Mathematics
Statics and Dynamics
Statics and dynamics are two branches of mechanics that deal with the study of objects at rest and in motion, respectively. Statics focuses on the equilibrium of forces acting on stationary objects, while dynamics examines the causes of motion and the effects of forces on objects in motion. These concepts are fundamental in understanding the behavior of physical systems.
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8 Key excerpts on "Statics and Dynamics"
eBook - PDFEngineering Mechanics
Statics
- James L. Meriam, L. G. Kraige, J. N. Bolton(Authors)
- 2018(Publication Date)
- Wiley(Publisher)
Substantial By Duke.of.arcH - www.flickr.com/photos/dukeofarch/Getty Images, Inc. Structures which support large forces must be designed with the principles of mechanics foremost in mind. In this view of Sydney, Australia, one can see a variety of such structures. S. Terry/Science Source Sir Isaac Newton 1 2 CHAPTER 1 Introduction to Statics contributions to the development of mechanics were also made by da Vinci, Varignon, Euler, D’Alembert, Lagrange, Laplace, and others. In this book we will be concerned with both the development of the principles of mechanics and their application. The principles of mechanics as a science are rigorously expressed by mathematics, and thus mathematics plays an important role in the application of these principles to the solution of practical problems. The subject of mechanics is logically divided into two parts: statics, which con- cerns the equilibrium of bodies under action of forces, and dynamics, which con- cerns the motion of bodies. Engineering Mechanics is divided into these two parts, Vol. 1 Statics and Vol. 2 Dynamics. 1/2 Basic Concepts The following concepts and definitions are basic to the study of mechanics, and they should be understood at the outset. Space is the geometric region occupied by bodies whose positions are described by linear and angular measurements relative to a coordinate system. For three- dimensional problems, three independent coordinates are needed. For two-dimensional problems, only two coordinates are required. Time is the measure of the succession of events and is a basic quantity in dy- namics. Time is not directly involved in the analysis of statics problems. Mass is a measure of the inertia of a body, which is its resistance to a change of velocity. Mass can also be thought of as the quantity of matter in a body. The mass of a body affects the gravitational attraction force between it and other bodies.
eBook - PDF- James L. Meriam, L. G. Kraige, J. N. Bolton(Authors)
- 2020(Publication Date)
- Wiley(Publisher)
Substantial By Duke.of.arcH - www.flickr.com/photos/dukeofarch/Getty Images, Inc. Structures which support large forces must be designed with the principles of mechanics foremost in mind. In this view of Sydney, Australia, one can see a variety of such structures. S. Terry/Science Source Sir Isaac Newton 1 2 CHAPTER 1 Introduction to Statics contributions to the development of mechanics were also made by da Vinci, Varignon, Euler, D’Alembert, Lagrange, Laplace, and others. In this book we will be concerned with both the development of the principles of mechanics and their application. The principles of mechanics as a science are rigorously expressed by mathematics, and thus mathematics plays an important role in the application of these principles to the solution of practical problems. The subject of mechanics is logically divided into two parts: statics, which con- cerns the equilibrium of bodies under action of forces, and dynamics, which con- cerns the motion of bodies. Engineering Mechanics is divided into these two parts, Vol. 1 Statics and Vol. 2 Dynamics. 1/2 Basic Concepts The following concepts and definitions are basic to the study of mechanics, and they should be understood at the outset. Space is the geometric region occupied by bodies whose positions are described by linear and angular measurements relative to a coordinate system. For three- dimensional problems, three independent coordinates are needed. For two-dimensional problems, only two coordinates are required. Time is the measure of the succession of events and is a basic quantity in dy- namics. Time is not directly involved in the analysis of statics problems. Mass is a measure of the inertia of a body, which is its resistance to a change of velocity. Mass can also be thought of as the quantity of matter in a body. The mass of a body affects the gravitational attraction force between it and other bodies.
