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Monitoring automatic machines
Mechanical Systems and Signal Processing (1992) 6(6), 517-534
MONITORING G.
AUTOMATIC MACHINES
DALPIAZ
AND
A.
MAGGIORE
LXZmrtimento di Zngegneria delle Costruzioni Meccaniche, Nucleari, Aeronautiche e di Metallurgia University of Bologna, Italy
(Received 7 August 1991, accepted 13 October 1991) In the context of full automation of manufacturing processes, it is important to develop predictive maintenance methodologies in every industrial field. This paper presents experimental tests carried out on some components of high-performance automatic machines. The aim is to assess and compare the effectiveness and reliability of different condition monitoring methodologies based on vibration analysis. A methodology based on recovery of the acceleration pattern of the final actuator of the mechanism under consideration seems to be particularly promising. It allows the working conditions to be judged, directly comparing the recovered output member acceleration with the functional requirements which, for this kind of mechanism, are mainly relative to the motion law. A lumped parameter model of the same mechanism is presented and its application for diagnostic purposes is shown.
1. INTRODUCTION
In-service condition monitoring is being applied to almost every field of industry. In many factories it is increasingly necessary to ensure the possibility of correct machine operation without continuous human supervision. A lot of plant and machine components are currently monitored by means of ever-improving but well-understood techniques. Wear debris and vibration analysis are nowadays the most widely employed techniques. Vibration analysis is used satisfactorily for many mechanical components. The vibration of the case is generally considered for monitoring purposes. Several processing and analysis techniques are applied for different components and aims. A variety of different parameters is usually employed for characterising machine health condition (e.g. RMS, kurtosis, the amplitude of frequency components characterising machine troubles). However, to correlate the value of case vibration parameters to machine condition may be difficult. In fact, the vibration limit levels relative to generic classes of machines and generic conditions are often the only allowable data (e.g. the Vibration Severity Criteria after IS0 IS 2372) [ 11. If more reliable alarm levels are needed and relationships between vibration parameters and specific faults are to be established, preliminary tests must be performed in production, following the development of actual faults or introducing artificial faults. The first case requires lengthy testing, the second might not give realistic results. Further difficulties can arise from result scattering due to individual differences between nominally equal machines: the same health condition may produce different case vibration levels. Most of the above-mentioned difficulties in assessing alarm levels and in correlating faults and case vibration parameters are due to the fact that case vibration is affected by the transfer function between the mechanism components where forces are applied and 517 OSSS-3270/92/060517+18 $08.00/O
@ 1992 Academic press Limited
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the transducer’s position on the case. In fact, the vibration sources are in the mechanism, whose dynamic behaviour is the result of the interaction between geometry, dynamic properties, applied forces and faults. As a consequence of machine working, forces are transmitted to the case and excite the case structure, whose response depends on its dynamic properties. Thus the effect of a transfer function between mechanism components and case transducers should be taken into account. Since wear and faults directly affect the mechanism motion and load, it would be more suitable to monitor these quantities rather than case vibration because the transfer function might partially or completely hide information about faults. Information about mechanism motion and load is of greater interest than case vibration: motion may be related to the machine function, as in the case of a cam mechanism designed to achieve a certain motion law; knowledge of overload due to the actual dynamic behaviour is important for safety. Therefore, monitoring mechanism motion or load permits us to judge whether they are acceptable for machine function and safety; thus preliminary tests in order to establish alarm levels are not required and individual differences in case response between nominally equal machines are without effect. In spite of the above-mentioned advantages, monitoring mechanism functional quantities (motion and load) is generally possible during preliminary tests by the customer, not in production. It is therefore necessary to obtain the measurements of interest from vibration signals picked up at the case. This can be carried out by experimentally estimating and storing the transfer functions which relate the considered quantities and the case vibration at the accelerometer locations. This operation has to be done, once and for all time, at the installation stage. In production the stored transfer functions are used as filter functions in order to recover the desired functional quantities from the case accelerometer signals [2-41. A mathematical model of a mechanism is a very useful tool in aiding understanding of the mechanism’s dynamic behaviour. It allows the design and functional parameters of a mechanism to be related to the dynamic behaviour [S]. Such a model may be used to identify vibration sources and study the connection between possible faults and alterations in dynamic behaviour. Since a model describes the mechanism behaviour in terms of functional quantities (e.g. motion of members, internal forces), it would be particularly useful to associate the model to the above-outlined recovery procedure [4,6]. In fact, when changes in recovered functional quantities are detected, the model might be used to diagnose which fault had occured and it might also forecast which effects are to be expected from the monitored functional quantities should different faults occur. This paper deals with vibration analysis methods applied to an automatic machine in order to monitor the components of the cam and other mechanisms for reciprocal or intermittent motion. The most common fault developing in this kind of mechanism is wear on the cam surface or in the kinematic pairs of the follower system, with a consequent increase in backlashes. This produces more intense internal shocks and stronger deviation of the motion law from the theoretical pattern, with higher accelerations and dynamic overloads. First, the main results obtained from the application of commonly-used vibration analysis techniques are reported and discussed. Secondly, the model of one particular cam mechanism is described, its use for understanding the dynamic behaviour is outlined, and its results are compared with the experimental ones in order to prove its capability in forecasting the effects of backlash increase. Finally, application of the filter function method to the same mechanism is illustrated, its capability in recovering the output member motion from the case vibration signal experimentally is shown, and its advantages in assessing the actual function condition are considered.
MONITORING
2. APPLICATION 2.1.
EXPERIMENTAL
AUTOMATIC
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automatic machine composed of subassemblies, each containing a cam or other mechanism for reciprocal or intermittent motion was investigated. Tests were performed on a number of different subassemblies. In general, the following results refer to all the tests, but are illustrated with reference to a particular subassembly, also considered for modeling (section 3) and for the advanced monitoring procedure (section 4). The mechanism is shown schematically in Fig. 1 and is made up of a cylindrical cam and a follower system composed of two rollers, a lever, a small connecting rod and a translating output member. An
,Connecting
rod 7
r’Tronsloting
member
Y3
Figure 1. Mechanism scheme.
The only faults considered are inherent to mechanisms for reciprocal or intermittent motion, e.g. wear on cam surface, increased backlash in follower bearings. Faults in transmission components, such as gears, shaft bearings, etc., were not considered, because these components are usually over-dimensioned and rarely, if ever, develop or cause faults. Tests were perfomed by varying working parameters (e.g. speed) and health conditions, as both natural and artificial faults were considered. In-field monitoring, however, has not yet been achieved. In the case of the mechanism in Fig. 1, tests were carried out at different cam speeds (diagrams presented in this paper refer to the maximum speed, 500 rpm), both in sound conditions and with artificially increased backlash between cam and rollers. Though, in the latter condition, the backlash was about four times greater than in the first, it was still judged acceptable for production, but the subassembly needs to be inspected more frequently. Accelerometers and acoustic emission transducers were mounted on the case of each subassembly; the basic experimental equipment is illustrated in Fig. 2; the acceleration of the output member was not taken into consideration in this first research phase. Accelerometer signals were analysed in the time, frequency and amplitude domains. Some of the basic analyses were: Time Synchronous Average (TSA) and its Fourier Transform (average was carried out with the help of a one-per-cam revolution tachometer signal collected using an inductive proximity probe); Power Spectral Density (PSD); Amplitude Probability Density ( APD), also used to evaluate some amplitude statistical parameters, such as standard deviation and kurtosis. Signals from acoustic emission (AE) transducers were first demodulated, i.e. low frequency content was extracted in order to obtain signals in a frequency range of mechanical significance. Demodulated acoustic emission (DAE) signals were then analysed in a similar manner to the accelerometer signals..
520
G. DALPIAZ AND A. MAGGIORE Accelerometer (outout member)
4
Accelerometer (case)
A.E. transducer (case)
53
53
Amplifier
’Peripheral equipment
I
Figure 2. Experimental equipment.
2.2. RESULTS AE is widely used for monitoring purposes, e.g. in the field of manufacturing processes. In the present case no satisfactory results were obtained by this technique; as a matter of fact, variations in the DAE signal with mechanism health conditions, if any, were never clearly distinguishable, so it was impossible to find a consistent relationship between them. The TSA was able to detect a fault from the accelerometer signal but in general with less sensitivity and more difficulty in interpretation than other options. It can be useful to diagnose a fault, localising it in the cycle of motion of the mechanism, but it is not recommended as a general health monitoring technique for the mechanisms under consideration. The APD proved to correlate quite well to mechanism health conditions. Standard deviation seemed to be a possible pointer for condition monitoring, even if not very sensitive and not always reliable; kurtosis did not appear to correlate to health conditions. PSD was the clearest ad most reliable option for monitoring purposes. The drawback is some difficulty in correlating the PSD pattern with the machine health condition. However, a particularly high sensitivity to machine condition was observed in some frequency ranges. These depend on the type of subassembly and the measurement point on the case, and are not affected by cam speed. Thus they correspond to resonance frequencies of the mechanism or of the case structure. The mean value of the PSD in the highest amplitude range was then evaluated. It was found to discriminate between intact and faulty condition when the same individual mechanism was considered. As an example, Fig. 3 compares the PSDs measured on the subassembly of Fig. 1 for sound condition (a) and after the bac&ash between cam and follower was artificially increased (b). A clear-cut difference between the two conditions is observed in the range 130 to 220 Hz. The PSD mean value in the same range increases from 0.05 to O-27 m’/s3. Teats carried out on some nominally equal machines, working in production for about the same period, have shown that a subassembly in a particular machine has a PSD signature which is quite stable in time and, as previously mentioned, consistent with mechanism health conditions. Subassemblies of the same type in dierent machines show resonances in the same ranges, but of rather different signature and amplitude; thus health
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(Hz)
Figure 3. Power spectral density of case acceleration; 18 averages. Cam revolution speed 500 rpm. (a) Sound conditions. (b) Artificially increased backlash between cam and follower.
conditions can be assessed only by comparing PSDs of the same machine and not by considering absolute limit values. As a further result, a transducer mounted on a particular subassembly is, in practice, sensitive only to the condition of that subassembly, not to the faults developing in another.
3. MODELING
3.1.
DESCRIPTION
OF THE
THE MECHANISM
MODEL
A mathematical model of the mechanism of the particular subassembly of Fig. 1 was built in order to study its dynamic behaviour for diagnostic purposes. It is a lumped parameter model [Fig. 4(a)] with 3 dof, obtained by applying the component element method [7,8]. In order to simulate the mechanism’s behaviour, the model takes nonlinearities into account, namely backlashes, stiffness dependence on the position of the mechanism and Coulomb friction in the prismatic pair. The model concerns only the cam mechanism, not the mechanical transmission driving the cam; in fact the transmission influence can be neglected and due to its high inertia the cam assumed to rotate at constant angular speed. The input is then the motion imposed by the cam, the output the motion of the extremity of the translating member. With reference to Figs 1 and 4(a), the co-ordinates z, and z, , taken on the arc of radius R,,represent respectively, the displacement imposed by the cam motion and the actual displacement of the lower extremity of the lever. Irrother words, z. is the known excitation and would be equal to the displacement imposed on the lower lever extremity if cam-roller backlash and member compliance were neglected. It is computed as a function of cam rotation (or time, as cam speed is constant), on the basis of the profile data with the addition of small random deviations, accounting for machining errors on the cam surface. The co-ordinate x2 is taken on the arc of radius R2 and represents the displacement of the upper extremity of the lever itself; y2 is the same displacement, but reduced to the translating member axis; the coupling
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(b)
+-Ya -
Xl
Figure 4. (a) Original and (b) reduced model scheme.
ratio T between y, and x2 depends on the mechanism position and more precisely on x2 itself; for this reason the link between these co-ordinates is represented by a template profile in the model scheme of Fig. 4. Finally, y, represents the motion of the translating member extremity. The correspondence between the model elements and the actual mechanism components is now described again with reference to Figs 1 and 4(a). Mass ml concerns the lower part of the lever and the rollers; m, takes account of the upper part of the lever and a portion of the connecting rod mass; m3 is the mass of the translating components, including the other portion of the connecting rod. Spring k: represents the stiffness of the cam rib and the Hertzian contact; the last stiffness depends on the contact force [9] and must be evaluated instantaneously. Spring k; takes the stiffness of the lower part of the lever and the roller pins into account; k,” is the bending stiffness of the lever pin; stiffness ky is relative to the lever upper part and the lever-rod pin; finally, k, takes the stiffness of the rod-translating member pin into account as well as the effect of bending of the translating member. The latter stiffness depends on the mechanism’s position. Two backlashes are introduced into the model: g: is the backlash between cam rib and rollers, g2 takes into account the backlashes in the two rod bearings. The coefficients of the viscous dampers are taken proportional to the corresponding stiffnesses; dampers are intended to represent the structural damping as well as other damping. When there is no cam-roller contact, coefficient c: is computed differently in order to represent the lubricant squeeze effect in the backlash [lo]: c; = 12&b’(d’/4h’)“‘,
(1)
where 7’ is the lubricant dynamic viscosity, b’ and d’ the roller length and diameter respectively, h’the decreasing instantaneous minimum film thickness. Finally, a Coulomb friction force, Ff, applied to the translating member (mass m3), is considered, taking the
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friction force due to both the prismatic pair and the lip seals into account (see [7,8]): Ff = -sign (j&p
* N+ Fls).
where ~1and N are, respectively, the coefficient of friction and the instantaneous normal force in the prismatic pair, and Fls is the seal friction force. In order to write the differential equations of motion, it is first opportune to reduce the model, eliminating the constant coupling ratio i = R,/ R, between the upper and lower extremity of the lever. The coupling ratio between the co-ordinates y, and x2, 7 = dy2/dx2, has not been eliminated, as it depends on x2 itself. The reduced model is shown in Fig. 4(b): all the .quantities relating to the lower lever extremity are reduced to the upper extremity and the three spring-damper elements between the first and second mass are condensed into one. The expressions of the reduced co-ordinates and quantities can be derived easily (see e.g. [lo]): x,=z,*
x, = z, * i.,
i*,
g, = g; * i;
k, = k:/ i2;
1/k2=i2/k;+(1+i)2/k;+l/k$‘;
c1 = ci/ i2;
1/c,=i2/c;+(1+i)2/c~+1/c:l.
m, = m:/i2; (2)
The equations of motion of the reduced model are: ml% = Fe1+ FV1- (Fe2+ FV2), m,jl, = Fe2+ FU2- 7 - (Fe, + FuJ),
(3)
m& = Fe3+ F,, + Ff, where Fej and Fuj (j = 1, 2, 3) are respectively the spring and damper forces, computed through the following expressions: F,, = -k,
if x1 -x0< -g,/2;
(x1-xo+g,/2)
if Ix1- x,,I< g,/2;
Fe1 = 0
if x,-x0>g,/2;
Fe,=-k,(x,-x,-g,/2) Fe2
=
Fe3 = 0
+2(x2-x,);
if
b3-y2i
<
Fe3=-k3(y3-y2-g2/2)
82/2;
ify3-y2>g2/2;
F,, = -cl&
-z&,0);
Fu2= -c2(i2-i,); Fv3
=
-c3(j3
-32).
As previously mentioned, the damper coefficients are taken proportional to the related stiffnesses (Cj= q - kj, j = 1, 2, 3), except c, when backlash g, is traversed (no cam-roller contact). In this case coefficient c: is obtained through the squeeze relationship (1) and reduced coefficient c, is computed [see equations (2)]. Numerical integration of the equations of motion (3), knowing instantaneous displacement x0 imposed by the cam, makes it possible to obtain actual motion y3 of the translating output member. The finite difference method is used [7]. For velocities the following backward difference expression is adopted: tik = (3~~ -4wk_, + ~~_~)/2At,
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where wk, w&i and wk_2 indicate the generic displacement respectively at the present and the two previous steps and At is the time step interval. As a consequence of this choice, each of the motion equations (3) can be solved independently. In fact all the forces on the right-hand side depend only on the known displacements at the present and previous steps, so the present inertia force on the left-hand side can be directly computed from each equation and mass acceleration ii)kobtained. Finally, by formulating tik as the Central difference tii;,= (Wk+,-2wk +
Wk_,)/(dt)‘,
displacement wk+i at the next step can be obtained. The present values of the model parameters which depend on the mechanism position are computed at each integration step. In order to obtain stability of the numerical integration and good accuracy, a suitably small time step interval At must be chosen. By extension of the conditions used by Levy and Wilkinson [7] for sdof systems, the following conditions for mdof systems were adopted [ 81: At <0+31/&,,,,
At<0*25/(q*
ll;,,),
where a,,,,, is the highest natural frequency (rad/s). It is estimated with reference to a simplified system obtained from the model of Fig. 4(b), neglecting damping, friction and backlashes and reducing all the masses and stitiesses to the translating member. As this system has variable parameters, due to the variability of both stiffness k3 and coupling ratio T, only instantaneous values of the natural frequencies can be computed. The maximum value of the third natural frequency is therefore taken as R,,,. 3.2. MODEL VALIDATION AND RESULTS Experimental tests were carried out first in order to adjust the values of the model parameters then to validate the model results in different operating conditions. The output motion was picked up experimentally by mounting an accelerometer on the translating member. Seven different values of cam speed and two different values of cam-roller backlash (as mentioned in section 2.1) were considered. The values of model masses and stiffnesses have been computed on the basis of the mechanism member dimensions. Stiffnesses were then adjusted in order to obtain a better match for experimental results. The values of proportionality constant q between damper coefficients and stiffnesses, seal friction force F,, and coefficient of fraction p have been inferred through observation of the experimental acceleration. Finally, the backlash values have been obtained from the design tolerances and direct measurements. Table 1 shows the values of the model parameters, as well as the corresponding minimum and maximum values of the natural frequencies (f~j, j = 1, 2, 3), computed as mentioned in section 3.1. Good agreement between numerical and experimental results was generally attained. Figures 5(a), (b) and 6(a), (b) compare the output acceleration patterns in the time domain for sound conditions and increased backlash, respectively. The patterns correspond exactly to one machine cycle (one cam revolution) and begin at the bottom of the output member stroke, where it dwells. The cam speed is SOOrpm. Numerical patterns agree satisfactorily with experimental ones both in shape and maximum values. Alterations due to the augmented backlash are clearly observable both in numerical (comparing Figs 5(a) and 6(a)] and experimental [Figs 5(b) and 6(b)] patterns. In the case of larger backlash, higher oscillations around the theoretical pattern arise. The origin of these oscillations are the internal shocks taking place when the roller hits the cam surface. This occurs at the beginning of the motion, after the initial backlash is traversed, and during
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TABLE
Values of the model parameters and natural frequencies
4 m2
m3
4 6 k; k"
k: 4 b’ d’
0.501 0.215 2.165 50* 37.8 110 3.2 6.5+ ll.lt 5.8 - lo-’ 13.2 32.0
* Neglecting t Depending
1)’
30 30 125 60 10
g{ (normal) g : (increased) BF’ IS CL
0.2
i= RJR, 0.85 f 135+ 921+ 1693+
T = dy,/dxz
(mm) (mm)
?In2
fn3
1-616 1 *OOt 145t 1271t 1700t
W
IF; i!
the Hertzian stiffness (variable). on the mechanism position.
-4oo_]/ 0
20
40
60 Time
80
100
120
( ms)
Figure 5. Comparison between (a) model, (b) directly measured and (c) recovered acceleration patterns of the output member, corresponding to one machine cycle; cam revolution speed 500 rpm; sound conditions.
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-4ooJ
I
I
20
40:
I
I
601
801
I
-400?-] 0
Time
100
120
( ms)
Figure 6. Comparison between (a) model, (b) directly measured and (c) recovered acceleration patterns of the output member, corresponding to one machine cycle; cam revolution speed 500 rpm; increasedbacklash.
the motion, when the inertia force direction reverses, causing the backlashes to be traversed and the contact side between rollers and cam rib to change (see [8]). The consequences of the latter phenomenon are clearly observable in the case of large backlash [Figs 6(a), (b)] at about 40 and 90 ms. Superimposed high frequency vibrations of quite large amplitude are observed in experimental patterns [Figs 5(b), 6(b)]. Their causes have not been investigated up until now. It was observed that similar vibrations are also obtained in the numerical patterns [Figs 5(a), 6(a)], adding small random deviations to the theoretical cam profile, as mentioned above. Figures 7(a), (b) and 8(a), (b) p resent the amplitude spectra obtained by Fourier transforming the time patterns considered above. The frequency range is limited to 500 Hz, as the most important components are in this range. The first three cam revolution harmonics are outside the diagram amplitude range, as the scale is set with the aim of showing the other components more clearly; however, the amplitude of the low frequency components is very close in numerical and experimental spectra, as they describe only the theoretical acceleration pattern. Excluding these components, both numerical and experimental spectra present other ranges of high amplitude, corresponding to natural frequencies of the mechanism, as they are independent of the operating speed. The numerical spectra [Figs 7(a), 8(a)]
527
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I
I
200
300
4bo
Frequency
(Hz 1
(a)
15--,
lO---
”
2 g .‘0 & E
15 10
5
s 0
15
10
5 0
b-
lb0
1
AI
I
500
Figure 7. Comparison between (a) model, (b) directly measured and (c) recovered acceleration amplitude spectra of the output member; cam revolution speed 500 rpm; sound conditions.
present only one range of high amplitude around 140 Hz, corresponding to the first natural frequency of the model. The other two natural frequencies (see Table 1) are not excited. The experimental spectra [Figs 7(b), 8(b)] show two resonance ranges: the first corresponds to the numerical one (the frequency is slightly higher, the amplitude lower), the second (275 to 290 Hz) is not present in the numerical spectra. Other components not particularly high in amplitude, spreading from about 1000 to 4000 Hz, correspond to the above-mentioned high frequency vibrations. From the above comparison, the model is able to describe the actual mechanism behaviour in a limited frequency range, comprising the first natural frequency. However, the simulation seems to be effective for diagnostic purposes, as shown in the case of increased backlash. In fact the preeminent deviations of the experimental time patterns [Figs 5(b), 6(b)] from the theoretical ones are oscillations having a period of about 7 ms, corresponding to the first natural frequency (about 140 Hz). Their amplitude is strongly influenced by the backlash size. Only secondary effects occur in the time patterns, due to other frequency components, in particular components in the range 275 to 290 Hz [Fig. 8(b)] and so the model produces satisfactory simulation. Such a model could be a useful tool in monitoring and diagnostics, as it can help in relating motion law changes and faults. For example, suppose that an output acceleration pattern like that in Fig. S(b) is measured for a machine in sound conditions and, after a
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1
I
(b)
Frequency
1
(Hz)
Figure 8. Comparison between (a) model, (b) directly measured and (c) recovered acceleration amplitude spectra of the output member; cam revolution speed 500 rpm; increased backlash.
certain period of operation, this pattern is modified to that in Fig. 6(b), which means that high steps, due to internal shocks, arise when acceleration direction reverses, with subsequent high oscillations. The use of the model can help to diagnose the cause of this modification. In fact a similar modification can be obtained in model output acceleration by changing only the value of the cam-roller backlash gi from 30 to 125 pm, as shown in Figs S(a) and 6(a). The wear of the cam or roller pins is thus the probable cause of the acceleration alteration. The general use of the model is shown schematically in Fig. (Diagnosis)
(Forecast)
Figure 9. Relationships between faults, motion law and case vibration.
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9: when alterations
in motion are detected, the model can be used to diagnose which fault has occurred; the model can also forecast which effects are to be expected when different faults take place. 4. FURTHER 4.1.
