To Calculate Natural Frequency At Three Different Load Condition To Attain A Flat Ride Bump Oscillation Of A Vehicle Using MATLAB scripting.

OBJECTIVE

To calculate natural frequency at three different load condition attain a flat ride bump oscillation for a given vehicle dimension using MATLAB scripting.

ABSTRACT

Ride frequency, derived straight from natural frequency, is the suspension frequency found in cars. This is how fast the suspension travels up and then back down to the same point when you

drive over a bump. If cars did not have shocks/dampers, the springs would continue to

bounce up and down at this rate for quite some time. By examining the suspension frequency, we

are able to fairly accurately predict the handling characteristics of the suspension and how it will

'react' to driver inputs, road surface feedback, brake/acceleration loading, and downforce loading.

The first step in choosing spring stiffness is to choose your desired ride frequencies, front and rear. A ride frequency is the undamped natural frequency of the body in ride. The higher the frequency, the stiffer the ride. So, this parameter can be viewed as normalized ride stiffness.

Based on the application, there are ballpark numbers

to consider.

0.5-1.0Hz - Passenger cars, typical OEM

1.0-1.5Hz - Typical lowering springs

1.5-2.0Hz - Rally Cars

1.5-2.5Hz - Non-Aero racecars, moderate downforce

Formula cars

2.5-3.5Hz - Moderate downforce racecars with up to

50% total weight in max downforce

Two examples are shown below plots of what happens with unequal ride frequencies front and rear as the car hits a bump. In Figure 1, we can see the undamped vertical motion of the chassis with the front ride frequency higher than the rear. The first period is the most dominant on the car when looking at frequency phase, due to effects ofdamping.

Fig.1 Ride frequency out of phase

The out of phase motion between front and rear vertical motion, caused by the time delay between when the front wheel and rear wheel hit the bump, is accentuated by the frequency difference. A result of the phase difference is pitching of the body. To reduce the pitch induced by hitting a bump, the rear needs to have a higher natural frequency to “catch up” with the front, as shown in Figure 2. This notion is called producing a “flat ride”, meaning that the induced body pitch from road bumps is minimized. For a given wheelbase and speed, a frequency split front to rear can be calculated to minimize pitching of the body due to road bumps. A common split is 10 – 20% front to rear.

Fig.2 Ride frequency In phase

PROCEDURE

1. Input all the given parameters such as sprung, unsprung weight , motion ratio, wheelbase,tire stiffness, front target natural frequency etc.
2. Using speed and wheelbase time lag between front and rear frequency is calculated, starting with the basic equation from physics, relating natural frequency, spring rate, and mass:

Formulae used

`f=1/(2*pi*sqrt(kr/(mass)`

Where, F = NATURAL FREQUENCY (HZ)

Kr = RIDE RATE (N/M)

M = MASS (KG)

3. MATLAB scripting for flat ride bump oscillation at three different load conditions.

clear all 
close all
clc
%Input Parameters
%Total weight in three load conditions
Total_curb_weight=1500;%Kg
Total_design_weight=1825;%Kg
Total_gross_vehicle_weight=2000;%Kg
%Weight distribution front/rear
front_weight_distribution_curb=60*Total_curb_weight/100;%Kg
front_weight_distribution_design=55*Total_design_weight/100;%Kg
front_weight_distribution_GVW=52*Total_gross_vehicle_weight/100;%Kg
rear_weight_distribution_curb=40*Total_curb_weight/100;%Kg
rear_weight_distribution_design=45*Total_design_weight/100;%Kg
rear_weight_distribution_GVW=48*Total_gross_vehicle_weight/100;%Kg
%Front unsprung mass at three load conditions
unsprung_mass_front_curb=90;%Kg
unsprung_mass_front_design=90;%Kg
unsprung_mass_front_GVW=90;%Kg
unsprung_wt_front=90;%Kg
%Rear unsprung mass at three load conditions
unsprung_mass_rear_curb=60;%Kg
unsprung_mass_rear_design=60;%Kg
unsprung_mass_rear_GVW=60;%Kg
unsprung_wt_rear=60;%Kg
%Front sprung mass at three load conditions
sprung_mass_front_curb=(front_weight_distribution_curb-unsprung_mass_front_curb);%Kg
sprung_mass_front_design=(front_weight_distribution_design-unsprung_mass_front_design);%Kg
sprung_mass_front_GVW=(front_weight_distribution_GVW-unsprung_mass_front_design);%Kg
%Rear sprung mass at three load conditions
sprung_mass_rear_curb=(rear_weight_distribution_curb-unsprung_mass_rear_curb);%Kg
sprung_mass_rear_design=(rear_weight_distribution_design-unsprung_mass_rear_design);%Kg
sprung_mass_rear_GVW=(rear_weight_distribution_GVW-unsprung_mass_rear_GVW);%Kg

