*2.2. Constant Twist Angle Control*

The principle of constant twist angle control is that it maintains a pre-set constant static twist angle of the PTTO [31]. This is ensured by a control system which adapts the overpressure of the gaseous medium in the PTTO to the static load torque in order to maintain the pre-set constant twist angle. Constant twist angle control is suitable for tuning mechanical systems with fan characteristics. The principle of operation is explained in more detail below [32].

The transmitted static load torque *Mstat* and dynamic stiffness *kdyn* [N·m·rad−1] of the PTTO by a given constant twist angle ϕ*const* (for simplicity neglecting the rubber shell torque *MG* which is relatively small compared to the total transmitted torque [13]) can be generally considered as proportional to the overpressure *pTC* in the compression space of the PTTO corresponding to the given constant twist angle (see Equation (1)).

Thus, the static load torque can be expressed as

$$M\_{\text{stat}} = \mathbf{a}\_{\left(q\_{\text{comst}}\right)} \cdot p\_{T\mathbb{C}} \tag{2}$$

and dynamic torsional stiffness as

$$k\_{dyn} = \mathbf{b}\_{(\varphi\_{count})} \cdot \mathcal{p}\_{T\mathcal{C}\_{\prime}} \tag{3}$$

where *<sup>a</sup>*(ϕ*const*) [N·m·Pa−1] and *<sup>b</sup>*(ϕ*const*) [N·m·rad−1·Pa−1] are factors whose values depend on the constant twist angle ϕ*const* only.

The transmitted static load torque corresponding to the fan characteristics (of the propeller) can be expressed as

$$M\_{\rm stat} = \mathfrak{c}\_f \cdot \omega^2,\tag{4}$$

where *cf* [N·m·rad−2·<sup>s</sup>2] is a constant and ω [rad·s<sup>−</sup>1] is the angular speed of the propeller shaft.

From the equality of Equations (2) and (4) for the static load torque *Mstat*, then expressing *pTC* and putting it into Equation (3), the dynamic torsional stiffness will be

$$k\_{dyn} = \mathbf{b}\_{(q\_{count})} \cdot \frac{c\_f}{\mathbf{a}\_{(q\_{count})}} \cdot a^2. \tag{5}$$

Considering a two-mass torsional oscillating mechanical system with dynamic torsional stiffness *kdyn* and equivalent mass moment of inertia *IRED* [kg·<sup>m</sup>2], the natural angular frequency of this system will be 

$$
\Omega \bullet = \sqrt{\frac{k\_{\text{dyn}}}{I\_{RED}}} = \sqrt{\frac{\mathbf{b}\_{(\text{\text{\textquotedblleft}cond)} \cdot \mathbf{c}\_f}{\mathbf{a}\_{(\text{\textquotedblleft}cond)} \cdot I\_{RED}}}} \cdot \boldsymbol{\omega} = \mathbf{C}\_{(\text{\textquotedblleft}cond)} \cdot \boldsymbol{\omega} \tag{6}
$$

where *<sup>C</sup>*(ϕ*const*) [-] is a factor depending on the given constant twist angle ϕ*const* only. This means that the natural frequency of the mechanical system will be proportional to the angular speed of the propeller shaft, and the value of factor *<sup>C</sup>*(ϕ*const*) can be properly set by selecting a suitable constant twist angle ϕ*const*. This is illustrated with an interference diagram of a two-mass torsional oscillating mechanical system using the PTTO with linear characteristics in Figure 4, where ω is the angular speed of the shaft, ω*e1* and ω*e2* are the excitation frequencies resulting from the periodically alternating load torque and Ω*0(*ϕ*const1* ... *8)* are the natural frequencies corresponding to the pre-set constant twist angles ϕ*const1* < ϕ*const2* < ... < ϕ*const8*. The intersections of the excitation frequencies with the natural frequency represent resonances at which the transmitted load torque reaches its maximum value. The case of resonance should be avoided in the operating speed range, as it can be dangerous in terms of a high alternating load torque.

**Figure 4.** Interference diagram of a mechanical system with constant twist angle control.

According to Figure 4, the course of natural frequencies consists of three parts. The first part, the *sub-regulatory area*, is a horizontal line beginning at zero speed when the load torque is zero (resulting from the fan characteristics) and the PTTO is inflated to the minimum operating pressure. At a certain speed, the torque twists the PTTO to the selected constant twist angle, and this is the start of the second part representing the *regulatory area* of the PTTO. In the regulatory area, the pressure in the PTTO is regulated to a value where the PTTO has the desired constant twist angle. The regulatory area ends at a speed when the pressure reaches the maximum operating value. This point is the start of the *over-regulation area*. After this point, the pressure has its maximum operating value. In the presented case, the optimum constant twist angle is ϕ*const4*, where the torsional natural frequency is farthest from the excitation frequencies, i.e., farthest from the resonance in the regulatory area of the PTTO.

Results of theoretical analyses in marine propulsion mechanical systems using constant twist angle control presented, for example, in [31,33] show that this type of control ensures proper tuning in terms of torsional vibration and has considerable potential for future application.

