Introduction to Nonlinear Aeroelasticity

By Grigorios Dimitriadis

Introduces the latest developments and technologies in the area of nonlinear aeroelasticity

Nonlinear aeroelasticity has become an increasingly popular research area in recent years. There have been many driving forces behind this development, increasingly flexible structures, nonlinear control laws, materials with nonlinear characteristics, etc. Introduction to Nonlinear Aeroelasticity covers the theoretical basics in nonlinear aeroelasticity and applies the theory to practical problems.

As nonlinear aeroelasticity is a combined topic, necessitating expertise from different areas, the book introduces methodologies from a variety of disciplines such as nonlinear dynamics, bifurcation analysis, unsteady aerodynamics, non-smooth systems and others. The emphasis throughout is on the practical application of the theories and methods, so as to enable the reader to apply their newly acquired knowledge.

Key features:

• Covers the major topics in nonlinear aeroelasticity, from the galloping of cables to supersonic panel flutter.
• Discusses nonlinear dynamics, bifurcation analysis, numerical continuation, unsteady aerodynamics and non-smooth systems.
• Considers the practical application of the theories and methods.
• Covers nonlinear dynamics, bifurcation analysis and numerical methods.
• Accompanied by a website hosting Matlab code.

Introduction to Nonlinear Aeroelasticity is a comprehensive reference for researchers and workers in industry and is also a useful introduction to the subject for graduate and undergraduate students across engineering disciplines.

• Language: English
• Publisher: Wiley
• Release date: Mar 10, 2017
• ISBN: 9781118756461

Book preview

1

Introduction

Nonlinear aeroelasticity is the study of the interactions between inertial, elastic and aerodynamic forces on engineering structures that are exposed to an airflow and feature non‐negligible nonlinearity. There exist several good textbooks on linear aeroelasticity for aircraft (Bisplinghoff et al. 1996; Fung 1993; Hodges and Alvin Pierce 2002; Wright and Cooper 2015). Dowell (2004) even includes chapters on nonlinear aeroelasticity and stall flutter, while Paidoussis et al. (2011) discusses a number of nonlinear aeroelastic phenomena occurring in civil engineering structures. However, there is no introductory text that presents the methodologies of nonlinear dynamics and applies them to a wide range of nonlinear aeroelastic systems. The present book aims to fill this gap to a certain degree. The subject area is vast and mutlidisciplinary and it would be impossible to fit every aspect of it in a textbook. The main omission is high fidelity numerical simulation using Computational Fluid Dynamics and Computational Structural Dynamics solvers; these methodologies are already the subject of a dedicated text (Bazilevs et al. 2013). The aerodynamic models used in this book are analytical, empirical or based on panel methods while the structural models are either analytical or make use of series solutions.

The book is introductory but it assumes knowledge of structural dynamics, aerodynamics and some linear aeroelasticity. The main linear aeroelastic phenomena of flutter and static divergence are discussed in detail because they can affect nonlinear behaviour, but the present work is by no means a complete text on linear aeroelasticity. Unsteady aerodynamic modelling is used throughout the book and discussed in Chapters 8, 10 and in the Appendix. However, again this book is not a complete reference on unsteady aerodynamics, linear or nonlinear. On the other hand, nonlinear dynamics and bifurcation analysis are presented in great detail as they do not normally feature in most undergraduate or even graduate Aerospace and Mechanical Engineering courses. The emphasis of all discussions is on the application rather than the rigorous derivation of the theorems; there already exist several classic textbooks for the latter (Kuznetsov 1998; Guckenheimer and Holmes 1983). More application‐based works on nonlinear dynamics also exist (e.g. Strogatz 1994) but they address a wide range of physical, chemical, biological, accounting models, to name a few, whereas the present book concentrates exclusively on aeroelastic phenomena.

Nonlinear aeroelasticity has become an increasingly popular research area over the last 30 years. There have been many driving forces behind this development, including faster computers, increasingly flexible structures, automatic control systems for aircraft and other engineering products, new materials, optimisation‐based design methods and others. Aeroelasticians have acquired expertise from many different fields in order to address nonlinear aeroelastic problems, mainly nonlinear dynamics, bifurcation analysis, control theory, nonlinear structural analysis and Computational Fluid Dynamics. The main applications of nonlinear aeroelasticity lie in aeronautics and civil engineering but other types of structure are also concerned, such as bridges and wind turbines.

In classical linear aeroelasticity, the relationships between the states of a system and the internal forces acting on them are always assumed to be linear. Force‐displacement diagrams for the structure and lift or moment curves for the aerodynamics are always assumed to be linear, while friction is neglected and damping is also linear. As an example, consider a torsional spring that provides a restoring moment M when twisted through an angle ϕ. Figure 1.1a plots experimentally measured values of ϕ and M. Clearly, the function M(ϕ) is not linear but, if we concentrate in the range , the curve is nearly linear and we can curve fit it as the straight line , where K is the linear stiffness of the spring.

2 Graphs depicting moment of torsional spring (top) and lift curve (bottom), both displaying an ascending line and circles representing linear curve fit and measured data, respectively.

Figure 1.1 Linearised load‐displacement diagrams

Figure 1.1b plots the aerodynamic lift coefficient acting on a wing placed at an angle α to a free stream of speed U, defined as

where l is the lift force per unit length, ρ is the air density and c is the chord. The curve cl(α) is by no means linear but, again, if we focus in the range , we can curve fit the lift coefficient as the straight line , where is the lift curve slope. An aeroelastic system featuring the spring of Figure 1.1a and the wing of Figure 1.1b will be nonlinear but, if we ensure that ϕ and α never exceed their respective linear ranges for all operating conditions, then we can treat the system as linear and use linear analysis to design it. In nonlinear aeroelasticity, the angles ϕ and α will always exceed their linear ranges and therefore we must use nonlinear analysis, both static and dynamic, in order to design the system.

Nonlinear dynamics is the field of study of nonlinear ordinary and partial differential equations, which in this book model aeroelastic systems. Unlike linear differential equations, nonlinear equations have no general analytical solutions and, in some cases, several different solutions may coexist at the same operating conditions. Furthermore, nonlinear systems can have many more types of solution than linear ones. The operating conditions of an aeroelastic system are primarily the free stream airspeed and the air density (or flight altitude), while the Reynolds number, Mach number and mean angle of attack can also be important. As these system parameters vary, the number and type of solutions of the nonlinear equations of motion can change drastically. The study of the changing nature of solutions as the system parameters are varied is known as bifurcation analysis. In this book we will use almost exclusively local bifurcation analysis, which means that we will identify individual solutions and track their nature and their intersections with other solutions for all the parameter values of interest.

A wide variety of nonlinear aeroelastic phenomena will be investigated, from the galloping of cables to the buckling and flutter of panels in supersonic flow and from stall flutter to the limit cycle oscillations of finite wings. We will also briefly discuss transonic aeroelastic phenomena but we will not analyse them in detail because such analysis requires high fidelity computational fluid and structural mechanics and is still the subject of extensive research. The equations of motion treated in this book are exclusively ordinary differential equations; whenever we encounter partial differential equations we will first transform them to ordinary using a series solution. It is hoped that the book will contribute towards the current trend of taking nonlinear aeroelasticity out of the research lab and introducing it into the classroom and in industry.

1.1 Sources of Nonlinearity

Traditionally, a lot of effort has been devoted to designing and building engineering structures that are as linear as possible. Despite this effort, nonlinearity, weak or strong, has always been present in engineering systems. In recent years, increasing amounts of nonlinearity have been tolerated or even purposefully included in many applications, since nonlinear analysis methods have progressed sufficiently to allow the handling of nonlinearity at the design stage. Furthermore, nonlinearity can have significant beneficial effects, for example in shock absorbers and suspension systems.

In this book we will only consider nonlinearities that are present in aeroelastic systems. Since aeroelasticity is of particularly importance to the fields of aeronautics, civil engineering and energy harvesting, we will limit the discussion of nonlinearity to these application areas. The nonlinear functions that are most often encountered in these systems have three main sources:

the structure,

the aerodynamics and

the control system.

The structural nonlinearities of interest occur during the normal operation of the underlying engineering system. Nonlinearities appearing in damaged, cracked, plastically deformed and, in general, off‐design systems are beyond the scope of this book. The most common forms of nonlinearity appearing in structures are geometric (caused by large deformations), clearance (i.e. freeplay, contact and other non‐smooth phenomena), dissipative (i.e. friction or other nonlinear damping forces) and inertial (of particular interest in rotors and turbomachinery).

Aerodynamic nonlinearities arise from the existence of either unsteady separated flow or oscillating shock waves or a combination of the two (e.g. shock‐induced separation). Separation‐induced nonlinearity can affect all aeroelastic systems, although bluff bodies such as bridges, towers and cables are always exposed to it. Shock‐induced nonlinearity is of interest mostly to the aeronautical industry. It should be noted that aerodynamic nonlinearity is inertial, dissipative and elastic.

Engineering structures are increasingly designed to feature passive and/or active control systems. These systems can either aim to stabilise the structure (e.g. suppress or mitigate unwanted vibrations) or to control it (e.g. aircraft automatic flight control systems). Passive systems can be seen as parts of the structure and therefore included in the structural nonlinearity category (if they are nonlinear). Active systems, however, can feature a number of prescribed and incidental nonlinearities that can be turned off by running the structure in open loop mode. These nonlinear functions are in a category of their own and can take many forms, such as deflection and rate limits on actuators or nonlinear control laws. Furthermore, control actuators always feature a certain amount of freeplay, which is usually strictly limited by airworthiness regulations.

One more source of nonlinearity can be external stores on aircraft that carry them (mainly military aircraft). Stores such as external fuel tanks, bombs and missiles can cause store‐induced oscillations, particularly at transonic flight conditions. However, the mechanisms behind these oscillations are still not fully understood and the relevant analyses usually involve computational fluid‐structure interaction. Consequently, these phenomena will not be discussed further in this book. Human operator‐related nonlinearities (pilot, driver, rider etc.) will not be considered either.

1.2 Origins of Nonlinear Aeroelasticity

Some of the first investigations of nonlinear aeroelasticity concerned stall flutter and started just after WWII. For example, Victory (1943) reported that the airspeed at which wings undergo flutter decreases at high incidence angles, while Mendelson (1948) attempted to model this phenomenon. Rainey (1956) carried out a range of wind tunnel experiments of aeroelastic models of wings and noted the parameters that affect their stall flutter behaviour. It was quickly recognised that, in order to analyse stall flutter, the phenomenon of unsteady flow separation known as dynamic stall needed to be isolated and studied in detail. Bratt and Wight (1945) and Halfman et al. (1951) carried out two of the first experimental studies of the unsteady aerodynamic loads acting on 2D airfoils oscillating at high angles of attack. They were to be followed by a significant number of increasingly sophisticated experiments, covering a wide range of airfoil geometries, Reynolds numbers, Mach numbers and oscillation amplitudes and frequencies. The phenomena of dynamic stall and stall flutter are discussed in Chapter 8.

