Application of a Numerical Procedure for the Dynamic Analysis of Plane Aeronautical Structures

Journal of Spacecraft and Rockets

2000

A numerical procedure, which lies between Rayleigh-Ritz and the nite element method so far applied only to aeronautical structures in both two-and three-dimensional space, is utilized for the dynamic analysis of periodic truss space structures, typical of appendices of arti cial satellites. The structural components are unidimensional tube-type elements with circular cross sections. They are considered as beam elements, for which a complete model for the internal strain energy has been utilized. Three different numbers of component bays of these structures are assumed in the numerical applications, and the obtained results are analyzed. The advantages of the proposed method as the number of the component bays increases, particularly with respect to the CPU time, are pointed out comparing the obtained results with the ones of a classical nite element method numerical program, such as MSC/NASTRAN. Nomenclature A = beam cross-sectional area E = modulus of elasticity E f 0 = parameter containing the modulus of elasticity e e , e de , e ded = occurring integrals determined by the local describing functions G = shear rigidity modulus G t 0 , G d = parameters containing the shear rigidity modulus g (n) ia ib ic , g (r ) ia ibic , = global describing functions g ja j b jc (s ) coef cients of the generic variable S n J f = exural moment of inertia J t = torsional moment of inertia K = stiffness matrix L = length of the generic beam L = nondimensional length of the generic beam L 0 = reference length l ( Ib ) nie , l ( Ib ) rie , l ( Ib) s j e = local describing functions coef cients of the generic variable S n in the I b th beam M = mass matrix N = number of Lagrangian degrees of freedom N a , N b , N c= number of global describing functions along the axes X , Y , and Z , respectively N el = number of local describing functions P e , P de , = occurring mixed integrals determined P ed , P ded by the global and local describing functions coupling P p , P d p , P d pd = occurring integrals obtained by the global describing functions q i = generic Lagrangian degree of freedom R i j = rotation matrix element connecting the axis x i with the axis X l j S d a = parameter containing the modulus of elasticity S m t , S m f , S m a = parameters containing the mass density and J t , J f , and A, respectively S n = generic independent variable = kinetic energy U, V , W ; = nondimensional displacements along U 1 , U 2 , U 3 the axes of the main reference system U l , V l , W l ; = nondimensional displacements along U l1 , U l2 , U l3 the axes of the local reference system a = axial strain energy f = exural strain energy T = total strain energy

Section: Mathematical Modelmentioning

confidence: 99%

Numerical Procedure for the Dynamic Analysis of Truss Space Structures

Journal of Spacecraft and Rockets

2000

“…where the coefficients t l t3 in Equation (15) depend on t l t , t m t , t n t as follows: and the relation between l t3 and l t , m t , n t can be obtained by Equations (9), (11), (13), (14) and (16). Consequently the stiffness matrix elements k ij , which are proportional to the integral of the product between the cubic power of the thickness t o T and some structural operators applied to W (15) , taking into account Equation (15), can be expressed as:…”

Section: The Stiffness and Mass Matricesmentioning

confidence: 99%

“…A numerical procedure (15)(16)(17) , which arises from the Rayleigh-Ritz method (18)(19) , and which utilises polynomial power series expansions both for the flexural displacement and for the thickness behaviour function, is introduced and developed.…”

Section: Introductionmentioning

confidence: 99%

“…It is necessary to compute the integral of the generic term X l Y m over the whole structure dimensionless surface S d , to evaluate the stiffness matrix elements k ij and for other forthcoming computations: It can be determined by utilising the natural reference system X,η, where 0 ≤ η ≤ 1 (16,17) . The coordinate Y is connected with X and η by a nonlinear algebraic relation (16) .…”

mentioning

confidence: 99%

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Polynomial series expansion for optimisation of wing plane structures in idealised critical flutter conditions

2005

Aeronaut. j.

