1 Introduction

Alongside the simulated process behaviour at the point of machining, the description of the dynamic compliance of a machine system is the decisive parameter for the simulation of a process-machine interaction (PMI) [6]. Modelling the machine’s compliance is often done using frequency response functions which are measured metrologically, as described for example in [8, 15, 20, 25]. Within the whole frequency spectrum required to depict the machine dynamics, multiple causes and effects in the measurement chain lead to a determination of the compliance behaviour which is riddled with uncertainty. Using these measurement results as input parameters, the quality of the results of downstreamed simulation applications is compromised e.g. the simulation of process stability in processing operations. Methods for the analytical error estimation of the metrologically obtained compliance behaviour are explained in [2, 16, 22, 24]. In several projects of the priority programme 1180 “Prediction and influence of the process-machine-interaction”, the question is discussed, in which way a variation of the reliability of the measurement data influences the depiction of the dynamic machine behaviour in stability simulation. A similar problem concerning the quantitative significance of frequency response functions was already discussed in [14]. In [1] both practical errors in the modal analysis, e.g. the alignment of the sensor, as well as possible sources of errors in the downstreamed signal processing are explored. Referring to this in [21] statistical uncertainty in the measurement of the modal parameters in relation to relative and absolute excitation was discussed.

An interlaboratory comparison test which was initiated and coordinated by the WZL at RWTH Aachen University, involving a total of nine university institutes (Fig. 1), was intended to determine and demonstrate potential effects and scale of influence on the metrological measurement of structural dynamic behaviour. The basis for this is the testing of an integral steel work piece, which was metrologically investigated regarding its dynamic compliance behaviour by all of the mentioned university institutes. Similar interlaboratory comparison tests were carried out back in the late 80s such as those on aluminium components described in [7], giving a comparison with the state of the art concerning the measurement technique for depicting the dynamic compliance behaviour at that time.

Fig. 1
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Interlaboratory comparison test in DFG priority program 1180—ProWeSP

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2 Construction of the demonstration work piece

The demonstration work piece (Fig. 2) was designed for the purposes of the interlaboratory comparison test, such that its dynamic compliance behaviour reproduces two definite and separated mode shapes. To keep the measurement results comparable for different sampling frequencies of the measurement systems used, both mode shapes were designed strongly damped. This means that the resonance peaks are formed across a broad frequency range and the absolute value of amplitude peaks can be mainly measured independent of the resolution of the measured frequency spectrums.

Fig. 2
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Construction of the demonstration work piece

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The dynamic structural behaviour of the demonstration work piece is represented by the use of two mass elements. Both masses are linked to each other and to the mounting plate by asymmetrically aligned flexible joints, which are used to determine the rigidity properties of the oscillating system. The flexible joints are an integral part of the work piece and as such they were milled with this out of a solid block of material.

Before carrying out the comparison test, preliminary frequency response measurements using an impulse hammer showed that the structural damping of the work piece coming predominantly from the flexible joints was inadequately dimensioned. Even after several seconds of measurement duration the amplitude of the structural vibration did not decrease to an acceptable level, so that the frequency response measurement could initially not be realized under ideal test conditions. Alongside the metrological investigation of the demonstration work piece, the dynamic compliance behaviour was being tested by simulation. For this purpose, the demonstration work piece was modelled in the FE-simulation environment I-DEAS®. Using the derived rigidity matrices of the work piece structure the calculation of the frequency response function was carried out in the multibody simulation system ADAMS®. For centric excitement of the demonstration work piece the frequency spectrum shows a clear resonance frequency of 76 Hz, due to a tilt of the upper mass element around the x-axis. The tilting movement is superimposed by a shift of the lower mass element relative to the mounting plate along the z-axis. If the work piece is excited eccentrically, a further resonance point can be detected at 185 Hz, additionally to the mode shape at 76 Hz. This mode shape is caused by a torsional oscillation of the upper mass element around the y-axis.

Based on the measurement and simulation results so far, the structural damping of the demonstration work piece was actively increased, in order to provide a damping behaviour of the structural oscillations suitable for metrological investigation. For this purpose, both mass elements as well as the mounting plate were additionally coupled to each other by a damping plate mounted onto the right side of the demonstration work piece. An elastomeric material was clamped between the damping plate and the mass elements to increase the damping effect of the structure.