The effects of structural nonlinearity were first investigated by Woolston et al. (1955, 1957) and Shen (1959). They both set up aeroelastic systems with structural nonlinearity and solved them using analog computers. The systems included 2D airfoils with nonlinear springs, wings with control surfaces and buckled panels in supersonic flow. Such systems have been explored ever since, using increasingly sophisticated mathematical and experimental methods. They are in fact the basis of nonlinear aeroelasticity and will be discussed in detail in the present book. Two‐dimensional airfoils with nonlinear springs will be analysed in Chapters 2 to 7, panels in supersonic flow will be presented in Chapter 9 and 3D wings in Chapter 10.

Wind tunnel experiments on nonlinear aeroelastic systems with nonlinear springs have been carried out since the 1980s, notably by McIntosh Jr. et al. (1981); Yang and Zhao (1988); Conner et al. (1997). These works provided both valuable insights into the phenomena that can be encountered in nonlinear aeroelasticity and a basis for the validation of various modelling and analysis methods. The focus of the present book is the application of nonlinear dynamic analysis to nonlinear aeroelasticity. Modelling will be discussed in the last three chapters, as well as in the Appendix.

Shen (1959) was one of the first works to apply the Harmonic Balance method to nonlinear aeroelasticity. This method was first presented in the West by Kryloff and Bogoliuboff (1947) and has since become one of the primary analysis tools for nonlinear dynamic systems undergoing periodic oscillations. We will use several different versions of the Harmonic Balance technique throughout this book.

One of the first studies to apply elements of bifurcation theory to nonlinear aeroelastic systems was carried out by Price et al. (1994). They used stability boundaries, Poincaré sections and bifurcation diagrams to analyse the behaviour of a simple 2D mathematical nonlinear aeroelastic system with structural nonlinearity. Aside from the Hopf bifurcation, they also observed period‐doubling bifurcations and chaotic responses. Bifurcation analysis is used throughout the present book but most of the bifurcations typically encountered in nonlinear aeroelasticity are discussed in detail in Chapter 5.

Alighanbari and Price (1996) were the first to use numerical continuation in nonlinear aeroelasticity. Numerical continuation (Allgower and Georg 1990) is a set of mathematical methods for solving nonlinear problems that have static or periodic dynamic solutions. Continuation methods are strongly linked to bifurcation analysis, as they very often start evaluating solutions at bifurcation points. Such methods will be presented in detail in Chapter 7 and used in all subsequent chapters.

Towards the end of the 1990s, Friedmann (1999) identified nonlinear aeroelasticity as a major research direction in his paper on the future of aeroelasticity. Lee et al. (1999) published a lengthy and authoritative review of past and current nonlinear aeroelastic research, describing all major advances in both understanding and methodologies. A few years later, the nonlinear aeroelasticity chapter by Dowell (2004) provided an extensive description of nonlinear aeroelastic phenomena encountered in flight and in benchmark aeroelastic wind tunnel models and summarised the state of the art.

Thirteen years later, there has been a significant increase in the research and application of nonlinear aeroelasticity. Transonic aeroelastic phenomena, the highly flexible structures of High Altitude Long Endurance aircraft, aeroelastic tailoring, gust loads acting on nonlinear aircraft, wind turbine aeroelasticity and high‐fidelity fluid structure interaction have all become major areas of research. Major national and international research projects have addressed such issues and the results are slowly starting to be applied in industry. Given this wealth of activity in the field, it was felt that an introductory text in nonlinear aeroelasticity is missing from the literature. It is hoped that the present book will come to fill this gap, providing a basis for understanding nonlinear aeroelastic phenomena and methodologies on relatively simple systems and preparing the reader for more advanced work in state‐of‐the‐art applications.

References

Alighanbari H and Price SJ 1996 The post‐hopf‐bifurcation response of an airfoil in incompressible two‐dimensional flow. Nonlinear Dynamics10(4), 381–400.

Allgower EL and Georg K 1990 Numerical Continuation Methods: An Introduction. Springer‐Verlag, New York.

Bazilevs Y, Takizawa K and Tezduyar TE 2013 Computational Fluid‐Structure Interaction: Methods and Applications. John Wiley & Sons, Ltd, Chichester, UK.

Bisplinghoff RL, Ashley H and Halfman RL 1996 Aeroelasticity. Dover Publications, New York.

Bratt JB and Wight KC 1945 The effect of mean incidence, amplitude of oscillation, profile and aspect ratio on pitching moment derivatives. Reports and Memoranda No. 2064, Aeronautical Research Committee.

Conner MD, Tang DM, Dowell EH and Virgin L 1997 Nonlinear behaviour of a typical airfoil section with control surface freeplay: a numerical and experimental study. Journal of Fluids and Structures11(1), 89–109.

Dowell EH (ed.) 2004 A Modern Course in Aeroelasticity, 4th edn. Kluwer Academic Publishers.

Friedmann PP 1999 Renaissance of aeroelasticity and its future. Journal of Aircraft36(1), 105–121.

Fung YC 1993 An Introduction to the Theory of Aeroelasticity. Dover Publications, Inc.

Guckenheimer J and Holmes P 1983 Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer‐Verlag, New York.

Halfman RL, Johnson HC and Haley SM 1951 Evaluation of high‐angle‐of‐attack aerodynamic‐derivative data and stall‐flutter prediction techniques. Technical Report TN 2533, NACA.

Hodges DH and Alvin Pierce G 2002 Introduction to Structural Dynamics and Aeroelasticity. Cambridge University Press, Cambridge, UK.

