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Viewing 1 to 30 of 37654
2017-07-10
Technical Paper
2017-28-1953
Tushar Narendra Puri lng, Lalitkumar Ramujagir Soni lng, Sourabh Deshpande
The infliction of rigorous emission norms across the world has made the automobile industry to focus and dwell upon researches to reduce the engine emissions of diesel engine. Variation in injection timing has better influence on reduction of engine emissions. This paper deal with numerical simulation of 4-stroke, single cylinder, naturally aspirated, direct injection diesel engine running at 1640 RPM using CONVERGE_STUDIO CFD tool. As the piston and bowl geometry considered in this work is symmetric only 45 degree sector engine model considered for simulation over 360 degree complete engine model. To study the combustion and inside flow physics taking place inside engine cylinder more accurately and to reduce computational time, simulation from 20 bTDC during compression stroke up to 140 aTDC in the power stroke is considered as available in the literature.
2017-06-17
Journal Article
2017-01-9077
Zaimin Zhong, Junjie Li, Shuihua Zhou, Yingkun Zhou, Shang Jiang
Abstract Description of PMSM torque in high accuracy is critical and previous work for its further research. However, the traditional linear torque model fails to describe its non-ideal characteristics of practical working. This paper presents a generalized torque model of PMSM based on flux linkage reconstruction. In synchronous rotating space coordinates, flux linkage were reconstructed through Fourier series expansion and bivariate polynomial. Based on this model, a precise PMSM torque ripple description and corresponding suppression method were developed. Current feed-forward compensation and the rotor field oriented control were applied in torque ripple suppression. Simulation and experimental results both show that the model not only accurately describes the nonlinear variation of PMSM torque in different working conditions, but also can be used to suppress PMSM torque ripple effectively.
2017-06-17
Journal Article
2017-01-9550
David Neihguk, M. L. Munjal, Arvind Ram, Abhinav Prasad
Abstract A production muffler of a 2.2 liter compression ignition engine is analyzed using plane wave (Transfer Matrix) method. The objective is to show the usefulness of plane wave models to analyze the acoustic performance (Transmission Loss, TL) of a compact hybrid muffler (made up of reactive and dissipative elements). The muffler consists of three chambers, two of which are acoustically short in the axial direction. The chambers are separated by an impervious baffle on the upstream side and a perforated plate on the downstream side. The first chamber is a Concentric Tube Resonator (CTR). The second chamber consists of an extended inlet and a flow reversal 180-degree curved outlet duct. The acoustic cavity in the third chamber is coupled with the second chamber through the acoustic impedances of the end plate and the perforated plate.
2017-06-05
Technical Paper
2017-01-1867
Mustafa Tosun, Mehdi Yildiz, Aytekin Ozkan
Abstract Structure borne noises can be transmitted to interior cabin via physical connections by gearbox as well as other active components. Experimental Transfer Path Analysis (TPA) Methods are utilized to investigate main paths of vibrations which are eventually perceived as noise components inside the cabin. For identifying the structure and air borne noise transfer paths in a system, Matrix Inversion (MI), Mount Stiffness (MS), Operational Transfer Path Analysis (OTPA) and Operational Path Analysis with Exogenous Inputs (OPAX) Methods exist. In this study, contribution ranking of transmission paths from active system components through the physical connections into the interior cabin are investigated by MI and OPAX Methods and finally a comparison of them is presented based on the accuracy of obtained results. The modifications are applied on dominant transfer paths which are determined by the mentioned methods above, respectively.
2017-06-05
Technical Paper
2017-01-1907
Yang Wang, Yong Xu, Xiao Tan
Abstract The vibration isolation performance of vehicle powertrain mounting system is mostly determined by the three-directional stiffness of each mount block. Because of the manufacturing tolerance and the coupling effect, the stiffness of mounts cannot be maintained stable. The purpose of this study was to find out the way to optimize the stiffness of mounts via the design of experiments (DOE). According to the DOE process, a full factorial design was implemented. The z-direction stiffness of three mount blocks in the mounting system was selected as the three analysis factors. The maximum and the minimum stiffness of each mount block within the manufacturing tolerance were selected as the two levels. The measured vibration of vehicle body under certain loading case was selected as the response factor. After eight times of experiment, the DOE parameters were analyzed with statistical methods.
