Spring Rates, Wheel Rates, Motion Ratios and Roll Stiffness

Motion Ratio Analysis

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.
G=
E 2 (1 + )
Spring Rate Calculations

Coil Spring Calculations:
If
Steel is used:
E = 30,000,000 psi
1,500,000 d 4 K 3 D N
Spring Rate Calculations

Torsion Bar Rates:

The deflection rate at the free end is
F


d 4G
32 L r
2
L
k
r
d
The deflection rate at the wheel can now be found through analysis of the motion ratio
Spring Rate Calculations
TL JG
J
T
JG L
T
L
d
L = Bar Length d = Bar Diameter r = lever arm length
d4
32
32 L
r
d 4G

Let the deflection at the end =
r
Spring Rate Calculations
Spring Rate Calculations

Coil Spring Calculations: G d4 K= 64 R3 N
K = Spring Rate in lbs/in G = Modulus of rigidity d = Spring Wire Diameter R = Mean Radius of the Spring N = Number of Active Coils Squared and Ground Ends -1.75 turns Squared or Closed Ends ---Plain Ends -0.5 turns Plain ends Ground -1.0 turns
Motion Ratio Analysis

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.
Figure 1
Motion Ratio Analysis

From the simple lever system a number of relationships can be drawn.
a FB= FA b b dB= dA a
a 2 F A FB = k b k a B A dB b b dA a
Typical Leaf Spring
Typical deflection behavior:
Typical Leaf Spring
Typical Path behavior on deflection
Motion Ratio Analysis
Motion Ratio Analysis
Motion Ratio Analysis
Spring Rates, Wheel Rates, Motion Ratios and Roll Stiffness
Dr. Richard Hathaway, P.E. Professor
Mechanical and Aeronautical Engineering
Spring Rate Calculations

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.

Torsion Bar Calculations:
If
Hale Waihona Puke Steel is used:
E = 30,000,000 psi
2,200,000 d 4 K L r2
L = Bar Length r = lever arm length d = Bar Diameter
Typical Leaf Spring

Torsion Bar Rates:
Since T = F x r &
r
L d
T


d G
4
F xr
32 L


d 4G
32 L
r
r
Then the deflection rate at the free end is found
F


d 4G
32 L r
2
k
Spring Rate Calculations