Torsional Analysis

What is Torsional Vibration Analysis

In a reciprocating compressor system that includes the driver,coupling and compressor, torsional vibration refers to angular oscillations along the driver shaft, coupling, and compressor crankshaft drive train. These oscillations are caused by vibratory torques acting on the compressor and drive shaft. The figure below illustrates the torsional vibration of a motor-driven compressor drive train, highlighting the vibratory behavior and stress distribution across the system's components.

Torsional Vibration Analysis (TVA) investigates the torsional dynamic behavior of the rotating compressor drive train as a result of forced vibration. This analysis involves calculating torsional natural frequencies (TNFs), torsional vibration levels, vibratory torques, and dynamic stresses within the compressor system.

Torsional Vibration of Motor-Driven Compressor Drive Train

Why Perform Torsional Vibration Analysis

Performing a TVA is essential for identifying and mitigating the risks of excessive torsional vibrations, particularly during resonance. Neglecting this analysis can lead to:

Coupling failure
Cracking of the motor shaft and compressor crankshaft
Excessive vibration velocity at the compressor's auxiliary end may potentially lead to lubrication pump failure
Engine damper failure, and more

Coupling Failure Caused by Excessive Torsional Vibration

Torsional Vibration Analysis Methodology

Torsional Vibration Analysis Procedure

Below is a flowchart outlining the procedure for TVA of reciprocating compressor packages.

Information collection involves gathering the necessary input data for the analysis, including the mass-elastic data of the compressor, coupling, engine or motor, as well as the operating conditions being evaluated.
Generation of discrete model simulates the physical compressor system as a mathematical torsional model, where lumped masses (disks) are connected by equivalent torsional springs. The moments of inertia and torsional stiffness between the disks are typically provided by equipment manufacturers or calculated using Finite Element Analysis (FEA). The figures below depict both the physical motor-driven compressor system and the corresponding torsional analysis model.

Motor-Driven Compressor System

Torsional Analysis Model

This model generation also includes calculating the driving torques of the motor or engine and the torque demands of the compressor based on the specified operating conditions, which provide the torsional excitations for the TVA.

Compressor Torque Demands

Engine Driving Torque

Motor Driving Torque

Undamped natural frequency calculation determines the undamped torsional natural frequencies (TNFs) of the compressor system, resulting in a Campbell diagram, as shown in the left figure below. This diagram highlights torsional resonance speeds by displaying the intersection points between the TNFs and the harmonics of the compressor operating speeds. The right figure below illustrates the corresponding mode shapes for the first three modes.

Campbell Diagram

Model Shapes of Compressor System

Forced response analysis calculates the torsional responses, including angular velocity, deflection, and vibratory torque for each lumped mass in the compressor system by applying the driving torque and compressor torque demands to the model. The system’s damping and phase angles are also taken into account in this calculation.

Analysis result assessment involves evaluating the analysis results against specified guidelines. The following table outlines the torsional analysis results to be assessed for achieving a suitable torsional vibration control design.

Damper	Vibration Resonance	RPM Vibration	Mean Torque	Vibratory torque	Fatigue Stress	Angular Deflection	Heat Loads
System	✔	✔
Engine			✔	✔	✔	✔	✔
Motor			✔	✔	✔	✔
Coupling			✔	✔		✔	✔
Compressor			✔	✔	✔	✔

Solution accomplished identifies an appropriate torsional vibration control design that brings all analysis results within the specified allowable limits. If the results do not meet the criteria, redesigning the torsional control measures is needed. This could involve using a different coupling, adding a flywheel, applying internal damper or detuners, or modifying the cylinder configuration, etc. This redesign process is repeated until all torsional analysis results fall within the allowable limits specified by API 618 and equipment vendors. Once this is achieved, an analysis report will be prepared and submitted for review.

API 684 Torsional Analysis Logic Diagram

The API 684 Standard addresses the purpose and basic method of torsional analysis, the considerations in torsional modeling, the analysis procedure and presentation of analysis results. The figure below, sourced from API 684, provides a flowchart for torsional analysis which is presented to assist in the design process of rotating shaft systems from a torsional dynamics perspective. The numbers in each diagram box of the flowchart correspond to the relevant clauses in the API 684 Standard.

API 684 Torsional Analysis Logic Diagram

Torsional Vibration Analysis Guidelines

(1) Resonance Guidelines

According to API 618 (6_th Ed.) Clauses 6.7.2 and 6.7.4, the torsional natural frequencies of the drive system (motor or engine) and the compressor (including couplings and gear units) shall remain at least 10% away from any compressor operating speed and 5% from any multiples up to the tenth. For motor-driven compressors, these frequencies should also be separated by 10% and 5% from the first and second multiples of the electrical power frequency. Synchronous motor-driven compressors must adhere to Clause 7.1.2.10 of API 618 (6_th Ed.).

If torsional resonances fall within these limits and cannot be adjusted, a stress analysis shall be performed to confirm that the resonances do not pose a risk to the driver-coupling-compressor system.

