1. Theoretical Mechanics STATICS KINEMATICS * Navigation: Right (Down) arrow – next slide Left (Up) arrow – previous slide Esc – Exit Notes and Recommendations: ruschev@tu-plovdiv.bg Technical University of Sofia Branch Plovdiv

2. Details of Lecturers Course Lecturers: Prof. Sonia Tabakova Room Number: 4237, Faculty of Mechanical Engineering Email: sonia@tu-plovdiv.bg Office Hours: 14.00-16.00 Tuesday Assoc. Prof. Dechko Ruschev Room Number: 3226, Faculty of Mechanical Engineering Email: ruschev@tu-plovdiv.bgruschev@tu-plovdiv.bg Office Hours: 12.00 -14.00 Wednesday

3. COURSE GOALS (i)To introduce students to basic concepts of force, couples, moments and mechanical motion in two and three dimensions. (ii)To develop analytical skills relevant to the areas mentioned in (i) above. (iii)To provide the methodology of modeling and solving of practical problems in the area of the Statics and Kinematics with engineering application.

4. Course Content Introduction to Mechanics. Fundamental Concepts. Introduction to Statics. Axioms of Statics. Two and Three - Dimensional Force systems. Equilibrium. Distributed Forces. Structures. Kinematics of a Particle. Description of Motion in Vector and Cartesian Approach. Rectilinear Kinematics. General Curvilinear Motion. Relative Motion of a Particle. Planar Kinematics of a Rigid Body. Three-DimensionaI Kinematics of a Rigid Body.

5. Bibliography Meriam J.L., L.G. Kraige, Engineering Mechanics - vol.1 Statics, vol. 2 Dynamics, 5 th edition., John Wiley & Sons, 2002; Hibbeler R.C., Engineering Mechanics - Statics, Dynamics,12 th edition, McMillan, New York, 2009; Beer F.P., E. R. Johnston Jr., D. F. Mozurek, E. R. Eisenberg, Vector Mechanics for Engineers (Statics and Dynamics), 9 th edition, McGraw-Hill, 2009; Ruina A, R. Pratap, Introduction to Statics and Dynamics, Pre-print for Oxford University Press, January 2002.

6. Lecture 1 Idealizations. Models or idealizations are used in mechanics in order to simplify application of the theory. Here we will consider three important idealizations. A particle is a body of negligible dimensions. In the mathematical sense, a particle is a body whose dimensions are considered to be near zero so that we may analyze it as a mass concentrated at a point. We often choose a particle as a differential element of a body. We may treat a body as a particle when its dimensions are irrelevant to the description of its position or the action of forces applied to it. For example, the size of the earth is insignificant compared to the size of its orbit, and therefore the earth can be modeled as a particle when studying its orbital motion. When a body is idealized as a particle the principles of mechanics reduce to a rather simplified form since the geometry of the body will not be involved in the analysis of the problem. Rigid body. A rigid body can be considered as a combination of a large number of particles in which all the particles remain at a fixed distance from one another, both before and after applying a load. This model is important because the material properties of any body that is assumed to be rigid will not have to be considered when studying the effects of forces acting on the body. In most cases the actual deformations occurring in structures, machines, mechanisms are relatively small and the rigid-body assumption is suitable for analysis. For instance, the calculation of the tension in the cable, which supports the boom of a mobile crane under load is essentially unaffected by the small internal deformations in the structural members of the boom. For the purpose, then, of determining the external forces which act on the boom, we may treat it as a rigid body. Mechanical system - a system of elements (particles and/or rigid-bodies) that interact on mechanical principles.

7. Lecture 1 FORCE ON A PARTICLE. RESULTANT OF TWO FORCES Experimental evidence shows that two forces P and Q acting on a particle A can be replaced by a single force R which has the same effect on the particle. This force is called the resultant of the forces P and Q and can be obtained, as shown, by constructing a parallelogram, using P and Q as two adjacent sides of the parallelogram. The diagonal that passes through A represents the resultant. This method for finding the resultant is known as the parallelogram law for the addition of two forces.

8. Lecture 1 Sample Problem 1 The two forces act on a bolt at A. Determine their resultant. SOLUTION: Graphical solution - construct a parallelogram with sides in the same direction as P and Q and lengths in proportion. Graphically evaluate the resultant which is equivalent in direction and proportional in magnitude to the diagonal. Trigonometric solution - use the triangle rule for vector addition in conjunction with the law of cosines and law of sines to find the resultant. Analytical solution

9. Lecture 1 Sample Problem 1 Contd. Graphical solution - A parallelogram with sides equal to P and Q is drawn to scale. The magnitude and direction of the resultant or of the diagonal to the parallelogram are measured, Graphical solution - A triangle is drawn with P and Q head-to-tail and to scale. The magnitude and direction of the resultant or of the third side of the triangle are measured,

10. Lecture 1 Sample Problem 1 Contd. Trigonometric solution - Apply the triangle rule. From the Law of Cosines, From the Law of Sines,

11. Lecture 1 Sample Problem 1 Contd. Analytical solution

12. Lecture 1 Sample Problem 2 A barge is pulled by two tugboats. If the resultant of the forces exerted by the tugboats is a 5000-lb force directed along the axis of the barge, determine : ( a ) the tension in each of the ropes knowing that a  = 45°, ( b ) the value of  for which the tension in rope 2 is minimum.

13. Lecture 1 SOLUTION (a) Tension for a  = 45°. The triangle rule can be used. We note that the triangle shown represents half of the parallelogram shown above. Using the law of sines, we write: With a calculator, we first compute and store the value of the last quotient. Multiplying this value successively by sin 45° and sin 30°, we obtain:

14. Lecture 1 SOLUTION (b) Value of  for minimum T 2. To determine the value of  for which the tension in rope 2 is minimum, the triangle rule is again used. In the sketch shown, line 1-1’ is the known direction of T 1. Several possible directions of T 2 are shown by the lines 2-2’. We note that the minimum value of T 2 occurs when T 1 and T 2 are perpendicular. The minimum value of T 2 is Corresponding values of T 1 and  are: