velocity diagrams 1

When we are dealing with a RIGIDmember, the velocity of one point on that member relative to another point on that member is very special, because:

The distance betweenthe points cannot change

Velocity DiagramsA new technique

Lecture Outline

An Example Mechanism

VelocityDiagrams – why learn them?

VelocityDiagrams – A new technique

An Example

Analysis of a velocity diagram

Conclusion

Elastic bands – deform easily

Steel girders – much more rigid

The velocity of B relative to Amust be perpendicular to theline AB

If this were not so, there would be a component of the velocity of B relative to A along AB – the length would thus change

If this were not so, there would be acomponent of the velocity of B relative to Aalong AB – the length would thus change

If this were not so, there would be a component of the velocity of B relative to A along AB – the length would thus change

A

B

Velocity of B relative to A

Velocity DiagramsMotion of a Rigid Body

Lecture Outline

An ExampleMechanism

VelocityDiagrams – whylearn them?

VelocityDiagrams – Anew technique

An Example

Analysis of a velocity diagram

Conclusion

The velocity of A relative to Bmust also be perpendicular to the line AB.

The member AB is still of constant lengthThe member AB is still of constant lengthThe member AB is still of constant length

A

B

Direction of relative velocity

Velocity DiagramsMotion of a Rigid Body

Lecture Outline

An ExampleMechanism

VelocityDiagrams – whylearn them?

VelocityDiagrams – Anew technique

An Example

Analysis of a velocity diagram

Conclusion

The key point is:There can be no component of relative velocity (of one end with respect to the other end) in the direction of the line

joining any two points on a rigid member

Velocity DiagramsMotion of a Rigid Body

Lecture Outline

An Example Mechanism

VelocityDiagrams – why learn them?

VelocityDiagrams – A new technique

An Example

Analysis of a velocity diagram

Conclusion

Question: Find the velocity of the piston for the position shown

A

B

O

� = 300 rpm

We usually draw velocities relative to ground and relative to other points on the mechanism

Velocity DiagramsAn Example: The slider crank mechanism

Lecture Outline

An ExampleMechanism

VelocityDiagrams – whylearn them?

VelocityDiagrams – Anew technique

An Example

Analysis of a velocity diagram

Conclusion

A

B

O

� = 300 rpm

First step: The velocity of B relative to ground is known

(Make sure you convert �� to radians per second before you calculate this velocity)

Velocity DiagramsAn Example: The slider crank mechanism

Lecture Outline

An ExampleMechanism

VelocityDiagrams – whylearn them?

VelocityDiagrams – Anew technique

An Example

Analysis of a velocity diagram

Conclusion

AB

O

� = 300 rpm

The direction of the velocity of A relative to ground is known - It must be horizontal

Its magnitude is unknown at present…

Velocity DiagramsAn Example: The slider crank mechanism

Lecture Outline

An ExampleMechanism

VelocityDiagrams – whylearn them?

VelocityDiagrams – Anew technique

An Example

Analysis of a velocity diagram

Conclusion

We will now consider a simple velocity diagram based on the slider crank mechanism

The velocity diagram proceeds as follows

Velocity DiagramsAn Example: The slider crank mechanism

Lecture Outline

An Example Mechanism

VelocityDiagrams – why learn them?

VelocityDiagrams – A new technique

An Example

Analysis of a velocity diagram

Conclusion

The velocity of B relative to ground is drawn

AB

O

� = 300 rpm

b

Remember: Start with what you knowWe know the velocity at the point B from the question o

Velocity DiagramsAn Example: The slider crank mechanism

Lecture Outline

An ExampleMechanism

VelocityDiagrams – whylearn them?

VelocityDiagrams – Anew technique

An Example

Analysis of a velocity diagram

Conclusion

The direction of A relative to the ground is known

AB

O

� = 300 rpm

bWe refer to this line as the “a” line

o

Velocity DiagramsAn Example: The slider crank mechanism

Lecture Outline

An ExampleMechanism

VelocityDiagrams – whylearn them?

VelocityDiagrams – Anew technique

An Example

Analysis of a velocity diagram

Conclusion

AB

O

� = 300 rpm

b

o

Velocity DiagramsAn Example: The slider crank mechanism

Lecture Outline

An ExampleMechanism

VelocityDiagrams – whylearn them?

