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- 1. Spring Rates, Wheel Rates, Motion Ratios and Roll Stiffness Appendix 1 ME 470 Vehicle Structural Design Dr. Richard Hathaway, P.E., Professor Mechanical and Aeronautical Engineering
- 2. Spring Rate Calculations
- 3. Spring Rate Calculations <ul><li>Coil Spring Calculations: </li></ul><ul><li>K = Spring Rate in lbs/in G = Modulus of rigidity </li></ul><ul><li>d = Spring Wire Diameter R = Mean Radius of the Spring </li></ul><ul><li>N = Number of Active Coils </li></ul><ul><li>Squared and Ground Ends -1.75 turns </li></ul><ul><li>Squared or Closed Ends ---- </li></ul><ul><li>Plain Ends -0.5 turns </li></ul><ul><li>Plain ends Ground -1.0 turns </li></ul>
- 4. Spring Rate Calculations <ul><li>Coil Spring Calculations: </li></ul><ul><ul><li>If Steel is used: E = 30,000,000 psi </li></ul></ul>
- 5. Spring Rate Calculations <ul><li>Torsion Bar Rates: </li></ul>L = Bar Length d = Bar Diameter r = lever arm length Let the deflection at the end = L r d
- 6. Spring Rate Calculations <ul><li>Torsion Bar Rates: </li></ul>L r d Then the deflection rate at the free end is found Since T = F x r &
- 7. Spring Rate Calculations <ul><li>The deflection rate at the free end is </li></ul>L r d The deflection rate at the wheel can now be found through analysis of the motion ratio
- 8. Spring Rate Calculations <ul><li>Torsion Bar Calculations: </li></ul><ul><ul><li>If Steel is used: E = 30,000,000 psi </li></ul></ul>L = Bar Length d = Bar Diameter r = lever arm length
- 9. Typical Leaf Spring
- 10. Typical Leaf Spring Typical deflection behavior:
- 11. Typical Leaf Spring Typical Path behavior on deflection
- 12. Motion Ratio Analysis
- 13. Motion Ratio Analysis
- 14. Motion Ratio Analysis <ul><li>Spring Position </li></ul><ul><li>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 (K w ). The rate of the spring itself is called the spring rate (K s ). 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. </li></ul>
- 15. <ul><li>Wheel Rate - Location Dependent. </li></ul><ul><ul><li>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. </li></ul></ul>Motion Ratio Analysis Figure 1
- 16. <ul><li>From the simple lever system a number of relationships can be drawn. </li></ul>Motion Ratio Analysis
- 17. <ul><li>Motion Ratio in the Road Vehicle. </li></ul><ul><ul><li>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. </li></ul></ul>Motion Ratio Analysis Figure 2
- 18. <ul><li>Using the previous analysis and Figure 2, the following apply. </li></ul><ul><ul><li>The above analysis assumes minimal camber change at the wheel. </li></ul></ul><ul><ul><li>The motion ratio can be determined experimentally and the measured distance ratio squared for an accurate value. </li></ul></ul>Motion Ratio Analysis
- 19. Suspension Roll Stiffness
- 20. Suspension Roll Stiffness <ul><li>ROLL STIFFNESS due to wheel Rates: </li></ul><ul><ul><li>The roll stiffness (K φ ) 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. </li></ul></ul><ul><li> </li></ul>Note: t is equal to the wheel track if the wheel rates are used
- 21. <ul><li>The torque to rotate the chassis about the roll axis is shown in the following equation. </li></ul><ul><li>For equal spring rates, left and right the above equation reduces to the following. </li></ul>Suspension Roll Stiffness
- 22. <ul><li>The roll stiffness is then as shown below. </li></ul><ul><li>For roll stiffness in N-m/Deg </li></ul>Suspension Roll Stiffness K = Individual wheel rate (N/m) t = track width (m)
- 23. <ul><li>In English units this can be reduced to Lb-Ft/Deg </li></ul>Suspension Roll Stiffness T = track width (in) K = Individual Wheel Rate (lb/in)
