3.1 STATE OF STRESS AT A POINT [LO1]

When a body is subjected to external forces, its behavior depends on the magnitude and distribution of forces and properties of the body material. Depending on these factors, the body may deform elastically or plastically, or it may fracture. The body may also fail by fatigue when subjected to repetitive loading. Here we are primarily interested in elastic deformation of materials.

In order to establish the concept of stress and stress at a point, let us consider a straight bar of uniform cross-section of area A and subjected to uniaxial force F as shown in Figure 3.1. Stress at a typical section A - A′ is normally given as σ = F/A. This is true only if the force is uniformly distributed over the area A, but this is rarely true. Therefore, definition of stress must be considered by progressively reducing the area until it is small enough such that the force may be considered to be uniformly distributed.

To understand this, consider a body subjected to external forces P1, P2, P3, and P4 as shown in Figure 3.2. If we now cut the body in two pieces,

Internal forces f1, f2, f3, etc. are developed to keep the pieces in equilibrium. Now consider an infinitesimal element of area ΔA Dat the cut section and let the resultant force on the element be Δf.

Inertial Effects for a Rigid Body. The primary focus is description of the angular momentum of a rigid body, and the laws and associated rotational equations of motion that relate this quantity to the forces that are exerted. The opening treatment of a system of particles leads to identification of the basic laws. The first example explains why a moment is required to change the orientation of a rotation axis. These concepts are extended to general rigid bodies and naturally lead to definitions for moments of inertia and products of inertia, followed by a discussion of the physical significance of these properties. Systems that execute a variety of rotations are the subject of examples, in which physical explanations are provided for the analytical results. The chapter closes with treatment of rigid bodies moving freely. Axisymmetric bodies, such as a projectile in flight, are treated first. The following treatment discusses asymmetric bodies, such as a block. The “tennis racket” theorem is derived. The chapter ends with application of computational tools, both mathematical and graphical, that provide a description of the overall rotation of asymmetric bodies.

Relative Motion. This chapter develops tools for describing the motion of a point in three dimensions. The concept of a rotation matrix is covered in detail, including how to use transformation matrices to describe the result of a sequence of rotations about body-fixed and/or space-fixed axes. These concepts are used to describe angular velocity and acceleration. The primary tool is a mathematical procedure, accompanied by a sequence of steps to implement it. A parallel approach based on graphical depiction of vectors provides physical insight and the ability to verify results derived by the formal procedure. The last section treats motion that is viewed from a moving reference frame, with emphasis on what an Earth-based observer sees. These concepts are used to understand the flow of fluid particles in a hurricane, as well as how the motion of a Foucault pendulum is influenced by the rotation of the Earth.

Particle motion. The development derives expressions for velocity and acceleration in terms of rectangular coordinates, followed by polar and spherical coordinate systems. A novel derivation of formulas for general curvilinear coordinates is provided in an online supplement. The focus then shifts to situations in which the path of a particle is known. This calls for the use of path variables, also known as tangent-normal components. Formulas are derived for paths that are twisted in three dimension, such as a rollercoaster. The closing topic is the method of mixed kinematical descriptions, which addresses situations in which aspects from two different descriptions are used to facilitate solutions of challenging problems.

Newton Euler Dynamics. A general procedure for formulation of the Newton-Euler equations of motion for rigid bodies and systems of rigid bodies. These advanced techniques are used, for example, to explain the interaction of a bicycle and its rider, including the gyroscopic effects caused by the inertia of the spinning wheels. Numerous examples address the various ways in which engineering systems move. The principles of work-energy, linear momentum, and angular impulse-momentum principles for bodies in spatial motion are developed and various examples are presented. An interesting example uses the properties of free motion to show how the rotation of an (American) football changes when it is suddenly deflected during the course of its flight. The closing section covers variable mass systems. The general principles are applied to determine the position versus time of a steerable full-scale rocket, as it is affected by variable mass, air resistance, and variation of the gravitational field.

Constrained Generalized Coordinates. When generalized coordinates are related kinematically, they are said to be constrained. These constraints are imposed by constraint forces that appear in any set of equations of motion. Constraints are nonholonomic if they are defined by differential equations that cannot be integrated to derive an algebraic form. Lagrange’s equations are derived for systems with constrained generalized coordinates using the concept of the configuration space. Physical restrictions are manifested in the configuration space as constraint surfaces. The associated constraint force is normal to this surface and is captured with a Lagrange multiplier. Several examples consider nonholonomic systems, while others explore systems that could be described by unconstrained coordinates, but constrained generalized coordinates simplify the analysis. A variety of computational algorithms are presented for solving the coupled set of Lagrange and constraint equations. These algorithms are applied to representative problems, one of which is a disk rolling over a flat surface. The closing section addresses systems with dry friction.

The preface provides a summary of the changes made in this edition of the text.

Fundamentals. The opening is a review of vector algebra and calculus concepts that are vital for the remainder of the book, including dot and cross products and derivatives of vector expressions. Attention is given to the use of matrix notation as an alternative to in-line notation for algebraic and computational operations. Newton’s laws for a particle initiate study of kinetics concepts, notably equations of motion, work-energy, and linear and angular momentum. Basic computational techniques are presented and applied to solve the differential equation of motion for a model rocket. The chapter closes with short biographies of many of the individuals whose work is the foundation of this book.

Introduction to Lagrange’s Equations. The static virtual work principle is used to introduce the concepts of generalized coordinates, constraint equations, and generalized forces. Lagrange’s equations are derived by application of the principle of dynamic virtual work. The derivation is restricted to unconstrained generalized coordinates, that is, situations in which there are no kinematic conditions that relate them. Lagrange’s equations are shown to lead to coupled second order, usually nonlinear, ordinary differential equations of motion. Numerous examples derive these equations for a diverse set of systems. One of these examples uses Lagrange’s equations to derive the period for precession of the equinoxes due to the gravitational pull of the Sun and the Moon. A Matlab template is presented that can be used to solve general equations of motion. It is applied to analyze the rotation of a balanced free gyroscope. The Newton-Euler equations are employed to explain the extreme sensitivity of this system to the initial conditions. This example leads to discussion of the various ways the gyroscopic effects are employed in inertial guidance systems.

Kinematics of Constrained Rigid Bodies. Moving reference frames and Euler angles are the basic tools for describing the three dimensional movement of multi-body systems. Particular attention is paid to the connections between bodies, and how to treat the restrictions that connections place on how the system may move. Connections are shown to impose constraints on the motion, which are enforced by a corresponding constraint or reaction force. The chapter ends by showing how these tools can be applied to the case of bodies that roll relative to one another.

Further Concepts. This chapter covers alternative ways of deriving equations of motion. The first derivation is Routh’s method for eliminating excess generalized coordinates. Then Hamilton’s equations, which replace generalized velocities with generalized momenta, are derived. Kane’s equations, which replace generalized velocities with quasi-velocities, are shown to have a general form. A sequential procedure is provided to expedite derivation of Kane’s equations for a specific system. Examples use these method to derive the equations of motion, then compare the merits of that analysis to Lagrange’s equations. The last section is devoted to derivation of field equations that govern the displacement of flexible bodies. The calculus of variations is introduced as a tool for solving some classic mathematics problems. Then that capability is used to derive Hamilton’s principle. Application of this principle to a stretched cable, and a straight bar that undergoes extensional, torsional, or flexural deformation leads to identification of field equations and boundary conditions. The Ritz series approximation method is then presented, as well as its specialization to the finite element method.