- #1

- 699

- 5

## 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.

## Homework Equations

## 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?