WEEK 1 Dynamics of Machinery

Transcription

1 WEEK 1 Dynamics of Machinery References Theory of Machines and Mechanisms, J.J. Uicker, G.R.Pennock ve J.E. Shigley, 2003 Makine Dinamiği, Prof. Dr. Eres SÖYLEMEZ, 2013 Uygulamalı Makine Dinamiği, Jeremy Hirschhorn, Çeviri: Prof.Dr. Mustafa SABUNCU,

2 Course Policy Two Midterms (20%) +Homework (10%) + 1 Final (50%) 2

3 A mechanism is a device which transforms motion to some desirable pattern and typically develops very low forces and transmits little power. A machine typically contains mechanisms which are designed to provide significant forces and transmit significant power Some examples of common mechanisms are a pencil sharpener, a camera shutter, an analog clock, a folding chair, an adjustable desk lamp, and an umbrella. Some examples of machines which possess motions similar to the mechanisms listed above are a food blender, a bank vault door, an automobile transmission, a bulldozer, and a robot. 3

4 Degree of freedom of a rigid body: The degree of freedom (DOF) of a rigid body is the number of independent parameters that define its configuration Six independent parameters are required to define the motion of the ship. An unrestrained rigid body in space has six degrees of freedom: three translating motions along the x, y and z axes and three rotary motions around the x, y and z axes respectively. Degree of freedom a kinematic pair: The degrees of freedom (DOF) of a kinemtaic pairs defined as the number of independent movements it has. 4

5 Theory of machines is separated into two section Dynamics is also separated into two section Statics: is that branch of theory of machines which deals with the forces and their effects, while the machine parts are rest. Dynamics: is that branch of theory of machines which deals with the forces and their effects, while acting upon the machine parts in motion. Kinematics: Kinematic analysis involves determination of position, displacement, rotation, speed, velocity, and acceleration of a mechanism. Kinetics: It is that branch of theory of machines which deals with the inertia forces which arise from the combined effect of the mass and motion of the machine parts. Kinetics analysis will be used for this lecture. 5

6 Newton's Three Laws of Motion: 6

7 F = ma m F = ma F + F = ma a = 1 2 F net m 7

8 Law 3: Reaction is always equal and opposite to action; that is to say, the actions of two bodies upon each other are always equal and directly opposite. 8

9 Rigid Body: is that body whose changes in shape are negligible compared with its overall dimensions or with the changes in position of the body as a whole, such as rigid link, rigid disc..etc. Links: are rigid bodies each having hinged holes or slot to be connected together by some means to constitute a mechanism which able to transmit motion or forces to some another locations. 9

10 FORCE AND MOMENT VECTORS A force is characterized by its magnitude and direction, and thus is a vector. In an (x, y)-plane the force vector, F, can be represented in different forms The characteristics of a force are its magnitude, its direction, and its point of application. The direction of a force includes the concept of a line along which the force is acting, and a sense. Thus a force may be directed either positively or negatively along its line of action. 10

11 Two equal and opposite forces along two parallel but noncollinear straight lines in a body cannot be combined to obtain a single resultant force on the body. Any two such forces acting on the body constitute a couple. The arm of the couple is the perpendicular distance between their lines of action, shown as h in the Figure, and the plane of the couple is the plane containing the two lines of action. M = R F =. A BA or M h F A or M = R F A BA M = R sin θ F= hf. A ( ) BA 11

12 FORCES IN MACHINE SYSTEMS A machine system is considered to be a system of an arbitrary group of bodies (links), which will be considered rigid. We are involved with different types of forces in such systems. a) Reaction Forces: are commonly called the joint forces in machine systems since the action and reaction between the bodies involved will be through the contacting kinematic elements of the links that form a joint. The joint forces are along the direction for which the degree-of-freedom is restricted. e.g. in constrained motion direction All lower pairs and their constraint forces: (a) revolute or turning pair with pair variable θ 12

13 (b) prismatic or sliding pair with pair variable z (c) cylindric pair with pair variables θ and z (d) screw or helical pair with pair variables θ and z 13

14 (e) planar or flat pair with pair variables x, z, and θ. (f) spheric pair with pair variables θ, φ and ψ 14

15 Reaction Forces 15

16 b) Physical Forces : As the physical forces acting on a rigid body we shall include external forces applied on the rigid body, the weight of the rigid body, driving force, or forces that are transmitted by bodies that are not rigid such as springs or strings attached to the rigid body. weight external forces Spring force 16

