Half Car Modeling

AIM: Owing to the wide separation of the natural frequency of the sprung parts from that of the unsprung parts, the bounce and pitch motions of the vehicle body and the wheel motions exist almost independently. The sprung parts of a vehicle weigh 9.79 kN (2200 lb), its center of gravity is 106.7 cm (42 in) behind the front axle, and the wheelbase is 228.6 cm (90 in). The combined stiffness of the springs of the front suspension is 24.52 kN/m (140 lb/in) and that of the rear suspension is 26.27 kN/m (150 lb/in). The radius of gyration of the sprung parts about a horizontal transverse axis through the center of gravity is 102.6 cm (40.4 in).

1. Calculate the natural frequencies of the pitch and bounce motions of the vehicle body.

2. Also determine the locations of the oscillation centers.

GIVEN DATA:

Parameter	Value	Units
Sprung mass	9.79 / 2200	kN / lb
Center of gravity	106.7 / 42	cm / in
Wheelbase	228.6 / 90	cm / in
Front spring stiffness	24.52 / 140	kN/m / lb/in
Rear spring stiffness	26.27 / 150	kN/m / lb/in
Radius of gyration	102.6 / 40.4	cm / in

GOVERNING EQUATION:

Moment of inertia: `I_y = m_s*r_y^2`

Constants:

`D_1 = (k_f + k_r)/m_s`

`D_2 = (b*k_r - a*k_f)/m_s`

`D_3 = (a^2*k_f + b^2*k_r)/I_y`

`D_4 = ((a+b)^2*k_f*k_r)/(m_s*I_y)`

Natural frequencies:

Bounce: `w_(n_1) = sqrt(((D1+D3) - sqrt((D1+D3)^2 - 4*D4)) / 2)`
Pitch: `w_(n_2) = sqrt(((D1+D3) + sqrt((D1+D3)^2 - 4*D4)) / 2)`

Oscillation centers:

Bounce: `l_1 = D_2/(w_(n_1)^2 - D_1)`
Pitch: `l_2 = D_2/(w_(n_2)^2 - D_1)`

SOLUTION:

Programming Language: MATLAB

Code:

% Input data:

W_s = 9.79;         % Sprung weight (kN)
a = 106.7;          % Center of gravity is behind the front axle (cm)
l = 228.6;          % Wheelbase (cm)
k_f = 24.52*10^3;   % Front spring stiffness (kN/m to N/m)
k_r = 26.27*10^3;   % Rear spring stiffness (kN/m to N/m)
r_y = 102.6;        % Radius of gyration (cm)


% Solution:

m_s = W_s*10^3/9.81;   % Sprung mass (kN to kg)
b = l-a;               % Center of gravity is behind the rear axle (cm)
I_y  = m_s*r_y^2;      % Moment of inertia (kg.cm^2)

D1 = (k_f + k_r)/m_s;
D2 = (b*k_r - a*k_f)/m_s;
D3 = (a^2*k_f + b^2*k_r)/I_y;
D4 = ((a+b)^2*k_f*k_r)/(m_s*I_y);

% Natural frequencies:
w_n1 = sqrt(((D1+D3) - sqrt((D1+D3)^2 - 4*D4)) / 2)   % Bounce (rad/sec)
w_n2 = sqrt(((D1+D3) + sqrt((D1+D3)^2 - 4*D4)) / 2)   % Pitch (rad/sec)

% Oscillation centers:
l_1 = D2/(w_n1^2 - D1)   % Bounce (cm)
l_2 = D2/(w_n2^2 - D1)   % Pitch (cm)

Output:

ANSWERS:

Parameter	Value	Units
Bounce natural frequency	6.9794	rad/sec
Pitch natural frequency	8.1187	rad/sec
Bounce oscillation center	-269.2332	cm
Pitch oscillation center	39.099	cm