Analysis and Design Open Oscillatory Systems with Forced Harmonic Motion (2)

Following to articles of “Analysis and Design Open Oscillatory Systems with Forced Harmonic Motion (1)” (https://emfps.blogspot.com/2014/02/analysis-and-design-open-oscillatory.html) and “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), the purpose of this article is, to find out the characteristics of unknown open oscillatory system which is under forced harmonic motion in the moment of close to resonance. In this case, three models have been designed. One model presents the results for two independent variables “m”, “b” and  another model gives us the results for three independent variables “m”, “b” and “ω''/ω” and the third model shows us the results for higher than a specific velocity.

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: