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# How to size a motor?

This article is dedicated to motor sizing. Althrough it is dedicated to vehicleics and electrical motors, it can easily be extended to any vehicle or other kind of motors, including combustion engines.

You'll find on this website an online calculator for sizing motors. This calculator is based on the equation detailed in the following of this post.

On the following, we explains how to calculate:

## Hypotheses

Let’s consider a wheeled vehicle that climb up an inclinated plane:

The wheeled vehicle illustrated on the above figure have the following specifications:

• $$m$$: weight in [$$Kg$$],
• $$D$$: wheel diameter in [$$m$$],
• $$v$$: maximum velocity of the vehicle in [$$m.s^{-1}$$],
• $$a$$: maximum acceleration of the vehicle in [$$m.s^{-2}$$],
• $$\alpha$$: angle of the greater positive slope to climb up in [$$rad$$],
• $$R$$: gear ratio between the motor and the wheel,
• $$\eta$$: efficiency of the gear box,
• $$N$$: number of motor.

Spécifications to determine :

• $$\tau_{wheel}$$: torque on wheel shaft in [$$N.m$$],
• $$\tau_{motor}$$: torque on motor shaft in [$$N.m$$],
• $$\omega_{wheel}$$: angular velocity of wheel shaft in [$$rad.s^{-1}$$ and $$rpm$$],
• $$\omega_{motor}$$: angular velocity of motor shaft in [$$rad.s^{-1}$$ and $$rpm$$],
• $$P_{motor}$$: power of the motor in [$$W$$].

In the case of multi-motors vehicle, we will assume in the following that the drive power is equally distributed on each motor.

## Angular velocities

Let's first calculate the angular velocity of the wheel:

$$\omega_{wheel} [rad.s^{-1}]=\frac{2v}{D}$$

The angular velocity of the motor shaft is given by:

$$\omega_{motor} [rad.s^{-1}]=R \times \omega_{wheel} =R \times \frac{2v}{D}$$

The angular velocities converted in rotations per minutes are given by: $$\begin{array}{r c l} \omega_{wheel} [rpm] &=& \frac{60.v}{D.pi} \\ \omega_{motor} [rpm] &=& R . \frac{60.v}{D.pi} \end{array}$$

## Torque on wheels shaft

Torque is more tricky to calculate. The fundamental principle of dynamics gives us:

$$\sum \vec{F_i} = m.\vec {a}$$

Forces acting on the vehicle are gravity and actuator. The force inducted by the gravity is $$\vec {F_{Gravity}}=m.\vec{g}$$. When projected on the direction of motion (see figure above), the force inducted by the gravity is given by:

$$\| \vec{F_g} \|=m.g.sin(\alpha)$$

The fundamental principle of dynamics can be rewritten as:

$$m.a = \|\vec{F_m}\| + \|\vec{F_g}\| = \|\vec{F_m}\| - m.g.sin(\alpha)$$

The force required for the motion $$\|\vec{F_m}\|$$ can be deducted :

$$\|\vec{F_m}\|=m.a + m.g.sin(\alpha)=m.(a + g.sin(\alpha))$$

Thus, the torque on wheel's shaft is given by:

$$\tau_{wheel} = \|\vec{F_m}\| \times \frac{D}{2}= \frac{m.D.(a + g.sin(\alpha))}{2.N}$$

## Torque on motor shaft

Considering the gear box specifications, the equation becomes:

$$\tau_{motor} = \frac {\tau_{wheel}}{R.\eta} = \frac{m.D.(a + g.sin(\alpha))}{2.N.R.\eta}$$

## Motor power

The power produced by the motor is given by the product of the angular speed by the torque on the motor's shaft. We can thus calculate the motor's minimal power:

$$P_{motor}=\omega_{wheel}.\tau_{motor}=\frac{m.v.(a + g.sin(\alpha))}{N.\eta}$$