FILTER
MONITORING
PROCEDURE
FUNCTION
failure are practically impossible in the kind AS mechanical breakage or catastrophic of machines under consideration, improper functioning of the mechanism, with consequent poor production needs to be prevented. Thus the best monitoring strategy would be to estimate directly the actual motion law in order to establish whether its deviations from the theoretical one are acceptable or not. Knowledge of the motion law allows the evaluation of the actual inertia forces, generally the most important in this type of mechanism, and permits direct judgment as to whether the actual motion law is acceptable or not from this point of view. Motion law is related to the bearing forces exciting the case vibrations, but these also depend on the case structure response. As an example, in the case of the subassembly of Fig. 1, high case vibrations take place in the same frequency range as mechanism oscillations (around 140 Hz) and case vibration amplitude is sensitive to backlash size, as shown in Fig. 3. However, in testing subassemblies of a different type, it is possible that high case vibrations do not generally correspond to high mechanism vibrations, due to the low response of the case structure in the mechanism resonance range. Consequently, low case vibration sensitivity to faults may result. The acceptability limits for case vibration cannot be established directly but need suitable tests both in production or by using artificially-worn components, taking into account that individual differences between nominally equal machines might be very influential, as mentioned in Section 2.2. For the above-mentioned reasons, as well as the possibility of relating faults and motion law through a mathematical model of the mechanism (Section 3.2.), it is preferable to monitor output member motion rather than case vibration. Unfortunately, this is possible only during preliminary tests by the customer, not in production. Thus a reliable technique, able to derive the actual motion of the output member from the signal achieved by the accelerometer on the case, is required. This recovery operation can be achieved using a suitable filter function. Figure 9 shows this idea schematically. The same idea was applied to gearboxes in order to recover gear torsional vibration from case vibration [3,4]. In the case of the mechanism shown in Fig. 1, the motion of the output member can be considered to be the response of the mechanism to the curve of motion manufactured on the cam working surface. Caso vibration can also be considered as the response of the structure to the same excitation, even if the path is more complicated because the exciting inertia forces arise in the mechanism; they are, however, obviously related to the curve of motion on the cam surface. It can be concluded that two transfer functions exist between the cam surface on the one hand and the translating output member and vibrating case on the other. The ratio between these two transfer functions represents the filter function, that is, the connection between the output motion and case vibration. In practice, the desired connection was achieved experimentally by evaluating the Frequency Response Function (FRF) H,,,, between the output member and the case. It is defined by the relationship: F,=K,;L, where F, and F,,, are the Fourier transforms of the case and output member acceleration,
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respectively. Strictly speaking, the FRF can be defined only for a linear, time invariant system, so it cannot be defined for the system under consideration. However, since the purpose is not system identification but signal recovery, the use of the FRF in order to obtain a filtering function is allowable [2] and related results are reliable, as will be shown later. Evaluation of the FRF was carried out on a machine in sound condition. Two accelerometers were mounted, one on the translating output member, the other on the case (next to the lever bearing), and their signals were acquired at seven different angular speeds, ranging from 260 to 500 rpm. The autopower spectra of the two signals were evaluated over 140 averages (20 averages for each speed) in the range O-5000 Hz. The Hv estimation [ 1l] of the FRF was then evaluated considering output member acceleration as input and case acceleration as output. The resulting FRF is shown in Fig. 10(b) in the range to 1000 Hz. The related coherence function [Fig. 10(a)] is good, except in the low frequency range from 0 to about 20 Hz. The desired filter function G,, was then obtained as the complex inverse of the FRF: G,, = I/J-L,. This operation may lead to a nonsense result if H,, has a very low amplitude at some frequencies, It does not occur in the present case, but it does in others. A possible way to solve this difficulty is data smoothing, as low amplitude is generally confined to very narrow bands.
I.0
0.5
0.0
400
600
Frequency
(Hz)
Figure 10. (a) Coherence function and (b) magnitude of the frequency response function between output member and case acceleration; sound conditions.
The filter function Gem allows recovery of the output member acceleration spectrum complex multiplication of the case acceleration spectrum F, by the filter
F, through
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function itself: F,,,= G,,- F,. As an example Fig. 11 shows the measured case spectrum (a) and the corresponding recovered output member spectrum (b) for the mechanism of Fig. 1 with large backlash in the range to 1000 Hz and containing the meaningful components. Although both spectra present the resonance range around 140 Hz they have, on the whole, very different shapes. In order to validate the proposed technique, the recovered and directly measured spectra of the output member acceleration were compared. Due to bad coherence in the low frequency range, disagreement was expected. Figures 12 and 13 show that the technique is actually unable to restore lower cam revolution harmonics properly. However, these are practically insensitive to mechanism conditions, as shown by the measured spectra of Figs 12(a) and 13(a), relative to different backlash size. The reason is that the first harmonics describe only the theoretical pattern of output acceleration, while the superposed dynamic effects, influenced by the mechanism conditions, correspond to a higher frequency range. As a consequence, the output acceleration spectrum could still be correctly estimated in all conditions by recovering components higher than the fourth revolution harmonic through the case vibration spectrum and the filter function, and by copying lower components from an output acceleration spectrum previously measured and stored in any condition. Applying this procedure brought very satisfactory agreement between recovered and directly measured output acceleration. Figures 5(b), (c) and 6(b), (c) compare time patterns, corresponding to one cam revolution exactly, for the two different backlash sizes and the same cam speed considered above. Figures 7(b), (c) and 8(b), (c) present the corresponding spectra. In the case of large backlash [Fig. 8(b), (c)], the recovered spectrum presents a lower amplitude level in the range 275 to 290 Hz. The consequence in the time domain is that the recovered pattern [Fig. 6(c)] is slightly smoother and with I.0
_ 0.5 (u u) 2 g
0.0 (b
1 8 8
40
2c
C
1
I
200
.
1
L
I
I
400
600
t
I
800
l(
10
Frequency (Hz) Figure 11. Amplitude spectrum of (a) the measured case acceleration and (b) recovered output member acceleration; cam revolution speed 500 rpm; increased backlash.
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20 0
0
40
60
Frequency
(Hz)
100
80
Figure 12. (a) Measured and (b) recovered amplitude spectrum of output acceleration; cam revolution speed 500 rpm; sound conditions.
60 40 Lic 20 5
-
0, 0
II I 20
A
IA 40 Frequency
,.I 60
h
-I*
80
. 100
(Hz)
Figure 13. (a) Measured and (b) recovered amplitude spectrum of output acceleration; cam revolution speed 500 rpm; increased backlash.
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less steep steps at the internal shocks (about 40 and 90 ms), with respect to the directly measured one [(Fig. 6(b)]. In spite of these differences, the recovered patterns are able to distinguish clearly between the two conditions (different backlash size) and are, therefore, a useful estimation of the actual patterns for monitoring purposes. 4.2. OUTPUT MOTION MONITORING On the basis of the above considerations and results, the following monitoring methodology can be proposed. For each individual machine in sound condition, the filter function relating the accelerations of a point on the case and of the output member, is evaluated experimentally and stored for every subassembly to be monitored. The output member spectrum in the range corresponding to the first revolution harmonics is also stored at the same time. During operation, acceleration of the selected point on the case is picked up, its frequency spectrum computed and the spectrum of output member acceleration estimated through the filter function (copying low frequency components from the experimental spectrum stored in sound conditions). Finally, inverse Fourier transforming, the time pattern acceleration of the output member can be recovered and compared with the functional limits. 5. CONCLUSIONS
A methodology for condition monitoring of cam and other mechanisms for reciprocal or intermittent motion is proposed. It allows recovery of the output motion from the case acceleration measurement. Thus, it is possible to directly judge whether the actual output motion is acceptable with respect to the mechanism function. Contrary to the usual methodologies based on the analysis of case vibration, the proposed methodology (a) does not involve difficult, long-lasting tests for preliminary establishment of warning levels and (b) is not affected by differences in the structural response between nominally equal machines. Although the last consideration has not been experimentally proved until now, it is well-founded, since the filter function, evaluated for each individual subassembly, takes individual differences in transfer function into account. Further tests on different subassemblies and in different conditions are required in order to establish the general effectiveness of the method. A numerical simulation of the mechanism dynamic behaviour was also presented and its application in diagnostics outlined. ACKNOWLEDGEMENT
This work was partially supported by a grant from the Italian Ministry of University and Research (MURST). REFERENCES 1. P. H. MAEDEL,
2. 3. 4. 5. 6.
JR. and R. L. ESHLEMAN 1988 Shock and Vibrution Handbook, Chap. 19: Vibration Standards. New York: McGraw-Hill. R. H. LYON 1987 Machinery Noise and Diagnostics. Boston: Butterworths. K. UMEZAWA, H. HOUJOH and H. MAKI 1988 JSME International Journal 31, 588-592. Estimation of the vibration of in-service gears by monitoring the exterior vibration. G. DALPIAZ and U. MENEGHEITI. NDT&E InternationaL Monitoring fatigue cracks in gears (to be published). A. 0. ANDRISANO and G. DALPIAZ 1990 Atti X Congr. Naz. AZMETA, Piss, Italy, pp. 633-638. A MDOF model for the dynamic analysis of drives with Geneva mechanism. (In Italian.) G. DALPIAZ and U. MENEGHE’ITI 1990 Condition Monitoring-Froceedings of the 3rd Zntemutional Conference. Detection and modelling of fatigue cracks in gears, pp. 73-82. London: Elsevier.
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7. S. LEVY and J. P. D. WILKINSON 1976 The Component Element Method in Dynamics. New York: McGraw-Hill. 8. A. RIVOLA and G. DALPIAZ 1992 DIEM, Universita’ di Bologna. Dynamic analysis of a mechanism for automatic machine. (In Italian.) 9. K. L. JOHNSON 1985 Contact Mechanics, pp. 129-131. Cambridge: Cambridge University Press. 10. M. P. KOSTER 1974 Vibrations of Cam Mechanisms. London: MacMillan. 11. R. ALLEMANG, D. BROWN and R. ROST 1988 Proceedings of 13th International Seminar on Modal Analysis, Katholieke Universiteit Leuven, Basic Course, Part I, pp. 52-79. Frequency response function estimation.
