Rotating shaft vibration monitor

A non-contact system and method for measuring torsional wave propagation along a rotating shaft as well as static angular deflection due to a constant applied torque. The relative twist angle between two cross-sectional portions of the rotating shaft separated by a predetermined distance is detected over time and then the axial variation of torque along the rotating shaft is determined as a function of relative twist angle. Detection is carried out electro-optically using a pair of photodetectors and bands of reflective marks adhered on the circumferential surfaces of the two cross-sectional portions. Axial variation of torque is determined by determining variations in the phase difference between pulse trains output by the photodetectors.

FIELD OF THE INVENTION 
This invention relates generally to the monitoring of torque in rotating 
machinery. In particular, the method and apparatus of the invention are 
useful in detecting and measuring torsional vibrations in pump shafts. 
BACKGROUND OF THE INVENTION 
Shaft torque in rotating machinery is an important quantity needed for the 
design and monitoring of a system. In the past, the torque was calculated 
using approximate methods on the basis of rough measurement of the 
produced or consumed power. These methods provided approximate estimates 
of the average torque value under constant pump conditions, but were 
insufficient to provide the local variation of the torque along the shaft 
(i.e., torsional wave propagation) during transient conditions, e.g., when 
the load, torque or speed of rotation of the shaft is changed. Recently, 
instruments such as torque meters have been used to measure the shaft 
torque. 
The majority of conventional torque meters use either of two principal 
methods for data collection. One type is based on strain gauges placed on 
the shaft with data collection through slip rings. The other type uses 
strain gauges on the shaft in conjunction with radio or magnetic 
transmitters for the data collection. Both of these methods entail 
significant alteration of the shaft and require open space on the shaft 
which is normally unavailable. Some torque meters require cutting of the 
shaft and insertion of the transducers. Electrical noise can be induced in 
the data transmission, leading to unacceptable errors. The insertion of 
mechanical sensing and data collection may itself reduce the useful life 
of the rotating part due to additional weight, misalignment and wear. 
Therefore, a need exists for a less complicated, more accurate method of 
measuring the torque in rotating shafts which does not need a long free 
span and can be applied at different locations along the shaft. 
Torsional vibrations in shafts of circular cross section are described by 
methods that are known in continuum mechanics. In particular, the angle 
.theta. through which a general cross section rotates about its 
equilibrium position obeys the differential equation: 
##EQU1## 
where J(z) is the polar moment of inertia of the circular shaft (either 
hollow or solid), E.sub.s is the shear modulus, .rho. is the density of 
the shaft, and g.sub.O is the acceleration of gravity. The torque T is 
related to the angular displacement by: 
##EQU2## 
Therefore, measurement of the angular displacement variation along the 
shaft, in general, implies deduction of the torque in the shaft. 
For shafts which have uniform or slowly varying properties, the bracketed 
term in the above wave equation is small and can be neglected. This is the 
simplest case, with solutions that are traveling waves or standing waves 
(a superposition of two traveling waves). Such solutions can be expressed 
mathematically as: 
##EQU3## 
Therefore, the vibrational torque is: 
EQU T(z,t)=-ik.theta..sub.O E.sub.s J(z) (e.sup.i(.omega.t-kz) 
.+-.e.sup.-i(.omega.t-kz)) 
whose real part is: 
EQU Re[T(z,t)]=2k.theta..sub.O E.sub.s J (z) sin(.omega.t-kz) 
The amplitude coefficient .theta..sub.O is determined by the driving 
function amplitude operating on the end of the shaft. 
These relationships imply that dynamic torque can be inferred from 
measurements of the properties of torsional wave propagation in the shaft. 
If the shaft is very nonuniform, then interpretation of the measurements 
is more complicated, yet feasible using series solutions of the more 
general differential equation. For example, a continuously variable shaft 
may possess a polar moment of inertia described by: 
##EQU4## 
The wave equation for this case becomes: 
##EQU5## 
whose solution is known in terms of tabulated infinite series, called 
zero-order Hankel functions of the second kind: 
EQU .theta.(z,t)=.theta..sub.O H.sup.(2).sub.O (kz)e.sup.i.omega.t 
If a time-varying torque of amplitude T.sub.O is applied at z=z.sub.O, the 
amplitude of the angular displacement is determined by: 
EQU T(z-z.sub.O,t)=T.sub.O e.sup.i.omega.t =kE.sub.s J.sub.O .theta..sub.O 
H.sup.(2).sub.1 (kz.sub.O)e.sup.i.omega.t 
from which we find: 
##EQU6## 
The torque in the shaft is then: 
##EQU7## 
For short-wavelength vibrations, the Hankel function becomes: 
##EQU8## 
so the torque approaches a phase-shifted traveling wave propagating toward 
positive z: 
##EQU9## 
EQU z.sub.O .ltoreq.z.ltoreq.Z.sub.max ; kZ&gt;&gt;1 
This result shows that torque is measurable as a traveling wave, even 
though the shaft is nonuniform. 
