Analysis of ground vibration transmission in high precision equipment by Frequency Based Substructuring
G. van Schothorst1 , M.A. Boogaard2 , G.W. van der Poel1 , D.J. Rixen2
1 Philips Innovation Services, Department Mechatronics Technologies
High Tech Campus 7, 5656 AE Eindhoven, The Netherlands
e-mail: Gert.van.Schothorst@philips.com
2
University of Technology, Faculty 3mE, Dep. Precision and Microsystems Engineering
Mekelweg 2, 2628 CD Delft, The Netherlands
Abstract
Machines with high accuracy that are sensitive to ground vibrations are generally designed using crude assumptions on the dynamic properties of the floor where they are placed. The effect of dynamic coupling
between floor dynamics and machine dynamics is generally omitted. This also holds for the prediction of
machine accuracy in the design phase, based on expected or measured ground vibration spectra. In this
paper, experimental dynamic substructuring methods are exploited to predict machine vibrations in situations where the machine is placed on a non-rigid, dynamic floor. More specifically, a new Transfer Path
Analysis is demonstrated based on the Frequency Based Substructuring technique for the case that ground
vibration levels are measured for free interface conditions. The method can be seen as the dual counterpart
to earlier presented approaches [1], where the disturbance vibrations have been measured in fixed interface
conditions (so-called blocked forces or equivalent forces). After proper coupling of the machine model with
the experimental characteristics of the floor dynamics, these ground vibrations are translated into machine
vibrations. The method is demonstrated on a practical implementation. A simplified experimental model,
similar to the dynamics present in high-precision machines, has been built and measured. From experimental
work performed, some lessons regarding the applicability of this method will be presented. In conclusion,
using the Frequency Based Substructuring method, more accurate performance prediction of (high precision)
equipment on factory floors is made possible, potentially saving costly and conservative design choices in
the machine design.
1 Introduction
It is well known that floor vibrations are one of the main disturbance sources for high precision equipment
[2, 3]. Especially for equipment in the semiconductor industry, like photolithographic machines, but also in
the field of metrology (e.g. electron microscopes), the impact of floor vibrations on the machine accuracy is
generally determining the dynamic architecture of the system [4]. This may imply design choices like the
selection of the vibration isolation system, machine frame mass, machine frame support stiffness and internal
machine dynamics characteristics.
Although there is ample attention for the role of the floor vibration level, often specified in terms of socalled VC-curves (Vibration Criteria) [3], the attention for the dynamic behavior of the floor itself, possibly
in combination with the machine dynamics, is only gradually growing. In precision engineering practice,
the design of the machine is often guided by dynamic modeling of the machine itself, either lumped mass
approach or using finite elements. The floor is assumed either infinitely stiff, or in better cases approximated
with a finite stiffness at the interface locations between machine and floor.
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One of the reasons behind is, that the equipment development industry has less (access to) knowledge of
building dynamics, and moreover it is not a priori known on which floor the machine will be placed. Nevertheless, from a dynamics point of view, it is straightforward to see the importance of including the effect
of floor dynamics in the machine dynamics analyses during the design. Firstly, the machine dynamics will
alter due to the fact that the machine is placed on a non-rigid floor. And secondly, the floor dynamics, and
therewith the effective floor vibration level, will alter due to the placement of a (heavy) machine on the floor.
Therefore, like the widespread use of VC-curves to specify floor vibration levels, it is considered to be useful
to have methods to further quantify and possibly specify floor dynamics, preferably based on experimental
data, in order to predict the coupled machine dynamics behavior. This paper intends to present and demonstrate a method, which allows for incorporation of floor dynamics in the analysis of high precision equipment
accuracy.
In section 2, the selected approach will be motivated and worked out in more detail, including a summary
of the underlying theory. After that, section 3 describes how the approach is applied to a simplified but
experimental test case. The results are discussed in section 4 and section 5 finally gives the conclusions.
