An elliptical hoop of negligible thickness with a vertical orientated semi-major axis a and a horizontal orientated semi-minor axis b, which has a mass per unit length of p, is mounted on a circular peg to which is attached a Carte- sian xy coordinate system in a vertical orienta- tion as shown in the accompanying sketch (the gravity g ellipse and peg are not rigidly joined to each other and are simply in kinematic contact with each other with negligible friction), such that the physical system lies within a flat xy plane. The ellipse is then allowed to freely oscillate in a to-and-fro rocking motion by a small angle 0, such that the gravitational acceleration g acts in a downwards direction that is parallel to the ellipse semi-major axis. 1.3 a Considering the above system determine the following information clearly showing all steps and appropriate reasoning: 1.4 Y Utilizing the previously derived equation of motion determine the corresponding natural angular frequency wn for the Simple Harmonic Motion (SHM) of the vibrating system; Explain what would happen to the engineering dynamics of the vibration system if the ellipse were in an initial configuration where the semi-major axis a was horizontal and the semi-minor axis b was vertical and why this would occur using appropriate mathematical and physical reasoning. I
An elliptical hoop of negligible thickness with a vertical orientated semi-major axis a and a horizontal orientated semi-minor axis b, which has a mass per unit length of p, is mounted on a circular peg to which is attached a Carte- sian xy coordinate system in a vertical orienta- tion as shown in the accompanying sketch (the gravity g ellipse and peg are not rigidly joined to each other and are simply in kinematic contact with each other with negligible friction), such that the physical system lies within a flat xy plane. The ellipse is then allowed to freely oscillate in a to-and-fro rocking motion by a small angle 0, such that the gravitational acceleration g acts in a downwards direction that is parallel to the ellipse semi-major axis. 1.3 a Considering the above system determine the following information clearly showing all steps and appropriate reasoning: 1.4 Y Utilizing the previously derived equation of motion determine the corresponding natural angular frequency wn for the Simple Harmonic Motion (SHM) of the vibrating system; Explain what would happen to the engineering dynamics of the vibration system if the ellipse were in an initial configuration where the semi-major axis a was horizontal and the semi-minor axis b was vertical and why this would occur using appropriate mathematical and physical reasoning. I
International Edition---engineering Mechanics: Statics, 4th Edition
4th Edition
ISBN:9781305501607
Author:Andrew Pytel And Jaan Kiusalaas
Publisher:Andrew Pytel And Jaan Kiusalaas
Chapter8: Centroids And Distributed Loads
Section: Chapter Questions
Problem 8.38P: The equation of the catenary shown is y = 100 cosh (x/100) where x and y are measured in feet (the...
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Transcribed Image Text:An elliptical hoop of negligible thickness with
a vertical orientated semi-major axis a and a
horizontal orientated semi-minor axis b, which
has a mass per unit length of p, is mounted
on a circular peg to which is attached a Carte-
sian xy coordinate system in a vertical orienta-
tion as shown in the accompanying sketch (the gravity g
ellipse and peg are not rigidly joined to each
other and are simply in kinematic contact with
each other with negligible friction), such that
the physical system lies within a flat ry plane.
The ellipse is then allowed to freely oscillate in
a to-and-fro rocking motion by a small angle 0,
such that the gravitational acceleration g acts
in a downwards direction that is parallel to the
ellipse semi-major axis.
1.3
a
1.4
b
Considering the above system determine the following information clearly showing all steps and
appropriate reasoning:
Y
Utilizing the previously derived equation of motion determine the corresponding
natural angular frequency wn for the Simple Harmonic Motion (SHM) of the
vibrating system;
Explain what would happen to the engineering dynamics of the vibration system
if the ellipse were in an initial configuration where the semi-major axis a was
horizontal and the semi-minor axis b was vertical and why this would occur using
appropriate mathematical and physical reasoning.
x
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