Quarter Car Modeling

AIM: The sprung parts of a passenger car weigh 11.12 kN (2500 lb) and the unsprung parts weigh 890 N (200 lb). The combined stiffness of the suspension springs is 45.53 kN/m (260 lb/in) and that of the tires is 525.35 kN/m (3000 lb/in) [Use the mass and stiffness as is – no need to divide]

1. Determine the two natural frequencies of the bounce motions of the sprung and unsprung masses.

2. Calculate the amplitudes of the sprung and unsprung parts if the car travels at a speed of 48 km/h (30 mph) over a road of a sinewave form with a wavelength of 9.15 m (30 ft) and an amplitude of 5 cm (2 in)

GIVEN DATA:

Parameter	Value	Units
Sprung mass	11.12 / 2500	kN / lb
Unsprung mass	890 / 200	N / lb
Suspension spring stiffness	45.43 / 260	kN/m / lb/in
Tire stiffness	525.35 / 3000	kN/m / lb/in
Speed	48 / 30	kmph / mph
Wavelength	9.15 / 30	m / ft
Amplitude	5 / 2	cm / in

GOVERNING EQUATION:

`f_s = 1/(2pi) sqrt(frac(k_s . k_t) (k_s + k_t) / (m_s)`

`f_(us) = 1/(2pi) sqrt(frac(k_s + k_t)(m_(us)))`

`f = v/lambda`

`omega = 2 pi f`

`Z_s/Z_o = frac(C_s . C_(us) .omega^2 + (k_s . C_(us) + k_t . C_s) . omega + k_s . k_t)(m_s . m_(us) . omega^4 + (m_s . C_s + m_s . C_(us) + m_(us) . C_s) . omega^3 +(k_s . m_(us) + m_s . (k_s + k_t) + C_s . C_(us)) . omega^2 + (k_s . C_(us) + k_t . C_s) . omega + k_s . k_t)`

`Z_(us)/Z_o = frac(m_s .C_(us) . omega^3 + (C_s . C_(us) + m_s . k_t). omega^2 + (k_s . C_(us) + k_t . C_s). omega + ks . k_t)(m_s . m_(us) . omega^4 + (m_s . C_s + m_s . C_(us) + m_(us) . C_s) . omega^3 +(k_s . m_(us) + m_s . (k_s + k_t) + C_s . C_(us)) . omega^2 + (k_s . C_(us) + k_t . C_s) . omega + k_s . k_t)`

SOLUTION:

Programming Language: MATLAB

Code:

% Input data:

W_s = 11.12*10^3;   % Sprung weight (kN to N)
W_us = 890;         % Unsprung weight (N)

k_s = 45.53*10^3;   % Spring stiffness (kN/m to N/m)
k_t = 525.35*10^3;  % Tire stiffness (kN/m to N/m)

v = 48*5/18;        % Speed (km/hr to m/sec)
lambda = 9.15;      % Wavelength (m/cycle)
Z_o = 5;            % Amplitude (cm)


% Solution:

m_s = W_s/9.81;     % Sprung mass (kg)
m_us = W_us/9.81;   % Unsprung mass (kg)

f_s = 1/(2*pi)*sqrt((k_s*k_t)/(m_s*(k_s+k_t)))
f_us = 1/(2*pi)*sqrt((k_s+k_t)/m_us)

f = v/lambda;   % Frequency (Hz)
w = 2*pi*f;     % Omega (rad/sec)

% Ignoring damping
Num1 = k_s*k_t;
Den1 = (m_s*m_us*(w^4)) + ((k_s*m_us) + (m_s*(k_s + k_t)))*(w^2) + (k_s*k_t);
ratio_1 = Num1 / Den1;   % Z_s/Z_o
Z_s = ratio_1 * Z_o      % Sprung amplitude (cm)

Num2 = (m_s*k_t*w^2) + (k_s*k_t);
Den2 = (m_s*m_us*w^4) + ((k_s*m_us) + (m_s*(k_s + k_t)))*w^2 + (k_s*k_t);
ratio_2 = Num2 / Den2;   % Z_us/Z_o
Z_us = ratio_2 * Z_o     % Unsprung amplitude (cm)

Output:

ANSWERS:

Parameter	Value	Units
Sprung natural frequency	0.9676	Hz
Unsprung natural frequency	12.6250	Hz
Sprung amplitude	1.5094	cm
Unsprung amplitude	4.6595	cm