2 Approach to incorporate ground vibrations
2.1
Motivation for the chosen approach
High precision motion systems can be represented by a simplified machine model like in figure 1. The
machine is connected to the floor with a finite stiffness; the big frame generally has a significant mass. A
guided motion system (small moving mass, ms ) is driven by an actuation system, which has its reaction
forces f exerted on the machine frame. A third part of the machine (mr ), which is connected with a finite
stiffness (relatively compliant spring) to the machine frame, acts as a reference frame, used to measure the
position U of the motion system relative to it.
kf
mr
f
U
ms
mf rame
kmount
kmount
Figure 1: Schematic representation of simplified machine model
When analyzing the effect of ground vibrations on machine performance during system design, obviously
with more sophisticated models of the machine to be designed than depicted above, there are a few rather
simplistic approaches for incorporating the contribution of the floor, as indicated in figure 2.
• Infinitely stiff floor. This approach is only valid if the machine is relatively light weight and the machine connection stiffness is relatively low as compared to the (local) floor stiffness. This is generally
not true for high precision equipment.
• Floor stiffness added to machine model. This approach is widely used in precision engineering practice. It takes the finite stiffness of the floor into account, which improves the prediction of machine
vibration modes due to the fact that the floor stiffness is in the same order of magnitude as the internal
stiffness of the machine, and sometimes even lower. An advantage of this method is that in many cases
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it is possible to experimentally identify the (local) floor stiffness with a hammer impact measurement.
However, the method still does not account for specific dynamic characteristics of the floor, which
may especially be of importance if the internal floor resonances are in the same frequency region as
the machine resonances.
• Single Degree of Freedom oscillator to represent floor dynamics. From literature on floor dynamics in
relation to machine performance [5], it is clear that a floor is better characterized with a mass, spring
and damper than with only a spring. Although this is not (yet) common practice in precision engineering, it will be shown that this rather simple approach would already improve prediction performance
for machine vibrations.
U
U
f
kmount
U
f
kmount
f
kmount
kmount
kf loor
kf loor
kmount
kmount
mf loor
kf loor
cf loor
Figure 2: State of the art approaches for incorporating floor vibrations in machine modeling: 1) imposing
floor vibrations from an infinitely stiff floor (left), 2) taking finite floor stiffness into account (middle), 3)
single degree of freedom dynamic model of floor (right)
Although from the approaches presented here, the third method is preferred, it still requires a proper quantification of the floor dynamics, e.g. by experimental identification or by specification to end-users. Furthermore, it still does not incorporate more complex floor dynamics including multi degree of freedom mode
shapes.
Therefore, a more generic approach is chosen to cope with the coupling between the machine dynamics and
the floor dynamics, as schematically depicted in figure 3. In this figure, the solid black rectangle represents
the floor dynamics (i.e. is a dynamic system in itself) and the upper part represents the machine dynamics.
Note furthermore, that a more generic approach will not restrict the machine model to the usual structure of
high precision equipment as sketched in figure 1, but will also be applicable to other systems like medical
imaging equipment, metrology systems, robots and other positioning devices that may suffer from floor
vibrations.
U
f
f loor
Figure 3: Schematic representation of machine on the floor
As explained in the following sections in more detail, we utilize the method of Dynamic Substructuring to
properly describe the dynamic coupling between the machine dynamics and the floor dynamics. Starting
point is firstly, that a machine model is assumed to be available, generally as a result of dynamic modeling
during the machine design phase. The machine model is presumed to have free interface conditions, i.e.
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the machine is modeled without a floor. In case of existing equipment or prototypes, these models may
be obtained from (modal analysis) measurements, although special attention is required to get proper measurements. Especially with heavy machines, it is hardly possible to obtain free interface measurements; the
machine should be softly suspended for that. An alternative is to measure the machine dynamics on an extremely rigid floor, or to use (piezo-electric) shakers with impedance sensors. Secondly, the floor dynamics
are generally not available as a numerical model, so it is assumed to be obtained from experimental responses
(measured without the machine present). Note here, that the experimental responses from the floor is typically obtained as free interface responses, as it will be impractical to provide a rigid fixation of the building
at the interface location. As a third ingredient in the chosen approach, it is assumed that the floor vibration
level of the building is characterized with power spectrum measurements before the machine is installed.
