*2.1. Pneumatic Flexible Shaft Couplings*

At our department, new types of pneumatic flexible shaft couplings are being developed [15]. The dynamic torsional sti ffness of these couplings can be continuously changed by changing the air pressure in their pneumatic flexible elements. They also show less changes in properties due to

the aging of the material of the elastic elements, because the elastic elements have low rigidity and most of the load is transmitted by the compressive force of the air [13]. Pneumatic flexible shaft couplings can be used for tuning torsional oscillating mechanical systems in two main ways [13–15]:


Several methods of continuous tuning using pneumatic tuners of torsional oscillations were proposed and patented [26]. The two main continuous tuning methods, which were also practically implemented in laboratory conditions, are *extremal control* [27] and *constant twist angle control* [13].

### 2.1.1. Used Pneumatic Tuner of Torsional Oscillations

In the presented research, a tangential pneumatic tuner of torsional oscillations with fully interconnected flexible elements with a *4–2*/*70–T–C*-type designation was used (Figure 1). This type of designation means that the PTTO has 4 double-bellows flexible elements with an outer diameter of 70 mm, the elements are placed tangentially and their compression spaces are fully interconnected. This PTTO consists of two identical hubs (1) connected by pneumatic flexible elements (2). The individual pneumatic flexible elements are mounted between triangular consoles (3). Under loading torque, one pair of opposing pneumatic flexible elements is stretched and at the same time the other pair is equally compressed, allowing the PTTO to transmit torque in both directions of rotation. The compression space of the PTTO is fully interconnected by polyamide hoses with a diameter of 6 mm (4). Its filling with gaseous medium is realized through a rotary air supply, which is a part of one of the connecting flanges of the PTTO. Each flexible element includes two pneumatic screw connections (5).

**Figure 1.** Pneumatic tuner of torsional oscillations type *4–2*/*70–T–C*.

In order to react flexibly with the variable load torque generated directly during the operation of the mechanical system, it is necessary to ensure that the pressure in its entire compression space changes as quickly and evenly as possible. This is ensured by the full interconnection of pneumatic elements compression spaces. The speed and uniformity of the filling of the compression space can be influenced by increasing the inner diameter of the connecting hoses too, but at the expense of reducing the resistance to the air flow between the pneumatic flexible elements. In practice, however, this means a reduction in the damping properties of the PTTO itself [28,29].

Generally, the static load torque of pneumatic couplings *Mstat* [N·m] at a twist angle ϕ [rad] can be expressed as [30]

$$M\_{\rm stat} = M\_{G(q)} + p\_T \cdot (S\_{\rm c} \cdot r)\_{(q)'} \tag{1}$$

where *MG(*ϕ*)* [N·m] is the pneumatic flexible element rubber shell torque, *pT* [Pa] is the overpressure in the pneumatic flexible elements of the coupling, *Se* [m2] is the effective area of the coupling's compression space and *r* [m] is the distance of the center of the effective area *Se* from the coupling's axis. Expression *Se*·*<sup>r</sup>* [m3] is then the static moment of the effective area to the coupling's axis. Rubber shell torque and static moment of effective area are expressed as a function of the twist angle ϕ. The parameters *MG* and *Se*·*<sup>r</sup>* can be determined from measured static load torque—static load characteristics (Figure 2a) and overpressure (Figure 2b) depending on the twist angle ϕ at different initial overpressures. The full procedure of obtaining these parameters is described in [30].

**Figure 2.** Results of *4–2*/*70–T–C*-type pneumatic tuner of torsional oscillations static measurements: (**a**) static load characteristics; (**b**) overpressure.

In Figure 3, the resulting rubber shell torque *MG* (Figure 3a) and effective area of the coupling compression space *Se*·*<sup>r</sup>* (Figure 3b) in graphical and equation form are shown [28].

**Figure 3.** Static parameters of *4–2*/*70–T–C*-type pneumatic tuner of torsional oscillations: (**a**) rubber shell torque; (**b**) effective area of the coupling compression space.

The model shows good agreemen<sup>t</sup> with the actual PTTO, as the difference between the calculated and measured values of the static load torque does not exceed 5% [28]
