Theoretical Mechanics Practicums

Transcription

1 + ALEKSANDRAS STULGINSKIS UNIVERSITY Faculty of Agricultural Engineering Department of Mechanics Eglė Jotautienė Theoretical Mechanics Practicums KAUNAS - AKADEMIJA 2012

2 UDK 629.1: Eglė Jotautienė Theoretical Mechanics Practicums Reviewers: Assoc. Prof. dr. Sigitas Petkevičius, Department of Mechanical Engineering; Lector dr. Rytis Skominas, Department of Civil Constructions. Approval: 28 August 2012, Meeting of the Department of Mechanical Engineering, Minutes No. 01. Technology in the field of Environmental Engineering Sciences (General direction of Engineering) and Water Engineering (Direction of Civil Engineering) branches of study committee meeting, , Minutes No.18 Eglė Jotautienė, 2012 Aleksandras Stulginskis University,

3 CONTENTS PREFACE 4 Problem 1. Statics of Solid Body Problem 2. Analysis of Trusses Problem 3. Body Equilibrium with Friction Force Problem 4. Determine the Support Reaction in Three Dimensions Problem 5. Body Movement and Rotation References

4 Preface Mechanical science evolution is closely related to the development of the productive events of human history. With the advancement of technologies, the requirements for engineers are continuously increasing. Many technical issues developing now require the first-year students to become proficient in basic subjects such as mathematics, physics, theoretical mechanics, strength of materials, etc. Theoretical mechanics for engineering study students is not only the basic study objective, but it is also a tool to be applied in practice. It is important to acquire the fundamental background be able to apply the concepts of mechanics and analysis techniques in practice because it is not easy to apply the mathematical formulations to practical problems. This Practicums book is designed for students of engineering profile. 4

5 The tasks are for the following topics: Force composition and decomposition; System equilibrium of intersecting force; Parallel force system equilibrium; Problem 1. Statics of Solid Body Creation of planar force system equilibrium equations and definition forces, acting on the support. A force F, gravitational weight G force, moment M and the distributed force with intensity q are acting on the homogeneous beam (see Figure 1, 2). Determine the reactions at beam supports when dimensions are given in meters and load kilo newton. Task data and schema are presented in Table 1 and Figure 1, 2. Table 1. Task data [6,7] Variant F kn G kn M kn m q k/nm α degree , ,5 30 Methodological advice. In order to perform the task you need to know the static axioms, force projections in axis, moment of force about point, supports reactions, principles of the resulting equilibrium equations. In calculating the reaction of supports, the first step is to indicate the reaction forces in the drawing, second - to choose the coordinate axes, and then present the calculation scheme. The planar in force system requires the three equations of equilibrium [1,2]: F = 0 ; F = 0 ; M = x y 0 0. (1.1) 5

6 Solving these equations three parameter are obtained. For the verification, the moment equilibrium equation for the next point also can be solved. If the problem is solved correctly, the same answer will be obtained. 6

7 Fig. 1. Task schema [6,7] 7

8 Fig. 2. Task schema [6,7] 8

9 Problem 2. Analysis of Trusses The tasks are for the following topics: Stress evaluation of truss using the method of joins; Stress evaluation of truss using the method of sections. Forces F 1, F 2, F 3 are acting on a planar truss (see Figure 3, 4). Determine the reactions at truss supports, the forces in each member of the truss using the method of joins and in three members using the method of sections (the parameters of members are presented in Table 2) Dimensions given in meters, and loads kilo newtons. Task data and schema are presented in Table 2, Figure 3, 4. Table 2. Task data[6,7] Variant F 1 F 2 F 3 a h α number of kn kn kn m m degree members ,8, ,6, ,11, ,6, ,5, ,9, ,6, ,4, ,7, ,10, ,7, ,5, ,5, ,5, ,9, ,6, ,9, ,4, ,5, ,7,8 Methodological advice. First step is to find the reactions of truss supports. For this purpose, you should apply the three equilibrium equations: F = 0 ; F = 0 ; M = x y 0 0. Solve for the three unknown member forces and verify their correct direction. Forces in each member of the truss may be obtained by applying the two force equations of joint equilibrium ( F = 0 ; F = 0. ) [3,4]. Continue to analyse each of x the other joints, where again it is necessary to choose a joint having at most two unknowns and at least one known force. The forces in the truss members are determined by the method of sections y 9