Kryloff N and Bogoliuboff N 1947 Introduction to Nonlinear Mechanics (a Free Translation by S. Lefschetz). Princeton University Press, Princeton, NJ.

Kuznetsov YA 1998 Elements of Applied Bifurcation Theory, 2nd edn. Springer, New York Berlin Heidelberg.

Lee BHK, Price SJ and Wong YS 1999 Nonlinear aeroelastic analysis of airfoils: bifurcation and chaos. Progress in Aerospace Sciences35(3), 205–334.

McIntosh Jr. SC, Reed RE and Rodden WP 1981 Experimental and theoretical study of nonlinear flutter. Journal of Aircraft18(12), 1057–1063.

Mendelson A 1948 Effect of aerodynamic hysteresis on critical flutter speed at stall. Research Memorandum RM No. E8B04, NACA.

Paidoussis MP, Price SJ and de Langre E 2011 Fluid Structure Interactions: Cross‐Flow‐Induced Instabilities. Cambridge University Press, Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo, Delhi, Dubai, Tokyo, Mexico City.

Price SJ, Lee BHK and Alighanbari H 1994 Poststability behavior of a two‐dimensional airfoil with a structural nonlinearity. Journal of Aircraft31(6), 1395–1401.

Rainey AG 1956 Preliminary study of some factors which affect the stall‐flutter characteristics of thin wings. Technical Note TN 3622, NACA.

Shen SF 1959 An approximate analysis of nonlinear flutter problems. Journal of the Aerospace Sciences26(1), 25–32.

Strogatz SH 1994 Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering. Perseus Books, Cambridge, MA.

Victory M 1943 Flutter at high incidence. Reports and Memoranda No. 2048, Aeronautical Research Committee.

Woolston DS, Runyan HL and Andrews RE 1957 An investigation of effects of certain types of structural nonlinearities on wing and control surface flutter. Journal of the Aeronautical Sciences24(1), 57–63.

Woolston DS, Runyan HL and Byrdsong TA 1955 Some effects of system nonlinearities in the problem of aircraft flutter. Technical Report NACA TN‐3539, NACA.

Wright JR and Cooper JE 2015 Introduction to Aircraft Aeroelasticity and Loads 2nd edn. John Wiley & Sons, Ltd, Chichester, UK.

Yang ZC and Zhao LC 1988 Analysis of limit cycle flutter of an airfoil in incompressible flow. Journal of Sound and Vibration123(1), 1–13.

2

Nonlinear Dynamics

2.1 Introduction

This chapter will introduce the subject of nonlinear dynamics and will discuss some of its most important concepts. The analysis will focus on dynamic systems with a single degree of freedom, such as the linear harmonic oscillator and the galloping oscillator. Although the concepts addressed in this chapter can be found in other textbooks on nonlinear dynamics, the context is aeroelastic. Furthermore, the information presented here will be used in later chapters in order to analyse more realistic aeroelastic systems with many degrees of freedom and various nonlinearities.

Some of the important concepts to be discussed are common to both linear and nonlinear dynamical systems. Examples are fixed points, the phase plane, response trajectories and stability. As linear systems have analytical solutions, they will be preferred to nonlinear ones for the introduction of such concepts. The early parts of the chapter are therefore mostly devoted to linear dynamics; readers already familiar with the subject should read these sections as a revision but also as a familiarisation with the terminology that will be encountered throughout the book.

The main part of the chapter concerns nonlinear dynamics. Concepts such as multiple solutions, bifurcations and limit cycle oscillations will be introduced on simple nonlinear systems. The emphasis is not on the calculation of solutions of the equations of motion; such calculations will be presented in the next chapter. The focus here is on the characterisation of the types of response that can be observed and on simple methods for qualitative or approximate analysis.

2.2 Ordinary Differential Equations

In this chapter, we will examine nonlinear Ordinary Differential Equations (ODE) of the form

(2.1)

where x(t) is the vector of system states, t is time, f is a vector of nonlinear functions, q is a vector of system parameters and the overdot denotes differentiation with respect to time. The states x are functions of time, t, while the parameters q are constants. The overdot denotes differentiation with respect to time, that is . Notice that the system described by equation 2.1 is autonomous, that is, there is no external excitation force. The equations are completed by a set of initial conditions .

Any linear or nonlinear unforced ODE can be written in the form of equation 2.1. To demonstrate this fact, consider the equation of motion of the damped linear harmonic oscillator

(2.2)

where y(t) is the oscillator’s instantaneous displacement, m the mass, d the linear damping coefficient and k the linear stiffness coefficient. The system defined be equation 2.2 has a single degree of freedom (DOF), the displacement of the oscillator, y. By defining and , the equation of motion becomes

(2.3)

(2.4)

Defining , , where T denotes transposition, we obtain

which is an expression of the form of equation 2.1, whereby the functions f are linear. Notice that the second order and first order formulations of the linear harmonic oscillator are equivalent. The variables x1(t) and x2(t) are known as the system states and denote the system’s velocity and displacement responses, respectively.

From here on in, the term system response will be used to denote the form of x(t) for all times starting from , up to . The objective of nonlinear dynamics is the calculation of the response and its evolution as the initial conditions and parameters change values. It should be stressed that, unlike linear systems, nonlinear equations can have multiple solutions. A global analysis of a nonlinear dynamic system consists in the calculation of all the solutions of the system at all parameter values of interest. In contrast, a local analysis follows one solution of the system as the parameters vary.

