## A

Neutral surfacc lor bcitJiiig dcfleclionsi in r diavlion Figure 2.24 Schematic of beam for bending dynamics. Figure 2.25 Schematic of differential beam segment. cross-sectional neutral axes associated with pure bending in and normal to the plane of the diagram in Fig. 2.24. For simplicity, however, we will only consider uncoupled bending in the x-y plane, thus excluding initially twisted beams from the development. The bending deflections are denoted by i> ( ', 0 in the v direction. The x...

## Aei

3 (i + j - l)(i + j)(l + + j) (2 + + )( 3 + + j) Similarly, the elements of the mass matrix can be found as pi 30240 + 28512( + j) + 9672( 2 + j2) + 1392( 3 + f) + 72( 4 + ) + 200400' + 4520(rj + ij2) + 320( 3j + if) + 520i2j2, P2 (l + 0(2 + 0(3 + Oj(l + J)(2 + )(3 + j)( 3 + + j) x (4 + i + j)(5 + i + j)(6 + i + j)(7 + i + j). The fact that the governing equation is derivable from energy is reflected in the symmetry of M and T . Results for free vibration (i.e., with 0) are given in Table 2.7....

## Al

0.2 0.4 0.6 0.8 q Figure 3.3 Relative change in lift caused by aeroelastic effect. encountered. Moreover, the structure will not tolerate infinite deformation, and failure will take place at some finite value of 0. When the system parameters are within the bounds of validity for linear theory, another fascinating feature of this problem emerges. One can invert the expression for 6 to obtain

## B

Figure 2.15 Elastically restrained end of a beam, or, for uniform beams Thus, the specific condition to be satisfied is X'(l) 0. An elastic constraint (see Fig. 2.15) can be treated as a linearly elastic torsional spring attached to the end of the beam. The twisting moment at the beam end must be equal and opposite to the spring reaction for any finite deflection of the end so that

## C

Figure 3.7 Detailed view of the sting-mounted wing. Two static aeroelastic equilibrium equations can now be written for the determination of Op and 6m- Using Eqs. (3.3) and (3.4) for the lift and pitching moment, the force equilibrium equation can be written as qSCLa(ar+0F + 0M)- W-F0 0, (3.15) and the sum of moments about the trailing edge yields qScCMac + qSCLa(ar + 0F + 0M)(c - xac) - W(C - xcg) - M0 0. (3.16) Substitution of Eqs. (3.14) into Eqs. (3.15) and (3.16), simultaneous solution for...

## Ci r

Thus, the string displacement can be written as a sum of the contributions from the odd-indexed modes. Recall that neither the excitation loading nor the initial conditions excite the even-indexed modes. Thus, v(x, t) J2 HAt)< Pi x) i l,3, When the forcing frequency coincides with one of the natural frequencies, an interesting situation results. Take a typical term in the series solution ofEq. (2.129), and consider only its time-dependent part, for example, When co & ), , the term becomes...

## Contents

2.1.2 Standing Wave (Modal) Solution 2.1.3 Orthogonality of Mode Shapes 2.1.6 Generalized Equations of Motion 2.2 Uniform Beam Torsional Dynamics 2.2.3 Example Solutions for Mode Shapes and Frequencies 2.3 Uniform Beam Bending Dynamics 2.3.4 Example Solutions for Mode Shapes and Frequencies 2.4 Approximate Solution Techniques 3.1.4 Wall-Mounted Model for Application to Aileron Reversal

## E

It should be noted from this solution that the modal contributions to the total displacement significantly decrease as the mode number (the index i) increases. This can be observed from the dependence of F, on i. This is characteristic of almost all structural dynamics response problems and thus permits a truncation of the infinite sum to a finite number of the lower-order modes. It also should be noted that the above solution indicates that the string will vibrate forever, with the string...

## Aj Cj Fi A A

Close inspection of this total string displacement indicates that at any given instant the transverse deflection is represented by summation over a denumerably infinite set of shapes. Each shape is of indeterminate amplitude and is associated with a particular eigenfunction these shapes are also called mode shapes in the field of structural dynamics. They will be represented here by < > , U ). Thus, for transverse deflection of a string the mode shapes may be written as or any constant...

