Past Paper 2

Question 1

what kind of vehicles would need minimal tyre force and why

Answer 1

• sports and racing cars
• because road handling, especially around corners, is important

Question 2

if you had a racing car subjected to large variations of downforce, what would you want to increase about it

Answer 2

• the sprung stiffness

Question 3

where would a typical car sit on a contoured plot of body acceleration against working space, with contours of spring stiffness and tyre force

Answer 3

• the main point is that the spring stiffness would be between 20 and 30 kN/m
• the damping would be around 2-3kNs/m for extra info

Question 4

what causes the effect of wheelbase filtering

Answer 4

• the time delay between the road input to the front and rear axles
• this results in multiple peaks and troughs in the transfer function of the two DOF pitch-plane model

Question 5

explain the mechanism of wheelbase filtering

Answer 5

• consider a massless, suspensionless vehicle with wheelbase L
• the wheelbase travels along a sinusoidal profile with wavelength λ
• the wheelbase filters out pitch excitation when L = Nλ and N is an integer
• the wheelbase filters out bounce excitation when L = (N+0.5)λ and N is an integer
• filtering out the excitation means that their respective gains are 0

Question 6

what is the expression for the natural frequency of the bounce mode f_bounce and pitch mode f_pitch if the excitation of the mode is minimized

Answer 6

• f = U/λ in general
• wheelbase filters out bounce excitation when L = (N+0.5)λ
• so f_bounce = U(N+0.5) / L
• wheelbase filters out pitch excitation when L = Nλ
• so f_pitch = UN / L

Question 7

when trying to find the natural frequencies of a system, you want the natural modes of vibration. what are the two conditions for natural modes

Answer 7

• undamped: c = 0
• free vibration: z_r = 0

Question 8

say you have matrices in the form Mx_dd + Kx = 0. when you take laplace transforms and rearrange it, what is the target expression

Answer 8

• (M^-1K - w^2)x(jw) = 0

Question 9

how do you finally solve for the natural frequencies with (M^-1K - w^2)x(jw) = 0

Answer 9

• by finding the determinant of the M^-1*K - w^2 matrix
• keep in mind the diagonal 0s dont get subtracted by w^2

Question 10

how do you then find the corresponding mode shapes

Answer 10

• by finding the eigenvectors of the matrix for each natural frequency
• input the frequency, and the matrix multiplied by the main variables matrix = 0
• then figure out what has to be 0 and what can be 1 (usually 1)

Question 11

what are the two base expressions for frequency f

Answer 11

• f = v/λ = w/2pi

Question 12

when doing those very long and scary looking laplace transform questions, what is the key relationship you need to remember to get started

Answer 12

• E[ ] = int[S_y(w) dw = int[ |H_xy(w)|^2*S_x(w)
• its the linking of two expressions in the datasheet

Question 13

in the case of roll-plane analysis, what does it suggest if youre told to find the natural frequencies and mode shapes at low speeds

Answer 13

• that y_dot is small so the dampers can be ignored
• so when you draw the simplified versions of the 6 DOF model you dont include the damper in between the sprung and unsprung masses

Question 14

for the case of roll stiffness T_s = 2k_s*S^2, what is S

Answer 14

• S = T = 0.75m usually
• unless stated otherwise

Question 15

when finding the natural frequencies in the roll-plane analysis, being told that the speed is low so dampers can be ignored, you can draw 4 modes (sprung mass vertical, unsprung mass vertical, unpsprung mass roll and sprung mass lateral+roll). what are the general expressions for the natural frequencies youll be using in these cases

Answer 15

• for the vertical modes, w_n = sqrt(k/m)
• for the roll and lateral modes, w_n = sqrt(tao/I)
• tao are the roll stiffness equations
• I is the moment of inertia

Question 16

what is something unique you have to remember about the unsprung mass roll mode

Answer 16

• because there are two sets of springs in the system
• you add the suspension roll stiffness T_s and the axle-ground roll stiffness T_t to get total roll stiffness

Question 17

what are the unique things (plural) you have to remember about the sprung mass lateral/roll model

Answer 17

• the sprung mass is drawn like a vertical beam
• the bottom of it has a horizontal spring attached to a vertical wall on the LHS
• the middle of it has a curved spring attached to a horizonal wall on the RHS
• the stiffness of k_lat is so large that it’s assumed to be rigid
• I in w_n = sqrt(tao/I) is I_s + m(hs - hr)^2, parallel axis theorem

Question 18

for roll-plane analysis, if n_c = 0.2 and U = 30, what does the freuency f equal

Answer 18

• because U = fλ and λ = 1/n, f = Un
• so 30*0.2 = 6 Hz

Question 19

relating to the expression z_p,dd = z_s,dd + p*θ_s,dd what is the significance of the frequency calculation using f = Un (use the 6 Hz as a frame of ref. for explanation)

Answer 19

• at any given speed, at n_c, the frequency you get determines which mode is primarily expressed
• when U = 30 so f = 6 Hz, frequencies below 6 Hz has the bounce excitation mainly dominate
• above 6 Hz, both bounce and roll excitations contribute decent amounts
• but generally, at high speeds only the bounce modes are excited

Question 20

if the speed U = 1 now, how would that change the ratio of p*θ_s,dd / z_p,dd

Answer 20

• at U = 1, f = 0.2 Hz
• this means that the bounce and roll motions make similar contributions across the frequency range
• this shows that, generally, that ratio INCREASES with DECREASING speed

Question 21

what is the approximation of r_2 - r_1 considering their expressions

Answer 21

• r_2 - r_1 = 2yε

Question 22

if you do a question and find that the lateral tracking error y = 0, what suggestion would you give for improving the design of the rigid bogie

Answer 22

• increase the track clearance by 2aθ