Analysis and Design Open Oscillatory Systems with Externally Forced Harmonic Motion (3)

 Following the article of Analysis and Design Open Oscillatory Systems with Forced Harmonic Motion (2), the purpose of this article is to present a model for finding the characteristics of an oscillatory system with forced harmonic motion where the acceleration will be equal zero. In this case, there is an interesting point that is related to an important difference between SHM and an oscillatory system with forced harmonic motion.

If an object is oscillating under simple harmonic motion, its linear velocity will be zero at the highest and lowest points where we have maximum displacement which is named the amplitude (A).On the other hand; at the maximum level of the displacement (x = A), the acceleration has also its maximum magnitude while at the middle (x = 0), acceleration is zero due to stop at those points in order to change direction while velocity gains its maximum magnitude at the equilibrium point (x = 0). At the extreme ends (x = A), when we have the maximum force and kinetic energy, the acceleration has its maximum magnitude. Therefore, the maximum of acceleration magnitude in simple harmonic motion occurs at maximum displacement (A) and acceleration at the middle is zero when we have the displacement equal to zero just like the below diagrams: 

But there is a different story about the oscillatory systems with forced harmonic motion. In this case, at the some points where the displacement is not zero (x = a), we have the acceleration equal to zero. Please see below diagrams:

Now, the question is: What are the characteristics of an oscillatory system with forced harmonic motion where we have the acceleration equal to zero at the point of x = a?

Below model is able to answer above question:

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 for the acceleration equal to zero which are the characteristics of driven oscillatory system. 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:I18, we have outputs which are the characteristics for driven oscillatory system responding to the acceleration equal to zero.

You can see below clip as the example for this model:

All researchers and individual people, who are interested in having this model, don’t hesitate to send their request to below addresses: