Principles of Aeroelasticity

Rama B. Bhat

• Publisher: CRC Press
• Year: 2018
• ISBN:
  9781498724784
• Edition: 1
• Topic: Sciences physiques
• Subtopic: Mécanique

Introductory Guide on the Design of Aerospace Structures

Developed from a course taught at Concordia University for more than 20 years, Principles of Aeroelasticity utilizes the author's extensive teaching experience to immerse undergraduate and first-year graduate students into this very specialized subject. 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. He also provides approximate methods for solving divergence, flutter, response and stability of structures, and addresses non-aeroelastic problems in other areas that are similar to aeroelastic problems.

Topics covered include:

• The fundamentals of vibration theory
• Vibration of single degree of freedom and two degrees of freedom systems
• Elasticity in the form of an idealized spring element
• Repetitive motion
• Flutter phenomenon
• Classical methods, Rayleigh-Ritz techniques, Galerkin's technique, influential coefficient methods, and finite element methods
• Unsteady aerodynamics, and more

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1

Introduction

In studying the mechanics of systems, we always start with the equilibrium of objects under the action of external forces. When the system is in motion, we also employ Newton’s laws of motion, which relate the motion with the cause of motion, which is the force. We start the study with simple systems such as particles, systems of particles, and rigid bodies, which are systems of particles held rigidly together. When we extend the study of dynamics of systems to include the elastic property of the systems also, we lump such elasticity in the form of idealized spring elements and analyze the repetitive motion of the system about its equilibrium position, which is vibration.

In reality, the mechanical systems that we come across, particularly in engineering applications, are not that simple. The mass is distributed over the entire system and so is the property of elasticity. These are termed as continuous systems. In addition to the mass and elasticity, the system also possesses a distributed damping property. In aerospace applications, the weight of the system is very crucial, and we always try to minimize the weight. Naturally, as we try to minimize the weight, the structure becomes more flexible and the strength considerations are extremely important.

When the external forces acting on the mechanical system are independent of the deformation in the system, we can employ the techniques in the classical theory of elasticity in order to study the deformation of the system under the action of the external forces. However, in aerospace applications, the aerodynamic loads acting on the structure depend on the structural deformations also. As an aircraft flies against the wind, the wind load on the aircraft wing will depend on the attitude angle of the wing heading against the wind. The wind load will deform the wing torsionally, and this increases the attitude angle with a consequent increase in the wind load. This situation may lead to an unstable situation leading to the failure of the wing to withstand the wind loads.

We have seen flags and boat sails fluttering in the wind. Tall palm trees sway back and forth in steady winds, as do chimney stacks. High-tension electrical transmission lines with very long spans oscillate slowly in steady winds and start galloping if the wind velocity is quite large. Flexible pipes carrying fluids can start humming due to its oscillations even when the flow velocity is constant. Suspension bridges are subject to buffeting oscillations in steady winds. All of these fall into the same class of phenomena, which are aeroelastic in nature.

Aeroelasticity is the study of the effect of aerodynamic forces on elastic bodies. In the area of elasticity or vibrations, the loads are assumed to be unaffected by the deformation or motion of the structure. However, the aerodynamic forces depend critically on the attitude of the body relative to the flow. The elastic deformation plays an important role in determining the external loading itself.

Stability of a structure in wind is an important consideration in aeroelasticity. For a given configuration of the elastic body, the aerodynamic force increases rapidly with the wind speed; however, the elastic stiffness is independent of the wind. Hence, there may exist a critical wind speed at which the structure becomes unstable. Such instability may cause excessive deformations and may lead to the destruction of the structure. This is termed a divergence failure.

Flutter stability is another important consideration in aeroelasticity where the amplitude of time-dependent structural deformations under dynamic conditions continues increasing, leading to structural failure.

In the ensuing discussion on aeroelasticity, Chapter 1 provides some ideas on the elementary aerodynamics in order to get an understanding of the types of loads acting on the aircraft structure. Then a simple analysis to illustrate the phenomenon of divergence is carried out on a one degree of freedom model of an aircraft wing. A complete analysis of an aircraft structure must consider the fuselage, wings, tail structure, etc. and can only be carried out using a finite element analysis. However, our interest is limited to the study of the interaction of aerodynamic forces and the elastic structure, and this is predominantly seen in the wings. Hence, we consider the fuselage as a rigid support and consider the wings as cantilever appendages fixed to the fuselage. Since the divergence phenomenon is predominantly due to the torsional response of the wings, we illustrate this phenomenon on a single degree of freedom model of the wing where the torsional stiffness of the wing is idealized in the form of a torsional spring attached at the elastic axis of the wing. The effect of adding a control surface to the wing is illustrated on the same simple model.

Before starting to discuss flutter, fundamentals of vibration theory are introduced using single degree of freedom and two degrees of freedom systems. Subsequently, flutter phenomenon is discussed using a two degrees of freedom system model for the aircraft wing. Dynamic response of aircraft wings under wind loads, in general, is discussed after this.

An aircraft structure is a flexible, continuous system in reality. At this stage, the wing is treated as a continuous system and the motion is described by differential equations. Exact methods and approximate methods of analyzing these differential equations are discussed. Classical methods, Rayleigh-Ritz techniques, Galerkin’s technique, influence coefficient method, and finite element methods are discussed.

Some nonairfoil types of aeroelastic problems are discussed in Chapter 10. Pipes with fluid flow and plates with fluid flow are some examples.

2

Elementary Aerodynamics

General Concepts

Whenever wind blows past a stationary obstacle, the obstacle is subjected to a force. A blowing wind is able to rustle the leaves on a tree, while a strong wind can even uproot a tree. Stationary structures such as chimney stacks, tall buildings, bridges, electrical transmission lines, and broadcast transmission towers are subjected to forces from blowing winds.

An object moving in atmosphere has to move against a stationary mass of air, which prevents it from moving. The moving object has to overcome this resistance. This resistance from the stationary mass of air is similar to the force exterted by blowing wind on stationary obstacles. Hence, objects falling through atmospheric air, automobiles traveling on highways, and airplanes moving through ambient masses of atmospheric air are all subject to wind loads.

In order to understand the types of loads acting on aircraft structures, we will have to understand the behavior of air flow over obstacles.

Stagnation Point

Whenever air flows around a body, part of it goes to one side and the rest to the other side. The streamline that follows the flow division (that divides on the upstream side of the body and joins again on the downstream side) is called the dividing streamline. At the point of division, the velocity of air relative to the body is zero. This point is called the stagnation point.

Ideal-Flow Theory

Ideal-flow theory stipulates that the fluid is incompressible and nonviscous. A differential equation fundamental to the ideal-flow theory is Laplace’s equation. Many useful insights in aeroelasticity can be obtained using this theory. When speeds are small compared to the speed of sound in a fluid, the fluid can be considered incompressible. Except in a small layer (called “boundary layer”) close to the ...