An alternative is, that a vibration criterion (e.g. VC-D) is taken as input spectrum for the coupled system
analysis.
The ultimate goal of the proposed approach is, to translate the measured ground vibration level (obtained
in free response of the floor dynamics, i.e. without machine present), into machine response, taking the
coupling between machine and floor dynamics into account.
2.2
Dynamic Coupling
Dynamic Substructuring allows analyzing of complex systems by analyzing its substructures. The description of the substructures can be in three different domains, the physical domain, the modal domain and the
frequency domain. The technique is equivalent for each domain. In this paper only the frequency domain
will be discussed. For more information on all three domains see [6].
To apply Dynamic Substructuring to the total system as shown in figure 3, the system will be divided into two
substructures. One substructure will represent the machine and the other will represent the floor, as shown in
figure 4. The region between these two substructures represents the interface, which will be discussed into
more detail. The mathematical background of the approach is worked out below.
um
o
um
i
U
fim
um
c
um
c
λ
λ
f loor
Figure 4: Schematic representation of machine on the floor
Consider the total system as given by
  


um
fim
Yiim Yiom Yicm 0
0
i
um  Y m Y m Y m 0
 fm 
0 
 o   oi

o
oo
oc

um  Y m Y m Y m 0
f m + g m 

0
=
 c   ci
c
c 
co
cc

 f 
 f
f 
 uc 
0
0
0
Yccf Ycif   fc + gc 
0
0
0 Yicm Yiim
ufi
fif
(1)
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Where the superscript m indicates that these entries are related to the machine and f indicates that these
entries are related to the floor. The subscript i denotes that these entries are related to the input DOF, o
denotes that these entries are related to the output DOF and c denotes that these entries are related to the
coupling DOF, as indicated in figure 4. The forces f are the externally imposed forces and g are the reaction
forces related to the coupling between these substructures, in figure 4 indicated with the blue arrows.
For the substructures in equation 1 to act like the total system, it must fulfill two conditions. The first
condition is known as the compatibility condition, which ensures that the distance between the interfaces
of the coupled substructures should be zero. The second condition is known as the equilibrium condition,
which ensures that the interface between each substructure is in equilibrium. Mathematically this is given by
f
um
c − uc = 0
gcm + gcf = 0
(2)
Because the reaction forces should be equal, but with an opposite sign, a Lagrange multiplier λ is used,
so gcf equals −λ and gcm equals +λ. Next the third and fourth row of equation 1 are substituted into the
compatibility condition and solved for λ. Now λ can be substituted into equation 1 and the total system can
be solved when it is assumed that all the transfer functions and applied forces are known. For this application
either or both the applied forces and the transmissibility function related to the floor are not known, so these
will be omitted here. These will be dealt with in the next section. So finally the dynamic coupling is given
by
 m
 
 m
0 ui
−1 fim
m m
m 
m
f
m
m
m

um
0 Ycc + Ycc
Yci Yco Ycc fo 
(3)
=Y f −Y
o
m
fcm
I
uc
Note that from this equation it follows that the dynamic coupling now only depends on the machine and the
interface flexibility of the floor.
2.3 Transfer Path Analysis
For some applications it is not possible to either measure the operational forces or the transmissibility from
the point where the operational forces are applied to the interface. To still be able to calculate the response
of the total system to these forces, a technique called Transfer Path Analysis (TPA) can be used.
In 2010 De Klerk and Rixen [1] published a TPA method based on a fixed interface. For this method
the interface of one substructure is fixed and the reaction forces are measured while this substructure was
operational. When this substructure is then assembled to a second substructure and the reaction forces are
applied as external forces on the interface, it can be shown that the response of the second substructure to the
applied reaction force is equivalent to the response of the coupled system to the original operational forces.
The response of the substructure where the excitation is located will however be different. This method was
developed to analyze how the gear forces of the differential of a car would change the noise inside the car.
