**Daniel Fritzinger**Senior Design Engineer

Dana Towing Group

Dana Corp.

Albion, Ind.

Edited by Kenneth Korane

For fairly simple, low-performance vehicles such as go-carts, golf carts, toy cars, and wagons, designers often choose between single and double, four-bar linkages for the steering system. (See *Designing a better steering system, *Sept. 26, 1996, and *How to design a double four-bar steering system, *Dec. 9. 1999, MACHINE DESIGN.)

The rack-and-pinion design provides another option. As with any steering system, the designer's goal is to have all four wheels pivot about the same turning point. In more analytical terms, when the vehicle is turning, centerlines drawn through both front axles must intersect a line extending through the rear-axle centerline at the same point.

One steering arrangement often used in rack-and-pinion designs restricts movement of the tie rod to only one axis left and right. The tie rod cannot move through an arc as in four-bar linkages, so the spindle lever arm usually has slots that permit sliding motion between the tie rod and spindle lever.

**Mechanism analysis**In this type of system, the designer typically defines the centerto-center length of the tie rod,

*T*; center-to-center distance between the front axle pivots,

*D*; distance between the tie-rod centerline and the front-axle pivot centerline,

*P*; and the distance between front and rear axle centerlines,

*M*. For this application the designer must also define

*x*, the linear side-to-side movement of the tie rod that turns the vehicle.

Let's assume that the tie rod moves to the right, causing the vehicle to turn right. First calculate^{1}^{, }the angle between the spindle lever and front axle. Also assume this value is the same for both the left and right-spindle assemblies.

^{1 }= tan ^{1 }(((DT) / 2) / *P*) + 90°.

Next, calculate the vehicle's turning radius at the rear-axle center. In actual practice the designer should determine two turning radii, one based on the geometry of the inner steering assembly, the other based on the outer steering assembly. In a properly designed steering system, the length of the vehicle's turning radius calculated for the inner steering assembly will equal the length of the turning radius calculated for the outer steering assembly. In other words, the inner and outer assemblies will both cause the vehicle to assume the same turning radius. All wheels are in harmony and a smooth turn results.

Let's first work with the inner steering assembly. Calculate 2 and 3 after the tie rod moves a distance *x *to the right.

^{2 }= tan ^{1 }((((DT) / 2) *x*) / *P*) + 90°

and

^{3 }= 1 2.

Next solve for the distance between the turning-radius pivot point and the inner spindle lever, *Hi*, which along with *M*and 3 form a right triangle.

*Hi *= *M */ tan 3.

Finally, solve for *Ri*, the turning radius based on the inner steering assembly,

*Ri *= 0.5D + *Hi*.

**Outer steering assembly**To ensure proper steering design, perform the same series of calculations for the outer steering assembly. First determine 4 and 5 after the tie rod moves a distance

*x*to the right.

^{4 }= 90° tan ^{1 }((((TD) / 2) *x*) /*P*)

and

^{5 }= 4 1.

Next solve for *Ho*, the distance between the turning-radius pivot point and the outer spindle lever.

*Ho *= *M */ tan 5.

Finally, solve for the turning radius based on the outer steering assembly,

*Ro *= *Ho * 0.5D.

The inner and outer turning radii will be equal for a properly designed steering system. Use a computer program to speed the design process and verify that the difference between the two turning radii is at or near zero. In addition, recommended practice is to graphically verify the design as a final check.

**Articles and computer programs **

Previous articles in MACHINE DESIGN on steering geometry include *Designing a better steering system, *which deals with single, four-bar linkages and *How to design a double four-bar steering system. *These articles can be found at www.machinedesign2.com/turnstyle.php?ID=44 www.machinedesign2.com/turnstyle.php?ID=45

In addition, the author has written Microsoft Excel computer programs for each type of steering mechanism. The rack-and-pinion program, for instance, performs all the required calculations, based on user inputs, and compares inner and outer turning radii to ensure a properly designed system. Request a copy via e-mail at danfritz@fwi.com