No longer available |Learn more- (Author)
- 2014(Publication Date)
- Academic Studio(Publisher)
When the pressing force is removed the spring attains its original state. 2. Statics Example of a beam in static equilibrium. The sum of force and moment is zero. Statics is the branch of mechanics concerned with the analysis of loads (force, torque/moment) on physical systems in static equilibrium, that is, in a state where the relative positions of subsystems do not vary over time, or where components and structures are at a constant velocity. When in static equilibrium, the system is either at rest, or its center of mass moves at constant velocity. The study of moving bodies is known as dynamics, and in fact the entire field of statics is a special case of dynamics. ________________________ WORLD TECHNOLOGIES ________________________ By Newton's first law, this situation implies that the net force and net torque (also known as moment of force) on every body in the system is zero. From this constraint, such quantities as stress or pressure can be derived. The net forces equalling zero is known as the first condition for equilibrium, and the net torque equalling zero is known as the second condition for equilibrium. Solids Statics is used in the analysis of structures, for instance in architectural and structural engineering. Strength of materials is a related field of mechanics that relies heavily on the application of static equilibrium. A key concept is the center of gravity of a body at rest: it represents an imaginary point at which all the mass of a body resides. The position of the point relative to the foundations on which a body lies determines its stability towards small movements. If the center of gravity exists outside the foundations, then the body is unstable because there is a torque acting: any small disturbance will cause the body to fall or topple. If the center of gravity exists within the foundations, the body is stable since no net torque acts on the body.
eBook - PDF- Pierluigi Zotto, Sergio Lo Russo, Paolo Sartori(Authors)
- 2022(Publication Date)
- Società Editrice Esculapio(Publisher)
Dynamics of Point-like Particles 4.1 Introduction Kinematics addresses the study of the motion of bodies using a merely descriptive ap- proach. Dynamics introduces the cause and effect relationship, i.e. it studies the causes of motion, called forces † , and the methods that allow to forecast the motion they generate starting from their knowledge. The assumption of dynamics is that the motion of a body is determined by an interaction with other bodies, which is mediated by a force. The formulae relating the forces acting on a body to its motion, or better to the variation of its motion, are called equations of motion. In other term, the problem of dynamics is the solution of the equations of motion of a body by considering its interaction with the rest of the universe. 4.2 Newton’s Laws The classical model of Mechanics, discipline including kinematics, dynamics and statics of bodies, is described by Newton’s theory (1642–1727). The model presumes that the mechanical processes that happen in nature are the conse- quence of three fundamental laws (or principles), obtained as a synthesis of centuries of experimental measurements of uncountable observers ‡ . • Newton’s First Law or Law of Inertia Any body keeps its state of motion unless external causes intervene to modify it. The index of a body state of motion is its velocity in a given reference system. If we define as free particle a body which is not subject to any interaction, i.e. to any force, the law establishes that a free particle can move only with constant vector velocity, i.e. with constant magnitude and direction of velocity. The modification of the state of motion, i.e. a change in velocity, is possible only by an external intervention, i.e. only if a force is applied to the particle. The law is obvious for rest, a state of null velocity, which changes only if an action is performed on the body.
eBook - PDF- Dean Karnopp(Author)
- 2016(Publication Date)
- CRC Press(Publisher)
For example, in vehicle stability discussions, one often encounters the concepts of static and dynamic stability. The first section of this chapter will consider these concepts in a general context. Static stability considerations arise logically in the studies of aircraft in Chapter 9 while dynamic stability studies typically must be used when studying ground vehicles. Chapter 9 will provide an argument why the use of the special case of static stability is often sufficient for aircraft while the more general concept of dynamic stabil-ity is necessary for most other vehicles. 3.1 Static and Dynamic Stability The basic ideas of stability and instability are often introduced by consider-ing simple physical systems that exhibit stability or instability in an intuitive way when displaced from an equilibrium position. For example, a marble lying on a surface and acted upon by gravity can illustrate the concept of static stability . Three cases are illustrated in Figure 3.1, distinguished by the surface the marble is resting on that can be flat, curved downward, or curved upward. The idea of the first case in Figure 3.1 is that a marble resting on a flat, horizontal surface could stand still at any point along the surface. In this case every point on the surface is an equilibrium point. The marble would neither tend to return to any point if moved a distance away nor would it tend to move further from the original point. Any point on the surface is thus called an equilibrium point of neutral static stability . In the second case in Figure 3.1, the marble rests at the top of a hill. The marble could remain there if it were exactly at the top of the hill, but if it were to be moved even a small distance toward either side, it would tend to roll Neutral stability Statically unstable Statically stable FIGURE 3.1 Static stability of a marble resting of a surface. 43 Stability of Motion further away from the equilibrium point down the hill.