This method is commonly known as the blocked force method.
Because it is quite impractical to fix the interface of the floor rigidly, another method should be used. Recently Rixen et al. [7] published an analogous method, based on the free interface. So instead of a fixed
interface, the interface is now completely free and instead of reaction forces, the free vibrations are measured.
Reconsider the system of equation 1. When the third and fourth row are substituted in the compatibility
condition and the Lagrange multiplier λ is substituted for the applied forces on the interface, the following
equation is obtained
 m  m
  m
Yii Yiom
Yicm
0
ui
fi
m Ym
m
um




Y
Y
0
fom 
oo
oc
 o  =  oi



(4)
 0  Y m Ycom Yccm + Yccf −Y f   λ 
ci
ci
ufi
fif
0
0
−Yicf
Yiif
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Note that there are no externally applied loads on the interface. These can however be applied easily, see
[7]. Next suppose that there is an imposed excitation, which can either be a force or a displacement, on the
internal DOF of the floor. Then the last row of equation 4 can be eliminated. Which in the case of an imposed
force results in
 m   m
  m
ui
Yii Yiom
Yicm
fi
m
m
m
 um




Yoc
fom 
(5)
= Yoi Yoo
o
f f
f
m
m
m
λ
Yci fi
Yci Yco Ycc + Ycc
Let us now define a new problem where there is a displacement differential δc is imposed on the interface
DOF. The compatibility condition is then defined as
f
um
c − uc = δc
(6)
In other words, the distance between the interface of the substructures is no longer zero, but a certain interface
gap is imposed. In this alternative problem, comparing to 4, the total set of equations is given by
 m  m
  m
Yii Yiom
Yicm
0
ui
fi
m
m
um
 Yoim Yoo
 fom 
Y
0
oc
o
 =
 
 δc  Y m Ycom Yccm + Yccf −Y m   λ 
ci
ci
0
ufi
0
0
−Yicf
Yiif
(7)
From equation 5 and 7 it can be easily seen that these are equivalent for what concerns the machine, when
the interface gap δc is equal to Ycif fif . If the same force is applied on the internal DOF of the floor when the
interface is free, it is easily shown that the free vibrations at the interface are equal to Ycif fif . In other words,
with reference to figure 5, if the free substructure A has a vibration level at the interface due to internal
forces like depicted in the left part of the figure, then the resulting vibrations of the coupled system, as far as
substructure B is concerned, are equal to those of the system depicted at the right, where the corresponding
interface gap displacements are imposed between the two substructures at the interface.
mB
2
mB
2
uB
o
k1B
δ̄c
mA
2
f¯iA
mB
1
uA
c
k2A
mA
1
k1A
uA
i
mA
2
f¯iA
uB
c
uA
c
k2A
mA
1
k1A
uB
o
k1B
≡
mB
1
uB
c
δ̄c
mA
2
uA
c
equ
uA
i
equ
k2A
uA
i
mA
1
k1A
Figure 5: Equivalence of interface gap with internal floor disturbance: Free interface vibrations (left), original
system (center) and equivalent system (right).
This basically summarizes the approach to predict machine vibrations under the presence of floor vibrations,
as they are measured under free interface conditions.
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3 Application to test case
For the test case described in Section 3.1, as a first step the floor dynamics have been characterized in absence
of the test set-up, as described in the Section 3.2. After that, the test-case was placed and the coupled response
was measured, as reported in Section 3.3. Finally, it will be described in Section 3.4 how the vibration levels
have been measured, first of the free floor vibrations and second of the test case placed on the floor with the
same disturbance, on which a Transfer Path Analysis was applied.
