281.INTRODUCTION1.1 AeroelasticityAeroelasticity is an important subset of Fluid-Structure Interaction encompassing those physical phenomena for which aerodynamic, elastic, and inertial forces influence each other and interact in a significant way. Fluid-structure interaction is where a moving gas or fluid interacts with a structure in such a way that the deformations induced by the medium are of a magnitude such that the flow field is affected. Fluid Structure Interaction is usually a dynamic process. Modern airplane structures are not completely rigid, and aeroelastic phenomena arise when structural deformations induce changes on aerodynamic forces. The additional aerodynamic forces from some sort of perturbation cause increase in the structural deformations, which lead to greater aerodynamic forces. These interactions may become smaller until a condition of equilibrium is reached, or may diverge catastrophically. Aeroelasticity can generally be divided into two fields of study: Static aeroelasticity Dynamic aeroelasticityStatic aeroelasticityStatic aeroelasticity studies the interaction between aerodynamics and elastic forces on an elastic structure. Mass properties are not significant in the calculations of this type of phenomena, since inertial forces are completely excluded from such analysis.Dynamic aeroelasticityDynamic aeroelasticity studies the interactions among unsteady aerodynamic, elastic, and inertial forces.1.2 Aeroelastic flutterAn example of dynamic aeroelastic phenomena is flutter, in which the flexibility and inertia of the structure play an essential part in the dynamic stability of the total fluid-structure system. It occurs when a structural system, under flow conditions beyond some threshold (critical) value of the flow parameter (viz. critical dynamic pressure), is driven into unstable, and self-excited oscillations due to unsteady aerodynamic forces from the flow. Flutter is basically a phenomenon of unstable oscillations in a flexible structure. Beyond the critical flow conditions, the onset of flutter instability is recognised by the exponential increase in the vibration amplitudes of the structural system with time (Figure 1.1). Aircraft structures that function as lifting surfaces are prone to flutter instability due to their interaction with the aerodynamic flow. The critical flow condition that leads to the onset of flutter is called the Flutter Boundary of the structure. The flutter boundary of an aerospace vehicle is a characteristic design parameter that is very important for practical design of its lifting surfaces.The mechanism of flutter can be explained from the physics of energy flow in the total fluid-structure system. Under sub-critical flow conditions, the structural oscillations in the

TimeDisplacement

Converging oscillation

(i)

Displacement

Time(ii)

Displacement

Time(iii)

Figure 1.1. Nature of dynamic response (displacement) of a structural system subjected to aerodynamic flow. For free stream flow velocities below a critical value, , the oscillations are stable, as shown in (i). At the critical flow velocity, the oscillations are un-damped, as shown in (ii). For velocities above this critical value, , the oscillations are unstable, as shown in (iii).aerodynamic flow are stable, and thus damped out since the net aerodynamic power flow over any oscillation cycle is less than what the structure actually dissipates out. At the flutter boundary (of critical flow velocity), the aerodynamic power input equals the dissipated power, and steady oscillations, of constant amplitudes, occur. Beyond this critical flow condition the aerodynamic power input exceeds the dissipated power in each cycle, leading to increase of the vibration amplitudes in time.1.3 Prediction and cureBesides aircraft structures, various other structural systems, like long span bridges, chimneys, tall buildings etc. are prone to flutter instability. To ensure safety of these structures against aerodynamic loads, it is necessary that they are designed to withstand severe wind conditions. It is thus essential that the flutter boundaries of these structures are estimated, and it is ensured that these are well above the worst aerodynamic loads that these structures are likely to encounter.Prediction of the flutter boundary of a structure subjected to aerodynamic loads is essential to ensure its safety against flutter. This involves making a mathematical model of the structure (say an aircraft) with appropriate inertial and stiffness distributions. This idealized structure is then analyzed with appropriate aerodynamic load simulations using various aerodynamic theories.If a structural system is prone to flutter instability, or the safety margin is quite low, appropriate cure for the problem can be prescribed through some ingenious redistribution of the inertial and stiffness properties so that an increase in the flutter velocity can be achieved. This kind of practice requires a reliable knowledge of the effects of changes of the various system properties upon the flutter boundary.1.4 Timoshenko Beam:Timoshenko beam theory which is a higher order beam theory, is known to be superior in predicting the transient response of the beam over the Euler Bernoulli beam model. In neglecting the contribution of shearing deformation the Euler-Bernoulli beam theory (EBT) requires that plane sections remain plane and perpendicular to the neutral axis after deformation. Consequently, this theory is best suited for thin or slender beams as shear strains have a considerable influence on the deformation of thick beams. A more accurate representation of beam flexure which allows for the inclusion of shear strains present in isotropic beams and more suited for thick beam analysis is the Timoshenko beam theory.The Euler-Bernoulli beam theory (EBT) frequently used for the analysis of isotropic beams, which have extensive use in engineering structures, describes beam kinematics completely in terms of flexural deformation. In neglecting the contribution of shearing deformation the EBT requires that plane sections remain plane and perpendicular to the neutral axis after deformation. Consequently, this theory is best suited for thin or slender beams as shear strains have a considerable influence on the deformation of thick beams. A more accurate representation of beam flexure which allows for the inclusion of shear strains present in isotropic beams and more suited for thick beam analysis is the Timoshenko beam theory. This theory, a first order shear deformation theory (FSDT), relaxes the normality assumption of plane sections evident in the EBT. By allowing for the inclusion of a constant through thickness shear strain, it violates the no-shear boundary condition at the top and bottom horizontal beam surfaces, requiring a problem dependent shear correction factor. Since Timoshenko Beam theory is known to be superior in predicting the transient response of the beam, it can yield better results and sometimes entirely new flutter modes and flutter velocities. Since flutter is a dynamic aeroelastic phenomena, Timoshenko beam theory constitutes an improvement over Euler Bernoulli theory in predicting the onset of flutter.1.5Sensitivity Study:Sensitivity analysis of nonlinear problems is a well-known topic in aeronautical engineering. The work of Bindolino and Mantegazza (1987),Murthy and Haftka (1988), Issac et al. (1995) are just a few of the contributions that can be found in the literature.

2LITERATURE REVIEWThe earliest study of flutter seems to have been made by Lanchester [1], Bairstow and Fage [2] in 1916. In 1918, Blasius made some calculations after the failure of the lower wing of Albatross D3 biplane. But the real development of the flutter analysis had to wait for the development of Non-stationary airfoil theory by Kutta and Joukowsky.Glauret [4,5] published data on the force and moment acting on a cylindrical body due to an arbitrary motion. In 1934, Theodorsen.s [6] exact solution of a harmonically oscillating wing with a flap was published. The torsion flutter was first found by Glauret in 1929. It is discussed in detail by Smilg [7] Several types of single degree of freedom flutter involving control surfaces at both subsonic and supersonic speeds have been found [8,9], all requiring the fulfillment of certain special conditions on the rotational axis locations, the reduced frequency and the mass moment of inertia.Pure bending flutter is possible for a cantilever swept wing if it is heavy enough relative to the surrounding air and has a sufficiently large sweep angle [10]. The stability of more complicated motions can be determined by calculating the energy input from the airstream. The bending torsion case in an incompressible fluid has been calculated by J.H.Greidanus and the energy coefficient in Bending-Torsion oscillations has been given [11]The use of .Quasi-steady. Aerodynamic theory for the flutter analysis of the wings and excellent treatises in the field of aeroelasticity are given by Y.C.Fung [12], E.H.Dowell [13,14], L.Mirovitch [15] and others.In the typical wing whose elastic axis (locus of shear centers) and mass axis (locus of center of gravity) do not coincide, the nature of oscillations is always coupled flexure-torsion. A vast literature exists on the flexure-torsion problem of engineering structures. Evins [16] has given comprehensive details about vibration fixture transducers and instrumentation. Bisplinghoff and H.Ashley [17] has described the elastic characteristics shape and inertial idealization.A new method for determining mass and stiffness matrices from modal test data is described by Alvin and Paterson [18]. This method determines minimum order mass and stiffness matrices, which is used to determine the optimum sensor location. Dugundji [19] examined panel flutter and the rate of damping. The problem of two and threedimensional plate undergoing cyclic oscillations and aeroelastic instability is investigated by Dowell [13,14].Abott [20] has suggested a technique for representing the shape of the aerofoil through analytical relations. The coupled flexure-torsion vibration response of beam under deterministic and random load is investigated thoroughly by Eslimy and Sobby [21] by use of normal mode method. The exact determination of coupled flexure-torsion vibration characteristics of uniform beam having single cross section symmetry is studied by Dokumaci [22]. At present, subsonic flight is a daily event and supersonic and hypersonic flights are a reality. Now aeroelastic analysis has become an organic part of the design.

