Mechanical Systems III

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Mechanical Systems III

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Drawing a F.B.D. using d’Alembert’s principle

• d’Alembert’s Principle :

The sum of the differences between the forces acting on a system and the time derivatives of the moments of the system itself along a virtual displacement con sistent with the constraints of the system, is zero.

• One can transform an accelerating rigid body into an equivalent static system

by adding the so-called “inertial force” and “inertial moment”

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Mechanical Impedances

( ) ( )

M

( ) Z s F s

 X s

( ) ( )

F s  KX s

( ) ( )

F s  CsX s

( )

2

( )

F s  Ms X s

K

( )

Z s  K

C

( )

Z s  Cs ( )

2

Z

M

s  Ms

Find TF without writing the differential equation!

Let’s define Mechanical Impedance as

[Sum of impedances] X(s) = [Sum of applied forces]

Problem Solving Technique:

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Example: Two mass system

1 1 1 1 2 ( 2 1 ) 1 1 2 ( 2 1 )

m x    k x  k x  x  c x   c x   x 

1 2 2 ( 2 1 ) 2 ( 2 1 )

m x    f k x  x  c x   x 

Using Mechanical Impedances Using differential equation

m

₁

m

₂

2

1 1 2 1 2 1 2 2 2

( m s  ( c  c s )  ( k  k )) X s ( )  ( c s k X s  ) ( )

2

1 2 2 2 2 2 1

( m s  c s k X s  ) ( )  F s ( ) (  c s k X s  ) ( )

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2 2

0 0

1 1

2 2

L L

KE   y dm 

r

  y   dy dm r   dy

2 2 2 3

2

2 2

0 0

0

1 1 1

2 2 2 3

L

y x

L

x y

KE x dy y dy

L L L

 

  

    

 

    

Mass of an infinitesimal element of thickness dy

 : mass density per unit length of the material

1

2

2 y dm 

^r

y x y

 L

 

2 2

1 1

2 3 2 3

m

r

KE      L    x         x 

3

r e

m  m 

  

  Kinetic Energy of the rod

Rewrite the KE in the form of the total mass of the rod Kinetic Energy of infinitesimal element

Rewrite the KE using

Energy Method for Deriving Equivalent Mass of Elastic Elements

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Equivalent masses of common elements

3 3

k 4Ewh

 L

3

4

3

k Ewh

 L

3 3

16Ewh k  L

4 T

32

k GD L

 

e

0.38

d

m  m

e

0.23

d

m  m

e

0.5

d

m  m

e

m

d

m

d

m

d

e d

/ 3

I   I I

I

d

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Example: Wheel-axle system with bearing damping

2 T I     T RF  c 

mx    f c x c   x c   x   kx

Example: Pneumatic Door Closer

w 2 s

I  I  I

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Example: System with displacement input

N ote that ultimately motion is generated by a force and that this force must be great e nough

.

( )

mx   k y   x cx 

( )

T T

I    k     c  