Driveline Dynamics

and

Setting Rear End Pinion Angle

1956 Dodge Plymouth Chassis Pinion Angle
Introduction

There is a lot of misinformation and vague factual information both on the internet and in printed magazines/articles regarding driveline dynamics and setting rear end pinion angle, so I hope in the article I can clear up the process and provide useful examples for those restoring or modifying their driveline and suspension. This article focuses on u-joint drivelines with leaf spring, four-link, and ladder bar configurations, but the matching of slopes and angles applies to ball-and-trunnion drive shafts as well. Many resources throw around vague “rules” such as the engine/transmission must be set at or below 3° negative slope, u-joint operating angles must be set at 1° and absolutely never over 3°, rear axle wrap on leaf spring cars ranges from 5° – 10°, and a bunch of other unexplained and unproven opinions. When I first began messing around with suspension and swapping rear ends that required new spring perches and therefore a new pinion angle, I found myself swimming in this vague and conflicting information. Throughout the years working in a body shop and building multiple custom chassis, I have gained an understanding that might be worthwhile to others. I summarize the basics and how to deal with less-usual situations that I have never seen covered in popular publications on the subject.

Terminology

Let’s start with some terminology for all the different considerations when setting up the driveline:

Output Shaft Angle: The angle of the transmission output shaft in relationship to level ground. This measurement can be taken off the front of the crankshaft pulley, the face of the transmission output shaft flange, or the flat face of the slip yoke installed on the output shaft.

Pinion Angle: The angle of the rear end pinion in relationship to level ground. This measurement can be taken off the face of the pinion flange.

Drive Shaft Angle: The angle of the drive shaft in relationship to level ground. This measurement is taken off the bottom or top of the drive shaft tube.

Slope: The slope of the angle is either positive or negative based off an X/Y axis (Figure 1). In Figure 1, line “A” is a positive slope and line “B” is a negative slope.

U-joint Operating Angle: The angle the u-joint will operate at, which is comprised of both the output shaft and drive shaft angles for the front u-joint and the pinion and drive shaft angles for the rear u-joint. I will discuss calculating this angle later.

Axle Wrap: The degree that the pinion/axle rotates up under load. This rotation varies depending on the type of rear suspension and is extremely difficult to calculate but can be limited or preset when using 4-link, ladder bars, traction bars, and other types of mechanical stops.

Intake Angle: The angle of the top of the intake manifold carburetor mounting pad from front to back in relationship to level ground. Note that on carbureted intake manifolds, this angle is not the same as the output shaft angle, which I discuss later.

Driveline Pinion Angle Slope
Figure 1: Driveline Slope
Principles of Driveline Angles

In summary, the purpose of properly setting up the pinion and u-joint operating angles is to remove harmonic vibration that results from the u-joints rotating at different speeds and to keep u-joint operating angles within their rated limits. Multiple chassis and suspension components influence these angles including type of rear suspension, rear leaf spring/coil spring arch/height, rear leaf spring rate, the elevation of the rear axle in relationship to the chassis, the elevation of the output shaft in relationship to the chassis, the angle of the pinion, and the angle of the output shaft. Tire size and vehicle rake does not impact the front and rear u-joint operating angles once set, although rear tire size does impact drive shaft rpm at a certain vehicle speed needed for calculating the angles, which I discuss more later.
While it may not seem obvious, if the front and rear u-joints are set at different operating angles and/or the output shaft and pinion set at different slopes, the u-joints rotate at different speeds and cause vibration along with eating up engine power through friction (Figure 2 is a good video demonstration). At this point, most articles on the subject start throwing around angles such as “no more than 3°” without considering all the variables. I’ll provide some general rules before moving on to real-life examples:

Rule A: Output Shaft and Pinion Slopes Must Match at Normal Driving Conditions
The “at normal driving conditions” is almost always left out of the discussion. In almost all cases, the output shaft will be at a negative slope (remember Figure 1 slope definitions). Along with crankshaft harmonics, the engine and transmission are angled down in the back to help force air pockets in the cooling system to the radiator rather than creating an air pocket in the block/heads/intake. The pinion slope must match the output shaft slope being careful not to get confused with if the pinion points toward the ground (nose down) or toward the floor pan (nose up). I’ve read on many forums people telling others that since the output shaft is pointing toward the ground, the pinion must point toward the ground to match the slope, which is completely incorrect. Referring to the lines in Figure 1, examine Figure 3. If the output shaft and pinion slopes must match, then a negative slope output shaft requires the pinion be set with a negative slope (nose-up). Pinion nose-down is in fact a positive slope and would only be the correct slope if the output shaft pointed up toward the floor pan.
Why did specifically say “at normal driving conditions” in this rule? When setting pinion slope, one must build in wiggle room for axle wrap when the axle is under the load it will see under normal driving conditions. This load will vary depending on the car’s purpose. For example, a full drag car should be calculated for constant hard acceleration with the maximum amount of constant axle wrap, whereas a street car should be calculated for more of a constant cruising load at the most often used MPH after the maximum axle wrap at takeoff has decreased to a lower angle. Whether a race or street configuration, the pinion slope with the car at rest is not of concern; it’s the pinion slope under load that matters. For example, if the necessary pinion angle under load is low enough of a negative slope (nose up), the pinion might need to be set at a positive slope (nose down) when the car is at rest so that when the car is driving down the road at 60 MPH the axle wrap rotates the pinion up to the desired negative slope (nose up). If the pinion angle under load is high enough of a negative slope (nose up), the pinion might be set at a negative slope (nose up) when the car is at rest. We’ll get into these calculations shortly.