The system response is the solution of the complete equations of motion 2.1. The static solutions, or fixed points, of the system are the solutions of

(2.5)

or, equivalently,

(2.6)

The fixed points are denoted by xF and are constant in time. Clearly, as and , fixed points are also solutions of equations 2.1. In some cases, the fixed points can represent the steady‐state response of the system, that is, the value reached by x(t) as . In other cases, a general system response will never subside to any one of the fixed points. Finally, if has no real solutions, there will be be no fixed points.

2.3 Linear Systems

In order to demonstrate some basic concepts of dynamic behaviour we will first consider the solution of the damped linear harmonic oscillator of equations 2.3 and 2.4. The equations can be written in matrix form as

(2.7)

where

(2.8)

The solution of equations of the form of 2.7 subject to the initial conditions are well known but will be derived here in detail because similar mathematical treatments will be used for the analysis of nonlinear systems in later chapters.

We will first evaluate the fixed points of the damped linear harmonic oscillator, that is the solution of

(2.9)

For a general A, these equations have one solution, . Therefore, autonomous linear systems have a single fixed point, the origin. As mentioned earlier, is a static solution of the complete equations of motion 2.7. Furthermore, if , the system response will be for all times.

In order to evaluate solutions of equations 2.7 from non‐trivial initial conditions, we will try to separate the variables x and t. To achieve this separation, we will make use of the matrix exponential function; the matrix exponential of A is defined as

(2.10)

where I is the unit matrix of the same size as A. This series always converges for a matrix A with finite entries. Arnold (1992) discusses in detail the definition and properties of the matrix exponential.

Equation 2.7 is written as

and pre‐multiplied by the matrix exponential of , that is, , giving

(2.11)

Using definition 2.10, it is straightforward to show that

Consequently, the left hand side of equation 2.11 is the time derivative of , so that

As the left hand side is only a function of time, this equation can be integrated in time from 0 to t, yielding

(2.12)

Consequently, the general solution becomes

or, after pre‐multiplying both sides by eAt,

(2.13)

Note that this general solution includes the fixed point; if we set , then for all times.

Expression 2.13 is the complete general solution of equation 2.7 but contains a matrix exponential. A more useful version of the solution can be obtained by eigenvalue decomposition. Consider the decomposition of matrix A into

where V is the matrix containing the eigenvectors of A as its columns and L is a diagonal matrix containing the eigenvalues of A in its diagonal. Now we note that

Substituting these results in the definition of the matrix exponential 2.10, and noting that , we get

Furthermore, the properties of the eigenvalue decomposition dictate that the eigenvectors of At are equal to the eigenvectors of A, while the eigenvalues of At are equal to Lt. Therefore,

(2.14)

For a system with n states, taking advantage of the fact that L is diagonal, equation 2.13 becomes

(2.15)

where vi is the ith eigenvector of A (i.e. the i th column of V), λi is its i th eigenvalue (i.e. the i th element of the diagonal of L) and bi is the ith element of the vector .

The eigenvalues of matrix A are the solutions to , which is a polynomial equation of order equal to the number of states n. This equations is usually referred to as the characteristic polynomial equation. For the particular case of the matrix A of equation 2.8, the eigenvalues are the solutions of the second order characteristic polynomial

(2.16)

and are given by

(2.17)

(2.18)

The eigenvector matrix is equal to

and the vector b is given by

so that the general solution becomes

(2.19)

(2.20)

The behaviour of x1(t) and x2(t) then depends exclusively on the initial conditions and the eigenvalues of A. As the eigenvalues are functions of m, d and k, we will investigate how the system response changes with these system parameters.

2.3.1 Stable Oscillatory Response

Choose the parameter values m=2 Kg, d=4 Ns/m, k=400 N/m and the initial conditions x1(0)=0 m/s, x2(0)=0.1 m. The system response, calculated from equations 2.19 and 2.20 is plotted in Figure 2.1. The calculation is carried out by means of Matlab code lindampharm.m. In fact, all examples in Sections 2.3.1 to 2.3.5 can be solved using this code, after inserting the corresponding parameter values.

2 Graphs illustrating the stable oscillatory response of harmonic oscillator. Top graph displays a fluctuating wave with dashed line. Bottom graph displays a spiral with arrowheads.

Figure 2.1 Stable oscillatory response of harmonic oscillator

Figure 2.1(a) plots the variation of x1(t) and x2(t) against time. The response is oscillatory but its amplitude decays exponentially with time towards the fixed point, ; this behaviour is usually referred to as damped response in the dynamics literature. Figure 2.1(b) plots x1(t) against x2(t), in what is known as a phase plane plot. The solution appears as a trajectory, winding clockwise from the initial condition at the far right to the centre of the phase plane, the point (0,0). The arrows in Figure 2.1(b) denote the direction of the motion.

The phase plane plot reveals that the point (0,0) attracts the system trajectory. This point has already been identified as the system’s fixed point. Fixed points of linear systems can be classified into different types, depending on the eigenvalues of matrix A. In this case, the eigenvalues are given by , that is, they are complex and have negative real parts. The fixed point is classified as a stable focus. The term ‘stable’ denotes that the fixed point attracts the trajectories and the term ‘focus’ denotes that the system response is oscillatory. Furthermore, a stable focus is characterised by the fact that the eigenvalues are complex conjugate with negative real part.

