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Unformatted text preview: ent to the other (rate of stretch/compression).
The displacement of each spring is the component of the displacement of the bar at the
attachment point of the spring, in the direction of the spring. Just as before, this is found via a
projection. First, the four unit vectors are quantified:
̂ cos
̂ cos ̂ sin ̂ ̂ To find the displacements of points
velocities of and are found: and ̂ , which are the attachment points of the springs, the
̂ / / ̂ ̂
̂ sin ̂ 3
2
̂ ̂ ̂ ̂
3
2
̂ Now, to obtain the displacements, replace ‘dot’ terms with ‘Δ’ terms, and the velocity with
Δ ̅ . This could be seen by eliminating the time increment
and changing the infinitesimal
change ̅ with a finite change Δ ̅ : Δ̅
Δ̅
However, since Δ̂
3
Δ̂
2 Δ̂ is measured relative to its ‘zero’ position, Δ can be replaced by :
Δ̅ ̂ Δ̅ 3
2
̂ ̂ These expressions are useful for the springs. The full velocities, given prior, are used for the
dampers. Now:
̂ ∙Δ ̅ Δ
Δ ̂ ∙Δ ̅ 3
sin
2 cos
̂ ∙Δ ̅ Δ
Δ
Now the generalized force
external forces is cos 0
3
2 ̂ ∙Δ ̅ must be found. Recall that the total virtual work ∙ due to ̅ where represents the external forces and ̅ represents a virtual displacement, which is found
by taking a virtual derivative of the position vector ̅ (the same rules as regular derivatives
apply!). In this case, only one external force, , acts on the system. The position vector relating
to i s
̂
̅ 3
4
̂ ̂ 3
4
̂ (verify this on your own using geometry). A virtual derivative of the above is therefore:
̅ ̂ 3
4
̂ Note that this same expression could have been found by first finding the velocity of point , and
then replacing ‘dot’ terms with ‘ ’ terms. The virtual work is therefore 3
4
The general form of virtual work is
∑
Comparing this form with the virtual work obtained gives
3
4
Taking all derivatives gives the following equation of motion
9
4 1
4 sin 3 3
4 sin cos Example 3:
Cons...
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This note was uploaded on 07/01/2013 for the course ME 4189 taught by Professor I.green during the Fall '12 term at Georgia Tech.
 Fall '12
 I.Green
 Strain

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