February 9, 2012
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An Animated Approach to Plotting Profiles
Animated simulations of trochoid spiral and epitrochoidal spiral curves are an elegant and useful means of displaying geometrical properties.
Trochoid Spiral (TS) and Epitrochoidal Spiral (ETS) curves and special flash animations were developed by the author in 2004 and 2005 for nonintersecting and intersecting right angle axis gear drives involving R/L (rotary-to-linear) and L/R motion conversion, and for parallel shaft R/R motion conversion, respectively.
The design of TS and ETS profile gears is based on cylindrical tooth followers in a toothed engagement with members of spiral groove type face gears. TS gearing represents a simple, limiting case of ETS gearing. TS and ETS curves represent pitch curves of corresponding gear profiles.
We distinguish three types each of TS and ETS curves as L-TS (looped), Z-TS (zero offset) and P-TS (prolate) types as in Figure 1(A), and L-ETS, Z-ETS, and P-ETS, as illustrated in Figure 2(B).
As in the case of cutting traditional hypotrochoid, epitrochoid, and trochoid gears, CNC gear cutting technology (milling, wire EDM, etc.) represents an excellent choice for high-precision TS and ETS parts for operation and service features that can seldom be matched by other gear cutting types. This also translates into low parts cost, short lead time for prototype and pilot production quantities, and for tooling.
Parametric equations for TS curves are as follows:
Equation 1: x = e q cos(q) - w sin(q)
Equation 2: y = e q sin(q) + w cos(q)
where π=3.1415, -2πe≤q≤2πe and q is the independent variable in radians, e is the radius of pitch circle for the sample curves shown in dashed line with center at x=y=0 in Figure 1. The sample curves depend on the particular value of design constant w=y for x=q=0.
Parametric equations for ETS curves are as follows:
Equation 3: x= -(b sin(q) - Q sin(b q)) n e
Equation 4: y= (b cos(q) - Q cos(b q)) n e
where -2p eæqæ2p e and q is the independent variable in radians, e is the radius of pitch circle for the three sample curves as shown as dashed lines in Figure 2(A) with center at x=y=0, while n designates the number of rollers in the follower, and b and Q are constants depending on the particular value of w=y for x=q=0, just like in the examples above for TS curves. For n=16 for the ETS curves in Figure 2(A), the following information applies:
Calculation of the gear ratio for TS and ETS gears is simple: one full turn of the gear corresponds to one linear or circular pitch, P, respectively, as P=2pe, where e designates the pitch circle radius for the gear.
As an example, a given ETS gear train uses a follower of n=10 rollers for a gear ratio, R, as, R=-n:1=-10:1, where the negative sign indicates that the gear and follower rotate in opposite directions.
ACADSvF (animated CAD Solutions version F by Trogetec, using flash by Macromedia) animation for %%plotting1005fig1%% and Figure 2 (using eight increments of 45-degree gear rotation as shown) is provided on the Web site listed below. The corresponding composite plots in Figures Figure 1(B) and Figure 2(B) serve as transition-type visuals between Figure 1(A) and Figure 2(A) on the one hand, and the corresponding flash animations on the other. For monitor screenshots of the shockwave flash animations we refer to Figure 3 and Figure 4.
This animation technique was primarily developed for the new TS, ETS, and other gear categories, for design studies by individuals or small groups of design/mfg/QC engineers, and for executive project reviews, when the use of visuals, prototypes, and demo units can be of paramount importance for the best results.
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