Looking at equation 2.17 and 2.18, the eigenvalues can only be complex if . In this case, the two equations can be re‐written as

where . Using Euler’s formula and the properties of the scalar exponential, it is easy to show that, for example,

We can now define a frequency and a damping factor for the time response of the system. The oscillation frequency is known as the damped natural frequency, given by

(2.21)

where is the undamped natural frequency and is the damping ratio. As d, m and ωn are all positive, . A positive damping ratio implies that the system is stable. Equation 2.20 can be re‐written as

where a1 and a2 are real coefficients that depend on the initial conditions. Also note that the natural frequency and damping ratio can be written as

(2.22)

(2.23)

where ℜ denotes the real part of a complex number. The equation of motion of the damped linear harmonic oscillator can be rewritten in terms of the natural frequency and damping ratio as

(2.24)

2.3.2 Neutral Oscillatory Response

Now change the value of the damping coefficient to . Equations 2.19 and 2.20 again give the system response, plotted in Figure 2.2. In this case, the response has constant amplitude and the fixed point does not attract the solution any more. The phase plane trajectory is a circle, still winding in a clockwise direction. The fixed point has become a centre, which means that it neither attracts nor repels the trajectories, it is simply their centre. Such points are defined by the fact that the system eigenvalues are purely imaginary and conjugate. Indeed, in this case the eigenvalues are equal to .

2 Graphs illustrating the neutral oscillatory response of harmonic oscillator, displaying a fluctuating line with dashed line (top) and a circle with arrowheads (bottom).

Figure 2.2 Neutral oscillatory response of harmonic oscillator

For neutrally stable systems the size of the circle depends on the initial conditions. Initial conditions far from the fixed point will give large circles and vice versa. It could be argued that the system admits an infinite number of solutions, just as was mentioned earlier for nonlinear systems. However, the character and frequency of all these solutions is identical; only the amplitude changes. Since , the damping ratio is also equal to zero and the system oscillates at the undamped natural frequency. Equation 2.20 can be re‐written as

(2.25)

2.3.3 Unstable Oscillatory Response

Change the value of the damping coefficient to , while keeping all the other parameters constant. The response of the system is now plotted in Figure 2.3. As usual, the solution is oscillatory but its amplitude grows exponentially with time. The phase plane trajectory winds in a clockwise direction around the fixed point but moves away from it. The fixed point is now an unstable focus, pushing away the solution. Unstable foci are defined by the fact the eigenvalues of matrix A are complex conjugate and have positive real parts. In this example the eigenvalues are equal to .

2 Graphs illustrating the unstable oscillatory response of harmonic oscillator, displaying fluctuating wave with dashed line (top) and a spiral with arrowheads (bottom).

Figure 2.3 Unstable oscillatory response of harmonic oscillator

All the solutions shown in the examples above are oscillatory but only the neutral response is periodic. In fact, as the value of d decreases, the character of the response changes from decaying oscillatory to periodic to diverging oscillatory. The mechanism for this change is the transformation of the eigenvalues from complex with negative real parts to purely imaginary to complex with positive real parts.

Looking at equations 2.17 and 2.18, the stability of the linear harmonic oscillator can be determined without actually calculating the eigenvalues. In fact, the real parts of the eigenvalues will be negative if all the coefficients of the equation of motion are non‐zero and have the same sign. If one of the coefficients has a different sign to the other two, then the system will be unstable. These statements can be verified by testing them on equations 2.17 and 2.18. They have been generalised and extended in the Routh–Hurwitz stability criteria (see e.g. Wright and Cooper 2015), which consider systems with any number of degrees of freedom. Since the value of d is negative, the damping ratio will also be negative. Therefore, a negative damping ratio implies instability but the system will still oscillate at the damped natural frequency of equation 2.21.

Example 2.1 Linear aeroelastic galloping

Galloping is an aeroelastic instability that can affect slender structures exposed in a fluid flow. Common examples of structures that are prone to galloping are electric power lines and bridge stay cables. Most slender bluff structures can gallop, irrespective of the cross‐sectional shape (circles, squares and rectangles are usually studied). A 2D cross‐section of one such structure is shown in Figure 2.4. A rectangle of height h and mass m is exposed to a uniform free stream with airspeed U and density ρ and can oscillate in the plunge direction, y, restrained by an extension spring of linear stiffness k, which represents the flexural stiffness of the slender structure. The motion of the rectangle is also resisted by a linear dashpot with damping constant d, representing the damping of the complete 3D structure. The flow of air around the rectangle causes an aerodynamic force fy(t) in the y direction. It also causes a force in the direction of the free stream but it is aeroelastically irrelevant as the system does not have a degree of freedom in this direction.

Schematic of two-dimensional rectangular cylinder with a plunge degree of freedom depicting k and d, with double-headed arrow labeled h and two arrows (rightward and downward) labeled U and y, y˙, fy.

Figure 2.4 Two‐dimensional rectangular cylinder with a plunge degree of freedom

Appendix A.1 and Section 8.3 discuss the development of linear and nonlinear mathematical models for the galloping oscillator. The linear equation of motion is given by

(2.26)

where is a real constant coefficient. The aerodynamic term is a linear function of and is therefore a damping force, additional to the structural damping term. See Section 8.3 for more details about the value of . Here, we are only demonstrating how the total damping in an aeroelastic system can take negative values. Equation 2.26 is an aeroelastic version of the damped linear harmonic oscillator. Assuming that m and k are both positive and that the total damping is low, the response of the square cylinder is stable oscillatory as long as

that is, the total damping is positive. If , there is a critical value of the airspeed, Ug, at which the total damping becomes equal to zero,

and the response is neutral oscillatory. Finally, for all airspeeds , the response becomes unstable oscillatory. This phenomenon is known as galloping. A nonlinear form of equation 2.26 will be treated later in this chapter.

