Ride Modeling

AIM: An independent front suspension of a passenger car carries a mass (sprung mass) of 454.5 kg (or an equivalent weight of 1000 lb). 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), estimate the lowest vehicle speed at which the tire may lose contact with the road. Use the plot shown below. GIVEN DATA: Parameter Value Units Sprung mass 454.5 / 1000 kg / lb Suspension spring rate 22 / 125 kN/m / lb/in Unsprung mass 45.45 / 100 kg / lb Equivalent tire stiffness 176 / 1000 kN/m / lb/in Damping ratio 0.3 - Wavelength 5 / 16.4 m / ft Amplitude 5 / 2 cm / in

GOVERNING EQUATION:

Static tire deflection: `delta_s = frac((m_s + m_us)*9.81)(k_t)`

For tire to lose contact,

Dynamic tire deflection `>=` Static tire deflection

`delta_d >= delta_s`;

As the graph represents dynamic tire deflection ratio, divide both sides of above equation by `Z_o`;

`delta_d/Z_o = delta_s/Z_o`

`(Z_o - Z_2)/( Z_o) = delta_s/Z_o`

`(Z_o - Z_2)/( Z_o) = 0.0279/0.05 = 0.5579`

From the graph, the frequency corresponding to the dynamic tire deflection ratio of `0.5579` is `6 Hz`

Thus, lowest vehicle speed at which the tire may lose contact with the road:

`v = f*lambda`

`v = 6 * 5 = 30 m/sec`

`v = 30*(18/5) = 108 (km)/(hr)`

SOLUTION:

Programming Language: MATLAB

Code:

% Input data:

m_s = 454.5;      % Sprung mass (kg)
m_us = 45.45;     % Unsprung mass (kg)
k_s = 22*10^3;    % Spring stiffness (kN/m to N/m)
k_t = 176*10^3;   % Tire stiffness (kN/m to N/m)
C_s = 0.3;        % Damping ratio
lambda = 5;       % Wavelength (m)
Z_o = 5;          % Amplitude (cm)


% Solution:

% Static tire deflection:
delta = (m_s + m_us)*9.81/k_t;   % m

% Dynamic tire deflection ratio:
ratio = delta/(Z_o*0.01);

% Frequency corresponding to tire deflection ratio:
f = 6;   % Hz (cycle/sec)

% Lowest vehicle speed at which the tire may loose contact with the road:
v = f*lambda   % m/sec
v1 = v*(18/5)  % km/hr

Output:

ANSWERS:

Parameter	Value	Units
Frequency corresponding to tire deflection ratio	6	Hz (cycle/sec)
Lowest vehicle speed at which the tire may loose contact with the road	30 108	m/sec km/hr