- 24. <ul><li>The total roll stiffness K is equal to </li></ul>Suspension Roll Stiffness K F = Front Roll Stiffness K R = Rear Roll Stiffness K (devices) = Stabilizer etc contributions
- 25. Lateral Spring Center Position
- 26. <ul><li>The Spring Center to Cg distance (x) at either end of the vehicle is important. </li></ul>Lateral Spring Center Position Which reduces to
- 27. <ul><ul><li>Then from </li></ul></ul><ul><li>The spring center to cg distance (x) is positive (to right of cg) if </li></ul>Lateral Spring Center Position
- 28. <ul><li>The location of the Cg from the inside wheel centerline, distance l l , at each axle can be found from the scale weights at each wheel location. </li></ul><ul><li>Then by substitution into equation 1 yields equation 6 indicating the distance between the spring center (sc) and the center of gravity (cg). </li></ul>Lateral Spring Center Position
- 29. Roll Stiffness (Asymmetric Chassis) <ul><li>Roll stiffness should be calculated using the distance from the instantaneous spring center to each of the wheel locations. </li></ul><ul><ul><li>The spring center location from the left tire centerline is as shown. </li></ul></ul><ul><ul><li>Therefore the roll stiffness for asymmetric springing is, </li></ul></ul>
- 30. Roll Stiffness (Asymmetric Chassis) <ul><li>Recall, for equal spring rates, </li></ul>Then by substitution becomes,
- 31. <ul><li>Example: </li></ul><ul><li>Symmetric Setup: </li></ul><ul><li>LR w = 175 lb/in RR w = 175 lb/in </li></ul><ul><li>Track = 68 inches </li></ul>Roll Stiffness
- 32. <ul><li>Example: </li></ul>Roll Stiffness Asymmetric Setup: LR w = 200 lb/in RR w = 175 lb/in Asymmetric Setup: LR w = 200 lb/in RR w = 150 lb/in Note: Avg = 175 lb/in Track = 68 inches
- 33. <ul><ul><li>The rotational stiffness of the rear axle (k ax ) due to the tire stiffness is </li></ul></ul><ul><ul><li>The rotational stiffness of the rear springs and rear stabilizer bar are </li></ul></ul>Suspension Roll Stiffness k t = tire stiffness (N/m) t r = rear track width k ax = Rotational stiffness (N-m/deg) k s = spring stiffness (N/m) t s = rear spring spacing k b = Rear stabilizer bar (N-m/deg) k r susp = Rotational stiffness (N-m/deg)
- 34. <ul><li>The moment produced on the rear axle due to the tire stiffness is </li></ul><ul><li>The moment produced on the rear axle due to the springs and anti-roll bar is </li></ul>Suspension Roll Stiffness a = Axle roll angle c = Chassis roll angle
- 35. <ul><li>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. </li></ul><ul><li>The rotational stiffness of the rear axle due to the tire stiffness is </li></ul>Suspension Roll Stiffness <ul><li>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. </li></ul>mr = motion ratio
- 36. <ul><li>Combining chassis roll rate with the tire contribution </li></ul>Suspension Roll Stiffness
- 37. Anti-Roll (Stabilizer) Bar Analysis
- 38. Anti Roll Bar Analysis <ul><li>The deflection rate at the free end of a torsion bar. </li></ul> The deflection rate at the wheel can now be found through analysis of the motion ratio previously defined. L r d
- 39. Anti Roll Bar Analysis <ul><li>The deflection rate at the wheel is based on the motion ratio. </li></ul>r 1 = length of the attachment arm r 2 = the pivot to attachment length <ul><li>The Roll stiffness has previously been defined as </li></ul>
- 40. Anti Roll Bar Analysis <ul><li>The Roll stiffness has previously been defined as </li></ul><ul><li>The stabilizer bar contribution to roll stiffness is now </li></ul>
- 41. The end! Thank You

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