2017-06-05
Technical Paper
2017-01-1908
Rong Guo, Jun Gao, Xiao-kang Wei, Zhao-ming Wu, Shao-kang Zhang
Abstract The statement of the engine shake problem is presented through comparing the quarter vehicle models with the rigid-connected and flexible-connected powertrain which is supported on the body by a rubber mount. Then the model is extended by replacing the rubber mount as a hydraulic engine mount (HEM) with regard to the inertia and resistance of the fluid within the inertia track. Based on these, a full vehicle model with 14 degree of freedoms (DOFs) is proposed to calculate the engine shake, which consists of 6 of the powertrain, 1 of the fluid within the inertia track of the HEM, 3 of the car body and 4 of the unsprung mass. Simulation analysis based on the proposed model is implemented, through which the conclusion is drawn that the HEM has great influence on the body and seat track response subjected to front wheel inputs, compared with the rubber mount.
2017-06-05
Technical Paper
2017-01-1753
Jack Hall Riddle, Ya-Juan Bemman, Tom Frei, Sihui Wu, Ishang Padalkar
Abstract Demands for engines to operate at low-frequency firing order are increasing in the automotive market. This requirement is driven by consumer and regulatory demand for vehicles which are more efficient in the use of fuel. As a result, engine and transmission technologies have been developed which permit operation of engines with fewer cylinders at increasingly low RPM’s. The resulting low frequency exhaust noise is more difficult to attenuate than in vehicles in years past. At the same time, vehicles often have less packaging space for mufflers, when larger volume would otherwise be needed to attenuate at lower frequencies. A further challenge is the demand for increasingly refined performance sounds from the exhaust systems of premium cars despite the technical obstacles involved in even maintaining sound quality. Finally, legally permissible sound levels are decreasing in some markets. These market and regulatory demands require new solutions.
2017-06-05
Technical Paper
2017-01-1752
Kapil Gupta, Arun Choudhary, Rakesh Bidre
Abstract At present, a Dual Mass Flywheel (DMF) system is widely known to provide benefits on driveline induced noise, vibration and drivability over a Single Mass Flywheel (SMF). A well-tuned DMF provides nice isolation of torsional vibrations generated in periodic combustion process of automobile IC engines. Similarly, a torsional vibration damper mounted on driveline component reduces the torsional excitation and results a lower torsional vibration at driveline components. Noise and vibration issues like boom noise and high vibrations at low engine RPM range drive are often resulted due to high engine firing order torsional excitation input to the driveline. More often, this becomes one of the most objectionable noise and vibration issues in vehicle and should be eliminated or reduced for better NVH performance. A 4 cylinder, 4 stroke small diesel engine equipped with SMF is found to have high engine firing order torsional excitation.
2017-06-05
Technical Paper
2017-01-1755
Frank C. Valeri, James T. Lagodzinski, Scott M. Reilly, John P. Miller
Abstract Hybrid powertrain vehicles inherently create discontinuous sounds during operation. The discontinuous noise created from the electrical motors during transition states are undesirable since they can create tones that do not correlate with the dynamics of the vehicle. The audible level of these motor whines and discontinuous tones can be reduced via common noise abatement techniques or reducing the amount of regeneration braking. One electronic solution which does not affect mass or fuel economy is Masking Sound Enhancement (MSE). MSE is an algorithm that uses the infotainment system to mask the naturally occurring discontinuous hybrid drive unit and driveline tones. MSE enables a variety of benefits, such as more aggressive regenerative braking strategies which yield higher levels of fuel economy and results in a more pleasing interior vehicle powertrain sound. This paper will discuss the techniques and signals used to implement MSE in a hybrid powertrain equipped vehicle.
2017-06-05
Technical Paper
2017-01-1768
Yong Xu
Abstract The NVH performance is one of the most important concerns in vehicle development. For all-wheel drive (AWD) vehicles and rear-wheel (RWD) drive vehicles, prop shaft is a major transmission component which may cause various NVH problems. This paper focuses on the vehicle NVH problems caused by the second order excitation force of prop shaft. In order to control the NVH performance of the prop shaft efficiently and fundamentally, this work first studied the rotation kinematical characteristics of prop shaft. Then a rigid-elastic coupling model of vehicle driveline was built with the theory of multi-body dynamics. With this model, the sensitive factors that may affect the second order excitation force were investigated. This paper also describes a case study to verify the conclusions which are revealed from the theoretical calculation and the simulation.