After improving the design of the demonstration work piece no further simulation of the damped compliance behaviour was carried out, as the parameterization of the damping behaviour in the flexible joints is generally associated with a large degree of uncertainty [1, 21]. The dominant eigenfrequencies of the damped structure as well as the mode shapes which occurred were determined metrologically at 140 and 370 Hz (Fig. 3). For the metrological investigation a piezoelectric shaker was used. To mount the shaker it was braced between the demonstration work piece and an external support, so that the structure was statically preloaded. Thus, variations in frequency and amplitude can be recognised compared to the absolute excitation with the impulse hammer without static preload. The metrological testing was carried out following the procedures presented in the standard works [3, 11, 20, 25, 26].

Fig. 3
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Pre-determined compliance behaviour of the dampened demonstration work piece

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3 Participants’ metrology/measurement systems

3.1 Mounting the demonstration work piece

The way in which the demonstration work piece is set up has a significant impact on its static compliance. Due to the setup, one or more mode shapes are created, where the demonstration work piece oscillates around the mounting/fixing point. The frequency and damping of these mode shapes largely depend on the geometry and arrangement of the clamping devices as well as the clamping load. As a predefined test setup which applied for all participants the component had to be mounted on a rigid plate with at least four clamps or screws. This specification was still not precise enough allowing the participants to create significantly different setups. The different mounting configurations are shown in Fig. 4. These mounting configurations all have in common that they are fixed using the four corners of the demonstration work piece base plate.

Fig. 4
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Schematic diagram of the component mountings carried out

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3.2 Measurement equipment used

For measurement of the vibration response of the component mainly piezoelectric acceleration sensors from various manufacturers such as PCB, Bruel&Kjaer, Kistler and Endevco were used. The frequency ranges of the used sensors were all from 1 to 5 Hz or above thus comprising the investigated frequency range of 2 kHz. The sensitivities range from 10 to 500 mV/g. For excitation various impact hammers with integrated force sensors, for example Kistler Type 9726A and Bruel&Kjaer Type 8202, as well as electrodynamic and piezo actuated shakers (MB Dynamics Modal 50, Tira TV5220, PI Type P242-10) were used.

Also for signal analysis the participants used different hardware. On the one hand, universal frontends such as LMS-Scadas III, Bruel&Kjaer Type 3560C or Focus frontends with integrated AD-converter, analogue low pass filter and ICP supply unit were used as well as setting up the measurement chain with individual components.

To mount the sensors on the demonstration work piece magnets, wax and instant adhesive (cyanoacrylates) were used resulting in differences in the rigidity of the joints in the contact between the sensors and the structure. A sensor fixed with wax or instant adhesive shows a higher degree of contact rigidity than one attached using a magnetic mount. The effects of different contact rigidities can, however, be ignored in this case as the contact eigenfrequencies are above 5 kHz for all of these mounting methods. The influence of different thicknesses of wax layers had already been examined in [12]. Other deleterious influences in the fixing, mounting and calibrating of acceleration sensors are presented in [1, 4, 9, 17, 19, 23].

To excite the demonstration work piece impulse hammers and electro-dynamic, electro-hydraulic and piezoelectric exciters were all used. The impulse hammers have an integrated piezoelectric sensor to measure the force of the impact. With electro-dynamic or electro-hydraulic exciters, piezoelectric force sensors or strain gauges are applied, too.

When the structure was excited by the impulse hammer, hammer tips of different hardness were involved. Among others, a special impulse pendulum was used, where a mechanism coupled by a torsion spring allows an easy-to-reproduce level of force to be applied (Fig. 5, right). This made it possible to avoid the irregularities of force and direction of strike that occur using the “hand-held” hammer excitation.

Fig. 5
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Test structure with piezo shaker and impulse pendulum

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When electro-dynamic absolute exciters were used, these were linked to the structure using an axially rigid, laterally flexible rod (stinger). The relative excitation employing the piezo shaker was done by fixing this to the demonstration work piece using an angle clamp (Fig. 5, left). This allowed the influence of different static pre-loads to be tested. Excitation using the shaker has several advantages over pulsed excitation by means of the hammer. For example, non-linearities in the structure can be detected through excitation with a random burst signal. Sinus sweep and stepped sinus excitations on the other hand are suitable for checking the linearity of a structure and pinpointing the excitation amplitude dependent non-linearities.