%Input parameters for all different load condition
k_t=245000;%N/m,tire rate for all condition
front_mr=0.97;
rear_mr=0.85;

%Lag time
wheelbase=2.7;%meters
speed=26.82;%meter/sec
t_wb=wheelbase/speed;%sec

%front sprung mass natural frequency at halfload/design load
sprung_freq_front=1.15;%Hz
%Rear sprung mass natural frequency at halfload/design load
sprung_freq_rear=1/(1/sprung_freq_front-t_wb);%Hz

%Rear spring rate
corner_sprung_wt_rear_design=(sprung_mass_rear_design/2);
k_r_rear=(2*pi*sprung_freq_rear)^2*corner_sprung_wt_rear_design;%N/m,Ride rate rear,(freq=1/2*pi*sqrtk_r_rear/mass)
k_s_rear=-k_t*k_r_rear/((k_r_rear*(rear_mr^2))-(k_t*(rear_mr^2)));%N/m (Rear spring rate)

%Front spring rate
corner_sprung_wt_front_design=(sprung_mass_front_design/2);
k_r_front=(2*pi*sprung_freq_front)^2*corner_sprung_wt_front_design;%N/m,Ride rate front,(freq=1/2*pi*sqrtk_r_front/mass)
k_s_front=-k_t*k_r_front/((k_r_front*(front_mr^2))-(k_t*(front_mr^2)));%N/m (Front spring rate)

%Front sprung mass natural frequency at curb load condition
corner_sprung_wt_front_curb=(sprung_mass_front_curb/2);
front_natural_freq_curb=(sqrt(k_r_front/corner_sprung_wt_front_curb)/2)/pi;%Hz , front frequency at curb

%Front sprung mass natural frequency at GVW load condition
corner_sprung_wt_front_GVW=(sprung_mass_front_GVW/2);
front_natural_freq_GVW=(sqrt(k_r_front/corner_sprung_wt_front_GVW)/2)/pi;%Hz , front frequency at GVW

%Rear sprung mass natural frequency at curb load condition
corner_sprung_wt_rear_curb=sprung_mass_rear_curb/2;
rear_natural_freq_curb=(sqrt(k_r_rear/corner_sprung_wt_rear_curb)/2)/pi;%Hz , rear frequency at curb

%Rear sprung mass natural frequency at GVW load condition
corner_sprung_wt_rear_GVW=sprung_mass_rear_GVW/2;
rear_natural_freq_GVW=(sqrt(k_r_rear/corner_sprung_wt_rear_GVW)/2)/pi;%Hz , rear frequency at GVW

%Front unsprung mass natural frequency at curb,halfload,GVW
k_r_front_unsprung=k_s_front*(front_mr^2)+(k_t);%N/m
corner_unsprung_wt_front=unsprung_wt_front;%Kg
front_unsprung_natural_freq=(sqrt(k_r_front_unsprung/corner_unsprung_wt_front)/2)/pi;%Hz ,front unsprung frequency 

%Rear unsprung mass natural frequency at curb,halfload,GVW
k_r_rear_unsprung=k_s_rear*(rear_mr^2)+(k_t);%N/m
corner_unsprung_wt_rear=unsprung_wt_rear;%Kg
rear_unsprung_natural_freq=(sqrt(k_r_rear_unsprung/corner_unsprung_wt_rear)/2)/pi;%Hz ,rear unsprung frequency 

