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An in-depth analysis of mechanical systems, focusing on the kinetic energy of mass in translation and rotation, the conservation of energy in simple, conservative systems, and the derivation of the equation of motion for a spring-mass system. It also covers the calculation of the maximum values of position and velocity, and the effect of gravity on the equation of motion. Examples of a pendulum and a vibrating system.
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Virginia Tech© D. J. Inman
The potential energy of mechanicalsystems
U
is often stored in “springs”
(remember that for a spring
F=kx
)
0
0
2
0
1
x
x
spring
U
F dx
kx dx
kx
=
=
=
∫
∫
k
x
0
x=
Virginia Tech© D. J. Inman
2/
0
0
0
2
spring
U
F dx
kx dx
kx
=
=
=
∫
∫
The kinetic energy of mechanical systems
T
is due to the
motion of the “mass” in the system
2
2
1
1
,
2
2
trans
rot
T
mx
T
J
θ
=
=
&
&
Mass
Spring
k
x
Mass
Spring
x=
Virginia Tech© D. J. Inman
(
)
2
2
1
1
(
)
0
2
2
0
Since
cannot be zero for all time, then
0
d
d
T
U
mx
kx
dt
dt
x mx
kx
x
mx
kx
=
=
⇒
=
=
&
&
&&
&
&&
Mass
Spring
U
max
=
1 2
kA
2
T
max
=
1 2
m
(
ω
n
A
)
2
If the solution is given by
x(t)
=
A
sin(
ωωωω
t+
φ φ
φ φ
) then the maximum
potential and kinetic energies can be used to calculate the naturalfrequency of the system
Virginia Tech© D. J. Inman
2
2
Since these two values must be equal
1 2
kA
2
=
1 2
m
(
ω
n
A
)
2
⇒
k
=
m
ω
n
2
⇒
ω
n
=
k
m
Solution continued
2
Rot
2
max
2
2
2
2
max
2
2
max
n
n
n
n
θ
θ
ω
ω
ω
ω
Virginia Tech© D. J. Inman
max
2
max
max
max
2
2
2
2
n
ω
ω
2
n
Effective mass
Example 1.4.2 Determine the equation ofmotion of the pendulum using energy
θ
l
Virginia Tech© D. J. Inman
θ
m
2
l
m
J
=
2
2
2
Virginia Tech© D. J. Inman
n
The effect of including the mass of
the spring on the value of the frequency.
Virginia Tech© D. J. Inman
s
l
m
equilibriu
static
and
FBD,
from
,
0
=
∆
−
k
mg
Virginia Tech© D. J. Inman
13/
0
2
2
1
(
)
2
1
2
spring
grav
U
k
x
U
mgx
T
mx
=
∆ +
= −
=
&
2
2
Now use
(
)
0
1
1
(
)
0
2
2
(
)
d
T
U
dt
d
mx
mgx
k
x
dt
mxx
mgx
k
x x
=
⇒
−
∆ +
=
⇒
−
∆ +
&
&&&
&
&
Virginia Tech© D. J. Inman
0 from staticequilibiurm
(
)
(
)
(
)
0
0
mxx
mgx
k
x x
x mx
kx
x k
mg
mx
kx
⇒
−
∆ +
⇒
∆ −
=
⇒
=
&&&
&
&
&
&&
&
1
4
24
3
&&
Gravity does not effect the
equation of motion or the naturalfrequency of the system for a linearsystem as shown previously with aforce balance.
Example 1.4.7 Derive the equation of motion
of a spring mass system via the Lagrangian
2
2
1
1
and
2
2
T
mx
U
kx
=
=
&
Here
q
=
x
,
and and the Lagrangian becomes
2
2
mx
kx
Virginia Tech© D. J. Inman
16/
Equation (1.64) becomes
(
)
Longitudinal motion
A
is the cross sectional
area (m
2
)
Virginia Tech© D. J. Inman
E
is the elastic
modulus (Pa=N/m
2
)
is the length (m)
k
is the stiffness (N/m)
k
l
l
l
compute the frequency of a shaft/mass
system {
J
= 0.5 kg m
2
}
( )
( )
0
( )
( )
0
M
J
J
t
k
t
k
t
t
J
=
⇒
=
⇒
=
∑
&&
&&
&&
From Equation (1.50)
Figure 1.
Virginia Tech© D. J. Inman
4
10
2
2
4
2
,
32
For a 2 m steel shaft, diameter of 0.5 cm
(
10
N/m )[
(0.
10
m) / 32]
(2 m)(0.5kg m )
p
n
p
p
n
GJ
k
d
J
J
J
GJ
J
−
⇒
=
=
=
⇒
×
×
=
=
⋅
=
l
l
2 rad/s
d
= diameter of wire
2
R=
diameter of turns
n
= number of turns
x
(
t
)= end deflection
Virginia Tech© D. J. Inman
x
(
t
)
G
= shear modulus of
spring material
k
=
Gd
4
64
nR
3
Allows the design of springs
to have specific stiffness
