- #1
Beyond Aphelion
I am working on an overly simplified roller-coaster design but I have found a problem in defining a banked helix track piece.
Essentially I want to be able to track the lateral, normal, and tangential g-forces acting on some point mass, so I need to obtain equations for the tangent and normal accelerations at any given point on the helix.
It would follow that a helix defined by the curve:
\overline{r}(t) = acos(t)\hat{i} + asin(t)\hat{j} + bt\hat{k}
\overline{T} = \frac{d\overline{r}}{ds} = \frac{d\overline{r}/dt}{ds/dt} = \frac{V}{|V|}
= \frac{1}{\sqrt{a^{2} + b^{2}}}\left[-asin(t)\hat{i} + acos(t)\hat{j} + b\hat{k}\right]
\overline{N} = \frac{d\overline{T}/dt}{|d\overline{T}/dt|} = -cos(t)\hate{i} - sin(t)\hat{j}
- has a defined principle unit normal vector. However, what if I want to be able to define how steeply the helix is banked ? Essentially, I want to rotate the normal and binormal principle unit vectors so as to reduce as much lateral g-force as possible or eliminate it entirely. My first inclination is that I should introduce a k-hat component (constant if I want the bank angle to be constant or otherwise if not). But does throwing that element into the equation invalidate some other assumptions I have made?
Not knowing the complications that may arise from this problem, I'd like to just define a constant bank angle, but if it is doable, I wonder if changing the bank angle as a function of time/distance would complicate this beyond my level of comprehension.
Another assumption would be that the unit tangent vector would remain the same for any given helix no matter how the bank is defined. However, as the bank is offset, components of acceleration will be transferred between a_t and a_n. Would this even transfer acceleration into my new binormal direction? How would this effect the radius of curvature? I'm just not sure how to go about modeling that.
I don't necessarily have to have a concrete equation, but it would be nice. I have MATLAB at my disposal. Any push in the right direction would be helpful.
Thanks.
Essentially I want to be able to track the lateral, normal, and tangential g-forces acting on some point mass, so I need to obtain equations for the tangent and normal accelerations at any given point on the helix.
It would follow that a helix defined by the curve:
\overline{r}(t) = acos(t)\hat{i} + asin(t)\hat{j} + bt\hat{k}
\overline{T} = \frac{d\overline{r}}{ds} = \frac{d\overline{r}/dt}{ds/dt} = \frac{V}{|V|}
= \frac{1}{\sqrt{a^{2} + b^{2}}}\left[-asin(t)\hat{i} + acos(t)\hat{j} + b\hat{k}\right]
\overline{N} = \frac{d\overline{T}/dt}{|d\overline{T}/dt|} = -cos(t)\hate{i} - sin(t)\hat{j}
- has a defined principle unit normal vector. However, what if I want to be able to define how steeply the helix is banked ? Essentially, I want to rotate the normal and binormal principle unit vectors so as to reduce as much lateral g-force as possible or eliminate it entirely. My first inclination is that I should introduce a k-hat component (constant if I want the bank angle to be constant or otherwise if not). But does throwing that element into the equation invalidate some other assumptions I have made?
Not knowing the complications that may arise from this problem, I'd like to just define a constant bank angle, but if it is doable, I wonder if changing the bank angle as a function of time/distance would complicate this beyond my level of comprehension.
Another assumption would be that the unit tangent vector would remain the same for any given helix no matter how the bank is defined. However, as the bank is offset, components of acceleration will be transferred between a_t and a_n. Would this even transfer acceleration into my new binormal direction? How would this effect the radius of curvature? I'm just not sure how to go about modeling that.
I don't necessarily have to have a concrete equation, but it would be nice. I have MATLAB at my disposal. Any push in the right direction would be helpful.
Thanks.
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