Question: How can I use a loop command ?

Hello everyone, 

I am seeking for help!

You can freely access the mentioned file in any of the following links:

https://www.mdpi.com/2076-3417/9/15/2996

https://www.mdpi.com/2076-3417/9/15/2996/pdf

https://www.researchgate.net/publication/334711617_Transverse_Vibration_of_Clamped-Pinned-Free_Beam_with_Mass_at_Free_End

I have a/L a parameter. Give it a value and then beta can be calculated as in Table 1. The first five lowest physically possible values are listed only, but the number of beta-s tend to infinity because of the sine/cosine functions. (22) is clearly a nonlinear equation. As such, in general, it can't be solved analytically. (Of course, e.g., sin(x)=0 is also nonlinear but has analytical solution, but (22) is much more complicated.)
At first step, I give a/L  a value, 0.6 , then plot the left side of (22) and try to read the zeros. Here, beta on the horizontal axis is the independent variable. This method is really slow and inaccurate to get all the results of Figure 2. Instead, I should find the roots numerically by using a built-in solver in Maple.

Once I am done with the above, I should add a do/for loop for the parameter a/L. HOWEVER, I DO NOT KNOW HOW TO DO.

In conclusion, the main question is: How could I plot the graph using MAPLE in Figure 2 considering the transcendental Equation (22)?

Best Regards, 

Wallyson Thomas