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Experimental modal analysis using a tracking interferometer
G Model
CIRP-1242; No. of Pages 4 CIRP Annals - Manufacturing Technology xxx (2014) xxx–xxx
Contents lists available at ScienceDirect
CIRP Annals - Manufacturing Technology jou rnal homep age : ht t p: // ees .e lse vi er . com /ci r p/ def a ult . asp
Experimental modal analysis using a tracking interferometer Christian Brecher (1), Stephan Ba¨umler, Alexander Guralnik * Laboratory for Machine Tools and Production Engineering (WZL), RWTH Aachen University, Germany
A R T I C L E I N F O
A B S T R A C T
Keywords: Vibration Measurement Machine tool
Experimental modal analysis is carried out for machine tools and in various other industries. It is used for dynamic characterization, diagnostics, condition monitoring, design optimization and as comparison data for computation results. 3D-scanning laser Doppler velocimeters and three-axial accelerometers dominate the experimental modal analysis today. This paper presents a method for modal analysis using a tracking interferometer as well for geometric measurement as for simultaneous 3D-vibration measurement with a single laser beam. A holistic measurement procedure is developed, its performance and robustness against external influences is investigated and a benchmark against accelerometers is carried out at a machining center. ß 2014 CIRP.
1. Introduction Dynamic properties of machine tools and production equipment have constantly been an important topic within CIRP for decades. Vibrations that lead to considerably reduced productivity, increased tool wear, short necessary service intervals, reduction of product quality and increase of production set-up expenses make experimental investigation of machine dynamics necessary [1]. The rapid growth in use and capabilities of simulation fuels the expanding need for experimental investigation for model verification [2]. Particularly, the high uncertainty in interface modeling and the lack of knowledge of damping properties results in the inevitable need for verification of simulation results with experimental data [3]. With the spreading use of modal analysis of production machines there is a need for rapid, efficient and versatile investigation methods. Most often a modal analysis of machine tools is carried out using three-axial accelerometers. For a medium sized machining center up to three hundred points may be measured corresponding to nearly one thousand degrees of freedom. The corresponding time distribution is shown in Fig. 1a. A rough geometrical wireframe model of the investigated machine is built up in step
1. Extraction of this geometric model out of CAD-data remains rather an exception and the precision of the geometry measured with a tape measure is usually poor. For experimental modal analysis of production machines usually only a few accelerometers are used making successive relocation of the sensors necessary to cover all model points. After fixing the accelerometer to the structure the user determines the sensor orientation in machine coordinates, enters it into a modal analysis software and finally waits for the vibration response to be measured before relocating the sensor again. This procedure, as shown in Fig. 1b, results in the actual net vibration measurement time being only a small part of the overall time expense. 3D-laser Doppler velocimeters (LDV) are used in automotive and airspace industries to measure open sheet surfaces. A 3D-LDV comprises three units with a uniaxial velocimeters and a beam aiming device each. Focusing all three beams on the same point a 3D-LDV measures vibration in three beam directions and transforms it to machine coordinates. This device quickly covers large surfaces in a contactless measurement and simplifies the model generation. With the three laser beams and a necessary minimum angle between them the requirement for a large corridor of sight constrains the application in housed structures as machine tools. An example of in-process measurement in machine tools with a 3D-LDV is given in [5]. 2. Modal analysis with single beam measurement technique
Fig. 1. Time distribution for a modal analysis of a machining center [4].
* Corresponding author.
Tracking interferometer (TI) are usually used for geometric measurements in large part assembly, freeform scanning and for calibration of robots and machine tools [6]. These devices are equipped for 3D-measurement in space, but do not have sufficient precision and sampling rate for vibration measurement on machine tools. The modal investigation procedure using a TI is introduced in [7]. There, the geometry measurement as well as the vibration measurement is carried out using a single modified TI. For the
http://dx.doi.org/10.1016/j.cirp.2014.03.131 0007-8506/ß 2014 CIRP.
Please cite this article in press as: Brecher C, et al. Experimental modal analysis using a tracking interferometer. CIRP Annals Manufacturing Technology (2014), http://dx.doi.org/10.1016/j.cirp.2014.03.131
G Model
CIRP-1242; No. of Pages 4 C. Brecher et al. / CIRP Annals - Manufacturing Technology xxx (2014) xxx–xxx
2
Table 1 FRAC comparison for sensor benchmark.
Fig. 2. Geometric model generation and single beam vibration measurement technique [7].
model generation the standard technique with two angle and absolute distance measurement is applied (Fig. 2a). This fulfills the low accuracy requirements of geometry measurement. The single beam vibration measurement technique uses the interferometer to measure vibration in beam direction. Lateral movement is measured using a two dimensional position sensing photo diode (PSD) (Fig. 2b). To achieve a sufficient sampling rate the analog PSD- and interferometer signals were sampled with auxiliary hardware with up to 30 kHz. These technique can resolve the micron and submicron vibration movement. Resolution is more important in this context then precision. 3. Considerations of uncertainty Every measurement result is biased by certain influencing factors. The Ishikawa-diagram in Fig. 3 clusters these factors. Influences of absolute distance measurement, reflector behavior and signal noise have been discussed in [4,7,11]. This paper concentrates on environment effects and rotatory axes precision. Human
accessibility
vibrations
temperature external light
relector
cross-talk
interferometer
beam length
Absolute distance measurement
error propagation
Environment
Method
DAQ-hardware rotatory axes
Device
Fig. 3. Clustering of factors influencing the measurement result.
3.1. Uncertainties in vibration measurement
Amplitude [μm/N]
Fig. 4 shows a benchmark of a frequency response of a machining center measured with a 1D-LDV, a displacement transducer (LVDT), an accelerometer as well as with the PSD and the interferometer of the TI. Toward the lowest frequencies the accelerometer has a low coherence where other sensors show very
Coherence [-] Phase [°]
LVDT
Accel.
PSD
Interf.
96.66
98.84
98.11
97.37
98.84
good measurement results. Older publications show acceptable PSD measurement results up to only 200 Hz [7], meanwhile the PSD shows a good performance up to 1000 Hz. Table 1 shows the frequency response assurance criterion (FRAC) for the benchmark measurement in Fig. 4. The FRAC can be calculated according to formula (1) [8]. It shows the equality between the frequency response functions (frf). As reference-frf serves the coherence weighted complex average calculated according to formula (2). FRACðHA ; HB Þ ¼
Href ¼
P 2 i g i Hi P 2 i
gi
T H H 2 B A ðHAT HA Þ ðHBT HB Þ i ¼ fA; B; :::g
0.2
(1)
(2)
i is the measurement with sensor i, H is the frequency response function, g2 is the coherence, T is the transposed, * is the Hermitian. The TI measurement devices show a performance comparable to the conventional sensors. Uncertainties in vibration measurement may arise through environmental influences. The actual wavelength l depends on environmental factors according to formulae (3)–(6) [9]:
l¼
l0 n; . . . ; n0
¼ 1:00027
1
Measurement Result
Position sensing diode
signal noise
air turbulances
physical properties
ground ixation
operating errors
LDV
(3)
dn dn ¼ 9:3 107 K 1 ) n dT
Object
capability
Sens. FRAC (%)
(5)ddnp ¼ 2:7 107 mbar
ðDT ¼ 15KÞ¼ ˆ 0:0014%
) dnn
dn dn 1 ¼ 0:4 107 mbar ) n de
(4)
ðD p ¼ 134 mbarÞ¼0:0036% ˆ ðDe ¼ 26:7 mbarÞ¼0:0001% ˆ
(6)
T is the air temperature, p is the air pressure, n is the refractive index of air, e is the partial water pressure. The derivative values are valid around the standard condition and the differences are calculated for typical shop floor environment variations. Though relevant e.g. for position measurement of machine tool axes, for vibration measurement environmental influences of slow variation rates are irrelevant and the sensors can be dispensed, if this leads to lower equipment costs or easier measurement set ups. A higher relevance is assigned to short term environment variation within the beam path and to external light. Fig. 5 shows the frf of a small size demonstrator machine in cross bed design, with HSK 32 tool interface, a massive tool dummy and axes travel ranges of X 560 mm, Y 240 mm and Z 200 mm. The machine was excited by a piezoelectric shaker in body diagonal direction. During
0.1 0.05
— LDV — LVDT — Accelerometer
0.02 0.01 0 -90
— PSD — Interferometer
-180 1 0.5 0
logarithmic linear 3
10
30
100
200300 00
4001000
600
800
1000
Frequency [Hz]
Fig. 4. Benchmark of vibration measurement.
Fig. 5. Influences of extraneous light and air turbulences.
Please cite this article in press as: Brecher C, et al. Experimental modal analysis using a tracking interferometer. CIRP Annals Manufacturing Technology (2014), http://dx.doi.org/10.1016/j.cirp.2014.03.131
G Model
CIRP-1242; No. of Pages 4 C. Brecher et al. / CIRP Annals - Manufacturing Technology xxx (2014) xxx–xxx
3
ψ [°] / Δψ [mm/m]
5.9/-.5
40
detailed view
* set point deviation -2.8/-.4
20
-.4/1.8
0 -2.3/1.0
-20
-·- reference __ mean deviation - - 3σ-variance
40
20
100
80
60
40
φ [°] / Δφ [mm/m]
a)
ψ - 3.9 [°] / Δψ [mm/m]
deviation of angles
60
20
0
40
b)
20
0
-20
-40
φ - 72.4 [°] / Δφ [mm/m]
Fig. 6. Linearity of the lateral displacement measurement. Fig. 8. Angular measurement uncertainty of TI.
Geometric measurement uncertainty of TI can be examined according to VDI guide line 2617-10 [10]. However, this guideline seeks an evaluation of whether or not the TI precision is acceptable. This work shows an overall characterization of deviations. The distance measurement precision has been shown in [10]. Angle precision is demonstrated here. Following investigations have been conducted on a machine tool as reference, the deviations are calculated in three steps. First the TI position, secondly the TI orientation and thirdly the angle deviation are calculated. Fig. 7a shows the measured points, best-fit TI position and residual distance deviations. The position and deviations were calculated from machine tool position data and the interferometric relative distance measurement. The calculated unknowns are initial laser beam length and the three position coordinates. Fig. 7b shows the TI position and the best-fit orientation as well as resulting 3D-deviations of point positions. The calculated unknowns are the three orientations angles as well as the zero elevation angle deviation. calculation of TI position and initial beam length Z [mm]
calculation of TI orientation and zenit axes zero position
Dy ¼ r sin u sinc Dy ¼ r sin D’ cos c
(7)
D’ ¼ a sinðsinu tancÞ
(8)
c is the elevation, u is the rectangularity deviation, Dy is the tangential deviation, Dw is the azimuth correction. After numeric correction of zero elevation angle and the rectangularity deviation according to formula (8), the angle measurement precision can be recalculated as shown in Fig. 9. Fig. 9 shows significantly smaller errors than Fig. 8. 60
-.4/-.6
40
detailed view
* set point deviation
-·- reference __ mean deviation - - 3σ-variance
40
-.02/-.4
20
20
.06/-.04
0
-20
a)
deviation of angles
ψ - 3.9 [°] / Δψ [mm/m]
3.2. Uncertainties in geometric measurement
around 3.98 elevation and around 72.48 azimuth. The variance is very low and the deviation unfold a very comprehensive picture that suggests a rectangularity deviation between the two rotatory axes. The rectangularity deviation u can be calculated to 9.87 mm/m. As the rectangularity deviation of the TI rotatory axes is known the measurements can be corrected according to formulae (7) and (8). Here, the tangential error resulting from rectangularity deviation is corrected by the azimuth axes.
ψ [°] / Δψ [mm/m]
the measurement the TI was blinded by a flash light and a red laser pointer. The air within the beam path was brought to turbulences using the depicted fan and the industrial blow-dryer, the latter producing an airflow of above 200 8C. The fan and the blow-dryer blew during vibration measurement into the laser beam path. The measurement distance was app. 1.5 m. The differences of the graphs lie nevertheless within the measurement uncertainty. Fig. 6 shows a test set up and results of the linearity examination of lateral measurement. A reflector is attached to a machine tool. The TI aims the reflector in origin position and holds this orientation. The machine tool positions in a square pattern of 0.05 mm in an area of 1.5 mm 1.5 mm. At each point the position is registered by the PSD and two inductive displacement sensors (LVDT) that serve as references. As can be seen from the analysis plot there is a circle of 1 mm diameter with non-linearity of 2% or less and a circle of 1.5 mm diameter with non-linearity of 3% or less. For typical vibrations amplitude during modal analysis this region can be regarded is sufficiently large.
-.05/.08
100
80
60
40
φ [°] / Δφ [mm/m]
20
b)
0
40
20
0
-20
-40
φ - 72.4 [°] / Δφ [mm/m]
Fig. 9. Angle deviation after numeric correction.
4. Case study For the case study the machining center in Fig. 10a is chosen. This is a traveling column medium sized housed milling machine. Within the case a model analysis was conducted with the conventional method, i.e. tape measure for model generation and a single three-axial accelerometer for vibration measurement,
Z [mm]
-100
-100
-300
-300
* set position TI -200 deviation of -200 Y [mm] distance -400 -400 X [mm] a) -600 x 1000
* set position
-200 3D-deviation Y [mm] -400x 100
b)
TI
-600
-400
-200 X [mm]
Fig. 7. Calculation of angular measurement uncertainty.
Fig. 8a shows the calculated angular measurement uncertainty of all measured points. Fig. 8b demonstrates the deviation and the 3s-variance for the ‘central cross’, i.e. exclusively the points
Fig. 10. Machining center and investigation time expenditure.
Please cite this article in press as: Brecher C, et al. Experimental modal analysis using a tracking interferometer. CIRP Annals Manufacturing Technology (2014), http://dx.doi.org/10.1016/j.cirp.2014.03.131
G Model
CIRP-1242; No. of Pages 4 C. Brecher et al. / CIRP Annals - Manufacturing Technology xxx (2014) xxx–xxx
4
Fig. 12. CrossMAC-matrix for modal analysis of the machining center.
