Simple harmonic motion. Impulse and momentum. Work and kinetic energy. Moment of a force. Vector product of two vectors. Moments of components of a force. Distributed forces. Equivalent force system in three dimensions. Co-planar force system. Equilibrium in three dimensions. Triple scalar product. Internal forces. Fluid statics. Stability of floating bodies. Types of motion. Relative motion between two points on a rigid body. Velocity diagrams. Instantaneous centre of rotation. Velocity image.

## Mechanical engineering principles, second edition

Acceleration diagrams. Accel- eration image. Simple spur gears. Epicyclic motion. Compound epicyclic gears. Rotation about a fixed axis. Moment of inertia of a body about an axis. Work and energy for system of particles. Kinetic energy of a rigid body. Potential energy. Non-conservative systems. The general energy principle. Summary of the energy method. The power equation. Virtual work. Moment of momentum. Conservation of momentum. Impact of rigid bodies. Deflection of fluid streams. The rocket in free space. Illustrative example. Equations of motion for a fixed region of space.

One-degree-of-freedom systems Introduction. Free vibration of undamped sys- tems. Vibration energy. Levels of vibration. Free vibration of a damped system. Phase-plane method. Response to simple input forces. Periodic excitation. Work done by a sinusoidal force.

Response to a sinusoidal force. Moving foundation. Rotating out-of-balance masses.

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Estimation of damping from width of peak. Section B. Two-degree-of-freedom systems Free vibration. Coupling of co-ordinates. Normal modes. Principle of orthogonality. Forced vibra- tion. Position-control system. Block- diagram notation. System response. System errors. Stability of control systems. Frequency response methods.

Prob- lems. Finite rotation. Angular velocity. Differentiation of a vector when expressed in terms of a moving set of axes. Dynamics of a particle in three-dimensional motion. Motion relative to translating axes. Motion relative to rotating axes. Kinematics of mechanisms.

## ISBN 13: 9780340568316

Kine- tics of a rigid body. Moment of force and rate of change of moment of momentum. Rotation about a fixed point of a body with an axis of symmetry. One-dimensionulcontinuum Introduction. One-dimensional con- tinuum. Elementary strain. Particle velocity. Ideal continuum. Simple tension.

### Description

Equation of motion for a one-dimensional solid. General solution of the wave equation. The control volume. Equation of motion for a fluid. Continuity for an elemental volume. Two-and three-dimensionalcontinua Introduction. Pure shear. Plane strain. Plane stress. Rotation of reference axes. Principal strain. Principal stress. The elastic constants. Strain energy. Section C. Applicationsto bars and beams Introduction.

Compound column. Torsion of circular cross-section shafts. Shear force and bending moment in beams. Stress and strain distribution within the beam. Deflection of beams. Area moment method. Discussion exam- ples. Appendices 1 Vector algebra, 2 Units, 3 Approximate integration, 4 Conservative forces and potential energy, 5 Properties of plane areas and rigid bodies, 6 Summary of important relationships, 7 Matrix methods, 8 Properties of structuralmaterials, Answers to problems, Index, 5.

Preface This book covers the basic principles of the Part 1,Part 2 and much of the Part 3 Engineering Mechanics syllabuses of degree courses in engineering. The emphasis of the book is on the principles of mechanics and examples are drawn from a wide range of engineering applications. The order of presentation has been chosen to correspond with that which we have found to be the most easily assimilated by students. Thus, although in some cases we proceed from the general to the particular, the gentler approach is adopted in discussing first two-dimensional and then three-dimensional problems.

The early part of the book deals with the dynamics of particles and of rigid bodies in two-dimensional motion. Both two- and three- dimensional statics problems are discussed. Vector notation is used initially as a label, in order to develop familiarity, and later on the methods of vector algebra are introduced as they naturally arise.

Vibration of single-degree-of-freedom systems are treated in detail and developed into a study of two-degree-of-freedom undamped systems. An introduction to automatic control systems is included extending into frequency response methods and the use of Nyquist and Bode diagrams. Three-dimensional dynamics of a particle and of a rigid body are tackled, making full use of vector algebra and introducing matrix notation.

It is becoming common to combine the areas usually referred to as mechanics and strength of materials and to present a single integrated course in solid mechanics. To this end a chapter is presented on continuum mechanics; this includes a study of one-dimensional and plane stress and strain leading to stresses and deflection of beams and shafts.

Also included in this chapter are the basic elements of fluid dynamics, the purpose of this material is to show the similarities and the differences in the methods of setting up the equations for solid and fluid continua. It is not intended that this should replace a text in fluid dynamics but to develop the basics in parallel with solid mechanics. Most students study the two fields independently, so it is hoped that seeing both Lagrangian and Eulerian co-ordinate sys- tems in use in the same chapter will assist in the understanding of both disciplines.

There is also a discussion of axial wave propagation in rods The fluid mechanics sections The student may be uncertain as to which method is best for a particular problem and because of this may be unable to start the solution. Each chapter in this book is thus divided into two parts. The first is an exposition of the basic theory with a few explanatory examples. The second part contains worked examples, many of which are described and explained in a manner usually reserved for the tutorial.

