This continues a discussion of the accident involving the Kennedy Airport to Queens light-rail train on a test run. The train consisted of three cars. Each of the first two cars contained eight 2,000-lb (Wb) unrestrained concrete blocks on a plywood floor, and the third car was empty. The cars weighed 55,000 lbs (Wc) each when empty, and were 10 ft 6 inches wide.
We are doing a preliminary analysis to see if suspicions that sliding blocks caused the accident are reasonable and to determine factors that might be further analyzed. In Part 1, we made several assumptions and discovered that sliding of the blocks was a reasonable possibility at the speed of operation of the train. This justifies a closer examination of this possibility using more exact parameters. It also provides motivation for this Part 2 in which we will consider the possibility of sliding blocks causing the derailment.
One factor that could cause derailment is tilting (rolling) of the train due to centrifugal force as the train goes around the curve (we assume the radius of curvature r = 955 ft). If the train tilts enough, the wheels on one side would leave the track creating an unstable situation, possibly leading to derailment. Since a loaded car is more likely to roll due to centrifugal forces, we will consider a single loaded car, and assume that there are no restraining moments or vertical or lateral forces between the cars within a limited range of relative motion. This should give a reasonable first approximation to the motion. We also assume that the center of mass (CM) of all the blocks is a distance "s" inside the outside rail, as shown in the figures, and that there is no superelevation (tilt) of the track. There will be two vertical force equivalents, N1 and N2, and two lateral force equivalents, R1 and R2 at the wheels on the track on each side of the train. These can be determined in terms of the speed of the train. N1 is of most interest since N1 = 0 defines the situation when the inward wheels begin to lift off the track (impending rolling). Summing moments about point A gives the relationship defining N1 in terms of the speed (V).
(2t) N1 + (Wc/g) (V²/r) (hc) + 8 (Wb/g) (V²/r) (hb)
= (t) Wc + 8 (s) Wb
where "g" is the gravitational acceleration. Notice that as V increases, N1 decreases until at the critical speed, N1 = 0. Assuming that hc= 40 inches, hb = 42.4 inches, and t = s = 28.3 inches, we set N1 = 0, and obtain V = 100 mph. This is probably well beyond the maximum speed of the train. Since in Part 1 we determined that the blocks would slip at around 70 mph, it is obvious that the blocks would slide before the train would tilt, and that tilting was probably not the reason the load shifted. There could have been a small rolling effect due to "give" in the suspension, allowing the blocks to slide at a slightly lower speed.
Since the blocks would have shifted before tilting of the train, they would have been at the outside wall of the train. Using the equation above to determine the speed for tilting in this configuration, we set s = -14.4 ft (assuming the blocks are cubes and that the wall of the train is 6 inches thick). In this configuration the CG of the blocks is now outside the track producing a tipping moment rather than a stabilizing one. Thus, we find a lower speed V = 81 mph for tipping due to centrifugal forces. If the train is capable of this speed, we would have to consider the possibility of centrifugal forces contributing to the derailment.
Besides the centrifugal forces, the impact of the blocks when they strike the side of the cars would produce an impulsive force also tending to roll the train. Since this occurs at a lower speed than tipping, it may have been the initiating factor. We will examine this aspect further in our next installment.