Designing a Transfer Track You are hired as a consultant to design a magnetic transfer track to connect proposed maglev tracks for a new train system. For standard diesel trains, a transfer track is a section of track to make a transition from two existing straight line tracks. Consider the problem of designing a transfer track connecting the point (0,0) to (L, H) to make a transition from an existing track straight line track to a parallel straight line track with H the distance between the tracks. Such a problem becomes essentially find a curve c(u) = (Lu, F(u)) with -∞ ≤ u≤ ∞ with the coordinates measured in miles, and F(u) of the form F(u) = 0 if u ≤0, F(u) = P(u) if 0
Designing a Transfer Track You are hired as a consultant to design a magnetic transfer track to connect proposed maglev tracks for a new train system. For standard diesel trains, a transfer track is a section of track to make a transition from two existing straight line tracks. Consider the problem of designing a transfer track connecting the point (0,0) to (L, H) to make a transition from an existing track straight line track to a parallel straight line track with H the distance between the tracks. Such a problem becomes essentially find a curve c(u) = (Lu, F(u)) with -∞ ≤ u≤ ∞ with the coordinates measured in miles, and F(u) of the form F(u) = 0 if u ≤0, F(u) = P(u) if 0
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![Designing a Transfer Track
You are hired as a consultant to design a magnetic transfer track to connect proposed maglev tracks for a
new train system. For standard diesel trains, a transfer track is a section of track to make a transition from
two existing straight line tracks.
Consider the problem of designing a transfer track connecting the point (0,0) to (L, H) to make a transition
from an existing track straight line track to a parallel straight line track with H the distance between the
tracks. Such a problem becomes essentially find a curve c(u) = (Lu, F(u)) with -∞ ≤ u≤ ∞ with the
coordinates measured in miles, and F(u) of the form
F(u) = 0 if u ≤0, F(u) = P(u) if 0 <u<1, F(u) = H if u ≥ 1,
so that F(u) is continuous, and has continuous slope and continuous curvature. This can be done by seeking
at 5th degree polynomial for P(u) with P(0) = 0, P′(0) = 0, P″(0) = 0, P(1) = H, P'(1) = 0 and P" (1) = 0.
Choose L and H in units of kilometers with 1 ≤ L ≤ 3 and 1 ≤ H ≤ 2.
• Determine the location(s) on the curve where the curvature is maximized.
• Determine the location(s) on the curve where the curvature is zero. These are the points of inflection
of the curve and places where all acceleration is tangential, so the engineer of the train can go as fast
as possible.
• Suppose the normal acceleration for the train must be under 3 (km/hr)/sec = 10,800 (km/hr²) or else
the train will not able to stay on the track. With such an assumption determine the speed limit on
the transfer track. [Train accidents are normally a result of going too fast around corners causing the
train to go off the track.]
• Change L and H, and see how different values change the speed limit.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6abec00c-0df5-4c95-bf6e-fd605cb914b4%2F5d8e6686-c969-4190-a96c-f7cb7941079a%2F28q1nzj_processed.png&w=3840&q=75)
Transcribed Image Text:Designing a Transfer Track
You are hired as a consultant to design a magnetic transfer track to connect proposed maglev tracks for a
new train system. For standard diesel trains, a transfer track is a section of track to make a transition from
two existing straight line tracks.
Consider the problem of designing a transfer track connecting the point (0,0) to (L, H) to make a transition
from an existing track straight line track to a parallel straight line track with H the distance between the
tracks. Such a problem becomes essentially find a curve c(u) = (Lu, F(u)) with -∞ ≤ u≤ ∞ with the
coordinates measured in miles, and F(u) of the form
F(u) = 0 if u ≤0, F(u) = P(u) if 0 <u<1, F(u) = H if u ≥ 1,
so that F(u) is continuous, and has continuous slope and continuous curvature. This can be done by seeking
at 5th degree polynomial for P(u) with P(0) = 0, P′(0) = 0, P″(0) = 0, P(1) = H, P'(1) = 0 and P" (1) = 0.
Choose L and H in units of kilometers with 1 ≤ L ≤ 3 and 1 ≤ H ≤ 2.
• Determine the location(s) on the curve where the curvature is maximized.
• Determine the location(s) on the curve where the curvature is zero. These are the points of inflection
of the curve and places where all acceleration is tangential, so the engineer of the train can go as fast
as possible.
• Suppose the normal acceleration for the train must be under 3 (km/hr)/sec = 10,800 (km/hr²) or else
the train will not able to stay on the track. With such an assumption determine the speed limit on
the transfer track. [Train accidents are normally a result of going too fast around corners causing the
train to go off the track.]
• Change L and H, and see how different values change the speed limit.
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