Unsprung Secrets 
Solutions
Version 2.4

Copyright 2015 by Paul Niquette. All rights reserved.


B-rig
                            IllumnationThe Unsprung Secrets puzzle seeks an explanation for why C-rigs damage roadways in California more than B-rigs in Brittany.  The secrets must be in their designs

In either case a truck-wheel deflected by the roadway requires a momentary change in its tangential -- and therefore its angular -- velocity.  Such changes are resisted by inertia, and not all truck-wheels are the same in that regard.  Three categories come to mind...
Steering Wheel SW.............................. same configuration for both rigs
Drive Wheel DW................ duals on
B-rigs and tandem-duals on C-rigs
Trailer Wheel TW...tandem-duals 
on C-rigs and tridem-singles on B-rigs
So far we have treated all wheels as independent of one another with respect to their angular rotations.  That's appropriate for both SWs and TWs, which are simply 'coasting' on the roadway surface.  However, as essential features of the tractor's drive train, DWs deliver tractive effort to the roadway via shear forces in their respective contact patches.
Inasmuch as our suspicion has to do with where the rubber meets to road, we must note the fact that the C-rig tractor has twice as many drive-wheel contact patches as the B-rig, which means less damage not more, ceteris parabus.
Drive-Train
A Review of the Hardware 

For a pleasant review of this subject and others, solvers are invited to enjoy
this animated video of a drive-train design which is typical of the tractor on a B-rig...

Animated Suspension

Non-drive wheels, SWs and TWs, go along spinning at whatever tangential speed complies with the local profile of the roadway.  Each SW or TW can be individually accelerated or decelerated by the roadway pavementDrive-wheels DWs are a different matter entirely.  For producing tractive effort DWs must be 'torqued' by the truck's drive-train, which means applying twisting forces through their respective axles.  
Oh sure, to accommodate turns and roadway curves, a drive-axle must be split to allow an outside DW to rotate more rapidly than the corresponding inside DW.  That does not, however, give DWs complete 'independence' from one another.  That will be addressed as a separate topic (Differential Differences).  For now, let's postulate straight roadway and regard drive-axles as not split.
Drive-Train: B-rig vs C-rig
Real Difference in Design 


As is obvious to anyone who sees typical camions plying the routes in Brittany, a
B-rig tractor has only one drive-axle.  Meanwhile, on the roadways in California the C-rig, which is best known by the sobriquet "18-wheeler," is seen to have two drive-axles.  Here is a cutaway of the drive-train on a C-rig tractor...

C-rig Drive Train

In the video we saw how a single drive shaft on a B-rig delivers tractive effort as torque from the engine to its single drive-axle via its single differential.  As shown in the cutaway above, the C-rig tractor has two drive-axles, two differentials, and a second drive shaft in between.  That mechanical configuration assures that both drive-axles are provided with equal torques produced by the engine.  That seems like a good idea (for large values of "seems").
On a C-rig tractor traveling at a constant speed vX along a perfectly flat roadway, all four dual-wheels will have the same tangential velocity vX and angular velocity both left and right: ωL1 = ωR1= ωL2= ωR2 = vX / rWWith a 3:1 ratio in the differentials, both drive shafts must have the same angular velocity ωD, which is three times faster than the DWs.  At a highway speed of 60 mph (vX = 88 fps), 42-inch DWs (3.5 ft in diameter) will be turning at 480 rpm and the drive shaft coming out of the gearbox spins at 1,440 rpm such that ωD = 150 radians/second. 

Same for the 'offset gearbox' in the cutaway above.  The more common expression is 'transfer case', which is a necessary feature in four-wheel drive trucks and passenger vehicles wherein gearing or a drive-belt must be used to facilitate other than 1:1 ratio for accommodating different-size wheels front and rear.  To address that requirement, a
'center differential' is often integrated into the transfer case.  Meanwhile, vehicles with all-wheel drive so popular in California can seldom be seen in Brittany, which may explain the lack of 'washboarding' observed by the author even on unpaved roads.
Most significant to the Unsprung Secrets solution is the fact that the two drive-axles on a C-rig tractor are not independent.  The drive-train coerces the angular velocities of all DWs to be the same regardless of any local disparities in the roadway.  By definition, if angular velocities are the same and the wheel radii are the same, then during a given time interval, the circumferential displacements of all DWs must be identical to one another.

Identical circumferential displacements of C-rig DWs can result in scuffing action at contact patches located on segments of roadway that do not have identical profiles.

