Analyzing the Effects of Battery Position on ##C_m-\alpha## Graph

AI Thread Summary
The addition of a counterweight, such as a battery, shifts the center of mass (CoM) of the model plane further back, which reduces the longitudinal distance (l_t) and increases the height of the center of gravity (hc). This change affects the slope of the C_m-α curve, potentially decreasing it, as indicated by the equations from Anderson's Intro to Flight. The impact on the slope's significance is uncertain due to the opposing effects of a larger h and a smaller V_H. The resulting graph could show either a lower equilibrium angle with reduced lift or a higher flight angle with increased lift, depending on the configuration. Further analysis is needed to confirm the reasoning and implications of these changes on flight performance.
Leo Liu
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I am trying to determine how the addition of a counterweight affects the ##C_m-\alpha## (in longitudinal direction) graph for a model plane project, where the counterweight can be considered as the battery.

Screenshot 2022-11-28 at 11.18.08 PM.png

Screenshot 2022-11-28 at 11.27.45 PM.png

Suppose we try to place the battery at the back of the current CoM. The new CoM will be at further back, reducing ##l_t## and increasing ##hc##. According to Anderson't Intro to Flight, the equation is as follows:
this will decrease the slope of the ##C_m-\alpha## curve:

Screenshot 2022-11-28 at 11.22.45 PM.png

Now I am wondering if it will have a significant impact on the slope; ##h## is larger, while ##V_H## is smaller. Yet I am not sure which change is more significant to the moment slope.

From the equation (7.24) we can see that at $L_{wing}=0$, ##V_H## will become smaller due to a smaller ##l_t##, increasing the value of ##C_{m\,(L=0)}##:
Screenshot 2022-11-28 at 11.18.48 PM.png

where
Screenshot 2022-11-28 at 11.31.49 PM.png
.

I think the graph of the line will look like either one of the added lines. The blue one will result in a lower equilibrium angle, thus providing a lower ##C_L##; yet the red line will allow a higher flight angle, thus producing more lift.
Screenshot 2022-11-28 at 11.39.36 PM.png

Could someone please give a bit more insight into this and tell me if my reasoning is correct?
 

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