eng1040 lec05

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


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




Forces to consider

Free Body Diagrams

Example


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




Forces to consider

Free Body Diagrams

Example


5

In this example which is a rigid body and which is a particle?

Lecture Outline




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




Forces to consider

Free Body Diagrams

Example


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




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




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




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




Forces to consider

Free Body Diagrams

Example

Forces to consider: Gravity

11

Lecture Outline




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




Forces to consider

Free Body Diagrams

Example

Fn

Forces to consider: Normal Force

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Lecture Outline




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

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Forces to consider: Friction

Nfr FF

• Kinetic friction always acts against the motion of the object.

Lecture Outline




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.

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• Typically, µs > µk

• We need bigger force to overcome a still object than to overcome a moving object

Lecture Outline




Forces to consider

Free Body Diagrams

Example

16


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




Forces to consider

Free Body Diagrams

Example

17


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




Forces to consider

Free Body Diagrams

Example

18


NFF Nstaticxmastaticfr 6.72,,


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




Forces to consider

Free Body Diagrams

Example

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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




Forces to consider

Free Body Diagrams

Example

20


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




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!

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Forces to consider: Spring stiffness

xkF

Lecture Outline




Forces to consider

Free Body Diagrams

Example

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Forces to consider: Spring stiffness

Mass

FBD:

Compressed springExtended spring

Mass

FBD:

Lecture Outline




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




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.

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amF

Lecture Outline




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.

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A single mass example

x

ynF

FrF

gmy

x

Lecture Outline




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:

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x

ynF

FrF

gm

xx amF

yy amF

y

x

x

y

+

y

x+

xx amF

yy amF

Lecture Outline




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?

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Lecture Outline




Forces to consider

Free Body Diagrams

Example

Free Body Diagrams

What are the forces acting on this body?Lecture Outline




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:

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nFxk

gm

y

Lecture Outline




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.

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Lecture Outline




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




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

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