Rayleigh had proposed a formulation for dynamic analysis of beams where Euler beam model is maintained for stiffness considerations, but rotator inertia has been taken into account. This formulation yielded results that have lower errors than those of the Euler beam model without rotational inertia considerations. However at high frequencies, the Rayleigh beam still showed severe deviations .Timoshenko realised the source of error, and took a ingenious step to create a beam model, now popularly known as Timoshenko beam[23]. In his formulation, he maintained the rotatory inertia terms as had been proposed by Rayleigh, but discarded the Euler beam modelThe development of structural and finite element models of the Timoshenko beam theory has been the subject of numerous papers in the literature[24-29]. The exact, 4 x 4 stiffness matrix of the Timoshenko beam is derived either using the methods of structural analysis or finite element formulations. The shear locking is due to the inconsistency of the interpolation used for w and , or equivalently, not satisfying the requirement that the shear strain xz = (dw/dx) + is element-wise constant for element-wise constant values of EI. Often, the Timoshenko finite element models are based on equal interpolation of w and and use reduced-order integration to evaluate the stiffness coefficients associated with the transverse shear strain and full integration for all other coefficients. Others have used so-called consistent interpolation based on the recovery of correct constraints in the thick beam limit (Prathap & Babu [30]; Shi & Voyiadjis [31]; Rakowski [32]; Reddy [33]). Although such elements do not experience locking, they do not lead to the two-node super-convergent element. Friedman & Kosmatka [34] and Reddy [35] and Reddy et al [36] have independently developed the two-node super-convergent element using the exact solution of the homogeneous form of the Timoshenko beam equations. Hermite cubic interpolation of w and interdependent quadratic interpolation of 4, was used in developing the element that has the super-convergence character for static problems. Reddy[37] has provided a excellent treatise on the dynamic behaviour of all the Timoshenko beam finite elements. A tool to find the gradient of a dynamic type constraint variable as a function of design parameters has wide applications in complex engineering problems. Rogers [38] deduced an expression for the derivative of eigenvalues and eigenvectors with respect to an arbitrary parameter of a dynamic system, which can be represented mathematically by a linear, constant coefficient differential equation. By using the expressions, a set of increments in the design variables may be selected to yield the desired improvements in the system characteristics of interest.Adelman and Haftka [39] surveyed the methods applicable to the calculation of structural sensitivity derivatives for finite element modelled structures and discussed literature published on four main topics: derivatives of static response (displacement and stresses), eigenvalues and eigenvectors, transient response and derivatives of optimum structural designs with respect to problem parameters. The survey also includes a number of methods developed in non-structural fields such as control and physical chemistry, which are directly applicable to structural formulations.Ringertz [40] has studied the optimal design problem of a wing in incompressible flow with aeroelastic constraints. The weight of a cantilever wing is minimized using the thickness of the composite face sheets as design variables subject to constraints on flutter and divergence speed. A doublet-lattice panel method is used for computation of unsteady aerodynamic loads. Ringertz discusses several difficulties with optimization of eigenvalues of un-symmetric and complex matrices.A methodology for carrying out analytical sensitivity analysis of the flutter phenomenon in long span bridges has been discussed by Jurado and Hernandez [41]. A nonlinear eigenvalue problem for the calculation of flutter instability has been modelled and is further used for the sensitivity analysis of flutter instability with respect to key chosen design variables, moments of inertia of the bridge deck. Testing of these derivatives has been performed through centred differences method. They have done detailed studies on Great Belt, Vasco da Gama and Old Tacoma Bridge based on the presented method.

3.THEORETICAL FORMULATION OF THE TIMOSHENKO BEAMReddy[37] has presented superconvergent finite element model for static problemsusing two alternative approaches: (1) assumed-strain finite element model of the conventional Timoshenko beam theory, and (2) assumed-displacement finite element model of a modified Timoshenko beam theory.The displacement field of the Timoshenko beam theory for the pure bending case is, , (3.1)Where w is the transverse deflection and x, the rotation of a transverse normal line about the y axis. The strains and stresses of the Timoshenko beam theory are (3.2) (3.3)The equilibrium equations of the beam are....(3.4) (3.5)where q(x) is the distributed transverse load, E Young's modulus, G the shear modulus, A the area of cross section, I the moment of inertia, and Ks the shear correction factor.3.1 Displacement Finite Element Models:3.1.1 The General Model:The displacement finite element model of the Timoshenko beam theory is constructed using the principle of minimum total potential energy, or equivalently, using the weak form....(3.5) (3.6)

q(x) x

waMeaMeb12 Va VbheFigure 3.1 Typical finite element with force degress of freedom

Suppose that w and x are approximated as (3.7) where(W j,j ) are the nodal values of (w, x) and j()(x) ( = 1, 2) are the associated interpolation functions. Substitution of (7) for w and x, and w = i(1) and x = i(2) into (3.5) yields the finite element model(3.8)whereKij11 = `Kij12 = Kij22 =

(3.9)

3.1.2 Reduced Integration element (RIE):For a linear interpolation of w and x and exact evaluation of the integrals of (3.9),(3.8) takes the form (3.10)Where(3.11a) (3.11b) In the thin beam limit, i.e 0, the first and third equations of (3.10) imply the following relation among (W1,W2,1,2) :(3.12) Which is equivalent to the Kirchoff constraint x +dw/dx =0 (or shear strain xz=0).The second and fourth equations of (10), in view of (12), yield the constraint(3.13) This is equivalent to dx/dx =0, which is an incorrect condition to satisfy as it forces the curvature and hence bending energy to zero. Thus, (3.10) in an effort to satisfy the constraints (3.12) and (3.13), will yield the trivial solution W1=W2=1=2=0 (i.e., the element locks).The Kirchoff condition (3.12) suggests that w and x be interpolated such that dw/dx is a polynomial of the same order as x. If w is approximated using a linear polynomial ( a minimum requirement), then x should be a constant. Since the minimum continuity requirement on x is also linear, it follows that w be approximated using a quadratic polynomial. This is a consistent interpolation. Unless the weak form of Timoshenko beam theory is modified, we have no alternative but to use a quadratic approximation for w and linear for x and use full integration to evaluate the coefficient matrices to obtain an element that does not experience locking. However, if one approximates both w and x with linear polynomials but treats x as a constant in the evaluation of the shear strain,it will also yield the stiffness matrix. This procedure is known in the literature as reduced integration of the shear stiffness. It amounts to evaluating the second term of Kij22 in (9) using one-point integration as opposed to two-point integration required to exactly evaluate the integral. The element equations of the reduced integration element are (3.14) This element is designated as the reduced integration element (RIE) by Reddy (1993). Alternate derivation of the element without using the reduced integration concepts will be presented in the sequel. In the thin beam limit, the element equations reduce to only one constraint, namely the Kirchhoff condition in (3.12). While the element does not lock, it does not yield exact displacements at the nodes for the static problems, and often a sufficient number of elements sis needed to obtain accurate deflections.3.1.3 Consistent interpolation element ( CIE)As suggested earlier, if we use a quadratic approximation of w and linear approximation of , (8) reduces to a 5 x 5 system of equations. By eliminating the mid-side degree of freedom associated with w, we can reduce the 5 x 5 system to the following 4 x 4 system of equations (Reddy 1993; Reddy 1999): (3.15) Where(3.16) and i(2) are the quadratic interpolation functions. Here the subscript c is used for the centre node of the element. Note that the element has the same stiffness matrix as the reduced integration element but a different load vector. The load vector is equivalent to that of the Euler-Bernoulli beam element. In fact, for constant q, the load vector in (3.15) is identical to that of the Euler-Bernoulli beam element. 3.1.4 Interdependent interpolation element (lIE)The next choice of consistent interpolation is to use cubic for w and quadratic for x. This will lead to a 7 7 system of equations. The displacement degrees of freedom associated with the interior nodes (three in total) can again be condensed out, for the static case, to obtain a 4 4 system of equations. Here we will not consider it further. Instead, we consider the Hermite cubic interpolation of w and a related quadratic approximation of . These sets of interpolation functions were derived by Reddy (1997) using the exact solution of (3.3) and (3.4) for q = 0. The resulting finite element is termed the interdependent interpolation element (IIE).To develop the interdependent interpolation element, we assume an approximation of the form(3.17)(3.18)Where i(1) and i(2) are the approximation functions,(3.19)(3.20)Here is the nondimensional local coordinate(3.21)When = 0, i(1) reduces to the usual Hermite interpolation functions i and i(2) to di/dx. Substitution of (17) into (5) yields the finite element model,(3.22)Where(3.23)(3.24)And Q1= Va , Q2 = Ma , Q3=Vb , and Q4 = Mb. Equation (3.22) has the explicit form,(3.25) This element leads to the exact nodal deflections in static analyses for any distribution of the transverse load q(x) and element-wise constant bending stiffness E1 and shear stiffness GAKs . Therefore, the element is said to be superconvergent. In the thin beam limit, (3.25) reduces to the Euler-Bemoulli beam equations, and no additional constraints are implied by the system.3.2The assumed strain-displacement (ASD) models3.2.1 General finite element modelHere we develop the finite element model based on a variational form in which the displacements (w, x) and strains (Kxx, Yxz) are treated as independent field variables. The variational statement associated with this mixed formulation is given by the stationarity of the following functional (see Oden & Reddy 1982, p. 116, equation (4.115)):

