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ride frequency calculation

sloopy

sloopy

2-digit whp
Joined
May 25, 2020
Location
North Texas
hello. ive ordered coilovers from ben and im trying to decide which spring rates i want to go with. i want to do some ride frequency calculations to help me choose a spring which will give the car sporty handling while remaining reasonably comfortable for the street.

i found this formula for ride frequency:
f = sqrt(k / m) / 2pi
where k is wheel rate in newtons per meter and m is mass in kilograms

to do this, i need the motion ratios of the front and rear axles and i need the weight over each axle. i know the rear motion ratio is 1.5 * spring rate. for the front, ive heard it is about 0.9 * spring rate. that would mean the steering axis is leaning at a considering im using ben's offset strut mounts which add 3 degrees of caster, this should increase that motion ratio slightly as will any positive adjustment of the camber. im also using ben's quick steer roll correctors, but since im lowering the car the recommended 1.5-2", this should have practically no impact on the motion ratio i believe.

as for the weight over each axle, i wasnt able to find much info besides hearing weight figures from 2900 lbs to 3100 lbs and a supposed 55/45 front to rear split. this might mean about 1650 lbs for the front axle, or 825 lbs for a front wheel, and 1350 lbs for the rear axle, or 675 lbs for a rear wheel.

given the factory front springs from lesjofors are 87.4 lbs/in and a motion ratio of 0.9, that would be a 78.66 lb/in wheel rate. using the ride frequency formula (and converting the units), that comes out to an estimated 0.97 hz front ride frequency. if we do the same for the rear given a 121lb/in spring rate, the wheel rate would be 181.5 and the ride frequency 1.62 hz. the numbers seem a bit weird to me.

as for a target ride frequency, i found this chart which shows:
an e46 m3 coupe has a ride frequency of 1.27 and 1.53 front to rear
an e46 m3 convertible has a ride frequency of 1.77 and 1.75 front to rear
https://www.fatcatmotorsports.com/FCM_Ride_Harmony_BMW_E46_M3_Online/FCM_Ride_Harmony_BMW_E46_M3_Online.htm

for the 240, a front spring rate of 150 lbs/in should produce a ride frequency of 1.27 hz assuming the 0.9 motion ratio and 825 lb corner weight. a front spring rate of 175 lbs/in would produce a ride frequency of 1.37 hz. and a front spring rate of 200 lbs/in would produce a ride frequency of 1.46 hz.

has anyone measured the front motion ratios for their modified cars? and does anyone have alternative weights i might use? i have a 1992 240 wagon naturally aspirated, ive replaced the battery with a 5 lb lithium, and ive replaced the really heavy factory exhaust header with a very light tubular one. also, is there really such a huge difference in ride frequency between an e46 m3 coupe and convertible? are they trying to compensate for the more flexible chassis in the convertible and they wanted to maintain the same performance as the coupe?
 
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I don't know if those spring rates are right for the E36. The convertible is a lot heavier than the coupe. Not twice the weight though.

I'm not sure what the front motion ratio is for the 240, but most cars are very close to 1:1, perhaps 1:1.02 at most, so I'd be inclined not to worry about it. I should imagine there is a greater error margin in the measured spring rates, which incidentally there is a thread on here:
http://forums.turbobricks.com/showthread.php?t=325739

I make the rear motion ratio to be closer to 1.24. The axle is 565mm from the pivot and the spring ~700mm. 700/565=1.24

I'm sure there is a thread somewhere with corner weights too. My 240 saloon is F616kg, R600kg, but is not stock.
 
There is no motion ratio in front, but there is loss due to strut angle. I usually assume wheel rate to be between 90% and 95% of spring, since the angle varies with different included angles and camber settings. i.e. I don't worry about it.

The rear MR is close to 1.25, but more importantly, the rear motion ratio SQUARED is in fact close to 1.5:1. The distance AND the pressure change, which is why the simple MR is not what you use.

From real world 245 spring experience, 200#/in fronts are a great start, anything less is not worth using. In the rear, 125#/in is a bit lower after the MR^2, 187.5#/in. That should work if you want close to equal hz numbers.
I personally would start at 250 front, 150 rear. it would still be a bit roly poly with stock bars, less so with a 19mm sedan rear bar and a 22-23mm front.
 
is loss due to strut angle not the same as a motion ratio? although i guess its more complicated than a static number because the ratio changes as the suspension compresses. i dont understand why you would square the rear motion ratio.

i eventually decided on 250 lb/in front springs and 150 lb/in rear springs. im running koni sports, 23mm front bar and 19mm rear bar. ill try the 16mm rear bar and no rear bar to see how those feel.
 
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With a MacPherson strut type suspension there are two things that contribute to the difference between the wheel rate and the spring rate, the wheel offset from the strut (Motion ratio) and the strut inclination. Luckily, in most cases the bottom of the strut ends up inside the wheel, so there is very little contribution of the wheel offset and the inclination of the strut is very close to vertical anyway. Hence not being too much of an issue. It’s not like the strut is laid down at 45 degrees.
As mikep says, the motion ratio applies to both the force and the displacement of the spring/damper, which is why you need to square it to get the wheel rate (rate being a combination of both force and displacement) from the spring rate, or vice versa.
 
crogthomas said:
With a MacPherson strut type suspension there are two things that contribute to the difference between the wheel rate and the spring rate, the wheel offset from the strut (Motion ratio) and the strut inclination. Luckily, in most cases the bottom of the strut ends up inside the wheel, so there is very little contribution of the wheel offset and the inclination of the strut is very close to vertical anyway. Hence not being too much of an issue. It’s not like the strut is laid down at 45 degrees.
As mikep says, the motion ratio applies to both the force and the displacement of the spring/damper, which is why you need to square it to get the wheel rate (rate being a combination of both force and displacement) from the spring rate, or vice versa.
Click to expand...
The wheel offset from the strut does not affect wheel rate. all force is transferred through the ball joint.

To clarify MR vs MR^2:
If the spring on a lower control arm (think american car or amazon) is at the halfway point between the control arm bushing and the ball joint, the Motion Ratio (MR) is 0.5. The wheel has half as much pressure as the spring, while the wheel moves twice as far as the spring.
So if the spring pressure (on the seat) increases 1000lb force per 1 inch compression, the wheel pressure (on the ground) increases 500lb force per 2 inches compression. Divide 500 by 2, the wheel rate is 250lbf/in.
0.5 MR squared is 0.25.
 
I changed my springs a little while back. My front springs are 200 lb/in and the rear springs are progressive 131-188 lb/in from lesjofors' sport spring kit for 240 wagons. Koni sports set to full stiff all around. 23 mm front and 19 mm rear bars. I calculated the front ride frequency to be about 1.55 hz. The car rides great. It feels nimble, but it doesn't throw you around too much on bumpy roads either. It's not harsh at all especially with the meaty 225/60R15 tires. Compared to the 250/150 lb/in front/rear spring combo, it feels like the car pitches much less over bumps. Actually it feels almost totally neutral depending on the bump and the speed. The car also feels safely balanced. It has a tendency to understeer, but you can overcome it with gas or brake if you're motivated. It's dead stable at 110 mph (my maximum speed) and I like to wonder how it would feel with an undertray :eek:)
 
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Because of this thread I re-thought my WRX wagon suspension, and increased the rear rate a bit with King springs. Still using a 22 mm front, 20mm rear.
 
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