3.1
Description of test case
With the schematic machine model as indicated in figure 1 in mind, and considering that the free moving
body of the guided motion system is not the primary interest in this study, a test case has been designed which
basically simulates the machine. A schematic representation is shown in figure 6. This test case consists of
uro
ref.
kt
frame
km
cm
fi
cm
km
mount
Figure 6: Schematic representation of test case
three rigid bodies, one represents the mount which will be coupled to the floor, the second represents the
frame on which the reaction forces are applied and the third represents the reference frame. In this setting,
the main variables of interest are the vibration levels uro of the reference, while incorporating the dynamic
behavior of the floor, as a result of two disturbances:
• (reaction) forces on the machine frame. The vibration levels due to this disturbance are represented as
a frequency response function, by the accelerance of the system.
• (free) vibration levels of the floor. The vibration levels due to this disturbance are represented as a
power spectrum, given the presence of a certain floor disturbance.
The realized experimental set-up for the test case is depicted in figure 7. In this setup the frame is represented
by a large granite block of 1150 kg which is suspended from the floor by four compressed disc springs. The
total stiffness is about 107 N/m, which results in a vertical vibration mode at 15 Hz. The second mass, a block
of 110 kg, represents the reference. This mass is suspended on passive air mounts, which should provide a
suspension frequency of about 6 Hz.
Although experimental models (describing the multi DOF dynamics by frequency responses) of the realized
test set-up have been obtained based on accelerance measurements on an extremely rigid floor, these models
had limited quality for a number of reasons, like non-linearity in the disc springs of the suspension to the
floor. More details can be found in [8]. To demonstrate the applicability of the methods presented in section
2, the dynamics of the test set-up will be represented by a relatively simple numerical model in the remainder
of this section. The model is chosen like depicted in figure 6, but excluding the mass for the reference frame.
The parameters (mass, damping and stiffness) have been fitted on experimental responses, when the test setup was placed on an extremely rigid floor. Although the model has vertical translations and two tilt rotations
as degrees of freedom, only the vertical translations will be analyzed further in the next sections.
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Figure 7: Realized experimental set-up of test case
3.2
Experimental characterization of floor dynamics
To experimentally determine the response of a floor, there are several techniques available. In this case it
is chosen to obtain the response of the floor from impact measurements, using a large impact hammer from
PCB, type 086D50. A building typically has a lot of damping, so a couple of things should be taken care of
when performing impact measurements.
Because of the large damping, the frequency resolution might be a problem. The response in this case lasted
less than half a second, resulting in a frequency resolution of 2. For a typical factory floor the eigenfrequency
is between 10 and 20 Hz, so a much better frequency resolution is needed. When longer measurement blocks
are used, more background noise is measured. If this noise is harmonic, for instance caused by a pump, it
will appear as a resonance peak.
To minimize the response caused by the background noise, an exponential window can be used. Generally
an exponential window is known to increase the measured damping. For a floor measurement, another
problem occurs. Because floor measurements are always done in-situ, there are no rigid body modes, so
the acceleration at 0 Hz should be zero and the response for low frequencies is also very small. For the
measurement system used in this case, even at AC settings, there was still a small constant signal present.
Without an exponential window, this would not cause a problem, only a peak at 0 Hz. When an exponential
window is applied, this signal now depends on time. In the response function this will appear as if there are
rigid body modes. By removing the constant before the window is applied, this can be easily solved. Finally
a response as shown in figure 8 is obtained.
Besides the measured floor accelerance, figure 8 also gives the approximative floor characteristics according
to the methods discussed in section 2.1. It is clear, that the low frequency behavior of the floor is well
approximated by the stiffness characteristic (in this case a stiffness of 7.7·107 N/m), while the first resonance
of the floor at 12 Hz can be approximated by the SDOF oscillator response. On the other hand, it is also clear
that more modes are present in the floor (e.g. 15 Hz) that are not captured by the SDOF oscillator.
3.3 Experimental results of coupled response
To experimentally validate the method as explained in section 2.2, the model as described in section 3.1 is
used (see also figure 6). For this validation, the translational response of the frame caused by the input force
fi is predicted using the model and validated with experimental responses from the realized test case. To
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accelerance (ms−2 /N )
10−3
measurement
linearized stiffness
SDOF response
10−4
10−5
10−6
10−7
1
10
frequency (Hz)
50
Figure 8: Free interface floor dynamics measurement. Experimental accelerance of floor (solid); fitted single
degree of freedom oscillator response (dotted); approximate linearized stiffness (dashed)
obtain a representative model, the test case was first placed on a very rigid floor, on which the test case is
assumed to behave as if it is fixed at the mounts. On this measurement, the parameters for the model are
fitted.
