Abstract

We present two methodologies for studying periodic oscillations and wave transmission in periodic continuous systems with strongly nonlinear supporting or coupling stiffnesses. The first methodology is based on a nonsmooth transformation of the spatial variable and eliminates the singularities (generalized functions) from the governing partial differential equation of motion. The resulting smooth partial differential equation is then analyzed asymptotically. This method is used to study localized nonlinear normal modes (NNMs) of an infinite linear string supported by a periodic array of nonlinear stiffnesses. A second methodology is developed to study primary pulse transmission in a periodic system composed of linear layers coupled by means of strongly nonlinear stiffnesses. A piecewise transformation of the time variable is introduced, and the scattering of the primary pulse at the nonlinear stiffnesses is reduced to solving a set of strongly nonlinear first-order ordinary differential equations. Approximate analytical and exact numerical solutions of this set are presented, and the methodology is employed to study primary pulse transmission in a system with clearance nonlinearities.

Introduction

Flexible periodic structures such as truss aerospace systems, periodically supported or stiffened shells and plates, and turbine cyclic assemblies are very common in engineering applications. Periodic structures are composed of a number of identical (or near identical) coupled substructures.

Abstract

The problem of an elastic distributed system coupled with a moving oscillator, often referred to as the “moving-oscillator” problem, is studied in this chapter. The problem is formulated using a “relative displacement” model. It is shown that, in the limiting case, the moving-mass problem is recovered. The coupled equations of motion are recast into an integral equation, which is amenable to solution by both iterative and direct numerical procedures. The response of a string with a moving oscillator is studied using the direct numerical method.

Introduction

Perspective

The prediction of the dynamic response of a distributed elastic system that supports one or more translating elastic subsystems has been a fundamental problem of interest for well over a century. Interest in this problem originates in structural engineering for the design of railroad tracks, railroad bridges, and highway bridges, wherein the accurate calculation of loads is essential for reliable design and accurate life prediction (Stokes, 1883; Ting and Yener, 1983; Tan and Shore, 1968). It has been observed that, as a structure is subjected to moving loads, the dynamic deflection and stresses can be significantly higher than those observed in the static case. Hence, strict design criteria are now required as structural engineers become more aggressive in the use of long, flexible spans in cable-stayed and suspension bridges and compliant bearings in highway bridges to accommodate environmental loads.

Abstract

A new analytical method is presented for modeling and analysis of stepped cylindrical shells and cylindrical shells stiffened by circumferential rings. Through use of the distributed transfer functions of the structural systems, various static and dynamic problems of cylindrical shells are systematically formulated. With this transfer function formulation, the static and dynamic response, natural frequencies and mode shapes, and buckling loads of general stiffened cylindrical shells under arbitrary external excitations and boundary conditions can be determined in exact and closed form. The proposed method is illustrated on a Donnell–Mushtari shell and compared with the finite element method and other modeling techniques.

Introduction

Cylindrical shells are the basic element in many structures and machines and therefore have been extensively studied in the past; for instance, see Donnell (1933), Soedel (1981), Irie et al. (1984), Yamada et al. (1984), Sheinman and Weissman (1987), Koga (1988), Thangaratnam et al. (1990), Huang and Hsu (1992), Heyliger and Jilani (1993), Birman (1993), and Miyazaki and Hagihara (1993). The static and dynamic problems of cylindrical shells are often complicated by engineering design in which a cylindrical shell is composed of a finite number of serially connected shell segments and/or stiffened by circumferential rings. For such complex structural systems, numerical methods are usually adopted.

Abstract

Distributed structures are often coupled to external components in engineering applications: The recording head in disk and tape drives, the guide bearing in circular and band saws, and the payload in cable transport systems are a few examples. Because the transient response of a distributed structure is characterized by complex multiple wave scattering at an external component, this behavior has rarely been explored. Classical transient analyses by finite difference, finite element, and modal expansion approaches use spatial discretization. As a result, some discontinuities in the interaction force between a distributed structure and an external component, a characteristic of multiple wave scattering, are not predicted in the normal application of these methods. A new transient analysis is developed in this chapter for the response of distributed structures interacting with discrete components. The transient response of a time-varying, cable transport system is analyzed first. The transient and steady-state responses of constrained translating strings are obtained next. Transient behaviors of cables transporting dynamic payloads and translating, magnetic tape-head systems are thoroughly investigated. Application of the method to the classical piano string response under a hammer strike avoids key limitations of the existing standing and traveling wave methods.

