Newton-Raphson solution for damped oscillation

[alphanov]

1. The oscillation amplitude of a damped system is given by:

x=-8e^0.5θ sin3θ​

Where θ is in radians
Using the Newton-Raphson iteration method, determine the value of t, near to 5.2 correct to 4 significant figures, when the amplitude is zero.




2. Newton's equation

r_2=r_1-f(r_1 )/(f^' (r_1 ) )​

3. Populating Newton's equation with the given function:

r_2=r_1-(-8e^0.5θ sin3θ)/(-4e^0.5θ sin⁡(3θ)-24e^0.5θ cos⁡(3θ))​

My quandry is that the question requires the time (t) when the amplitude is zero (near to 5.2) whereas the expression is couched in angular movement. I cannot, for the life of me, think how to express the expression in terms of time! I can manage with the rest of the problem!

Thanks in anticipation!

 

[lippyka]

The oscillation amplitude of a damped system can be expressed in terms of time by solving the equation for θ: θ = arcsin(x/(-8e^0.5t))/3Then you can plug this into Newton's equation and iterate until you reach the desired t value.