MONITORING G.
AUTOMATIC MACHINES
DALPIAZ
AND
A.
MAGGIORE
LXZmrtimento di Zngegneria delle Costruzioni Meccaniche, Nucleari, Aeronautiche e di Metallurgia University of Bologna, Italy
(Received 7 August 1991, accepted 13 October 1991) In the context of full automation of manufacturing processes, it is important to develop predictive maintenance methodologies in every industrial field. This paper presents experimental tests carried out on some components of high-performance automatic machines. The aim is to assess and compare the effectiveness and reliability of different condition monitoring methodologies based on vibration analysis. A methodology based on recovery of the acceleration pattern of the final actuator of the mechanism under consideration seems to be particularly promising. It allows the working conditions to be judged, directly comparing the recovered output member acceleration with the functional requirements which, for this kind of mechanism, are mainly relative to the motion law. A lumped parameter model of the same mechanism is presented and its application for diagnostic purposes is shown.
1. INTRODUCTION
In-service condition monitoring is being applied to almost every field of industry. In many factories it is increasingly necessary to ensure the possibility of correct machine operation without continuous human supervision. A lot of plant and machine components are currently monitored by means of ever-improving but well-understood techniques. Wear debris and vibration analysis are nowadays the most widely employed techniques. Vibration analysis is used satisfactorily for many mechanical components. The vibration of the case is generally considered for monitoring purposes. Several processing and analysis techniques are applied for different components and aims. A variety of different parameters is usually employed for characterising machine health condition (e.g. RMS, kurtosis, the amplitude of frequency components characterising machine troubles). However, to correlate the value of case vibration parameters to machine condition may be difficult. In fact, the vibration limit levels relative to generic classes of machines and generic conditions are often the only allowable data (e.g. the Vibration Severity Criteria after IS0 IS 2372) [ 11. If more reliable alarm levels are needed and relationships between vibration parameters and specific faults are to be established, preliminary tests must be performed in production, following the development of actual faults or introducing artificial faults. The first case requires lengthy testing, the second might not give realistic results. Further difficulties can arise from result scattering due to individual differences between nominally equal machines: the same health condition may produce different case vibration levels. Most of the above-mentioned difficulties in assessing alarm levels and in correlating faults and case vibration parameters are due to the fact that case vibration is affected by the transfer function between the mechanism components where forces are applied and 517 OSSS-3270/92/060517+18 $08.00/O
@ 1992 Academic press Limited
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the transducer’s position on the case. In fact, the vibration sources are in the mechanism, whose dynamic behaviour is the result of the interaction between geometry, dynamic properties, applied forces and faults. As a consequence of machine working, forces are transmitted to the case and excite the case structure, whose response depends on its dynamic properties. Thus the effect of a transfer function between mechanism components and case transducers should be taken into account. Since wear and faults directly affect the mechanism motion and load, it would be more suitable to monitor these quantities rather than case vibration because the transfer function might partially or completely hide information about faults. Information about mechanism motion and load is of greater interest than case vibration: motion may be related to the machine function, as in the case of a cam mechanism designed to achieve a certain motion law; knowledge of overload due to the actual dynamic behaviour is important for safety. Therefore, monitoring mechanism motion or load permits us to judge whether they are acceptable for machine function and safety; thus preliminary tests in order to establish alarm levels are not required and individual differences in case response between nominally equal machines are without effect. In spite of the above-mentioned advantages, monitoring mechanism functional quantities (motion and load) is generally possible during preliminary tests by the customer, not in production. It is therefore necessary to obtain the measurements of interest from vibration signals picked up at the case. This can be carried out by experimentally estimating and storing the transfer functions which relate the considered quantities and the case vibration at the accelerometer locations. This operation has to be done, once and for all time, at the installation stage. In production the stored transfer functions are used as filter functions in order to recover the desired functional quantities from the case accelerometer signals [2-41. A mathematical model of a mechanism is a very useful tool in aiding understanding of the mechanism’s dynamic behaviour. It allows the design and functional parameters of a mechanism to be related to the dynamic behaviour [S]. Such a model may be used to identify vibration sources and study the connection between possible faults and alterations in dynamic behaviour. Since a model describes the mechanism behaviour in terms of functional quantities (e.g. motion of members, internal forces), it would be particularly useful to associate the model to the above-outlined recovery procedure [4,6]. In fact, when changes in recovered functional quantities are detected, the model might be used to diagnose which fault had occured and it might also forecast which effects are to be expected from the monitored functional quantities should different faults occur. This paper deals with vibration analysis methods applied to an automatic machine in order to monitor the components of the cam and other mechanisms for reciprocal or intermittent motion. The most common fault developing in this kind of mechanism is wear on the cam surface or in the kinematic pairs of the follower system, with a consequent increase in backlashes. This produces more intense internal shocks and stronger deviation of the motion law from the theoretical pattern, with higher accelerations and dynamic overloads. First, the main results obtained from the application of commonly-used vibration analysis techniques are reported and discussed. Secondly, the model of one particular cam mechanism is described, its use for understanding the dynamic behaviour is outlined, and its results are compared with the experimental ones in order to prove its capability in forecasting the effects of backlash increase. Finally, application of the filter function method to the same mechanism is illustrated, its capability in recovering the output member motion from the case vibration signal experimentally is shown, and its advantages in assessing the actual function condition are considered.
MONITORING
2. APPLICATION 2.1.
EXPERIMENTAL
AUTOMATIC
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automatic machine composed of subassemblies, each containing a cam or other mechanism for reciprocal or intermittent motion was investigated. Tests were performed on a number of different subassemblies. In general, the following results refer to all the tests, but are illustrated with reference to a particular subassembly, also considered for modeling (section 3) and for the advanced monitoring procedure (section 4). The mechanism is shown schematically in Fig. 1 and is made up of a cylindrical cam and a follower system composed of two rollers, a lever, a small connecting rod and a translating output member. An
,Connecting
rod 7
r’Tronsloting
member
Y3
Figure 1. Mechanism scheme.
The only faults considered are inherent to mechanisms for reciprocal or intermittent motion, e.g. wear on cam surface, increased backlash in follower bearings. Faults in transmission components, such as gears, shaft bearings, etc., were not considered, because these components are usually over-dimensioned and rarely, if ever, develop or cause faults. Tests were perfomed by varying working parameters (e.g. speed) and health conditions, as both natural and artificial faults were considered. In-field monitoring, however, has not yet been achieved. In the case of the mechanism in Fig. 1, tests were carried out at different cam speeds (diagrams presented in this paper refer to the maximum speed, 500 rpm), both in sound conditions and with artificially increased backlash between cam and rollers. Though, in the latter condition, the backlash was about four times greater than in the first, it was still judged acceptable for production, but the subassembly needs to be inspected more frequently. Accelerometers and acoustic emission transducers were mounted on the case of each subassembly; the basic experimental equipment is illustrated in Fig. 2; the acceleration of the output member was not taken into consideration in this first research phase. Accelerometer signals were analysed in the time, frequency and amplitude domains. Some of the basic analyses were: Time Synchronous Average (TSA) and its Fourier Transform (average was carried out with the help of a one-per-cam revolution tachometer signal collected using an inductive proximity probe); Power Spectral Density (PSD); Amplitude Probability Density ( APD), also used to evaluate some amplitude statistical parameters, such as standard deviation and kurtosis. Signals from acoustic emission (AE) transducers were first demodulated, i.e. low frequency content was extracted in order to obtain signals in a frequency range of mechanical significance. Demodulated acoustic emission (DAE) signals were then analysed in a similar manner to the accelerometer signals..
520
G. DALPIAZ AND A. MAGGIORE Accelerometer (outout member)
4
Accelerometer (case)
A.E. transducer (case)
53
53
Amplifier
’Peripheral equipment
I
Figure 2. Experimental equipment.
