Some weird circular relationship

• I
• Runei
In summary, the mathematician is trying to find a way to model a circular rotating system in which the time it takes to turn an amount of angular velocity is dependent on the angular velocity. The equation is coupled with another equation which states that the time it takes to turn an amount of angular velocity is equal to the period of a square wave signal with an on-time of 1 and an off-time of 2.f

Runei

Hello there!

I'm currently doing some mathematical modelling at my work, and I have arrived at an interesting kind of circular relationship integral - and now I'm wondering about what to do.

The integral looks very innocent at first glance:
$$\theta_s = \int\limits_0^{t_1} \omega (t) dt$$
So, it's a circular rotating body, with some time-dependent angular velocity. However, the ##\theta_s## is a constant - a parameter we can design in the system.

But what makes the thing complex (at least in my head) is that ##t_1## is the time it takes for the body to rotate the amount ##\theta_s## - so the upper limit becomes dependent on the angular velocity also.

Can I create another integral that relates the time ##t_1## to ##\omega (t)## and ##\theta_s##?
Is Leibniz' rule the way to go?

And ##\omega (t)## is an unknown function, by the way.

You can consider it more abstractly. You have an arbitrary function y=f(x), and you want to find the value of x2 such that the area under the curve from x=x1 to x2 is a given value. Doesn't sound like there's any way to cast that than the form of integral you quote.

We can't uniquely determine ##\omega(t)## neither ##t_1## just by this integral equation, need more equations to uniquely determine them.

For example if you try ##\omega(t)=at+b## you ''ll find a ##t_1## that depends on a,b and ##\theta_s## (##at_1^2+2bt_1-2\theta_s+c=0##), if you try ##\omega(t)=sin(at)## you 'll find a totally different ##t_1=\frac{1}{a}arcos(a\theta_s+1)##.

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We can't uniquely determine ##\omega(t)## neither ##t_1## just by this integral equation, need more equations to uniquely determine them.
I don't think that is what Runei was trying to do. Rather, he was looking to turn the equation into the form ##t_1=F(\omega, \theta_s)##, where F is some functional, maybe an integral.

Thanks for the replies! The equation is coupled with another equation namely
$$\int\limits_0^{t_1}\tau_1(t)\omega(t)dt+\int\limits_0^{t_2}\tau_2(t)\omega(t)dt+\int\limits_0^{t_1}\tau_{in}\omega(t)dt = 0$$
The ##\tau_1(t)## and ##\tau_2(t)## are controllable functions - they can be chosen by design (EDIT: And they will have the opposite sign of ##\tau_{in}##). The function ##\tau_{in}## is a square wave signal (perhaps modulating another signal - but that is of lesser importance right now) with an on-time of ##t_1## and an off-time of ##t_2##. And as mentioned earlier, the ##t_1## is precisely the time it takes the system to turn an amount ##\theta_s##.

$$\theta_s = \int\limits_0^{t_1}\omega(t) dt$$

What I am basically trying to do is modelling the system in steady-state, where I know that the input torque will be a square wave, with duty cycle determined by the angular velocity as mentioned above.

I've been considering using a trapezoidal expansion of the integrals and solving the equations numerically, but I was wondering if they could be "massaged" even more.

Thanks again for the replies! :-)

I might be wrong but seems to me again that we can't determine uniquely ##\omega(t)##. Can choose "quite a random " ##\omega(t)## and then just solve for ##t_1## and ##t_2##. For example if we put ##\omega(t)=C## we see how everything is simplified and easy to determine ##t_1## and ##t_2## so that the two integral equations hold. But I might be wrong.

I think perhaps a third equation , even one not directly involving ##\omega(t)## but some equation like ##t_1+t_2=C## will narrow down our choices for ##\omega(t)##.

Well there are actually some more now that I think about it.
One thing I realized is that the integral with ##t_2## should probably have a lower bound being ##t_1## instead.
##\int\limits_0^{t_1}\tau_1(t)\omega(t)dt+\int\limits_0^{t_1}\tau_{in}(t)\omega(t)dt+\int\limits_{t_1}^{t_2}\tau_2(t)\omega(t)dt = 0##
##\theta_s = \int\limits_0^{t_1}\omega(t) dt##
The function ##\tau_{in}## will have a period ##T_{in}## and that period will be equal to ##t_2##. Furthermore, the function ##\tau_{in}## has it's duty cycle ##D## which means that
##t_1 = D\cdot T_{in}##

But there's more I see. The ##\omega(t)## is at any time related to the torque ##\tau_{net}## and moment of inertia ##I##, which means that we have in the period ##[0;t_1]##:
##\dot\omega(t)\cdot I = \tau_{net} = \tau_{in} + \tau_{1}##
And in the time period ##[t_1;t_2]##
##\dot\omega(t)\cdot I = \tau_{net} = \tau_{2}##

So I guess the integrals above could be rewritten as
##\int\limits_0^{t_1}\dot\omega(t)\omega(t) dt+\int\limits_{t_1}^{t_2}\dot\omega(t)\omega(t) dt = 0##
The moment of inertia can be removed since it's constant and can be multiplied out.

Well there are actually some more now that I think about it.
One thing I realized is that the integral with ##t_2## should probably have a lower bound being ##t_1## instead.
##\int\limits_0^{t_1}\tau_1(t)\omega(t)dt+\int\limits_0^{t_1}\tau_{in}(t)\omega(t)dt+\int\limits_{t_1}^{t_2}\tau_2(t)\omega(t)dt = 0##
##\theta_s = \int\limits_0^{t_1}\omega(t) dt##
The function ##\tau_{in}## will have a period ##T_{in}## and that period will be equal to ##t_2##. Furthermore, the function ##\tau_{in}## has it's duty cycle ##D## which means that
##t_1 = D\cdot T_{in}##

But there's more I see. The ##\omega(t)## is at any time related to the torque ##\tau_{net}## and moment of inertia ##I##, which means that we have in the period ##[0;t_1]##:
##\dot\omega(t)\cdot I = \tau_{net} = \tau_{in} + \tau_{1}##
And in the time period ##[t_1;t_2]##
##\dot\omega(t)\cdot I = \tau_{net} = \tau_{2}##

So I guess the integrals above could be rewritten as
##\int\limits_0^{t_1}\dot\omega(t)\omega(t) dt+\int\limits_{t_1}^{t_2}\dot\omega(t)\omega(t) dt = 0##
The moment of inertia can be removed since it's constant and can be multiplied out.
Isn't ##\int \dot\omega(t)\omega(t).dt## just ##[\omega^2(t)]/2##?