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| Re: Calculation (36565) | |||
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Re: Calculation |
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Posted by Jeff H. on Thu Jan 6 20:53:25 2005, in response to Re: Calculation, posted by H.S.Relay on Thu Jan 6 08:11:15 2005. Sorry, there's some physics and math in advance of this sentence:When the car is rounding a curve, its inertia wants to take it in a direction tangent to the curve. The "steering" force is normal to the curve, and is pushing the car towards the inside of the curve. From the standpoint of a stationary observer, the car is accelerating radially, towards the center of the curve, and the magnitude of the acceleration is v**2/r. To an observer within the non-inertial reference frame of the car (i.e. a passenger ^H^H^H^H^H^H^H^H customer) that translates to a phantom "centripetal force" of m*v**2/r, where m is the mass of the person. The effect of super-elevation is to change the direction in which gravity acts relative to the person's reference frame. It introduces a force of m*g*sin(x), where g is the gravitational constant (32 ft/s/s or 9.8m/s/s) and x is the angle of super-elevation. This reduces the feeling of being thrown towards the outside of the curve as perceived by the moving observer (who is actually accelerating towards the inside of the curve). The stationary observer sees that the component of gravity acting in the radial direction is M*g*sin(x) and that force is acting to drag the car towards the inside of the curve. (M being the mass of the car). This reduces the total radial force which the track must exert on the car. When g*sin(x)=v**2/r, the force of gravity provides 100% of the steering and there is no thrust on the track and no perceptible sense of being thrown observed by the passenger. We can simplify using the small angle approximation and say that E=56.5*sin(x), where E is the super-elevation of the outer rail above the inner, in inches (56.5 being the track gauge, YMMV in Pennsylvania). Then we get the formula: E=(56.5/32)*(v**2/r) with v in ft/s and r the radius in feet. Sample calculation: At 30MPH (45fps), a 400 foot radius curve would require 8.9 inches of super-elevation to be balanced. This is considered an excessive number, as 8 inches is generally the most lift that one would want to attempt. Therefore, a lower speed limit would be recommended. At 20MPH, the required super-elevation would be only 3.9" |
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