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The work done by all the forces acting on a system, during a small virtual displacement is ZERO. Definition A virtual displacement is a small displacement of ...
Typology: Study notes
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z
r
y
3
particle in free motion in spacehas 3 degrees of freedom
x
thus we need to resolve in two or three dimensions – choice of method of resolution
needs to be made. Also need to introduce all internal forces on free-body diagrams– these usually disappear when the final equation of motion is found.
a routine method – no need to take arbitrary decisions.
T
Free body diagram mg
1(- a
2(b
1
2(b
1
2
x
A
& &
=
F
A
M
(acceleration)
D’Alembert’s Principle Consider a rigid mass, M, with force F
A
applied
From Newton’s 2
nd
law of motion
x
M
Ma
F
A
&
&
=
=
or
0
=
−
x
M
F
A
&
&
k
x
m
Example 1Mass/Spring System
Here number of degrees of freedom =1Co-ordinate to describe the motion is xNow consider free-body diagram, at some time t
mm
R (reactive force)
inertial force
mg (gravity force)
restoring force kx
x
m
& &
−
(
)
δ
δ
1
1
q
Q
W
δ
δ
=
General one degree of freedom systemIf q
1
is the co-ordinate used to describe the movement then the general form of
δ
W is as follows:
we call
q
Q
1
From principle of virtual work
0
0
1
1
1
=
∴
=
=
Q
Q
W
q
δ
δ
θ
Example 2Simple pendulum
2
&
ml
−
& &
ml
−
This is another one degree of freedom system.During a virtual displacement,
δθ
, the virtual work done is
δ
W =
)
sin
(
θ
θ
mg
ml
−
−
&
&
0
=
δθ
l
δ
W = 0 (PVW)
or
0
mg
P
(inertial force – tangential)
(inertial force –radial)
m
Example 3 – two degrees of freedom systemfree-body diagrams.
k
x
1
m
k
x
2
m
m
m
kx
1
-mx
1
k(x
2
-x
1
)
-mx
2
n
2
1
qn
q
q
,...,
,
2
1
(
)
n
i
q
i
≤
≤
1
qi
δ
qi
i
δ
δ
i
Q
i
Q
i
q
0
=
=
qi
i
Q
W
δ
δ
qi
δ
i
.
,...,
2
,
1
n
i
=
2 1
n
Inertial Forces (See also Handout)The position of the i
th
particle of mass, in the system, is, in general, related to the
n generalised co-ordinates, and time (if the constraints are independent of time)then the position of the i
th
particle depends only on the n generalised co-
ordinates. Thus
t q q q r r
n
i
i
,
,...,
,
2
1
r
r
=
(1)
Now we suppose that the system is in motion and that we represent the inertialforce on the i
th
particle (using D’Alembert’s Principle) as
i
i
r
m
&
& r
−
(2)
We now give the system an arbitrary virtual displacement – this can berepresented in terms of generalised co-ordinates by
n
δ
δ
δ
2
1
. The virtual
displacement of the I
th
particle can be represented by
r
r
δ
(3)
and the virtual work done by the inertia force on the I
th
particle is simply
r
r
m
r
&
&
r
).
(
−
(4)
(note that this is a scalar product). From this result we get the total virtual work as
i
i
i
i
I
r
r
m
W
r
&
&
r
δ
δ
).
(
∑
−
=
(5)
Using equation (1) we have
j
n
i
j i
i
q
q
r
r
δ
δ
⋅
∂
∂
=
∑
=
1
r
r
(6)
Hence
j
n j
j i
i
i
i
I
q
q
r
r
m
W
⋅
∂
∂
⋅
−
=
∑
∑
=
1
)
(
r
& & r
(7)
and re-arranging
(
)
∂ ∂
⋅
−
=
∑
∑
=
i
i j
i
i
n j
j
I
r r
r
m
q
W
& & r
1
δ
δ
(8)
However, the generalised inertial forces, Q
j
, are effectively defined by
j
n j
I j
i
i
i
I
δ
δ
∑
∑
=
1
(9)
Comparing (8) and (9) we have
∑
i
j i
i
i
I j
(10)
It is shown in the handout notes that
(
)
j
j
I j
(11)
