Info Lzl
Figure 2.45 Approximate fundamental frequency for a clamped-free beam with a particle of mass mi attached at .v ri. 22. Consider a clamped-free beam to which is attached at spanwise location x tr a particle of mass unit. Using a two-term Ritz approximation based on the functions in Eq. 2.311 , plot the approximate value for the fundamental natural frequency as a function of r for 1. I discovered that with increasing load, the angle of incidence at the wing tips increased perceptibly. It...
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...
Gj
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...
Info Qvq
It can be verified by applying Cramer's method for their solution that a nontrivial solution only exists if the determinant of the coefficients is equal to zero. This is typical of all nontrivial solutions to homogeneous, linear, algebraic equations, and here yields sinh2 al - sin2 af - cosh af cos af 2 0, 2.247 one obtains the characteristic equation as simply cos a cosh af 1 0. 2.249 One cannot extract a closed-form exact solution for this transcendental equation. However, numerical solutions...
Info Dju
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...
A 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...
Ei Qau
For the clamped-free boundary conditions C 0 ' ' - ' 0, this equation has a known analytical solution that yields a divergence dynamic pressure of The minus sign implies that this bending divergence instability only takes place for forward-swept wings, that is, where A lt 0. Examination of Eqs. 3.76 illustrates that there are two ways in which the sweep influences the aeroelastic behavior. One is the loss of aerodynamic effectiveness as exhibited by the change in the second term of the torsion...
Info Mve
The pilot of the airplane succeeded in landing with roughly two-thirds of his horizontal tail surface out of action some others have, unfortunately, not been so lucky The flutter problem is now generally accepted as a problem of primary concern in the design of current aircraft structures. Stiffness criteria based on flutter requirements are, in many instances, the critical design criteria. There is no evidence that flutter will have any less influence on the design of aerodynamically...
I 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....
V 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...
Ei Aaw
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...
Info Amz
The damping coefficients gi, and gs have representative values from 0.01 to 0.05 depending on the structural configuration. Most early analysts who incorporated this type of structural damping model into their flutter analyses specified the coefficient values a priori with the intention of improving the accuracy of their results. It was Scanlan and Rosenbaum 1948 who suggested that the damping coefficients be treated as unknown together with co. In this instance the subscripts on g can be...
Info Fya
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...
Info Dwf
Theodorsen's Unsteady Thin-Airfoil Theory Theodorsen 1934 derived a theory of unsteady aerodynamics for a thin airfoil undergoing small oscillations in incompressible flow. The lift contains both circulatory and noncirculatory terms, whereas the pitching moment about the quarter-chord is entirely noncirculatory. According to Theodorsen's theory, the lift and pitching moment are given
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...
Info Bvm
1 Bisplinghoff, R. L., H. Ashley, and R. L. Halfman, Aeroelasticity, Addison-Wesley Publishing Co., Inc., 1955. 2 Chen, Y., Vibrations Theoretical Methods, Addison-Wesley Publishing Co., Inc., 1966. 3 Fry'ba, L., Vibration and Solids and Structures under Moving Loads, Noordhoff International Publishing, 1972. 4 Hurty, W. C., and M. F. Rubenstein, Dynamics of Structures, Prentice-Hall, Inc., 1964. 5 Kalnins, A., and C. L. Dym, Vibration Beams, Plates and Shells, Dowden, Hutchinson and Ross,...
Vkeas
Figure 4.7 Comparison between p and p-k methods of flutter analysis for a twin-jet transport airplane. From Hassig 1971 Fig. 2, used by permission. one can compute A ifei . Using this new matrix in Eq. 4.75 leads to another set of ps, so that Continual updating of the aerodynamic matrix in this way provides an iterative scheme that is convergent for each of the roots, with negative y being a measure of the modal damping. The earliest presentation of this technique was offered by Irwin and...
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...