How to select/compute spring rates for any car

[rattus]

Here is a Matlab script I whipped up to calculate spring rates / suspension frequencies.

I adopted the forumlas from various websites (and my undergrad physics textbook) and I'm not 100% sure I have things right (I'm waiting for my copy of RCVD to show up). The vehicle constants are rough approximates for my 69 Firebird.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%
vehicle_weight = 3400; %total vehicle weight
f_weight_dist = 0.54; %front weight percentage
f_wheel_unsprng = 100; %front wheel unsprung weight (per wheel)
r_wheel_unsprng = 200; %rear wheel unsprung weight (per wheel)
f_mr = 8./16; %approx. front motion ratio from LCA
r_mr = 0.9; %approx. rear motion ratio (live axle)
f_sprate = 850; % given front spring rate (lb/in)
r_sprate = 250; % given rear spring rate (lb/in)
f_damp_ratio = 0.35;
r_damp_ratio = 0.35;
f_freq = 1.7; %Hz
r_freq = f_freq + 0.15.*f_freq; %Hz, rear freq is 15% greater than front freq
wheelbase = 100; %approx. (inches)
avgspeed = 80; % mph

% convert to SI units
fmass = ((vehicle_weight.*f_weight_dist)./2-f_wheel_unsprng).*0.45359237;
rmass = ((vehicle_weight.*(1 - f_weight_dist))./2-r_wheel_unsprng).*0.45359237;
f_sprate = f_sprate.*175.126835;
r_sprate = r_sprate.*175.126835;

% calculate wheel rates
f_wheel_rate = f_sprate.*(f_mr.^2);
r_wheel_rate = r_sprate.*(r_mr.^2);

% calculate undamped spring rates given frequencies
Ksf = 4.*(pi^2).*fmass.*(f_freq^2)./(f_mr^2)./175.126835
Ksr = 4.*(pi^2).*rmass.*(r_freq^2)./(r_mr^2)./175.126835

% calculate undamped frequencies given spring rates
f_freq_undamp = (1./(2.*pi)).*sqrt(f_wheel_rate./fmass)
r_freq_undamp = (1./(2.*pi)).*sqrt(r_wheel_rate./rmass)

%calculate damped frequencies given spring rates
f_freq_damp = f_freq_undamp.*sqrt(1-f_damp_ratio.^2)
r_freq_damp = r_freq_undamp.*sqrt(1-f_damp_ratio.^2)

% calculate damped time constants
f_tau = (2.*pi).*f_freq_undamp.*f_damp_ratio;
r_tau = (2.*pi).*r_freq_undamp.*r_damp_ratio;

% whip up time independent variables
tf = linspace(0,f_tau,1000);
f_to_r_timlag=wheelbase./(avgspeed*63360/3600);%time lag from bump front to rear
tr = tf-f_to_r_timlag;tr = tr.*(tr >= 0);%offset rear time, get rid of neg times

% plot damped trajectories
figure(1); clf;
plot(tf,exp(-f_tau.*tf).*sin(2.*pi.*f_freq_damp.*tf));hold on;
plot(tf,exp(-r_tau.*tr).*sin(2.*pi.*r_freq_damp.*tr),'r');hold off;
title('damped suspension trajectories');
xlabel('time (s)'); ylabel('amplitude (arbitrary units)');
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Results:

Ksf = 967 lb/in, front spring rate given 1.7Hz (undamped)
Ksr = 281 lb/in, rear spring rate given 2.0Hz (undamped)



f_freq_undamp = 1.6 Hz, front freq. given 850 lb/in spring rate (undamped)
r_freq_undamp = 1.8 Hz, rear freq. given 250 lb/in spring rate (undamped)

f_freq_damp = 1.4931 Hz, front freq. given 850 lb/in spring rate (damped 35% of critical)
r_freq_damp = 1.7280 Hz, rear freq. given 250 lb/in spring rate (damped 35% of critical)

Attached is a plot of the front and rear trajectories:

[Norm Peterson]

Looks pretty good, although 200 lbs per rear corner sounds a bit high unless you have a real beast of an axle. Are leaf springs really that heavy (remembering that perhaps a third to half of their weight is sprung)?

I ran a sanity check for one of my spreadsheets and got . . .

Ksf = 965 lb/in, front spring rate given 1.7Hz (undamped)
Ksr = 293 lb/in, rear spring rate given 2.0Hz (undamped)

Note that Ksr = 280 lb/in if I use 1.15*1.7 = 1.955 Hz instead of 2.0 Hz.

f_freq_undamp = 1.60 Hz, front freq. given 850 lb/in spring rate (undamped)
r_freq_undamp = 1.85 Hz, rear freq. given 250 lb/in spring rate (undamped)

f_freq_damp = 1.49 Hz, front freq. given 850 lb/in spring rate (damped 35% of critical)
r_freq_damp = 1.73 Hz, rear freq. given 250 lb/in spring rate (damped 35% of critical)

(Sorry for the crappy thumbnail resolution - it's easier to see as a full screen display and then expanded with the sizing button in the lower right hand corner.)

Norm

[rattus]

Cool, thanks Norm. After I posted, I thought that the rear unsprung weight should have been about half of what I entered. D. Pozzi has 110 lbs (per rear wheel, 220 total) unsprung weight listed on his web site.

[David Pozzi]

I forgot to include the weight of the wheels and tires in that calculation!
Here's some info:

REAR SUSPENSION
rear axle 12 bolt with drm brks, posi -complete 185
leaf spring - multi stock 32
Hotchkis type leaf closer to 40 - 45.
shock 3
Wheel and tire is around 50 lbs
sway bar is probably around 10 lbs

I think you'd take half the weight of the spring, bar, and shock.
somewhere around 287 lbs total depending on your exact wheel/tire weight.

[paul67]

would you not have to take the weight of the driver , and tire pressures ?

[Norm Peterson]

(1) Best results would consider the driver's weight and his own CG location. And the weight and CG of any other variable loading.

(2) Tire pressures will make a generally small difference, but this effect isn't very usable unless you have some idea of the correlation between inflation pressure and tire vertical/radial spring rate.


Norm

[silver69camaro]

Just for a reference, Racecar Engineering quotes:

1 Hz for Caddy rides
1.3Hz for sports cars
1.8Hz high performance road cars
2+Hz for racecars w/o aero aids

It also mentions that ground effect racecars could have attained up to 10Hz! Wow!

Anyway, my personal experience hasn't followed the above guidelines. I've done vehicles with a frequency slightly above 2.0, and it had a very pleasant ride with excellent handling (.95G).

[Bjkadron]

I know this is a REALLY old thread... But none of the links work. Does anyone have any new info or spreadsheet links?

Thanks!

Benjamin

[Norm Peterson]

What links?

All I see are thumbnail pics.

I probably have one version or other of my sheet on most of the computers I ever use, but nowhere that can be linked to directly.


Norm

[John Wright]

What links?

I guess he was speaking of David's link in the first few posts in the thread....

Sorry, the GeoCities web site you were trying to reach is no longer available.

[Bjkadron]

I guess he was speaking of David's link in the first few posts in the thread....

Yes that was it. So nobody knows of another one that I can get too? Or how I could possibly get my Grubby little hands on one of those spreadsheets? I suck at excell.

Thanks

[Norm Peterson]

If you PM me with an e-mail address, I can send out the latest version that's on this computer.

Forum e-mails don't generally permit attachments.


Norm