MCEN90038 Dynamics End of semester 1 2023 – EXAM
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MCEN90038 Dynamics
End of semester 1 2023 – EXAM
Exam Blue
General Instructions:
Follow the submission format instruc
ons published earlier in the LMS.
If an angle, direc
on, or distance is not described, you may define it yourself. However, make sure to state what the declared variables represent clearly in you submission.
Iner
a tensor component values must be stated and clearly indicated in the problem setup (this part should be done in your handwriten submission). However, your calcula
ons may treat them as
general terms. Example: bIro(G)d= I0 l 
| where I0 = 
. But the calcula
on can use I0 instead
of the expression.
Describe all your frames, rota
on matrices and relevant posi
on vectors in the handwriten part.
System 1 [35 points]
An x-ray device is shown inFigure 2(a). We will simplify the structure for the analysis, dividing the system into 2 links (Figure 2(b)).
Link 1 (black) from the base to point B, rotates about the z0-axis with angular velocity ȧ . Assume Its centre of mass (G1) to be located at a distance 
from point A in the direc
on indicated in the figure. The inclina
on e is a constant.
Link 2 (blue) rotates about point C, with angular velocity Ḟ . Its centre of mass (G2) is located at a distance D from point C .
Assume the iner
a tensors of the 2 links, in their body centred frames, are:
a) Determine the 
me deriva
ve of the linear momentum of link 2 in the frame atached to it.
b) Determine the 
me deriva
ve of the angular momentum of link 2 in the frame atached to it.
c) Draw the free-body diagram of the different components of the system.
d) Determine the Newton-Euler equa
ons of link 2.
e) Determine the equa
ons of mo
on of the system.
f) Assume the 
me deriva
ves of the linear and angular momenta of link 1 to be known.
Determine the torque necessary to keep the system rota
ng with constant angular velocity ȧ =
业0 and Ḟ = 业1 .
System 2 [30 points]
The figure shows a simplified model of a simulated amusement park ride. A person will sit inside the cylindrical cabins (green), which rotate (angle e) and translate about the radial axis of the connec
ng central structure. Assume empty cabins for this problem.
Assume the central point of the central connec
on is at point M and the distance from M to the centre of mass of each cylinder is described by L . The central structure rotates about the ver
cal axis (z0) with angular velocity Ω . Assume all cylindrical cabins will rotate and translate simultaneously and in phase.
Hollow Cylinders (inset figure): Inner radius r1, outer radius r2, density p, inner length a, height b
a) Determine the 
me deriva
ve of the linear momentum of cylinder A in the frame moving with the central structure.
b) Determine the 
me deriva
ve of the angular momentum of cylinder A in the frame following the rota
on of the cylinder itself.
c) Draw the free-body diagram of the different components of the system.
d) Determine the Newton-Euler equa
ons of cylinder A. You may choose the frame.
e) Determine the equa
ons of mo
on of the system.
f) Determine the torque necessary to keep Ω constant.
System 3 [35 points]
A sphere of mass M and radius R can freely rotate about sha
AB, of length L. You must define your angle of rota
on. The sha
is connected at joint A via an ideal pin. The system precesses about the ver
cal with constant angular velocity Ω .
The mass of the sha
is m1 and it can be considered as a thin rod.
a) Determine the 
me deriva
ve of the linear momentum of sphere in the frame atached to the sha
.
b) Determine the 
me deriva
ve of the angular momentum of sphere in the frame atached to the sha
.
c) Draw the free body diagram of the different components of the system.
d) Determine the Newton-Euler equa
ons of the sphere. Now assume the sphere collides with the top wall.
f) Draw the impulsive free-body diagram of the different components of the system. Include the representa
on of the forces in your chosen frame of reference,
g) Determine the impulse momentum equa
ons of the sphere and the expression of the coefficient of res
tu
on.
2023-07-13