2017-06-05
Journal Article
2017-01-1770
Wallace Hill, Dennis Kinchen, Mark A. Gehringer
Abstract This paper describes the development of an analytical method to assess and optimize halfshaft joint angles to avoid excessive 3rd halfshaft order vibrations during wide-open-throttle (WOT) and light drive-away events. The objective was to develop a test-correlated analytical model to assess and optimize driveline working angles during the virtual design phase of a vehicle program when packaging tradeoffs are decided. A twelve degree-of-freedom (12DOF) system model was constructed that comprehends halfshaft dynamic angle change, axle torque, powertrain (P/T) mount rate progression and axial forces generated by tripot type constant velocity (CV) joints. Note: “tripot” and “tripod” are alternate nomenclatures for the same type of joint. Simple lumped parameter models have historically been used for P/T mount optimization; however, this paper describes a method for using a lumped parameter model to also optimize driveline working angles.
2017-06-05
Technical Paper
2017-01-1769
Onkar Gangvekar, Santosh Deshmane
Abstract In today’s automobile market, most OEMs use manual transmission for cars. Gear Shifting is a crucial customer touch point. Any issue or inconvenience caused while shifting gears can result into customer dissatisfaction and will affect the brand image. Synchronizer is a vital subsystem for precise gear shifting mechanism. Based on vehicle application selection of synchronizer for given inertia and speed difference is a key factor which decides overall shift quality of gearbox. For more demanding driver abuse conditions like skip shifting, conventional brass synchronizers have proved inadequate for required speed difference and gear inertia, which eventually results into synchronizer crashing and affects driving performance. To increase synchronizer performance of multi-cone compact brass synchronizer, a ‘Grit blasting process’ has been added. These components tested with an accelerated test plan successfully.
2017-06-05
Journal Article
2017-01-1772
Yawen Wang, Xuan Li, Guan Qiao, Teik Lim
Abstract The prediction and control of gear vibration and noise has become very important in the design of a quiet, high-quality gearbox systems. The vibratory energy of the gear pair caused by transmission error excitation is transmitted structurally through shaft-bearing-housing assembly and radiates off from exterior housing surface. Most of the previous studies ignore the contribution of components flexibility to the transmission error (TE) and system dynamic responses. In this study, a system level model of axle system with hypoid gear pair is developed, aiming at investigating the effect of the elasticity of the shafts, bearings and housing on TE as well as the contribution of flexible bearings on the dynamic responses. The load distribution results and gear transmission errors are calculated and compared between different assumptions on the boundary conditions.
2017-06-05
Journal Article
2017-01-1774
Fabio Luis Marques dos Santos, Tristan Enault, Jan Deleener, Tom Van Houcke
Abstract The increasing pressure on fuel economy has brought car manufacturers to implement solutions that improve vehicle efficiency, such as downsized engines, cylinder deactivation and advanced torque lock-up strategies. However, these solutions have a major drawback in terms of noise and vibration comfort. Downsized engines and lock-up strategies lead to the use of the engine at lower RPMs, and the reduced number of cylinders generates higher torque irregularities. Since the torque generated by the engine is transferred through flexible elements (clutch, torsional damper, gearbox, transmission, tire), these also impact the energy that is transferred to the vehicle body and perceived by the driver. This phenomenon leads to low frequency behavior, for instance booming noise and vibration. This paper presents a combined test and CAE modelling approach (1D/3D) to reverse engineer a vehicle equipped with a CPVA (centrifugal pendulum vibration absorber).
2017-06-05
Technical Paper
2017-01-1773
Jing Yuan
Abstract The dual phase twin synchronous drive has been developed for belt noise reduction. Two identical synchronous belts are arranged parallel side by side with one tooth staggered against other by the half pitch offset. The noise cancellation effect is achieved as one belt tooth engagement coincides with the other belt tooth dis-engagement. A center flange is used as a divider to prevent the belts contacting each other along the axial direction during their entrance and exit of the sprocket. An overall 20 [dBA] noise reduction has been achieved with the dual phase twin belt drive compared to an equal width single belt counterpart. The vibration amplitude of the hub load is also reduced which is directly correlated to the structural borne noise. Comparing to the related dual phase helical tooth belt, the dual phase twin belt is superior in torque carrying capability; and is on par with noise mitigation.
