separation of lateral and torsional shaft vibration via a...

Separation of lateral and torsional shaft vibration via a multiple sensor Time Interval Measurement System M.W. Trethewey1, M.S. Lebold2

1 Penn State University, Department of Mechanical and Nuclear Engineering University Park, PA 16802, USA e-mail: [email protected] 2 Penn State University, Complex Systems Monitoring and Automation Department Applied Research Laboratory University Park, PA 16802, USA

Abstract A common torsional vibration sensing method for rotating equipment is the Time Interval Measurement System (TIMS). The method utilizes the time passage of discrete intervals from an incremental geometric encoder (i.e., gear) on a rotating shaft. Ideal measurement conditions consist of a constant shaft running speed, an encoder with identical segments and no transverse motion of either the shaft centerline or the encoder passage sensor. In practice, these ideal conditions are rarely achieved as the shafts of large rotating equipment can experience lateral vibration due to imbalance and/or misalignment. The lateral shaft vibration can affect the TIMS torsional encoder passage times and hence results in measurement errors. The work in this paper describes and evaluates a multiple encoder passage sensing and data processing algorithm to separate the torsion and lateral vibration. The multi-sensor TIMS method is then applied to an industrial pump to illustrate the separation capabilities of the method.

1 Introduction

Torsional vibration is important for analysis and diagnostics of rotating equipment. Applications are plentiful, including the automotive [1,2,3], and the electrical power industry [4,5,6]. As the measurement of torsional vibration on a rotating shaft is inherently more difficult than translational vibration, a significant body of work has focused on effective torsional measurement techniques. A variety of schemes have evolved including lasers [7,8], in line torque sensors [9], angular accelerometers [10] and time passage encoder based systems [11,12,13]. The time passage encoder based systems have gained popularity. The method uses a fixed angular encoding device that rotates with the shaft, such as a gear or optical rotary encoder. A transducer senses the passage of each encoder segment. Optical encoders use a light based system to sense the passage of a rotating grating. Gear type shaft encoder systems have used a number of transducers to sense the time passage of the segments, including, Hall Effect, fiber-optic reflective light intensity, inductive and capacitive sensors. The sensor output is a pulse train type signal in which the passage times vary as a function of the shaft rotation and the torsional oscillation. A digital processing method referred to as the Time Interval Measurement System (TIMS) uses a high-speed timer to capture the passage of each encoder segment. The encoder passage times are then compared to a reference and the time difference between the encoder and the reference signal is used to compute the torsional vibration. Because of the intricate instrumentation setup and computationally intensive processing, TIMS torsional vibration measurement is susceptible to artifacts and errors from; 1) reference signal corruption; 2) non uniform sampling intervals; 3) timer resolution, and 4) transverse shaft movement with respect to passage sensor. The work in this paper will discuss the effects of lateral shaft movement on TIMS computed torsional vibration time histories and spectra. The separation is based upon the geometric relationship between the lateral and torsional vibration to the captured encoder segment passage times. The relationship between

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the encoder/sensor geometry, torsional vibration and shaft transverse movement is examined for one sensor. Three passage sensors are then placed circumferentially around the encoder. Extending the relationships result in three simultaneous equations which equate the TIMS array from each sensor independently to the shaft movement in two orthogonal directions and the elastic shaft dynamic twist at an instant in time. The equations can subsequently be algebraically solved to separate the transverse and torsion shaft movement. The solution requires data processing in the following sequence: 1) the encoder passage times from the multiple sensors are each processed separately, via a traditional TIMS demodulation method. Note the incremental time bases for each sensor are different being determined by when the encoder passage segments are detected. 2) The three arrays are then resampled to place all on a common discrete time basis; 3) the simultaneous equations are solved at each time increment to estimate the shaft transverse and torsional displacement. Spectral processing methods are subsequently applied to the arrays for analysis and feature identification purposes. In the following sections, the requisite data processing and capabilities of the multi-sensor TIMS method are discussed. The method is applied to a 10 meter industrial vertical pump to illustrate practical implementation. Single sensor results are first presented showing the presence of possible corrupting artifacts. Application of the multi-sensor method show improved torsion results and estimates of lateral shaft movement in two orthogonal directions.