2.2. RESULTS AE is widely used for monitoring purposes, e.g. in the field of manufacturing processes. In the present case no satisfactory results were obtained by this technique; as a matter of fact, variations in the DAE signal with mechanism health conditions, if any, were never clearly distinguishable, so it was impossible to find a consistent relationship between them. The TSA was able to detect a fault from the accelerometer signal but in general with less sensitivity and more difficulty in interpretation than other options. It can be useful to diagnose a fault, localising it in the cycle of motion of the mechanism, but it is not recommended as a general health monitoring technique for the mechanisms under consideration. The APD proved to correlate quite well to mechanism health conditions. Standard deviation seemed to be a possible pointer for condition monitoring, even if not very sensitive and not always reliable; kurtosis did not appear to correlate to health conditions. PSD was the clearest ad most reliable option for monitoring purposes. The drawback is some difficulty in correlating the PSD pattern with the machine health condition. However, a particularly high sensitivity to machine condition was observed in some frequency ranges. These depend on the type of subassembly and the measurement point on the case, and are not affected by cam speed. Thus they correspond to resonance frequencies of the mechanism or of the case structure. The mean value of the PSD in the highest amplitude range was then evaluated. It was found to discriminate between intact and faulty condition when the same individual mechanism was considered. As an example, Fig. 3 compares the PSDs measured on the subassembly of Fig. 1 for sound condition (a) and after the bac&ash between cam and follower was artificially increased (b). A clear-cut difference between the two conditions is observed in the range 130 to 220 Hz. The PSD mean value in the same range increases from 0.05 to O-27 m’/s3. Teats carried out on some nominally equal machines, working in production for about the same period, have shown that a subassembly in a particular machine has a PSD signature which is quite stable in time and, as previously mentioned, consistent with mechanism health conditions. Subassemblies of the same type in dierent machines show resonances in the same ranges, but of rather different signature and amplitude; thus health
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(Hz)
Figure 3. Power spectral density of case acceleration; 18 averages. Cam revolution speed 500 rpm. (a) Sound conditions. (b) Artificially increased backlash between cam and follower.
conditions can be assessed only by comparing PSDs of the same machine and not by considering absolute limit values. As a further result, a transducer mounted on a particular subassembly is, in practice, sensitive only to the condition of that subassembly, not to the faults developing in another.
3. MODELING
3.1.
DESCRIPTION
OF THE
THE MECHANISM
MODEL
A mathematical model of the mechanism of the particular subassembly of Fig. 1 was built in order to study its dynamic behaviour for diagnostic purposes. It is a lumped parameter model [Fig. 4(a)] with 3 dof, obtained by applying the component element method [7,8]. In order to simulate the mechanism’s behaviour, the model takes nonlinearities into account, namely backlashes, stiffness dependence on the position of the mechanism and Coulomb friction in the prismatic pair. The model concerns only the cam mechanism, not the mechanical transmission driving the cam; in fact the transmission influence can be neglected and due to its high inertia the cam assumed to rotate at constant angular speed. The input is then the motion imposed by the cam, the output the motion of the extremity of the translating member. With reference to Figs 1 and 4(a), the co-ordinates z, and z, , taken on the arc of radius R,,represent respectively, the displacement imposed by the cam motion and the actual displacement of the lower extremity of the lever. Irrother words, z. is the known excitation and would be equal to the displacement imposed on the lower lever extremity if cam-roller backlash and member compliance were neglected. It is computed as a function of cam rotation (or time, as cam speed is constant), on the basis of the profile data with the addition of small random deviations, accounting for machining errors on the cam surface. The co-ordinate x2 is taken on the arc of radius R2 and represents the displacement of the upper extremity of the lever itself; y2 is the same displacement, but reduced to the translating member axis; the coupling
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(b)
+-Ya -
Xl
Figure 4. (a) Original and (b) reduced model scheme.
ratio T between y, and x2 depends on the mechanism position and more precisely on x2 itself; for this reason the link between these co-ordinates is represented by a template profile in the model scheme of Fig. 4. Finally, y, represents the motion of the translating member extremity. The correspondence between the model elements and the actual mechanism components is now described again with reference to Figs 1 and 4(a). Mass ml concerns the lower part of the lever and the rollers; m, takes account of the upper part of the lever and a portion of the connecting rod mass; m3 is the mass of the translating components, including the other portion of the connecting rod. Spring k: represents the stiffness of the cam rib and the Hertzian contact; the last stiffness depends on the contact force [9] and must be evaluated instantaneously. Spring k; takes the stiffness of the lower part of the lever and the roller pins into account; k,” is the bending stiffness of the lever pin; stiffness ky is relative to the lever upper part and the lever-rod pin; finally, k, takes the stiffness of the rod-translating member pin into account as well as the effect of bending of the translating member. The latter stiffness depends on the mechanism’s position. Two backlashes are introduced into the model: g: is the backlash between cam rib and rollers, g2 takes into account the backlashes in the two rod bearings. The coefficients of the viscous dampers are taken proportional to the corresponding stiffnesses; dampers are intended to represent the structural damping as well as other damping. When there is no cam-roller contact, coefficient c: is computed differently in order to represent the lubricant squeeze effect in the backlash [lo]: c; = 12&b’(d’/4h’)“‘,
(1)
where 7’ is the lubricant dynamic viscosity, b’ and d’ the roller length and diameter respectively, h’the decreasing instantaneous minimum film thickness. Finally, a Coulomb friction force, Ff, applied to the translating member (mass m3), is considered, taking the
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friction force due to both the prismatic pair and the lip seals into account (see [7,8]): Ff = -sign (j&p
* N+ Fls).
where ~1and N are, respectively, the coefficient of friction and the instantaneous normal force in the prismatic pair, and Fls is the seal friction force. In order to write the differential equations of motion, it is first opportune to reduce the model, eliminating the constant coupling ratio i = R,/ R, between the upper and lower extremity of the lever. The coupling ratio between the co-ordinates y, and x2, 7 = dy2/dx2, has not been eliminated, as it depends on x2 itself. The reduced model is shown in Fig. 4(b): all the .quantities relating to the lower lever extremity are reduced to the upper extremity and the three spring-damper elements between the first and second mass are condensed into one. The expressions of the reduced co-ordinates and quantities can be derived easily (see e.g. [lo]): x,=z,*
x, = z, * i.,
i*,
g, = g; * i;
k, = k:/ i2;
1/k2=i2/k;+(1+i)2/k;+l/k$‘;
c1 = ci/ i2;
1/c,=i2/c;+(1+i)2/c~+1/c:l.
m, = m:/i2; (2)
The equations of motion of the reduced model are: ml% = Fe1+ FV1- (Fe2+ FV2), m,jl, = Fe2+ FU2- 7 - (Fe, + FuJ),
(3)
m& = Fe3+ F,, + Ff, where Fej and Fuj (j = 1, 2, 3) are respectively the spring and damper forces, computed through the following expressions: F,, = -k,
if x1 -x0< -g,/2;
(x1-xo+g,/2)
if Ix1- x,,I< g,/2;
Fe1 = 0
if x,-x0>g,/2;
Fe,=-k,(x,-x,-g,/2) Fe2
=
Fe3 = 0
+2(x2-x,);
if
b3-y2i
<
Fe3=-k3(y3-y2-g2/2)
82/2;
ify3-y2>g2/2;
F,, = -cl&
-z&,0);
Fu2= -c2(i2-i,); Fv3
=
-c3(j3
-32).