Introduction

Distributed structures are usually coupled to external dynamic components in engineering applications.

Abstract

This chapter illustrates the use of continuum models in control design for stabilizing flexible structures. A 6-degree-of-freedom anisotropic Timoshenko beam with discrete nodes where lumped masses or actuators are located provides a sufficiently rich model to be of interest for mathematical theory as well as practical application. We develop concepts and tools to help answer engineering questions without having to resort to ad hoc heuristic (“physical”) arguments or faith. In this sense the paper is more mathematically oriented than engineering papers and vice versa at the same time. For instance we make precise time-domain solutions using the theory of semigroups of operators rather than formal “inverse Laplace transforms.” We show that the modes arise as eigenvalues of the generator of the semigroup, which are then related to the eigenvalues of the stiffness operator. With the feedback control, the modes are no longer orthogonal and the question naturally arises as to whether there is still a modal expansion. Here we prove that the eigenfunctions yield a biorthogonal Riesz basis and indicate the corresponding expansion. We prove mathematically that the number of eigenvalues is nonfinite, based on the theory of zeros of entire functions. We make precise the notion of asymptotic modes and indicate how to calculate them.

Introduction

Local deformations in the contact region must be accounted for in the analysis in order to accurately predict the contact force history. The indentation, defined as the difference between the displacement of the projectile and that of the back face of the laminate, can be of the same order as or larger than the overall displacement of the laminate. One could consider the projectile and the structure as two solids in contact and then analyze the impact problem as a dynamic contact problem. However, this approach is computationally expensive and cannot describe the effect of permanent deformation and local damage on the unloading process. The unloading part of the indentation process can be modeled only using experimentally determined contact laws. To predict the contact force history and the overall deformation of the target, a detailed model of the contact region is not necessary. A simple relationship between the contact force and the indentation, called the contact law, has been used by Timoshenko (1913) to study the impact of a beam by a steel sphere. This approach has been used extensively since then and is commonly used for the analysis of impact on composite materials.

Although the impact event is a highly dynamic event in which many vibration modes of the target are excited, statically determined contact laws can be used in the impact dynamics analysis of low-velocity impacts because strain rate and wave propagation effects are negligible with commonly used material systems. Hunter (1957) calculated the energy lost by elastic waves during the impact of a sphere on an elastic half-space.

Introduction

Composite materials are used extensively in sandwich constructions, and compared with skin-stiffener panels, sandwich construction presents definite advantages: improved stability, weight savings, ease of manufacture, and easy repairs. Using composite materials instead of aluminum for the facesheets results in higher performance and lower weight, even though composite materials are more susceptible to impact damage. Composite sandwich construction has been used extensively for side skin panels, crown skin panels, frames, and longerons in the Boeing 360 helicopter (Llorente 1989). Compared to skin-stringer aluminum construction, the use of sandwich structures lead to an 85% reduction in the number of parts, a 90% reduction in tooling costs, and a 50% reduction in the number of work-hours needed to fabricate the helicopter. In spite of these advantages, the problem of impact on sandwich structures has received only minimal attention.

Contact laws for sandwich structures are significantly different that those for monolithic laminates. With sandwich structures, the indentation is dominated by the behavior of the core material, which becomes crushed as transverse stresses become large. To predict the contact force history and the overall response of the structure to impact by a foreign object, mathematical models should account for the dynamics of the projectile, the dynamics of the structure, and the contact behavior. Failure modes involved in impact damage of sandwich structures and the influence of the several important parameters will be discussed. We will also examine the effect of impact damage on the residual properties of the structure.