Fig. 11. Eigen modes of the machining center.
differences of Eigen vector components of actually not vibration points and result hence in low crossMAC values. and with the single beam technique using a modified TI. 71 points were measured. For the single beam technique a set of nine measurement reflectors was used. The TI was replaced once. One point was measured with an accelerometer as the visibility was not given from either of these two measurement positions. Fig. 10b shows the time expenditure and distribution for the different tasks of the modal analysis. Many of the structure points were not accessible from outside the machine, so the operator had to climb in the machine to attach the accelerometer or the reflectors and even for some measurements with the tape measure. As the operator moves the accelerometer from point to point on the one hand but replaces the whole set of reflectors on the other hand, he climbs the machine a lot less often for the single beam technique. This results in a considerable time saving and in a less intensive physical effort. The physical effort for model generation is dispensed due to parallelization of the processes. Despite the adequate care in determination of accelerometer orientation the direction had to be corrected at three positions determining the time expenditure in the data check phase. Fig. 11 displays the Eigen modes of the machining center calculated from TI measurement results. Low-frequency modes are tilting modes of the entire machine and axes sliding modes. Midfrequency modes are mostly structural parts deformation. The two modes at highest frequencies are spindle modes. The deformation patterns calculated from accelerometer measurements are hardly distinguishable from the depicted ones. For this reason a quantitative comparison of the Eigen modes is presented in Fig. 12 as a crossMAC matrix. The crossMAC values are calculated according to formula (9) [8].
!
! T ! 2 ca cTI
!
crossMACðca ; cTI Þ ¼
!T
!
!T
!
(9)
ðca ca Þ ðcTI cTI Þ
c is the eigen vector, a is the accelerometer measurement. For modes up to approximately 500 Hz the crossMAC values remain mostly above 70%. This reflects a good correspondence between the different measurements. The two spindle modes, though very similar in the animated pictures, show very low crossMAC values. This is partly due to the circumstance that spindle modes have very low participation, i.e. most of the structure hardly vibrates in the spindle mode. Random differences in measurements and fitting procedures result in considerable
5. Summary This research work deals with the development of an investigation method for conducting experimental modal analysis with the single beam technique using a modified TI. The measurement performance of the current device was shown to be comparable to conventional sensor measurements of a machine tool up to 1000 Hz. The developed velocity demodulation technique allows high resolution and high velocity interferometric measurement using standard data acquisition hardware. Within a cause–effect diagram the relevant factors with influence on vibration and geometric measurement were summarized. A part of these effects was studied in detail, particularly the air temperature, pressure and humidity, air turbulances and extraneous light effects, as well as PSD-linearity and angle measurement precision. The case study of modal analysis of a machining center reveals the advantages of the single beam technique. References [1] Brecher C, Esser M, Witt S (2009) Interaction of Manufacturing Process and Machine Tool. Annals of the CIRP 58(2):588–612. [2] Ewins DJ (2000) Modal Testing Theory Practice and Application, Research Studies Press LTD, Baldock. [3] Brecher C, Fey M, Ba¨umler S (2013) Damping Models for Machine Tool Components of Linear Axes. Annals of the CIRP 62(1):399–402. [4] Brecher C, Ba¨umler S, Wissmann M, Guralnik A (2013) Single Beam Method for Three-Axial Vibration Measurement. in Blunt L, Knapp W, (Eds.) Procedures of the 10th LAMDAMAP, Euspen, pp. 39–48. [5] Loehe J, Zaeh MF, Roesch O (2012) In-Process Deformation Measurement of Thin-Walled Workpieces. Procedia CIRP 1:546–551. [6] Schwenke H, Knapp W, Haitjema W, Weckenmann H, Schmitt A, Delbressine RF (2008) Geometric Error Measurement and Compensation of Machines – An Update. Annals of the CIRP 57(2):660–675. [7] Brecher C, Guralnik A, Ba¨umler S (2011) Measurement of Structure Dynamics Using A Tracking-Interferometer. Production Engineering Research and Development 6:89–96. [8] Allemang R (2003) The Modal Assurance Criterion–Twenty Years of Use and Abuse. Sound and Vibration ;(August)V. [9] Kohlrausch F (1996) Praktische Physik, 2nd ed., vol. 2. Teubner Verlag, Stuttgart. [10] VDI 2617-10 (2011) Genauigkeit von Koordinatenmessgera¨ten, Kenngro¨ßen und deren Pru¨fung, Annahme- und Besta¨tigungspru¨fung von Lasertrackern . VDI/ VDE 2617 Blatt 10:2011. [11] Guralnik A, Brecher C, Schwenke H, Wissmann M, Paluszek H, Berk D (2012) Untersuchung der dynamischen Strukturverformung von Werkzeugmaschinen mit Tracking-Interferometern. 16. GMA/ITG-Fachtagung Sensoren und Messsysteme, AMA Science Protal386–396.