(3.26)Where(3.27) The first variation of R yields the weak forms(3.28)(3.29)(3.30)(3.31)Let the variables (w, x,xx,xz) be approximated as(3.32)where (Wj, j, j, j) are the nodal values of (w,x ,xx,xz) and j()(x) ( =1, 2, 3, 4) are the associated interpolation functions whose choice is yet to be made. Substituting (3.32) into (3.28)-(31), we obtain the following finite element model:(3.33)Where(3.34)Couple of observations are in order concerning the finite element model in (3.33). We note that [A] is a vector {A} when xz is approximated as a constant, 0. In addition, the first equation of (33) has the form(3.35) when w is interpolated using quadratic or higher-order polynomials. The nonzero entries correspond to the deflection degrees of freedom at node 1 and node m. For linear interpolation of w, we have m = 2 and (3.35) is alright. However, when m > 2, (3.35) implies that Fi = 0 for i = 2, ---, m - 1, which, in general, is not true. Thus, either the distributed load is zero or it is converted to generalized point forces at the end nodes through Hermite cubic polynomials. In the latter case, the force components can be added to Va and Vb and the moment components to Ma and Mb at nodes 1 and m respectively.3.2.2 ASD-LLCC elementFor linear (L) interpolation of (w, x) and constant (C) representation of (xx, xz), and for constant values of E1 and GAKs, the element equations become (m = n = 2 and p=q=l)(3.36) (3.37) (3.38) (3.39)Solving (3.38) and (3.39) for 0 and 0 and substituting into (3.36) and (3.37) (i.e condensing out 0 and 0), we arrive at the following 3x4 system of equations,(3.40) where = EI/GAKsh2. These are exactly the same equations obtained in the displacement formulation with the linear interpolation of w and x and using one-point Gauss quadrature to evaluate the shear stiffnesses, i.e., the reduced integration element (RIE). Thus, the assumed strain-displacement formulation eliminates the need for reduced integration concepts.3.2.3 ASD-HQLC element Suppose that the distributed load is represented using(3.41) A Lagrange or Hermite cubic interpolation of w, quadratic interpolation of x, linear interpolation of xx, and constant representation of xz yields the equations(3.42)(3.43)(3.44)(3.45)where the end nodes of the element are designated as '1' and '2', and the middle node as 'c', and the interior nodal degrees of freedom associated with w are omitted as they do not contribute to the equations. Solving (3.44) for {} and (3.45) for 0, substituting the result into (3.42) and (3.43), and eliminating c, we obtain(3.46)(3.47)Adding (3.46) and (3.47), we obtain(3.48) The stiffness matrix is the same as that of the superconvergent element derived by Reddy (1997); however, the load vector is different. It is the same when either the applied load q is element-wise uniform or the load vector is computed using (3.24) with i given by (3.19).It should be noted that the degree of the polynomial interpolation used for w does not enter the equations presented in all the models discussed in this section. However, the load representation implies that w be interpolated with Hermite cubic polynomials or i(1) of (3.19). It can be shown that the use of the interdependent interpolations of (3.19) and (3.20) for w and x also results in (3.48).

3.3Two-component form of the Timoshenko beam theory 3.3.1 Theoretical formulationThe displacement and mixed formulations of the conventional Timoshenko beam theory yield the superconvergent stiffness matrix only when higher-order interpolations of w and x are used. In contrast, the Euler-Bernoulli beam element is superconvergent for the lowest admissible interpolation, namely, the Hermite cubic interpolation. In this section, it is shown that the superconvergent element can be developed with the lowest admissible interpolation of various displacement components. This requires a reformulation of the Timoshenko beam theory in terms of the bending and shear components of the transverse deflection. The two-component form of the transverse deflection was discussed by Anderson (1953), Miklowitz (1953), Huffington (1963), and Krishna Murty (1970) for beams, and Miklowitz (1960), Chow (1971), Bhashyam & Gallagher (1984), Reddy (1987), Lim et al (1988), and Senthilnathan et al (1988) for plates. Assume displacement field of the form(3.49)where wb and ws denote the bending and shear components, respectively, of the total transverse deflection w (see Reddy 1999), and ~x denotes the shear rotation, in addition to the bending rotation, of a transverse normal about the y axis. The strains and the stressstrain relations are given by(3.50)(3.51)The principle of virtual displacements yields the following Euler-Lagrange equations:(3.52)(3.53)(3.54)Where M(x) and Q(x) are the bending and shear force resultants,(3.55)(3.56)3.2.2 Finite element modelThe finite element model of the modified Timoshenko beam theory can be developed using the standard steps. The first step is to write the weak forms of the three equations over a typical element. We have(3.57)Where (3.58)(3.58)From the weak form (57) , it is clear that x and ws can be interpolated using the Lagrange interpolation and wb using Hermite interpolation. The lowest admissible functions are linear for x and ws and cubic for wb. However, the condition that the shear force be element wise constant for element wise constant values of EI in turn requires that ws be quadratic.Let (x,wb,ws) be interpolated as(3.59)where i, Wis and Wib denote the nodal values of x, ws and wb, respectively, i(1) and i(2) are linear and quadratic interpolation functions, respectively, and i are the Hermite cubic interpolation functions (m = 2, n = 3, p = 4). Substituting the interpolations (59) into the weak form (57), we obtain the following finite element model:(3.60)or simply(3.61)where the stiffness matrix [KR] is of the order 9x9.The coefficients of various matrices and vectors in (60) are defined by

(3.62) The element equations (60) are not suitable for practical use. The reason is that we only know the total displacement w = wb + ws and not its bending and shear parts separately. This is also true about the total rotation (x = -wbx + x). Hence, it is necessary to recast the element equations (60) in terms of the physical nodal variables.3.3.3 Reduction of equationsHere we select specific interpolation functions and evaluate the element matrices. For the choice of linear interpolation functions for i(1), quadratic interpolation functions for i(2), and Hermite cubic interpolations functions for i (the minimum polynomials required by the weak form), we obtain (see figure 2)(3.63) (3.64)(3.65)

(3.66)where(3.67)Wcs denotes the value of ws and is the specified transverse load at the centre node of the element. Note that the finite element equations associated with the second equation in (3.60) is split into a pair of equations for convenience.

As noted earlier, it is necessary to combine the two components of the transverse deflection as well as the rotation into total deflection and rotation. This amounts to rewriting the algebraic equations (60) to obtain a model solely in terms of the total deflection w = wb + ws and rotation x = --wb, x + x at the element nodes. First we condense out Wcs using the second equation of (66). We have(3.68)where . Substituting (3.68) into (3.63) and (3.66), we obtain

(3.69)(3.70) where i denote the total generalized displacements,

(3.71) Adding (64) to (70) and (65) to (69), we find

(3.72) (3.73) Now combining (3.72) and (3.73), we arrive at(3.74) Where (3.75)Equation (3.74) is the same as (3.25)3.4Finite element models for dynamic analysis3.4.1 Weak forms and finite element modelsFor the dynamic case, the weak forms in (3.5), (3.28) and (3.29), and (3.57) (which correspond to the displacement and mixed finite element models of the conventional Timoshenko beam theory and the displacement model of the modified Timoshenko beam theory) must be modified to read(3.76)(3.77)(3.78) (3.79)respectively, where(3.80) being the mass density of the material.For the dynamic case, the finite element models in (8), (33), and (60) take the followingforms.

Reduced integration element (RIE):(3.81)Interdependent interpolation element (IIE):(3.82) Assumed strain-displacement model (ASD):(3.83) Two-component theory displacement finite element model:(3.84)Where(3.85)(3.86)3.4.2 Mass matricesBecause of the presence of the second time derivative terms and , it is not possible to algebraically manipulate the equations, as was done in the static case for CIE, ASD-HQLC , and finite element model based on two component form of Timoshenko beam theory. Recall that for RIE (linear or quadratic ),IIE, and ASD-LLCC, no algebraic manipulations were necessary. Therefore, these elements are directly applicable to dynamic analysis. For the finite element model based on the two-component form of the Timoshenko beam theory, one may select a mass matrix to go with the superconvergent (SCE) stiffness matrix for the dynamic analysis. The explicit forms of the finite element calculations for the RIE,IIE and SCE are summarized below. Reduced integration element(RIE): For linear interpolation of w and x , the finite element equations are given by(3.87) For quadratic interpolation of both w and x, the element matrices are of order 6x6 for pure bending case.Interdependent interpolation element (IIE):For this case, the stiffness matrix and load vector are given in (25) . The mass matrix[M] of (85) consists of several parts as given below. (3.88)Finite element model with superconvergent stiffness matrix (SCE): Although the superconvergent form of the stiffness matrix can be derived using various approaches, only the interdependent interpolation element formulation is readily extendable to the dynamic case. The other formulations do not permit the algebraic manipulations with the mass terms in place. Hence, one may choose a mass matrix to go with (48) and(74). There are several choices (i) use the same mass matrix as in (88) , (ii) use the mass matrix of the euler-bernoulli beam element, or (iii) use the mass matrix of the IIE element with =0(hence, =1). The first choice reduces the formulation to IIE, the second and third choices are the same because of the relationship between i(1), i(2) and i. Thus for the dynamic case, the finite element model in(74) takes the form(3.89) Where (3.90)Note that when is set to zero in mass as well as stiffness matrices, the equations of IIE and SCE are reduced to those of the Euler-Bernoulli beam element.

4.MATHEMATICAL FORMULATION FOR THE FLUTTER PROBLEM4.1Introduction to Flutter AnalysisThe mathematical formulation of subsonic flutter analysis of a typical subsonic wing is presented. For low speed subsonic aircrafts, the wings are usually un-swept or the sweep angle will usually be very small. A typical subsonic wing is shown in Figure 4.1. For aerodynamic reasons, a typical low speed subsonic wing is characterized by high aspect ratio (semi-span/mean chord) and a straight or nearly straight configuration. This fact is advantageous for structural analysis of the wing using a simple beam model, despite the complex arrangement of the constituent structural elements. Each wing is assumed to behave like a cantilever, supported at the axis of connectivity of the two wings, inside the fuselage. The present analysis is limited to the clean wing, i.e., the ailerons are not involved in the analysis. The method is demonstrated using a simple wide cantilever beam with uniform rectangular cross-section.

Elastic AxisInertia AxisXY

Figure 4.1 (a) A typical subsonic wing

+ ve + ve M+ ve h+ ve LXOXcmcXZ

Figure 4.1 (b) A typical airfoil section showing heave and pitch degree of freedom

Before proceeding to flutter analysis, it is required to define various matrices involved in the equation of motion viz., inertia matrix, , structural and aerodynamic stiffness matrices, and and aerodynamic damping matrix, . Considering the strip theory of aerodynamics, equation of motion of a Timoshenko beam element is presented with respect to the assumed affirmative directions of generalized coordinates (heave and pitch ) and external generalized forces (aerodynamic lift and aerodynamic moment ).. Another way of defining matrices are through finite element formulation of a beam element with both bending and torsion degree of freedom. 4.2Formulation of equations of motion for a wide rectangular cantilever beam of uniform cross sectionThe equation of motion of a uniform cantilever beam is derived using two approaches; 1) through analytical Timoshenko beam formulation and 2) through finite element beam formulation with bending and torsion degree of freedom. Figure 4.2 shows a uniform cantilever beam.

wXYlElastic axis and Inertia axistV

Figure 4.2 Wide cantilever beam of uniform cross sectionAn un-swept cantilever wing having a straight elastic axis perpendicular to the fuselage, which is assumed to be fixed in space, is considered. The wing deformation can be measured by a bending deflection h in Z-Y plane and a rotation about the elastic axis, h being positive upward and is assumed positive if the leading edge up. The chord wise displacement is neglected. The frame of reference is chosen as shown in Figure 2.2, with the Y-axis coinciding with the elastic axis. Let l be the semi-span of the wing, w be the width and t be the thickness of rectangular cross section of the beam model. Let be the distance between the centre of mass and the elastic axis at any section, positive if the former lies behind the latter (here, since the beam model cross section is uniform throughout the span and it is rectangular in geometry, throughout the semi-span). Let c be the chord length and be the distance of the elastic axis after the leading edge. In a steady flow of speed V, the wing will have some elastic deformation, which is however, of no concern to the problem of flutter.4.3Formulation of equations of motion for a wide rectangular cantilever beam of uniform cross section by means of continuous beam model4.3.1 Euler Bernoulli beamConsider a differential beam element. Let h and be the deviations from its equilibrium state, and let the inertia, elastic and aerodynamic forces correspond also to the deviations from the [steady-state values; then, for small disturbances, the principle of superposition holds, and we have the following equations of motion for the Euler Bernoulli beam,(Y.C.Fung [12],Dowell E.H. [2]).(4.1)(4.2)for 0 < y < Lwhere EI and GJ are the bending and torsional rigidity of the wing, m and I are the mass and mass moment of inertia about the elastic axis of the wing section at y, per unit length along the span, L and M are the aerodynamic lift and moment per unit span, respectively. The aerodynamic lift and moment at the flexure point, which act on a symmetrical airfoil, according to quasi-steady strip theory is given by,(4.3)(4.4)Moreover the aerodynamic analysis is subject to the quasi-steady assumption, which implies that only the instantaneous deformation is important and the history of motion may be neglected. Aerodynamic damping is incorporated through heave and pitch velocities as suggested by Fung [12] based on the aerodynamic strip theory. The lift gradient for the infinite thin airfoil in a two-dimensional incompressible flow is . A corrected set of expressions for the lift and moment coefficients (about support point) with compressibility effects considered is given by,(4.5)where is the free stream Mach number. Here asound represents the isentropic velocity of sound in a gas, given by the expression , where is the isentropic index (specific heat ratio) of the gas, R is the corresponding gas constant, and T is the gas temperature. For air, = 1.4, and R=287 J kg-1 K-1. Thus for the free stream flow of air, of assumed ambient temperature T=T=288.16 K, the velocity of sound is asound = 340.26 m/s. The effective angle of attack, , for the computation of these steady aerodynamic forces is given by the following expression,(4.6)4.3.2 Timoshenko Beam Analytical formulationFor Timoshenko beam formulation equation (4.1) becomes(4.7)Where is the density of the beam material, A is the cross section area, , is the Timoshenko shear coefficient, depends on the geometry and J=I, which is the rotatory inertia.. Normally, = 5 / 6 for a rectangular section . Equation (4.2) remains the same in Timoshenko beam formulation.Using equations (4.3), (4.4) and (4.6), equations (4.7) and (4.2), can be rewritten as(4.8))

(4.9)The displacements and are subject to the boundary conditions(4.10)When and equal to zero, equations (4.8) and (4.9) reduce to two independent equations, one for and one for . The terms involving and indicate inertia and aerodynamic couplings.Equations (4.8) and (4.9) are linear equations with constant coefficients. For such a (coupled) system, the solution can be written in the form , and (4.11)where is generally complex. Introducing equation (2.10) into equations (2.7) and (2.8) and dividing throughout by , we obtain the ordinary differential equations,(4.12a)(4.12b)where and .The boundary conditions retain the same form except that and are replaced by and , respectively and partial derivatives of and with respect to y by total derivatives. No closed form solution of equation (4.12) (for values of ) is possible; hence an approximate solution is used. It will prove instructive to examine the effect of airflow speed on the parameter that gives the stability condition of the system. Before the stability analysis of the system, the free vibration characteristics are investigated.4.4Formulation of equations of motion for a wide rectangular cantilever beam of uniform cross section using finite element beam modelA combined bending-torsion Timoshenko beam model is considered for the analysis. Hence the two node beam element has three degrees of freedom per node ( and ). Required element consistent stiffness and inertia matrix for the combined bending-torsion beam element can be obtained by coupling the individual consistent stiffness and inertia matrices for bending and torsion elements. Figure 6.1 shows combined bending-torsion beam element with the corresponding nodal degrees of freedom. The consistent element stiffness and inertia matrices are as given below,Intermediate Interpolation Element:(4.13)

(4.14)

Reduced Integration Element:

(4.15)(4.16)(4.17)(4.18)Where material properties like is constant throughout the beam and since the beam is uniform in cross section (rectangular) other properties like and also remain same. For a rectangular cross section, torsion constant, and area moment of inertia with respect to X axis (passes through the of cross section centroid), . Mass per unit length of the beam, density of the material () area of cross section (A = wt). Mass moment of inertia, .

