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December 30th, 2018, 02:45 AM   #1
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Thought problem differential equation

Does anybody know how to solve the following differntial equation?

$\displaystyle ω^2⋅r+dω/dt⋅r+dr/dt⋅ω=g$

where w and r are a function of time so ω(t) and r(t). g is a constant

This equation came from a thought problem of mine which I will now explain since I still do not know how to approach the problem.

Basically I want to design a spiral track in which a vehicle will accelerate. However the total acceleration needs to be constant.

The vehicle will have 2 acceleration components, a radial component ($\displaystyle a_r$) and a change in speed component ($\displaystyle a_s$). Since we want constant acceleration: ar+as=constant

From circular motion we know that $\displaystyle a_r=ω^2r$
where r is the radius, and ω

Further we know that v=ω∗r, therefore:
$\displaystyle a_s=d/dt(ωr)$

For constant acceleration $\displaystyle a_{total}=a_r+a_s=ω^2r+d/dt(ωr)=constant$

Further usefull information:
Length of the track covered at a certain time
l=v∗t=ωrt

With all this the radius will be a function of time, r(t)
and the angular velocity will be a function of time, ω(t)

Boundary conditions:
r(0)=0
ω(0)=0

Now I would like to find a possible set of solutions to create a track by implementing end boundary conditions that i choose e.g. $\displaystyle v(t_{end})=100[m/s]$
$\displaystyle and l(t_{end})=5000[m] $
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December 30th, 2018, 10:28 AM   #2
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What I would do is parameterize your curve.

From that parameterization the acceleration components are easily derived and then you will have a much less general differential equation.

As you've written it $\omega(t),~r(t)$ are much too general to do anything with.
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