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# ENG1040 Lec05

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### ENG1040 Lec05

1. 1. Faculty of Engineering ENG1040 Engineering Dynamics Particle vs Rigid Bodies Free body diagrams Dr Lau Ee Von – Sunway Lecture 5 ENG1040 – Engineering Dynamics
2. 2. Lecture Outline Lecture Outline Particles vs Rigid Bodies Review : Newton’s 2nd Law Relating reality to theory • • • • • Particle vs Rigid bodies Review of Newton’s second law Free body diagrams Forces to consider An Example Forces to consider Free Body Diagrams Example 2
3. 3. Particles vs Rigid Bodies Lecture Outline Particles vs Rigid Bodies Review : Newton’s 2nd Law When we apply Newton’s Laws of Motion we need to be clear on a number of matters. • What is a particle? We can treat a body as a particle if: Relating reality to theory • Its geometry is irrelevant to its motion Forces to consider • Motion is through its mass centre Free Body Diagrams Example • Rotation is neglected / irrelevant A particle is a body of negligible dimensions For particle dynamics coverage, see Hibbeler Ch. 12-14 3
4. 4. Particles vs Rigid Bodies Lecture Outline Particles vs Rigid Bodies Review : Newton’s 2nd Law Relating reality to theory Forces to consider Free Body Diagrams When we apply Newton’s Laws of Motion we need to be clear on a number of matters. • What is a rigid body? We can treat a body as a rigid body if: • Geometry is relevant to its motion, but it experiences negligible changes in shape • Motion can be in rotation or translation • Motion depends on where forces are applied Example For rigid body dynamics coverage, see Hibbeler Ch. 16-18 4
5. 5. Particles vs Rigid Bodies Lecture Outline In this example which is a rigid body and which is a particle? Particles vs Rigid Bodies Review : Newton’s 2nd Law Relating reality to theory Forces to consider Free Body Diagrams Example 5
6. 6. Kinetics Lecture Outline Particles vs Rigid Bodies Review : Newton’s 2nd Law Relating reality to theory Forces to consider Free Body Diagrams Example In this unit, we will use Newton’s Laws as a model governing the motion of mechanical devices. He deduced three laws of motion. The Second Law
7. 7. Relating reality to theory Lecture Outline How do we transfer what we see in reality to our mathematical model? Particles vs Rigid Bodies Review : Newton’s 2nd Law Relating reality to theory Forces to consider F2 Free Body Diagrams F1 Example Answer: using a free body diagram (FBD)! mg 7
8. 8. Distributed Loading Lecture Outline Particles vs Rigid Bodies Review : Newton’s 2nd Law Relating reality to theory Forces to consider • Equivalent Point Forces: • Often we have a distributed load. Usually, we can approximate a distributed load with an equivalent point force. • For example, if someone sits on a see-saw, their mass is distributed across the chair they are sitting on. However we could approximate them as a point mass. Free Body Diagrams Example m1 g m2 g 8
9. 9. Distributed Loading Lecture Outline • Sometimes, we cannot Particles vs Rigid Bodies Review : Newton’s 2nd Law Relating reality to theory Forces to consider Free Body Diagrams Example A yacht mast is subject to distributed loading. It may not be appropriate to replace the distributed force with a point force (depending on what results we need) 9
10. 10. Forces to consider: Gravity • Gravity affects all masses (and is referred to as a body force) • In FBD, the body force (i.e. weight) always act downwards Lecture Outline Particles vs Rigid Bodies Review : Newton’s 2nd Law Relating reality to theory At sea level, the force due to gravity is defined as: F Forces to consider Free Body Diagrams Example m g Typically (in SI units) the acceleration due to gravity is defined at sea-level as: g 9.81 m s 2 10
11. 11. Forces to consider: Gravity Lecture Outline In FBD, the body force (i.e. weight) always act downwards Particles vs Rigid Bodies Review : Newton’s 2nd Law Relating reality to theory Forces to consider Free Body Diagrams Example 11