21

Figure 4.3 Combined bending-torsion beam elementThe nodal displacement vector for the ith element is The element stiffness and inertial matrices can be assembled to form the global stiffness and inertial matrices and [M] respectively.

4.4.1Aerodynamic force vectorHere, while formulating the aerodynamic forces, damping from viscous effects and unsteady aerodynamic flows are taken into account. The aerodynamic damping is incorporated through heave and pitch velocities as suggested by Fung [12] based on the aerodynamic strip theory. The aerodynamic lift and moment at the flexure point, which act on a symmetrical airfoil, according to strip theory, is given by equations (4.3) and (4.4) respectively. (Section 4.3.1)(4.3)(4.4)where, , represents a corrected set of expressions for the lift and moment coefficients (about support point) with compressibility effects considered (through Prandtle-Glauert correction factor) and denotes the effective angle of attack, for the computation of the steady aerodynamic forces.andThese lift and moment values correspond to unit span case. In our investigation each element is of length and hence the corresponding force (lift and moment) at each node (for each element) can be calculated by multiplying the unit span value with respective. Let be the aerodynamic moment about X axis due to lift force. In this report this moment force is assumed to be zero; i.e., effect of aerodynamic bending moment is ignored. Therefore, for the ith element,

Element force vector,

(4.19)In the assembled form, the equation of motion can be represented as (4.20)where [M] is the inertia matrix and is the structural stiffness matrix. Matrices and can be termed as aerodynamic stiffness and aerodynamic damping matrices respectively. If there are n elements, [M], , and will be of the size 3(n+1) 3(n+1). The nodal displacement vector will be of the size 3(n+1) 1. After imposing the cantilever boundary condition, the size of [M], , and reduces 3n3n and the size of will be 3n1.4.5State-space methodThis method modifies a second order ordinary differential equation into a first order ordinary differential equation. The equation of motion in modal domain, for the uniform beam can be represented as (4.21)where, , , and represents generalized mass, aerodynamic damping, stiffness and aerodynamic stiffness matrices respectively and is the vector of the natural coordinates. Suppose the generalized matrices are truncated to m modes. Equation (2.42) can now be expressed in the state space form as (4.42)where and Using , the above equation can be rewritten as or (4.43)where is the identity matrix.The following eigenvalue problem can be defined for the nontrivial solution of equation (4.43) (4.44)4.6Theodorsens function and the p-k method of analysis The Jones formula [12] for the frequency dependent Theodorsens complex function C(k) is used here to introduce the phase difference between the aerodynamic loading and the response. This is achieved by updating the aerodynamic matrices [A] and [DA] by multiplying these by the function C(k), (4.45)

where is the non-dimensional reduced frequency obtained from the imaginary part of the eigenvalue . Convergence in k values for each modal branch is achieved through an iterative method for a given flow velocity. The flow chart for the above p-k algorithm is presented in Figure 4.4. Here the updated aerodynamic matrix and the updated aerodynamic damping matrix are given as and (4.46)4.7Stability conditionsCase 1. At subcritical flow velocities in the presence of damping, all the eigenvalues are complex, = r ii = (), with negative real parts, r < 0, indicating that the net effective damping is positive, (since >0), leading to stable oscillations, characterized by decrease in amplitude with time. The imaginary parts of the eigenvalues give the circular frequencies (i= in rad/s) of the associated branches from the two modes, while the real parts give the time dependence of the amplitudes. Case 2. Beyond a critical velocity, (), the real part of at least one of the complex eigenvalues, = r ii = (), becomes positive, i.e. r = > 0. This indicates that beyond this critical velocity, the net damping is negative, leading to unstable oscillations, characterized by increase in amplitude with time. At the critical (flutter) velocity () i.e. at the flutter boundary, the real part of the eigenvalues vanishes, (r = 0), indicating purely simple harmonic motion, without any net damping at all.Case 3. Divergence is indicated by the condition that the imaginary part of vanishes, i.e. i==0, when the corresponding real part is positive.

Go to next modal branch for convergence of next rootModal branch for first rootYesNoCompute stiffness and mass matricesRead current flow velocityInitialize Theodorsens function with C(k) = 1Update aerodynamic matrix = C(k) x [A] and aerodynamic damping matrix = C(k) x Find eigenvalues: Converged eigenvalues for mode at corresponding flow velocityUpdated flow velocity if all roots have convergedFind C(k)is k=k previous

Figure 4.4 Algorithm for p-k-method.

4.8 Flutter analysis of the wing:

Flutter analysis of the wing is also carried out using the same elementary beam model. The quasi-steady aerodynamic theory is used to obtain the aerodynamic forces interacting with the structure. First the problem is solved taking one bending mode and one torsion mode as a first estimate and then the result has been improved taking higher modes.The flutter speed obtained for the present configuration has shown to be very high and also that the wing is very stiff. Hence the stiffness of the wing is reduced by reducing the modulus of elasticity and correspondingly the modulus of rigidity. The results obtained for the wing with reduced stiffness parameters are typical for subsonic flutter.The eigenvalue is a continuous function of the air speed U. When U is not zero, but infinitesimally small, the exponent is no longer pure imaginary but complex, = + i. Of course, to investigate this case, we must return to the non-self adjoint system. It can be shown that for sufficiently small U and for (dCL / d) < 2 , the wing is losing energy to the surrounding air, so that the motion is damped oscillatory, and hence asymptotically stable. The clear implication is that is negative. As U increases, can become positive, so that at the point at which changes sign, the motion ceases to be damped oscillatory and becomes unstable. The air speed corresponding to = 0 is known as critical speed and denoted by Ucr. There are many critical values of U but, because in actual flight U increases from an initially zero value, the lowest critical value is the most important. One can distinguish between two critical cases, depending on the value of . When = 0 and = 0 the wing is said to be in critical divergent condition. When = 0 and 0 the wing is said to be in critical flutter condition.The above qualitative discussion can be substantiated by a more quantitative analysis. To this end, we must derive and solve the complete non-self adjoint eigenvalue problem.we obtain the eigenvalue problem[K + U2H + UL + 2M ] a = 0 ----------(4.47),, , and represents generalized mass, aerodynamic damping, stiffness and aerodynamic stiffness matrices respectively and is the vector of the natural coordinates. Suppose the generalized matrices are truncated to m modes. These expressions are substituted in the eqn. (2.19) for flutter analysis of tapered beamThe chord length of the wing is assumed to be varying linearly along the length (i.e., from Root to Tip). The chord length (cr) for each element is taken at the middle of each section (Table 3.3). These chord lengths are substituted in the above expressions, which are in turn substituted in Eqs (4.47).

5 NUMERICAL RESULTSBased on the Timoshenko beam finite element formulations given in chapter 4, a MATLAB code was written for the free vibration analysis and flutter analysis of aircraft wing. The results have been validated using a standard package NASTRAN and compared with the results predicted by the Euler Bernoulli formulation. Further the results of some parametric studies have been presented. 5.1 Free vibration analysis results5.1.1 Uniform beam In this section, to ascertain the correctness of the formulation, a bench mark problem of an uniform cantilever beam is solved.Numerical data:The following properties of the cantilever beam are used for the analysis:Length =0. 5mWidth = 0.1mThickness = 0.003mYoungs Modulus of elasticity = E = 71 * 109 N/m2Shear Modulus of rigidity = G = 26 * 109 N/m2Density of the material = s = 2722.77 kg/m3Density of air = = 1.225 kg/m3

U SHEAR CENTER AND CENTROID x

x 0.1m0.5m 0.003mELASTIC AXIS AND INERTIA AXIS y

Fig 5.1 (a) Planar view of uniform wingFig-5.1 (b) Sectional view of uniform wing

The natural frequencies of typical uniform beam with the above properties are as shown in the following Tables 5.1, 5.2, 5.3 and 5.4. Table 5.1 shows the comparison of natural frequencies between Linear RIE formulation and analytical models. The values of bending frequencies predicted by the Rayleigh beam model, which considers only rotary inertia but not shear deformation are also presented. From the comparison of all the analytical formulations it is seen that rotary inertia mainly contributes to the reduction in flexural frequencies of the beam. Shear deformation effects are small in case of the first few bending modes. It can be seen that the frequencies of the RIE formulation converge very slowly and in case of some higher frequencies the values predicted by the linear RIE formulation are higher than that predicted by the corresponding Euler Bernoulli beam element (Table 5.3). Linear RIE elements which use linear shape functions for interpolating the longitudinal displacement w, require large no. of elements to converge to the frequency predicted by the analytical model. The low value of Shear rigidity justifies the less pronounced changes in bending frequencies between the analytical models. As shear deformation effect on the torsion was not considered in this analysis, the torsional mode frequencies are the same as that of the Euler Bernoulli beam elements. (Table 5.3)

