Spring Rates, Wheel Rates, Motion Ratios and Roll Stiffness

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Spring Rates, Wheel Rates, Motion Ratios and Roll
Stiffness

• Dr. Richard Hathaway, P.E.
• Professor
• Mechanical and Aeronautical Engineering

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Spring Rate Calculations
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Spring Rate Calculations

• Coil Spring Calculations
• 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

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Spring Rate Calculations

• Coil Spring Calculations
• If Steel is used E 30,000,000 psi

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Spring Rate Calculations

• Torsion Bar Rates

L Bar Length d Bar Diameter r
lever arm length
Let the deflection at the end ?
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Spring Rate Calculations

• Torsion Bar Rates

L
Since T F x r
d
r
?
Then the deflection rate at the free end is found
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Spring Rate Calculations

• The deflection rate at the free end is

L
d
r
?
The deflection rate at the wheel can now be found
through analysis of the motion ratio
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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
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Typical Leaf Spring
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Typical Leaf Spring
Typical deflection behavior
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Typical Leaf Spring
Typical Path behavior on deflection
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Motion Ratio Analysis
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Motion Ratio Analysis
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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.

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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
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Motion Ratio Analysis

• From the simple lever system a number of
  relationships can be drawn.

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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.

Figure 2
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Motion Ratio Analysis

• 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.

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Suspension Roll Stiffness
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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
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Suspension Roll Stiffness

• 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.

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Suspension Roll Stiffness

• The roll stiffness is then as shown below.
• For roll stiffness in N-m/Deg

K Individual wheel rate (N/m) t track width
(m)
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Suspension Roll Stiffness

• In English units this can be reduced to Lb-Ft/Deg

T track width (in) K Individual Wheel Rate
(lb/in)
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Suspension Roll Stiffness

• The total roll stiffness K? is equal to

K ? F Front Roll Stiffness K ?R Rear
Roll Stiffness K ?(devices) Stabilizer etc
contributions
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Lateral Spring Center Position
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Lateral Spring Center Position

• The Spring Center to Cg distance (x) at either
  end of the vehicle is important.

Which reduces to
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Lateral Spring Center Position

• Then from
• The spring center to cg distance (x) is positive
  (to right of cg) if

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Lateral Spring Center Position

• 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.
• Then by substitution into equation 1 yields
  equation 6 indicating the distance between the
  spring center (sc) and the center of gravity
  (cg).

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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,

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Roll Stiffness (Asymmetric Chassis)

• Recall, for equal spring rates,

Then by substitution becomes,
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Roll Stiffness

• Example
• Symmetric Setup
• LRw 175 lb/in RRw 175 lb/in
• Track 68 inches

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Roll Stiffness

• Example

Asymmetric Setup LRw 200 lb/in RRw 150
lb/in Note Avg 175 lb/in Track 68 inches
Asymmetric Setup LRw 200 lb/in RRw 175
lb/in
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Suspension Roll Stiffness

• 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)
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Suspension Roll Stiffness

• 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
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Suspension Roll Stiffness

• 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

mr motion ratio

• 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.

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Suspension Roll Stiffness

• Combining chassis roll rate with the tire
  contribution

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Anti-Roll (Stabilizer) Bar Analysis
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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.
?
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Anti Roll Bar Analysis

• The deflection rate at the wheel is based on the
  motion ratio.

r1 length of the attachment arm r2 the pivot
to attachment length

• The Roll stiffness has previously been defined as

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Anti Roll Bar Analysis

• The Roll stiffness has previously been defined as

• The stabilizer bar contribution to roll
  stiffness is now

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The end!

• Thank You