D2 Report
D2 – Forced, Damped Vibrations with Rotating Out-of Balance Excitation (Norwood Apparatus)
Introduction
This lab report was aimed at investigating out-of-balance rotating masses in single degree of freedom systems, paying specific attention to obtaining plots of amplitude against angular velocity. This was done using a specific setup of apparatus (Detailed below) and our findings were plotted in the form of graphs and also, we investigated the experiment in a way which allowed us to produce multiple Lissajouse figures which were key in allowing us to investigate the variation of phase with angular velocity.
This was all necessary to investigate as sometimes objects/structures such as cars will experience motions which bring them close to their natural frequency of vibration, be it sinusoidally or otherwise which causes the amplitude experienced by the object to reach its maximum which can have significant effects on that object and the factors affecting it during oscillation.
Theory
There are numerous equations involved in this experiment all of which are important in different ways and are listed below:
Equation 1
The first equation of motion for the carriage is:
mx.. + cx. + kx = m0eω2sinωt
Where:
m – Mass of carriage
c – Damping coefficient
k – Spring stiffness
ω – Frequency of input vibrations (Rads/s)
m0 – Mass of the out-of-balance rotating masses
e – Eccentricity of these masses
This is the standard equation of motion for single degree of freedom systems that are generally in the setup of the following diagram:
However, this specific experiment is with respect to rotating out-of-balance excitation in the system. This means that the right hand side of the equals in the equation is what the equation of motion of this single degree of freedom system equates to with respect the initial conditions/conditions placed in this experiment, e.g. the mass
Introduction
This lab report was aimed at investigating out-of-balance rotating masses in single degree of freedom systems, paying specific attention to obtaining plots of amplitude against angular velocity. This was done using a specific setup of apparatus (Detailed below) and our findings were plotted in the form of graphs and also, we investigated the experiment in a way which allowed us to produce multiple Lissajouse figures which were key in allowing us to investigate the variation of phase with angular velocity.
This was all necessary to investigate as sometimes objects/structures such as cars will experience motions which bring them close to their natural frequency of vibration, be it sinusoidally or otherwise which causes the amplitude experienced by the object to reach its maximum which can have significant effects on that object and the factors affecting it during oscillation.
Theory
There are numerous equations involved in this experiment all of which are important in different ways and are listed below:
Equation 1
The first equation of motion for the carriage is:
mx.. + cx. + kx = m0eω2sinωt
Where:
m – Mass of carriage
c – Damping coefficient
k – Spring stiffness
ω – Frequency of input vibrations (Rads/s)
m0 – Mass of the out-of-balance rotating masses
e – Eccentricity of these masses
This is the standard equation of motion for single degree of freedom systems that are generally in the setup of the following diagram:
However, this specific experiment is with respect to rotating out-of-balance excitation in the system. This means that the right hand side of the equals in the equation is what the equation of motion of this single degree of freedom system equates to with respect the initial conditions/conditions placed in this experiment, e.g. the mass