eBook - PDFEngineering Mechanics
Statics, SI Edition
- Andrew Pytel, Jaan Kiusalaas(Authors)
- 2016(Publication Date)
- Cengage Learning EMEA(Publisher)
1 The Flemish mathematician and engineer Simon Stevinus (1548–1620) was the first to demonstrate resolution of forces, thereby establishing the foundation of modern statics. Science Source 1 Introduction to Statics 1.1 Introduction 1.1a What is engineering mechanics? Mechanics is the branch of physics that considers the action of forces on bodies or fluids that are at rest or in motion . Correspondingly, the primary topics of mechanics are Statics and Dynamics. Engineering mechanics is the branch of engineering that applies the principles of mechanics to mechanical design (i.e., any design that must take into account the effect of forces). Engineering mechanics is an integral component of the education of engineers whose disciplines are related to the mechanical sciences, such as aerospace engineering, architectural engineering, civil engineering, and mechanical engineering. Copyright 2017 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 2 CH A P T E R 1: Introduct ion to S tat ics 1.1b Problem formulation and the accuracy of solutions Your mastery of the principles of engineering mechanics will be reflected in your ability to formulate and solve problems. Unfortunately, there is no simple method for teaching problem-solving skills. Nearly all individuals require a considerable amount of practice in solving problems before they begin to develop the analytical skills that are so necessary for success in engineering. For this reason, a relatively large number of sample problems and homework problems are placed at strategic points throughout this text.
eBook - PDF- Soumitro Banerjee(Author)
- 2005(Publication Date)
- Wiley(Publisher)
Thus, following Newton, a simple methodology of understanding the dynamics of any system emerged; just look for the opposing tendencies in a system since motion or change in any system is the result of these opposing tendencies. Example 2.1 Consider the mass-spring system in Fig. 2.1. The externally applied force is F . Let the position of the mass m be measured such that it has value zero at the unstretched position of the spring, and is positive in the direction of the applied force. Dynamics for Engineers S. Banerjee 2005 John Wiley & Sons, Ltd 12 THE NEWTONIAN METHOD Frictionless surface k m q F Figure 2.1 The mass-spring system of Example 2.1. F kq mq .. Figure 2.2 The free body diagram of the system of Example 2.1 following D’Alembert’s principle. For understanding the balance of forces, one draws what is known as the free body diagram (FBD) – where the forces acting on each mass point are shown separately. In this system, there is only one mass point, and its FBD is shown in Fig. 2.2. When the elongation of the spring is q , the force on the mass exerted by the spring is kq , where k is the spring constant. Therefore the total force on the mass is F − kq . Owing to this force the mass moves, and the rate of change of momentum acts in opposition to the applied force. Equating these two opposing tendencies, we get d dt (m ˙ q) = F − kq or m ¨ q + kq − F = 0. This is the differential equation that describes the dynamics of the mass. 2.1 The Configuration Space The positional status of any system can be uniquely specified with the help of a few real numbers. If we have one particle (or a solid body), its position can be specified with a vector r consisting of three real numbers representing the three coordinates. If we have two solid bodies making up a system, we would need two vectors r 1 and r 2 consisting of six position coordinates. Likewise, for a system with N mass points, one requires N vectors r j (j varying from 1 to N ) consisting of 3N coordinates.
eBook - PDF- Bogdan Skalmierski(Author)
- 2013(Publication Date)
- Elsevier(Publisher)
CHAPTER 3 Statics 3.1 Equations of equilibrum We shall now examine the conditions which are satisfied by forces when a body remains at rest. If a body is at rest both the linear momentum p and the angular momentum K are simultaneously equal to zero. On this basis we can tell that its being at rest entails the disappearance of derivatives dp = 0 dt and dK dt = 0. By the principles of linear and angular momentum, the disappearance of the derivatives (1) and (2) brings about the disappearance of the sum of the forces and of the moments of external forces acting on the body. Thus in the considered case, the system of vector equations, called the equations of equilibrium, is satisfied. This system of equations is the basis for the solution of all statical problems: P i(1) = 0, ( 3 ) ri c Pi(1 ) = 0. (4) Equations (3) and (4) give the fundamental theorem on equilibrium of forces. A body will remain in equilibrium when such conditions of support are satified that ensure the standstill of the body. In this situation, the applied forces, together with the reactions, satisfy (1) (2) 96 STATICS Ch. 3 the conditions of equilibrium, i.e., satisfy equations (3) and (4). In other words: If a body remains in equilibrium, then the sum of external forces and the sum of moments of these forces about an arbitrary point in space are both simultaneously equal to zero. The converse is not true. Note should be taken of the linear character of equations (3) and (4), by virtue of which in solving them use can be made of superposition of solutions. 3.2 Couple Let us imagine two forces in two parallel straight lines l and m (Fig. 3.1). If the line 1 together with the force is translated until the two lines super-pose, then the lengths of the two vectors can be compared. When these lengths are identical but the vectors have opposite senses, we are dealing with what is called a couple.
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