Introduction to structural dynamics and aeroelasticity download

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The basic questions addressed are dynamic stability and response of fluid structural systems as revealed by both linear and nonlinear mathematical models and correlation with experiment. The use of scaled models and full scale experiments and tests play a key role where theory is not considered sufficiently reliable.

In this new edition the more recent literature on nonlinear aeroelasticity has been brought up to date and the opportunity has been taken to correct the inevitable typographical errors that the authors and our readers have found to date.

The early chapters of this book may be used for a first course in aeroelasticity taught at the senior undergraduate or early graduate level and the later chapters may serve as the basis for a more advanced course, a graduate research seminar or as reference to provide an entree to the current research literature. Geared toward advanced undergraduates and graduate students, this outstanding text was written by one of the founders of bioengineering and modern biomechanics.

It offers unusually thorough coverage of the interaction of aerodynamic forces and elastic structures. It has also proven highly useful to designers and engineers concerned with flutter, structural dynamics, flight loads, and related subjects.

An introductory chapter covers concepts of aerodynamics, elasticity, and mechanical vibrations. Chapters 2 through 11 survey aeroelastic problems, their historical background, basic physical concepts, and the principles of analysis.

Chapters 12 through 15 contain the fundamentals of oscillating airfoil theory and a brief summary of experimental results. Each chapter is followed by a bibliography, and illustrations and 20 tables illuminate the text. Each chapter is followed by a bibliography, and illustrations and 20 tables illuminate the text.

The text represents carefully developed course materials, beginning with an introductory chapter on matrix algebra and methods for numerical computations, followed by a series of chapters discussing specific aeronautical applications. In this way, the student can be guided from the simple concept of a single-degree-of-freedom structural system to the more complex multidegree-of-freedom and continuous systems, including random vibrations, nonlinear systems, and aeroelastic phenomena.

Among the various examples used in the text, the chapter on aeroelasticity of flight vehicles is particularly noteworthy with its clear presentation of the phenomena and its mathematical formulation for structural and aerodynamic loads.

Ideal for coursework or self-study, this detailed examination introduces the concepts of aeroelasticity, describes how aircraft lift structures behave when subjected to aerodynamic loads, and finds its application in aerospace, civil, and mechanical engineering.

The book begins with a discussion on static behavior, and moves on to static instability and divergence, dynamic behavior leading up to flutter, and fluid structure interaction problems. It covers classical approaches based on low-order aerodynamic models and provides a rationale for adopting certain aeroelastic models. The author describes the formulation of discrete models as well as continuous structural models.

For instance, the structural deformation of wings can cause the quasi-static airloads to differ substantially from those of a more rigid vehicle. This includes a variety of unstable behaviors, both quasi-static and dynamic, which can shorten the life of the aircraft, or even destroy it. ASD researchers are developing computational methodologies which incorporate these aspects, and are utilizing this technology to explore the vitally important effects of structural flexibility and dynamics.

They have also developed a methodology, under Wright Laboratory funding, to couple existing rigid CFD codes and CSM codes to examine static aeroelastic problems. As before, inertial forces contribute to [M], there are contributions from the inertial forces to [C] and [K] when there is a rotating coordinate system, and damping also contributes to [C].

Finally, because aeroelastic loads, in general, depend on the displacement and its time derivatives, aerodynamics can contribute terms to [M], [C], and [K].

Now, we illustrate how the approximating functions are actually used. Substituting Eq. Results for free vibration i. As with the Ritz method, we see monotonic convergence from above and accuracy comparable to that achieved via the Ritz method. However, unlike the Ritz method, we do not always obtain results for free-vibration problems that converge from above.

Consider again a clamped-free beam. Although this equation of motion is somewhat more complicated, it is only a second-order equation. Thus, a much simpler 3. Moreover, the results presented in Table 3.

Schematic of a nonuniform beam with distributed twisting moment per unit length 3. The name derives from the breaking of a structure into a large number of small elements, modeling them approximately, and connecting them together appropriately. Because of this way of discretizing the geometry, it is possible to accurately capture modeling details that other methods cannot.

It typically makes use of polynomial shape functions over each of the finite elements into which the original structure is broken. Equations based on the finite element method have the same structure as Eq.