As previously mentioned, the damper coefficients are taken proportional to the related stiffnesses (Cj= q - kj, j = 1, 2, 3), except c, when backlash g, is traversed (no cam-roller contact). In this case coefficient c: is obtained through the squeeze relationship (1) and reduced coefficient c, is computed [see equations (2)]. Numerical integration of the equations of motion (3), knowing instantaneous displacement x0 imposed by the cam, makes it possible to obtain actual motion y3 of the translating output member. The finite difference method is used [7]. For velocities the following backward difference expression is adopted: tik = (3~~ -4wk_, + ~~_~)/2At,
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where wk, w&i and wk_2 indicate the generic displacement respectively at the present and the two previous steps and At is the time step interval. As a consequence of this choice, each of the motion equations (3) can be solved independently. In fact all the forces on the right-hand side depend only on the known displacements at the present and previous steps, so the present inertia force on the left-hand side can be directly computed from each equation and mass acceleration ii)kobtained. Finally, by formulating tik as the Central difference tii;,= (Wk+,-2wk +
Wk_,)/(dt)‘,
displacement wk+i at the next step can be obtained. The present values of the model parameters which depend on the mechanism position are computed at each integration step. In order to obtain stability of the numerical integration and good accuracy, a suitably small time step interval At must be chosen. By extension of the conditions used by Levy and Wilkinson [7] for sdof systems, the following conditions for mdof systems were adopted [ 81: At <0+31/&,,,,
At<0*25/(q*
ll;,,),
where a,,,,, is the highest natural frequency (rad/s). It is estimated with reference to a simplified system obtained from the model of Fig. 4(b), neglecting damping, friction and backlashes and reducing all the masses and stitiesses to the translating member. As this system has variable parameters, due to the variability of both stiffness k3 and coupling ratio T, only instantaneous values of the natural frequencies can be computed. The maximum value of the third natural frequency is therefore taken as R,,,. 3.2. MODEL VALIDATION AND RESULTS Experimental tests were carried out first in order to adjust the values of the model parameters then to validate the model results in different operating conditions. The output motion was picked up experimentally by mounting an accelerometer on the translating member. Seven different values of cam speed and two different values of cam-roller backlash (as mentioned in section 2.1) were considered. The values of model masses and stiffnesses have been computed on the basis of the mechanism member dimensions. Stiffnesses were then adjusted in order to obtain a better match for experimental results. The values of proportionality constant q between damper coefficients and stiffnesses, seal friction force F,, and coefficient of fraction p have been inferred through observation of the experimental acceleration. Finally, the backlash values have been obtained from the design tolerances and direct measurements. Table 1 shows the values of the model parameters, as well as the corresponding minimum and maximum values of the natural frequencies (f~j, j = 1, 2, 3), computed as mentioned in section 3.1. Good agreement between numerical and experimental results was generally attained. Figures 5(a), (b) and 6(a), (b) compare the output acceleration patterns in the time domain for sound conditions and increased backlash, respectively. The patterns correspond exactly to one machine cycle (one cam revolution) and begin at the bottom of the output member stroke, where it dwells. The cam speed is SOOrpm. Numerical patterns agree satisfactorily with experimental ones both in shape and maximum values. Alterations due to the augmented backlash are clearly observable both in numerical (comparing Figs 5(a) and 6(a)] and experimental [Figs 5(b) and 6(b)] patterns. In the case of larger backlash, higher oscillations around the theoretical pattern arise. The origin of these oscillations are the internal shocks taking place when the roller hits the cam surface. This occurs at the beginning of the motion, after the initial backlash is traversed, and during
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TABLE
Values of the model parameters and natural frequencies
4 m2
m3
4 6 k; k"
k: 4 b’ d’
0.501 0.215 2.165 50* 37.8 110 3.2 6.5+ ll.lt 5.8 - lo-’ 13.2 32.0
* Neglecting t Depending
1)’
30 30 125 60 10
g{ (normal) g : (increased) BF’ IS CL
0.2
i= RJR, 0.85 f 135+ 921+ 1693+
T = dy,/dxz
(mm) (mm)
?In2
fn3
1-616 1 *OOt 145t 1271t 1700t
W
IF; i!
the Hertzian stiffness (variable). on the mechanism position.
-4oo_]/ 0
20
40
60 Time
80
100
120
( ms)
Figure 5. Comparison between (a) model, (b) directly measured and (c) recovered acceleration patterns of the output member, corresponding to one machine cycle; cam revolution speed 500 rpm; sound conditions.
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-4ooJ
I
I
20
40:
I
I
601
801
I
-400?-] 0
Time
100
120
( ms)
Figure 6. Comparison between (a) model, (b) directly measured and (c) recovered acceleration patterns of the output member, corresponding to one machine cycle; cam revolution speed 500 rpm; increasedbacklash.
the motion, when the inertia force direction reverses, causing the backlashes to be traversed and the contact side between rollers and cam rib to change (see [8]). The consequences of the latter phenomenon are clearly observable in the case of large backlash [Figs 6(a), (b)] at about 40 and 90 ms. Superimposed high frequency vibrations of quite large amplitude are observed in experimental patterns [Figs 5(b), 6(b)]. Their causes have not been investigated up until now. It was observed that similar vibrations are also obtained in the numerical patterns [Figs 5(a), 6(a)], adding small random deviations to the theoretical cam profile, as mentioned above. Figures 7(a), (b) and 8(a), (b) p resent the amplitude spectra obtained by Fourier transforming the time patterns considered above. The frequency range is limited to 500 Hz, as the most important components are in this range. The first three cam revolution harmonics are outside the diagram amplitude range, as the scale is set with the aim of showing the other components more clearly; however, the amplitude of the low frequency components is very close in numerical and experimental spectra, as they describe only the theoretical acceleration pattern. Excluding these components, both numerical and experimental spectra present other ranges of high amplitude, corresponding to natural frequencies of the mechanism, as they are independent of the operating speed. The numerical spectra [Figs 7(a), 8(a)]
527
MONITORING AUTOMATIC MACHINES ,
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I
I
200
300
4bo
Frequency
(Hz 1
(a)
15--,
lO---
”
2 g .‘0 & E
15 10
5
s 0
15
10
5 0
b-
lb0
1
AI
I
500
Figure 7. Comparison between (a) model, (b) directly measured and (c) recovered acceleration amplitude spectra of the output member; cam revolution speed 500 rpm; sound conditions.
present only one range of high amplitude around 140 Hz, corresponding to the first natural frequency of the model. The other two natural frequencies (see Table 1) are not excited. The experimental spectra [Figs 7(b), 8(b)] show two resonance ranges: the first corresponds to the numerical one (the frequency is slightly higher, the amplitude lower), the second (275 to 290 Hz) is not present in the numerical spectra. Other components not particularly high in amplitude, spreading from about 1000 to 4000 Hz, correspond to the above-mentioned high frequency vibrations. From the above comparison, the model is able to describe the actual mechanism behaviour in a limited frequency range, comprising the first natural frequency. However, the simulation seems to be effective for diagnostic purposes, as shown in the case of increased backlash. In fact the preeminent deviations of the experimental time patterns [Figs 5(b), 6(b)] from the theoretical ones are oscillations having a period of about 7 ms, corresponding to the first natural frequency (about 140 Hz). Their amplitude is strongly influenced by the backlash size. Only secondary effects occur in the time patterns, due to other frequency components, in particular components in the range 275 to 290 Hz [Fig. 8(b)] and so the model produces satisfactory simulation. Such a model could be a useful tool in monitoring and diagnostics, as it can help in relating motion law changes and faults. For example, suppose that an output acceleration pattern like that in Fig. S(b) is measured for a machine in sound conditions and, after a
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1
I
(b)
Frequency
1
(Hz)
Figure 8. Comparison between (a) model, (b) directly measured and (c) recovered acceleration amplitude spectra of the output member; cam revolution speed 500 rpm; increased backlash.
certain period of operation, this pattern is modified to that in Fig. 6(b), which means that high steps, due to internal shocks, arise when acceleration direction reverses, with subsequent high oscillations. The use of the model can help to diagnose the cause of this modification. In fact a similar modification can be obtained in model output acceleration by changing only the value of the cam-roller backlash gi from 30 to 125 pm, as shown in Figs S(a) and 6(a). The wear of the cam or roller pins is thus the probable cause of the acceleration alteration. The general use of the model is shown schematically in Fig. (Diagnosis)
(Forecast)
Figure 9. Relationships between faults, motion law and case vibration.
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9: when alterations
in motion are detected, the model can be used to diagnose which fault has occurred; the model can also forecast which effects are to be expected when different faults take place. 4. FURTHER 4.1.
