Spring Rates, Wheel Rates, Motion Ratios and Roll StiffnessSession 5 :

1 Spring Rates, Wheel Rates, Motion Ratios and Roll StiffnessSession 5 Jim Ziech
Adjunct Professor
Mechanical and Aeronautical Engineering

Spring Rate Calculations :

2 Spring Rate Calculations

Coil Springs :

3 Coil Springs Fig. 21.3 p.762 Milliken

Spring Rate Calculations :

4 Spring Rate Calculations Coil Spring Calculations:
K s= Spring Rate in lbs/in G = Modulus of rigidity
d = Spring Wire Diameter R = Mean Radius of the Spring
N = Number of Active Coils µ = Poissons Ratio
Squared and Ground Ends -1.75 turns
Squared or Closed Ends ----
Plain Ends -0.5 turns
Plain ends Ground -1.0 turns

Spring Rate Calculations :

5 Spring Rate Calculations Coil Spring Calculations:
If Steel is used: E = 30,000,000 psi D = mean coil diameter

Spring Rate Calculations :

6 Spring Rate Calculations Torsion Bar Rates: L = Bar Length
d = Bar Diameter
r = lever arm length Let the deflection at the end = ?

Spring Rate Calculations :

7 Spring Rate Calculations Torsion Bar Rates: L r d ? Then the deflection rate at the free end is found Since T = F r &

Spring Rate Calculations :

8 Spring Rate Calculations The deflection rate at the free end is L r d ? The deflection rate at the wheel can now be found through analysis of the motion ratio

Spring Rate Calculations :

9 Spring Rate Calculations Torsion Bar Calculations:
If Steel is used: E = 30,000,000 psi L = Bar Length d = Bar Diameter
r = lever arm length

Typical Leaf Spring :

10 Typical Leaf Spring

Typical Leaf Spring :

11 Typical Leaf Spring Typical deflection behavior:

Typical Leaf Spring :

12 Typical Leaf Spring Typical Path behavior on deflection

Leaf Spring Calculations :

13 Leaf Spring Calculations Single Leaf Spring Rate: Where t = leaf thickness
b = leaf width
l = leaf half length Muli-Leaf Spring Rate: Where n = number of leaves
n’ = number leaves at ends

Motion Ratio Analysis(Installation Ratio) :

14 Motion Ratio Analysis(Installation Ratio)

Motion Ratio Analysis :

15 Motion Ratio Analysis

Motion Ratio Analysis :

16 Motion Ratio Analysis Spring Position
The displacement relationship between the spring and the wheel determines the actual rate the wheel works against for any spring rate. This displacement relationship may be defined as a motion ratio. The rate at the wheel is defined as the wheel rate (Kw). The rate of the spring itself is called the spring rate (Ks). The displacement relationship is a function of both spring position on the load carrying member and the angular orientation of the spring to that member.

Motion Ratio Analysis :

17 Wheel Rate - Location Dependent.
The spring position is important as it defines the mechanical advantage which exists between the wheel and the spring. Figure 1 depicts a spring acting on a simple lever. Motion Ratio Analysis Figure 1

Motion Ratio Analysis :

18 From the simple lever system a number of relationships can be drawn. Motion Ratio Analysis

Motion Ratio Analysis :

19 Motion Ratio in the Road Vehicle.
The motion ratio describes the displacement ratio between the spring and the centerline of the wheel. The motion ratio squared times the spring rate gives the wheel rate. Motion Ratio Analysis Figure 2

Motion Ratio Analysis :

20 Using the previous analysis and Figure 2, the following apply.
The above analysis assumes minimal camber change at the wheel.
The motion ratio can be determined experimentally and the measured distance ratio squared for an accurate value. Motion Ratio Analysis

Suspension Roll Stiffness :

21 Suspension Roll Stiffness

Suspension Roll Stiffness :

22 Suspension Roll Stiffness ROLL STIFFNESS due to wheel Rates:
The roll stiffness (Kf) can be determined using elementary analysis techniques. If the wheel rates (K) are determined and the spring spacing (t) is known then the roll stiffness relationship to spring stiffness follows. Note: t is equal to the wheel track if the wheel rates are used

Suspension Roll Stiffness :