As composite materials are used more extensively, a constant source of concern is the effect of foreign objects impacts. Such impacts can reasonably be expected during the life of the structure and can result in internal damage that is often difficult to detect and can cause severe reductions in the strength and stability of the structure. This concern provided the motivation for intense research resulting in hundreds of journal and conference articles. Important advances have been made, and many aspects of the problem have been investigated. One would need to study a voluminous literature in order to get an appreciation for this new research area. After writing three comprehensive literature reviews on the topic of impact on composite materials, I felt that there was a need to present this material in book form. The study of impact on composite structures involves many different topics, including contact mechanics, structural dynamics, strength, stability, fatigue, damage mechanics, and micromechanics. Impacts are simple events with many complicated effects, and what appears as a logical conclusion in one situation seems to be completely reversed in another. This variety and complexity have kept me interested in this area for several years, and I hope to communicate that to the readers.

This book attempts to present this new body of knowledge in a unified, detailed, and comprehensive manner. It assumes only a basic knowledge of solid mechanics and the mechanics of composite materials and can be used as a text for a graduate-level course or for self-study.

Introduction

The expression composite repairs usually refers to one of three areas of extensive research activity: (1) repair of a damaged composite structure, (2) repair of aging (metallic) aircraft with composite patches, (3) and repair of civil engineering structures using composite reinforcement. Each one of those areas has received considerable attention. In this chapter we focus on the repair of impact-damaged composite structures.

Once damage is detected and the effects on the residual properties of the structure have been estimated, a decision must be made as to whether this composite part should be repaired or replaced. There are cases where damage cannot be repaired. For example, highly stressed members may not have sufficient strength after repair. Three types of repairs are possible:

1. Large damages that reduce the load-carrying capability of the component below the ultimate load must be repaired immediately. In time, temporary field repairs using bolted metal patches must be replaced by permanent shop repairs. Permanent repairs can be either bolted or bonded repairs with precured composite parts. Another option, called laminate repair or cocured repair, consists of simultaneously curing the reinforcing patches and the bonded joints.

2. Minor damages such that the part can sustain the ultimate load must be repaired within a defined period. Measures must be taken to prevent water and airstream ingress and to prevent damage propagation before the repair takes place.

3. […]

Introduction

Having studied the dynamics of impact, damage development, and damage prediction methods, the next area of interest is the effect of damage on the mechanical properties of laminated composite structures. This is often called the study of damage tolerance since it refers to the experimental determination or the numerical prediction of the residual mechanical properties of the damaged structure. Many organizations have developed guidelines or requirements for residual strength after impact. For example, U.S. Air Force draft requirements for damage tolerance for low-velocity impacts are that laminates should maintain a minimum design strength after impacts with 100 ft-lb kinetic energy by a 1-in.-diameter hemispherical indenter or after impacts resulting in a 0.10 in. dent, whichever is less severe (Schoeppner 1993).

An understanding of damage tolerance can be gained through experiments and available models for predicting residual properties. The general trend for the residual strength of laminated composites with impact damage is that, for low initial kinetic energy levels, the strength is not affected since little or no damage is introduced. As damage size increases, the strength drops rapidly and then levels off. The effects of impact damage on the residual strength in tension, compression, shear, and bending have been investigated at length and follow the same general trend. Experimental techniques, general results obtained from experiments, and models for predicting residual properties are presented in this chapter.

Compressive Strength

The large number of articles devoted to understanding the compression after impact behavior of composite materials (Table 6.1) is evidence of the importance of the topic and the level of effort focused on it.

Introduction

Impact damage usually follows some very complex distributions, and it may not be possible to reconstruct the entire sequence of events leading to a given damaged state. For low-velocity impacts, damage starts with the creation of a matrix crack. In some cases the target is flexible and the crack is created by tensile flexural stresses in the bottom ply of the laminate. This crack, which is usually perpendicular to the plane of the laminate, is called a tensile crack. For thick laminates, cracks appear near the top of the laminate and are created by the contact stresses. These cracks, called shear cracks, are inclined relative to the normal to the midplane. Matrix cracks induce delaminations at interfaces between adjacent plies and initiate a pattern of damage evolution either from the bottom up or from the top down. Therefore, while it is possible to predict the onset of damage, a detailed prediction of the final damage state cannot realistically be achieved.

Two types of approaches are used for predicting impact damage. The first type attempts to estimate the overall size of the damaged area based on the stress distribution around the impact point without considering individual failure modes. The general idea is that impact induces high stresses near the impact point and that these localized stresses initiate cracks, propagate delaminations, and eventually lead to the final damage state. Section 5.2 describes how this approach is used for predicting damage size for thick laminates, which behave essentially as semi-infinite bodies. In Section 5.3, the approach is applied to thin laminates, which are modeled as shear deformable plates.

This chapter deals broadly will the subject of impact resistance, which is the study of damage induced by foreign object impact in a laminate and the factors affecting it. Methods for predicting impact damage are discussed in Chapter 5, and the study of damage tolerance – that is, the effect of impact damage on the stiffness, strength, fatigue life, and other properties of the laminate – will be presented in Chapter 6.

An understanding of impact damage development, the failure modes involved, and the various factors affecting damage size has been gained through extensive experimental studies. In this chapter, several of the most commonly used impact test procedures will be discussed. Experimental techniques for impact damage detection and detailed mapping of the damage zone after impact are reviewed. Of the many different techniques discussed in the literature, some are nondestructive, and others are destructive. Some techniques are used extensively, and others have seen only limited applications. A few experimental techniques have been developed to observe damage development during the impact event. While not attempting to give an exhaustive description of the techniques used, the objectives are to briefly describe each one and to give a general idea of what the most commonly used techniques are.

Understanding the process of impact damage initiation and growth and identifying the governing parameters are important for the development of mathematical models for damage prediction, for designing impact resistant structures, and for developing improved material systems. The basic morphology of impact damage, its development, and the parameters affecting its initiation, growth, and final size will be described.

During the life of a structure, impacts by foreign objects can be expected to occur during manufacturing, service, and maintenance operations. An example of in-service impact occurs during aircraft takeoffs and landings, when stones -and other small debris from the runway are propelled at high velocities by the tires. During the manufacturing process or during maintenance, tools can be dropped on the structure. In this case, impact velocities are small but the mass of the projectile is larger. Laminated composite structures are more susceptible to impact damage than a similar metallic structure. In composite structures, impacts create internal damage that often cannot be detected by visual inspection. This internal damage can cause severe reductions in strength and can grow under load. Therefore, the effects of foreign object impacts on composite structures must be understood, and proper measures should be taken in the design process to account for these expected events. Concerns about the effect of impacts on the performance of composite structures have been a factor in limiting the use of composite materials. For these reasons, the problem of impact has received considerable attention in the literature. The objective with this book is to present a comprehensive view of current knowledge on this very important topic.

A first step in gaining some understanding of the problem is to develop mathematical models for predicting the force applied by the projectile on the structure during impact. In order to predict this contact force history, the model should account for the motion of the structure, the motion of the projectile, and the local deformations in the contact zone.

Introduction

A first step in gaining some understanding of the effect of impact by a foreign object on a structure is to predict the structure's dynamic response to such an impact. Predictions are made using a mathematical model that appropriately accounts for the motion of the projectile, the overall motion of the target, and the local deformations in the area surrounding the impact point. In general, details of the local interaction between the projectile and the target are not needed to predict the contact force history. A particular beam, plate, or shell theory is selected, and the local deformation in the through-the-thickness direction, which is not accounted for in such theories, is included through the use of an appropriate contact law. Contact laws relate the contact force to the indentation, which is the difference between the displacement of the projectile and the displacement of the target at the point of impact. With most low-velocity impacts, small amounts of damage are introduced in a small zone surrounding the impact point, and the dynamic properties of the structures usually are not affected by the presence of damage. Therefore, impact dynamic analyses generally do not attempt to model damage as it develops during the impact event.

The choice of a particular structural theory must be based on careful consideration of the effect of complicating factors such as transverse shear deformation and rotary inertia. For example, for beams, one can select to use the Bernoulli-Euler beam theory, the Timoshenko beam theory, or one of several other available beam theories.