As presented in terms of Newton's laws of motion, our notion of (dynamic) equilibrium is that “the sum of the forces is equal to mass times acceleration” applied to a free-body diagram. This does not generalize well to large, complex systems (with many DoFs), particularly if they are nonlinear. The concepts of virtual work and virtual energy allow us to present a formulation for the equilibrium of discrete systems that does not entail the use of summing forces on free-body diagrams. This formulation, usually referred to as analytical mechanics, generalizes very nicely for the many-body systems we wish to explore. It is quite versatile because it is applicable to nonlinear, nonconservative, and dynamic problems alike and is very suitable for numerical implementation. Furthermore, our complex systems, by necessity, involve approximations in the structural modeling, and our formulation is quite compatible with this necessity.

We first develop the principle of virtual work and then, through D'Alembert's principle, apply it to dynamic systems. The resulting governing equations are general enough to handle nonlinear systems. The different types of nonlinear contributions are then clarified. The work and energy ideas of this chapter are covered extensively in references 52, 54 and 90, and the style of presentation follows reference 30.

Principle of Virtual Work

Our idea of a structure is an object with multiple parts. We use the multimember structure of Figure 2.1 to illustrate some of the basic concepts in structural mechanics.

A plate is an extended body where one of the dimensions is substantially smaller than the other two. The plates in three-dimensional (3D) thin-walled structures (called shells and folded-plate structures) can support both in-plane and out-of-plane loading. Furthermore, because the plates are thin, they lend themselves to approximation – while the structure may be three-dimensional, the local behavior is two-dimensional under plane stress. Plates in flexure are the two-dimensional equivalent of beams, and classical plate theory is the equivalent of the Bernoulli-Euler beam theory, whereas the in-plane or membrane behavior of plates is analogous to that of rods. We consider both of these separately.

We first develop the strong formulation for plates and present their spectral forms. It turns out, however, that very few strong-form solutions are available for arbitrary BCs, but there is one class with periodic BCs where very useful results can be obtained. We concentrate on these.

Flexural Behavior of Flat Plates

We develop a thin-plate model (called classical plate theory) that does not take the transverse shear deformation into account – this is therefore the plate counterpart of the Bernoulli-Euler beam. We could develop a model (called the Mindlin plate theory) that takes the shear deformation into account; however, it is worth keeping in mind that if shear effects in the plate might be important, then the plate should be modeled using the 3D solid Hex20 element.

We investigated the dynamic equilibrium of various structural systems in the previous chapters. Now we consider a very important property of systems in equilibrium, namely, their stability. Our intuitive notion of stability is that small perturbations of a system (which are always present in real situations) only cause the system to have small motions about its (dynamic) equilibrium configuration. If the small perturbations cause motions that are excessive, we talk of an instability of the motion. Some situations where the stability of the motion arises are aeroelastic flutter, whirling of shafts, rotating saw blades and computer disks, belt drives, galloping of power lines, and control of structures.

Underlying this is the idea of a dynamic equilibrium position; although the system is in motion, it is stable in the sense that any disturbance eventually dies down, and the system returns to that dynamic state. This is a concept we make clear at the beginning.

In truth, instability is a nonlinear dynamic phenomenon because we are talking of structural behavior changing – the system is in one state and then moves to another. This new state could be near or far (in the phase-plane sense), and that informs the nature of the instability. Thus we also need to make clear the difference between the initiator of the instability and where the system goes.

Some Preliminary Stability Ideas

Central to the idea of instability is the concept of an unfolding parameter.

Our solution method of choice is to discretize all problems and solve the resulting simultaneous equations numerically. Sometimes deeper insights into a problem can be obtained by solving the continuous problem directly. This is not always feasible, does not generalize very well, and typically is restricted to linearized systems, but when it can be accomplished, the results can be very rewarding. This chapter therefore develops this aspect of dynamic analysis.

The first task is to derive adequate dynamic models to describe continuous systems; we use Hamilton's principle in conjunction with the Ritz method to derive these models in a consistent rational way. Section 3.1 developed the energies for a number of structural components. Hamilton's principal is used to convert these energy representations into a set of governing differential equations plus the associated boundary conditions. This is called the strong formulation of problems.

The derived models are in the form of a system of partial differential equations. Partial differential equations are notoriously difficult to solve in general; we introduce spectral analysis as a powerful tool for simplifying and solving problems arising in the analysis of continuous systems. In essence, dynamic problems are reduced to a series of pseudostatic problems, and thus they are amenable to the solution procedures that are standard for static problems.

Strong Formulation of Problems

The strong formulation of a problem comprises the set of governing equations plus the appropriate geometric and natural boundary conditions.

Real structures, of course, are not just a collection of springs and concentrated masses as modeled in the two preceding chapters; they are assembled systems comprising members such as panels and beams that have continuously distributed properties of mass and elasticity. Our analytical mechanics formulation is now extended to include these systems by replacing them with a discretized representation. The process of replacing a continuous body with a discrete representation is inherently approximate, but the principle of virtual work in conjunction with the Ritz method allows us to do this in a rational way. Furthermore, the approach adopted has the built-in capability to achieve any desirable level of accuracy. The ultimate computer implementation of the approach is in the form of the finite-element (FE) method.

The finite-element method is covered extensively in references such as 5, 18, 22, 30, 96, and 99 so only the aspects most relevant to the dynamics of structures are covered here. The principal structural types covered are frames and solids; structures such as plates and shells are rendered as special cases of three dimensional (3D) solids. The frame is modeled as having the separate actions of axial extension, bending, and torsion, and a FE model is developed for each of these. The dynamics of a general solid is modeled with the very versatile Hex20 element.

