###### BY ANDREA FUSI, FRENCESCO GRASSO, ALESSANDRO USSI AND ALESSANDRO BAYLON

*This article was originally published in the May issue of *COMPRESSORtech2*. Get every issue in your inbox/mailbox and access to our digital archives with a free subscription.*

Reciprocating compressors generate considerable torque variations that cause torsional vibrations into the drive train that, if excessive, could cause the components of the train to fail (e.g., coupling failure, Figure 1).

If the rotor has a superimposed torsional vibration over the steady rotation, the flux linkages, produced by the induction motor, start to oscillate. This phenomenon modifies the electromagnetic fields across the air gap between stator and rotor, generating an additional torque, which acts in conjunction with the rotor’s harmonic excitations. The air gap effect influences the natural frequencies of the electromechanical system. This means that the electric behavior of the motor has a significant role in torsional vibration analysis and is necessary to accurately evaluate its contribution.

For this reason, a simplified approach was developed [1, 2] to include the air gap effect inside the study of the torsional analysis, adding one spring and one damper in the mechanical model obtained from the equivalent shaft of the rotor.

The API 618 standard code for reciprocating compressors recommends that the “torsional natural frequencies of the complete driver-compressor system (including couplings and any gear unit) shall not be within 10% of any operating shaft speed and within 5% of any multiple of operating shaft speed in the rotating system up to and including the 10th multiple.”

Regarding to the electrical part of the system, torque variations induce a current pulsation on the power supply line, which can produce problems like transformers overheating, electromagnetic interference and flickering. To prevent these risks, API codes recommend that current variations shall not exceed respectively 40 or 66% of the full load current, when an induction or synchronous motor is adopted [3].

To satisfy the API standard, the traditional simulation is performed in two steps, separating the mechanical and the electrical part of the system. In fact, torsional vibration analysis (TVA) is usually carried out by the packager and, based on its results, source current pulsation analysis (SCPA) is performed by the electric motor manufacturer.

To satisfy the API standard, the traditional simulation is performed in two steps, separating the mechanical and the electrical part of the system. In fact, torsional vibration analysis (TVA) is usually carried out by the packager and, based on its results, source current pulsation analysis (SCPA) is performed by the electric motor manufacturer.

In this paper, a new approach to performing dynamical simulations of reciprocating compressors driven by an induction motor is presented, consisting in the simultaneous simulation of the complete electromechanical system and performed using an internally developed multiphysics model tool in MATLAB.

#### Induction motor air-gap effect on torsional vibrations and current pulsations

The absorbed torque of a reciprocating compressor can be represented by a variable and constant term:

**C=C _{m} +C(t)**

Induction motors react to these torque fluctuations with their own air-gap torque variations. This phenomenon can be intuitively understood by observing the induction motor torque-speed characteristic curve (Figure 2).

These torque variations have an important effect on system torsional dynamics and must be included into the lumped parameter model used to calculate torsional natural frequencies. Knop [1] and Hauptman [2] reported a method to model this effect by adding a spring and a damper to the original equivalent shaft. The spring and the damper value are easily obtainable from two simplified equations (shown in Figure 3 for a system with only 2° of freedom [DOF]) that require only few common electric motor parameters (number of poles, breakdown torque, slip, etc.):

- where
*T*is the Electric Motor Time Constant [_{L}*s*],*T*is the Rated Torque [_{R}*Nm*],*T*is the Breakdown Torque [_{B}*Nm*] and*S*is the Rated Slip._{R}

Jordan, et al. [4, 5], presented more detailed equations that consider also negative damping effect near the supply frequency, which represents not an energy dissipation but a phase shift between excitation and vibration, causing an increase of vibration amplitudes (Figure 4).

CST has developed a new method to perform TVA and SCPA together by means of a multi-physics model in MATLAB environment, thus considering the reciprocal effect that the mechanical and electrical parts have on each other. Figure 5 shows a scheme of the model with a 2 DOF mechanical system and a three-phase induction motor.

This method guarantees a more accurate evaluation of the dynamic interaction between electrical and mechanical system in steady-state conditions and allows the simulation of transient conditions (start, stop, short circuits, voltage re-closure) evaluating the electromagnetic torque amplification and current pulsations. This allows the packagers to identify the best selection of the main items of the system, in accordance to API 618 standard code, from the earliest stages of the project.

Case study

A reference case of a compression train composed of a two-cylinder reciprocating compressor, flywheel, highly flexible coupling and induction motor (nr. 6 DOF) has been studied. Table 1 lists the parameters of the system model, while Table 2 lists the electrical parameters taken from the induction motor data sheet. Table 3 shows the influence of the air gap effect over the value of the natural frequencies, simulated with the simplified approach and with the multi-physical model in MATLAB that CST developed. As it is well known, first and second natural frequencies fluctuate, when the air gap effect is included, while the higher frequencies (related to mechanical side of the model) do not change. When the electric motor model is included, induction motor self-excited torques appear at vibratory frequencies near the supply one (“negative damping” effect).

As mentioned, the electromechanical model allows simultaneous solving of TVA and SCPA in the early stage of the job, thus avoiding possible later expensive modifications in terms of time and money.

This CST tool allows transient analysis. These kinds of investigations are not always strictly required by the standards, but they can provide some useful information in any case.