## Ssl

Mi xi ' - y -z, . < V dqi dqi dqi, 3Xj dy 3it , Mj ( Xi - + ' fr- + Z - ) Sq2 dq2 dq2 dq2 v dx> , v dW , 7 dz' 1 Xj ---h Yi---hZ, 8q2 Now let us shift our attack on the problem and consider the kinetic energy of the system. This is We next calculate the time derivative of for which the chain rule gives

## Ei

(' . 1 > (' . > (1+ '+ ) and the mass matrix is _2mi 3( 2 + f ) + lij + 23( + j) + 40 _ U ij (i + 1 ) (I + 2) (j + 1 ) (j + 2) ( + j + 3) (I + j + 4) ( + j + 5) 4 imi Kij - , -, > 1 , , (i, j 1,2 ) (2.302) Results from this calculation are given in Tables 2.5 and 2.6 for the first two modes. It is clear that these results are much better than those obtained with the clamped-free beam modes. It is not unusual for polynomial functions to provide better results than those obtained with beam...

## EI o o fV o

And so the boundary condition can now be written as EIX( 0) -Ica4a4X 0). (2.231) At the right end the sign is changed as the result of the bending moment sign convention to yield EIX(l) Ica4u4X'(l). (2.232) We note here that when the attached mass is idealized as a particle, then Ic 0 and the moment boundary condition reduces to be the same as indicated above for the translational mass, that is, the bending moment is zero. Figure 2.31 Schematic of rotational inertia end conditions. v W Figure...

## I

The generalized equations of motion become F0l(O(-l) ('odd). The corresponding general solutions are A sin(fw, r) + Bj cos(& > , ) (i even), Ai sin(& ) f) + Bj cos(coit) + C, ( ' odd). Consider the finite initial displacement fi,sin + Y. (B, +C,)sin (2.139) This last equality will be multiplied by sin( j it x ) and integrated over the length of the string to yield . 4lrx . (jltx fl . (iirx . (j irx in I - I sin I - dx > R. I sin I - I sin I - I , h sin (4 A) sin ( A) dx Bj sin , ) dx...

## Cdi

Figure 2.26 Schematic of pinned end condition. the beam. There are four such states of practical interest 1. deflection v(x, t) X(x)Y(t), 3. bending moment M(x. t) EI x. t) EIX(x)Y(t), 4. shear V(jt, t) - 70(jt, r) -EIX'(x)Y(t). It should be noted when relating these beam states that the positive convention for deflection and slope is the same at both ends of the beam. In contrast, the shear and bending moment sign conventions differ at opposite beam ends as illustrated by the free-body...

## J

To prove that the mode shapes obtained for the uniform string problem are orthogonal, an individual modal contribution given by is substituted into the governing differential equation (wave equation) to obtain Because the generalized coordinate is a simple harmonic function for the homogeneous solution, Hi - -< Hi, (2.38) and the wave equation becomes Ttfixm) - -m(pi(x)a> Hi(t), (2.39) Ttf(x) -m< l> i(x)a> f. (2.40) If this procedure is repeated by substituting the j th modal...

## Twh

The time-dependent portion of the solution for these rigid-body type motions is seen to be aperiodic. This means that natural frequencies for both rigid-body modes are zero. The two arbitrary constants contained in Y(t) can be evaluated from the initial rigid-body displacement and velocity associated with the translation and rotation. Thus, the complete solution for the free-free beam bending problem can now be written in terms of all of its modes as This example provides a convenient vehicle...

## Info

The even-indexed modes will not be excited because their generalized forces are also zero. For the odd-indexed modes the general solution to their equation of motion is Aj sin (wit) + Bi cos (w t) + C, sin(& > f). (2.120) It may be noted that the first two terms correspond to the homogeneous portion of the solution, whereas the third term represents the particular solution. In this example, the particular solution has the same form of time dependence as the generalized force. To evaluate...

## J o

This integral value, M, , is called the generalized mass of the z'th mode. These relations thus demonstrate that the mode shapes for the string, which is fixed at both ends, form an orthogonal set of functions. The above development is for a string of constant mass per unit length and constant tension force. It is important to note that it can readily be generalized for nonuniform mass per unit length. In more involved developments for beam bending and torsional deformation, the structural...

