Statement of the problem
In many situations of oscillatory motions, we have to
control the point of the resonance to prevent a huge collapse of system. On the
other hand, sometimes we need to know the width of the resonance curve and the
characteristics of the system such as natural frequency to increase the
velocity of the system by using a forced harmonic motion. In fact, the question
is: What is the amount of the force and angular frequency for a forced harmonic
motion where the amplitude and velocity of an oscillatory system will
significantly increase? These two models are able to answer to this question.
Model (1): The
results for two independent variables “mass” and “linear damping constant”
Please be informed that here we encounter with seven independent
variables and we have to process all these variables simultaneously for solving
the nonlinear equations. For first try, I breakdown the problem and start my
analysis by using the most important independent variables which are “m” and “b”
and also utilizing the method mentioned in article of “A Model to Solve a
System of SHM Equations” (https://emfps.blogspot.com/2018/08/a-model-to-solve-system-of-shm-equations_23.html) accompanied by some tricks in Microsoft excel. I think this is the
easiest way. This model says us what are maximum displacement and velocity for
a defined range of “m” and “b” close to the point of the resonance.
Below figure as well as shows you the components of
this model:
The components of above model are as follows:
1. In right side on cells K6:M6, we have inputs
including given data
of “ω″/ω”, “Fm”, and “t”.
2. In left side on cells H6:I7, we have other inputs
including lower and upper ranges for independent variables of “m” and “b” to
reach the answers which are the solution for driven oscillatory system close to the point of the resonance.
Here, there are lower and upper ranges which are changed by click on cell A2
and also this change will again go back by click on cell B2 (Go & Back).
3. On cells I9:I17, we have outputs
which are maximum displacement and velocity for driven oscillatory system close to the point of the resonance.
Note: On I9 and I10, we have maximum
displacement and velocity for a
defined range of “m” and “b”. Please do not consider them as the amplitude of
harmonic motion. On cell I16 and I17, we have the amplitude of displacement (A)
and the amplitude of velocity (Av).
You can see below clip as the examples for this model:
Model (2): The
results for three independent variables “mass”, “linear damping constant” and “ω″/ω”
In this model, there are the defined ranges
for “m”, “b” and “ω″/ω” which are our three independent variables and by
changing the forced harmonic motion (Fm) and time (t), we are willing to know what are the characteristics of a
unknown oscillatory system that give us the maximum displacement and velocity
close to the point of the resonance.
Below figure as well as shows you the components of
this model:
The components of above model are as follows:
1. In right side on cells L6:M6, we have inputs
including given data
of “Fm”, and “t”.
2. In left side on cells H6:J7, we have other inputs
including lower and upper ranges for independent variables of “m”, “b” and “ω″/ω” to reach the answers which are the solution for driven
oscillatory system close
to the point of the resonance. Here, there are lower and upper ranges which are
changed by click on cell A2 and also this change will again go back by click on
cell B2 (Go & Back).
3. On cells J9:J17, we have outputs
which are maximum displacement and velocity for driven oscillatory system close to the point of the resonance.
Note: On J9 and J10, we have maximum
displacement and velocity for a
defined range of “m”, “b” and “ω″/ω”. Please do not consider them as the amplitude
of harmonic motion. On cell J16 and J17, we have the amplitude of displacement
(A) and the amplitude of velocity (Av).
You can see below clip as the examples for this model:
Model (3): The
results of three independent variables “mass”, “linear damping constant” and “ω″/ω” for higher than a specific velocity
In this model, we want to design an unknown oscillatory system for
a specific velocity. We can have all characteristics of the system for focus,
lower and higher than a specific velocity. Here, I consider it for a higher
than specific velocity. Since there are many answers, I have only fixed 15
answers for this model.
Below figure as well as shows you the components of
this model:
As you can see, all components of this model are the similar to
model (2). Only I have added15 absolute value for velocities more than a
specific velocity and also on cell M7, we have specific amount of velocity.
You can see below clip as the examples for this model:
Data Analysis and Conclusion
One of the most important applications of above models
is to create the value from data analysis. Please see below results:
Above table shows us that there are many answers for a constant
velocity. But if you carefully focus on the results, you will find out an
interesting property about driven oscillations and resonance.
What is this property?
If an oscillatory system has very low angular frequency (ω) and
linear damping constant (b) in the conditions close to the point of the
resonance and in zero time (t =0), surprisingly by increasing the mass, the
displacement (x) will go up and by
decreasing the mass, the displacement will go down. In this situation, the
amplitude for any changes on mass will stay the constant forever.
All researchers and individual people, who are interested in having
these models, don’t hesitate to send their request to below addresses:
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