2 Torsional Time Interval Measurement System (TIMS)

2.1 Single Sensor TIMS

A time interval measurement system used for torsional vibration is seen in Figure 1. The transducer output is a high or low voltage depending on whether it detects “white” or “black” regions of the transducer segments. If the shaft rotates at a constant rate in the absence of torsional vibration a periodic carrier wave is produced. When torsional vibration occurs, it causes a fluctuation in the rotating shaft speed. The output signal is a carrier wave modulated by the torsional vibration. A time interval measurement approach is used to demodulate the torsional vibration from the carrier wave [14]. The technique is based on the encoder segments’ passage times with respect to a stationary transducer. A triggering sensor is used to record the times when the reference signal has a zero value with either a positive or negative slope. Note the term “zero crossing” is used for descriptive purposes. An array containing the zero crossing times (tref) may be computed by:

( ) .Nn1),s(fNnntshaft

ref ≤≤= (1)

When negative slope zero crossing detection is applied to the encoder signal an array (tencoder) indicative of respective passage times is created. The difference between the encoder and reference zero crossing times are depicted in Figure 2 and computed by equation 2.

Figure 1: Digital Time Interval torsional vibration measurement schematic.

Angular encoder

Transducer

Conditioning amplifier

Timer/Counter

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( ) ( ) ( ) (seconds), ntntnt refencoder -=∆ (2) The angular variation due to the vibration, in degrees, can then be calculated from the time difference array:

( ) ( ) ( ).reesdeg360fntn shaft∆θ = (3)

Equation 3 can be used to create a discrete array containing the torsional vibration amplitude at the respective time corresponding to the passage of each encoder segment. Since the shaft is assumed to rotate at a constant speed and use an identical segment encoder, the array is sampled on a uniform angular increment basis. With the assumption that the time interval associated with the torsional vibration is much smaller than that due to the shaft rotation, a constant time sampling interval corresponding to the angular increments is:

.Nft shaftsample =∆ (4) By combining equations 3 and 4, the shaft torsional vibration sampled with a constant interval time basis over one revolution becomes:

( )( ) ( ),nnt θθ = (5) where:

( ) .ntnt sample∆= (6)

Equation 5 produces a discrete time representative of the elastic torsional response.

2.2 Multiple Sensor TIMS

Although implementation of the measurement process in Section 2.1 is theoretically feasible, the procedure has several potential problems when applied to mechanical equipment. Potential errors result

Figure 2: Time passage sensor signal in relation to the reference.

DYNAMICS OF ROTATING MACHINERY 1375

from differences in the implementation than the assumptions used in the derivation. The derivation assumes a constant shaft running speed, identical equally spaced encoder segments, a stationary sensor and a shaft rotating strictly about its center point [11, 12, 15, 16]. In practice many of the assumptions are violated and introduce erroneous artifacts into the TIMS computed torsional signal. This can be particularly true on large rotating equipment when imbalance can induce significant lateral shaft bending displacements. The artifact of the lateral shaft movement on the torsional signal is illustrated in the following development. Consider an encoder with identical increments rotating at a constant speed as depicted in Figure 3. Whenever a leading edge of the encoder passes the sensor the time is recorded. If the shaft center is stationary only the rotational movement affects the recorded passage times. The geometric depiction of one leading edge in relation to the sensor is shown in Figure 4. Figure 5 depicts the geometry of an encoder leading edge with a lateral shaft movement (x). The shaft movement prevents the leading edge from being correctly triggered. An incremental rotation, ΔΘ, is required to position the leading edge near the sensor to trigger the time passage recording as depicted in Figure 6. When the torsional response is computed it contains a superposition of the actual torsion, Θ, and an artifact induced from the shaft movement in the x direction, ΔΘx.

𝜃� = 𝜃 + ∆𝜃𝑥 (7)

Figure 3: Depiction of toothed encoder and time passage sensor.

Figure 4: Geometric depiction of an encoder leading edge in relation to sensor location.


Figure 7: Geometry relationship between lateral shaft movement, x, and induced torsional artifact ΔΘ.

Figure 5: Geometric depiction of an encoder leading edge with lateral shaft movement in relation to sensor location.

Figure 6: Geometry relationships of encoder leading edge trigger with shaft movement.


Timer/Counter

Signal Amplifier & Analog/TTL Converter

Hall Effect transducer

Hall Effect transducer

Toothed Encoder Wheel

Figure 8. Instrumentation for 3 three probe torsion and translational vibration measurement.