23 The torque to rotate the chassis about the roll axis is shown in the following equation.
For equal spring rates, left and right the above equation reduces to the following. Suspension Roll Stiffness

Suspension Roll Stiffness :

24 The roll stiffness is then as shown below.
For roll stiffness in N-m/Deg [57.3 deg./rad] K = Individual wheel rate (N/m) t = track width (m) Suspension Roll Stiffness

Suspension Roll Stiffness :

25 In English units this can be reduced to Lb-Ft/Deg Suspension Roll Stiffness T = track width (in) K = Individual Wheel Rate (lb/in)

Suspension Roll Stiffness :

26 The total roll stiffness K? is equal to K ? F = Front Susp. Roll Stiffness K ?R = Rear Susp. Roll Stiffness
K ?(devices) = Stabilizer Bar etc. contributions (auxiliary roll stiffness) Suspension Roll Stiffness

Lateral Spring Center Position :

27 Lateral Spring Center Position Note: Not SAE axis system

Lateral Spring Center Position :

28 The Spring Center to Cg distance (x) at either end of the vehicle is important if unequal left/right spring rates are used. Create vertical displacement of y and sum moments about left tire patch. Which reduces to Lateral Spring Center Position

Lateral Spring Center Position :

29 Then from
The spring center to cg distance (x) is positive (to right of cg) if Lateral Spring Center Position

Lateral Spring Center Position :

30 The location of the Cg from the inside wheel centerline, distance ll, at each axle can be found from the scale weights at each wheel location if Cg is offset left/right.
Then by substitution into equation 1 yields equation 6 indicating the distance between the spring center (sc) and the center of gravity (cg). Lateral Spring Center Position

Roll Stiffness (Asymmetric Chassis) :

31 Roll Stiffness (Asymmetric Chassis) Roll stiffness should be calculated using the distance from the instantaneous spring center to each of the wheel locations.
The spring center location from the left tire centerline is as shown.
Therefore the roll stiffness for asymmetric springing is,

Roll Stiffness (Asymmetric Chassis) :

32 Roll Stiffness (Asymmetric Chassis) Recall, for equal spring rates, Then by substitution becomes,

35 The rotational stiffness of the rear axle (k? ax) due to the tire stiffness is
The rotational stiffness of the rear springs and rear stabilizer bar are kt = tire stiffness (N/m)
tr = rear track width
k? ax = Rotational stiffness (N-m/deg) ks = spring stiffness (N/m)
ts = rear spring spacing
k?b = Rear stabilizer bar (N-m/deg)
k?r susp = Rotational stiffness (N-m/deg) Suspension Roll Stiffness

Suspension Roll Stiffness :

36 The moment produced on the rear axle due to the tire stiffness is
The moment produced on the rear axle due to the springs and anti-roll bar is ?a = Axle roll angle ?c = Chassis roll angle Suspension Roll Stiffness

Suspension Roll Stiffness :

37 If no stabilizer bar is present the front suspension springs and the tire stiffness can be combined as a series system of springs to determine an equivalent ride rate.
The rotational stiffness of the rear axle due to the tire stiffness is If a stabilizer bar is present, the front springs and the stabilizer bar act together (parallel) to contribute to the stiffness, this is then translated to the tires. mr = motion ratio Suspension Roll Stiffness

Suspension Roll Stiffness :

38 Combining chassis roll rate with the tire contribution Suspension Roll Stiffness

Anti-Roll (Stabilizer) Bar :

39 Anti-Roll (Stabilizer) Bar

Anti Roll Bar Analysis :

40 Anti Roll Bar Analysis The deflection rate at the free end of a torsion bar. ? The deflection rate at the wheel can now be found through analysis of the motion ratio previously defined.

Anti Roll Bar Analysis :

41 Anti Roll Bar Analysis The deflection rate at the wheel is based on the motion ratio between the wheel and the bar end r2 versus r1. r1 = length of the attachment arm r2 = the pivot to attachment length The Roll stiffness has previously been defined as

Anti Roll Bar Analysis :

42 Anti Roll Bar Analysis The Roll stiffness has previously been defined as The stabilizer bar contribution to roll stiffness is now

The end! :

43 The end! Thank You

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