Vehicle dynamics-Ride modelling

Given Inputs:

An independent front suspension of a passenger car carries a mass (sprung mass) of 454.5 kg (or an equivalent weight of 100 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.Car is traveling on a sinusoidal road profile with a wavelength of 5 m (16.41 ft) and an amplitude of 5 cm (2 in).

Aim:

To find the lowest speed of vehicle at which tire may lose contact with road.

Notations used:

`delta_s`= Static deflection

`delta_d`= Dynamic deflection

`m_s`= Sprung mass

`m_{us}`= Unsprung mass

`lambda`= Wavelength of road profile

`k_s`= Spring stiffness

`k_t`= Tire resistance

`z_o`= Amplitude of road profile

Numerical computation using concept of Ride modelling:

Tire static deflection=`"Total force"/"Total stiffness"`

Hence `delta_s=((m_s+m_u)*9.81)/k_t`

We get static deflection as `delta_s`= 0.0279m

Dynamic tire deflection `delta_d=z_0-z_2`

Tire will lose contact when:

Dynamic deflection`>=` Static deflection

So we get`z_0-z_2>=delta_s`

On simplifying we get:

`(z_0-z_2)/z_0>=delta_s/z_0`

`(z_0-z_2)/z_0>=0.0279/0.05=0.5574`

So tire loses contact when tire dynamic deflection ratio`>=0.5574`

Frequency corresponding to dynamic ratio of 0.5574 is 5Hz.

So velocity can be obtained by`v=f(lambda)`

`v=5(5)=25m/s`

Minimum velocity so that vehicle donot lose contact with road is`25m/s`

Computation using Matlab:

% inputs

m_s=454.5;     % sprung mass(kg)
m_u=45.54;     % unsprung mass(kg)
k_s=22*10^3;   % spring stiffness(N/m)
k_t=176*10^3;  % tire stiffness(N/m)
e=0.3;         % damping ratio
wl=5;          % wavelength
z0=5*10^-2;    % amplitude(m)

%static tire deflection
delta_s=(m_s+m_u)*9.81/k_t;

%static tire deflection ratio
delta_d_ratio=delta_s/z0;

%corresponding frequency in Hz for dynamic tire deflection ratio as per given graph
f=5;  

% lowest speed
v=f*wl;

fprintf('Lowest velocity to lose contact with ground is %.0f m/s\n',v)

Results: