Introduction

In this chapter, more advanced problems of finite deformation (geometric nonlinearity) of shells and plates are considered. Initially, Donnell's and Novozhilov's nonlinear theories for doubly curved shells with constant curvature are presented. Then, the classical theory for thin shells of arbitrary shape is presented, which makes use of the theory of surfaces. Composite, sandwich and innovative functionally graded materials are introduced in the next section. In order to deal with these special materials and with moderately thick shells, nonlinear shear deformation theories are introduced. These theories, formulated for shells, can easily be modified to be applied to laminated, sandwich and functionally graded plates by setting the surface curvature equal to zero. Finally, the effect of thermal stresses is addressed.

Doubly Curved Shells of Constant Curvature

A doubly curved shell with rectangular base is considered, as shown in Figure 2.1. A curvilinear coordinate system (O; x, y, z) having the origin O at one edge of the panel is assumed; the curvilinear coordinates are defined as x = ψ Rx and y = ϑ Ry, where ψ and θ are the angular coordinates and Rx and Ry are principal radii of curvature (constant); a and b are the curvilinear lengths of the edges and h is the shell thickness. The smallest radius of curvature at every point of the shell is larger than the greatest lengths measured along the middle surface of the shell.

Introduction

Thin-walled circular cylindrical shell structures containing or immersed in flowing fluid may be found in many engineering and biomechanical systems. There are many applications of great interest in which shells are subjected to incompressible or subsonic flows. For example, thin cylindrical shells are used as thermal shields in nuclear reactors and heat shields in aircraft engines; as shell structures in jet pumps, heat exchangers and storage tanks; as thin-walled piping for aerospace vehicles. Furthermore, in biomechanics, veins, pulmonary passages and urinary systems can be modeled as shells conveying fluid.

Circular cylindrical shells conveying subsonic flow are addressed in this chapter; they lose stability by divergence (which is a static pitchfork bifurcation of the equilibrium, exactly as buckling) when the flow speed reaches a critical value. The divergence is strongly subcritical, becoming supercritical for larger amplitudes. It is very interesting to observe that the system has two or more stable solutions, related to divergence in the first or a combination of the first and second longitudinal modes, much before the pitchfork bifurcation occurs. This means that the shell, if perturbed from the initial configuration, has severe deformations causing failure much before the critical velocity predicted by the linear threshold. In particular, for the case studied in this chapter, the system can diverge for flow velocity three times smaller than the velocity predicted by linear theory, indicating, in this case, the necessity of using a nonlinear shell theory for engineering design.

Introduction

Most of studies on large-amplitude (geometrically nonlinear) vibrations of circular cylindrical shells used Donnell's nonlinear shallow-shell theory to obtain the equations of motion, as shown in Chapter 5. Only a few used the more refined Sanders-Koiter or Flügge-Lur'e-Byrne nonlinear shell theories. The majority of these studies do not include geometric imperfections, and some of them use a single-mode approximation to describe the shell dynamics.

This chapter presents a comparison of shell responses to radial harmonic excitation in the spectral neighborhood of the lowest natural frequency computed by using five different nonlinear shell theories: (i) Donnell's shallow-shell, (ii) Donnell's with in-plane inertia, (iii) Sanders-Koiter, (iv) Flügge-Lur'e-Byrne and (v) Novozhilov theories. These five shell theories are practically the only ones applied to geometrically nonlinear problems among the theories that neglect shear deformation. Donnell's shallow-shell theory has already been used in Chapter 5, and the numerical results presented there are used for comparison. Shell theories including shear deformation and rotary inertia are not considered in this chapter. The results presented are based on the study by Amabili (2003).

Energy Approach

The elastic strain energy of the shell is given by equation (1.141), in which the expressions of the middle surface strain-displacement relationships and changes in curvature and torsion must be inserted according to the selected nonlinear shell theory.

The preceding developments suffice to treat systems that are described by a finite number of degrees of freedom. They are not directly applicable if a system is best modeled as a flexible continuum, in which bodies deform and also have mass. One cannot compartmentalize kinetic and potential energy in such systems, because a mass element also stores strain energy. Consequently, concepts like generalized coordinates become problematic. The derivation of principles that can be used to model continuous media is the first priority for this chapter.

Another focus here is exploration of alternative formulations for deriving equations of motion for discrete systems. Derivation of these formulations has received considerable attention for more than a century and a half. Those efforts were motivated by a desire to seek simpler equation forms, either from the perspective of ease of implementation or ease of solution. We consider a few formulations, but extensive discussions may be found in the works by Greenwood (2003) or Papastavridis (1998, 2002).