What keeps the computational effort from being overly burdensome is that the matrices have a narrow-banded structure, which allows specialized software to be used in solving the equations of motion that takes advantage of this structure, reducing both memory and floating-point operations and resulting in significant computational advantages. Here, we present only a simple outline of the method as applied to beams in torsion and in bending, leaving more advanced topics such as plates and shells to textbooks devoted to the finite element method, such as those by Reddy and Zienkiewicz and Taylor Application to Beams in Torsion.

Here, we use the finite element method to analyze the behavior of a nonuniform beam in torsion. Regardless of how finite elements are derived, however, for a sufficiently fine mesh, the results should approach the exact structural behavior. This development encompasses both forced response and free vibration.

Consider a clamped-free beam subjected to a distributed torque r x, t as depicted in Fig. Note that the x coordinate is along the beam. The strain energy 3. In the finite-element approach, the beam is divided into n elements, as shown in Fig. Although there is no requirement to make the elements of constant stiffness, we do so for convenience.

Relaxation of this assumption is left as an exercise for readers see Problem Within element i, the torsional stiffness is assumed to be a constant, GJ i.

Introducing this approximation into the strain energy, Eq. Note that we could add the potential energy of springs attached to ground at any nodes to represent elastic restraints. The contribution of the applied torque r x, t comes into the analysis through the generalized force, which may be extracted from the virtual work, given by Eq. The boundary condition at the free end i. Therefore, it need not be taken into account in a solution by this approach.

This has the effect of removing the first row and column from each of the matrices [M] and [K], and the first row from [D]. Given the approximation of the twist field in Eq. First, as noted previously in the discussion of the Ritz method see Section 3. Second, they are banded; that is, the nonvanishing entries are concentrated around the diagonals of the matrices.

Third, [M] is positive definite and [K] is at least positive semidefinite. In the absence of rigid-body modes, [K] is positive definite because it results from the computation of the strain energy of the structure, itself a positive-definite quantity when rigid-body motion is excluded. For example: 1. For this, we do not need the mass matrix [M]. Finally, the forced response of the structure may be determined by numerical integration of Eqs.

Complex structures including entire aircraft can be modeled with the finite element method. The resulting discretized equations are similar to Eq. As the complexity of the model increases, the various arrays increase in size. For the most general types of models, such as those based on three-dimensional brick elements, hundreds of thousands of degrees of freedom or more may be required to accurately model a complete wing structure.

The convergence is monotonic, and the answers are evidently upper bounds. Application to Beams in Bending. As another example of applying the finite element method, we next turn to its application to beams in bending. The theory of bending for beams was presented in terms of strain energy, kinetic energy, and virtual work; this framework is sufficient for constructing a finite-element model for nonuniform beams in bending. Again, strictly for simplicity, we assume the bending stiffness and mass per unit length to be constants, respectively equal to EI i and mi within element i.

We consider a beam loaded with a distributed force per unit length f x, t and a distributed bending moment per unit length q x, t as shown in Fig. As with the beam in torsion, we now develop the stiffness matrix from the strain energy. Schematic of a nonuniform beam with distributed force and bending moment per unit length 3. The four cubic polynomials in Eq. This way, the element degrees of freedom are displacements or rotations at the ends of the element.

With this pattern in mind, it is a straightforward matter to expand the matrix to an arbitrary number of elements.

This pattern is the same as that of the stiffness matrix, so it is also a straightforward matter to expand the mass matrix to an arbitrary number of elements. Finally, the contributions of the applied distributed force and bending moment are determined using the virtual work.

If we interpolate both f x, t and q x, t in the same way that r x, t was treated for torsion, viz. Because the approach is based on the Ritz method, only the geometric boundary conditions need to be satisfied.

The accuracy of finite elements for beam bending is illustrated in Problem Moreover, the approximation techniques of the Ritz method, the Galerkin method, and the finite element method were introduced. This sets the stage for consideration of aeroelastic problems in Chapters 4 and 5. The static-aeroelasticity problem, addressed in Chapter 4, results from interaction of structural and aerodynamic loads.

These loads are a subset of those involved in dynamic aeroelasticity, which includes inertial effects. One aspect of dynamic aeroelasticity is flutter, which is discussed in Chapter 5, where it is shown that both the modal representation and the modal approximation methods apply equally well to both types of problems.

Use of a table of integrals may be helpful. Considering Eq. Transverse vibration of the string is restrained at its midpoint by a linear spring with spring constant k. The spring is unstretched when the string is undeflected. As a check, derive the equation taking into account the spring through the potential energy instead of through the generalized force.

Make a plot of the behavior of the lowest natural frequency versus the value of the concentrated inertia. Consider a clamped-free beam undergoing torsion: a Prove that the free-vibration mode shapes are orthogonal, regardless of whether the beam is uniform. Problems 10—12 Consider a clamped-free beam undergoing bending: a Prove that the free-vibration mode shapes are orthogonal, regardless of whether the beam is uniform.

Consider a uniform beam with the boundary conditions shown in Fig. Check your results versus those given in Fig. Suggestion: use Eq. Normalize the mode shapes to have unit deflection at the free end and determine the generalized mass for the first five modes. As a sample of the mode shapes, the first elastic mode is plotted in Fig. Consider the beam in Problem Plot the mode shapes, normalizing them to have unit deflection at the free end.

Consider a beam that at its left end is clamped and at its right end is pinned with a rigid body attached to it. Ignoring those terms that are time dependent they would die out in a real beam because of dissipation , plot the tip displacement versus the number of mode shapes retained in the solution up to five modes.

Show the static tip deflection from elementary beam theory on the plot. This part of the problem illustrates how the modal representation can be applied to staticresponse problems. Applying the Ritz method, write the equations of motion for a system that consists of the beam plus identical rigid bodies attached to the ends, where each body has a moment of inertia IC and mass mc.

Use as assumed modes those of the exact solution of the free-free beam without the attached bodies, obtained in the text. Note the terms that provide inertial coupling. Use a rigid-body rotation plus the set of clamped-free modes as the assumed modes of the Ritz method. Answer: See Tables 3.

Repeat Problem 17 using a set of polynomial admissible functions. Use one rigid-body mode x and a varying number of polynomials that satisfy all the boundary conditions of a clamped-free beam. Answer: See Table 3. Table 3. Rework Problem 19 using Eq. Using a two-term Ritz approximation based on the functions in Eq. Consider a clamped-free beam undergoing coupled bending and torsion.

Answer these questions: How do the signs of d and K affect the frequencies? How do they affect the predicted mode shapes? Repeat Problem 23 using an appropriate power series for bending and for torsion.

Develop a finite-element solution for the static twist of a clamped-free beam in torsion, accounting for linearly varying GJ x within each element. Compare results for the tip rotation caused by twisting, with identical loading and properties i. Note that the results in Section 3. Answer: The results do not change; see Table 3.

Compare [M] and [K] matrices obtained for the case developed in the text i. The resulting torsion caused the wings to collapse under the strain of combat maneuvers.

Fokker in The Flying Dutchman, Henry Holt and Company, The field of static aeroelasticity is the study of flight-vehicle phenomena associated with the interaction of aerodynamic loading induced by steady flow and the resulting elastic deformation of the lifting-surface structure. These phenomena are characterized as being insensitive to the rates and accelerations of the structural deflections. There are two classes of design problems that are encountered in this area. The first and most common to all flight vehicles is the effects of elastic deformation on the airloads, as well as effects of airloads on the elastic deformation, associated with normal operating conditions.

These effects can have a profound influence on performance, handling qualities, flight stability, structural-load distribution, and control effectiveness.

The second class of problems involves the potential for static instability of the lifting-surface structure to result in a catastrophic failure. Simply stated, divergence occurs when a lifting surface deforms under aerodynamic loads in such a way as to increase the applied load, and the increased load deflects the structure further—eventually to the point of failure. Such a failure is not simply the result of a load that is too large for the structure as designed; instead, the aerodynamic forces actually interact with the structure to create a loss of effective stiffness.

This phenomenon is explored in more detail in this chapter. The material presented in this chapter is an introduction to some of these static aeroelastic phenomena. To illustrate clearly the mechanics of these problems and yet maintain a low level of mathematical complexity, relatively simple configurations are considered. The first items treated are rigid aerodynamic models that are elastically mounted in a wind-tunnel test section; such elastic mounting is characteristic of most load-measurement systems.

The second aeroelastic configuration to be treated is Static Aeroelasticity Figure 4. Planform view of a wind-tunnel model on a torsionally elastic support a uniform elastic lifting surface of finite span.

Its static aeroelastic properties are similar to those of most lifting surfaces on conventional flight vehicles. Expressions for the aeroelastic pitch deflections are developed for these simple models that, in turn, lead to a cursory understanding of the divergence instability. Finally, we briefly return to the wallmounted model in this section to consider the qualitatively different phenomenon of aileron reversal.