## Jo i even but

The zero initial velocity requires that 0) 1,(0)0 W - f> A,. sin (y ) - 0. (2.142) Multiplication by sin(jjrx ) and integration will result in determining that A,- 0 for all . These results can be summarized by noting that 0 for all even values of except C, 1 -cos(oV) ( odd). (2.144) The Cj s can be determined by substitution of the odd generalized coordinates back into the equations of motion, MiCiOjj Fo(-l) 1 (t > 0). (2.145) so that the complete string displacement becomes zcos(tt> 4...

## Mkd

A similar relation holds for partial derivatives of with respect to < 72 and q2. Hence Eq. (A.9) can be written Since < 71 and < 72 are independent coordinates they can be varied arbitrarily. Hence, we can conclude that dt S iy) dqi 1 ' dq ' dq ' dq The right-hand side of Eq. (A.20) is the work done by all the forces on the system when the coordinates of the z'th particle undergo the small displacement Sx,, by,, and Sz, due to changes < 71 and Sq2 in the generalized coordinates < 71...

## Kde r

Thus, GJ X'( )Y(t) -kX(i)Y(t), (2.165) The reader should verify that the same type of boundary condition at the other end would yield where the sign change comes about by virtue of the switch in direction noted above for a positive twisting moment. Here we consider a beam with a rigid body attached to its right end (see Fig. 2.16). This rigid body has a mass moment of inertia about the beam elastic axis denoted by < ., which contributes a concentrated...

## L

The term Sv(x, f) represents the virtual displacement, which can be written in terms of the generalized coordinates and mode shapes as where < 5 ,(f) is an arbitrary increment in the ,'th generalized coordinate. Thus, the virtual work becomes Identifying the generalized force as one finds that the virtual work reduces to The loading F(x, f ) in the above development is a distributed load with units of force per unit length. If instead this loading is concentrated at one or more points, say as...

## N

Figure 2.6 Example initial shape of wave. Figure 2.6 Example initial shape of wave. We have used Eq. (2.75) to reduce the range ofmotion, which was initially oo < x < +oo, down to the range 0 < x < I, our physical space (i.e., where the string is actually mounted). The initial string shape is given in Fig. 2.6. At subsequent times the string shape will appear as shown in Fig. 2.7. The absolute distance each of the half shapes has traveled at time t is denoted by x. Note that the faint...

## P

Obtain the equations of motion for the system shown in Fig. A.2. The bar is weightless. Solution The coordinates Xi and x2 can be taken as generalized coordinates. Take as the zero datum the configuration for which the bar is horizontal and the K it H--ii, P W.vi - Wx2 + - t (jc2 - xi )2 , 3A _ W . 3A _ W . dK _ dK _ o W-k(x 2-xi), - W + k (x2 - xi). 3, 'i dX2

## Q

Figure 4.19 Plot of estimated value of i2j 2versus U (bwn) using the p-k method with Theodorsen aerodynamics (symbols) and the p method with the aerodynamics of Peters et al. (lines) for a -l 5,e -1 10, a, 20, r2 6 25, and< x 2 5. Figure 4.19 Plot of estimated value of i2j 2versus U (bwn) using the p-k method with Theodorsen aerodynamics (symbols) and the p method with the aerodynamics of Peters et al. (lines) for a -l 5,e -1 10, a, 20, r2 6 25, and< x 2 5.

## Sm it sm sm jor

A set of mode shapes for any problem. Completeness also implies that there can be no missing terms between the lowest and highest ones used in any series. 4. The set of functions must be linearly independent. This means that ai(pi(x) 0 > a, 0 for all i. (2.286) A set of functions that satisfies all these criteria is said to be admissible. By use of the series approximation, we have reduced a problem with an infinite number of degrees of freedom to one with n degrees of freedom. Instead of...

## Tmat at

This transcendental equation has a denumerably infinite set of roots that cannot be found in closed form. However, as many of these roots as desired can be found using numerical procedures found in commercially available software packages such as Mathematica, Maple, or MATLAB. These roots of Eq. (2.198) will be functions of and the first four such roots are plotted versus C in Fig. 2.22. Denoting these roots by a,, with i 1.2 one obtains the corresponding natural frequencies From the plots (and...

