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Generalized force expression for this "airplane" oscillator system?

Physics Asked on December 23, 2020

I am reading a book on aircraft dynamics (Blakelock Aircraft and Missiles) where the author presents the following system on page 408:

enter image description here

Lagrange’s equation is given by:

$$
frac{d}{dt}left(frac{partial T}{partialdot q_j}right)-frac{partial T}{partial q_j}
+frac{partial F}{partialdot q_j}+frac{partial U}{partial q_j}=Q_j,~j=1,2,dots,n,
$$

where $T$ is the kinetic energy, $U$ is the potential energy (including internal strain energy), $F$ is one half the rate at which energy is dissipated, and $Q_j$ is the generalized external force acting on the $j$th station, with $q_j$ representing the $j$th station’s generalized coordinate.

In this case, the author splits each force $P_j$ in the above diagram into a gravitational force $P_{g_j}$ and an aerodynamic force $P_{a_j}$. The gravitation force is of course included in the potential energy term, whereas the aerodynamic force is included in $Q_j$. Furthermore, because the system is symmetric, only half of it is considered. With this in mind, the author says that:

$$
Q_0=P_{a_0}+P_{a_1}+P_{a_2},~
Q_1=P_{a_1},~
Q_2=P_{a_2}.
$$

My question is why is the generalized force $Q_0$ equal to the sum of all the aerodynamic forces? I don’t see how the center station (the airplane fuselage) is different from the other stations. In particular, I would have said that $Q_0=P_{a_0}$.

One Answer

"My question is why is the generalized force Q0 equal to the sum of all the aerodynamic forces?"

if you start with the positions of $p_0~,p_1~,p_2$ you obtain

$$p_0=q_0~,p_1=q_0+q_1~,p_2=q_0+q_2$$

with :

$$vec{R}=begin{bmatrix} p_0 p_1 p_2 end{bmatrix}$$

the generalized coordinate vector $vec{q}$

$$vec{q}=begin{bmatrix} q_0 q_1 q_2 end{bmatrix}$$

and the applied force vector $vec{F}$

$$vec{F}=begin{bmatrix} F_1 F_2 F_3 end{bmatrix}$$

you obtain the generalized force vector $vec{Q}$

$$vec{Q}=J^T,vec{F}$$

where J is the Jacobi matrix $$J=frac{partial vec R}{partial vec q}= left[ begin {array}{ccc} 1&0&0 1&1&0 1&0&1end {array} right] $$

$Rightarrow$

$$vec{Q}=left[ begin {array}{c} {F_1}+{ F_2}+{ F_3} {F_2} {F_3}end {array} right] ~surd $$

Correct answer by Eli on December 23, 2020

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