Mechanical Vibration in Rotating Shaft

ENGR 213 Fall 2024
This project will be online on Monday October 21 at 10 am. Marked on 100
Due date and time: November 4, 10 AM
Project to be done in teams of maximum 3 students, minimum 2.
Submit the solution online via the Moodle website of the course.
Accepted format: pdf, jpeg, jpg.
You are required to submit the complete solution of your project with explanation when needed. No
need for a cover page, but make sure your name and your partner’s names are clearly indicated on the
first page.
No question will be answered on the project. It is your responsibility to seek the information based on
what is given.
Context:
In this project, you will be asked to work on mechanical
vibration in a system made of a circular disk attached at
the end of a shaft, both shaft and disk inside a medium
providing a damping proportional to the angular velocity.
The system can be modelled based on a mass-spring and
damper system. This system can be useful as a first
introduction to either rotating shaft in water (propeller), or
as to model part of a rotating shaft in oil. Of course, the
model describe here is merely the introduction.
Important concept to consider:
Moment of inertia: The moment of inertia can be understood as the inertia of the rotating body, or the
physical phenomena that prevent the motion (for example, the inertia in a linear motion is the mass of
the object; for rotation, the geometry of the object is also important). It will be represented by the letter
I, and you will have to work with the moment of inertia about the central axis of the rotation.
Shear modulus: In the current application, it is the pressure needed to produce an elastic displacement
along the plane perpendicular to the axis of rotation. We will use the letter G to represent it and it will
be a constant.
Polar surface moment of inertia: This is a geometric value based on the cross section of the shaft. The
symbol used will be J.
The global motion of the object will be described using the angular position θ. The general equation you
will consider to start your study will be Newton’s second law in angular motion, thus
∑? ⃗ =??
Where ? ⃗ represent the torque applied on the shaft.
For the system at hand, consider the mass (disk) attached to a vertical shaft that fix at the top end and
free to rotate along the vertical z axis. The mass is in a liquid that impart a damping that is 4 times the
angular velocity of rotation.
To describe the system, you can use the following data:
Shaft:
L = 0.5 m , r = 0.01 m
G = 79.6 GPa
J is the polar moment of inertia of the shaft (cross-section) – use ??4
Mass:
Radius: R = 0.5 m , thickness: 0.2 m , density: 5000 kg/m3
I is moment of inertia about the center, use 1
2
Angular motion:
�
�?2
Use the angle position ?(?) to describe the motion, with ? and ?
Elastic torque: −??
?
?
Damping torque: −4 ?
2
, r is the radius of the shaft
a) Start with Newton’s second law for rotation to build a differential equation describing the
system at hand. /15
b) Find the general solution of the system. /35
c) Find the particular solution if at time 0, the initial angle is ?
6
and the initial angular velocity is 0.
/30
d) Assuming you change the mass by a sphere with the same radius and density, and that the
damping torque is now −3 ?, find the difference in the frequency of oscillation between this
scenario and the one studied before? (you will need to find the mass moment of inertia of the
sphere) /20