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Where relevant, different methods for solving the same problem are compared and difficulties arising with certain techniques are pointed out. Each chapter ends with a series of problems for solution. These are graded in such a way as to build up the confidence of students as they proceed. Answers are given. Numerical problems are posed using SI units, but other systems of units are covered in an appendix.

The intention of the book is to provide a firm basis in mechanics, preparing the ground for advanced study in any specialisation. The applications are wide-ranging and chosen to show as many facets of engineering mechanics as is practical in a book of this size. We are grateful to The City University for permission to use examination questions as a basis for a large number of the problems. Thanks are also due to our fellow teachers of Engineering Mechanics who contributed many of the ques- tions.

July H. Statics is a special case where there is no motion. The combined study of are in common use. The position of a point is defined only in relation to some reference axes. In three- dimensional space we require three independent co-ordinates to specify the unique position of a point relative to the chosen set of axes. One-dimensional systems If a point is known to lie on a fixed path - such as a straight line, circle or helix - then only one number is required to locate the point with respect to some arbitrary reference point on the path.

This is the system used in road maps, where place B Fig. Unless A happens to be the end of road R, we must specify the direction which is to be regarded as positive. This system is often referred to as a path co-ordinate system.

Two-dimensionalsystems If a point lies on a surface - such as that of a plane, a cylinder or a sphere - then two numbers are required to specify the position of the point. For a plane surface, two systems of co-ordinates a Cartesian co-ordinates. In this system an orthogonal grid of lines is constructed and a point is defined as being the intersection of two of these straight lines. In Fig. Figure 1. In this system Fig.

If the surface is that of a sphere, then lines of latitude and longitude may be used as in terrestrial navigation. In this system the position is specified by the distance of a point from the origin, and the direction is given by two angles as shown in Fig. U Figure 1. This is a simple extension of the two-dimensional case where a third axis, the z-axis, has been added.

The sense is not arbitrary but is drawn according to the right-hand screw convention, as shown in Fig. This set of axes is known as a normal right-handed triad. This is an extension of the polar co-ordinate system, the convention for positive 8 and z being as shown in Fig. It is clear that if R is constant then the point will lie on the surface of a right circular cylinder. It is also true that spherical co-ordinates could be used in a problem involving motion in a straight line not passing through the origin 0 of the axes; however, this would involve an unnecessary complication.

It follows that some other line drawn to a convenient scale can also be used to re resent the In Fig. These are called free vectors. Hence in mechanics a vector may be defined as a line segment which represents a physical quantity in magnitude and direction. There is, however, a restriction on this definition which is now considered.

Now, since the position magnitude and is in the required direction. Conversely, if a physical quantity is a vector then addition must satisfy the parallelogram law. The important physical quantity which does not obey this addition rule is finite rotation, because it can be demonstrated that the sum of two finite Figure 1. The modulus, written as , is the size of the vector and is always positive. In this book, vector magnitudes may be positive or negative.

Components of a vector Any number of vectors which add to give another vector are said to be components of that other vector. Usually the components are taken to be orthogonal, as shown in Fig. It is convenient to represent a vector by a single symbol and it is conventional to use bold-face type in printed work or to underline a symbol in manuscript.

This is done by introducing a unit vector e which has unit Figure 1. Hence the components of A Fig. The direction cosine, I, is defined as the cosine of the angle between the vector and the positive x-axis, i. Notice that because A and B are free vectors. Scalarproduct of two vectors The scalar product of two vectors A and B sometimes referred to as the dot product is formally defined as IA 1 IB 1cos0, Fig.

This definitionwill later be seen to be useful in the description of work and power. If B is a unit vector e, then 1'8 that is the scalar component of A in the direction of e. A surveying instrument at C can measure distance and angle. Relative to the fixed x-, y-, z-axes at C, point A is at an elevation of 9. The body of the instrument has to be rotated about the vertical axis through 41" from the x direction in order to be aligned with A. Corresponding values for point B are 1. Determine a the locations of points A and B in Cartesian co-ordinates relative to the axes at C, b the distance from A to B, and c the distance from A to B projected on to the horizontal plane.

If the location vector from A to C is -2,0,4 m, find the position of point C and the position vector from B to C. Solution A simple application of the laws of vector addition is all that is required for the solution of this problem. Referring to Fig. Determine the scalar component of the vector OP in the direction B to A and the vector component parallel to the line AB. Solution To determine the component of a given vector in a particular direction, we first obtain the unit vector for the direction and then form the dot product between the unit vector and the given vector.

This gives the magnitude of the component, otherwise known as the scalar component. Points C and D are located at 1,2,4 m and 2, -1,1 m respectively. Determine its unit vector. Write PQ as a vector. Determine the position of B relative to the origin of the co-ordinate 1. Determine the position vector a from A to B and b from B to A. Determine the position vector from P to Q 1. The scalar or dot product involves the angle Determine the 1. The location of an aircraft in location of C.