Table 5.1 Natural frequency comparison for wide cantilever beam of uniform cross section between Reduced Integration Element and analytical modelsType andMode no.Linear RIE Element Results in HzAnalytical Euler Bernoulli beamAnalytical Rayleigh beam Results in HzAnalytical Timoshenko BeamResults in Hz

N=10N=20N=40N=80

1 bending9.9105179.9015819.8993189.89875489.898860479.8988082709.898808272

2 bending63.5663962.4045562.1175662.04602662.035099662.0330494162.03304985

3 bending186.4138176.7054174.3802173.80515173.700075173.684002173.684012

4 bending393.3687352.427343.1035340.82585340.382821340.321109340.321182

1 torsion92.7584592.6869892.6691192.66464892.663159892.663159892.6631598

2 torsion280.5683278.633278.1503278.02967277.989479277.989479277.989479

3 torsion475.2967466.2983464.0604463.50188463.315799463.3157991463.3157991

4 torsion681.5952

656.839

650.6863

649.15280648.642118648.6421188648.6421188

Table 5.2 Natural frequency comparison for wide cantilever beam of uniform cross section between Interdependent Interpolation Element and analytical models Type andMode no.IIE Element Results in HzAnalytical Euler Bernoulli beamAnalytical Rayleigh beam Results in HzAnalytical Timoshenko BeamResults in Hz

N=10N=20N=30N=40

1 bending9.8987989.8987919.8987919.8987919.898860479.8988082709.898808272

2 bending62.0340462.0321962.032162.0320962.035099662.0330494162.03304985

3 bending173.7235173.6826173.6804173.6801173.700075173.684002173.684012

4 bending340.6319340.3306340.3139340.3111340.382821340.321109340.321182

1 torsion92.7584592.68698 92.6737592.6691192.663159892.663159892.6631598

2 torsion280.5683278.633278.2754278.1503277.989479277.989479277.989479

3 torsion475.2967466.2983464.64464.0604463.315799463.3157991463.3157991

4 torsion681.5952

656.839

652.2786

650.6863

648.642118648.6421188648.6421188

Table 5.2 shows the comparison of natural frequencies between Intermediate Interpolation Element (IIE) formulation and the analytical models. The IIE element frequencies converge rapidly and good agreement can be seen with the frequency values predicted by the analytical Timoshenko beam model. The superconvergent two node IIE is superior in predicting the flexural mode frequencies although it does not represent the pure shear frequencies accurately(Reddy [37]). The torsional mode frequencies are the same as that of the Euler Bernoulli beam elements. (Table 5.3). Table 5.3 shows the comparison of natural frequencies between the Euler Bernoulli beam formulation and the analytical models. Table 5.4 shows the results obtained from MSC Nastran beam element CBEAM with 10 elements. The NASTRAN beam elements also yield frequencies which are lower than the Euler Bernoulli beam formulation

Table 5.3 Natural frequency comparison for wide cantilever beam of uniform cross section between Euler Bernoulli Beam and analytical modelsType andMode no.Euler Bernoulli Beam element Results in HzAnalytical Euler Bernoulli beamAnalytical Rayleigh beam Results in HzAnalytical Timoshenko BeamResults in Hz

N=10N=20N=30N=40

1 bending9.8988699.8988619.8988619.8988619.898860479.8988082709.898808272

2 bending62.0371562.0352362.0351362.0351162.035099662.0330494162.03304985

3 bending173.7443173.7029173.7006173.7003173.700075173.684002173.684012

4 bending340.7072340.4041340.3871340.3842340.382821340.321109340.321182

1 torsion92.7584592.6869892.6737592.6691192.663159892.663159892.6631598

2 torsion280.5683278.633278.2754278.1503277.989479277.989479277.989479

3 torsion475.2967466.2983464.64464.0604463.315799463.3157991463.3157991

4 torsion681.5952

656.839

652.2786

650.6863

648.642118648.6421188648.6421188

Table 5.4 Natural frequency for wide cantilever beam of uniform cross section using MSC NastranType and Mode noFEM Nastran Beam Results in Hz

1 bending9.88733

2 bending61.78090

3 bending172.5161

4 bending342.1795

1 torsion92.63934

2 torsion277.3469

5.1.2 The Aircraft wing

A Typical discretization of the aircraft wing (FE model and Aerodynamic model) are shown in Figs 5.3 and 5.4.. The aircraft wing and empennage are modeled using Timoshenko beam elements. The beam formulation is suitably adapted to account for the bending-torsion coupling in the normal modes, due to offset of the shear center from the centroid. Numerical dataThe numerical data used for the actual wing and also for the wings with reduced stiffness parameters are as shown below.Case (1): Actual wing:Youngs Modulus of elasticity = E = 72 * 109 N/m2Poissons ratio = = 0.3Shear Modulus of rigidity = G = 27.69 * 109 N/m2Case (2): With reduced stiffness parameters:E* = 0.1E and G* = 0.1GYoungs Modulus of elasticity = E* = 7.2 * 109 N/m2Poissons ratio = = 0.3Shear Modulus of rigidity = G* = 2.769 * 109 N/m2

Fig 5.3 An FE model of the aircraft wing

Fig 5.4 Aerodynamic model of the aircraft wingTable 5.5 Mass distribution and mass densities of the beam elementRef [42] PD ST 0314Sl. No.Ele L inm (lr)C/S Area in mx 10-6(Ar)Density inKg/m3(s) rMass per unit lengthKg/m (mr)

10.35012871.09309.30119.82

20.31512037.07803.9893.94

30.28512318.09609.27118.37

40.30011400.09784.31111.54

50.32515438.07654.20118.16

60.315 7348.17968.5558.55

70.315 6826.9 49346.43336.88

80.325 6470.2 41645.03269.45

90.325 5314.3 44344.80235.66

100.325 5110.9 40029.30204.58

110.325 4947.1 34392.09170.14

120.325 4817.5 29235.0814.84

130.325 4153.8 26966.23112.01

140.325 3939.4 22792.8089.79

150.325 3455.0 17246.9759.58

160.350 3259.6 9963.2832.47

170.350 3132.6 6385.2020.00

180.300 3153.5 6178.5719.48

190.210 2803.0 9275.7725.99

200.350 2557.7 6660.9517.04

210.370 2009.6 7063.6614.20

220.370 2119.7 6485.0713.75

Total wing mass = 762.6 kgTable 5.6Sectional properties and aerodynamic chord lengths of elements of the wingRef [42] PD ST - 0314Sl.No.Ele L in mIzz in m4x 10-4Iyy in m4x 10-4J in m4x 10-4Chord length(m)

10.3504.047735.398070.31852.402

20.3153.294929.814045.87632.326

30.2853.124929.562025.74002.256

40.3002.838823.311015.00452.191

50.3253.762223.3330 9.03012.120

60.3151.568011.6677 7.17002.047

70.3151.415910.4640 5.67901.975

80.3251.2581 9.3388 5.28151.902

90.3250.9428 6.8012 4.37701.828

100.3250.8215 6.4647 3.81301.754

110.3250.7414 6.0293 3.33001.680

120.3250.6706 5.4199 2.75251.606

130.3250.5127 4.1906 2.36401.532

140.3250.4474 3.6573 2.02501.458

150.3250.3453 2.8600 1.63351.384

160.3500.2958 2.3924 1.31101.307

170.3500.2537 2.0252 1.05151.227

180.3000.2295 1.8596 0.87301.153

190.2100.1901 1.5036 0.74701.095

200.3500.1525 1.2544 0.58651.031

210.3700.1036 0.7833 0.41250.949

220.3700.9409 0.6224 0.29400.865

Table 5.7 Shear center position w.r.t. Centroidal axis Ref [42] PD ST - 0314Sl.No.Ele L in mZG in mx 10-3YG in mx 10-3

10.350-355.43036.300

20.315-432.248-137.681

30.285-379.560-199.990

40.300-434.200-278.100

50.325-161.000-9.200

60.315 -1.64839.906

70.315-9.04011.800

80.325-6.3337.034

90.325-8.275-3.881

100.325-19.418-0.664

110.325-18.830-3.497

120.325-10.42513.077

130.325-16.588-4.718

140.325-14.829-5.609

150.325-6.369-12.955

160.350-0.028-10.495

170.350-2.4688.142

180.3002.517-8.635

190.2101.100-7.300

200.350-1.173-6.608

210.370-0.851-5.770

220.370-1.952-5.314

The above numerical data are used for the analysis of the subsonic wing. The wing is visualized as a collection of stepped beam elements, each having its respective properties as shown in the above tables. The natural frequencies obtained for the wing for each case are given below.

