eng1040 lec05
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Faculty of Engineering
ENG1040 – Engineering Dynamics
ENG1040Engineering Dynamics
Particle vs Rigid BodiesFree body diagrams
Dr Lau Ee Von – Sunway
Lecture 5
Lecture Outline
2
Lecture Outline
Particles vs Rigid Bodies
Review :Newton’s 2nd Law
Relating reality to theory
Forces to consider
Free Body Diagrams
Example
• Particle vs Rigid bodies• Review of Newton’s second law• Free body diagrams• Forces to consider• An Example
Particles vs Rigid Bodies
3
• What is a particle?
A particle is a body of negligible dimensions
We can treat a body as a particle if:
• Its geometry is irrelevant to its motion
• Motion is through its mass centre
• Rotation is neglected / irrelevant
For particle dynamics coverage, see Hibbeler Ch. 12-14
When we apply Newton’s Laws of Motion we need to be clear on a number of matters.
Lecture Outline
Particles vs Rigid Bodies
Review :Newton’s 2nd Law
Relating reality to theory
Forces to consider
Free Body Diagrams
Example
Particles vs Rigid Bodies
4
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
• What is a rigid body?
For rigid body dynamics coverage, see Hibbeler Ch. 16-18
When we apply Newton’s Laws of Motion we need to be clear on a number of matters.
Lecture Outline
Particles vs Rigid Bodies
Review :Newton’s 2nd Law
Relating reality to theory
Forces to consider
Free Body Diagrams
Example
Particles vs Rigid Bodies
5
In this example which is a rigid body and which is a particle?
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.
Kinetics
A particle acted upon by an unbalanced force F experiences an acceleration a that has the same direction as the force and a magnitude that is directly proportional to the force.
The Second Law
Lecture Outline
Particles vs Rigid Bodies
Review :Newton’s 2nd Law
Relating reality to theory
Forces to consider
Free Body Diagrams
Example
Relating reality to theory
How do we transfer what we see in reality to our mathematical model?
7
mg
1F2F
Answer: using a free body diagram
(FBD)!
Lecture Outline
Particles vs Rigid Bodies
Review :Newton’s 2nd Law
Relating reality to theory
Forces to consider
Free Body Diagrams
Example
Distributed Loading
• 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.
8
gm1 gm2
Lecture Outline
Particles vs Rigid Bodies
Review :Newton’s 2nd Law
Relating reality to theory
Forces to consider
Free Body Diagrams
Example
Distributed Loading
• Sometimes, we cannot
9
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)
Lecture Outline
Particles vs Rigid Bodies
Review :Newton’s 2nd Law
Relating reality to theory
Forces to consider
Free Body Diagrams
Example
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
10
At sea level, the force due to gravity is defined as:
Typically (in SI units) the acceleration due to gravity is defined at sea-level as:
gmF
2 81.9 smg
Lecture Outline
Particles vs Rigid Bodies
Review :Newton’s 2nd Law
Relating reality to theory
Forces to consider
Free Body Diagrams
Example
Forces to consider: Gravity
11
Lecture Outline
Particles vs Rigid Bodies
Review :Newton’s 2nd Law
Relating reality to theory
Forces to consider
Free Body Diagrams
Example
In FBD, the body force (i.e. weight) always act downwards
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.
This models the fact that the mass does not fall through the surface.
Newton’s 3rd law shows why it is needed.
Lecture Outline
Particles vs Rigid Bodies
Review :Newton’s 2nd Law
Relating reality to theory
Forces to consider
Free Body Diagrams
Example
Fn
Forces to consider: Normal Force
13
Lecture Outline
Particles vs Rigid Bodies
Review :Newton’s 2nd Law
Relating reality to theory
Forces to consider
Free Body Diagrams
Example
In FBD, the normal force, Fn exists whenever there is contact between two surfaces.
When a mass is no longer in contact, Fn = 0
• The effect of friction is modeled as:
• is the coefficient of friction (material dependent)
• Fn is the normal force
14
Forces to consider: Friction
Nfr FF
• Kinetic friction always acts against the motion of the object.
Lecture Outline
Particles vs Rigid Bodies
Review :Newton’s 2nd Law
Relating reality to theory
Forces to consider
Free Body Diagrams
Example
Static vs kinetic friction:
• 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.