SUMMARY OF THE INVENTION 
The invention lies in a non-contact method for measuring torsional wave 
propagation along a rotating shaft as well as static angular deflection 
due to a constant applied torque. The method is carried out using an 
apparatus which is simple and easy to implement on existing or new shafts. 
The method requires no alteration of the shaft configuration and 
accurately and remotely measures the properties of torsional waves 
propagating along rotating shafts, such as those used in pumps. The method 
of the invention enables measurement of phase velocity (or wave number) as 
well as torque for propagating or standing torsional waves. 
The spatial variation of the torque in the shaft can be deduced using 
electro-optical methods. Digitized electro-optical data is employed to 
control noise and to directly measure torsional vibration frequencies and 
amplitudes without direct contact devices, such as strain gauges. 
Further, the unique properties of digital electronics systems are used to 
store, analyze and display the frequency and pulse-train data obtained 
while the shaft is turning. As a result, the frequency, wave number and 
torque can be computed and stored over a period of service time, thereby 
enabling the history of torsional shaft vibrations to be tracked and 
providing the basis for predicting potential pump failures of the system 
or changes of system characteristics. 
Thus, the invention eliminates the existing disadvantages of conventional 
torque meters and provides a means for collecting torsion data with a high 
accuracy heretofore unattainable with conventional torque measuring 
devices.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT 
The method and apparatus of the invention are described below with 
reference to a uniform shaft model for the sake of simplicity and 
convenience. However, it is understood that the same apparatus is 
applicable to a wide variety of shaft configurations. In particular, more 
sophisticated data processing and analysis methods can be employed for 
nonuniform cases. 
As shown in FIG. 1, the rotating shaft 2 has affixed to it two thin 
circumferential bands 4 and 4', separated by a distance L. Preferably the 
bands are taped or otherwise adhered on the shaft. Each band is inscribed 
with an array of parallel lines or marks circumferentially distributed 
around the periphery of the shaft when the band is mounted thereon. It is 
convenient, although not necessary, that the inscribed marks on each band 
be distributed at equal intervals and that the marks on one band line up 
with their counterparts on the other band. The lines or marks have an 
optical property different than that of the spaces between the notches or 
marks, e.g., the lines or marks are reflective and the intervening spaces 
are not reflective. 
In the preferred embodiment, each band is illuminated with a respective 
light source 5 and 5'. Light reflected from the marks of each band is 
received by a pair of photodetectors 8 and 8' by way of corresponding 
focusing lenses 6 and 6' arranged in proximity to the bands. The lenses 
are designed to focus the reflected light from the lines or marks onto the 
photodetector windows. Each photodetector outputs a respective pulse of 
electric current to channels A and B in response to impingement of a 
reflected pulse of light on its detecting surface. 
As the shaft rotates at angular velocity .OMEGA., the bands reflect light 
pulses that are separated in time by an interval: 
##EQU10## 
where R is the shaft outer radius and d is the notch or mark spacing. If 
there is no torsional displacement, then the pulses will be in phase in 
channels A and B, i.e., their time sequences overlap when displayed on a 
dual-channel oscilloscope 12. For a steady rotational speed, the outputs 
of the two channels can be synchronized and displayed as standing wave 
patterns on display 16. 
The controller 10 is a special-purpose data acquisition device, well-known 
in the art, that serves to acquire, filter and digitize the data, e.g., 
the time between pulses, so that the computer 14 can perform digital 
analysis of the pulse trains as required. As long as there is steady 
rotation without torsional vibrations, the computer function is simply to 
monitor the status of the standing pulse trains. 
In the event of simple torsional vibrations in the shaft, the respective 
wave patterns (denoting variation in twist angle .theta. over time) are no 
longer in phase. In this case, the wave patterns for the twist angle 
.theta. oscillate about the "rest" position of the wave patterns, which is 
recorded in the computer. A snapshot in time then shows the phase shift of 
the wave patterns for the two channels relative to each other and relative 
to the zero-displacement wave pattern. By sampling the vibration at either 
band location at a sufficiently high rate, the frequency of the torsional 
wave propagating down the shaft can be computed. The phase shift between 
bands provides the information necessary to compute wave number, thereby 
allowing calculation of the phase velocity of the wave, from which the 
shear modulus of the shaft can be calculated. As shown above, the 
torsional wave amplitude is directly related to the local torque and the 
geometric and material properties of the shaft, so torque can be inferred 
from the measured torsional wave amplitude. 