When using the shaker to excite the structure it must be taken into account that the structural behaviour can be changed, for example by the attachment of the rod. This can lead to problems in evaluation with multi-reference procedures where multiple reference measurements are carried out using a single shaker. Depending on the connection of the shaker there may be frequency shifts. In addition, when using a rod to transfer forces it must be ensured that no momentum or shear force is introduced to the structure due to the mode shapes of the rod itself. These additional stimulating forces will not be captured by most uni-axial force sensors.

For the processing and evaluation of the measured signals, again very different measurement equipment was used by the participants in the test. Measuring chains made up of individual amplifiers, analogue filters and A/D-convertors were configured as well as measurement cards, and front-ends were being used, where two or more of these functions are combined. The margins of error applied to the measurement signal during its processing and A/D-transforming are discussed and estimated in [1, 3, 5, 10, 13, 18].

4 Results/discussion

4.1 Presentation of the measurement results

In Fig. 6 the measured frequency response function (FRF) and phase responses for centric excitation are shown with a detailed view of the dominant mode shapes. The average frequency response function here shows the compliance of a two mass oscillator, whose parameterization corresponds to the mean values of the metrologically measured modal parameters of the demonstration work piece. Figures 6, 7 and 8 demonstrate the deviations in FRF measurement of the interlaboratory comparison test. In first instance, the results are presented without specifying the institutes responsible for the measurement or the boundary conditions influencing the results e.g. excitation method or mounting of the work piece. The influences causing these deviations are discussed in the following paragraphs.

Fig. 6
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Frequency response function and phase responses for centric excitation

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Fig. 7
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Eigenfrequencies measured

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Fig. 8
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Damping factors derived from the measurements

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Figures 7 and 8 depict the influence of measurement equipment and test setup on the eigenfrequencies and damping ratios measured by the different institutes. The damping values were determined in frequency domain using evaluation functions of the signal processing software. The dynamic compliances in the resonance peaks are not shown in a separate figure, since the deviations are similar to the deviations of the damping values (Fig. 8). It can be seen that the characteristic parameters which are influenced by the system damping show much wider variations than the inertia and rigidity dependent values of the eigenfrequencies. This statement is also true for the measurement results using an eccentric excitation of the demonstration work piece. The variation of the eigenfrequency of the first mode shape is clearly higher than that of the second one (Fig. 7). In determining the compliance magnitudes there are significantly higher variations for the second eigenfrequency. Figure 7 shows that for one measurement with eccentric excitation the eigenfrequency of the second mode differs from the other measurements and has been determined at approximately 270 Hz. This measurement has been taken as an outlier and has not been considered for the calculation of mean values and standard deviations.

The metrological investigations were carried out by qualified technical staff, so that measurement errors (e.g. attaching sensors, incorrect connections or calibration of sensors, etc.) can be neglected to the greatest possible extend. Statistically occurring sources of error were reduced by each participant by averaging several measurements, so that the deviations shown in Figs. 7 and 8 can be seen as being primarily systematic errors. These are partly due to the use of different measurement equipment and different measurement methods. Despite a common specification of the constraints for carrying out the measurements, there could still be differences in the mounting of the demonstration work piece, the way the excitation force was applied as well as other influences which could change the dynamic behaviour of the work piece.

4.2 Investigation of metrological factors

Different metrological factors have an influence on the acquired measurement data. The characteristic of the dominant mode shapes of the demonstration work piece are used as the basis for evaluating the scale of the impact of metrological factors. To keep the presentation of the results to a reasonable size, this will be restricted to some selected constraints. All the following results were acquired by centric excitation of the work piece using an impact hammer mounted to the impact pendulum presented in Sect. 3. For some of the factors investigated no noticeable influence could actually be detected. For example, carrying out measurements using different acceleration sensors of different type and from various suppliers all gave almost identical frequency response functions. Also the duration of measurement does not influence the result when the number of acquired samples remains the same and when the impulse shaped force as well as the structural response are totally captured. For continuous excitation as well as for the impulse shaped structural excitation, the duration of measurement only affects the resolution of the recorded frequency spectrums when the sampling frequency is kept constant. As shown in Sect. 2, the damping properties of the demonstration work piece were deliberately set up in a way that the damping behaviour allows a clean capture of the amplitude peaks even with a fairly coarse frequency resolution.