17 c) Friction or Resisting Force: The friction force is the force exerted by a surface as an object moves across it or makes an effort to move across it. F = R = ( ) µ F friction F tan φ = µ = friction F µ: coefficient of friction 23 d) Inertia Forces are the forces due to the inertia of the rigid bodies involved. F i a F i F = ( ) i ma = ma m: mass, Kg a=acceleration, m/s 2 F=force, N 17

18 Inertia Torque i F = ma G T i = I α G m: mass, Kg a=acceleration, m/s 2 F=force, N T=Torque, Nm I= Moment of Inertia, kgm 2 18

19 We shall be using SI systems of units. A list of units relevant to the course is given below SI System of units 19

20 Moment of Inertia of a Mass For a body of mass m the resistance to rotation about the axis AA is I = = m + r2 m + r3 m + 2 r r dm = mass moment of inertia The radius of gyration for a concentrated mass with equivalent mass moment of inertia is I k m k = = 2 Moment of inertia with respect to the y coordinate axis is 2 2 I y ( 2 ) z x = r dm = + dm Similarly, for the moment of inertia with respect to the x and z axes, ( 2 2 ) I = y + z dm In SI units, x I = kg m I = x + y dm r dm = z ( ) I m ( 2 ) 20

21 Moments of Inertia of Common Geometric Shapes 21

22 FREE-BODY DIAGRAMS A free-body diagram is a sketch or drawing of the body, isolated from the rest of the machine and its surroundings, upon which the forces and moments are shown in action. A free body diagram shows all forces of all types acting on this body 22

23 Slider Crank Mechanism external force and torque, F 4 and T 2 All frictions are neglected except for the friction at joint 14 23

24 STATIC EQUILIBRIUM: A body is said to be in static equilibrium if under a set of applied forces and torques its translational (linear) and rotational accelerations are zeros (a body could be stationary or in motion with a constant linear velocity). Planar static equilibrium equations for a single body that is acted upon by forces and torques are expressed as F F = = 0 x Fy = 0 0 M z = M = 0 24

25 Two-force member: If only two forces act on a body that is in static equilibrium, the two forces are along the axis of the link, equal in magnitude, and opposite in direction.. If an element has pins or hinge supports at both ends and carries no load in-between, it is called a two-force member Two force and one moment member: A rigid body acted on by two forces and a moment is in static equilibrium only when the two forces form a couple whose moment is equal in magnitude but in opposite sense to the applied moment 25

26 Three-force member: If only three forces act on a body that is in static equilibrium, their axes intersect at a single point. A special case of the three-force member is when three forces meet at a pin joint that is connected between three links. When the system is in static equilibrium, the sum of the three forces must be equal to zero. For example, if the axes of two of the forces are known, the intersection of those two axes can assist us in determining the axis of the third force. 26

27 Let the force F A be completely specified. And the line of action of F B and the point of application of F C be known. When the moment equilibrium equation is written for the sum of moments about the point of intersection of the line of action of F A and F B (point O), since M O =0, the moment of F C about O must be zero, or the line of action of the force F C must pass through point O. The magnitudes of the forces can then be determined from the force and moment equilibrium equations. O their axes intersect at a single point. F = 0 27

28 Example: Find all the pin (joint) forces and the external torque M 12, that must be applied to link 2 of the mechanism (static). AO 2 =6 m, AB=18 m, BO 4 =12 m ve BQ=5 m GRAPHIC SOLUTION =120 N M 12 28

29 F 34 =33.1 N F 14 =89 N M 12 =183 N.m 29

30 ANALYTIC SOLUTION we sum moments about point O 4. Thus m m m m N N m m N N N N N 30

31 from the free-body diagram of link 2 m.n 31

32 Example: For the mechanism shown A 0 A= a 2 = 80, AB= a 3 =100, B 0 B= a 4 =120, A 0 B 0 = a 1 = 140, AC= b 3 = 70, BC=80 and B 0 D= b4=90 mm. When θ 12 =60 0, from kinematic analysis θ 13 =29,98 0, θ 14 = Two forces F 13 =50 N < and F 14 = 100 N < are acting on links 3 and 4 respectively 32

33 The free-body diagrams of the moving links are shown (STATIC). 33

34 The three equilibrium equations for link 4 are: 34

35 There are four unknowns in three equations, therefore the equations obtained from one free-body diagram is not enough to solve for the unknowns. Equations 1 and 2 can be used to solve for G 14x and G 14y, only when F 34xy and F 34y are determined. The three equilibrium equations for link 3 must also be written (note that F 34 and F 43 are of equal magnitude). Where a= (using the cosine theorem for the triangle ABC). 35

36 Equations 4 and 5 can be used to determine F 23x and F 23y Equations 3 and 6 must be used simultaneously to solve for F 34x and F 34y. Substituting the known values into equations 3 and 6 results: 36

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