other boundary conditions, using appropriate Green functions determined either numerically or by experiment. The presented approximate method gives good results, the accuracy depending on the number of the used collocation points and on the number and precision of the collocation functions. ACKNOLEDGEMENT This article is an improved version of the science communication with the same title, presented in The 37 th edition of the Conference Caius Iacob on Fluid Mechanics and its Technical Applications, 6-7 November 07, Bucharest, Romania, (held at INCAS, B-dul Iuliu Maniu 0, sector 6), Section. Basic Methods in Fluid Mechanics. REFERENCES [] R. L. Bisplinghoff, H. Ashley, R. L. Halfman, Aeroelasticity, Reading, Massachusetts, Addison-Wesley Publishing Co. Inc., 955. [] A. Petre, Theory of the Aeroelasticity - Statics (in Romanian), Romanian Academy Publishing House, 966. [3] A. Petre, Theory of the Aeroelasticity Dynamic periodic phenomena (in Romanian), Romanian Academy Publishing House, 973. [4] V. Anghel, M. Stoia, An Integral Formulation of the Equation of Transverse Vibrations of the Rotating Beam (in Romanian), Studies and Researches in Applied Mechanic (S.C.M.A, Vol. 54, No. 5-6), pp , 995. [5] V. Anghel, M. Stoia, Analysis of the Coupled Bending-Torsion Vibrations of the Straight Blades Using an Integral Formulation of the Equations of Motion (in Romanian), 6 pages, XIX th National Conference of Solid Mechanic, Section B, Târgoviște, June -3, 995. [6] V. Anghel, Coupled Bending Vibrations Analysis of the Pretwisted Blades. An Integral Formulation Using Green Functions, Rev. Roum. Sci. Techn - Méc. Appl., Vol. 4, No. -, pp , 997. [7] G. Surace, V. Anghel, C. Mareș, Coupled Bending-Bending-Torsion Vibration Analysis of Rotating Pretwisted Blades. An Integral Formulation and Numerical Examples, Journal of Sound and Vibration, Vol. 06, No. 4, pp , October 997. [8] G. Surace, L. Cardascia, V. Anghel, Vibration of Rotating Beam with Tip Mass. A Formulation based on Greens Functions, Proceedings Vol. 4, pp , 5 th Pan American Congress of Applied Mechanic - PACAM V, S. Juan, Puerto Rico, -4 January, 997. [9] V. Anghel, M. Stoia, Transverse Vibrations Analysis of the Rotating Beam with Tip Mass by Using an Integral Formulation of the Equation of Motion, U.P.B Bulletin, Series D, Vol. 60, Nr. 3-4, pp. 3-4, 998. [0] V. Anghel, I. Trandafir, Transverse Vibrations Analysis of the Non-uniform Beam Using an Integral Formulation of the Equation for Motion, U.P.B Bulletin, Series D, Vol. 60, Nr. -, pp. 9-4, 998. [] V. Anghel, An Integral Formulation of the Equations of Coupled Bending-Torsion Vibrations of Thin Walled Open Profile Beams, Proceedings, Vol., pp. -6, IX th Conference on Mechanical Vibrations, Timișoara, 7-9 May, 999. [] G. Surace, L. Cardascia, V. Anghel, Free Vibration of Composite Rotating Beams - An Integral Method Based on Greens Functions, Proceedings -Volume two, pp , nd European Rotorcraft Forum and 3 th European Helicopter Association Symposium, Brighton, UK,7-9 September 996. [3] V. Anghel, C. Petre, An Application of Green Functions to Static Structural Response of Open Section Thin Walled Composite Beams, on CD-Rom, paper07.pdf, 3 pages, European Conference on Computational Mechanic, ECCM 99, Munich, 9 august - 3 september, 999. [4] V. Anghel, C. Petre, New Applications of Green s Functions in Static and Dynamic Response Analysis of Beams, ZAMM-Journal of Applied Mathematics and Mechanics-Zeitschript für Angewandte Mathematik und Mechanik, Vol. 8, Suppl. 4, pp , 00. INCAS BULLETIN, Volume 9, Issue 4/ 07

8 Viorel ANGHEL 0 [5] V. Anghel, C. Petre, A. Alecu, Static and Dynamic Response Analysis of Beams by Using Structural Influence Functions, on CD-rom, 0 pages, VIII th Conference on Numerical Methods in Continuum Mechanics, NMCM 000, Liptovsky Jan, Slovak Republic, September 9-4, 000. [6] V. Anghel, C. Petre, F. Frunzulică, A Comparison of Several Numerical Methods for Aeroelastic Analysis of Wings, paper 5D, 6 pages, Proceedings of the Annual Symposium of the Institute of Solid Mechanics, SISOM 00, Bucharest, Romania, 4-5 May, 00. [7] L. Li, X. Zhang, Y. Li, Analysis of Coupled Vibration Characteristics of Wind Turbine Blade Based on Green s Functions, Acta Mechanica Solida Sinica, Vol. 9, No. 6, pp , December, 06. [8] H. Han, D. Cao, L. Liu, Green s Functions for Forced Vibration Analysis of Bending-Torsion Coupled Timoshenko Beam, Applied Mathematical Modelling, Vol. 45, pp , May, 07. [9] Ghe. Buzdugan, E. Mihăilescu, M. Radeș, Vibration Measurement, Martinus Nijhoff Publishers, Dordrecht, 986. INCAS BULLETIN, Volume 9, Issue 4/ 07