The choice of a too low sampling frequency also leads to measurement errors. To avoid the so-called alias-effect the sampling frequency must be at least twice the highest frequency response contained in the signal being measured. To prevent the appearance of unwanted high-frequency signal content, this can be filtered out before the A/D-conversion using analogue low-pass filters. Figure 9 shows the frequency response function of the demonstration work piece for different sampling frequencies. The measured signal has not been low-pass filtered for the sake of demonstration. The sampling frequency of 512 Hz does not meet the requirement of the minimum sampling frequency in relation to the second eigenfrequency at 365 Hz. As a result, the frequency response is falsified. The frequency response function for a sampling frequency of 512 Hz was capped at the maximum frequency of the FFT of 256 Hz and is therefore not shown in the right-hand part of the figure. With a twice higher sampling frequency of 1,024 Hz there are only minor deviations in the amplitude for the second eigenfrequency. Once the sampling rate is four times higher there are almost no significant deviations between the measured frequency spectrums anymore.

Fig. 9
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Influence of the sampling frequency

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The magnitude of the force impulse has a decisive influence on the frequency response function being measured when exciting the work piece using an impulse hammer. This is particularly seen with the amplitude peak of the second eigenfrequency, Fig. 10. By increasing the maximum impulse force the dynamic compliance in the resonance peaks declines steadily. The same impact can be observed in the first resonance peak in an attenuated way.

Fig. 10
figure 10

Influence of the level of force impulse when exciting using an impulse hammer

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Figure 11 shows a similar impact on the frequency response functions for variation of common used hammer tips. The maximum of the pulsed force for each measurement was 1,000 N. The lower the rigidity of the hammer tip the longer the hammer and the object being measured remain in contact during a pulse. As a result of a longer contact time more energy is transferred in the lower frequency band of the vibration spectrum. Higher frequencies on the other hand are no longer excited by the pulse. The fact that the impact force has such an influence on the results leads to the conclusion that the demonstration work piece and/or the mounting behaves non-linear. This corresponds to the results of a comparison between shaker excitation and impact hammer excitation, Fig. 3. To ensure the comparability of the measurement results for all investigations shown, the impact pendulum described in Sect. 3 has been used, which allows to excite the structure by a reproducible force impulse level of 1,000 N in maximum.

Fig. 11
figure 11

Influence of the hammer tip on excitation using an impulse hammer

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As already mentioned in Sect. 3 the mounting of the demonstration work piece has a significant impact on the measured dynamic behaviour. The various forms of mounting tested and the frequency response functions related to each are shown in Fig. 12. Especially in the lower frequency band the influence is very strong. Using less than 3 fixing points leads to a dynamic compliance behaviour where the first eigenfrequency is no longer identifiable (Asp2, Asp4). The number of fixing points as well as the clamping load affects damping and stiffness properties of the mounting. This results in a shift of eigenfrequencies and magnitudes for the mounting version Asp3.

Fig. 12
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Influence of the demonstration work piece’s mounting

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4.3 Discussion of the results

Besides the relevance of these measurements for the interlaboratory comparison test presented here, these results also provide indications for evaluating possible sources of errors when measuring the dynamic compliance behaviour of complex structures in machine tools. The results of the systematic investigation of error sources in the measuring chain are shown in Fig. 13. In this figure the deviation of the amplitude peaks as well as the frequency shifts of the mode shapes are shown with dependencies on the influencing constraints. The figure shows that the dynamic compliance in the resonance points is subject to more scatter than their position in the frequency spectrum. The differences in the evaluated eigenfrequency in the first resonance mode are stronger compared to the frequency scatter of the second resonance point. The opposite is true for the dynamic compliance.

Fig. 13
figure 13

Influence of metrological conditions on the properties being measured

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The measurement results from the participants, which in part clearly diverged, cannot all be explained by the minor sources of error occasioned by the measurement chain alone. The frequency response functions acquired by the individual participants reflect the dynamic behaviour of a system which consists of the demonstration work piece itself, the mechanical excitation method, and the mounting. Changes in the behaviour of any one of these system components affect the behaviour of the entire system.