2.3.4 Stable Non‐oscillatory Response

We return to the linear harmonic oscillator of Section 2.3.1 and increase the value of the damping coefficient to . The resulting system response is plotted in Figure 2.5. Clearly, the form of both x1(t) and x2(t) is not oscillatory; the states approach the fixed point as but never cross the time axis.

2 Graphs illustrating the stable non-oscillatory response of harmonic oscillator, displaying two curves (top) and a concave up with arrowheads (bottom).

Figure 2.5 Stable non‐oscillatory response of harmonic oscillator

The phase plane plot of Figure 2.5(b) is markedly different from all the previous phase plane plots. In these earlier plots, the trajectory moved over all four quadrants of the phase plane. In Figure 2.5(b) the trajectory only moves in one of the quadrants before ending up on the fixed point. This behaviour is a defining characteristic of non‐oscillatory motion: the response never completes a full circuit around the fixed point. In the case of stable non‐oscillatory motion, the fixed point is known as a stable node. Nodes are defined by the fact that all eigenvalues of A are real and negative. In this case the eigenvalues are equal to and .

The term node can be visualised more clearly by plotting the responses from a number of different initial conditions in the phase plane. In Figure 2.6 the response of the overdamped linear harmonic oscillator is plotted from 22 different sets of initial conditions, ranging from to . All these trajectories tend towards the fixed point at the origin, following the direction of the eigenvector corresponding to the eigenvalue that is closest to zero. In this case, the eigenvalue closest to zero is and the corresponding eigenvector is so that the trajectories approach the fixed point along the line

Away from this line, the trajectories bend towards a direction parallel to the other eigenvector, , that is, parallel to the line

The fixed point is called a node because all of the trajectories meet there, approaching along the same line.

Graph illustrating phase plane plot of many response trajectories attracted to a stable node depicted by curves (solid line) having arrowheads, with a dashed line.

Figure 2.6 Phase plane plot of many response trajectories attracted by a stable node

In the theory of vibrations, the motion depicted is Figure 2.5 is known as overdamped. The term implies the fact that the damping coefficient is larger than the critical value required to render the eigenvalues real. From equations 2.17 and 2.18, this critical value is . For all higher values of d the damping ratio becomes greater than 1 and the concept of frequency no longer applies. Equations 2.22 and 2.23 can no longer be used.

2.3.5 Unstable Non‐oscillatory Response

Now change the value of the damping coefficient to . The eigenvalues become and . The corresponding eigenvectors are [1 0.1]T and [1 0.05]T. The phase portrait of this system is similar to the one shown in Figure 2.6 but the trajectories move away from the fixed point and the slope of the eigenvectors’ directions has changed. The fixed point is an unstable node, characterised by the fact that both the eigenvalues are real and positive. Although the eigenvalues are real, , and therefore the motion cannot be called overdamped.

There is another type of unsteady non‐oscillatory system. An example can be obtained from the linear harmonic oscillator by setting the parameters to , and . The eigenvalues of A are and . The corresponding eigenvectors are and [1 0.0759]T. The fixed point is known as a saddle, defined by the fact that the eigenvalues are real but of opposite sign.

The phase portrait for this system is shown in Figure 2.7. Any trajectories close to the line , which corresponds to the stable eigenvector will initially move towards the fixed point. However, as they approach the fixed point, they also approach the line , which corresponds to the unstable eigenvector. They are then pushed away towards infinity in this direction. Only trajectories that actually start on end up at the fixed point. Nevertheless, the system is considered to be unstable, since general trajectories will end up at infinity.

Graph illustrating phase plane plot of many response trajectories around a saddle depicted by curves (solid lines) having arrowheads, with 2 crossing dashed lines.

Figure 2.7 Phase plane plot of many response trajectories around a saddle

Example 2.2 Static divergence

Negative stiffness can occur in aeroelastic systems undergoing static divergence. The aerodynamic stiffness becomes greater than the structural stiffness and acts in the opposite direction. The critical condition is, of course, , whereby the system has one positive real and one zero eigenvalue.

Consider the pitching wing section of Figure 2.8, a 2D flat plate of mass m and chord c, free to pitch around its pitch axis, xf, in a free stream of speed U and density ρ. The pitch angular displacement is denoted by α and is restrained by a torsional spring with stiffness Kα. The moment of inertia of the wing around its flat plate is Iα. The equation of motion of this system is derived in Appendix A.5 and is given by

(2.27)

where and is the non‐dimensional distance between the flexural axis and half‐chord. Clearly, this aeroelastic equation of motion is a damped linear harmonic oscillator of the form of equation 2.2 with stiffness term equal to

. Thin airfoil theory predicts that the aerodynamic centre of a static wing section lies on its quarter‐chord c/4. The aerodynamic moment around the pitch axis, is stabilising (i.e. nose‐down) if the pitch axis lies in front of the aerodynamic centre (i.e. ) and destabilising (i.e. nose‐up) if . Furthermore, the pitching moment is equal to zero when the flexural axis lies on the aerodynamic centre, that is, .

Schematic illustration of two-dimensional wing section with a pitch degree of freedom, with a single torsional spring, 2 rightward arrows labeled Xf and C, and a dashed horizontal line labeled Kα.

Figure 2.8 Two‐dimensional wing section with a pitch degree of freedom

It follows that, if , the total stiffness is always positive and the fixed point is a focus. However, assuming that , the total stiffness becomes equal to zero when , that is, when the airspeed is such that the aerodynamic stiffness balances the structural restoring force. This phenomenon is known as static divergence. The critical static divergence airspeed is given by

(2.28)

All airspeeds equal to or higher than this value are unsafe. Aeroelastic divergence is therefore associated with a change in the character of the fixed point, from a stable focus to a saddle. As will be shown later in this chapter, the same change can occur in nonlinear systems. Aeroelastic divergence will be discussed more extensively later in this book.