12. 12. Forces to consider: Normal Force When a mass is placed on a surface, the surface exerts a force normal to the surface, pushing the mass outward from the surface. Lecture Outline Particles vs Rigid Bodies Review : Newton’s 2nd Law Relating reality to theory Forces to consider Free Body Diagrams Fn This models the fact that the mass does not fall through the surface. Example Newton’s 3rd law shows why it is needed. 12
13. 13. Forces to consider: Normal Force Lecture Outline Particles vs Rigid Bodies In FBD, the normal force, Fn exists whenever there is contact between two surfaces. Review : Newton’s 2nd Law Relating reality to theory Forces to consider Free Body Diagrams Example When a mass is no longer in contact, Fn = 0 13
14. 14. Forces to consider: Friction Lecture Outline • The effect of friction is modeled as: Particles vs Rigid Bodies F fr Review : Newton’s 2nd Law Relating reality to theory Forces to consider • FN is the coefficient of friction (material dependent) • Fn is the normal force Free Body Diagrams Example • Kinetic friction always acts against the motion of the object. 14
15. 15. Forces to consider: Friction Static vs kinetic friction: Lecture Outline Particles vs Rigid Bodies Review : Newton’s 2nd Law Relating reality to theory Forces to consider Free Body Diagrams Example • Typically, there are two friction coefficients: • A static friction coefficient (µs) which must be overcome for an object to start slipping, • A kinetic friction coefficient (µk) which must be overcome for an object to continue slipping. • Typically, µs > µk • We need bigger force to overcome a still object than to overcome a moving object 15
16. 16. Forces to consider: Friction Lecture Outline A note on static friction: Particles vs Rigid Bodies Review : Newton’s 2nd Law Relating reality to theory Forces to consider Free Body Diagrams Example The equation: F fr , static static FN Provides the maximum static friction force. This has to be overcomed in order to move an object. A lower force acting on the object will not cause it to move 16
17. 17. Forces to consider: Friction Lecture Outline Particles vs Rigid Bodies Determine what happens to the block when an external force, F of 25 N is applied. (µs = 0.74) W = mg Review : Newton’s 2nd Law m = 10kg Relating reality to theory Ffr Forces to consider Free Body Diagrams Example F = 25 N Fy ma y 0, (external force) FN FN mg 98 .1N Maximum static friction needed to move the block, F fr , static ,max F fr , static ,max static F FN 72 .6 N i.e. Block does not move 17
18. 18. Forces to consider: Friction Lecture Outline Particles vs Rigid Bodies Determine what happens to the block when an external force, F of 100 N is applied. (µk = 0.57) W = mg Review : Newton’s 2nd Law m = 10kg Relating reality to theory Fr Forces to consider Free Body Diagrams Example F = 100 N Fy ma y 0, (external force) FN FN mg 98 .1N Maximum static friction needed to move the block, F fr , static ,max F fr , static ,max static F FN 72 .6 N i.e. Block moves (slides) to the left 18
19. 19. Forces to consider: Friction Lecture Outline Particles vs Rigid Bodies Determine what happens to the block when an external force, F of 100 N is applied. (µk = 0.57) W = mg Review : Newton’s 2nd Law m = 10kg Relating reality to theory Forces to consider Free Body Diagrams Example Fr + Fx Fr (external force) FN ma x 100 F = 100 N ma x 100 ax k FN 4.4m / s 2 10 a x i.e. Block moves to the left with an acceleration of 4.4m/s2 19
20. 20. Forces to consider: Friction Lecture Outline Static friction vs kinetic friction for wheels Particles vs Rigid Bodies Review : Newton’s 2nd Law Relating reality to theory Forces to consider Free Body Diagrams Remember: Example We only use kinetic friction when two surfaces are sliding past each other. For rolling tyres (without slip), we use the static friction coefficient. 20
21. 21. Forces to consider: Spring stiffness Lecture Outline Particles vs Rigid Bodies Review : Newton’s 2nd Law Relating reality to theory Forces to consider Free Body Diagrams Example • Linear springs are modelled with Hooke’s Law (1660): F k x • k is the spring coefficient. • x is the displacement. • Direction of displacement is important! 25
