Needing a Flat Belt style Pulley transmission system for my turbine I found this book titled – A treatise on belts and pulleys By John Howard Cromwell published 1885.

I was just wanting a general outline of the style and dimensions for a pulley and this book pretty much will answer any question I have. Like many book from the period it is very wordy – I think they were probably paid by how many words were in the book. This blog gives me a chance to winnow out the bits that I need.

Geeze. Trying to make sense of this is fun – in a .. hitting your finger with a hammer feels better when you stop – kinda way.


I just realized that while I go about calculating the pulley size it is based on a pre-determined belt width and pulley dia – meaning I haven’t done any calculation on what horsepower that combination would transmit.

Rim Dimensions

The rim of a pulley intended to carry a flat belt is generally rounded (Figs. 48 and 49), in order that the belt may remain in the centre of the pulley-fact, instead of working to one side, as is the case with flat-faced pulleys. The amount of this rounding (s) may be taken equal to \frac{1}{20} the width of the belt.

For isolated pulleys the face-width B is taken some-what greater than the width of the belt b; often we take –
B = \frac{5}{4}b

B = Face Width
b = Belt Width

– Rim Edge Thickness –
The thickness k of the edge of the rim, or the thickness at the ends of the face-width, may be easily calculated from the formula –
k = 0.08 + \frac{B}{100}

k = Rim Thickness
B = Face Width

– Rounding at Pulley Face –
For the amount of rounding at the pulley-face


s = Rounding at the pulley face
b = Belt Width

The thickness of the rim at the center is therefore the rounding + thickness at the rim


k = Rim Thickness
s = Rounding at the pulley face


The thickness of a pulley-nave is given by the formula –
w = 0.4 + \frac{d}{6} + \frac{R}{50}
in which d represents the diameter of the shaft upon which the pulley is keyed, and R the radius of the pulley.
The length of the nave should not be taken less than
L = 2.50w
Often (in idle pulleys, for example) the length L is taken equal to the face-width B of the pulley.

Arms of Pulleys

Ordinarily the arms of pulleys have oval cross-sections, the diameter in the plane of the pulley being twice the smaller diameter.

Fig. 57 shows a cross-section of the arm but that was simply how to draw an oval that is half the width.

The axes of pulley-arms may be straight as in Fig. 58, curved as in Fig. 59, or double curved in the form of a letter S. Single-curved arms may be drawn in the following manner: Take (Fig. 59) the arc AE equal to 2/8 the arc EF, determined by the centre s of the arms at the rim of the pulley, and draw AO perpendicular to AO. From the centre D draw CD perpendicular to AO. From the centre D draw CD perpendicular to OE, and the point C of intersection of DC and OC is the centre for the curved axis of the arm.
– Number of Arms –
The number of arms (N) necessary for pulleys of different sizes may be determined by means of the formulaN = \frac{1}{2} (5+\frac{R}{b})N = Number of Arms
R = Pulley Radius
b = Belt Width
– Pulley Arm Widths –
The formula

h = 0.24 + \frac{b}{4} + \frac{R}{10N})

gives the greater diameter for the pulley-arms (width of the arms in the plane of the pulley). The diameter or width h is taken at the nave as shown in Fig. 58, and the width h, at the rim may be conveniently taken equal to


The width at right angles with the plane of the pulley is therefore

h_1 = \frac{2}{3} X h