2. (30 pts.) Motion of a sphere rolling in a channel
The figures below show a uniform rigid spherical ball B that rolls in a rigid channel A. The channel
is made of two flat surfaces A1 and A2 that are welded together at right angles. The point of B in
contact with A1 is denoted BA1 . The point of B in contact with A2 is denoted BA2 . Channel A is
attached by a revolute torque motor at No , a point fixed in a long thin rigid parallelepiped support
N (a Newtonian reference frame). B’s geometric center is denoted Bcm . The center of mass of A is
denoted Acm and is coincident with No .
A
A1
B
Ao
Bcm
A2
ay a2
a3
ny
az
nz
a1 = ax
B
Right-handed orthogonal unit vectors nx , ny , nz are fixed in N with
• nx directed horizontally right
• ny directed locally vertically upward
• nz directed along N ’s longest dimension
x
Quantity
Mass of B
Radius of sphere
Moment of inertia of A about Acm for az
Earth’s local gravitational acceleration
az measure of motor torque on A from N
ax measure of Bcm ’s position vector from Acm
Angle between nx and ax
ax measure of B’s angular velocity in N
ay measure of B’s angular velocity in N
az measure of B’s angular velocity in N
Symbol
m
R
A
Izz
g
Tz
x
θ
ωx
ωy
ωz
ax
Ao
ny
θ
No
nx
nz
A
N
Side view
(parallel to A’s axis of rotation in N )
Right-handed orthogonal unit vectors ax , ay , az are fixed in A with
• ax parallel to the line of intersection between A1 and A2
• az = nz
Right-handed orthogonal unit vectors a1 , a2 , a3 are fixed in A with
• a1 = ax
• a2 directed from BA2 to Bcm
• a3 directed from BA1 to Bcm
ay
Bcm
B
a3
A2
Bcm
ay
ax
Type
constant
constant
constant
constant
specified
variable
variable
variable
variable
variable
A1
BA1
BA2
az
A
a2
a1
No
45ο
End view
(initial) Value
0.2 kg
0.02 m
0.2 kg m2
9.8 m/sec2
specified
0◦
0.1 m
Not applicable
Not applicable
Not applicable
The inertia dyadic of B about Bcm can be expressed in terms of a scalar I B and the unit dyadic 1
as shown below. Complete the following equation for I B in terms of symbols in the table.
IB/Bcm = I B 1
IB =
Show when B rolls on A at both A1 and A2, ωx , ωy , ωz can be expressed in terms of x, ẋ, R, as
ωy = 0
ωx = 0
ωz = θ̇ −
√ ẋ
2
R
Simulate this system’s motion for 9 seconds. Use an integration step of 0.02 seconds and integrator
error tolerances of 1.0 x 10-8 . Start the system from rest (not moving) and use a control law of the
following form
Tz = k1 θ + k2 x + k3 θ̇ + k4 ẋ
where the “feedback control constants” are k1 = -0.5, k2 = 2.0, k3 = 0.0, k4 = 0.1.5
Calculate the system’s “total energy” (TotalEnergy) defined as the work done by the motor (WorkOfMotor),
the system’s kinetic energy, and the system’s gravitational potential energy. Output the following
quantities and complete the following graphs.
t seconds,
θ degrees,
x meters,
0.8
0.8
0.7
Work done by motor (Joules)
1
20
0.6
10
x (m)
Theta (degrees)
TotalEnergy Joules
25
15
5
0.4
0.2
0
0
-5
-10
-0.2
-15
-0.4
0
5
WorkOfMotor Joules,
1
2
3
4
5
Time (sec)
6
7
8
9
0.6
0.5
0.4
0.3
0.2
0.1
0
0
1
2
3
4
5
Time (sec)
6
7
8
9
These feedback control constants values do not correspond to a stable motion.
0
1
2
3
4
5
Time (sec)
6
7
8
9