10 where first you make a cut section where forces are to be determined. Before isolating an appropriate section, it may be necessary to determine truss external reactions. Three equilibrium equations are available to solve for the members forces at the cut section. 10

11 Fig. 3. Task schema [6,7] 11

12 Fig. 4. Task schema [6,7] 12

13 Problem 3. Body Equilibrium with Friction Force The tasks are for the following topic: Body equilibrium with friction force. The weight forces G 1 and G 2 are acting on the mechanical system (see Figure 5, 6). Determine the minimum (at variants 1-20, 25, 26, 29, 30) and the maximum (at variants 21 24, 27, 28) forces and the reactions at supports when the system is in equilibrium. The weight forces G 1, G 2, dimentions a, b, c, angle α and coefficient of static friction f are given in Table 3. In the variants 1-20, the friction force is evaluated only between the wheel and the slipper brake. Table 3. Task data [6,7] Variant G 1 G 2 a b c α f kn kn m m m degree , Methodological advice. Between the wheel and slipper brake (1-20 variants) the friction forces are acting. According to Coulomb law, the friction force F tr = fn [5]. Here N is the normal force, f is the coefficient of static friction. The mechanism must be disassembled into two bodies at friction contact points. Plot the separate parts, add the support reactions and the reactions of normal and friction forces. Write to each part three equilibrium equations and one friction balance equation. Solve for the unknown forces and verify their correct direction. 13

14 Fig.5. Task schema [6,7] 14

15 Fig. 6. Task schema [6,7] 15

16 Variant Problem 4. Determine the Support Reactions in Three Dimensions The tasks are for the following topic: Development of spatial force system equilibrium equations and definition of support reactions. The forces F 1, F 2 and weight G are acting on the construction (see Figure 7, 8). Determine support reactions of the construction and the magnitude of force F 2. Dimensions given in centimeters, and loads kilo newton. Task data are presented in Table 4. Table 4. Data[6,7] Forces kn Dimensions cm F 1 F 2 G a b c R r Methodological advice. In solving the problem, all forces acting on the mechanical system must be shown. Applying the six scalar equations of equilibrium, the three-dimensional equilibrium problem can be solved. For verification, the moment equilibrium equation for the new chosen axes can be solved. 16

17 Fig. 7. Task schema [6,7] 17

18 Fig. 8. Task schema [6,7] 18

19 Problem 5. Body Movement and Rotation The tasks are for the following topic: Investigation of point movement; Definition of point (body) kinematic parameter. A mechanism acted by weight (see Figure 9, 10) is moving under the given law of sliding motion. Determine speed, tangential and normal acceleration, absolute acceleration of point M located on the mechanism when given rod s of body 1. Task data and schema are presented in Table 5, Figures 9, 10. Variant Table 5. Task data [6, 7] Radii, cm R 2 r 2 R 3 r 3 x=f(t), cm; t, s s, m t t t t t t t t t t t t t t t t t t t Methodological advice. Solving problem of finding a mechanism to determine velocity of point M when the wheels are connected directly to wheels or to belt. For this purpose, you should apply the expressions [1, 5]: V=ωR; ω 1 /ω 2 =r 2 /r 1 ; here ω ; ω 1 ; ω 2 are the angular velocities; R 1, r 1, r 2 are the radii of the wheels. 19

20 Fig.9. Task schema [6,7] 20

21 Fig. 10. Task schema [6, 7] 21

22 References 1. Hibbeler R.C. Engineering mechanics: Statics&Dynamics. - Prentice Hall, Inc., USA p. 2. Sandor B.I. Engineering mechanics: Statics&Dynamics - Prentice Hall, Inc., USA p 3. Bedford A., Fowler W. Engineering mechanics. - Prentice Hall, Inc., USA p. 4. Meriam J. L., Kraige L. G. Engineering Mechanics: Dynamics. Fifth edition. John Wiley & Sons, Inc., USA p. 5. Beer F.P., Russell Johnston E. Jr. Vector Mechanics for Engineers. Statics&Dynamics. McGraw Hill Companies, Inc., USA p. 6. Jotautienė E., Bozys J. Teorinės mechanikos savarankiško darbo užduotys ir metodiniai patarimai. Statika. Kinematika. Dinamika. Akademija, p. 7. Teorinės mechanikos namų darbai ir metodiniai patarimai jiems atlikti. I dalis. Akademija, p. 22