All of these wing models are assumed to be rigid and twodimensional. That is, the airfoil geometry is independent of spanwise location, and the span is sufficiently large that the lift and pitching moment do not depend on a spanwise coordinate.

The support is flexible in torsion, which means that it restricts the pitch rotation of the wing in the same way as a rotational spring would. We denote the rotational stiffness of the support by k, as shown in Fig.

If we assume the body to be pivoted about its support O, located at a distance xO from the leading edge, moment equilibrium requires that the sum of all moments about O must equal zero. The treatment herein is restricted to incompressible flow, but compressibility effects may be taken into account by means of Prandtl-Glauert corrections to the airfoil coefficients. For this, the freestream Mach number must remain less than roughly 0.

If experimental data or results from computational fluid dynamics provide an alternative value, then it should be used. Using Eqs. Recalling the discussion of stability in Section 2. Such is the case when the moment of the lift about point O exceeds the restoring moment from the spring. Indeed, a plot of the latter is given in Fig. However, remember that there are limitations on the validity of both expressions.

Namely, the lift will not continue to increase as stall is encountered. When the system parameters are within the bounds of validity for linear theory, another fascinating feature of this problem emerges. As shown in the figure, the slope of this line also can be used to estimate qD. The form of this plot is of great practical value because estimates of qD can be extrapolated from data taken at speeds far below the divergence speed. This means that qD can be estimated even when the values of the model parameters are not precisely known, thereby circumventing the need to risk destruction of the model by testing all the way up to the divergence boundary.

A simplified version of this kind of model is shown in Figs. Also in Fig. Note the equal and opposite directions on the force F0 and moment M0 at the trailing edge of the wing in Fig. Figure 4.

Schematic of a sting-mounted wind-tunnel model 4. Detailed view of the sting-mounted wing Static Aeroelasticity Figure 4. The two linearly elastic struts have the same extensional stiffness, k, and are mounted at the leading and trailing edges of the wing.

As illustrated in Fig. However, divergence can be eliminated if the leading-edge spring stiffness is increased relative to that of the trailing-edge spring. This is left as an exercise for readers see Problem 5. Schematic of the airfoil section of a flapped two-dimensional wing in a wind tunnel is that the pilot cannot control the aircraft in the usual way.

There are additional concerns for aircraft, the missions of which depend on their being highly maneuverable. This loss in control effectiveness and eventual reversal is the focus of this section. Consider the airfoil section of a flapped two-dimensional wing, shown in Fig. Similar to the model discussed in Section 4. Conversely, the response is significantly affected by the aileron, as we now show. We can solve the response problem by substituting Eqs.

We then find the lift as follows: 1. Substitute Eq. Finally, substitute the lift coefficient into the first of Eqs. The second term is aeroelastic. Obviously, a stiffer k gives a higher reversal speed, and a model that is rigid in pitch analogous to a torsionally rigid wing will Static Aeroelasticity not undergo reversal. Now let us consider the effect of both numerator and denominator. Equations 4. Uniform unswept clamped-free lifting surface 4. These idealized configurations provide insight into the aeroelastic stability and response, but practical analyses must take into account flexibility of the lifting surface.

That being the case, in this section, we address flexible wings, albeit with simplified structural representation. Consider an unswept uniform elastic lifting surface as illustrated in Figs.

The lifting surface is modeled as a beam and, in keeping with historical practice in the field of aeroelasticity, the spanwise coordinate along the elastic axis is denoted by y.

The beam is presumed to be built in at the root i. The y axis corresponds to the elastic axis, which may be defined as the line of effective shear centers, assumed here to be straight. Recall that for isotropic beams, a transverse force applied at any point along this axis results in bending with no elastic torsional rotation about the axis. This axis is also the axis of twist in response to a pure twisting moment applied to the wing.

Because the primary concern here Static Aeroelasticity Mac' Figure 4. Cross section of spanwise uniform lifting surface is the determination of the airload distributions, the only elastic deformation that influences these loads is rotation due to twist about the elastic axis. That is, the airfoil geometry including incidence angle is independent of spanwise location, and the span is sufficiently large that lift and pitching moment are not functions of a spanwise coordinate.

In turning our attention to wings that can be modeled as isotropic beams, the incidence angle now may be a function of the spanwise coordinate because of the possibility of elastic twist. We need the distributed lift force and pitching moment per unit span exerted by aerodynamic forces along a slender beam-like wing. At this stage, however, we ignore the three-dimensional tip effects associated with wings of finite length; the aerodynamic loads at a given spanwise location do not depend on those at any other.

To simplify the calculation, the wing can be broken up into spanwise segments of infinitesimal length, where the local lift and pitching moment can be estimated from two-dimensional theory.

The angle of attack is represented by two components. Now, a static equation of moment equilibrium about the elastic axis Static Aeroelasticity can be obtained by equating the rate of change of twisting moment to the negative of the applied torque distribution.

This is a specialization of Eq. The boundary conditions for other end conditions for beams in torsion are given in Section 3. Such a function is plotted in Fig. Note that the character of the plot in Fig. It is of practical interest to note that the tip-twist angle may become sufficiently large to warrant concern about the structural integrity for dynamic pressures well below qD.

Because this instability occurs at a dynamic pressure that is independent of the right-hand side of Eq. This result implies that there are nontrivial solutions of the homogeneous equation for the elastic twist. In other words, even for cases in which the right-hand side of Eq. If the elastic axis is upstream of the aerodynamic center, then e 4.

This airload can be integrated over the vehicle to obtain the total lift, L. These equations can be used to find the torsional deformation and the resulting airload distribution for a specified flight condition. The calculation of the spanwise aeroelastic airload distribution is immensely practical and is used in industry in two separate ways.

First, it is used to satisfy a requirement of aerodynamicists or performance engineers who need to know the total force and moment on the flight vehicle as a function of altitude and flight condition. A second requirement is that of structural engineers, who must ensure the structural integrity of the lifting surface for a specified load factor N and flight condition. Such a specification normally is described by what is called a V-N diagram. For the conditions of given load factor and flight condition, it is necessary for structural engineers to know the airload distribution to conduct a subsequent loads and stress analysis.

From this, the distributions of torsional and bending moments along the wing can be found, leading directly to the maximum stress in the wing, generally somewhere in the root cross section.

Observe that the overall effect of torsional flexibility on the unswept lifting surface is to significantly change the spanwise-airload distribution. Because this elastic torsional rotation generally increases as the distance from the root i. In the other case, when N is specified by a structural engineer, the total lift i. The addition of lift in the outboard region must be balanced by a decrease inboard. Rigid and elastic wing-lift distributions holding total lift constant Static Aeroelasticity All of the preceding equations for torsional divergence and airload distribution were based on a strip-theory aerodynamic representation.

A slight numerical improvement in their predictive capability can be obtained if the two-dimensional lift—curve slope, a, is replaced everywhere by the total i. Although there is little theoretical justification for this modification, it alters the numerical results in the direction of the exact answer. Also, it is important to note that the lift distributions depicted in Figs.

In such a case, closed-form expressions such as those of Eqs. In this section, we examine the same physical phenomenon using a torsionally flexible wing model.

With the geometry and boundary conditions of the uniform, torsionally flexible lifting surface as before, we can derive the reversal dynamic pressure for a clamped-free wing. Two logical choices are presented regarding the defining condition.

One is to define reversal dynamic pressure as that dynamic pressure at which the change of total lift with respect to the aileron deflection is equal to zero. Another equally valid definition is to define it as the dynamic pressure at which the change in root-bending moment with respect to the aileron deflection is equal to zero. Assuming the weight to have a negligible effect on the reversal condition, the modified version of Eq.

Similarly, we may account for an aileron that does not extend over the entire length of the wing. This treatment can be generalized easily to consider the roll effectiveness of a complete aircraft model. Similar problems can be posed in the framework of dynamics, in which the objective is, say, to predict the angular acceleration caused by deflection of a control surface, or the time to change the orientation of the aircraft from one roll angle to another. Depending on the aircraft and the maneuver, it may be necessary to consider nonlinearities.

Here, however, only a static, linear treatment is included. Consider a rolling aircraft with unswept wings, the right half of which is shown in Fig. As shown in Fig. Some contributions to the lift and pitching moment are the same opposite on both sides of the aircraft; these are referred to as symmetric antisymmetric components. Separate problems can be posed in terms of symmetric and antisymmetric parts, which are generally uncoupled from one another.

In particular, we can treat the roll problem as an antisymmetric problem noting that all symmetric components cancel out in pure roll. Hence, we can discard them a priori. The last term, which represents the increment in the angle of attack from 4. Schematic of a rolling aircraft the roll rate p based on the assumption of a small angle of attack, also is explicitly antisymmetric.

Section of right wing with positive aileron deflection 4. Note that the three terms in Eq. The lowest value is associated the aileron reversal.