C-rig-Specific Phenomenon
Discovery of Shear Pulse

The sketch below depicts wheels on the tandem drive-axles of a C-rig tractor as they sequentially encounter a deflection yD in the roadway.  The vertical profile of the roadway is assumed to be the the gentle arc of a circle.  The distance between the axles S is held constant by the suspension system (typically S = 5 ft).  Before coming to the arc, yD = 0 which assures that xD = S and that the trailing drive-axle reaches any point on the roadway at a distance exactly S behind the leading drive-axle.

Derivation of
                                Shear Pulse 

For yD 0, the projected location of the leading drive-axle will be forced by the roadway pavement to travel a longer distance than the trailing drive-axle during the same interval of time tDWhile the trailing drive-axle is traveling xD = S, the leading drive-axle is traveling 2Rθ as shown in the sketch.  The theoretical difference in circumferential distance traveled by the two drive axles is given by δ = 2Rθ - S, where δ shear pulse
The expression shear pulse is an in situ coinage for Unsprung Secrets.
The graph above indicates that for a change in roadway slope of about 15%, the shear pulse approaches a quarter of an inch of scuffing action on the pavement surface.  For undamaged roadway, however, one would expect a much smaller shear pulse, such that the tire would not actually scrub the pavement surface but only deform elastically.  Meanwhile...

In reality δ = 0.  Always.  In order for δ 0, the two drive-axles would have to rotate different amounts during the same time interval, which is mechanically forbidden by Drive Shaft #2

Drive Shaft #2
Mechanical Coercion in the C-rig

The sketch below shows details about Drive Shaft #2, giving emphasis to the reality that instantaneous angular displacement φ is identical for the tandem drive-axles on a C-rig.

Drive Shaft
                                #2

The tandem drive-axles are unsprung, thus vertically independent of one another.  Best keep in mind, however, Drive Shaft #2 has absolute angular authority over the two drive-axles.  Think steel-on-steel. The drive-train design in a C-rig tractor assures equivalence in angular displacement φ for all values of φ.
That absolute angular authority is maintained at any speed of the truck vX
By the way, the shear pulse phenomenon is independent of tractive effort.  Any damage from δ 0 can occur even if the C-rig is coasting with its transmission in neutral.

Differential Differences
Unfair bullying' is sanctioned by tandem differentials

Sophisticated solvers know that the so-called differential on a roadway vehicle enforces mechanical averaging of angular rotations between its left DW and its right DW, which enables the vehicle to negotiating turns and roadway curves without scuffing action.

Roadway geometry balances the operation, such that the outside DW rotates through an angle φ + φ while the inside DW rotates through an angle φ -φ.  The average 'computed' by the differential will be [(φ +φ) + (φ -φ)]/2 = φ.
Vertical roadway deflections will not always be so polite. 
Let us suppose that a B-rig with its single drive-axle encounters a vertical deflection on the left-hand side of the roadway, which causes the left DW to rotate through an angle φ + φ while the right DW continues to rotate through angle φThe average 'computed' by the differential would be given by [(φ +φ) + φ]/2 = φ + φ/2.  If the B-rig continues with no change in speed, the prime-mover must merely accommodate a minor change in torque corresponding to half of the incremental angular displacement imposed by the vertical deflection in the roadway.

Let us now suppose that a C-rig with its tandem drive-axles encounters a vertical deflection on the left-hand side of the roadway.  The left DW on the leading axle must rotate through an angle φ + φ while the right DW on the leading axle continues to rotate through angle φThe average 'computed' by the differential on the leading axle would be given by [(φ +φ) + )]/2 = φ + φ/2That will not be possible.  Here's why...
Since the trailing axle has not yet reached the deflection in the roadway, both of its DWs are rotating through angle φThe average 'computed' by the differential on the trailing axle would be (φ + φ)/2 = φ.  But, golly, the two differentials are lashed together steel-on-steel by Drive-Shaft #2, and φ + φ/2 φ.
The conflict between the two differentials has nothing to do with the prime-mover.   The scuffing action will most likely occur under the left DW on the leading axle, as it is being bullied by the other three drive wheelsThen farther along, when the vertical deflection comes under trailing axle, the scuffing action will take place under the left trailing DW.



At this point we have ascertained that the drive-train design for the C-rig is over-constrained...

  • Simultaneous Circumferential Distance: No difference can be allowed among the tandem drive-wheels (δ = 0), despite any discrepancies in the roadway profiles, AND...
  • Instantaneous Angular Displacement: The value of φ is at all times identical for the tandem drive-axles, as constrained by Drive Shaft #2
...so something's gotta give.



Elastic Deformation
Strain in the Tire = Stress in the Pavement

As modeled above, a C-rig-specific phenomenon has been identified in the design of those tandem drive-axle
s.  Your puzzle-master has coined the expression shear pulse and applied the symbol δ.  The phenomenon most often takes the form of elastic deformation measured in units of distance (inches).  The sketch below illustrates shear strain in the tire resulting from shear pulse.