22. 22. Forces to consider: Spring stiffness Lecture Outline Extended spring Compressed spring Particles vs Rigid Bodies Mass Review : Newton’s 2nd Law Relating reality to theory Mass Forces to consider Free Body Diagrams FBD: FBD: Example 26
23. 23. Forces to consider: Damping Lecture Outline • Linear damping is modelled with the following equation: Particles vs Rigid Bodies F c v Review : Newton’s 2nd Law • c is the damping coefficient Forces to consider • v is the velocity Free Body Diagrams • All fluids provide some damping • Damping always acts against the direction of motion. Relating reality to theory Example 27
24. 24. Forces to consider • • • • • Lecture Outline Particles vs Rigid Bodies Review : Newton’s 2nd Law Relating reality to theory • Forces to consider In reality, all these forces (and possibly even more) have to be taken into account, and summed together form the LHS of Newton’s 2nd law: F Free Body Diagrams Example Gravity Normal Force Friction Spring stiffness Damping forces m a • In this unit, you will be told if you need to include a force. For example, you may be given a friction coefficient. • Usually gravity is always acting, and must always be included. 28
25. 25. A single mass example Lecture Outline Particles vs Rigid Bodies Review : Newton’s 2nd Law • Commence by defining the coordinate system. • It is usually better to define +ve in the direction of motion • Next draw a representation for the mass. • Finally draw all the forces acting on the mass. Relating reality to theory y m g Forces to consider Free Body Diagrams FFr x Example Fn x y 29
26. 26. A single mass example y • Finally, using our FBD, rewrite Newton’s 2nd law for the problem in each dimension: Lecture Outline Particles vs Rigid Bodies Review : Newton’s 2nd Law Relating reality to theory Forces to consider x FFr Fn y y y x mg x + + x Fx m ax Fx m ax Fy m ay Fy m ay Free Body Diagrams Example 30
27. 27. Dealing with multiple masses Lecture Outline Particles vs Rigid Bodies Review : Newton’s 2nd Law • Where there are multiple, independently moving masses, ISOLATE the body and draw one FBD for each mass. • Apply Newton’s 2nd law for each mass in isolation. • How many equations would you have for this system? Relating reality to theory Forces to consider Free Body Diagrams Example 31
28. 28. Free Body Diagrams Lecture Outline What are the forces acting on this body? Particles vs Rigid Bodies Draw simple diagrams (FBD) for each component (each isolated body) Review : Newton’s 2nd Law Relating reality to theory Forces to consider Free Body Diagrams Example T′ T′
29. 29. Dealing with elastic objects Lecture Outline Particles vs Rigid Bodies Review : Newton’s 2nd Law Relating reality to theory • In reality, all masses have some elastic properties. • We can assume there are no elastic properties if the body is rigid. • In order to model this, we separate out the forces acting on the mass: m g Forces to consider Free Body Diagrams Example y k x Fn 33
30. 30. Rotation Lecture Outline • Rotation of a mass adds further complexity to a problem. This will be considered from lecture 7. Particles vs Rigid Bodies Review : Newton’s 2nd Law Relating reality to theory Forces to consider Free Body Diagrams Example 34
31. 31. Free Body Diagrams Lecture Outline Particles vs Rigid Bodies Review : Newton’s 2nd Law Relating reality to theory Forces to consider Free Body Diagrams Example • Example • An elevator E has a mass of 500 kg and the counterweight at A as a mass of 150 kg. • If the elevator attains a speed of 10 m/s after it rises 40 m, determine the constant force developed in the cable at B. • Neglect the mass of the pulleys and cable.
32. 32. Conclusions • Free Body Diagrams (FBD) allow us to separate a difficult problem into a series of simpler problems • They also remind us of all the forces acting on all the components of a mechanical device. • They are essential to solving dynamics problems successfully 44