Please cite this article in press as: Brecher C, et al. Experimental modal analysis using a tracking interferometer. CIRP Annals Manufacturing Technology (2014), http://dx.doi.org/10.1016/j.cirp.2014.03.131
CIRP-1242; No. of Pages 4 CIRP Annals - Manufacturing Technology xxx (2014) xxx–xxx
Contents lists available at ScienceDirect
CIRP Annals - Manufacturing Technology jou rnal homep age : ht t p: // ees .e lse vi er . com /ci r p/ def a ult . asp
Experimental modal analysis using a tracking interferometer Christian Brecher (1), Stephan Ba¨umler, Alexander Guralnik * Laboratory for Machine Tools and Production Engineering (WZL), RWTH Aachen University, Germany
A R T I C L E I N F O
A B S T R A C T
Keywords: Vibration Measurement Machine tool
Experimental modal analysis is carried out for machine tools and in various other industries. It is used for dynamic characterization, diagnostics, condition monitoring, design optimization and as comparison data for computation results. 3D-scanning laser Doppler velocimeters and three-axial accelerometers dominate the experimental modal analysis today. This paper presents a method for modal analysis using a tracking interferometer as well for geometric measurement as for simultaneous 3D-vibration measurement with a single laser beam. A holistic measurement procedure is developed, its performance and robustness against external influences is investigated and a benchmark against accelerometers is carried out at a machining center. ß 2014 CIRP.
1. Introduction Dynamic properties of machine tools and production equipment have constantly been an important topic within CIRP for decades. Vibrations that lead to considerably reduced productivity, increased tool wear, short necessary service intervals, reduction of product quality and increase of production set-up expenses make experimental investigation of machine dynamics necessary [1]. The rapid growth in use and capabilities of simulation fuels the expanding need for experimental investigation for model verification [2]. Particularly, the high uncertainty in interface modeling and the lack of knowledge of damping properties results in the inevitable need for verification of simulation results with experimental data [3]. With the spreading use of modal analysis of production machines there is a need for rapid, efficient and versatile investigation methods. Most often a modal analysis of machine tools is carried out using three-axial accelerometers. For a medium sized machining center up to three hundred points may be measured corresponding to nearly one thousand degrees of freedom. The corresponding time distribution is shown in Fig. 1a. A rough geometrical wireframe model of the investigated machine is built up in step
1. Extraction of this geometric model out of CAD-data remains rather an exception and the precision of the geometry measured with a tape measure is usually poor. For experimental modal analysis of production machines usually only a few accelerometers are used making successive relocation of the sensors necessary to cover all model points. After fixing the accelerometer to the structure the user determines the sensor orientation in machine coordinates, enters it into a modal analysis software and finally waits for the vibration response to be measured before relocating the sensor again. This procedure, as shown in Fig. 1b, results in the actual net vibration measurement time being only a small part of the overall time expense. 3D-laser Doppler velocimeters (LDV) are used in automotive and airspace industries to measure open sheet surfaces. A 3D-LDV comprises three units with a uniaxial velocimeters and a beam aiming device each. Focusing all three beams on the same point a 3D-LDV measures vibration in three beam directions and transforms it to machine coordinates. This device quickly covers large surfaces in a contactless measurement and simplifies the model generation. With the three laser beams and a necessary minimum angle between them the requirement for a large corridor of sight constrains the application in housed structures as machine tools. An example of in-process measurement in machine tools with a 3D-LDV is given in [5]. 2. Modal analysis with single beam measurement technique
Fig. 1. Time distribution for a modal analysis of a machining center [4].
* Corresponding author.
Tracking interferometer (TI) are usually used for geometric measurements in large part assembly, freeform scanning and for calibration of robots and machine tools [6]. These devices are equipped for 3D-measurement in space, but do not have sufficient precision and sampling rate for vibration measurement on machine tools. The modal investigation procedure using a TI is introduced in [7]. There, the geometry measurement as well as the vibration measurement is carried out using a single modified TI. For the
http://dx.doi.org/10.1016/j.cirp.2014.03.131 0007-8506/ß 2014 CIRP.
Please cite this article in press as: Brecher C, et al. Experimental modal analysis using a tracking interferometer. CIRP Annals Manufacturing Technology (2014), http://dx.doi.org/10.1016/j.cirp.2014.03.131
G Model
CIRP-1242; No. of Pages 4 C. Brecher et al. / CIRP Annals - Manufacturing Technology xxx (2014) xxx–xxx
2
Table 1 FRAC comparison for sensor benchmark.