Determine the '" The dimensions Of a room at 6 m x 5 m x 4m' as location in Cartesian and cylindrical co-ordinates. A cable is suspended from the point P in the ceiling and a lamp L at the end of the cable is 1. Determine the Cartesian and cylindrical co-ordinates of the lamp L relative to the x-, y-, z-axes and also find 1. What is the expressions for the corresponding cylindrical unit component of this vector a in the y-direction and b vectors eR, eo and e, in terms of i, j and k see in a direction parallel to the line from A to B, where A Fig.

Displacement of aparticle If a particle occupies position A at time tl and at a later time t2 it occupies a position B, then the displacement is the vector 3 as shown in Fig. Having defined velocity and acceleration in a quite general way, the components of these quantities for a particle confined to move in a plane can now be formulated. It is useful to consider the ways in which a vector quantity may change with time, as this will help in understanding the full meaning of acceleration.

Since velocity is defined by both magnitude and direction, a variation in either quantity will constitute a change in the velocity vector.

## Variational Principles in Classical Mechanics - Second Edition

If the velocity remains in a fixed direction, then the acceleration has a magnitude equal to the rate The direction of a is not obvious and will not be The acceleration is equally easy to derive. Av Ai Ay Figure 2. For this case we can see triangle. In the limit, for small changes in time, and hence small changes in direction, the change in velocity is normal to the velocity vector. Integrating again, Figure 2. Figure 2. This analysis should be contrasted with the more direct approach in terms of path and polar The change in velocity resolved tangentially to the path is co-ordinates shown later in this chapter.

During this interval e, and ee In this caseFigure 2. The magnitude of Ae, for independent of r. This component is often called small AB is 1xAO since the magnitude of e, is the Coriolis component, after the French unity, by definition. The acceleration a can also be found from the chain rule, thus d dt Figure 2. The differentiation of rotating vectors is dealt with more fully in Chapter The path traced out by a point B on the rim of the wheel is complex, but the velocity and acceleration of B may be easily obtained by use of equations 2.

As before we consider the two simple cases. This quantity is also the component of acceleration tangential to the path, but it is not the total acceleration. Since 2. As an example: given the way in which a component of acceleration varies with displacement, determine the variation of speed with time. In such problems the sketching of appropriate graphs is a useful aid to the 2.

We have one-dimensional motion in path co-ordinates if we consider only displacement along the path; in polar co-ordinates we can consider only variations in angle, regarding the radius as constant. Let us consider a problem in path co-ordinates, Fig. This path could, of course, be a straight line. The distance moved along the path is s. Solution Rate-of-change-of-speed-time graph Fig. For such problems, however, the methods of section 2. Determine the speed the time taken. Solution a We are given a, in terms of s and require to find v, therefore we must use an expression relating these three parameters.

The constant- acceleration formulae are of course not relevant here. At the same time the distance travelled, s, is recorded with the following results see section 3. To find values of v at various 2. Corresponding values are given below and are plotted in Fig. The centre C of the wheel of radius 0.

The angular velocity of the wheel is constant and equal to 6 rads clockwise. Program outcome 1 Apply the learning objects to real engineering problems Program outcome 2. Hibbeler, R. McGill, D. Merriam, J. Pytel, A. Riley, W. Ali Kosar - kosara sabanciuniv. Prerequisites only for SU students. Common Outcomes For All Programs. Understand the world, their country, their society, as well as themselves and have awareness of ethical problems, social rights, values and responsibility to the self and to others. Understand different disciplines from natural and social sciences to mathematics and art, and develop interdisciplinary approaches in thinking and practice.

Think critically, follow innovations and developments in science and technology, demonstrate personal and organizational entrepreneurship and engage in life-long learning in various subjects. Communicate effectively in Turkish and English by oral, written, graphical and technological means. Take individual and team responsibility, function effectively and respectively as an individual and a member or a leader of a team; and have the skills to work effectively in multi-disciplinary teams. Common Outcomes ForFaculty of Eng. Possess sufficient knowledge of mathematics, science and program-specific engineering topics; use theoretical and applied knowledge of these areas in complex engineering problems.

Identify, define, formulate and solve complex engineering problems; choose and apply suitable analysis and modeling methods for this purpose. Develop, choose and use modern techniques and tools that are needed for analysis and solution of complex problems faced in engineering applications; possess knowledge of standards used in engineering applications; use information technologies effectively. Ability to design a complex system, process, instrument or a product under realistic constraints and conditions, with the goal of fulfilling specified needs; apply modern design techniques for this purpose.

Design and conduct experiments, collect data, analyze and interpret the results to investigate complex engineering problems or program-specific research areas. Knowledge of business practices such as project management, risk management and change management; awareness on innovation; knowledge of sustainable development. Knowledge of impact of engineering solutions in a global, economic, environmental, health and societal context; knowledge of contemporary issues; awareness on legal outcomes of engineering solutions; understanding of professional and ethical responsibility.

Familiarity with concepts in statistics and optimization, knowledge in basic differential and integral calculus, linear algebra, differential equations, complex variables, multi-variable calculus, as well as physics and computer science, and ability to use this knowledge in modeling, design and analysis of complex dynamical systems containing hardware and software components.