Vehicle Dynamics - Ride Modelling

AIM: To solve the given conceptual numerical question based on "Ride Modelling'-

Question: An independent front suspension of a passenger car carries a mass (sprung mass) of 454.5 kg (or an equivalent weight of 1000 Ib). The suspension spring rate is 22 kN/m (125 lb/in.). The mass of the tire wheel assembly (unsprung mass) is 45.45 kg (or an equivalent weight of 100 lb) and the equivalent tire stiffness is 176 kN/m (1000 lb/in.). The damping ratio of the suspension produced by the shock absorber is 0.3. If the car is traveling on a sinusoidal road profile with a wavelength of 5 m (16.4 ft) and an amplitude of 5 cm (2 in.)

OBJECTIVE: To estimate the lowest vehicle speed at which the tire may lose contact with the road.

GIVEN:

Sprung mass, `m_s = 454.5kg`

Unsprung mass, `m_(us) = 45.54kg`

Spring stiffness, `k_s = 22(kN)/m`

Tire stiffness, `k_t = 176(kN)/m`

Damping ratio, `zeta = 0.3`

Wavelength (road), `lambda = 5m`

Amplitude (road), `Z_0 = 5cm = 0.05m`

SOLUTION:

Tire static deflection = (total force)/(tire stiffness)

`=> delta_s = ((m_s+m_(us))*9.81)/k_t`

`delta_s = 0.0279m`

Dynamic tire deflection, `delta_d=Z_0-Z_2`

Tire will lose contact when

Dynamic tire deflection `>=` Static tire deflection

`Z_0 - Z_2 >= delta_s`

Dividing both sides by `Z_0`

`(Z_0-Z_2)/Z_0>=delta_s/Z_0`

`(Z_0-Z_2)/Z_0>= 0.0279/0.05=0.5574`

I.e., for tire to lose contact with the road,

Dynamic tire deflection ratio `>= 0.5574`

From the given graph, the corresponding frequency for the dynamic tire deflection ratio of `0.5574` is `5Hz`.

i.e, `f = 5Hz`

`therefore V = f*lambda = 5*5`

`V = 25m/s=90(km)/h`

MATLAB code:

clear all
close all
clc

%inputs
ms = 454.5;     %sprung mass (kg)
mu = 45.54;     %unsprung mass (kg)
ks = 22*10^3;       %spring stiffness (N/m)
kt = 176*10^3;      %tire stiffness (N/m)
zeta = 0.3;     %damping ratio
lambda = 5;     %road wavelength (m)
Z0 = 5*10^-2;       %amplitude (m)

%static tire deflection
delta_s = (ms+mu)*9.81/kt;

%dynamic tire deflection ratio
delta_d_ratio = delta_s/Z0;

%corresponding frequency for dynamic tire deflection ratio from the given graph
f = 5;      %(Hz)

%lowest speed for tire to lose contct with the ground
V = f*lambda        %speed (m/s)

OUTPUT:

FINAL ANSWER:

The lowest vehicle speed at which the tire may lose contact with the road is 25m/s = 90km/h.