## 1

Where the ellipsis refers to terms of third and higher degree in the spatial partial derivatives of u and v. Then, when all terms are dropped that are of third and higher degree in the spatial partial derivatives of u and v, the strain energy becomes P Tq I dx H-- - I I ) dx + ( dx + r Jo d dX + 2(1 +eo) Jo dX + Jo V Assuming (0) ( o) 0, one finds that the first term vanishes. Since perturbations of the transverse deflections are the unknowns in which we are most interested, and since...

## Ebb

Recall that the bending moment is proportional to the local curvature and so where EI may be regarded as the effective bending stiffness of the beam at a particular cross section. For homogeneous and isotropic beams, E is Young's modulus and I is the cross-sectional area moment of inertia about the z axis for a particular cross section. Substitution of Eq. (2.207) into Eq. (2.206) and of the resulting equation into Eq. (2.204) yields the partial differential equation of motion for a nonuniform...

## Keas

Figure 4.6 Comparison between p and k methods of flutter analysis for a twin-jet transport airplane. From Hassig (1971) Fig. 1, used by permission. Hassig applied the p and k methods of flutter analysis to a realistic aircraft configuration. By incorporating the same unsteady aerodynamic representation in each analysis he was able to make a valid comparison of the results. His observations are typified by Fig. 4.6 (which is his Fig. 1). It can be noted from this figure that not only is the...

## Ee fisin

I l,3, i l,3, 1 Multiplying both sides of this relation by sm.(jnx i) and integrating over x from 0 to I yields sin I ) sin I J ) tlx 0. (2.122) Applying the orthogonality property of the sine functions in the integrand indicates that The same procedure can be applied to the initial velocity where O V v v v v IJTX x,0) E E Uiwi + Ciw)sm ) 0. dt i l,3, i l,3, 1 Again this relation can be multiplied by sm(jjtx i) and integrated over the string length. The orthogonality property in this case...

## T iX

Because gravitational effects are being neglected, the potential energy of the string will consist of only strain energy caused by extension of the string. This can be expressed as and the original length is o- In order to pick up all of the linear terms in the generalized equations of motion, one must include all terms in the energy up through the second power of the unknowns. Taking the pertinent unknowns to be perturbations relative to the stretched but undeflected string, we can again write...

## Gj Y

X(x) 0 and Y(t) 0. (2.191) The general solutions to these equations can be written as X(x) ax + b, Y(t) ct + d. The arbitrary constants, a and b, in the spatially dependent portion of the solution can again be determined from the boundary conditions. For the present case of the free-free beam the conditions are Because both conditions are satisfied without imposing any restrictions on the constant b, this constant can be anything, which implies that the torsional deflection can be nontrivial...

## Ei Cj

With Eq. (3.107) one may determine the divergence dynamic pressure with sufficient accuracy to ascertain its trends versus sweep angle A and elastic coupling parameter k. The formula shows that there is a strong relationship between these two quantities. To illustrate the utility of the above analysis, let us first normalize qi> with the value it would have at zero sweep angle and zero coupling, namely, qD(l, so that The denominator's vanishing corresponds to infinite divergence dynamic...

## M V

Finally, considering the string as linearly elastic, one can write the tension force as a linear function of the elongation, so that where E A is the constant longitudinal stiffness of the string. This completes the system of nonlinear equations that govern the vibration of the string. In order for us to develop analytical solutions, these equations must be simplified. Let us presuppose the existence of a static equilibrium solution of the string deflection so that One then finds that such a...

## Oo

Y A, cos(i2f) - cos (a),, ) sin and where the Kronecker symbol Sjj 1 for i j and Sjj 0 for i j. 1. Consider a uniform circular rod of length I, torsional rigidity GJ, and mass moment of inertia per unit length pj. The beam is clamped at the end x 0 and it has a concentrated inertia Ic at its other end where x I. (a) Determine the characteristic equation that can be solved for the torsional natural frequencies for the case in which Ic pjIt,, where C is a dimensionless parameter. (b) Verify that...

## Deficiency Lift Function Theodorsen

The kinetic energy is also considerably simplified because of the orthogonality of the assumed modes and can be written as Vjdy ( 1,2 Ne y 1,2 Nw). Note the inertial coupling between bending and torsion modes reflected by the term involving A,j. The virtual work expression can be used to identify the generalized forces

## Ritz Method Aeroelasticity

Graduate level texts on structural dynamics. In those cases, the Ritz and Galerkin's methods give the same results when used with the same assumed mode functions. As we see here, however, Galerkin's method provides a viable alternative to the Ritz method in cases where the equations of motion are not of the form presented earlier in this chapter. In this chapter we have considered the free-vibration analysis and modal representation for flexible structures, along with methods for solving...

## At Jtan

Since N L W, the above expression can be divided by the vehicle weight to yield a relation for N in terms of ur and aT. This relation can then be solved simultaneously with the preceding expression for ar, Eq. 3.47 , in terms of uv and N. In this manner uv can be eliminated, providing either a relation that expresses ar in terms of N, given by or a relation that expresses uv in terms of N, These relations permit one to specify a constant ar and find N q or, alternatively, to specify a constant...

## G J

Here the first of Eqs. 3.81 and the final boundary condition from Eqs. 3.77 are used to derive the third boundary condition. The exact solution for Eqs. 3.83 and 3.84 has been obtained by Diederich and Budiansky 1948 . Its behavior is quite complex, having multiple branches, and it is not easily used in a design context. However, a simple approximation of one branch is presented next and compared with plots of the exact solution. Approximate Solution for Bending Torsion Divergence In view of...

## Static Aeroelasticity Solution For Divergence Dynamic Pressure Of Wing

Note that X2 and uv are independent of y since the wing is assumed to be uniform. The static aeroelastic equilibrium equation can now be written as - X20 - 2 ar Or . 3.48 The general solution to this linear ordinary differential equation is 0 A sin y B cos Xy - ar r . 3.49 Applying the boundary conditions, one finds that 0 0 0 B ar dr. where ' d dy. Thus, the elastic twist distribution becomes e ar r tan Xl sin y cos y - 1 . 3.51 Since 0 is now known, the spanwise lift distribution can be found...

## Aeroelasticity Efficiency Change As Q Approaches

Figure 3.26 Sweep angle for which divergence dynamic pressure is infinite for a wing with e 0.02 solid line is for GJ EI 1.0 dashed line is for GJ EI 0.25. In this chapter we have considered divergence and aileron reversal of simple wind tunnel models, torsional divergence and load redistribution in flexible beam representations of lifting surfaces, the effects of sweep on coupled bending-torsion divergence, and the role of aeroelastic tailoring. In all these cases, the inertial loads are...

## Beam Theory Wing

Making it evident that 1 0 is proportional to 1 q see Fig. 3.4 . Therefore, for a model of this type only two data points are needed to extrapolate the line down and to the left until it Figure 3.4 Plot of 1 6 versus 1 q. Figure 3.5 Schematic of a sting-mounted wind tunnel model. Figure 3.5 Schematic of a sting-mounted wind tunnel model. intercepts the 1 q axis at a distance 1 q gt from the origin. As can be seen from the figure, the slope of this line can also be used to estimate qD. The form...

## Tvr

Figure 3.12 Cross section of spanwise uniform lifting surface. Now, a static equilibrium equation for the elastic torsional rotation, 0, about the elastic axis can be obtained from the fundamental torsional relation where G J is the effective torsional stiffness and T is the twisting moment about the elastic axis. One can obtain an equilibrium equation by equating the rate of change of twisting moment to the negative of the applied torque distribution so that Recognizing that uniformity implies...

## Theodorsen Function

Where the generalized forces are given in Eqs. 4.22 . The function C k is a complex-valued function of the reduced frequency k, given by where W, k are Hankel functions of the second kind, which can be expressed in terms of Bessel functions of the first and second kind, respectively, as Hf k J k - iY k . The function C k is called Theodorsen's function and is plotted in Fig. 4.9. Note that C k is real and equal to unity for the steady case i.e., for k 0 . As k increases, one finds that the...

## Dew

Keeping in mind that these functions are not orthogonal. 13. Referring back to either Problem 11 or 12, starting with the virtual work of the aerodynamic forces as where L' and M' are the sectional lift and pitching moment expressions used to develop Eqs. 3.76 , and using the given deformation modes, find the generalized forces S , ' 1,2 N Nw Ng. As discussed in the text, generalized forces are the coefficients of the variations of the generalized coordinates in the virtual work expression....