2017-06-05
Technical Paper
2017-01-1778
Enrico Galvagno, Antonio Tota, Mauro Velardocchia, Alessandro Vigliani
Abstract This paper explores the potentiality of reducing noise and vibration of a vehicle transmission thanks to powertrain control integration with active braking. Due to external disturbances, coming from the driver, e.g. during tip-in / tip-out maneuvers, or from the road, e.g. crossing a speed bump or driving on a rough road, the torsional backlashes between transmission rotating components (gears, synchronizers, splines, CV joints), may lead to NVH issues known as clonk. This study initially focuses on the positive effect on transmission NVH performance of a concurrent application of a braking torque at the driving wheels and of an engine torque increase during these maneuvers; then a powertrain/brake integrated control strategy is proposed. The braking system is activated in advance with respect to the perturbation and it is deactivated immediately after to minimize losses.
2017-06-05
Journal Article
2017-01-1777
Thomas Wellmann, Kiran Govindswamy, Dean Tomazic
Abstract The automotive industry continues to develop new technologies aimed at reducing overall vehicle level fuel consumption. Powertrain and driveline related technologies will play a key role in helping OEM’s meet fleet CO2 reduction targets for 2025 and beyond. Specifically, use of technologies such as downsized engines, idle start-stop systems, aggressive torque converter lock-up schedules, wide-ratio spread transmissions, and electrified propulsion systems are vital towards meeting aggressive fuel economy targets. Judicious combinations of such powertrain and driveline technology packages in conjunction with measures such as the use of low rolling resistance tires and vehicle lightweighting will be required to meet future OEM fleet CO2 targets. Many of the technologies needed for meeting the fuel economy and CO2 targets come with unique NVH challenges. In order to ensure customer acceptance of new vehicles, it is imperative that these NVH challenges be understood and solved.
2017-06-05
Technical Paper
2017-01-1780
Yong Xu
Abstract In the design or match process of vehicle powertrain system, gearbox rattle is a common NVH problem which directly affects passengers’ judgment on the quality and performance of vehicle. During the development process of a passenger car, prototype vehicles have serious gear rattle problem. In order to efficiently and fundamentally control this problem, this work first studied the characteristics and mechanisms of the gearbox rattle. The study results revealed that the torsional vibration of powertrain system was the root cause of gearbox rattle. Then a simulation model of the full vehicle was built with the aid of Simulink® toolbox, which is a graphical extension to MATLAB® for modeling and simulation of variety of systems. With this model, the sensitivity analysis and parametrical optimization were performed, and the simulation results indicated that the dual-mass flywheel (DMF) was the best measure to control the rattle.
2017-06-05
Technical Paper
2017-01-1779
Xianwu Yang, Jian Pang, Lanjun Wang, Xiong Tian, Yu Tang
Abstract With drastically reduction of engine noise, the gear rattle noise generated by the impact between neutral gears inside transmission can be much easily perceived. It is well known that the torsional mode has a direct relationship with the transmission gear rattle noise. This paper establishes a torsional model of a front wheel drive automotive drivetrain, including clutch system, transmission box and equivalent load of a full vehicle, in AMESim software. The experimental engine speed fluctuations at different gears are used to excite the torsional model. The influences of several parameters, including flywheel inertia, clutch stiffness, clutch hysteresis and drive shaft stiffness, on the 2nd order (major engine firing order for a 4-cylinder-4-stroke engine) torsional resonant frequency and the 2nd order torsional resonant peak of the transmission input shaft are analyzed by changing them alternatively.
2017-06-05
Technical Paper
2017-01-1782
Jobin Puthuparampil, Henry Pong, Pierre Sullivan
Abstract Large-scale emergency or off-grid power generation is typically achieved through diesel or natural gas generators. To meet governmental emission requirements, emission control systems (ECS) are required. In operation, effective control over the generator’s acoustic emission is also necessary, and can be accomplished within the ECS system. Plug flow mufflers are commonly used, as they provide a sufficient level of noise attenuation in a compact structure. The key design parameter is the transmission loss of the muffler, as this dictates the level of attenuation at a given frequency. This work implements an analytically decoupled solution, using multiple perforate impedance models, through the transfer matrix method (TMM) to predict the transmission loss based on the muffler geometry. An equivalent finite element model is implemented for numerical simulation. The analytical results and numerical results are then evaluated against experimental data from literature.