Fig. 2. Geometric model generation and single beam vibration measurement technique [7].
model generation the standard technique with two angle and absolute distance measurement is applied (Fig. 2a). This fulfills the low accuracy requirements of geometry measurement. The single beam vibration measurement technique uses the interferometer to measure vibration in beam direction. Lateral movement is measured using a two dimensional position sensing photo diode (PSD) (Fig. 2b). To achieve a sufficient sampling rate the analog PSD- and interferometer signals were sampled with auxiliary hardware with up to 30 kHz. These technique can resolve the micron and submicron vibration movement. Resolution is more important in this context then precision. 3. Considerations of uncertainty Every measurement result is biased by certain influencing factors. The Ishikawa-diagram in Fig. 3 clusters these factors. Influences of absolute distance measurement, reflector behavior and signal noise have been discussed in [4,7,11]. This paper concentrates on environment effects and rotatory axes precision. Human
accessibility
vibrations
temperature external light
relector
cross-talk
interferometer
beam length
Absolute distance measurement
error propagation
Environment
Method
DAQ-hardware rotatory axes
Device
Fig. 3. Clustering of factors influencing the measurement result.
3.1. Uncertainties in vibration measurement
Amplitude [μm/N]
Fig. 4 shows a benchmark of a frequency response of a machining center measured with a 1D-LDV, a displacement transducer (LVDT), an accelerometer as well as with the PSD and the interferometer of the TI. Toward the lowest frequencies the accelerometer has a low coherence where other sensors show very
Coherence [-] Phase [°]
LVDT
Accel.
PSD
Interf.
96.66
98.84
98.11
97.37
98.84
good measurement results. Older publications show acceptable PSD measurement results up to only 200 Hz [7], meanwhile the PSD shows a good performance up to 1000 Hz. Table 1 shows the frequency response assurance criterion (FRAC) for the benchmark measurement in Fig. 4. The FRAC can be calculated according to formula (1) [8]. It shows the equality between the frequency response functions (frf). As reference-frf serves the coherence weighted complex average calculated according to formula (2). FRACðHA ; HB Þ ¼
Href ¼
P 2 i g i Hi P 2 i
gi
T H H 2 B A ðHAT HA Þ ðHBT HB Þ i ¼ fA; B; :::g
0.2
(1)
(2)
i is the measurement with sensor i, H is the frequency response function, g2 is the coherence, T is the transposed, * is the Hermitian. The TI measurement devices show a performance comparable to the conventional sensors. Uncertainties in vibration measurement may arise through environmental influences. The actual wavelength l depends on environmental factors according to formulae (3)–(6) [9]:
l¼
l0 n; . . . ; n0
¼ 1:00027
1
Measurement Result
Position sensing diode
signal noise
air turbulances
physical properties
ground ixation
operating errors
LDV
(3)
dn dn ¼ 9:3 107 K 1 ) n dT
Object
capability
Sens. FRAC (%)
(5)ddnp ¼ 2:7 107 mbar
ðDT ¼ 15KÞ¼ ˆ 0:0014%
) dnn
dn dn 1 ¼ 0:4 107 mbar ) n de
(4)
ðD p ¼ 134 mbarÞ¼0:0036% ˆ ðDe ¼ 26:7 mbarÞ¼0:0001% ˆ
(6)
T is the air temperature, p is the air pressure, n is the refractive index of air, e is the partial water pressure. The derivative values are valid around the standard condition and the differences are calculated for typical shop floor environment variations. Though relevant e.g. for position measurement of machine tool axes, for vibration measurement environmental influences of slow variation rates are irrelevant and the sensors can be dispensed, if this leads to lower equipment costs or easier measurement set ups. A higher relevance is assigned to short term environment variation within the beam path and to external light. Fig. 5 shows the frf of a small size demonstrator machine in cross bed design, with HSK 32 tool interface, a massive tool dummy and axes travel ranges of X 560 mm, Y 240 mm and Z 200 mm. The machine was excited by a piezoelectric shaker in body diagonal direction. During
0.1 0.05
— LDV — LVDT — Accelerometer
0.02 0.01 0 -90
— PSD — Interferometer
-180 1 0.5 0
logarithmic linear 3
10
30
100
200300 00
4001000
600
800
1000
Frequency [Hz]
Fig. 4. Benchmark of vibration measurement.
Fig. 5. Influences of extraneous light and air turbulences.
Please cite this article in press as: Brecher C, et al. Experimental modal analysis using a tracking interferometer. CIRP Annals Manufacturing Technology (2014), http://dx.doi.org/10.1016/j.cirp.2014.03.131
G Model
CIRP-1242; No. of Pages 4 C. Brecher et al. / CIRP Annals - Manufacturing Technology xxx (2014) xxx–xxx
3
ψ [°] / Δψ [mm/m]
5.9/-.5
40
detailed view
* set point deviation -2.8/-.4
20
-.4/1.8
0 -2.3/1.0
-20
-·- reference __ mean deviation - - 3σ-variance
40
20
100
80
60
40
φ [°] / Δφ [mm/m]
a)
ψ - 3.9 [°] / Δψ [mm/m]
deviation of angles
60
20
0
40
b)
20
0
-20
-40
φ - 72.4 [°] / Δφ [mm/m]
Fig. 6. Linearity of the lateral displacement measurement. Fig. 8. Angular measurement uncertainty of TI.
Geometric measurement uncertainty of TI can be examined according to VDI guide line 2617-10 [10]. However, this guideline seeks an evaluation of whether or not the TI precision is acceptable. This work shows an overall characterization of deviations. The distance measurement precision has been shown in [10]. Angle precision is demonstrated here. Following investigations have been conducted on a machine tool as reference, the deviations are calculated in three steps. First the TI position, secondly the TI orientation and thirdly the angle deviation are calculated. Fig. 7a shows the measured points, best-fit TI position and residual distance deviations. The position and deviations were calculated from machine tool position data and the interferometric relative distance measurement. The calculated unknowns are initial laser beam length and the three position coordinates. Fig. 7b shows the TI position and the best-fit orientation as well as resulting 3D-deviations of point positions. The calculated unknowns are the three orientations angles as well as the zero elevation angle deviation. calculation of TI position and initial beam length Z [mm]
calculation of TI orientation and zenit axes zero position
Dy ¼ r sin u sinc Dy ¼ r sin D’ cos c
(7)
D’ ¼ a sinðsinu tancÞ
(8)
c is the elevation, u is the rectangularity deviation, Dy is the tangential deviation, Dw is the azimuth correction. After numeric correction of zero elevation angle and the rectangularity deviation according to formula (8), the angle measurement precision can be recalculated as shown in Fig. 9. Fig. 9 shows significantly smaller errors than Fig. 8. 60
-.4/-.6
40
detailed view
* set point deviation
-·- reference __ mean deviation - - 3σ-variance
40
-.02/-.4
20
20
.06/-.04
0
-20
a)
deviation of angles
ψ - 3.9 [°] / Δψ [mm/m]
3.2. Uncertainties in geometric measurement
around 3.98 elevation and around 72.48 azimuth. The variance is very low and the deviation unfold a very comprehensive picture that suggests a rectangularity deviation between the two rotatory axes. The rectangularity deviation u can be calculated to 9.87 mm/m. As the rectangularity deviation of the TI rotatory axes is known the measurements can be corrected according to formulae (7) and (8). Here, the tangential error resulting from rectangularity deviation is corrected by the azimuth axes.
ψ [°] / Δψ [mm/m]
the measurement the TI was blinded by a flash light and a red laser pointer. The air within the beam path was brought to turbulences using the depicted fan and the industrial blow-dryer, the latter producing an airflow of above 200 8C. The fan and the blow-dryer blew during vibration measurement into the laser beam path. The measurement distance was app. 1.5 m. The differences of the graphs lie nevertheless within the measurement uncertainty. Fig. 6 shows a test set up and results of the linearity examination of lateral measurement. A reflector is attached to a machine tool. The TI aims the reflector in origin position and holds this orientation. The machine tool positions in a square pattern of 0.05 mm in an area of 1.5 mm 1.5 mm. At each point the position is registered by the PSD and two inductive displacement sensors (LVDT) that serve as references. As can be seen from the analysis plot there is a circle of 1 mm diameter with non-linearity of 2% or less and a circle of 1.5 mm diameter with non-linearity of 3% or less. For typical vibrations amplitude during modal analysis this region can be regarded is sufficiently large.
-.05/.08
100
80
60
40
φ [°] / Δφ [mm/m]
20
b)
0
40
20
0
-20
-40
φ - 72.4 [°] / Δφ [mm/m]
Fig. 9. Angle deviation after numeric correction.
4. Case study For the case study the machining center in Fig. 10a is chosen. This is a traveling column medium sized housed milling machine. Within the case a model analysis was conducted with the conventional method, i.e. tape measure for model generation and a single three-axial accelerometer for vibration measurement,
Z [mm]
-100
-100
-300
-300
* set position TI -200 deviation of -200 Y [mm] distance -400 -400 X [mm] a) -600 x 1000
* set position
-200 3D-deviation Y [mm] -400x 100
b)
TI
-600
-400
-200 X [mm]
Fig. 7. Calculation of angular measurement uncertainty.
Fig. 8a shows the calculated angular measurement uncertainty of all measured points. Fig. 8b demonstrates the deviation and the 3s-variance for the ‘central cross’, i.e. exclusively the points
Fig. 10. Machining center and investigation time expenditure.
Please cite this article in press as: Brecher C, et al. Experimental modal analysis using a tracking interferometer. CIRP Annals Manufacturing Technology (2014), http://dx.doi.org/10.1016/j.cirp.2014.03.131
G Model
CIRP-1242; No. of Pages 4 C. Brecher et al. / CIRP Annals - Manufacturing Technology xxx (2014) xxx–xxx
4
Fig. 12. CrossMAC-matrix for modal analysis of the machining center.
Fig. 11. Eigen modes of the machining center.
differences of Eigen vector components of actually not vibration points and result hence in low crossMAC values. and with the single beam technique using a modified TI. 71 points were measured. For the single beam technique a set of nine measurement reflectors was used. The TI was replaced once. One point was measured with an accelerometer as the visibility was not given from either of these two measurement positions. Fig. 10b shows the time expenditure and distribution for the different tasks of the modal analysis. Many of the structure points were not accessible from outside the machine, so the operator had to climb in the machine to attach the accelerometer or the reflectors and even for some measurements with the tape measure. As the operator moves the accelerometer from point to point on the one hand but replaces the whole set of reflectors on the other hand, he climbs the machine a lot less often for the single beam technique. This results in a considerable time saving and in a less intensive physical effort. The physical effort for model generation is dispensed due to parallelization of the processes. Despite the adequate care in determination of accelerometer orientation the direction had to be corrected at three positions determining the time expenditure in the data check phase. Fig. 11 displays the Eigen modes of the machining center calculated from TI measurement results. Low-frequency modes are tilting modes of the entire machine and axes sliding modes. Midfrequency modes are mostly structural parts deformation. The two modes at highest frequencies are spindle modes. The deformation patterns calculated from accelerometer measurements are hardly distinguishable from the depicted ones. For this reason a quantitative comparison of the Eigen modes is presented in Fig. 12 as a crossMAC matrix. The crossMAC values are calculated according to formula (9) [8].
!
! T ! 2 ca cTI
!
crossMACðca ; cTI Þ ¼
!T
!
!T
!
(9)
ðca ca Þ ðcTI cTI Þ
c is the eigen vector, a is the accelerometer measurement. For modes up to approximately 500 Hz the crossMAC values remain mostly above 70%. This reflects a good correspondence between the different measurements. The two spindle modes, though very similar in the animated pictures, show very low crossMAC values. This is partly due to the circumstance that spindle modes have very low participation, i.e. most of the structure hardly vibrates in the spindle mode. Random differences in measurements and fitting procedures result in considerable
5. Summary This research work deals with the development of an investigation method for conducting experimental modal analysis with the single beam technique using a modified TI. The measurement performance of the current device was shown to be comparable to conventional sensor measurements of a machine tool up to 1000 Hz. The developed velocity demodulation technique allows high resolution and high velocity interferometric measurement using standard data acquisition hardware. Within a cause–effect diagram the relevant factors with influence on vibration and geometric measurement were summarized. A part of these effects was studied in detail, particularly the air temperature, pressure and humidity, air turbulances and extraneous light effects, as well as PSD-linearity and angle measurement precision. The case study of modal analysis of a machining center reveals the advantages of the single beam technique. References [1] Brecher C, Esser M, Witt S (2009) Interaction of Manufacturing Process and Machine Tool. Annals of the CIRP 58(2):588–612. [2] Ewins DJ (2000) Modal Testing Theory Practice and Application, Research Studies Press LTD, Baldock. [3] Brecher C, Fey M, Ba¨umler S (2013) Damping Models for Machine Tool Components of Linear Axes. Annals of the CIRP 62(1):399–402. [4] Brecher C, Ba¨umler S, Wissmann M, Guralnik A (2013) Single Beam Method for Three-Axial Vibration Measurement. in Blunt L, Knapp W, (Eds.) Procedures of the 10th LAMDAMAP, Euspen, pp. 39–48. [5] Loehe J, Zaeh MF, Roesch O (2012) In-Process Deformation Measurement of Thin-Walled Workpieces. Procedia CIRP 1:546–551. [6] Schwenke H, Knapp W, Haitjema W, Weckenmann H, Schmitt A, Delbressine RF (2008) Geometric Error Measurement and Compensation of Machines – An Update. Annals of the CIRP 57(2):660–675. [7] Brecher C, Guralnik A, Ba¨umler S (2011) Measurement of Structure Dynamics Using A Tracking-Interferometer. Production Engineering Research and Development 6:89–96. [8] Allemang R (2003) The Modal Assurance Criterion–Twenty Years of Use and Abuse. Sound and Vibration ;(August)V. [9] Kohlrausch F (1996) Praktische Physik, 2nd ed., vol. 2. Teubner Verlag, Stuttgart. [10] VDI 2617-10 (2011) Genauigkeit von Koordinatenmessgera¨ten, Kenngro¨ßen und deren Pru¨fung, Annahme- und Besta¨tigungspru¨fung von Lasertrackern . VDI/ VDE 2617 Blatt 10:2011. [11] Guralnik A, Brecher C, Schwenke H, Wissmann M, Paluszek H, Berk D (2012) Untersuchung der dynamischen Strukturverformung von Werkzeugmaschinen mit Tracking-Interferometern. 16. GMA/ITG-Fachtagung Sensoren und Messsysteme, AMA Science Protal386–396.
Please cite this article in press as: Brecher C, et al. Experimental modal analysis using a tracking interferometer. CIRP Annals Manufacturing Technology (2014), http://dx.doi.org/10.1016/j.cirp.2014.03.131