Figure 7 illustrates the geometric relationship between the sensor mounting angle (α), shaft vertical movement (x), the trigger arifact torsion (ΔΘ), the encoder radius (r) and the distance between the projection of the shaft movement perpendicular to the sensor (Δεx). The geometric relationship between the variables is:

∆𝜀𝑥 = 𝑥 sin𝛼 = 𝑟 sin∆𝜃𝑥 (8) The approximation (sin β = β) can be used since ΔΘ is small yielding:

𝑥 sin𝛼 = 𝑟 ∆𝜃𝑥 (9) Solving for ΔΘx:

∆𝜃𝑥 = 𝑥𝑟

sin𝛼 (10)

As a positive motion in the x direction induces a retardation of the trigger time it is noted as a positive angle. A similar development can be applied for simultaneous shaft movement in both orthogonal directions x and y. For the shaft sensor geometry in Figure 7, a positive shaft movement in the y direction produces a premature leading edge passage time. Assembling the shaft movement torsional artifacts with equation 7, produces equation 11 which relates the TIMS computed torsional response to the actual torsional response and shaft lateral movement in two orthogonal directions at an instant in time (ti):

𝜃�(𝑡𝑖) = 𝜃(𝑡𝑖) − 𝑥 (𝑡𝑖)𝑟

sin𝛼 + 𝑦 (𝑡𝑖)𝑟

cos𝛼 (11)

θ(ti) is desired as the correputing effects induced from the transverse shaft movement have been removed. Consider the instrumentation implementation in Figure 8 which has three sensors located around the circumference of an encoder. Extending the previous derivation to the three sensors an equation similar to equation 11 can be developed producing three equations and three unknowns.

�𝜃�1(𝑡𝑛)𝜃�2(𝑡𝑛)𝜃�2(𝑡𝑛)

� = [𝑇]�𝜃(𝑡𝑛)𝑥(𝑡𝑛)𝑦(𝑡𝑛)

� (12)


0 0.005 0.01 0.015 0.02 0.025-0.02

0

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Sen

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0 0.005 0.01 0.015 0.02 0.025-0.02

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𝜃�1(𝑡𝑛) is the computed the response from transducer 1 by the TIMS algorithm, likewise for 𝜃�2(𝑡𝑛) and 𝜃�3(𝑡𝑛) respectively. [T] is a transformation matrix that relates the sensor positions to the encoder leading edge passage detection with lateral movement based upon equation 11 for each respective sensor, n represents the time increment index. The solution of equation 12 requires a presriptive set of measurement and preprocessing steps.

1. The TIMS torsional vibration method is applied to time passage arrays acquired from each of the three transducers; 𝜃�1(𝑡𝑖), 𝜃�2(𝑡𝑗) and 𝜃�3(𝑡𝑘). The three hypothetical responses are depicted in Figure 9. The solid lines represent hypothetical responses, while the discrete array marks represent those calculated by the passage of the encoder segments. Note the discrete time arrays for each response are different and irregular, albeit exaggerated in the figure for visualization purposes. The time array differences result from; 1) positioning of the passage sensors with respect to the encoder segments; 2) non-uniformity of the encoder segments; 3) the time variant torsional dynamics, and; 4) transverse dynamic responses.

2. To be able to utilize equation 12 to separate the lateral movement from the individual probe TIMS arrays they need to be on the identical time basis. Each of the probe arrays ( 𝜃�1(𝑡𝑖), 𝜃�2(𝑡𝑗) and 𝜃�3(𝑡𝑘)) are resampled to a common constant interval time basis. The resampled arrays are depicted in Figure 10.

3. Equation 12 is formed and solved at each time increment, resulting in arrays for the torsional response, θ(t), translational response, x(t), in the x direction as defined in [T], and the translational response, y(t), in the y direction as defined in [T].

Figure 9. Hypothetical TIMS responses calculated from three transducers located around a shaft line experiencing torsional and translational motion.


3. Application of TIMS to Industrial Rotating Equipment

3.1 Sensor Installation

The multi-sensor hardware instrumentation package is depicted in Figure 8. In large industrial retrofit applications the encoder is typically a toothed encoder installed at a convenient location along the shaft line. The sensing passage sensors must be located near the encoder. The implementation of both the encoder and sensors are usually specifically designed for the equipment on an individual basis. An example of an installation on a 5 meter vertical pump is shown in Figure 11. The toothed gear encoder and the Hall Effect time passage transducers are visible. The transducer mounts are designed to be adjustable, yet stiff in the tangential direction. The multiple sensors around the encoder are apparent.

Figure 12 shows a transducer bracket constructed to hold a Hall Effect transducer on a 10 meter nuclear reactor coolant pump. The attachment locations to the pump frame were very limited and dictated the cantilever type sensor bracket. Three sensors using the cantilever bracket were installed at 120 degree increments around the circumference of the toothed encoder.

Recall, fundamental premises of the incremental time sensing approach to measure torsional vibration are that; 1) transducer is remains stationary, and 2) the shaft rotates about its centerline without lateral movement. Given the scale of this rotating equipment lateral shaft motion is to be expected and bracket movement possible. Violating either of these assumptions can cause errors in the incremental time passage sensing and subsequently errors in the computed torsional vibration.