2017-06-05
Technical Paper
2017-01-1787
Jan Biermann, Adrien Mann, Barbara Neuhierl, Min-Suk Kim
Abstract Over the past decades, interior noise from wind noise or engine noise have been significantly reduced by leveraging improvements of both the overall vehicle design and of sound package. Consequently, noise sources originating from HVAC systems (Heat Ventilation and Air Conditioning), fans or exhaust systems are becoming more relevant for perceived quality and passenger comfort. This study focuses on HVAC systems and discusses a Flow-Induced Noise Detection Contributions (FIND Contributions) numerical method enabling the identification of the flow-induced noise sources inside and around HVAC systems. This methodology is based on the post-processing of unsteady flow results obtained using Lattice Boltzmann based Method (LBM) Computational Fluid Dynamics (CFD) simulations combined with LBM-simulated Acoustic Transfer Functions (ATF) between the position of the sources inside the system and the passenger’s ears.
2017-06-05
Technical Paper
2017-01-1791
David Neihguk, Shreyas Fulkar
Abstract Parametric model of a production hybrid (made up of reactive and dissipative elements) muffler for tractor engine is developed to compute the acoustic Transmission Loss (TL). The objective is to simplify complex muffler acoustic simulations without any loss of accuracy, robustness and usability so that it is accessible to all product development engineers and designers. The parametric model is a 3D Finite Element Method (FEM) based built in COMSOL model builder which is then converted into a user-friendly application (App) using COMSOL App builder. The uniqueness of the App lies in its ability to handle not only wide range of parametric variations but also variations in the physics and boundary conditions. This enables designers to explore various design options in the early design phase without the need to have deep expertise in a specific simulation tool nor in numerical acoustic modeling.
2017-06-05
Technical Paper
2017-01-1792
Magnus Knutsson, Erik Kjellson, Rodney Glover, Hans Boden
Abstract Increased demands for reduction of fuel consumption and CO2 emissions are driven by the global warming. To meet these challenges with respect to the passenger car segment the strategy of utilizing IC-engine downsizing has shown to be effective. In order to additionally meet requirements for high power and torque output supercharging is required. This can be realized using e.g. turbo-chargers, roots blowers or a combination of several such devices for the highest specific power segment. Both turbo-chargers and roots blowers can be strong sources of sound depending on the operating conditions and extensive NVH abatements such as resonators and encapsulation might be required to achieve superior vehicle NVH. For an efficient resonator tuning process in-duct acoustic source data is required. No published studies exists that describe how the gas exchange process for roots blowers can be described by acoustic sources in the frequency domain.
2017-06-05
Technical Paper
2017-01-1789
Rafael Veloso, Robert Fairbrother, Yasser Elnemr
Abstract The acoustics of automotive intake and exhaust systems is typically modeled using linear acoustics or gas-dynamics simulation. These approaches are preferred during basic sound design in the early development stages due to their computational efficiency compared to complex 3D CFD and FEM solutions. The linear acoustic method reduces the component being modelled to an equivalent acoustic two-port transfer matrix which describes the acoustic characteristic of the muffler. Recently this method was used to create more detailed and more accurate models based on a network of 3D cells. As the typical automotive muffler includes perforated elements and sound absorptive material, this paper demonstrates the extension of the 3D linear acoustic network description of a muffler to include the aforementioned elements. The proposed method was then validated against experimental results from muffler systems with perforated elements and sound absorptive material.
2017-06-05
Technical Paper
2017-01-1790
Vinayak H. Patil, Ravi Kumar Sara, T. R. Milind, Rodney C. Glover
Abstract Vehicle noise emission requirements are becoming more stringent each passing year. Pass-by noise requirement for passenger vehicles is now 74 dB (A) in some parts of the world. The common focus areas for noise treatment in the vehicle are primarily on three sub-systems i.e., engine compartment, exhaust systems and power train systems. Down- sizing and down- speeding of engines, without compromising on power output, has meant use of boosting technologies that have produced challenges in order to design low-noise intake systems which minimize losses and also meet today’s vehicle emission regulations. In a boosted system, there are a variety of potential noise sources in the intake system. Thus an understanding of the noise source strength in each component of the intake system is needed. One such boosting system consists of Turbo-Super configuration with various components, including an air box, supercharger, an outlet manifold, and an intercooler.