Rule B: Front and Rear U-joint Operating Angles Must Match within 1° at Normal Driving Conditions
Once again, “at normal driving conditions” is almost always left out of the discussion. In order for the front and rear u-joints to rotate at or very close to the same velocity and cancel out harmonic vibrations, they must be set within 1° of each other during normal driving conditions. Similar to my comments regarding the slopes matching, these angles may not necessarily match when the car is at rest but should within 1° when the car is at its most common operating load. At rest, a street car that requires 3° rear u-joint operating angle cruising on the highway at 60 MPH may end up having an operating angle of 4° at rest if the pinion slope is set lower than it will be under load to compensate for axle wrap.

Rule C: U-joint Operating Angles Must Not Exceed Their Drive Shaft RPM Design Limits
Not to be confused with engine RPM, u-joint are designed to operate up to a certain angle at a certain drive shaft RPM. There are two ways this RPM should be calculated: at the engine’s redline RPM and at the vehicle’s maximum speed.

Rule D: The Drive Shaft and U-joints Must Have some Angle Built In
None of these components should be dead level at 0°. While the lower the angle the better for limiting engine power waste through friction and for u-joint life expectancy, an angle of 1° minimum should be maintained for proper dynamics and balance.

Next, we’ll cover how to calculate all these angles and run through some real-life examples.

Figure 2: Drive Shaft Velocity with Varying U-joint Operating Angles
Driveline Output Shaft and Pinion Slopes
Figure 3: Output Shaft and Pinion Slopes
Calculating Carburetor, Output Shaft, Drive Shaft, Pinion, and U-joint Operating Angles

When setting up the driveline, the vehicle should be resting under its own weight with the amount of gas in the tank that it will normally have. The rear axle and output shaft elevations in relationship to the chassis should also be set exactly where they will normally rest. If you cannot crawl underneath the vehicle and you don’t have a drive-on lift or pit, place the vehicle on jackstands located under the rear axle and under the front control arms/axle to where the suspension is fully loaded and not hanging.

Carburetor Mounting Pad Angle:
Place a protractor across the carburetor mounting pad to get the angle from front to back of car (I recommend a 6″ digital protractor or torpedo level that reads 00.00° with a “hold” option). This angle with the rake of the vehicle should be as close to 0° as possible since it impacts carburetor float level. The pad is designed at a different angle than the crankshaft centerline so that when it is level, the crankshaft/output shaft will have some negative slope to them for the purpose of harmonics and cooling system air pockets as previous discussed. If necessary and possible, adjust the mounting pad angle to or close to 0° by shimming the transmission rear mount or modifying the transmission crossmember.

Front U-joint Operating Angle:
Take the angle measurement of the output shaft by placing the protractor vertically from top to bottom of the output shaft flange or slip yoke face or across the front of the crankshaft pulley. Write down this angle as the “output shaft” and note if it is a positive or negative slope (see Figure 1 for slope definitions). Contrary to what many online resources say, this angle does not necessarily have to be under 3°, so hold off on any adjustment.

Place the protractor on the bottom of the drive shaft running front to back, record this angle as the “drive shaft”, and note the slope negative or positive.

Now we get into the math. Referring to Figure 4, if both the output shaft and drive shaft slopes (not angles) are the same–whether that is a positive or negative slope–subtract the smaller number from the larger number and write this number down as the “front u-joint operating angle.” For example, if the output shaft is 5° negative slope and the drive shaft is 1° negative slope, the u-joint operating angle is 4° (5 – 1 = 4); if the slopes differ–such as one slope is negative and the other is positive–add the numbers to find the u-joint operating angle. For example, if the output shaft is 3° negative slope and the drive shaft is 2° positive, the u-joint operating angle is 5° (3 + 2 = 5). Do not use negative numbers for negative slopes. To clarify, it is incorrect to calculate this angle as -3 + 2 = -1.