Next this model is coupled to the floor measurements as obtained in section 3.2, which results in a prediction
of the response on this non-rigid floor. The actual response of the test case on this non-rigid floor is also
measured and provides a validation response. These three responses are shown in figure 9.
accelerance (ms−2 /N )
10−2
fixed
prediction
validation
10−3
10−4
2
5
10
frequency (Hz)
20
50
Figure 9: Machine dynamics accelerance response including coupling effects. Rigid floor (dotted), predicted
response on non-rigid floor (solid) and validated response on non-rigid floor (dashed).
From this figure it can be concluded that there is a lot of dynamic coupling around the first eigenfrequency
of the floor. Furthermore the amplitude of the first mode of the frame is less for the coupled response. It is
found that this technique is able to predict both these eigenfrequencies properly. For frequencies above 18
Hz, the predicted response is almost the same as the fixed response, which indicates that there is no dynamic
coupling above 18Hz. It was found that the experimental responses for the rigid floor and the non-rigid
floor are also equal, which indeed indicates that there is no dynamic coupling above 18 Hz. The difference
between the predicted and the measured response on the non-rigid floor (beyond 18 Hz) may be explained
by the mismatch between the fitted model and the experimental frequency responses of the test set-up.
3.4
Experimental results of Transfer Path Analysis
To validate the technique from section 2.3 (equation 7 and figure 5), both ingredients from the previous
two sections are needed, as well as the free vibration level of the (non-rigid) floor. To provide a constant
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disturbance for this measurement, a shaker with a small reaction mass was placed on the floor, near to test
case. This shaker was excited with a two-tone signal at 12 and 16 Hz.
To measure the floor vibrations, four accelerometers are placed next to the four mounts of the test case
and a time trace of 1600 seconds is recorded. The frequency resolution should of course be equal to the
frequency resolution of the other measurements, which is 0.25 Hz, so 400 blocks can be made from this
measurement. This measurement is done without the test case, to obtain the free vibration level δc , and with
the test case for a validation measurement. The response of all four signals is averaged to obtain only the
vertical translational vibrations. The free vibrations are used in equation 7 to predict the new vibration level
of the floor. The results are shown in figure 10.
10−6
power (m2 s−4 )
free
prediction
validation
10−7
10−8
10−9
10
11
12
13
14
15
16
frequency (Hz)
17
18
19
20
Figure 10: Validation of Transfer Path Analysis. Measured free floor vibrations (dotted), predicted vibrations
with coupling (solid) and validated vibrations with coupling (dashed)
From these results it appears that in the measurements there is hardly any change between the vibration
levels of the free interface floor and the coupled system where the test case is placed on the floor. The only
difference is at the two excited frequencies, which seems slightly amplified by the test case.
There is however a lot of difference predicted by the transfer path analysis method applied here. Only at 12
Hz the vibrations are predicted accurately, but at 16 Hz the vibrations are underestimated. Overall it can be
concluded that the method from section 2.3 is not yet validated with this experiment. A possible explanation
is that the floor may behave non-linear. The response of the floor has been obtained with an impact force
input of about 5 kN, whereas the forces in the validation experiment have been orders of magnitude smaller.
Apparently, the floor behaves much stiffer at these small signal levels, causing a strongly reduced impact of
the coupling between floor and machine dynamics due to the presence of the test case.
4
Comparison of floor coupling techniques
Although the proposed Transfer Path Analysis method could not yet be fully validated on the selected test
case, the experimental results on Frequency Based Substructuring (FBS) gave good confidence in being able
to predict the effect of floor dynamics on the coupled response [8]. Therefore, this method will now be
exploited to make a comparison between the methods described in section 2.1 at one hand, and the FBS
approach on the other hand. This will be done for dynamic coupling in section 4.1 and for the ground
vibrations in section 4.2.
