# Modeling a quad rotor vibrations

Dustinsfl

## Homework Statement

I have a system of five coupled ODEs. I believe I should be able to reduce it down to four coupled ODEs, but I am not sure how. This system is modeling a quad X rotor where the arms are viewed as end loaded cantilever beams with motors attached.

## The Attempt at a Solution

\begin{align}
m_{eq}\ddot{y}_1 &= k_{eq}(y_5 - y_1) + c(\dot{y}_5 - \dot{y}_1) + F_1(t)\tag{1}\\
m_{eq}\ddot{y}_2 &= k_{eq}(y_5 - y_2) + c(\dot{y}_5 - \dot{y}_2) + F_2(t)\tag{2}\\
m_{eq}\ddot{y}_3 &= k_{eq}(y_5 - y_3) + c(\dot{y}_5 - \dot{y}_3) + F_3(t)\tag{3}\\
m_{eq}\ddot{y}_4 &= k_{eq}(y_5 - y_4) + c(\dot{y}_5 - \dot{y}_4) + F_4(t)\tag{4}\\
m_t\ddot{y}_5 &= F_1(t) + F_2(t) + F_3(t) + F_4(t) \tag{5}
\end{align}

This system is modeling the vibrations of quadrotor due to the motors spinnings. Also, I am trying to determine the initial conditions too. If we assume at ##t = 0##, we have no displacement then ##y_i(0) = 0## and ##\dot{y}_i = a_i\delta(t)##. My thought on the initial velocity is at ##t = 0## the rotors start running at flying speed (instantly to make the problem easier). Would this be the correct way to model the initial velocity? Also, ##F_i(t) = c_i\cos(\omega t + \phi_i)## where ##c_i## are the amplitudes, ##\omega## the rad/sec of the rotors, and ##\phi## the phase offset.

Is equation (5) correct for modeling the movement of the center of the quadrotor?

## Answers and Replies

Dr.D
Do you have a picture showing the system and your coordinates?

Dustinsfl
Do you have a picture showing the system and your coordinates?

I am using z to denote the displacement of the body (where the arms connect) only. Also, I am viewing the arms as cantilever beams with an end load.

It is just a normal X quad rotor. I have, also, made some adjustments. I think the system should be of the form:
1. I noticed I neglected gravity. Would that be simple adding the term
##-m_{eq}y_i## to equations one to four and ##-m_bz##? Or is it more
involved?
2. Do I still need ##m_t\ddot{y}_5 = \sum_iF_i(t)## any more? Is it correct in this form?

Here is one of my thoughts for final system of ODE form. Is it correct?
\begin{align}
m_{eq}\ddot{y}_1 &= k_{eq}(z - y_1) + c(\dot{z} - \dot{y}_1) - m_{eq}y_1 +
F_1(t)\\
m_{eq}\ddot{y}_2 &= k_{eq}(z - y_2) + c(\dot{z} - \dot{y}_2) - m_{eq}y_2 +
F_2(t)\\
m_{eq}\ddot{y}_3 &= k_{eq}(z - y_3) + c(\dot{z} - \dot{y}_3) - m_{eq}y_3 +
F_3(t)\\
m_{eq}\ddot{y}_4 &= k_{eq}(z - y_4) + c(\dot{z} - \dot{y}_4) - m_{eq}y_4 +
F_4(t)\\
m_b\ddot{z} &= \sum_i\bigl[F_i(t) + k_{eq}(y_i - z) +
c(\dot{y}_i - \dot{z})\bigr] - m_bz\\
m_t\ddot{y}_5 &= \sum_iF_i(t) - m_ty_5\tag{6}\\
Y &= \frac{m_bz + \sum_im_{eq}y_i}{m_t}\tag{7}
\end{align}
Is equation (6) needed? It is modeling the overall displacement of the quadrotor. Equation (7) is my constraint.

Dr.D
It is just a normal X quad rotor.

I don't speak quad rotor, so I probably cannot help you without a figure,

Dr.D
I did not see a single one of those figures that identified with certainty the coordinate you are using. If you cannot be bothered to draw a figure, then please don't expect too much help.

Dustinsfl
I did not see a single one of those figures that identified with certainty the coordinate you are using. If you cannot be bothered to draw a figure, then please don't expect too much help.
I guess I need to break it down Barney style for you. There are 4 arms so y_i where i is 1,2,3,4 are the arm displacement. I then said the body I am using z. That means the equation which isn't a constraint must be the displacement of the whole quad rotor if we view it as a rigid body. Also the arms have the rotors so the forcing functions should have been a give away too. Equation 7 is a COM constraint equation. Are you sure you are a Dr of anything?

Dr.D
If I were interested in working this problem myself, I would not need to ask for help. I'm not interested in the problem personally, so anything I have said was intended to get you thinking straight. Without a good figure, you are not likely to ever get the problem in hand.

I see some problems with your formulation, but I'm not going to try to point them out to you. It would require a figure, and you evidently consider such a thing childish and beneath your dignity.

Don't worry too much about my Ph.D., even though it is almost 50 years old. It is still quite durable.

I don't think I have any more time for you.