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For matching speeds, Fig. 2-15 shows a gear mechanism where the shafts are assumed to be of infinite stiffness, and the masses of the gears are ignored. We will further assume that there is no power loss in the gears. Both gears must have the same linear speed at the point of contact. Therefore, their angular speeds are related by their respective radii r₁ and r₂ such that

(2-39)

and

(2-40)

Combining Eqs. (2-39) and (2-40),

(2-41)

where T₁ and T₂ are the torques at the ends of the gear mechanism, as shown in Fig. 2-15. Expressing T₁ and T₂ in terms of T_em and T_L in Eq. (2-41),

(2-42)

From Eq. (2-42), the electromagnetic torque required from the motor is

(2-43)

where the equivalent inertia at the motor side is

(2-44)

Fig. 2-15 Gear mechanism for coupling the motor to the load.