(2) Speed Variation Guidelines

As specified in API 618 (6_th Ed.) Clause 7.1.1.7, during the initial design phase (excluding motor-driven systems), the peak-to-peak speed variation of the system should not exceed 1.5% of the compressor's rated speed under full load or partial cylinder load conditions, especially when step unloading is involved. This guideline ensures stable operation and prevents excessive vibration or stress on the system.

(3) Allowable Vibration Limit and Stress Guidelines

Motor Shaft: The non-drive end torsional vibration amplitude shall comply with the manufacturer’s specifications, if available. The alternating stress in the motor shaft should be evaluated to ensure it remains within allowable limits using the Modified Goodman Criteria.
Engine Crankshaft: The calculated maximum vibratory stress on the engine crankshaft, thermal loads on the viscous damper, and the torsional vibration amplitude at the engine front end during operation shall all fall within the manufacturer’s specified allowable limits.
Coupling: The calculated maximum and minimum torques shall be within allowable limits as specified by a Modified Goodman diagram or the manufacturer’s torque limits. For flexible couplings, heat generation due to vibratory torque must also be considered, ensuring it remains within safe limits to prevent overheating or damage.
Compressor: The vibratory torques on the compressor drive stub and journal, and the auxiliary end velocity shall be within the allowable limits specified by the manufacturer.

Torsional Vibration Analysis Tools

There are many analytical tools available for Torsional Vibration Analysis (TVA) of compressor systems. At CCPGE, we use our in-house developed program for this purpose. This program combines both time-domain and frequency-domain analysis techniques and has been validated through real torsional analysis and site measurement projects. The frequency-domain analysis focuses on calculating torsional resonant frequencies, while the time-domain analysis provides more accurate torsional response results. The following figures present the vibratory torques acting on the coupling, calculated from both analyses.

Vibratory Torque Acting on the Coupling
(Time Domain Analysis Results)

Vibratory Torque Acting on the Coupling
(Frequency Domain Analysis Results)

Torsional Vibration Analysis Example

An example of torsional vibration analysis for a reciprocating compressor package is illustrated below. This analysis was performed using our in-house developed program in accordance with the API 618 Standard and the API 684 approach flow chart.

Compressor Model:	Ariel KBZ/4	Power:	2796 kW
Engine:	Caterpillar G3612	Speed:	800 – 1000 RPM
Number of Stages:	3	Flow Rate:	439 – 550 E3m3/D
Number of Cylinders:	4	Suction / Discharge Pressure:	0.41 MPag / 8.8 – 9.8 MPag

The vibratory torques acting on the coupling of the compressor system were calculated using combined time-domain and frequency-domain analysis techniques, as shown in the following figures. The time-domain analysis reveals the fluctuation of torques as the system transitions from rest to steady-state operating conditions, while the frequency-domain analysis highlights a significant increase in vibratory torque at resonance speeds.

Vibratory Torque Acting on the Coupling
(Time Domain Analysis Results)

Vibratory Torque Acting on the Coupling
(Frequency Domain Analysis Results)

The table below presents the calculated vibratory torques at the compressor drive stub and journal, along with their corresponding allowable limits for four operating cases selected based on the Load Step Selection criteria recommended by Ariel.

Case No	Mean Torque （Nm）	Calculated Drive Stub Vibratory Torque (Nm)	Allowable Drive Stub Vibratory Torque (Nm)	Calculated Journal Vibratory Torque (Nm)	Allowable Journal Vibratory Torque (Nm)	Assessment
1	25,276	33,304	59,660	39,802	59,660	Acceptable
2	25,912	33,504	59,660	40,180	59,660	Acceptable
7	24,862	30,755	59,660	36,382	59,660	Acceptable
9	24,107	22,185	59,660	22,314	59,660	Acceptable

The calculated torsional vibration velocities at the compressor auxiliary end across harmonic orders (1X to 9X) are also displayed in the following figure, with a red dashed line indicating the design limit of 40 RPM. All plotted values fall within this limit, confirming the safe operation of the compressor system. The Torsional-Lateral (T/L) curve, included as a marginal guideline to reduce high-frequency lateral vibrations, is plotted but serves as a recommendation rather than a mandatory mechanical limit.

Compressor Aux End Torsional Velocity

Additionally, the table below shows that the calculated damper thermal loads for the four selected cases remain within the allowable limits specified by the Caterpillar G3612 engine data sheet.

Case	Calculated Thermal Load(kW)	Thermal Load Limit	Assessment
1	1.20	8.24	Acceptable
2	1.21	8.24	Acceptable
7	1.26	7.79	Acceptable
9	1.85	6.89	Acceptable

The predicted torsional vibration amplitudes at the engine front end for the selected cases meet the manufacturer's requirements, with maximum amplitudes remaining within the specified limits for all orders.