Self Cite

A numerical procedure, which utilises polynomial power series expansions for the optimisation of multipanel wing structures in idealised critical flutter conditions, is introduced and developed. It arises from the Rayleigh-Ritz method and employes trial polynomial describing functions both for the flexural displacement and for the thickness variation over the multipanel surface. An idealised structural plate model, according to the Kirchhoff's theory, together with a linearised supersonic aerodynamic approach, are supposed. The classical Euler-Lagrange optimality criterion, based on variational principles, has been utilised for the optimisation operations, where by imposing the stationary conditions of the Lagrangian functional expression, a nonlinear algebraic equations system is obtained, whose solution is found by an appropriate algorithm. By utilising an iterative process it is possible to reach the reference structure critical conditions, with an optimised thickness distribution throughout the multipanel surface. The final part of the work consists in searching the minimum weight of the multipanel planform wing structure with optimised thickness profile vs the flutter frequency, considered as a variable imput parameter, for fixed flutter speed and equal to the critical one of the reference uniform structure. NOMENCLATURE EYoung's modulus f (a) aerodynamic distributed force over the multipanel structure surface f i trial describing functions of the dimensionless flexural displacement behaviour over the wing plane structure g l t trial describing functions of the nondimensional thickness behaviour over the wing plane structure h wing plate thickness I o second moment of inertial area per unit length of the reference plate structure k ij , k nj stiffness matrix elements k ij (l t3 ) generic components of the stiffness matrix elements k ij * matrix elements which take into account both the structural and the aerodymic forces presence L cantilever length of the wing structure m ij , m nj mass matrix elements m ij (l t ) , m nj (l t ) generic components of the mass matrix elements N whole number of the degrees of freedom N a , N b + 1 numbers of the coefficients and describing functions in the flexural displacement series expansions along the axes x and y, respectively N t whole number of the coefficients and describing functions in the nondimensional thickness series expansion N tx , N ty number of the coefficients and describing functions in the non-dimensional thickness series expansions along axes x and y, respectively N t3 whole number of the coefficients and describing functions in the series expansion of the nondimensional thickness cubic power N t3x , N t3y number of the coefficients and describing functions in the series expansion of the nondimensional thickness cubic power behaviour along the axes x and y, respectively N w whole number of the coefficients and describing functions in the flexural displacement series expansion P(x, y) generic point on the multipanel structure surface q dynamic pre...

“…First a numerical procedure [Tizzi 1994;1996;2003] which arises from the Rayleigh-Ritz method [Kantorovich and Krylov 1964, pp. 258-303;Mikhlin 1964, pp.…”

Section: Introductionmentioning

confidence: 99%

Influence of the nonlinear aerodynamic force components on a vibrating beam, simply supported or clamped at both ends, exposed to a high supersonic airflow along its axial direction

J. Mech. Mater. Struct.

2009

In this work we analyze the case of a vibrating beam, simply supported or clamped at both ends, under the effect of a high supersonic airflow along its axial direction. A complete aerodynamic model of the piston theory, which also takes into account the nonlinear components of the distributed aerodynamic transversal force, is used. The postcritical flutter behavior and its influence on the vibration state solutions of a fluttering beam without aerodynamic damping have been studied. This paper focuses particularly on the effects of these nonlinear aerodynamic forces on three frequencies, which are useful in characterizing the postcritical flutter solution set of the undamped beam in the whole frequency range: the minimum frequency, the frequency where the change of the modal shape with lower amplitude occurs, and the frequency corresponding to the solution with minimum amplitude of the vibration mode. Special attention has been given to the influence on the solution of the vibrating undamped beam with minimum modal amplitude, whose frequency is the most important among the three mentioned above; in fact, in the neighborhood of this particular solution, there exists the flutter state of the vibrating damped beam in limit cycle conditions. Three different schemes, two of them semianalytical (based on the classical and well known RayleighRitz and Galerkin methods) and one of them numerical (based on the finite element method), have been herein exploited, as in the author's previous papers, where beam flutter models with linear aerodynamic analysis were used. The good agreement between the results obtained by the three methods corroborates their effectiveness.More sophisticated models have been herein set up, considering that a more accurate analysis is necessary than in previous cases, where the aerodynamic numerical model was limited to within the framework of the quasisteady linearized piston theory, both for the coupling component between odd and even order vibrating modes, and for the aerodynamic damping component.The results obtained enable us to assess quantitatively the influence of these nonlinear aerodynamic forces on the postcritical beam flutter behavior, and particularly on the undamped beam solution with minimum amplitude of the vibration mode.