# equation for velocity vs displacement for nonlinear acceleration [closed]

An object starts out at t=0, s=0 and accelerates with A=sin(pi*t/T), where T is defined as (pi/2 * Vmax/Amax). Vmax is maximum velocity, and Amax is maximum acceleration. Solution is restricted to t=0 to t=T

The velocity equation is v(t) = 0.5Vmax(1-cos(pi*t/T))

What I am trying to solve is velocity as a function of displacement, v(s).

In other words, without having access to elapsed time, I am trying to generate an equation for velocity as a function of the observed displacement.

Is this possible?

Thanks

## closed as off-topic by Danu, Daniel Griscom, Gert, user36790, John RennieJan 25 '16 at 7:15

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## 1 Answer

Given the acceleration $a = \sin \left( \frac{\pi t}{T}\right)$, by integration you get

$$v(t) = \int_0^t a\,{\rm d}t = \frac{T}{2} \left(1-\cos\left(\frac{\pi t}{T}\right) \right)$$ $$s(t) = \int_0^t v\,{\rm d}t = \frac{T}{\pi^2} \left(\pi t - T \sin\left(\frac{\pi t}{T}\right) \right)$$

Since the last one cannot be inverted for $t(s)$, we can invert the first one to get $t(v)$ and use it in the second one

$$t(v) = \frac{T}{\pi} \cos^{-1} \left( 1- \frac{\pi v}{T} \right)$$

$$s(v) = \frac{T^2}{\pi^2} \cos^{-1} \left( 1- \frac{\pi v}{T} \right) - T \sqrt{ \frac{2 T v}{\pi^3} - \frac{v^2}{\pi^2} }$$

• The purpose of my query is an experimental attempt to operate a track mounted vehicle (powered by a variable frequency drive) which has a position measuring transducer, and to guide it to a stop at a specific position. I have already succeeded controlling it using constant acceleration, using v=SQRT(2AS). I am attempting to improve by reducing the jerk in the system with the acceleration I proposed. I had a typo in my original question in that the acceleration is Amax * sin(pi * t/T). However I need v(s), not s(v). I recognize my calculus is rusty. Thanks – MartyH Jan 25 '16 at 20:31
• $v(s)$ does not exists analytically. This is because $s(t)$ cannot be solved for $t$ since it is of the form $t=a \sin(t)$. Otherwise you could state $v(s) = v(t(s))$ from $v(t)$. – ja72 Jan 25 '16 at 20:37
• Thanks for the insight. Based on your input, I instead created a spreadsheet table of normalized s and v values. I can now accomplish the goal using a table lookup. – MartyH Jan 26 '16 at 21:28
• So my answer was useful. Great. – ja72 Jan 27 '16 at 3:15