Technology & Engineering
Equilibrium Equations
Equilibrium equations are mathematical expressions used to analyze the balance of forces and moments acting on a system. They are fundamental in engineering and physics for determining the stability and behavior of structures and objects. These equations are derived from the principles of statics and are essential for solving problems related to structural design and mechanical systems.
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4 Key excerpts on "Equilibrium Equations"
eBook - PDF- Harold Josephs, Ronald Huston(Authors)
- 2002(Publication Date)
- CRC Press(Publisher)
Finally, the expression equilibrium position for a mechanical system means a constant or steady-state solution of the governing equations of motion of the system. To illustrate these concepts, consider again the simple pendulum of Figure 14.2.1, here consisting of the concentrated mass (or bob) P connected to a pin supported by a light (massless) rod, as opposed to a string. Recall from Eq. (8.4.4) that the governing differential equation for this system is: (14.2.1) where as before θ describes the orientation of the pendulum, is its length, and g is the gravity constant. ˙˙ / sin θ θ + ( ) = g l 0 480 Dynamics of Mechanical Systems Observe that there are two equilibrium, or “constant,” solutions to Eq. (14.2.1). That is, (14.2.2) These solutions represent static equilibrium positions shown graphically in Figure 14.2.2. Intuitively, we would expect the solution θ = 0 to be stable and the solution θ = π to be unstable. That is, if we displace (or “perturb”) the pendulum a small amount away from θ = 0 and then release it, we would expect the pendulum to simply oscillate about θ = 0. Alternatively, if we perturb the pendulum by a small amount away from θ = π and then release it, we would expect the pendulum to fall increasingly further away from the equilibrium position. We can show this mathematically through an analysis of the governing equation (Eq. (14.2.1)). Let us designate “small” quantities by a superscript star or asterisk ( * ). Specifi-cally, a small displacement of the pendulum will be represented by the angle θ * . Hence, a perturbation or disturbance away from the equilibrium position θ = 0 may be expressed as: (14.2.3) By substituting from Eq. (14.2.3) into Eq. (14.2.1), we obtain: (14.2.4) Because θ * is small, we can approximate and represent sin θ * by θ * leading to the equation: (14.2.5) The solution of Eq. (14.2.5) may be expressed in the form: (14.2.6) FIGURE 14.2.1 The simple pendulum.
eBook - ePub- Robert DeHoff(Author)
- 2006(Publication Date)
- CRC Press(Publisher)
This thermodynamic externum principle (the entropy has an extreme value, a maximum) is then formulated mathematically. A set of equations, called the conditions for equilibrium, that describe the relationships that the internal properties of the isolated system must have when it achieves its equilibrium state are derived from this extremum principle. It is then demonstrated that although these conditions for equilibrium are derived for an isolated system, they are valid for any system at equilibrium whether or not it was isolated during its approach to that final condition.The strategy for obtaining these conditions for equilibrium is illustrated in this chapter by applying it to the simplest case: equilibrium in a unary, two-phase system.This strategy is applied repeatedly in subsequent chapters to derive the conditions for equilibrium in systems of progressively increasing sophistication. These equations are the basis for making practical calculations about the final resting condition of complex systems.The chapter ends with a collection of alternate statements of the criterion for equilibrium, each of which is useful in the description of systems with specific external constraints.5.1 INTUITIVE NOTIONS OF EQUILIBRIUM
The idea that a system changes toward some final ultimate condition and, once there, remains in that condition unless acted upon by some external agent is a familiar one. Such a system is said to come to equilibrium or approach its equilibrium state. The specific nature of that state depends upon the chemical and energy content of the system and that of its surroundings. A system comes to equilibrium in given surroundings. It may only be disturbed from its equilibrium state by changing the state of its surroundings.A simple application of this idea is illustrated in Figure 5.1
eBook - PDF- Paul Flowers, Klaus Theopold, Richard Langley, William R. Robinson(Authors)
- 2019(Publication Date)
- Openstax(Publisher)
INTRODUCTION CHAPTER 13 Fundamental Equilibrium Concepts 13.1 Chemical Equilibria 13.2 Equilibrium Constants 13.3 Shifting Equilibria: Le Châtelier’s Principle 13.4 Equilibrium Calculations Imagine a beach populated with sunbathers and swimmers. As those basking in the sun get too hot, they enter the surf to swim and cool off. As the swimmers tire, they return to the beach to rest. If the rate at which sunbathers enter the surf were to equal the rate at which swimmers return to the sand, then the numbers (though not the identities) of sunbathers and swimmers would remain constant. This scenario illustrates a dynamic phenomenon known as equilibrium, in which opposing processes occur at equal rates. Chemical and physical processes are subject to this phenomenon; these processes are at equilibrium when the forward and reverse reaction rates are equal. Equilibrium systems are pervasive in nature; the various reactions involving carbon dioxide dissolved in blood are examples (see Figure 13.1). This chapter provides a thorough introduction to the essential aspects of chemical equilibria. 13.1 Chemical Equilibria LEARNING OBJECTIVES By the end of this section, you will be able to: • Describe the nature of equilibrium systems • Explain the dynamic nature of a chemical equilibrium The convention for writing chemical equations involves placing reactant formulas on the left side of a reaction Figure 13.1 Transport of carbon dioxide in the body involves several reversible chemical reactions, including hydrolysis and acid ionization (among others). CHAPTER OUTLINE arrow and product formulas on the right side. By this convention, and the definitions of “reactant” and “product,” a chemical equation represents the reaction in question as proceeding from left to right. Reversible reactions, however, may proceed in both forward (left to right) and reverse (right to left) directions.
eBook - PDF- Jonathan A. Dantzig, Michel Rappaz(Authors)
- 2009(Publication Date)
- EPFL PRESS(Publisher)
CHAPTER 4 BALANCE EQUATIONS 4.1 I NTRODUCTION Chapter 3 described the equilibrium between phases of different composi- tion at various temperatures. The term equilibrium implies, in particular, that there is no variation of any quantity in either space or time. It is obvious, however, that solidification processes and specifically microstruc- ture development must include spatial and temporal evolution, which are usually called transport phenomena. In this chapter, we introduce the gov- erning equations for the transport of mass, solute, momentum and energy. These are often referred to as balance equations since they are obtained by applying the basic balance, i.e., that the rate of accumulation of any quantity within a volume is equal to the sum of inputs through the sur- face and direct production within the volume. This is a subject that has been thoroughly treated (see the chapter references for some examples), and we therefore include only as much as is needed to provide a basis for the material in later chapters. Key Concept 4.1: Constitutive relations and scaling The balance equations by themselves are correct, but incomplete, since there are more unknowns than equations. They must be supplemented by constitutive relations that describe the materials’ responses to ap- plied fields. Some common examples are Fourier’s law for heat conduc- tion and Fick’s law for diffusion of species. In many cases, there are terms in the equations that represent phenomena that are unimpor- tant for the solidification problems that we deal with in this book. We identify these terms using a process called scaling, and then usually neglect them in order to concentrate on the most important physical phenomena for our purposes. Our focus on the special forms needed for solidification leads us to examine balances at the solid-liquid interface, as well as in control volumes consisting of a mixture of both phases.
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