Vibration Amplitude vs. Harmonic Order at Engine Front End

Torsional Vibration Analysis of Screw Compressors

The methodology for torsional vibration analysis of screw compressor packages is generally similar to that used for reciprocating compressor packages, with the primary distinction being adherence to the API 619 Standard. This standard emphasizes the importance of avoiding torsional resonance between the system's natural frequencies and the excitation frequencies.

The pocket passing frequency (PPF) serves as the main excitation frequency for the torsional vibration analysis of screw compressor packages. It is calculated by multiplying the number of male rotor lobes by the male rotor's rotational speed in Hertz (Hz).

The figures below illustrate the physical system model, the calculated first three orders of torsional vibration modes, and the Campbell diagram, which indicates the potential torsional resonance frequencies for a screw compressor package.

Motor-Driven Screw Compressor Package

Model Shapes of Screw Compressor

Campbell Diagram

Torsional Vibration Analysis of Multi-Branch Systems

Multi-branch torsional vibration systems in compressor packages typically consist of a primary driving branch and two or more driven branches. Each branch may operate at different speed and possesses distinct dynamic characteristics, including moment of inertia, stiffness, and damping, resulting in different torsional natural frequencies.

The torsional behavior of one branch can significantly influence the behavior of others. For instance, if one branch encounters torsional resonance, vibrations can propagate through the couplings and junction point (usually the gear train) to affect the other branches. Such interactions lead to complex torsional vibration behavior, making the analysis of multi-branch systems particularly challenging.

To analyze a multi-branch torsional vibration system, the first step is to designate one branch (typically the driving branch) as the reference branch. Next, the speed ratios of the other branches relative to the reference branch are determined. This allows for the calculation of the equivalent moment of inertia and stiffness of each branch in relation to the reference branch. The torsional vibrations of the multi-branch system are then resolved by satisfying the geometrical constraints or compatibility conditions of each branch at their junction point.

The figures below illustrate the physical system model, the calculated Campbell diagram, and the vibratory torque response acting on one of the three couplings for two screw compressors driven by a motor through a gearbox, which is often found in screw compressor applications. The selection of three suitable couplings and the assessment of torsional vibration responses and vibratory torques acting on the components of system are achieved through the torsional vibration analysis of multi-branch systems.

Motor-Driven Screw Compressor Package

Campbell Diagram

Vibratory Torque Acting on the Coupling

Notes on Torsional Vibration Analysis

Single-Acting Operating Cases: Single-acting operating cases represent the worst-case scenario for torsional analysis due to a more dynamic torque demand. The increased torque variation amplifies vibratory torque on compressor crankshafts and couplings, leading to stronger vibrations, particularly at higher speeds and across varying pressure ranges. Although this complexity makes it more challenging to identify effective torsional vibration control measures, it is crucial to include these operating cases in the torsional analysis.
1/3 Penetration in TVA Modeling: The Shaft Penetration Factor (SPF) is crucial for determining the torsional stiffness and overall dynamic response of compressor systems. According to API guidelines, a 1/3 SPF significantly reduces the first torsional natural frequency while enhancing damping due to increased sliding motion between the shaft and hub. As SPF increases, a trade-off occurs between reduced stiffness and improved damping, which shall be carefully evaluated in TVA modeling to ensure accurate representation of the real system.
Effect of Motor Core Construction on Shaft Stiffness: The construction of the motor core significantly impacts shaft stiffness. Machined or welded webs can enhance stiffness by 10% to 40%, while keyed or shrunk-on laminations provide minimal structural support. Similarly, squirrel cage construction contributes little to stiffness unless rotor bars add additional rigidity. These variations complicate the accurate quantification of motor stiffness in torsional vibration analysis, necessitating accurate modeling. CCPGE is actively collaborating with the University of Calgary to develop advanced methods for improving motor shaft stiffness calculations by considering the effect of motor core construction.
Motor Fan Failure: The motor fan, located at the non-drive end of the motor and primarily used for cooling, is susceptible to failures induced by torsional vibrations. Torsional vibratory torque can lead to fan fatigue failure, especially if there is an imbalance in the fan's mass or mounting rigidity, or if the torsional natural frequencies of the motor fan align with the torsional excitation frequencies in compressor systems. Investigating the root causes of motor fan failure falls outside the normal scope of TVA and is the responsibility of the fan manufacturer. However, it would be beneficial to check the torsional resonance of the motor fan within the TVA framework.

Torsional Vibration Analysis

Quick Navigation

What is Torsional Vibration Analysis

Why Perform Torsional Vibration Analysis

Torsional Vibration Analysis Methodology

Torsional Vibration Analysis Procedure

API 684 Torsional Analysis Logic Diagram

Torsional Vibration Analysis Guidelines

Torsional Vibration Analysis Tools

Torsional Vibration Analysis Example

Torsional Vibration Analysis of Screw Compressors

Torsional Vibration Analysis of Multi-Branch Systems

Notes on Torsional Vibration Analysis

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