Hamburg Wheel-Track Test Equipment Requirements and Improvements to AASHTO T 324 (2016)

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78 9. APPENDIX C The following discussion presents the analytical solution of the wheel and metal-specimen interaction. Figure 43 shows the drawing of the metal specimen used in this study (curvature with radius R) with a HWT wheel (with radius r) placed over it at a distance of 𝛾𝛾𝑐𝑐 from the center. As can be seen from the figure, the wheel will come in contact with the metal specimen tangentially at the point 𝛾𝛾. Therefore, the rut depth reported by the machine LVDT will be less than the actual rut in the metal specimen at all points except the center. The following steps present the mathematical derivation to obtain the difference in rut depth reported by the machine LVDT and the impression of the metal specimen (𝛼𝛼0 − 𝛼𝛼𝑐𝑐). It should be noted that the center of the curvature of the metal specimen is at (0, R). 1. The equation of the circle with radius R is: 𝑥𝑥2 + (𝑦𝑦 − 𝑅𝑅)2 = 𝑅𝑅2 (1) Therefore, (𝑦𝑦 − 𝑅𝑅) = ±�𝑅𝑅2 − 𝑥𝑥2 (2) 2. Since we are dealing with only the bottom half of the circle (𝑦𝑦 − 𝑅𝑅) = −�𝑅𝑅2 − 𝑥𝑥2 (3) 3. Assume a 𝛾𝛾 𝑦𝑦 = 𝑅𝑅 − �𝑅𝑅2 − 𝛾𝛾2 = 𝛼𝛼 (4) 𝑦𝑦′ = + 𝛾𝛾 �𝑅𝑅2 − 𝛾𝛾2 = 𝛽𝛽 (5) 4. Use r and 𝛽𝛽 to find 𝛾𝛾𝑐𝑐 𝛾𝛾𝑐𝑐 = 𝛾𝛾 − 𝑟𝑟 × sin(𝜋𝜋𝑡𝑡𝑠𝑠−1 𝛽𝛽) (6) 5. Use R and 𝛾𝛾𝑐𝑐 to find 𝛼𝛼𝑐𝑐 𝛼𝛼𝑐𝑐 = 𝑅𝑅 − �𝑅𝑅2 − 𝛾𝛾𝑐𝑐2 (7)

79 6. Use r and ( 𝜃𝜃 = 𝜋𝜋𝑡𝑡𝑠𝑠−1𝛽𝛽 ) to find f 𝑓𝑓 = 𝑟𝑟 × cos(𝜃𝜃) (8) 7. Find 𝛼𝛼𝑅𝑅 and 𝛼𝛼0 𝛼𝛼𝑅𝑅 = 𝛼𝛼 + 𝑓𝑓 (9) 𝛼𝛼0 = 𝛼𝛼𝑅𝑅 − 𝑟𝑟 (10) Maximum speed location computation for the non-sinusoidal configuration The position of the wheel in the non-sinusoidal machine is described as follows: 𝑥𝑥 = 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟(𝜃𝜃) + �𝑙𝑙2 − 𝑟𝑟2𝑟𝑟𝑠𝑠𝑠𝑠2(𝜃𝜃) (11) where, θ = crank angle, r = radius of the crank circle, and l = length of the connecting rod. The speed of the wheel is obtained by taking the derivative of the position and is shown below: 𝑥𝑥′ = −𝑟𝑟𝑟𝑟𝑠𝑠𝑠𝑠(𝜃𝜃) − 𝑟𝑟2 sin(𝜃𝜃) cos (𝜃𝜃) �𝑙𝑙2 − 𝑟𝑟2𝑟𝑟𝑠𝑠𝑠𝑠2(𝜃𝜃) (12) The maximum value of speed is obtained by taking the derivative of speed and equating it to zero. i.e. 𝑥𝑥" = −𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟(𝜃𝜃) − 𝑟𝑟2�𝑟𝑟𝑟𝑟𝑟𝑟2(𝜃𝜃) − 𝑟𝑟𝑠𝑠𝑠𝑠2(𝜃𝜃)� �𝑙𝑙2 − 𝑟𝑟2𝑟𝑟𝑠𝑠𝑠𝑠2(𝜃𝜃) − 𝑟𝑟4𝑟𝑟𝑠𝑠𝑠𝑠2(𝜃𝜃)𝑟𝑟𝑟𝑟𝑟𝑟2(𝜃𝜃)��𝑙𝑙2 − 𝑟𝑟2𝑟𝑟𝑠𝑠𝑠𝑠2(𝜃𝜃)�3 = 0 (13) MATLAB software (MuPAD) was used to numerically solve this equation to obtain θ. The resulting θ was plugged back into the distance equation to obtain position. The position of the maximum velocity was thus found to be 0.61 in. from the midpoint of the track. It should be noted that the values of r and l used were 4.5 and 13.0 in., respectively.

80 Figure 43 Geometry of metal specimen and wheel Figure 44 Difference between the rut of the metal specimen and the LVDT reading

81 Figure 45 Details of the metal specimen (all dimensions are in inches)