Hereby, the machine dynamics will be represented by the model of figure 6, now including the additional
DOF for the reference frame. For the floor dynamics, the experimental floor characteristics from section 3.2
will be used.
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Coupled machine / floor model
In this section the three methods from section 2.1 will be compared to the method from section 2.2. From
the experimental results it was found that the Frequency Based Substructuring method was able to predict
the actual response fairly well [8] (see also figure 9), so this will be the reference case for what the actual
response will be and this will be referred to as the full coupling case. In this section the response is considered
from a force on the frame to accelerations of the reference frame, as shown in figure 6.
The fixed response of this model is very easily obtained by fixing the coupling DOF. For the method where
only the floor stiffness is taken into account, a spring is added for each physical coupling DOF, four in this
case, with a stiffness of 7.7 · 107 N/m, which is the same value as shown in figure 8. The three responses are
shown in figure 11.
accelerance (ms−2 /N )
10−2
fixed
floor stiffness
full coupling
10−3
10−4
2
5
10
frequency (Hz)
20
50
Figure 11: Transfer function in the machine, from frame to the reference: fixed response (dotted), only floor
stiffness (dashed) and actual response (solid)
From this figure it can be concluded that for this case the response of the machine has not improved, when
only the floor stiffness incorporated into the model. Because the floor stiffness is placed at each mount, the
total stiffness is four times the actual stiffness and therefore the response with the floor stiffness is almost
identical to the fixed response.
Next the mass and the damping of the floor is added to the model. It is assumed that the area under the
machine behaves rigidly for the frequency range of interest, so all four mounts are rigidly attached to the
rigid body, which represents the floor. The mass of the floor is around 14 · 103 kg and the damping is 10%,
such that a response as shown in figure 8 is obtained. The response of the machine with the floor modeled as
an SDOF oscillator, together with a fixed and actual response is shown in figure 12.
From this figure it can be concluded that for this case the SDOF approximation is a much better assumption.
The damping of the floor is apparently not completely well estimated with a viscous damping model, hence
the slightly deeper anti-resonance, but the three eigenmodes of this system are very well estimated. The peak
at 15 Hz originates from a second mode of the floor, which is not present in an SDOF approximation.
4.2 Transfer Path Analysis
Although the TPA method from section 2.3 is not validated with an experiment yet, it is still interesting to
see what the vibrations of the reference frame will be as predicted with this method, compared to a more
classical approach, directly imposed vibrations on the mounts of the machine. For this comparison the
machine model of figure 6 is used again, and a flat power spectrum is assumed for the floor. In the first
case, these vibrations are directly applied as imposed accelerations on the mounts of the machine and the
vibrations of the reference frame are computed. For the second case, these vibrations are used as the interface
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accelerance (ms−2 /N )
10−2
fixed
SDOF floor
full coupling
10−3
10−4
2
5
10
frequency (Hz)
20
50
Figure 12: Transfer function in the machine, from frame to the reference: fixed response (dotted), SDOF
floor (dashed) and actual response (solid)
gap δc and then the vibrations of the reference frame are computed. These two vibration levels, together with
the original free vibration level of the floor are shown in figure 13.
101
floor
direct
transmission
power (m2 s−4 )
100
10−1
10−2
10−3
1
10
frequency (Hz)
Figure 13: Modeled transmission of floor vibrations to a simple machine model. Free floor vibrations (dotted), directly applied vibrations (dashed) and transfer path analysis vibrations (solid)
From this figure it is clear that the directly applied vibrations do not take the coupled dynamics into account,
whereas the TPA method clearly shows the dynamic coupling in the vibration level. It can also be concluded
that when there is no dynamic coupling, both methods are equal.