15
Forces to consider: Friction
• Typically, µs > µk
• We need bigger force to overcome a still object than to overcome a moving object
Lecture Outline
Particles vs Rigid Bodies
Review :Newton’s 2nd Law
Relating reality to theory
Forces to consider
Free Body Diagrams
Example
16
Forces to consider: Friction
A note on static friction:
The equation:
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
Nstaticstaticfr FF ,
Lecture Outline
Particles vs Rigid Bodies
Review :Newton’s 2nd Law
Relating reality to theory
Forces to consider
Free Body Diagrams
Example
17
Forces to consider: Friction
NFF Nstaticxmastaticfr 6.72,,
F = 25 N (external force)m = 10kg
FN
Ffr
W = mg
Determine what happens to the block when an external force, F of 25 N is applied. (µs = 0.74)
Maximum static friction needed to move the block,
NmgFmaF Nyy 1.98,0
FF xmastaticfr ,, i.e. Block does not move
Lecture Outline
Particles vs Rigid Bodies
Review :Newton’s 2nd Law
Relating reality to theory
Forces to consider
Free Body Diagrams
Example
18
Forces to consider: Friction
NFF Nstaticxmastaticfr 6.72,,
F = 100 N (external force)m = 10kg
FN
Fr
W = mg
Determine what happens to the block when an external force, F of 100 N is applied. (µk = 0.57)
Maximum static friction needed to move the block,
NmgFmaF Nyy 1.98,0
FF xmastaticfr ,,i.e. Block moves (slides) to the left
Lecture Outline
Particles vs Rigid Bodies
Review :Newton’s 2nd Law
Relating reality to theory
Forces to consider
Free Body Diagrams
Example
19
Forces to consider: Friction
F = 100 N (external force)m = 10kg
FN
Fr
W = mg
Determine what happens to the block when an external force, F of 100 N is applied. (µk = 0.57)
i.e. Block moves to the left with an acceleration of 4.4m/s22/4.4
10100
100
sma
aF
maF
maF
x
xNk
xr
xx
+
Lecture Outline
Particles vs Rigid Bodies
Review :Newton’s 2nd Law
Relating reality to theory
Forces to consider
Free Body Diagrams
Example
20
Forces to consider: Friction
Static friction vs kinetic friction for wheels
Remember:
We only use kinetic friction when two surfaces are sliding past each other.
For rolling tyres (without slip), we use the static friction coefficient.
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):
• k is the spring coefficient.
• x is the displacement.
• Direction of displacement is important!
25
Forces to consider: Spring stiffness
xkF
Lecture Outline
Particles vs Rigid Bodies
Review :Newton’s 2nd Law
Relating reality to theory
Forces to consider
Free Body Diagrams
Example
26
Forces to consider: Spring stiffness
Mass
FBD:
Compressed springExtended spring
Mass
FBD:
Lecture Outline
Particles vs Rigid Bodies
Review :Newton’s 2nd Law
Relating reality to theory
Forces to consider
Free Body Diagrams
Example
27
Forces to consider: Damping
• Linear damping is modelled with the following equation:
vcF
• c is the damping coefficient
• v is the velocity
• All fluids provide some damping
• Damping always acts against the direction of motion.
Lecture Outline
Particles vs Rigid Bodies
Review :Newton’s 2nd Law
Relating reality to theory
Forces to consider
Free Body Diagrams
Example
Forces to consider
• Gravity• Normal Force• Friction• Spring stiffness• Damping forces
• 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:
• 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
amF
Lecture Outline
Particles vs Rigid Bodies
Review :Newton’s 2nd Law
Relating reality to theory
Forces to consider
Free Body Diagrams
Example
• 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.
29
A single mass example
x
ynF
FrF
gmy
x
Lecture Outline
Particles vs Rigid Bodies
Review :Newton’s 2nd Law
Relating reality to theory
Forces to consider
Free Body Diagrams
Example
A single mass example
• Finally, using our FBD, rewrite Newton’s 2nd law for the problem in each dimension:
30
x
ynF
FrF
gm
xx amF
yy amF
y
x
x
y
+
y
x+
xx amF
yy amF
Lecture Outline
Particles vs Rigid Bodies
Review :Newton’s 2nd Law
Relating reality to theory
Forces to consider
Free Body Diagrams
Example
Dealing with multiple masses
• 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?
31
Lecture Outline
Particles vs Rigid Bodies
Review :Newton’s 2nd Law
Relating reality to theory
Forces to consider
Free Body Diagrams
Example
Free Body Diagrams
What are the forces acting on this body?Lecture Outline
Particles vs Rigid Bodies
Review :Newton’s 2nd Law
Relating reality to theory
Forces to consider
Free Body Diagrams
Example
Draw simple diagrams (FBD) for each component (each isolated body)
T′ T′
Dealing with elastic objects
• 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:
33
nFxk
gm
y
Lecture Outline
Particles vs Rigid Bodies
Review :Newton’s 2nd Law
Relating reality to theory
Forces to consider
Free Body Diagrams
Example
Rotation
• Rotation of a mass adds further complexity to a problem. This will be considered from lecture 7.
34
Lecture Outline
Particles vs Rigid Bodies
Review :Newton’s 2nd Law
Relating reality to theory
Forces to consider
Free Body Diagrams
Example
Free Body Diagrams
• 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.
Lecture Outline
Particles vs Rigid Bodies
Review :Newton’s 2nd Law
Relating reality to theory
Forces to consider
Free Body Diagrams
Example
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
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