The simulation revealed that almost no torque oscillations are present when the steady state is reached. This fact can be explained with the use of a soft coupling that tends to soften the alternate torques. By observing Figures 6, 7 and 8, some considerations regarding the transient phase can be made:

- The torque oscillation at the startup remains within acceptable limits (2x nominal torque).
- The induction motor has a torque oscillation that occurs at line frequency (50 Hz) as expected. This justifies the API requirement to separate torsional natural frequencies from the first and second multiplies of the electrical power supply frequency by more than 10 and 5%, respectively.
- The coupling torque oscillations are not amplified when passing through the resonance (796 rpm, 2.8 s). This is due to the presence of damping in the system and to the short time in which the compressor passes through the resonance region.
- The maximum torque acting on the coupling occurs at the electric motor breakdown torque as expected.
- Starting from these, the current pulsations percentage for the reference case (estimated as per [6]) is 1.3%, so it is possible to affirm that they are allowable. Furthermore, it allows getting an estimation of the current pulsations much earlier in the project.

#### Conclusions

A new method to conduct torsional vibration and source current pulsation analysis has been developed. The study has been carried out considering the reciprocal effect that the mechanical and electrical parts have between each other.

For this purpose, a multi-physics model in Matlab environment has been developed to perform the TVA and the SCPA at the same time.

This new approach, theoretically more accurate than the ones in the literature, allows the simulation of not only steady state conditions but also transient ones (start, stop, short circuits, voltage re-closure) and estimates the current pulsations at an early stage during the project, which is also a design criterion of API 618.

In further works, the focus will be on development procedures that optimize the choice of parts, such as the coupling and the flywheel, and reducing the costs during the design phase. In addition, the focus will be also on the investigation of negative damping effects in reciprocating compressor trains with all-steel couplings, which are generally more sensitive to this phenomenon than the highly flexible ones.

**About the authors: ****Andrea Fusi** is an executive R&D manager at Compression Service Technology S.r.l. (CST). Contact him at: andrea.fusi@cstfirenze. com. **Francesco Grasso** is an assistant professor at Universitiá degli Studi di Firenze, Dept. of Telecommunications. Contact him at: Francesco.grasso@unifi.it. **Alessandro Ussi** is an instrument and control engineer at CST. Contact him at: alessandro.ussi@cstfi- renze.com. **Alessandro Baylon** is an R&D engineer at CST. Contact him at: alessandro.baylon@cstfirenze.com.

#### References:

- [1] Knop, “
*The Importance of Motor Dynamics in Reciprocating Compressor Drives*”, 8^{th}Conference of the EFRC, September 27^{th}/28^{th}, 2012, Dusseldorf - [2] G. Hauptmann, W.F. Eckert, B.C. Howes, “
*The Influence on Torsional Vibration Analysis of Electromagnetic Effects Across an Induction Motor Air-gap*”, Gas Machinery Conf. 2013 (Albuquerque MN, 6^{th}–9^{th}October 2013) - [3] API STD 618 Reciprocating compressors for petroleum, chemical, and gas industry services, 5
^{th} - [4] Jordan, J. Muller, H.O. Seinsch, “Uber elektromagnetische und mechanische Ausgleichsvorgange bei Drehstromantrieben”, AEG-Telefunken (1979).
- [5] [H. Jordan, J. Muller, H.O. Seinsch, “Uber Das Verhalten von Drehstromasynchronmotoren in drehelastischen Antrieben”, AEG-Telefunken (1980).
- [6] T. Joshi. “Current Pulsation Calculations of an Induction Motor Connected to a Reciprocating Compressor”, International Compressor Engineering Conference (1984)

#### Related works:

*New Method for Current Pulsation Calculation in Induction Machines Driving Reciprocating Compressor. **A Fusi**1 , F Grasso2, A Ussi1. *

*1. C.S.T. Compression Service Technology Srl, V. Giovanni del Pian dei Carpini n.1, 50127 Firenze – IT.*

*2. Department of Information Engineering – DINFO, University of Firenze, via S. Marta, 3, 50139 Firenze, IT*;

**International CAE Conference 2015, Lazise**

*New Harmonics Current Mitigation Technique in Induction Motor Driving Reciprocating Compressor. **A Fusi**1 , F Grasso2, A Ussi1 *

*1. C.S.T. Compression Service Technology Srl, V. Giovanni del Pian dei Carpini n.1, 50127 Firenze – IT.*

*2. Department of Information Engineering – DINFO, University of Firenze, via S. Marta, 3, 50139 Firenze, IT*;

**1**

^{st}IEEE International Symposium on Systems Engineering (ISSE) _ Rome, 2015*Torsional Vibration Analysis of Reciprocating Compressor Trains driven by Induction Motors **M Brunelli**1 , A Fusi2, F Grasso3, F Pasteur4, and A Ussi2. *

*1. BSIM srl, Corso Vittorio Emanuele II – 32, 10123 – Torino, IT.*

*2. C.S.T. Compression Service Technology Srl, V. Giovanni del Pian dei Carpini n.1, 50127 Firenze – IT.*

*3. Department of Information Engineering – DINFO, University of Firenze, via S. Marta, 3, 50139 Firenze, IT.*

*4. Siemens Industry Software S.r.l. Viale Piero e Alberto Pirelli, 10, 20126 Milano, IT*;

**Presented To The City University, London – Institution of Mechanical Engineers (London, 2015)**

*Electro-Mechanical Modelling of a Reciprocating Compression Train driven by Induction Motor. ***Andrea Fusi1, Francesco Grasso2, Alessio Sambataro2, Alessandro Baylon1, Luca Pugi2**. 1CST – Compression Service Technology Via G. Del Pian dei Carpini n.1, Florence, Italy *2University of Florence Dept. of Industrial Engineering Via di Santa Marta n.3, Florence, Italy* **Presented at the Torsional Vibration Symposium (Salzburg, Austria, 17-19 May 2017).**