One of the outcomes of these alternative formulations are conservation principles that sometimes can be used when the standard momentum and energy principles cannot be implemented. Such principles enable us to determine features of a system's response without solving equations of motion and also provide checks for computation solutions. Overall, the developments that follow are intended to enhance understanding of the basic concepts of analytical mechanics and to provide increased versatility to carry an analysis to completion.

It has been more than a decade since the second edition of Advanced Engineering Dynamics was published. Although I was pleased with that effort, my experience teaching dynamics with that book as a companion has given me insights that I either did not have or did not fully appreciate. I tried to satisfy multiple objectives as I wrote the present book. I wished to convey both physical and analytical understanding of the fundamental principles, and to expose the beauty of the discipline as a tightly woven sequence of concepts. I wanted to address the complexities of real-world engineering problems and explore the implications of dynamics for other subjects, but to do so in a manner that is accessible to those who come to it from a wide range of experiences. I wanted to provide a self-contained resource from which the motivated reader could learn directly. At first, I thought this book would just be a third edition of Advanced Engineering Dynamics, but as I progressed, I realized that the expanded scope and the amount of material that is either new or redone necessitated treating it as a new work.

The subject of dynamics is an interdisciplinary blend of physics, applied mathematics, computational methods, and basic logic. The least difficult aspect of the subject is the basic physical laws, most of which are at least a century old. A primary element that has moved the study of dynamics from natural philosophy to engineering is the development of powerful tools for describing motion and for solving equations of motion.

The previous chapter focused on describing and understanding the variability of angular momentum. We now apply those concepts to relate the motion of a system to the forces driving that motion. The formulation is based on the linear and angular momentum principles of Newton and Euler. These principles govern the motion of a single rigid body, but practical applications feature many bodies. In such situations, individual equations of motion may be written for each body. If one pursues such an analysis, careful attention must be given to accounting for the forces exerted between bodies, so the construction of free-body diagrams will play a prominent role in this chapter's development. As a supplement to this approach, a following section develops a momentum-based concept for systems of rigid bodies that sometimes can lead to the desired solution without considering all of the interaction forces. Ultimately, the energy-based concepts associated with Lagrange, whose development is taken up in the next chapter, provide a more robust alternative approach. However, they are mathematical in nature and afford little physical insight. For this reason, particular attention is given here to providing physical explanations for the results derived from the Newton–Euler formulation of equations of motion.

The concept of a rigid body is an artificial one, in that all materials deform when forces are applied to them. Nevertheless, this artifice is very useful when we are concerned with an object whose shape changes little in the course of its motion. In addition, it often is convenient to decompose the motion of a flexible body into rigid-body and deformational contributions.

Most engineering systems feature bodies that are interconnected. Each body must move consistently with the restrictions imposed on it by the other bodies. We refer to these restrictions as constraints. Constraint conditions are the kinematical manifestations of the reaction forces. Indeed, a synonym for reactions is constraint forces. A keystone of analytical dynamics, whose treatment begins in Chapter 7, is the duality of constraint forces and constraint conditions, which enable us to describe one if we know the other. However, in a kinematics analysis one is not concerned with the forces required to attain a specified state of motion.

GENERAL EQUATIONS

When an object is modeled as a rigid body, the distance separating any pair of points in that object is considered to be invariant. This approximation is quite useful because it leads to greatly simplified kinematical and kinetic analyses. Because the distance between points cannot change, any set of coordinate axes xyz that is scribed in the body will maintain its orientation relative to the body.

Chasle's theorem states that the general motion of a rigid body can be represented as a superposition of a translation following any point in a body and a pure rotation about that point. The kinematics tools we have developed provide the capability to describe these motions in terms of a few parameters. In this chapter we begin to characterize the relationship between forces acting on a rigid body and kinematical parameters for that body. The resultant of a set of forces may be regarded intuitively as the net tendency of the force system to push a body, so one should expect it to be related to the translational effect. Similarly, it is reasonable to expect that the resultant moment of a set of forces represents the rotational influence. We shall confirm and quantify these expectations.

From a philosophical perspective, the shift from statics, in which one equilibrates forces, to kinetics, in which the forces must match an inertial effect, is rather drastic. For a particle, Newton's Second and Third Laws are readily understood in this regard. However, the corresponding shift for the rotational effect will lead to effects associated with the angular momentum of a rigid body that sometimes are counterintuitive. This is especially true for those who try to examine spatial motion from a planar motion viewpoint. This chapter focuses on the determination and evaluation of angular momentum.