2017-06-05
Journal Article
2017-01-1796
Rick D. Dehner, Ahmet Selamet, Michael Steiger, Keith Miazgowicz, Ahsanul Karim
Abstract Ported shroud compressor covers recirculate low momentum air near the inducer blade tips, and the use of these devices has traditionally been confined to extending the low-flow operating region at elevated rotational speeds for compressors on compression-ignition (CI) engines. Implementation of ported shrouds on compressors for spark-ignition (SI) engines has been generally avoided due to operation at pressure ratios below the region where ported shrouds improve low-flow range, the slight efficiency penalty, and the perception of increased noise. The present study provides an experimental investigation of performance and acoustics for a SI engine turbocharger compressor both with a ported shroud and without (baseline). The objective of implementing the ported shroud was to reduce mid-flow range broadband whoosh noise of the baseline compressor over 4-12 kHz.
2017-06-05
Technical Paper
2017-01-1795
Ahmad Abosrea, Tamer Elnady
Abstract Flow-generated noise has recently received a lot of attention within the process of designing exhaust and intake systems. Flow-generated noise can limit the amount of sound reduction a muffler can introduce inside ducts. This is more important in the modern system design where mufflers are compact and the flow speeds become higher in different sections inside the muffler. In this paper, three measurement techniques are used to measure the flow-generated noise from a duct element. The first is based on calculating the sound power levels inside a reverberation room according to ISO 3741. The radiated noise is measured from the muffler body as a source of noise, then from the tail pipe as an active one-port source. The second is based on sound power measurements inside the ducts using the active two-port theory. The third is measuring the sound pressure radiation inside an anechoic room.
2017-06-05
Technical Paper
2017-01-1794
William Seldon, Jamie Hamilton, Jared Cromas, Daniel Schimmel
Abstract As regulations become increasingly stringent and customer expectations of vehicle refinement increase, the accurate control and prediction of induction system airborne acoustics are a critical factor in creating a vehicle that wins in the marketplace. The goal of this project was to improve the predicative accuracy of a 1-D GT-power engine and induction model and to update internal best practices for modeling. The paper will explore the details of an induction focused correlation project that was performed on a spark ignition turbocharged inline four-cylinder engine. This paper and SAE paper “Experimental GT-POWER Correlation Techniques and Best Practices” share similar abstracts and introductions; however, they were split for readability and to keep the focus on a single a single subsystem. This paper compares 1D GT-Power engine air induction system (AIS) sound predictions with chassis dyno experimental measurements during a fixed gear, full-load speed sweep.
2017-06-05
Technical Paper
2017-01-1793
William Seldon, Amer Shoeb, Daniel Schimmel, Jared Cromas
Abstract As regulations become increasingly stringent and customer expectations of vehicle refinement increase, the accurate control and prediction of exhaust system airborne acoustics are a critical factor in creating a vehicle that wins in the marketplace. The goal of this project was to improve the predicative accuracy of the GT-power engine and exhaust model and to update internal best practices for modeling. This paper will explore the details of an exhaust focused correlation project that was performed on a naturally aspirated spark ignition eight-cylinder engine. This paper and SAE paper “Experimental GT-POWER Correlation Techniques and Best Practices Low Frequency Acoustic Modeling of the Intake System of a Turbocharged Engine” share similar abstracts and introductions; however, they were split for readability and to keep the focus on a single a single subsystem.
2017-06-05
Technical Paper
2017-01-1800
Robert White
Abstract Several analytical tools exist for estimating a driveshaft’s critical speed, from simple elementary beam theory to sophisticated FEA models. Ultimately, nothing is better than a test, because no one will argue with the outcome from a well-designed measurement. Impact response measurements are easy, but they tend to over predict the critical speed. A test which sweeps the shaft speed up until failure is telling, but the speed causing failure is strongly dependent on even small amounts of variation in rotor unbalance. Waterfall plots of shaft displacement measurements offer the best indication of critical speed, however sometimes the resonance isn’t unmistakable or multiple resonances exist, making the critical speed unclear. A method less susceptible to system variation is offered here, fitting shaft orbit measurements to the theoretical single degree of freedom equation.
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