%Flat ride calculation for curb load condition
time=0:0.1:3.5;
ct=1;
for t =0:0.1:3.5;
front_amp_curb(ct)=exp(-t)*sin(2*pi*front_natural_freq_curb*t);
rear_amp_curb(ct)=exp(-t)*sin(2*pi*rear_natural_freq_curb*(t-t_wb));
ct=ct+1;
end
%Flat ride calculation for design/half load condition
ct=1;
for t =0:0.1:3.5;
front_amp_design(ct)=exp(-t)*sin(2*pi*sprung_freq_front*t);
rear_amp_design(ct)=exp(-t)*sin(2*pi*sprung_freq_rear*(t-t_wb));
ct=ct+1;
end
%Flat ride calculation for GVW load condition
ct=1;
for t=0:0.1:3.5;
front_amp_GVW(ct)=exp(-t)*sin(2*pi*front_natural_freq_GVW*t);
rear_amp_GVW(ct)=exp(-t)*sin(2*pi*rear_natural_freq_GVW*(t-t_wb));
ct=ct+1;
end

%Ratio of rear/front frequency at curb load condition
freq_ratio_curb=(rear_natural_freq_curb/front_natural_freq_curb);
%Ratio of rear/front frequency at half/design load condition
freq_ratio_design=(sprung_freq_rear/sprung_freq_front);
%Ratio of rear/front frequency at GVW load condition
freq_ratio_GVW=(rear_natural_freq_GVW/front_natural_freq_GVW);
%Ratio of rear/front frequency at unsprung condition
freq_ratio_unsprung=rear_unsprung_natural_freq/front_unsprung_natural_freq;
figure(1);
plot(time,front_amp_curb,'linewidth',1.5);
hold 
plot(time,rear_amp_curb,'linewidth',1.5);
grid on
xlabel('Time');
ylabel('Ride frequency');
legend('Front','Rear');
title('Curb Load Condition');
hold off

figure(2);
plot(time,front_amp_design,'linewidth',1.5);
hold 
plot(time,rear_amp_design,'linewidth',1.5);
grid on
xlabel('Time');
ylabel('Ride frequency');
legend('Front','Rear');
title('Design Load Condition');
hold off

figure(3);
plot(time,front_amp_GVW,'linewidth',1.5);
hold 
plot(time,rear_amp_GVW,'linewidth',1.5);
grid on
xlabel('Time');
ylabel('Ride frequency');
legend('Front','Rear');
title('GVW Load Condition');
hold off

RESULTS

• Recommended rear sprung mass frequency = 1.3006 Hz
• Front sprung mass natural frequency at half load =1.15 Hz
• Front spring rate =28086.35N/m
• Rear spring rate =39251.6 N/m
• Front sprung mass natural frequencies at curb=1.22 Hz
• Front sprung mass natural frequencies at GVW =1.127 Hz
• Rear sprung mass natural frequencies at curb =1.54 Hz
• Rear sprung mass natural frequencies at GVW =1.196 Hz
• Front unsprung mass natural frequency at all three load conditions =08.74 Hz
• Rear unsprung mass natural frequency at all three load conditions =10.74 Hz
• Rear/Front frequency ratio for curb load condition = 1.26
• Rear/Front frequency ratio for half load /design load condition =1.13
• Rear/Front frequency ratio for GVW load condition =1.06

BUMP OSCILLATION AT DIFFERENT LOAD CONDITIONS

fig.3 BUMP OSCILLATION PROFILE AT HALF LOAD/DESIGN LOAD CONDITION

fig.4 BUMP OSCILLATION PROFILE AT CURB LOAD CONDITION

fig.5 BUMP OSCILLATION PROFILE AT GVW LOAD CONDITION

CONCLUSION

As to increase the comfortability and pitching motion of the vehicle is minimized so as to achieve flat ride. Ride frequency of the suspension system plays a major role in the design of the vehicle. By considering this parameter the comfort level and performance level of the vehicle are equally maintained.