2.3.6 Fixed Point Summary

The previous discussion shows that, for linear systems, every type of response (stable or unstable, oscillatory or non‐oscillatory) is associated with a particular type of fixed point. Therefore, identifying the type of fixed point will directly give information about the stability of the system and the type of response to be expected. Table 2.1 summarises the different categories of fixed points and the corresponding responses.

Table 2.1 Summary of fixed points

Example 2.3 Can the fixed point of the pitching wing section of equation 2.27 ever become an unstable focus?

For the fixed point to be an unstable focus, the eigenvalues must be complex with positive real parts. Therefore,

where p and q are both real and positive. These eigenvalues are the solutions of the polynomial

or, after carrying out the multiplications,

Comparing this equation with the characteristic polynomial of equation 2.16, we obtain

Mass is the measure of inertia and, therefore, always positive. Consequently, for the fixed point to be an unstable focus, the damping coefficient must be negative and the stiffness coefficient must be positive. Applying these two conditions to the aeroelastic equation 2.27 we get

The first condition results in . Therefore, the damping of the quasi‐steady pitching wing section can be negative if the flexural axis is positioned between c/2 and 3/4c. It should be stressed that the conclusion above is due to the quasi‐steady aerodynamic assumptions. In fact, unsteady aerodynamic analysis shows that the fixed point of this system is a stable focus for but can become unstable if Iα is large and the flexural axis lies close to the leading edge or even in front of the leading edge. This instability is known as single‐degree of freedom flutter (Runyan 1951).

2.4 Nonlinear Systems

The general solution of linear systems (i.e. equations 2.13 or 2.15) gives a complete description of all system responses for all parameter values and initial conditions. Unfortunately, such solutions do not exist for the vast majority of nonlinear systems. In the next chapter, we will discuss how to obtain numerical solutions of nonlinear systems for a particular set of initial condition and parameter values. Nevertheless, a lot of interesting information about nonlinear systems can be obtained without actually solving for the time response. In this chapter we will discuss some mathematical tools available for obtaining estimates of the stability and response type of a nonlinear system.

Let us repeat the equation of motion of a general nonlinear ODE

The fixed points of this system can be obtained in the same way as for the linear system discussed in the previous section, that is, by setting and looking for the solutions of

(2.29)

This calculation is a nonlinear algebraic problem and can be solved using suitable analytic or numerical methods. Depending on the form of f, there may be zero, one or more solutions. It is even possible that the number of solutions may not be known. There may also be an infinite number of solutions.

Example 2.4 Calculate the fixed points of the simple pendulum

where θ is the instantaneous angle of the pendulum, l its length and g the acceleration due to gravity. In first order form the equation of motion becomes

where , and

Setting yields two independent equations, and . The first equation has an infinite number of solutions, while the solution of the second equation is simply 0. So the fixed points of the system are given by

for all values of the integer k (which are, of course, infinite). In practical terms only the fixed points occurring for are of interest since they define angles and . All the other fixed points coincide geometrically with one of these two.

The fact that nonlinear systems can have an unknown or infinite number of fixed points means that a lot of nonlinear system analysis is local in nature. In other words, a number of fixed points of interest are chosen and the solution of the system is examined in the neighbourhood of these fixed points.

2.4.1 Linearisation Around Fixed Points

For many nonlinear systems, local stability analysis can be carried out by linearising them around their fixed points. If we assume that f(x) is infinitely differentiable at xF, we can expand it as a Taylor series around xF, that is

(2.30)

where and

(2.31)

is known as the system’s Jacobian matrix,

(2.32)

for are the higher order derivatives of f with respect to x and the notation signifies that the derivatives are calculated at . As x approaches xF, the second and higher order terms become negligible compared to the first order term and we can approximate the nonlinear function as

(2.33)

in the neighbourhood of the fixed point. The equations of motion become

(2.34)

in the same neighbourhood, that is, they are linearised. Now, if we use the change of variable , the linearised equations become

(2.35)

that is, they are now linear homogenous first order equations of the form of expression 2.7, for which we have already determined the solution. This is a very important result because it means that the stability of a nonlinear system near its fixed points can be deduced by solving the linearised system. Furthermore, the stability of the fixed points can be determined by looking at the eigenvalues of the Jacobian . The system of equations 2.35 will be referred to as the underlying linear system around fixed point xF.

This mathematical treatment of nonlinear systems is known as linearisation and is by far the most widely used analysis method for such systems. In fact, it could be argued that most engineering analysis is a process of linearisation as engineering materials behave linearly within a narrow operating range. Linearisation allows us to expect that the stability of a nonlinear system around a fixed point is identical to the stability of the linearised system in the same neighbourhood. However, there are cases in which we cannot apply this principle. Here are some examples:

has no solutions, that is, there are no fixed points. No linearised system can be defined

f(x) is not differentiable at xF. The Jacobian and higher order derivatives cannot be evaluated.

. The stability of the nonlinear system depends on the higher order terms of the Taylor series, that is, it is impossible to linearise the system.

Despite these exceptions, linearisation is the ‘premier method for the determination of the stability’ of nonlinear systems around their fixed points (Chicone 1999).

Example 2.5 Linearise the simple pendulum around its fixed points.

From Example 2.4, the nonlinear function of the simple pendulum is

and its fixed points and . The Jacobian and higher derivatives are calculated by differentiating the nonlinear function and evaluating them at the fixed points. Starting with ,