Table 5.8 Natural frequencies of the subsonic wing case (1)Type andMode no.Euler Bernoulli beam elementresultsin HzRIE results in HzIIE results in HzFunction space approach Ref [32]Table 4.3PD ST-0314 3D model NASTRANresults in HzRef[32]Table 4.3PD ST-0314

1 bending7.21693467.1571227.1562437.1247.087

2 bending21.14096720.7772520.673520.78620.481

3 bending50.40369648.757848.0626348.53847.781

4 bending101.4167795.7776793.12571-

1 torsion56.83277756.7881656.8303356.38556.338

2 torsion121.05991120.9533120.9788-

3 torsion175.83192175.1956175.1329-

4 torsion249.07651241.5394241.8016-

The values of frequencies predicted by different methods are given in the table (5.8). The Function space approach frequencies correspond to the method developed by Mukerjee and Prathap for explaining the locking phenomena[ref]. It can be seen that both the Reduced Integration element and Intermediate Interpolation Element yield good agreement with the values predicted by function space approach and 3-d NASTRAN results. Thus the Timoshenko beam element finite element formulations constitute an improvement over Euler Bernoulli beam formulations in predicting the dynamic characteristics of a complex structure such as an aircraft wing. The element wise values are given in the table 5.9

Table 5.9 Element wise valuesSl.No.Ele L in m

10.3501.1125

20.3151.1955

30.2851.3535

40.3001.1991

50.3250.9999

60.3150.932

70.3150.9058

80.3250.7978

90.3250.7279

100.3250.6595

110.3250.6149

120.3250.5711

130.3250.5064

140.3250.466

150.3250.4101

160.3500.321

170.3500.2865

180.3000.3504

190.2100.6665

200.3500.2109

210.3700.1632

220.3701.4052

5.2 Flutter analysis5.2.1 Uniform beam:The flutter analysis of the uniform rectangular beam is carried out using the finite element analysis method. The shear center in the case of uniform section coincides with the centroid of the section. Hence there will be no dynamic coupling in the case of uniform beam. Results are then compared with that from MSC NASTRAN, where a finite element model of the structure is created using beam elements. Here the wing-like structure is discretized into 10 beam elements and strip theory of aerodynamics is used for load calculation, ignoring the Prandtl-Glauert correction factor for compressibility effects. And p-k method of flutter analysis with approximate value of Theodorsens function for unsteady effects (as suggested by R.T. Johns) is used for flutter prediction.

Flutter analysis is done by solving equation (4.44) for the wide cantilever beam. Generalized stiffness, , generalized mass,, and modal matrix, are obtained as explained before. The aerodynamic damping, , and aerodynamic stiffness, , matrices can be obtained as mentioned in section 2.4.1 (equation (2.35)), which use strip theory of aerodynamics, and use equation (2.40) to generalize those matrices. Later, the eigenvalue problem is defined using state-space method. The p-k method of solution is employed for flutter analysis. Iterative p-k method algorithm (Figure 2.5) is run for each mode at each velocity points until a convergence in reduced frequency is achieved. Converged eigenvalue at each velocity point can be represented as . In the forthcoming discussions, (circular frequency) represents the absolute value of the imaginary part of the eigenvalue ( is the frequency in Hz) and the corresponding damping factor is given by. Beyond the critical flutter velocity, the real part of one root is positive (among all that for different modes), i.e., the corresponding damping factor g > 0, indicating unstable oscillation. Figures 5.5, 5.6 and 5,7 depict the variation of damping, g, frequency, , with free stream velocity for different modes of the beam as obtained , by IIE formulation, RIE formulation and by commercial FEM software, NASTRAN (by strip theory ) respectively. Table 5.10 shows the flutter velocities and frequencies as predicted by the various elements and the NASTRAN beam model. It can be noticed that although very small in this case, the effect of reduction in bending mode frequency increases the flutter velocity and decreases the flutter frequency. This is because the reduction in bending mode frequencies cause an increase in the gap between bending and torsion mode frequencies in the frequency spectrum, thus leading to a increase in flutter velocity and a reduction in flutter frequency. The results of the NASTRAN beam element agree with the expected trend. Also since shear deformation effects are not pronounced on lower bending mode frequencies in the case of this beam, no appreciable difference is found between the results predicted by the different methods.

Vf = 147.061m/s

Figure 5.5 Variation of damping (g) and frequency () with free stream velocity for the case of RIE formulation for a continuous cantilever beam of uniform cross section in combined bending-torsion vibration.

Vf = 147.058 m/s

Figure 5.6 Variation of damping (g) and frequency () with free stream velocity for the case of IIE finite element formulation for a cantilever beam of uniform cross section in combined bending-torsion vibration.

Vf = 150 m/s

Figure 5.7 Variation of damping (g) and frequency () with free stream velocity for a cantilever beam of uniform cross section in combined bending-torsion vibration using MSC NASTRAN (strip theory of aerodynamics with p-k method of flutter analysis).Table 5.10Uniform beam Flutter resultsFlutter speed (m/s)` Flutter frequency (Hz)

N=5N=10N=20N=5N=10N=20

Euler Bernoulli beam element145.828146.806147.05746.23145.41345.192

RIE element145.978146.826147.06145.94145.34645.175

IIE element145.829146.807147.05846.23145.41245.191

MSC NASTRAN14942.7

Table 5.10 also shows the convergence characteristics of the various elements. It can be seen that the RIE elements slowly converge to the value predicted by the IIE element. This shows the delayed convergence characteristics of the RIE element in predicting the flutter boundary. The IIE element although showing a marginal increase in the flutter velocity validates with the free vibration frequency which doesnt show any significant difference between the Euler Bernoulli and the Timoshenko model. 5.2.2 Aircraft wingThe flutter analysis of the wing is also carried in the same way with the inclusion of effect of shear center offset. The clean wing modeled as having stepped beam elements is analysed as mentioned before. The velocity v/s the real part of eigen values and velocity v/s the imaginary part of eigen values are plotted from the complex eigen values obtained from the present analysis .The velocity v/s damping curves i.e., v-g curves and the velocity v/s frequency curves i.e., v-f curves can also be plotted. The relation between the damping (g) values and eigen values and the relation between the frequency (f) values and eigen values are as given below.If the eigen value obtained is = +i, where is the real part and is the imaginary part, then

x 2 and Hence the v-g and v-f curves can be plotted from the eigen values.The velocity v/s Real part and velocity v/s Imaginary part are also plotted for the wing with reduced stiffness parameters. Some of the typical graphs obtained are shown below in the Figs 3.5, 3.6, 3.7 and 3.8

Vf = 637.530 m/s

Figure 5.8 Variation of damping (g) and frequency () with free stream velocity for the case of RIE formulation for the aircraft wing in combined bending-torsion vibration.

Vf = 636.642m/s

Figure 5.9 Variation of damping (g) and frequency () with free stream velocity for the case of IIE formulation for the aircraft wing in combined bending-torsion vibration.

Te flutter speeds obtained from the graphs for the actual wing and for the wing with reduced stiffness parameters are shown in the following Table 5.9 and 5.10Table 5.11 Flutter results of the subsonic wing (actual wing)Beam ElementsFlutter speed (m/s)Flutter frequency (Hz)

Euler Bernoulli beam element635.80339.832

IIE element636.64239.603

RIE element637.53039.534

Vf = 201.864m/s

Figure 5.10 Variation of damping (g) and frequency () with free stream velocity for the case of RIE formulation for the aircraft wing (with reduced stiffness ) in combined bending-torsion vibration.

Vf = 201.287m/s

Figure 5.11 Variation of damping (g) and frequency () with free stream velocity for the case of RIE formulation for the aircraft wing (with reduced stiffness parameters) in combined bending-torsion vibration.Table 5.12Flutter results of the subsonic wing (with reduced stiffness parameters)

Beam ElementsFlutter speed (m/s)Flutter frequency (Hz)

Euler Bernoulli beam element201.01612.584

IIE element201.28712.533

RIE element201.86412.333

The flutter velocities and flutter frequencies of the various elements are given in Tables 5.11 and 5.12 . Here the effect of shear deformation on flutter frequency is more pronounced and Timoshenko beam elements show a increase in the flutter velocity and decrease in flutter frequency. The flutter modes correspond to the 3rd Bending and 1st Torsion mode frequencies. From table 5.8 , it can be seen that the 3rd bending mode frequencies significantly come down for Timoshenko beam elements. Thus there is a increased gap between the bending and torsion mode frequencies leading to increased flutter velocities and decreased flutter frequencies. The RIE predicts higher flutter velocities and lower flutter frequencies over IIE. This is because of the delayed convergence characteristics of the RIE.