Our method of choice for solving general problems is the finite-element (FE) method. The drawback of the method is that it does not directly give insights into the phenomena being studied. To help overcome this, Part II introduces additional analysis methods. The choice of analysis depends on the type of problem being considered; therefore, this chapter is an attempt to classify the problem types encountered in structural dynamics so as to set the context for the types of analyses introduced. We find it most useful to use the loading (both its variation in time and its variation in space over time) as the main discriminator between the different problem types and introduce appropriate analysis methods accordingly. In broad terms, we distinguish between low-frequency loads that cause large deformations and higher-frequency loads that cause nonlinear responses but no significant change of geometry. This facilitate using the linearized versions of problems as a scheme to classify them.

We begin by surveying an array of problems based on the types of loading. What they have in common is that there are interactions between one or more bodies. How these interactions are modeled, to a large extent, dictates the type of problem that results. This is followed by an analysis of conservative loads, and it is shown how these can be treated as part of the system.

This chapter is concerned with the formulation of the equations of motion (EoM) of simple systems. What is meant by simple is that the systems have just a single degree of freedom (SDoF) and does not imply that the underlying mechanics is simple or elementary in any way.

The concept of vibration is fundamental to understanding the dynamics of elastic structures. The study of vibration is concerned with the oscillatory motion of bodies; all bodies with elasticity and mass are capable of exhibiting vibrations. Resonant (or natural) frequencies are the frequencies at which a structure exhibits relatively large response amplitudes for relatively small inputs. Even if the excitation forces are not sinusoidal, these frequencies tend to dominate the response. In practice, large resonant responses are mitigated by the presence of damping and nonlinear effects. Damping is considered in this chapter, whereas the effects of nonlinearities are distributed throughout the other chapters. The use of Fourier analysis (or spectral analysis) as a means of describing time-varying behavior is essential to the study of structural dynamics, and this too is developed in this chapter.

Motion of Simple Systems

This section reviews the dynamics of elastic systems in the form of a spring-mass-dashpot. We restrict the emphasis to concepts that are used directly in this and later chapters. References 45, 81, and 83 are good sources for additional details on the material covered here.

When a structure experiences a localized disturbance over a short period of time, an impulse, the energy propagates throughout the structure as waves. When the structure is relatively large so that the waves do not have many interactions with the boundaries (within the observation time), then the behavior is amenable to wave analysis methods; that is, there is a definite space/time relationship for the location of the energy. On the other hand, when the structure is relatively small so that there are many wave reflections, then the behavior is dominated by the vibration characteristics and therefore amenable to spectral and modal analysis methods, as discussed earlier.

Waves can propagate in extended media; common examples are surface water waves and sound pressure waves. They can also propagate in slender members with traction-free lateral boundary conditions; simple examples are rods and beams. Section 7.2 developed spectral analysis tools for these types of members, and therefore we mostly concentrate our wave analysis on these slender members – only briefly do we consider waves in extended media. In a wave context, the slender members are referred to as waveguides.

Introduction to Wave Propagation

This section gives a general introduction to the area of stress-wave propagation. The intent is to establish some of the characteristics of waves that differentiate them from other dynamic behaviors such as vibration. The main idea developed is the distinction between dispersive and nondispersive waves.

Structures are to be found in various shapes and sizes for various purposes and uses. These range from the human-made structures of bridges carrying traffic, buildings housing offices, and airplanes carrying passengers all the way down to the biologic structures of cells and proteins carrying genetic information. Structural mechanics is concerned with the behavior of structures under the action of applied loads – their deformations and internal loads. We present, in the following chapters, versatile methods to tackle some of the most common (and most difficult) problems facing engineers in the analysis of structures. This volume specifically considers the situations where the loads vary in time such that inertia effects are important in computing the responses.

The modeling of the dynamic response of structures introduces many additional considerations not anticipated from a static analysis. It is therefore worth our while to say just a little about structural dynamics and its place in structural analyses. The subject of rigid-body dynamics treats physical objects as bodies that undergo motion without any change of shape.

Use of the computer is essential for dynamic structural analyses; therefore, it is important to have some understanding of how the computer is actually used to accomplish these analyses. This chapter introduces the basic computer methods and algorithms used in dynamic analyses; these are schemes for solving systems of equations, time integration of simultaneous equations, and eigenvalue problems. References 5, 27, and 88 consider many other aspects of using computers for dynamic structural analysis.

It is difficult to describe computer algorithms without describing the complete programming context. Therefore, most of the discussions presented here refer to the algorithms implemented in the finite-element programs SDframe/SDsolid, which are available on the associated website www.cambridge.org/doyle_structures_FEM and are reduced from the QED package [28]. The performance of the algorithms is estimated based on a hypothetical benchmark machine and problem. The machine operates at 1 Gflops (one-thousand million floating-point operations per second); small, medium, and large problems refer to systems of equations with 1,000, 10,000, and 100,000 unknowns, respectively. Algorithms appropriate for structural analysis should be suitably scalable across this range of problems.

Solving Large Systems of Equations

The primary computational task associated with matrix analysis of structures consists of solving a set of N simultaneous linear algebraic equations for N unknowns. For small-scale problems, a wide variety of schemes can be used, but for large systems where robustness and scalability are important attributes, the choice is rather limited.