Constant twist angle control can be ensured by a constant twist angle regulator which is a part of the PTTO [13,26], or ECTACS can be used.

### *2.3. Experimental Torsional Oscillating Mechanical System*

In order to verify the ability of our ECTACS to keep the twist angle of a PTTO at a pre-set constant value, the ECTACS was applied into an experimental torsional oscillating mechanical system.

The TOMS of the piston compressor drive (Figure 5) is made up of a 3-phase asynchronous electromotor, *Siemens 1LE10011DB234AF4-Z* (11 kW, 1500 RPM) *(1)*. Rotational speed of this electromotor is continuously vector-controlled by a frequency converter, *Sinamics G120C*. The electromotor drives a 3-cylinder piston compressor, *ORLIK 3JSK-75 (3)*, through a PTTO type *4-2*/*70-T-C (2)* (Figure 1). The compressor in this system acts as a load and torsional vibration exciter too. For proper tuning of the system, it is necessary to know the dynamic behavior under different operating conditions, which was experimentally investigated in [34].

Under standard conditions, the load torque of the system has no fan characteristics, and this means that this TOMS is not very suitable for tuning torsional vibrations with constant twist angle control, so our goal is not the tuning itself but only the verification of the ability to maintain the desired constant twist angle of the PTTO.

**Figure 5.** Experimental torsional oscillating mechanical system.

The whole drive is mounted on a rigid sprung frame. The compressed air from the compressor flows into an air pressure tank with a volume of 20 l. The air pressure in the tank is controlled by a throttling valve. So, the load of the TOMS can be adjusted. Maximum compressor output air overpressure is 800 kPa and its value is measured by a pressure sensor, *Danfoss MBS 3000*, with an overpressure measuring range of 0–1 MPa. The same type of pressure sensor is used for measuring the air pressure value in the compression space of the PTTO. The accuracy of the *MBS 3000* sensor with a metal membrane is 0.5% of its measuring range, i.e., 5 kPa (combined fault–nonlinearity, hysteresis and reproducibility). The supply of compressed air into the PTTO is ensured by a rotation supply *(4)*. The mechanical part of the experimental TOMS is described in more detail in [35].

The multifunctional electronic module (MFEM), marked with the number *(5)*, which is a part of an electronic constant twist angle control system, developed by us, has the following functions: Page: 7 Is the italics necessary?


### *2.4. Data Measuring and Processing*

The dynamic load torque transmitted by the PTTO causes mutual dynamic angular twisting of the driving and driven hubs of the PTTO. The measurement is based on determining the PTTO's twist angle time course. The PTTO is equipped with black–white tape, which is stuck to the circumference of the driving and driven hubs of the PTTO and they are scanned by a pair of optoelectronic sensors (Figure 6). According to our specific requirements, a pair of *Dewetron* optoelectronic sensors of type *SE-TACHO-PROBE-01* [36] (Figure 6) was used.

These sensors detect the reflection from the reflective black–white moving tape (Figure 6b). The sensors react to the edges between the black and white stripes. There is a distinct change of electrical output voltage at the moment the edge is crossed. These sensors can work with a maximum frequency of 10 kHz but the cleanness of the reflective tape, cleanness of the optical parts of the sensors and the sharpness of the edges between the black and white stripes must be excellent [36].

**Figure 6.** Optoelectronic sensors *Dewetron SE-TACHO-PROBE-01*: (**a**) sensor attached to a data cable; (**b**) pair of sensors mounted in the experimental torsional oscillating mechanical system.

The number of black and white stripe pairs for the driving and driven hubs should be equal and chosen with respect to the maximum twist angle of the PTTO and to the character of the transmitted load torque (especially its dynamic component). It is also important that the length of all the black and white stripes should be equal. The angle corresponding to length of a black and white stripe pair must be larger than the maximum twist angle of the PTTO. For the PTTO type *4-2*/*70-T-C*, the maximum twist angle is 11◦ which corresponds to a maximum of 32 pairs of stripes. Further, as the major harmonic component for a three-piston compressor is the 3rd harmonic component, it is advantageous to select the number of black and white stripe pairs as integer multiples of 3 to obtain the best results for the twist angle time course by the equal operation of all cylinders (theoretically "3 identical recorded curve portions in one revolution"). Therefore, 30 black–white stripe pairs for each hub of the PTTO have been selected. The edge-crossing times *ti* need to be measured, where the index *i* stands for the sample order number. In our case, only the black-to-white stripe edges are considered. The times *ti* are computed from the counted number of impulses from the counter of the MFEM microprocessor. One impulse represents a time of 1/14745600 s.

Considering the times *t1i* for the driving hub and the times *t2i* for the driven hub of the PTTO, the time delays Δ*ti* [s] can be computed using the following equation:

$$
\Delta t\_i = t\_{2i} - t\_{1i}.\tag{7}
$$

In the next step, the total twist angle ϕ*T* [rad] of the PTTO can be computed according to the following equation:

$$
\varphi\_{Ti} = \frac{\pi \cdot n \cdot \Delta t\_i}{30},
\tag{8}
$$

where *n* [min−1] is the immediate rotational speed of the mechanical system.

