Old Wives Tale, Belt Drive

Reprinted from CONTACT! Magazine, issue #72

Written by Dan Horton

Issue #70 includes an article about a marvelous scale P-51 built by Dan Hawken of Calgary.  Page 10 includes the statement "The reason I went to a belt drive was that the belt isolates the torsional vibrations coming from the propeller". 

With all due respect to Mr. Hawken, I contend that the statement is based on myth, printed and reprinted until accepted as fact. Belts have no magic properties, regardless of what you read on the internet or hear at the vendor's booth.    

System frequencies are a function of the system's inertias coupled by it's stiffnesses.  A simple torsional model of a belt drive with a two-plate frame would have at least 4 inertias (crank, flywheel and lower sprocket, upper sprocket, and prop), coupled by 3 stiffnesses (crank stub, belt, and propshaft).  In truth, the dynamic system has many more elements.  For every element (a stiffness and an inertia) there is a natural frequency.

The point is that the belt serves as a connecting stiffness, nothing more.  It makes a contribution to the system's fundamental frequency as well as having it's own natural frequency in concert with it's adjacent inertia.  The overall system or the individual element can be driven into resonance by a matching exciting frequency.  Exciting frequencies not matched can perhaps be considered as "isolated", but in practice piston engine output includes a whole range of exciting frequencies.  If you have a lot of elements, something resonates to some degree almost everywhere in the RPM range.  As a practical matter we concern ourselves with worst resonant frequencies by shifting an inertia or a stiffness.  To do that you must know what they are.

It isn't difficult to measure or estimate an inertia.  The same is true for the torsional stiffness of a simple shaft.  A connecting member like a belt or chain has a torsional stiffness equivalent derived from it's "stretch" and the arm of it's sprocket.

The torsional spring rate equivalent of the "average" belt (there are considerable variations between brands) is a non-linear curve, soft under initial loading and far stiffer as the tensile members in the belt backing are tensioned.  The shape of the curve is a function of both tooth shore hardness and the tensile material's properties and weave.

"Soft" is a relative term for toothed belts.  For example, an 8mm, 60mm wide Dayco RPP Panther has an elongation of 0.0015" per inch of belt span at 100 lbs tension, the highest value on the initial part of the curve.  If we figure a pitch diameter of 4" for the small sprocket and a 7" C to C distance, you have 0.0105" elongation at 33 ft-lbs of crankshaft torque, or about 0.3 degrees of sprocket rotation, or an equivalent torsional spring rate of about 6300 ft-lbs per radian.

The stiffer region?  The belt spring rate is nonlinear, so for this example I'll apply the published Rover V-8 torque (278 ft-lbs at 3000 RPM) to the belt data.  For the above sprocket, that's 834 lbs of belt load and 0.005 elongation, or about 47,780 ft-lbs per

radian.

For comparison, the torsional spring rate of a 4130 steel propeller shaft of 1.5" dia, 0.188" wall thickness, and 5.25" length would be 64,819 ft-lbs per radian.  At the stated mean torque the belt isn't a whole lot softer than the shaft.

Mean torque is an average.  Peak oscillating torque due to gas pressure variation is perhaps twice the mean, and varies with engine selection.  At twice the mean torque, the belt's equivalent torsional spring rate would be almost 60,000 ft-lbs per radian. Since stiffness relates to frequency, the belt would be an "isolator" to about the same degree as the propshaft.      

The RPP Panther probably has stiffness values in the upper range of available belts.  A Gates Poly-Chain is much higher.  Basic belts with fiberglass tensile members are probably lower, but I have no data for those.  The wide variation in equivalent torsional spring rates means you can't make a general statement about all belts.  You must know the spring rate equivalent of the belt you're using to draw any conclusions about it's stiffness contribution to frequency or isolation.  For the record,  also note vibratory effects and frequencies brought to the design by the inclusion of a belt rather than gears or chains.  An example might be flap or standing wave effects in the belt spans.

Now let's look at what we're isolating.  What torsional vibrations "come from the propeller"?  In a simple torsional model, the propeller is an inertia with no inherent capability to excite the system.  A more complex model does treat the prop as a series of inertias and stiffnesses, and indeed, blades do vibrate.  The blades can be driven into resonance from a number of sources, but they are usually not the source of a significant exciting torsional frequency.  There are a few exceptions, but we're usually trying to protect the prop from the engine, not the other way around.

There is no shortage of textbook and SAE material on the subject of torsional vibration.  Component data is available from the manufacturer.  There are established techniques to allow actual measurement of torsional behavior.  Isn't it about time we gave up trial and error (and old wives tales) to design our auto conversions?

Dan Horton