**1. Introduction**

106 Mechanical Engineering

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*Schrägverzahnten Antriebstechnik,* Heft 11, F. Vieweg und Sohn, Braunschweig,

A *machine* is a system realized by many parts with different functions, linked each other to reach a defined task. Depending on the task, a classification of machines equipped with moving parts can be done. In particular, one can distinguish:


The combination of a motor, a transmission and a mechanical user is the simplest form of machine.

In servo-actuated machines, the choice of the electric motor required to handle a dynamic load, is closely related to the choice of the transmission Giberti et al. (2011).

The choice of the transmission plays an important role in ensuring the performance of the machine. It must be carried out to meet the limitations imposed by the motor working range and it is subjected to a large number of constraints depending on the motor, through its rotor inertia *JM* or its mechanical speed and on the speed reducer, through its transmission ratio *τ*, its mechanical efficiency *η* and its moment of inertia *JT*.

This chapter critically analyzes the role of the transmission on the performance of an automatic machine and clarifies the strategies to choose this component. In particular, it is treated the general case of coupled dynamic addressing the problem of inertia matching and presenting a methodology based on a graphical approach to the choice of the transmission.

The identification of a suitable coupling between motor and transmission for a given load has been addressed by several authors proposing different methods of selection. The most common used procedure are described in Pasch et al. (1984), Van de Straete et al. (1998), Van de Straete et al. (1999), Roos et al. (2006). In these procedures, the transmission is

and it is characterized by a transmission ratio *τ* and a mechanical efficiency *η*.

*τ* minimum

Spur gear 750 1/6 low high 0.96 low Helical gears 50000 1/10 low high 0.98 low Worm gears 300 1/100 low high 0.80 medium Belt 200 1/6 high low 0.95 high

Toothed belt 100 1/6 medium low 0.90 low Linkages 200 1/6 medium medium 0.90 low

> *<sup>τ</sup>* <sup>=</sup> *<sup>ω</sup>out ωin*

where *ωin* and *ωout* are the angular velocity of the input and the output shafts respectively. This value characterizes the transmissions. If *τ <* 1 the gearbox is a *speed reducer*, while if

The mechanical transmission on the market can be subdivided into three main categories: transmissions with constant ratio *τ*, transmissions with variable ratio *τ* and transmissions

Since it is generally easier to produce mechanical power with small torques at high speeds, the transmission performs the task of changing the distribution of power between its extensive

> *<sup>μ</sup>* <sup>=</sup> *Tout Tin*

where *Tin*, and *Tout* are respectively the torque upstream and downstream the transmission.

*<sup>μ</sup>* <sup>=</sup> <sup>1</sup>

*<sup>τ</sup>* . (3)

that change the kind of movement (for example from linear to rotational).

It is possible to define the term *μ* as the *multiplication factor of force (or torque)*:

If the power losses in the transmission can be considered as negligible, it results:

and intensive factors to match the characteristics of the load.

common applications of mechanical transmissions.

Friction wheels

Trapezoidal belt

**2.1 The transmission ratio**

*τ >* 1 it is a *speed multiplier*.

**2.2 The mechanical efficiency**

The *transmission ratio τ* is defined as:

Power max (kW)

Table 1. Typical characteristics of mechanical transmissions.

Conventional mechanical transmissions with a constant ratio involve the use of *friction wheels*, *gear*, *belt* or *chains*. The choice of the most suitable transmission for a given application depends on many factors such as dimensions, power, speed, gear ratio, motor and load characteristics, cost, maintenance requirements. Table 1 gives an overview of the most

The Role of the Gearbox in an Automatic Machine 109

1/6 20 low medium 0.90 high

350 1/6 medium low 0.95 high

DimensionsCost EfficiencyLoad on

bearings

(1)

(2)

Fig. 1. A scheme of a simple transmission

approximated to an ideal system in which power losses are neglected, as the effects of the transmission inertia. Only after the motor and the reducer are selected, the transmission mechanical efficiency and inertia are considered to check the validity of the choice. Naturally, if the check gives a negative result, a new motor and a new transmission should be selected and the entire procedure has to be performed again. Differently, in Giberti et al. (2010a) the effects of transmission efficiency and inertia are considered since the beginning.

This chapter is structured as follows. Paragraph 2 gives an overview of main features of a mechanical transmission, while Par.3 describes the functioning of a generic machine when a given load is applied. Paragraph 4 describes the conditions, in terms of useful transmission ratios, for which a motor-load combination is feasible. Paragraph 5 gives the guidelines for the selection of both the gearbox and the electric motor, neglecting the effects of the transmission mechanical efficiency and inertia. Theoretical aspects are supported by a practical industrial case. Paragraphs 6 and 7 extend results previously reached considering the effects of the transmission mechanical efficiency and inertia. Finally conclusions are drawn in paragraph 8.

All the symbols used through the chapter are defined in Par.9.