From the total twist angle of the PTTO ϕ*<sup>T</sup>*, its static component ϕ*stat* can be computed as the mean value:

$$
\varphi\_{stat} = \frac{\sum\_{i=1}^{k} \varphi\_{Ti}}{k},
\tag{9}
$$

where *k* [-] is the number of samples, which has to be an integer multiple of the black–white stripe pairs number on the hub. In practice, the floating average method is used for this computation.

The computations according to Equations (7)–(9) are performed by the software part of our ECTACS in PC in real time. This way, the controlled variable (ϕ*stat*) for the ECTACS can be computed. It is very advantageous because the torsional vibration does not directly affect the control device like in the case of regulators directly built into the PTTO.

### *2.5. Description of the Constant Twist Angle Control System Function*

The goal of constant twist angle control is to maintain a pre-set static (mean) value of the twist angle by any given static load torque *Mstat* resulting from the current operating mode. Although it is possible to use a closed-loop PID control system with a static twist angle as a controlled variable and overpressure as a manipulated variable, based on the fact that the mathematical and physical model of the PTTO (presented in Section 2.1.1) is well known, it was decided to use a model-based adaptive control system [37]. This approach allows us to reach the desired value of the static twist angle more quickly, which is very important in terms of passing through the resonance as quickly as possible.

The static load torque of the PTTO at a periodically alternating load torque can be computed as

$$M\_{\rm stat} = M\_{\rm G\left(\varphi\_{\rm stat}\right)} + p\_{T^\cdot}(S\_{\rm t}r)\_{\left(\varphi\_{\rm stat}\right)^\*} \tag{10}$$

where *pT* is the mean overpressure in the PTTO, and the values of the rubber shell torque *MG* and static moment of effective area *Se*·*<sup>r</sup>* are computed from the static twist angle ϕ*stat*. The value of the twist angle is continuously measured and its mean value is computed by the control system.

The rubber shell torque *MG(*ϕ*stat)* and the static moment of effective area *Se*·*r(*ϕ*stat)* are computed as fifth degree polynomials:

$$M\_{\rm G\left(\varphi\_{\rm stat}\right)} = \sum\_{i=0}^{5} a\_{i} \cdot \varphi\_{\rm stat'}^{i} \tag{11}$$

$$(S\_{\mathfrak{c}} \cdot r)\_{\left(\varphi\_{\text{stat}}\right)} = \sum\_{i=0}^{5} b\_{i} \cdot \varphi\_{\text{stat}}^{i} \,. \tag{12}$$

After computing the static load torque according to Equation (10), the value of overpressure *pTC* needed for obtaining the desired constant twist angle ϕ*const*, using Equations (11)–(12), where the desired constant twist angle ϕ*const* is set instead of the static twist angle ϕ*stat*, can be expressed as follows:

$$p\_{TC} = \frac{M\_{\text{stat}} - M\_{G\left(\varphi\_{\text{cnot}}\right)}}{\left(S\_{\text{t}} \cdot r\right)\_{\left(\varphi\_{\text{cnot}}\right)}}.\tag{13}$$

After setting the new value of overpressure according to Equation (13), the value of the actual static twist angle is measured. The difference between the actual static twist angle ϕ*stat* and the desired static twist angle ϕ*const* can be expressed as

$$
\Delta \boldsymbol{\varrho} = \boldsymbol{\varrho}\_{\text{const}} - \boldsymbol{\varrho}\_{\text{stat}}.\tag{14}
$$

In the case where the achieved value of the mean twist angle lies outside the insensitivity range ϕ*ins*, <sup>Δ</sup>ϕ > ϕ*ins*, but inside the fine tuning range ϕ*FT*, <sup>Δ</sup>ϕ ≤ ϕ*FT*, fine tuning is used. The value of the needed overpressure is then computed as

$$p\_{\rm TC} = p\_{\rm T} + X \cdot \Delta \boldsymbol{\varrho} \cdot \mathbf{c},\tag{15}$$

where *X* [Pa·rad−1] is a derivation of *pTC* according to Equation (13) by angle ϕ*const* and *c* [-] is a constant factor. The value of the constant factor *c* should be selected in the range (0; 1.

Derivation *X* is then computed as

$$X = -\frac{\sum\_{i=1}^{5} i \cdot a\_{i^\*} \cdot \varphi\_{\text{stat}}^{i-1}}{\left(S\_{\varepsilon} \cdot r\right)\_{\left(\varphi\_{\text{stat}}\right)}} - \frac{\left(M\_{\text{stat}} - M\_{G\left(\varphi\_{\text{stat}}\right)}\right) \cdot \sum\_{i=1}^{5} i \cdot b\_{i^\*} \cdot \varphi\_{\text{stat}}^{i-1}}{\left(\left(S\_{\varepsilon} \cdot r\right)\_{\left(\varphi\_{\text{stat}}\right)}\right)^2}. \tag{16}$$

The flowchart of the constant twist angle control algorithm is shown in Figure 7.

**Figure 7.** Flowchart of constant twist angle control algorithm.

The parameters of the algorithm can be set via the software graphic user interface of the ECTACS on a PC.