3.2 Torsional and Lateral Motion Separation

Data was acquired from the 10 meter nuclear reactor coolant pump and processed using the TIMS algorithm. The spectrum from one sensor is shown in Figure 13. Multiple harmonic and potential resonant

0 0.005 0.01 0.015 0.02 0.025-0.02

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0 0.005 0.01 0.015 0.02 0.025-0.02

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Figure 10. Hypothetical TIMS responses in Figure 9 resampled to a constant interval time basis.


Figure 11. Torsional vibration instrumentation on a 5 meter vertical pump.

Figure 12. Bracket for torsional vibration sensor on a large electric motor driven pump.


Figure 14. Torsional and translational spectra from three probe processing algorithm; A) Torsion; B) x-axis translation; c) y-axis translation.

0 50 100 150 200 250 300 350 400 450 50010

-6

10-5

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10-3 TVA RCP 1 4, Theta

Frequency (Hz)

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peak

4Mar05 20:36 (S17R1)9Mar05 1:09 (S25R1)14Mar05 6:51 (S35R1)19Mar05 12:33 (S45R1)22Mar05 3:24 (S50R1)

Figure 13. Torsional vibration from one sensor on a constant speed vertical pump.


responses are clearly apparent. It is not possible by visual inspection to separate the actual torsional response from the artifact induced by lateral shaft or bracket movement.

The data from the three probes was procesed simultaneously using the multiple proble algorithm described in Section 2.2. The separated torsional and the orthogonal lateral motion spectra are shown in Figure 14. The torsional resonances at approximately 35 and 130 Hz are apparent in Figure 14A and not the lateral responses in Figures 14B or 14C. High amplitude lateral responses are apprent at 20 Hz and around 100 Hz. The separation capability of the algorithm becomes apparent when comparing to the single probe result in Figure 13.

The coordinate separation process greatly enhances the respective vibration features associated with the respective motion measurement direction. By further combining with the following information:

1. Torsional finite element model shaft line analysis; 2. Bump tests on the transducer mounting brackets; 3. Operational vibration, including shaft proximity sensor spectral data during nominal operation; 4. Pump and frame natural frequencies;

additional insight into the dynamic response of the electric motor driven pump can be found.

Inspection of the torsional spectrum in Figure 14A in conjunction with a finite element model results allows modes to be identified. The responses of modes 1 (30 Hz) and mode 3 (134) Hz are significant and well defined. However, the response of mode 2 (46 Hz) is less obvious. Also apparent are some unidentified responses (e.g., around 80 Hz), along with ever-present power mains components at 60 and 120 Hz.

Integration of this information with the intermediate spectral results (i.e., cross probe coherence functions, order spectra, etc.) allowed for a more comprehensive understanding of the captured vibration features. As an example, Figure 15 shows a typical multi probe TIMS computed lateral response, with the identification of several features marked resulting from the analysis. The effects of the transducer mount resonant motion can be more readily identified. Also the effects of both the shaft lateral movement and the pump frame motion are separated and identified. Without the multi-channel processing these responses

Figure 15. Translational spectra illustrating the identification of response features.


would only be artifacts contained in the computed TIM torsional response. Obviously, their appearance in the torsional spectrum would cloud the ability to interpret the results.

4 Summary

Torsional vibration is inherently more difficult to measure than translational motion; it is susceptible to a number of measurement and processing issues. High amplitude order content introduced from measurement and/or processing of torsional vibration signals can obscure the fixed frequency components. Corrupting order content can be produced in time interval measurement schemes from a variety of sources. Corrective algorithms have been implemented to alleviate transducer and speed related computational errors.

Line power mains harmonics and speed controller artifacts can make torsional vibration measurement with electrical motor drives difficult. The dynamic response of sensor mounting hardware, the shaft line and framework can add further corruption and artifacts. The measurement and processing of multi-channel TIMS sensors can be used to separate two dimensional lateral movement and torsional responses. While the processing is computationally intensive, the results can significantly enhance the ability to separate the respective directional dependent features.

Although torsional vibration is more difficult from both an instrumentation and computational perspective, the data is very useful in understanding the dynamics and responses of rotating equipment. Continued improvements to instrumentation, installation methods and processing algorithms should improve the capabilities and implementation costs.

Acknowledgements

This work was supported by the Electric Power Research Institute (EPRI Contract EP-P9801/C4961). The content of the information does not necessarily reflect the position or policy of the EPRI, and no official endorsement should be inferred.

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