There will always be deviations in measurement results when measuring the dynamic behaviour of a mechanical structure, which do not result from errors in the measurement chain but from a changing dynamic structural behaviour of the object being measured and the measurement environment. If the experience from the investigations made in the interlaboratory comparison test is transferred to the measurement of the dynamic compliance behaviour of machine structures, then it becomes clear that scattering of measurement results also arises from changes and uncertainties in the behaviour of the machine system itself.

Examples might be changes in machine behaviour depending on process loads, positioning of the tool centre point in the workspace, ambient temperature etc., which are ignored for the metrological description of the system dynamics in general. This has a particular effect on the results of simulations which are carried out using metrologically estimated results as input parameters.

5 Impact on simulation results

In the following, the results of the metrological investigations serve exemplary as input data for the simulation of the interactions between machine structure and process. The significance of the simulation results depends on the quality of the model parameters applied—and therefore on how representative the measurement results are to depict the actual machine behaviour. In this context, it is irrelevant whether divergences between simulation input data and actual system behaviour are reasonable in measurement uncertainty or result from the fact that the system behaviour in process differs from the idealised conditions in the measurement setup, e.g. as a result of the process-constrained working position, or due to actual process loads.

How such divergences between assumed and real structural behaviour can effect simulation results will be shown in the following example of a so-called stability lobe diagram (Fig. 14). In stability lobe diagrams the maximum axial depth of cut, at which a machining process can still run under stable process conditions, is shown over the spindle speed. As an example, the milling of a full slot using a face milling tool with four cutting edges was considered. To model the machine dynamics in the simulation of the process-machine interactions the dynamic behaviour of the machine structure was represented by the compliance of the demonstration work piece. Different measurement results determined by participants of the interlaboratory comparison test were used to quantify the impact on the simulation results. The specific response functions shown in Fig. 14 (on the top) are chosen randomly from measurements using an excitation by an impact hammer.

Fig. 14
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Calculated stability lobe diagrams

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The divergences between the results of frequency response measurements are reflected in the predicted stability limits. The variations in the metrologically determined amplitude of the compliance behaviour have a specific impact on the achievable maximum depth of cut. On the other hand, variations in the frequency value of the measured eigenfrequencies generally cause a shift of the chatter peaks along the investigated speed range. The strong scatter of the amplitude peaks in the frequency response functions measured by the participants of the interlaboratory comparison test is also reflected in the stability charts. The minimum depth of cut varies by a factor of about 4 for the different simulation results.

6 Conclusion/summary

Using a demonstration work piece as part of the presented interlaboratory comparison test, initially awareness shall be created about the existence of measurement divergences in the frequency response analysis. These divergences appear in the measurements of a work piece structure although with clearly defined dynamic behaviour and with the use of defined test constraints. Serving as input data for downstreamed simulation applications measurement results which are not sufficiently precise to depict the dynamic machine behaviour under process conditions lead to an amplification of measurement errors in the simulation results.

The still used methods of frequency response measurement and modal analysis were not designed for high precision parameterization of simulation models, but are used for approximate quantification of the dynamic machine behaviour. The interlaboratory comparison test carried out shows this problem as an example. Against this background it is today more important than ever to reduce the sources of errors in the metrological assessment of the dynamic structural behaviour of a machine tool to a minimum.

There are two major aspects of interest which future research works have to take into account. First, further research is needed to clarify how different methods for measuring data acquisition, data processing and force transmission affect the results of the frequency response measurement on real machine systems. Closely linked to this is also the question of new measurement procedures and methodologies that allow a process-oriented evaluation of the dynamic machine behaviour considering varying parameters of dynamic compliance during the machining process.

A second aspect is the question about the reproducibility of the structurally dynamic compliance behaviour of a machine tool given under the influence of different measuring conditions and test setups. The latter area of interest particularly needs discussion from the point of view of constantly changing machine behaviour under process forces and positioning within the working space, which generally cannot be replicated satisfactorily under idealised measuring conditions. In particular to simulate process-machine interaction it will be necessary in future to find suitable methods of modelling dynamic machine behaviour considering varying parameters of dynamic compliance during the machining process.