VelocityDiagrams – Anew technique

An Example

Analysis of a velocity diagram

Conclusion

The direction of A relative to the ground is known

AB

O

� = 300 rpm

bBecause the velocity at A is with respect to ground, it starts at ground (the point “O” in the diagram)

o

Velocity DiagramsAn Example: The slider crank mechanism

Lecture Outline

An ExampleMechanism

VelocityDiagrams – whylearn them?

VelocityDiagrams – Anew technique

An Example

Analysis of a velocity diagram

Conclusion

The direction of A relative to the ground is known

To complete the diagram we require the Velocity of A relative to B

AB is a rigid member. Its length cannot change!

What do we know therefore about the direction of the velocity of A relative to B?

The velocity of A relative to Bmust also be perpendicular to the line AB.

A

B

Direction of relative velocity

Velocity DiagramsAn Example: The slider crank mechanism

Lecture Outline

An ExampleMechanism

VelocityDiagrams – whylearn them?

VelocityDiagrams – Anew technique

An Example

Analysis of a velocity diagram

Conclusion

AB

O

� = 300 rpm

b

o

The direction of the velocity of A relative to B is knownThe magnitude is unknown

a

Velocity DiagramsAn Example: The slider crank mechanism

Lecture Outline

An ExampleMechanism

VelocityDiagrams – whylearn them?

VelocityDiagrams – Anew technique

An Example

Analysis of a velocity diagram

Conclusion

The direction of A relative to the ground is known

AB

O

� = 300 rpm

b

o

Note that it connects to the point b

Why is this?

a

Velocity DiagramsAn Example: The slider crank mechanism

Lecture Outline

An ExampleMechanism

VelocityDiagrams – whylearn them?

VelocityDiagrams – Anew technique

An Example

Analysis of a velocity diagram

Conclusion

The direction of A relative to the ground is known

AB

O

� = 300 rpm

b

o

Graphically, we can now determine the velocity of a relative to the ground.

(we can measure its length with a ruler)

a

Velocity DiagramsAn Example: The slider crank mechanism

Lecture Outline

An ExampleMechanism

VelocityDiagrams – whylearn them?

VelocityDiagrams – Anew technique

An Example

Analysis of a velocity diagram

Conclusion

The direction of A relative to the ground is known

AB

O

� = 300 rpm

b

o

We now have the required velocity diagram!

a

Velocity DiagramsAn Example: The slider crank mechanism

Lecture Outline

An ExampleMechanism

VelocityDiagrams – whylearn them?

VelocityDiagrams – Anew technique

An Example

Analysis of a velocity diagram

Conclusion

The direction of A relative to the ground is known

Note the following

VELOCITY DIAGRAMS GRAPHICALLY REPRESENT THE VECTOR SUMS OF VELOCITIES:

ie. VA = VB+VA rel. to B

Capital letters are used in the mechanism. The mechanism must be drawn to scale.

Lower case letters are used in the velocity diagram.

A suitable scale must also be used for the velocity diagram.

Lecture Outline

An Example Mechanism

VelocityDiagrams – why learn them?

VelocityDiagrams – A new technique

An Example

Analysis of a velocity diagram

Conclusion

The velocity diagram is NOT the mechanism diagram. They are totally different diagrams – although they are related.

Note again that:

An APPROPRIATE scale must be chosen for the velocity diagram – simple to calculate, and a diagram of a reasonable size – (postage stamp size is too small). You can select the scale.

Note Also:

Lecture Outline

An Example Mechanism

VelocityDiagrams – why learn them?

VelocityDiagrams – A new technique

An Example

Analysis of a velocity diagram

Conclusion

AB

O

� = 300 rpm

b

oTherefore we can have a larger velocity diagram.

In an exam, this would be easier to work with!

a

Velocity DiagramsAn Example: The slider crank mechanism

Lecture Outline

An ExampleMechanism

VelocityDiagrams – whylearn them?

VelocityDiagrams – Anew technique

An Example

Analysis of a velocity diagram

Conclusion

The direction of A relative to the ground is known

A complete understanding of this is essential before

attempting morecomplicated examples

Some features of the velocity diagram.

Lecture Outline

An Example Mechanism

VelocityDiagrams – why learn them?

VelocityDiagrams – A new technique

An Example

Analysis of a velocity diagram

Conclusion