Here’s where a lot of misinformation comes into play and requires some deeper discussion. By 99% of the information on the internet, the 4° u-joint operating angle will cause terrible vibrations and destroy the u-joint. Not necessarily, evident from the millions of lifted trucks and off-road vehicles safely and comfortably cruising down the highway with operating angles far above 4°. Drive shaft rpm is the critical factor in determining the safe u-joint operating angles, not a set number for all vehicles no matter the context. Figure 5 shows Spicer’s published maximum operating angles, but keep in mind these are actually conservative figures that even automotive manufacturers and drive shaft builders, particularly in trucks, do not always follow. There’s much to be learned here from the off-roaders among us since many of them run much more u-joint angle that 3° at highway speeds without vibration and premature u-joint failure. In practice, 10° seems to be the upper limit before harmonic vibrations according to multiple drive shaft builders I’ve consulted, but keep in mind that the higher the angle, the less drive shaft RPM the u-joint can handle and the more engine power is lost through friction.

Let’s use an example to help clarify how a 4° u-joint operating angle may work just fine. Let’s say for whatever reason the transmission cannot be shimmed and the rear end cannot be raised or lowered to change drive shaft angle. We are stuck with 4°. Looking at Figure 5, Spicer claims a 4.2° operating angle is good to 4,000 drive shaft rpm, so let’s use that number. What is the highest engine rpm the engine will ever practically see? For most A-blocks with stock or mild cams, maximum HP comes in by 5,500 rpm. For hotter A-blocks, 6,500. With the hottest up at 7,000. However, keep in mind that none of these vehicles will ever see their maximum engine rpm in the highest transmission gear unless it’s a land-speed car. It is reasonable that they would see this maximum rpm in second gear, so we will use the transmission’s second gear ratio for the following calculation:


Calculating Drive Shaft RPM from Engine RPM: Engine rpm divided by transmission gear ratio (second gear in our case). Let’s say our engine redlines at 6,500 rpm and we are running a TorqueFlite 727 with a 1.54:1 second gear. 6500/1.54 = 4,220 drive shaft rpm.

Calculating Drive Shaft RPM from Vehicle MPH: Now let’s check the drive shaft speed at the maximum vehicle speed. For this calculation, let’s say the car might see some 1/4 mile track time at 100 MPH:
Tire circumference in feet = (tire height x 3.14 pi) / 12 inches
Tire rotation per mile = 5280 feet per mile / tire circumference in feet
Drive shaft rotation per mile = Tire rotation per mile x rear axle gear ratio
Drive shaft RPM = vehicle MPH x rotation of drive shaft per mil / 60 minutes
For our example:
Tire circumference in feet = (26.5)(3.14) / 12 inches = 6.93 feet
Tire rotation per mile = 5280 feet per mile / tire circumference in feet = 5280/6.93 = 761.90 tire rotation per mile
Drive shaft rotation per mile = Tire rotation per mile x rear axle gear ratio = 761.90 x 3.54 = 2697.13 revolutions per mile
Drive shaft RPM = (rotation of drive shaft per mile)(vehicle MPH) / 60 minutes = (2697.13 x 100) / 60 = 4,495 revolutions per minute

Putting all this together, we’ve found that at the highest engine rpm of 6,500 in second gear, the drive shaft will spin at 4,220 rpm. At the highest vehicle speed with 26.5″ rear tires and 3.54:1 gears, the drive shaft will spin at 4,496 rpm. Looking at Figure 5, the maximum rpm for a u-join operating angle of 4° is around 4,000 rpm, so we are slightly outside Spicer’s range. If one wants to go off the Spicer limits, then the output shaft and/or drive shaft angles would need to be adjusted to decrease the u-joint operating angle, the engine redline would need to be decreased, or the tire size and rear gear would need to be changed. However, practical anecdotal evidence from professional drive shaft builders who have spun up hundreds of thousands of drive shafts combined suggests the Spicer maximum operating angles are kept purposefully low and in practice this 4° u-joint operating angle would work just fine and safely for this build without vibration or explosion.

Pinion Angle:
Back to our steps in setting pinion angle, let’s stay with our example of an output shaft at 5° negative slope and the drive shaft is 1° negative slope. The front u-joint operating angle is 4° (5 – 1 = 4).

If I am installing a new or swapped rear end with loose spring perch pads that need welding such as the Ford 8.8″ rear end in my 1956 Dodge coupe, I center the rear axle side to side, confirm proper wheelbase, and snug down the axle on the spring perch pads and leaf springs using the u-bolts with enough tension that I can still rotate the axle but it will stay where I put it. For existing axles whose spring perch pads are already welded, I use proper metal shims between the pad and leaf springs.