## Co

Here lw, lg, mw, and nig are defined in a manner similar to the quantities on the right-hand side of Eqs. 4.35 with the loads from Theodorsen theory, , 1 i-a l-2 i fl C fe li 2 a C k and the fundamental bending and torsion frequencies are Finally, the constant An 0.958641. It is clear that these equations are in the same form as the ones solved earlier for the typical section and that the influence of wing flexibility for this simplest two-mode case only enters in a minor way, namely, to adjust...

## Aeroelasticity Airfoil

Recall from Eq. 4.11 that the total displacement is a sum of all modal contributions. It is therefore necessary to consider all possible combinations of r and where F , can be lt 0, 0, or gt 0 and Q , can be 0 or 0. The corresponding type of motion and stability characteristics are indicated in Table 4.1 for various combinations of r and Q ,. Although our primary concern here is with regard to the dynamic instability of flutter for which Q-k 0, Table 4.1 shows that the generalized equations of...

## Strut Mounted Model For Aileron Reversal

However, unlike the previous example, one cannot make the divergence dynamic pressure infinite or negative thereby making divergence mathematically impossible by choice of configuration parameters because xac c lt 1. For a given wing configuration, one is left only with the possibility of increasing the sting bending stiffness to make the divergence dynamic pressure larger. A third configuration of a wind tunnel mount is a strut system as idealized in Figs. 3.8 and 3.9. The two linearly elastic...

## 2 Dof Static Airfoil Divergence Speed Formula

This is the same answer as one would obtain with an analysis similar to those of Chapter 3. For looking at flutter, we consider a specific configuration defined by a 1 5, e 1 10, i 20, r2 6 25, and a 2 5. The divergence speed for this configuration is Vd 2.828 or l , 2.828 b oi,, . Plots of the imaginary and real parts of the roots versus Figure 4.3 Plot of the modal frequency versus V for a 1 5, e 1 10, a 20, r2 6 25, and a 2 5 steady-flow theory . Figure 4.3 Plot of the modal frequency versus...

## Structural Dynamics

O students, study mathematics, and do not build without foundations The purpose of this chapter is to convey to the student a small, introductory portion of the theory of structural dynamics. Much of the theory to which the student will be exposed in this treatment was developed by mathematicians during the time between Newton and Rayleigh. The grasp of this mathematical foundation is therefore a goal that is worthwhile in its own right. Moreover, as implied by the above quotation, a proper use...

## Flutter Flight Envelope Below Sealevel

Figure 4.16 Flight envelope for typical Mach 2 fighter. Figure 4.16 Flight envelope for typical Mach 2 fighter. flutter does not appear to happen for any combination of a and r when the mass centroid, elastic axis, and aerodynamic center all coincide i.e., when e a 1 2 . Even if this prediction of the analysis is correct, practically speaking, it is very difficult to achieve coincidence of these points in wing design. Remember, however, that all these statements are made with respect to...

## Clamped Free Beam

Figure 2.16 Inertially restrained end of a beam. finite angular acceleration of the end. Therefore, T L t - GJ L t - -Ie j t, t , 2.168 GJ X' Y t -IcX l Y t . 2.169 From the functional form of Y t as established from the separation procedure, it can be noted that Substitution into the preceding condition yields GJX' l Y t a2 IcX i Y t , 2.171 Pip As above, the reader should verify that the same type of boundary condition at the other end would yield pIpX' 0 -a2IcX 0 . 2.173 2.2.3 Example...

## Vertical Blinds Phenomenon Swaying

A senior-level undergraduate course entitled Vibration and Flutter was taught for many years at Georgia Tech under the quarter system. This course dealt with elementary topics involving the static and or dynamic behavior of structural elements, both without and with the influence of a flowing fluid. The course did not deal with the static behavior of structures in the absence of fluid flow, because this is typically considered in courses in structural mechanics. Thus, the course essentially...

## Bending Torsion Flutter

Aeroelasticity is the term used to denote the field of study concerned with the interaction between the deformation of an elastic structure in an airstream and the resulting aerodynamic force. The interdisciplinary nature of the field can be best illustrated by Fig. 1.1, which depicts the interaction of the three disciplines of aerodynamics, dynamics, and elasticity. Classical aerodynamic theories provide a prediction of the forces acting on a body of a given shape. Elasticity provides a...