Engineering Expert Witness Blog

      Hold onto your hats, we’re going to deal with a lot of equations today!

      Last time we used flashbacks to previous blogs in this series to revisit key equations in our ongoing discussion of gear trains and torque.   We also introduced The Law of Conservation of Energy in conjunction with five equations that together demonstrate how when increasing torque by use of a simple gear train, we do so at the cost of speed.

      Those five equations are:

where, R is the gear ratio, N the number of gear teeth, n the gear’s rotational speed, T the torque, D the gear pitch radius, and HP is the horsepower transmitted by the gears.

      As we work the equations, keep in mind that our ultimate objective is to find a way to link together (1) and (2), the equations dealing with gear torque and speed.   Once we accomplish this we’ll see how increased torque is obtained at the cost of speed.   But because there are no common terms between equations (1) and (2), our first step is to develop one.

      Developing a link between equations (1) and (2) is a process that begins with combining equations (2), (3), and (4).

      The common terms in these three equations are T_Driving and T_Driven, so we’ll manipulate things in order to group them together.   We’ll substitute equation (3) for the T_Driving term in equation (2), and substitute equation (4) for the T_Driven term in equation (2).   We are now able to link all three equations to get:

{[HP_Driving ÷ n_Driving] × 63,025} ÷  {[HP_Driven ÷ n_Driven] × 63,025}

                                                                                        = D_Driving ÷ D_Driven         (6)

      Now let’s go a step further to simplify equation (6).   From equation (5) we know that the driving and driven gear horsepowers are equal.   So, in equation (6), the HP_Driving and HP_Driven cancel out, along with the two 63,025 terms, allowing us to arrive at equation (7):

{[HP_Driving ÷ n_Driving] × 63,025} ÷  {[HP_Driven ÷ n_Driven] × 63,025}

                                                                             = D_Driving ÷ D_Driven

      Next week we’ll use equation (7) to link together R, N, n, of equation (1) with D and T of equation (2) and in so doing disclose mathematically the tradeoff between torque and speed, then apply our findings to an example.

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