6 SENSITIVITY ANALYSISThe following section is dedicated for the development of an analytical expression for the derivative of flutter velocity with respect to the design parameters of the aircraft wing, which is modeled as a uniform cantilever beam. The design variables xd considered in this analysis are Lengthof the beam(l),Bending Rigidity (EI), Torsional Rigidity (GJ), Mass per unit length () and Inertia per unit length (Im).6.1 Analytical expression for the derivative of flutter velocity with respect to the design variables: Using the state space method, the eigenvalue eigenvector problem has been defined as,

(6.1)For a beam whose generalized matrices are truncated to m modes(6.2)

where [S] is a matrix composed of aerodynamic stiffness and aerodynamic damping matrices and is the vector of the natural coordinates. These matrices depend upon the free stream air velocity, V and reduced frequency parameter, k (); where is the imaginary part of the complex eigenvalue, = r ii =(- i). At the critical condition in which the instability of flutter begins, the real part of one of the complex eigenvalue becomes zero and hence for that particular mode, f = if. Subscript f denotes the values at flutter velocity.Hence the reduced frequency parameter at flutter speed can be written as,

`(6.3)

which gives,(6.4)At flutter boundary, equation (6.1) can be written as,

(6.5)where subscript f denotes respective values at critical (flutter) speed.Using equation (6.4), equation (6.5) can be rewritten as,

(6.6)The eigenvalue problem, defined in equation (6.1) can also be written as

(6.7)

where is known as left eigenvector of the eigenvalue problem. Since is a non symmetric matrix, both left and right eigenvectors ( and respectively) will be complex in nature.

Differentiating equation (6.5) with respect to the design variable, xd and pre-multiplying by the left eigenvector, ,

(6.8)The complex terms in the above equation is defined as,

(6.9)Equation (6.8) can be rewritten as,

(6.10)Now, the complex terms appearing in the above equation can be written as,

(6.11)Therefore equation (6.10) can be rewritten as,

(6.12)

where and are real numbers. Pre-multiplying equation (6.12) by the complex conjugate of , written as , gives,

(6.13)comparing the imaginary part of the equation gives,

(6.14)

similarly, pre-multiplying equation (6.13) by the complex conjugate of , written as , and comparing the imaginary parts results in,

(6.15)

6.2 Formulation of equations of motion for a wide rectangular cantilever beam of uniform cross section using Timoshenko finite element beam modelA combined bending-torsion Timoshenko beam model is considered for the analysis. Hence the two node beam element has three degrees of freedom per node ( and ). Required element consistent stiffness and inertia matrix for the combined bending-torsion beam element can be obtained by coupling the individual consistent stiffness and inertia matrices for bending and torsion elements. Figure 6.1 shows combined bending-torsion beam element with the corresponding nodal degrees of freedom. The consistent element stiffness and inertia matrices are as given below,(6.16)

(6.17)Where material properties like is constant throughout the beam and since the beam is uniform in cross section (rectangular) other properties like and also remain same. For a rectangular cross section, torsion constant, and area moment of inertia with respect to X axis (passes through the of cross section centroid), . Mass per unit length of the beam, density of the material () area of cross section (A = wt). Mass moment of inertia, .

21

Figure 6.1 Combined bending-torsion beam elementThe nodal displacement vector for the ith element is The element stiffness and inertial matrices can be assembled to form the global stiffness and inertial matrices and [M] respectively.

6.3.Expressions for of the ith beam element :(6.18)

(6.19)(6.20)(6.21)(6.22)

(6.23)

(6.24)Differentiating equation (6.24) partially with respect to Vf and kf gives,(6.25)

Now, the partial derivatives of with respect to the design variables are(6.26)(6.27)(6.28)The updated stiffness and damping matrices for the m dof aircraft wing at the flutter speed is given by,

(6.29)(6.30)

where C(kf) is given by, (6.31)Now, the partial derivatives of with respect to Vf and kf are be found to be,

(6.32)And also(6.33)(6.34)

Partial derivative of the aerodynamic matrix, with respect to can be derived as follows.

(6.35)

(6.36)

Now the partial derivative of with respect to can be found from,

(6.37)

(6.38)

where is given by,

(6.39)

Partial derivative of the aerodynamic matrix, with respect to can be derived as follows.(6.40)(6.41)

Now the partial derivative of with respect to can be found from(6.41)

(6.42)

where is given by,

(6.43)

6.4 Expressions for of the beam

Now, [] for the beam is given by,

(6.44)

The partial derivatives of with respect to xd, Vf and kf can be written as,

(6.45)

where is given by equation (5.25)

(6.46)

where and are given by equations (6.36) and (6.41) respectively.

(6.47)

where and are given by equations (6.38) and (6.42) respectively

6.5 Numerical evaluation of the derivative of flutter velocity with respect to the design variable by finite central difference methodFinite central difference method can be used to verify the gradient information, obtained analytically. The derivative can be found by using the formula as given below.

(6.48)

where the parameter , determines the level of accuracy for the obtained gradient value. In this particular case testing is carried out with = 1% and 5% of xd. However, this is a numerical method and it lacks precision and errors occur through truncation and rounding off.

6.6 Sensitivity Analysis Results6.6.1 Uniform beam:The sensitivity analysis of the uniform rectangular beam is carried out using the modal analysis method. The beam with the same properties as in section 5.1.1 is considered for the sensitivity analysis. The Gradients of the parameters are plotted in the table 6.1Table 6.1. Comparison of flutter velocity derivative for 2 DOF airfoil using different methodsParameterParameter ValueAnalytical MethodCentral Differnece Method

N=10=0.1

Lengthof the beam(l)0.5 m-238.0587-277

Bending Rigidity (EI)15.975 Nm2-0.2435-.25039

Torsional Rigidity (GJ)23.4 N m23.30323.31623

Mass per unit length ()0.816831 N/m23.710924.1175

The result indicate that flutter velocity gradients obtained from the analytical method match well for most parameters. There is a large deviation in length term. This occurs because all the terms involved in dynamic analysis critically depend on the length and hence higher order terms are required to accurately predict the flutter sensitivity. Length sensitivity is highly nonlinear. bending rigidity sensitivity is negative because as bending rigidity increases second bending mode frequency also increases leading to decrease in gap between the second bending and first torsional mode. Hence flutter velocity decreases. The bending rigidity sensitivity is observed to be small. This explains why the flutter velocities of this beam were not extremely sensitive to Timoshenko formulations as only the bending stiffness is altered. Torsional rigidity sensitivity is positive due to the same reason that an increase in torsional frequency will lead to increase in gap between the second bending and first torsional mode leading to increase in flutter velocity. Thus any effect on torsional frequencies affects flutter velocity in a significant way.

7 CONCLUSIONS7.1 Discussion of Results: The aeroelastic sensitivity analysis of subsonic flutter using Timoshenko beam finite element formulations indicates the critical effect of the frequency spectrum on the onset of flutter. Hence the flutter sensitivity depends on the particular bending and torsion modes in flutter.Timoshenko dynamic analysis of flutter is even more significant in case of real aircraft structures which have a higher torsional rigidity and hence flutter at higher bending modes. As Timoshenko dynamic analysis constitutes an improvement over the Euler Bernoulli model , it predicts more precise values of flutter velocities and flutter frequencies. Also the exact prediction of dynamic characteristics is crucial in control techniques aimed at active vibration control and active flutter control. From the free vibration and flutter analysis carried out in the previous sections, it is found that the two noded superconvergent Timoshenko beam element yields better results over the conventional Reduced Integration element as it converges faster than the RIE. Flutter analysis results also indicate better convergence characteristics of the superconvergent element (IIE).The flutter sensitivity analysis by analytical formulations yielded flutter gradients which are comparable with those obtained by central difference method. The sensitivities obtained further validate the earlier results of flutter analysis by Timoshenko beam finite element. 7.2Further scope :1. The present work is limited to the clean wing analysis that doesnt show flutter in the subsonic regime. However it is necessary to check if the wing with control surfaces is prone to subsonic flutter. The present method can be easily extended to determine flutter boundaries of wing with control surfaces2. The flutter analysis of the T-tail is critical from the point of design. The quasi-steady method can easily be extended to the T-Tail assembly consisting of Horizontal tail, Vertical tail, Rudder and Elevator. Since the aspect ratio of tail assembly surfaces is small , shear deformation effect play a significant role in determining flutter velocities.3. The present analysis can be extended to include shear deformation effects on torsional frequencies as the torsional rigidity sensitivity is found to be high in the present analysis.4. Sensitivity analysis can be extended to other parameters like shear centre offset, span of the beam and position of centre of mass and aerodynamic centre of the wing like structure.

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