Shear
                                    Strain Model
Whereas the real value of shear pulse is constrained (δ = 0), the theoretical value of shear pulse δ = 2Rθ - S can be used to characterize the magnitude of shear strain in the tire -- hey, and δ = 2Rθ - S can also be used to characterize the shear stress in the roadway pavement.  Exclamatory punctuation may be appropriate.
In strength of materials work, one generally treats stress as an independent variable, an empirical consequence of an applied force.  Meanwhile strain is a measurable deformation resulting from applied stress.  Here a conceptual reversal is needed: Given a geometrically derived strain (δ 0), what stress is being caused in the tire-tread?  Then too, what force can be estimated as acting on the pavement for each drive wheel?   That force can then be compared with, say, tractive effort. 
It is beyond the scope of the Unsprung Secrets puzzle to accomplish all that.  One might expect that even a small shear pulse deformation of a truck tire comprising rubber and steel belts might require significant circumferential forces, which then add and subtract from pavement shear stresses attributable to tractive effort. 

Do shear pulses that damage roadways offer any benefits to the trucking industry?

Transient State vs Steady State
Lingering thus Lengthening Shear Pulse

During the transition from a flat roadway to an upwardly curved profile,
2Rθ - S
0, and
the tandem drive-axles on a C-rig impose a shear pulse
δ
0
in the roadway pavement.  The shear pulse phenomenon arises in a
Transient State Zone along the roadway, wherein the value of R is changing.  Moreover, the magnitude of δ 0 lingers into a Steady State Zone, wherein R is held constant, as illustrated in this sketch...

Transient vs
                                Steady-State 

In a roadway hump, we see spatial forces of compression ('squeezing') in the pavement between the contact patches, increasing in the "Transient Zone" on the left and then decreasing in the "Transient Zone" ('stretching') on the right. 
The C-rig-specific phenomenon will have the same magnitude for a roadway dip.  Here's irony for you: The pavement would be experiencing spatial forces in compression between the contact patches -- same as for a hump.  Go figure. 
During the "Steady State Zone" of that particular roadway hump, both drive-axles, which are unsprung features of the drive train, are moving vertically along a constant radius of curvature R, first concave upward then inflecting to concave downward, all the while the pavement suffers constant longitudinal forces in compression from the C-rig's shear pulse δ. 
Think of drive-wheel tires on a C-rig as having 'analog memory', capable of storing the value of a shear pulse in one sense (squeezing) until it is 'erased' by a shear pulse in the opposite sense (stretching).  The technical term is elastic hysteresis.  Thus the most pronounced adverse effects attributable to shear pulse may be the accumulation of heat in the treads on drive-wheels, raising temperatures, and increasing wear -- all with no benefit for the transportation of the truck's payload.

Pavement Corrugation
Hot Pavements and Unloaded C-rigs

The photograph of a magnificent C-rig which was selected for illustrating the Unsprung Secrets puzzle, has been excerpted below.  It is being used here to draw attention to pavement defects in the near lane, which are variously called 'corrugation', 'rippling', or 'wash-boarding' by roadway authorities.  An image search on the web turns up many such cases on U.S. roadways. 

The condition may be common worldwide, although photographic evidence is not especially abundant.  Here in Brittany your puzzle-master has not observed corrugation, even on unpaved roads in the countryside. 
In studying Drive Shaft #2 we came across the Double Cardan Shaft, which is sketched below.  The objective is to achieve 'constant velocity' (CV) for the shaft -- more specifically to get rid of sinusoidal variations in the angular turning rates caused by Cardan U-joints.

Washboard

Experts know that successful CV depends on mechanically managing the planes of rotation for various values of the Drive Shaft Offset Angle γ.  Given that Drive Shaft #2 is rotating at three times the axle turning rate and that a truck wheel is typically 42 inches in diameter, one can calculate that spatial waves caused by Drive Shaft #2 would be inflicted by the tandem drive configuration at 42 π / 3 = 44 inches/wave (or sub-multiples 22 or 11 inches/wave).


 
Proposed Solution for the Unsprung Secrets Puzzle
Confirm the Explanation and Fix the Cause -- in That Order

In the absence of empirical evidence to support the various hypotheses set forth above, let us offer the solution to the Unsprung Secrets puzzle in the form of a proposed explanation...

C-rigs cause more damage to roadways than B-rigs because of one design feature...
tandem drive-axles.

Case in Point

Solvers are invited to provide comments on this proposal here.

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