FILTER
MONITORING
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failure are practically impossible in the kind AS mechanical breakage or catastrophic of machines under consideration, improper functioning of the mechanism, with consequent poor production needs to be prevented. Thus the best monitoring strategy would be to estimate directly the actual motion law in order to establish whether its deviations from the theoretical one are acceptable or not. Knowledge of the motion law allows the evaluation of the actual inertia forces, generally the most important in this type of mechanism, and permits direct judgment as to whether the actual motion law is acceptable or not from this point of view. Motion law is related to the bearing forces exciting the case vibrations, but these also depend on the case structure response. As an example, in the case of the subassembly of Fig. 1, high case vibrations take place in the same frequency range as mechanism oscillations (around 140 Hz) and case vibration amplitude is sensitive to backlash size, as shown in Fig. 3. However, in testing subassemblies of a different type, it is possible that high case vibrations do not generally correspond to high mechanism vibrations, due to the low response of the case structure in the mechanism resonance range. Consequently, low case vibration sensitivity to faults may result. The acceptability limits for case vibration cannot be established directly but need suitable tests both in production or by using artificially-worn components, taking into account that individual differences between nominally equal machines might be very influential, as mentioned in Section 2.2. For the above-mentioned reasons, as well as the possibility of relating faults and motion law through a mathematical model of the mechanism (Section 3.2.), it is preferable to monitor output member motion rather than case vibration. Unfortunately, this is possible only during preliminary tests by the customer, not in production. Thus a reliable technique, able to derive the actual motion of the output member from the signal achieved by the accelerometer on the case, is required. This recovery operation can be achieved using a suitable filter function. Figure 9 shows this idea schematically. The same idea was applied to gearboxes in order to recover gear torsional vibration from case vibration [3,4]. In the case of the mechanism shown in Fig. 1, the motion of the output member can be considered to be the response of the mechanism to the curve of motion manufactured on the cam working surface. Caso vibration can also be considered as the response of the structure to the same excitation, even if the path is more complicated because the exciting inertia forces arise in the mechanism; they are, however, obviously related to the curve of motion on the cam surface. It can be concluded that two transfer functions exist between the cam surface on the one hand and the translating output member and vibrating case on the other. The ratio between these two transfer functions represents the filter function, that is, the connection between the output motion and case vibration. In practice, the desired connection was achieved experimentally by evaluating the Frequency Response Function (FRF) H,,,, between the output member and the case. It is defined by the relationship: F,=K,;L, where F, and F,,, are the Fourier transforms of the case and output member acceleration,
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respectively. Strictly speaking, the FRF can be defined only for a linear, time invariant system, so it cannot be defined for the system under consideration. However, since the purpose is not system identification but signal recovery, the use of the FRF in order to obtain a filtering function is allowable [2] and related results are reliable, as will be shown later. Evaluation of the FRF was carried out on a machine in sound condition. Two accelerometers were mounted, one on the translating output member, the other on the case (next to the lever bearing), and their signals were acquired at seven different angular speeds, ranging from 260 to 500 rpm. The autopower spectra of the two signals were evaluated over 140 averages (20 averages for each speed) in the range O-5000 Hz. The Hv estimation [ 1l] of the FRF was then evaluated considering output member acceleration as input and case acceleration as output. The resulting FRF is shown in Fig. 10(b) in the range to 1000 Hz. The related coherence function [Fig. 10(a)] is good, except in the low frequency range from 0 to about 20 Hz. The desired filter function G,, was then obtained as the complex inverse of the FRF: G,, = I/J-L,. This operation may lead to a nonsense result if H,, has a very low amplitude at some frequencies, It does not occur in the present case, but it does in others. A possible way to solve this difficulty is data smoothing, as low amplitude is generally confined to very narrow bands.
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Figure 10. (a) Coherence function and (b) magnitude of the frequency response function between output member and case acceleration; sound conditions.
The filter function Gem allows recovery of the output member acceleration spectrum complex multiplication of the case acceleration spectrum F, by the filter
F, through
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function itself: F,,,= G,,- F,. As an example Fig. 11 shows the measured case spectrum (a) and the corresponding recovered output member spectrum (b) for the mechanism of Fig. 1 with large backlash in the range to 1000 Hz and containing the meaningful components. Although both spectra present the resonance range around 140 Hz they have, on the whole, very different shapes. In order to validate the proposed technique, the recovered and directly measured spectra of the output member acceleration were compared. Due to bad coherence in the low frequency range, disagreement was expected. Figures 12 and 13 show that the technique is actually unable to restore lower cam revolution harmonics properly. However, these are practically insensitive to mechanism conditions, as shown by the measured spectra of Figs 12(a) and 13(a), relative to different backlash size. The reason is that the first harmonics describe only the theoretical pattern of output acceleration, while the superposed dynamic effects, influenced by the mechanism conditions, correspond to a higher frequency range. As a consequence, the output acceleration spectrum could still be correctly estimated in all conditions by recovering components higher than the fourth revolution harmonic through the case vibration spectrum and the filter function, and by copying lower components from an output acceleration spectrum previously measured and stored in any condition. Applying this procedure brought very satisfactory agreement between recovered and directly measured output acceleration. Figures 5(b), (c) and 6(b), (c) compare time patterns, corresponding to one cam revolution exactly, for the two different backlash sizes and the same cam speed considered above. Figures 7(b), (c) and 8(b), (c) present the corresponding spectra. In the case of large backlash [Fig. 8(b), (c)], the recovered spectrum presents a lower amplitude level in the range 275 to 290 Hz. The consequence in the time domain is that the recovered pattern [Fig. 6(c)] is slightly smoother and with I.0
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Figure 12. (a) Measured and (b) recovered amplitude spectrum of output acceleration; cam revolution speed 500 rpm; sound conditions.
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Figure 13. (a) Measured and (b) recovered amplitude spectrum of output acceleration; cam revolution speed 500 rpm; increased backlash.
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less steep steps at the internal shocks (about 40 and 90 ms), with respect to the directly measured one [(Fig. 6(b)]. In spite of these differences, the recovered patterns are able to distinguish clearly between the two conditions (different backlash size) and are, therefore, a useful estimation of the actual patterns for monitoring purposes. 4.2. OUTPUT MOTION MONITORING On the basis of the above considerations and results, the following monitoring methodology can be proposed. For each individual machine in sound condition, the filter function relating the accelerations of a point on the case and of the output member, is evaluated experimentally and stored for every subassembly to be monitored. The output member spectrum in the range corresponding to the first revolution harmonics is also stored at the same time. During operation, acceleration of the selected point on the case is picked up, its frequency spectrum computed and the spectrum of output member acceleration estimated through the filter function (copying low frequency components from the experimental spectrum stored in sound conditions). Finally, inverse Fourier transforming, the time pattern acceleration of the output member can be recovered and compared with the functional limits. 5. CONCLUSIONS
A methodology for condition monitoring of cam and other mechanisms for reciprocal or intermittent motion is proposed. It allows recovery of the output motion from the case acceleration measurement. Thus, it is possible to directly judge whether the actual output motion is acceptable with respect to the mechanism function. Contrary to the usual methodologies based on the analysis of case vibration, the proposed methodology (a) does not involve difficult, long-lasting tests for preliminary establishment of warning levels and (b) is not affected by differences in the structural response between nominally equal machines. Although the last consideration has not been experimentally proved until now, it is well-founded, since the filter function, evaluated for each individual subassembly, takes individual differences in transfer function into account. Further tests on different subassemblies and in different conditions are required in order to establish the general effectiveness of the method. A numerical simulation of the mechanism dynamic behaviour was also presented and its application in diagnostics outlined. ACKNOWLEDGEMENT
This work was partially supported by a grant from the Italian Ministry of University and Research (MURST). REFERENCES 1. P. H. MAEDEL,
2. 3. 4. 5. 6.
JR. and R. L. ESHLEMAN 1988 Shock and Vibrution Handbook, Chap. 19: Vibration Standards. New York: McGraw-Hill. R. H. LYON 1987 Machinery Noise and Diagnostics. Boston: Butterworths. K. UMEZAWA, H. HOUJOH and H. MAKI 1988 JSME International Journal 31, 588-592. Estimation of the vibration of in-service gears by monitoring the exterior vibration. G. DALPIAZ and U. MENEGHEITI. NDT&E InternationaL Monitoring fatigue cracks in gears (to be published). A. 0. ANDRISANO and G. DALPIAZ 1990 Atti X Congr. Naz. AZMETA, Piss, Italy, pp. 633-638. A MDOF model for the dynamic analysis of drives with Geneva mechanism. (In Italian.) G. DALPIAZ and U. MENEGHE’ITI 1990 Condition Monitoring-Froceedings of the 3rd Zntemutional Conference. Detection and modelling of fatigue cracks in gears, pp. 73-82. London: Elsevier.
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7. S. LEVY and J. P. D. WILKINSON 1976 The Component Element Method in Dynamics. New York: McGraw-Hill. 8. A. RIVOLA and G. DALPIAZ 1992 DIEM, Universita’ di Bologna. Dynamic analysis of a mechanism for automatic machine. (In Italian.) 9. K. L. JOHNSON 1985 Contact Mechanics, pp. 129-131. Cambridge: Cambridge University Press. 10. M. P. KOSTER 1974 Vibrations of Cam Mechanisms. London: MacMillan. 11. R. ALLEMANG, D. BROWN and R. ROST 1988 Proceedings of 13th International Seminar on Modal Analysis, Katholieke Universiteit Leuven, Basic Course, Part I, pp. 52-79. Frequency response function estimation.