As I explained in detail above, the rules are that the pinion slope must match the output shaft slope and the rear u-joint operating angle must match the front u-joint within 1° under normal operating conditions. For our example, the output shaft slope is negative, so the pinion slope must be negative (nose up). The front u-joint operating angle is 4°, so I need to match that at normal operating conditions. Let’s say this car will likely see an average street cruising speed of 50 MPH around town, so the axle wrap won’t be much compared to wide open throttle. There is no precise way to determine axle wrap since no two cars are the same. The only way to tell would be to strap a protractor to the pinion case, strap a video-recording device to the underside of the vehicle facing the protractor, and go drive at 50 MPH. The amount of bouncing around in the suspension would likely give an inaccurate reading anyway, so we are left with approximation based off industry standard. After discussing axle wrap with multiple drive shaft builders, the consensus is that about 1° – 2° is the most a leaf spring rear end would see at 50 MPH with strong leaf springs. Old or soft leaf springs would see more and possibly excessive axle wrap. Four-link, ladder bar, and leaf springs with traction bars will have less axle wrap and can be limited through adjusting the components. These components should be inspected for how much flex they provide in the bars and bushings and an axle-wrap conclusion built off that inspection.

Here is yet another area where online resources can really screw someone up since many articles say to factor in a whopping 5° – 10° of axle wrap, which is a wild range considering the front and rear u-joint operating angles need to be within 1° of each other. In the case of our example, I need 4° of rear u-joint operating angle with the pinion at a negative slope (nose up) at 50 MPH. If I listened to those saying to account for 5° – 10° axle wrap, I’d need to set the pinion angle at 3° positive (nose down) so that when the axle wrapped up 7° my front and rear u-joint operating angles would match at 4° with the pinion at the matched negative slope. But what if the axle only wraps 4°? My pinion angle would be at 0° and the pinion at no slope, which would both not match the front u-joint and no longer be at the same negative slope as the output shaft–a scenario that would definitely cause harmonic vibrations. Following the drive shaft builders’ recommendations, I would set the pinion angle at 2° negative slope (nose up) with the assumption that the axle will wrap up between 1° and 3° that would keep me within the 4° plus or minus 1° rear u-joint operating angle at 50 MPH.

For another practical example, let’s say the output shaft is a negative slope and the front u-joint operating angle is 2°. In this situation, I would set the pinion angle to 0° expecting that axle wrap of 1° – 3° would bring the pinion angle up to a negative slope (nose up) that matches the output shaft and the rear u-joint operating angle would be within 1° of the front’s 2°.

Calculating U-joint Operating Angles
Figure 4: Calculating U-joint Operating Angles
Spicer U-joint Operating Angle Limits for Drive shaft RPM
Figure 5: Spicer U-joint Operating Angle Limits for Drive Shaft RPM
Testing

Once I triple-checked all my angles and math, I would tack-weld the spring perch pads at the four corners well to the axle housing and torque the u-bolts or if using shims torque the u-bolts. I’d test drive the car carefully through its normal operating range to check for vibrations. If I needed to make an adjustment, I would use shims between the spring perch pad and leaf spring to dial in pinion angle, take that new pinion angle measurement, remove the shims, cut the perch pad tack welds, reposition the pinion at the new angle, and weld the spring perch pads.

Conclusion

I hope that by going through the principles, math, and practical examples that driveline dynamics and pinion angle are clearer. After reading this article, we can see how making some modifications after setting u-joint operating angle and pinion angle will negatively impact the settings. For example, moving the rear axle elevation in relationship to the chassis will change both the drive shaft and pinion angles and therefore change the front and rear u-joint operating angles. One should plan on checking and resetting these angles if he adds leaf springs with more or less arch or longer/shorter coil springs, adds/removes lowering blocks, lengthens/shortens the shackles, or relocates the axle below or above the leaf spring. If someone installs a device to limit axle wrap (e.g. traction bars), he should ensure there is enough preset in the device to let the pinion angle rotate enough to align on the same slope as the output shaft and put the rear u-joint operating angle within 1° of the front u-joint’s since it was set taking into account axle wrap without a limiting device. Changing the front suspension elevation with different springs and changing tire height will not impact the drive shaft angle, u-joint operating angles, and pinion angle since all these components are fixed to the chassis and will change angle evenly together if the front of the chassis is lowered or raised in relationship the ground. Changing the rear tire height and/or gear ratio will impact the drive shaft RPM at vehicle speed, which may put u-joint operating angles outside of their limits.

While it would be a long list to thank those throughout my life who have helped hone my craft building and tuning chassis, I want to put in a plug for my friend Shawn Wood of Tom Wood’s Custom Drive Shafts who builds excellent drive shafts and ships them worldwide. His work with off-road vehicles and the conversations I’ve had with him on driveline dynamic experimenting we’ve both done and about particularly challenging situations have taught me a great deal through the years.