3. Railway Bogies

Question 1

what is the rolling radius r_1 and r_2 of railway wheels that have been displaced horizontally by y, with an effective conicity ε and radius of wheel in the central position r

Answer 1

• r_1 = r - y*tan(ε)
• r_2 = r + y*tan(ε)

Question 2

what is the no slip condition for railway wheels with angular velocity w

Answer 2

• u_1 = r_1*w
• u_2 = r_2*w

Question 3

if the span of the axle is 2d with the left wheel moving at speed u_2 and the right moving at speed u_1, what is the angular velocity at the central axle position θ_dot

Answer 3

• θ_dot = (u_1 - u_2) / 2d

Question 4

what is the horizontal wheel velocity y_dot

Answer 4

• y_dot = wrθ

Question 5

what is the natural frequency w_n of the railway bogie and where does it come from

Answer 5

• w_n = sqrt(w^2rtan(ε) / d)
• this comes from its SHM equation in the form y_doubledot + []*y = 0

Question 6

what is the kinematic wavelength of the oscillations λ

Answer 6

• λ = 2pisqrt(dr/tan(ε))

Question 7

what is the instantaneous radius R

Answer 7

• R = dr / yε

Question 8

what are the longitudinal creep velocities v_1 and v_2

Answer 8

• v_1 = u - d(θ_dot + γ_dot) - u*r_1/r
• v_2 = u + d(θ_dot + γ_dot) - u*r_2/r

Question 9

what is the net lateral force on the railway bogie Y

Answer 9

• Y = 2C(θ + y_dot/u)

Question 10

what is the net longitudinal force on the railway bogie X

Answer 10

• X = 0

Question 11

what is the net turning force at the centre of the railway bogie M_p

Answer 11

• M_p = 2dc[εy/R - d*θ_dot/u - d/R]

Question 12

what is the steady state condition for a rigid bogie

Answer 12

• y_dot = θ_dot = 0

Question 13

what is y for a rigid bogie

Answer 13

• y = dr/εR*(1 + a^2/d^2)

Question 14

what is the equation for the undamped SHM for rigid steady state bogie

Answer 14

• y_double_dot + [εdu^2/r(a^2 + d^2)]y = 0
• the sqrt of the y coefficient is w_n

Question 15

what is the corresponding hunting wavelength λ unexpanded

Answer 15

• λ = 2pi*u / w_n
• just input w_n into it and clean up