Any formulation of equations of motion requires characterization of the role of the physical restrictions that are imposed on a system's movement. These restrictions lead to kinematical relations between motion variables, and they also are manifested as reaction forces. When a system consists of interconnected bodies, the standard Newton–Euler formulation isolates individual bodies. The need to account for the kinematical constraints and corresponding reaction forces associated with each connection substantially enhances the level of effort entailed in deriving equations of motion.

The Lagrangian formulation implicitly recognizes the dual role of motion constraints. Indeed, it is recognition of this duality that has made it preferable to use the term constraint force rather than reaction. A primary benefit of the Lagrangian formulation is the ability to automatically account for constraint forces in the equations of motion. The formulation will allow us to treat connected bodies as a single system, rather than individual entities. The primary kinetic quantity for Lagrange's equations of motion is mechanical energy (kinetic and potential), whereas the Newtonian equations of motion are time derivatives of momentum principles.

The term analytical mechanics, which encompasses the developments of Lagrange, Hamilton, and many others who followed Euler, refers to the fact that the procedures that we shall develop are more mathematical than those of Newton and Euler. They also are more abstract. In fact, we often will find that features of the equations of motion, as well as of the physical responses predicted by those equations, are most readily explained in terms of Newton–Euler concepts.

This chapter develops some basic techniques for describing the motion of a point and therefore of a particle. The procedures we follow are driven not merely by how the point's motion is described, but also by the information we seek. Each formulation is based on describing vector quantities with respect to a different set of unit vectors. Which description is best suited to a particular situation depends on a variety of factors, but a primary consideration is whether the associated quantities, such as the coordinates, naturally fit the known aspects of the motion. Ultimately we will find that it might be beneficial to combine a variety of descriptions.

The various kinematical description that we use fall into two general classes. The one that might seem to be the most natural is extrinsic coordinates, which means that the description is extrinsic to knowledge of the path followed by the point. A simple case is rectangular Cartesian coordinates, for which the position is know in terms of distances measured along three mutually orthogonal straight lines representing reference directions. A variety of other extrinsic coordinate systems are in use. However, we begin by studying intrinsic coordinates, in which knowledge of the path is fundamental to the description of the motion. For example, the unit vectors for intrinsic coordinates are defined in terms of the properties of the path. For this reason, intrinsic coordinates are more commonly referred to as path variables.

Since ancient times many researchers have devoted themselves to predicting and explaining how bodies move under the action of forces. This is the scope of the subject of dynamics, which consists of two phases: kinematics and kinetics. A kinematical analysis entails a quantitative description of the motion of bodies without concern for what is causing the motion. Sometimes that is all that is required, as would be the case if we needed to ascertain the output motion of a gear system or linkage. More significantly, a kinematical analysis will always be a key component of a kinetics study, which analyzes the interplay between forces and motion. Indeed, we will see that the kinematical description provides the skeleton on which the laws of kinetics are applied.

A primary objective will be the development of procedures for applying kinematics and kinetics principles in a logical and consistent manner, so that one may successfully analyze systems that have novel features. Particular emphasis will be placed on three-dimensional systems, some of which feature phenomena that are counterintuitive for those whose experience is limited to systems that move in a plane. A related objective is development of the capability to address realistic situations encountered in current engineering practice.

The scope of this text is limited to situations that are accurately described by the classical laws of physics. The only kinetics laws we will take to be axiomatic are those of Newton, which are accurate whenever the object of interest is moving much more slowly than the speed of light.

The basic development of Lagrange's equations in the preceding chapter is suitable for many important engineering applications. However, up to now these equations have only been employed when the generalized coordinates constitute an unconstrained set. The first part of this chapter removes this limitation. For nonholonomic systems the use of constrained generalized coordinates is mandatory. However, it might be desirable to use constrained coordinates to analyze holonomic systems, as will be seen. This is the situation when the effect of sliding friction is an important feature, which will be treated in depth.

Regardless of whether one follows the Lagrangian or Newton–Euler approach, derivation of the differential equations of motion is only the first phase of a dynamic analysis. Solution of those equations to simulate a system's response is usually the ultimate objective. As several examples have already demonstrated, the equations of motion can be quite complicated, and therefore not amenable to analytical solution. The basic state-space approach to solving the differential equations of motion associated with holonomic systems was developed in Section 7.6. The occurrence of constraint equations and Lagrange multipliers requires modification of that formulation. The second part of this chapter develops and implements several numerical algorithms that may be used to solve the equations of motion governing constrained generalized coordinates.

LAGRANGE'S EQUATIONS–CONSTRAINED CASE

Lagrange's equations for unconstrained coordinates constitute a set of differential equations of motion whose number equals the number of generalized coordinates.