vehicle dynamics week 10E

Question:

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 (5+5 points)
2. Also determine the locations of the oscillation centers (5+5 points)

Matlab code:

clear all

close all

clc

ms = 9790/9.81; % Mass (conversion from N to kg)

l1= 1.067; % cog to front axle (m)

l = 2.286; % wheelbase (m)

kf = 24520; % Front stiffness (N/m)

kr = 26270; % Rear stiffness (N/m)

ry = 1.026; % Radius of gyration (m)

l2=l-l1

iy=ms*(ry)^2

d1=(kf+kr)/ms

d2=(kr*l2-kf*l1)/ms

d3=(1/iy)*(kf*(l1)^2+kr*(l2)^2)

wn1_sq=((d1+d3)/2)-(sqrt((((d1-d3)^2)/4)+((d2)^2/(ry)^2)))

wn2_sq=((d1+d3)/2)+(sqrt((((d1-d3)^2)/4)+((d2)^2/(ry)^2)))

wn1=(wn1_sq)^0.5

wn2=(wn2_sq)^0.5

lo1=(d2)/(wn1_sq-d1)

lo2=(d2)/(wn2_sq-d1)

steps:

-inputs are declared

-then distance from cg to rear axle caulated using l2=wheelbase-distance from cg to front axle

-then moment of inertia is calculated using iy=m_sprung*m_unsprung/2

Then d1 calculated using:d1=front stifness+rear stiffness /2

-then d2is calculated using :d2=(kr*12-kf*11)/m_sprung

-then d3 is calculated using d3=(1/iy)*(kf*(l1)^2+kr*(l2)^2)

-Then first natural frequency squared is calculated using wn1 wn1_sq=((d1+d3)/2)-(sqrt((((d1-d3)^2)/4)+((d2)^2/(ry)^2)))

Where ry is the radius of gyration

-then the second natural frequency squared is calculated using: wn2_sq=((d1+d3)/2)+(sqrt((((d1-d3)^2)/4)+((d2)^2/(ry)^2)))

-then the first and second natural frequencies are calculated by taking the square root of the previous equations

-then bounce oscillation center is calculated using: lo1=(d2)/(wn1_squared -d1)

-then pitch oscillation center is calculated using:

lo2=(d2)/(wn2_sqared-d1)

Results:

coupled frequencies:

wn1 = 6.9794 rad/s

wn2= 8.1187 rad/s

Bounce oscillation center lo1 = 2.6923m behind the cg

Pitch oscillation center lo2=0.3910m ahead of the cg

Files:

https://drive.google.com/file/d/1tj9mVZLI_Nmalg4tbzfyg9PN6IMLKxxD/view?usp=sharing