5
5.1
Conclusions and outlook
Conclusions
In current precision engineering practice, prediction of high precision equipment accuracy relies on rather
crude approximations of floor characteristics. In this paper, more advanced methods for incorporating the
coupling of the machine dynamics with (measured) floor dynamics have been presented, including implementation on an experimental test case. The method presented also allows for predicting machine vibration
levels on a specific floor, of which both the dynamic characteristics and the (free interface) vibration levels
are known. This approach is especially useful in engineering applications, where the machine is being designed while the floor characteristics, both dynamics and vibration levels, are known a priori, either from
measurements or by means of specification.
S UBSTRUCTURING AND COUPLING
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Related to the effect of coupling between machine dynamics and floor dynamics, it can be concluded that
coupling of (numerical) machine models with (experimental) floor characteristics, based on Frequency Based
Substructuring (FBS) methods, is successfully implemented. Comparing the different approaches available
for dealing with this coupling, more advanced methods will typically better approach the real coupling.
Assuming a rigid floor is only valid if the machine support stiffness is relatively low with respect to the
actual floor stiffness. Assuming a finite floor stiffness can be a better approximation, but will omit the
influence of any dynamics, that is generally present in non-rigid floors. A better approximation is then to
use a Single Degree of Freedom (SDOF) oscillator model for the floor, although that still omits other mode
shapes of the floor. In fact, full coupling via e.g. the FBS approach is the only way to correctly incorporate
floor dynamics in machine dynamics modelling for high precision applications. This is especially true for
applications where performance is determined by first machine resonances that are typically in the same
frequency range as floor dynamics resonances, e.g. from 10 - 20 Hz, which is quite realistic, like in the
presented test case.
Related to the prediction of machine vibrations based on measured (free interface) floor vibrations, it can be
concluded that the FBS approach provides a good framework to do so, with a so called Transfer Path Analysis
(TPA), which directly incorporates coupling effects. Although the method is succesfully implemented in
practice, the results could not yet be validated by real life experiments on the test case. A possible reason
for this is non-linearity, whereas the models have been obtained under different signal conditions (impact
measurement) than the validation experiment that was performed to apply the TPA method (steady state
multi-sine).
Despite the fact that the TPA method was not validated on the experimental test case at hand, the experimental
results of the FBS approach give good confidence, and it is recommended to further introduce the presented
approach in dynamic analysis of the effect of floor vibrations in high precision equipment.
5.2
Outlook
Based on the results presented in this paper, it is recommended to further validate the presented approach.
Firstly, that would imply a further analysis of possible non-linear behavior in either the machine dynamics
or the floor dynamics, by using different signal types and amplitudes. Secondly, an even more simplified test
case might be realized to validate the the Transfer Path Analysis method in practice.
Against the background of current precision engineering practice, where only the floor vibration level and
possibly floor stiffness is specified to predict machine accuracy, the presented work leads to the insight that
at least a Single Degree of Freedom oscillator model should be introduced to represent the floor dynamics.
If only the first resonance frequency and the effective modal mass of the floor are known, this would already
allow for a much more accurate prediction of the coupled dynamics. In more advanced applications, the
introduction of full coupling with experimental floor models by Frequency Based Substructuring will allow
for even more accurate performance prediction of (high precision) equipment on factory floors. This way,
one may potentially avoid costly and conservative design choices in the machine design.
References
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dynamics, Mechanical Systems and Signal Processing, Vol. 24, No. 6, (2010), pp. 1693 - 1710.
[2] E.I. Rivin, Vibration isolation of precision equipment, Precision Engineering, Vol. 17, No. 1, (1995), pp.
41-56.
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[4] R.M. Schmidt, G. Schitter, J. van Eijk, The Design of High Performance Mechatronics, Delft University
Press, Delft (2011).
[5] H. Amick, S. Hardash, P. Gillett, and R.J. Reaveley, Design of stiff, low-vibration floor structures, International Society for Optical Engineering (SPIE), Vol. 1619, (1992), pp. 180-191.
[6] D. de Klerk, D.J. Rixen and S.N. Voormeeren, General framework for dynamic substructuring: history,
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(submitted).
[8] M.A. Boogaard, Machines with high accuracy on factory floors, Master of Science thesis, TU Delft
(2012)