This point can be illustrated by a simple example. Assume that the torque 
is represented by a traveling wave propagating from left to right along 
the shaft: 
EQU T=A.sin[.phi.(t,z)] 
where the phase is a function of angular frequency .omega. and wave number 
k: 
EQU .phi.(t,z)=.omega.t-kz 
and the amplitude A is proportional to the peak angular displacement 
.theta..sub.O to be determined by measurement: 
EQU A=2kE.sub.s J(z).theta..sub.O 
Clearly, measurements of .theta..sub.O and k allow the direct calculation 
of torque amplitude, since J(z) and E.sub.s can be assumed known for any 
particular shaft. 
The frequency and wave number are given by: 
##EQU11## 
Therefore, a recording of the phase at either value of z corresponding to 
the band positions can be analyzed to obtain the frequency. Analysis of 
the phase difference between the respective data streams of the two 
channels gives k, from which the phase velocity c can be calculated in 
accordance with the equation: 
EQU c=.omega./k 
The digital data is acquired in a way to approximate the partial 
derivatives as follows: 
##EQU12## 
Here, the number of zero crossings, N, occurring in the time interval 
.DELTA.t=l..tau. at either location A or B defines the phase increment 
.delta..phi.. Thus, the phase velocity is approximated by: 
##EQU13## 
assuming that the angular velocity .OMEGA. of the shaft is constant. The 
circular frequency f is: 
##EQU14## 
The normalized, sampled data streams for channels A and B are as shown in 
FIGS. 2 and 3. The ordinate is the normalized or relative twist angle 
.theta. plotted relative to a base line representing the zero twist angle 
associated with a static torque. The abscissa in each plot is related to 
time in increments of .tau.. Channel B data is shifted slightly to the 
right of channel A data because of the phase lag generated as the 
torsional wave propagates the distance L (FIG. 1) along the shaft. This 
phase difference is denoted by [.phi.(B)=.phi.(A)] in the above equations. 
The sampling rate is sufficiently high to exclude aliasing for this 
example, so the frequency can be computed from either waveform (channel A 
or B). 
In a pure mode of vibration, the measured frequency could actually be any 
multiple of the fundamental torsion frequency. However, this redundancy 
can be eliminated on physical grounds, since the fundamental frequency can 
be approximated by analysis. If the shaft is vibrating in a superposition 
mode, involving the weighted sum of more than one normal mode, then the 
spectral character can be measured using this technique. 
The method and apparatus of the present invention can also be used to 
deduce the static deflection due to a constant applied torque, e.g., after 
transient shaft conditions have been damped over time. This is a special 
case of the more general disclosure given above, which may be important in 
specific applications. For example, if the shaft length L.sub.s is short 
and the excitation frequency is low, then the inequality: 
##EQU15## 
is satisfied. In this case, k is small compared to L.sub.s, so the angular 
displacement is given by: 
##EQU16## 
where T.sub.O is the applied torque and .theta..sub.O is a reference angle 
(that may be zero). In general, the moment of inertia can be represented 
by a power-series expansion of the form: 
##EQU17## 
for all z (distance along the shaft). The integral in the previous 
equation can then be evaluated term-by-term to obtain: 
##EQU18## 
If the shaft is uniform and homogeneous, then C.sub.O is unity and all 
other coefficients are zero in the series. For nonuniform shafts, the 
C.sub.i can be determined from the shaft configuration by measurement and 
analysis techniques that are well known. In the simplest case of a uniform 
shaft, the above power-series expansion implies that the angular twist 
increases linearly with distance from the point of the applied torque. 
Therefore, the relative displacements of the two inscribed bands on the 
shaft surface can be related to the applied torque(s) on the shaft as it 
rotates. This can be shown by inserting the equations for [J(z)].sup.-1 
and .theta.(z) into the general expression relating torque and deflection 
angle to obtain: 
##EQU19## 
where A, B and L refer to the configuration of FIG. 1. 
Since all electronic systems contain noise, the signals must be properly 
conditioned to minimize noise, especially in the mechanical vibrations 
spectrum. Means for accomplishing this are commonplace, especially in 
differential electronics systems. This allows frequency detection 
hardware, or software, to effectively and accurately measure the frequency 
of vibration. Such devices are available as standard additions to many 
commercially available digital oscilloscopes and/or computers. 
The detectable means adhered on the rotating shaft may comprise bands with 
reflective marks and nonreflective spaces therebetween or bands with 
nonreflective marks and reflective spaces therebetween. Alternative 
methods of generating the data, such as eddy-current probes, magnetic 
pick-ups, and infra-red illumination and detection, are equally suited for 
use in the invention. These and other variations and modifications of the 
disclosed preferred embodiment will be readily apparent to practitioners 
skilled in the art of position detection. All such variations and 
modifications are intended to be encompassed by the claims set forth 
hereinafter.