Abstract A vehicle start/stop engine system is investigated with focus on human perception of discomfort. The FRF between engine rotational acceleration around the neutral torque axis (NTA) and seat rail acceleration with the specific interest in start up and shut down events is estimated from measurements, using a shaker torque excitation. The inverted torque sensitivity formulates a frequency dependent target of powertrain torque related to seat rail acceleration. The main rotation direction for the transversally mounted engine is the pitch- direction, yielding high acceleration levels in both longitudinal x-, and vertical z- direction compared to the lateral y-direction. Specific weighting functions are used to normalize these accelerations to human perception. Acceleration levels in x-, and z-direction at the seat rail are similar after weighting curves have been applied. The amount of discomfort experienced by a person depends on vibration amplitude, frequency, direction and time duration. However, there is no evidence on in what way a specific time dependence affects the perceived comfort. The NTA around which the engine as a rigid body rotates is experimentally estimated and verified by comparing to the reference CAD NTA. The powertrain mounting system center of gravity is assumed to be known from the CAD model. The seat rail is found to be a more convenient measurement position to use in order to achieve reproducible results rather than the seat cushion. This, since the acceleration at the cushion also depends on several other factors such as person size, weight and body posture and also on the non-linear properties of the seat foam padding.
Transcript
Abstract A vehicle start/stop engine system is investigated with focus on human perception of discomfort. The FRF between engine rotational acceleration around the neutral torque axis (NTA) and seat rail acceleration with the specific interest in start up and shut down events is estimated from measurements, using a shaker torque excitation. The inverted torque sensitivity formulates a frequency dependent target of powertrain torque related to seat rail acceleration. The main rotation direction for the transversally mounted engine is the pitch-direction, yielding high acceleration levels in both longitudinal x-, and vertical z-direction compared to the lateral y-direction. Specific weighting functions are used to normalize these accelerations to human perception. Acceleration levels in x-, and z-direction at the seat rail are similar after weighting curves have been applied. The amount of discomfort experienced by a person depends on vibration amplitude, frequency, direction and time duration. However, there is no evidence on in what way a specific time dependence affects the perceived comfort. The NTA around which the engine as a rigid body rotates is experimentally estimated and verified by comparing to the reference CAD NTA. The powertrain mounting system center of gravity is assumed to be known from the CAD model. The seat rail is found to be a more convenient measurement position to use in order to achieve reproducible results rather than the seat cushion. This, since the acceleration at the cushion also depends on several other factors such as person size, weight and body posture and also on the non-linear properties of the seat foam padding.
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Acknowledgements I would like to thank all personnel at the Noise and Vibration laboratory at Saab Automobile AB and especially my supervisor Jan Weckner for the valuable guidance he has given me. He has suggested several approaches that helped me see things from new perspectives. I also appreciate the contribution from my supervisor Dr. Ulf Carlsson at MWL, KTH, in the form of technical and project structural advice. I would also like to express my gratitude to Dr. Per-Olof Sturesson for the technical and moral support given to me throughout this project. Last but not least I thank my mother for her unyielding support and encouragement and my grandfather who showed me to appreciate the beauty of nature from a physicist point of view. Rikard
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List of symbols CAD Computer Aided Design C.G. Center Of Gravity DFT Discrete Fourier Transform DOF Degree Of Freedom DSLA Dynamic Stiffness Loss Angle FRF Frequency Response Function HEM Hydraulic Engine Mount HSC Human Sensitivity Curves LHFM Left Hand Front Mount LHM Left Hand Mount LHRM Left Hand Rear Mount LTI Linear Time Invariant MTVV Maximum Transient Vibration Value NTA Neutral Torque Axis PMS Powertrain Mounting System PTP Peak To Peak RHM Right Hand Mount RMS Root Mean Square RPM Rotations Per Minute SEAT Seat Effective Amplitude Transmission TRA Torque Roll Axis (see NTA) VDV Vibration Dose Value Sxy Single sided cross spectrum Gxy Double sided cross spectrum
2 Theory ................................................................................................................................. 7 2.1 Human body vibration ........................................................................................................ 7
2.1.1 Frequency weighting ................................................................................................................. 7 2.1.2 Measurable quantities ............................................................................................................... 8
2.2 Powertrain mounting system designs ........................................................................14 2.2.1 Neutral torque axis (NTA) .................................................................................................... 14 2.2.2 Standard 3 point ....................................................................................................................... 15 2.2.3 NTA 3 point ................................................................................................................................. 15 2.2.4 NTA 4 point ................................................................................................................................. 15
2.3 Mounts ....................................................................................................................................16 2.3.1 Weight carrying mounts ........................................................................................................ 16 2.3.2 Torque struts ............................................................................................................................. 19 2.4 The start up and shut down event ........................................................................................ 19
2.4 Signal acquisition and analysis .....................................................................................21 2.4.1 Data recording ........................................................................................................................... 21 2.4.2 FRF estimation .......................................................................................................................... 22 2.4.3 Transient signal frequency content visualization ...................................................... 23
2.5 Torque sensitivity analysis .............................................................................................24 2.6 NTA estimation ....................................................................................................................24
2.6.1 Estimating the NTA by using Moore-Penrose pseudoinverse ............................... 26 2.7 Error consideration ...........................................................................................................27
3.2.1 Transient start/stop vibrations in a Saab 9-3 .............................................................. 28 3.2.2 Transient start/stop vibrations in a Saab 9-5 with a fine resolution rpm sensor ....................................................................................................................................................... 29 3.2.3 NTA estimation and FRF estimation using a known torque excitation ............. 29 3.2.4 Torque sensitivity measurement using a known torque excitation ................... 30
4 Results .............................................................................................................................. 31 4.1 Transient start/stop vibrations in a Saab 9-3 ..........................................................31
4.1.1 SEAT value .................................................................................................................................. 35 4.2 Transient start/stop vibrations in a Saab 9-5 with a fine resolution engine speed sensor ................................................................................................................................40 4.3 NTA estimation and FRF estimation using a torque excitation .........................45
4.3.1 NTA estimation ......................................................................................................................... 45 4.3.2 Transfer between powertrain rotational acceleration and seat rail acceleration ........................................................................................................................................... 47
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4.3.3 Transfer between powertrain rotational acceleration and steering wheel acceleration ........................................................................................................................................... 50 4.3.4 Input accelerance ..................................................................................................................... 52
Appendix A. Dimension of the VDV-value ............................................................... 59
Appendix B. Signal acquisition.................................................................................... 60 A.1 Anti-aliased A/D conversion (ADC) ..................................................................................... 60 A.2 Time window weighting ........................................................................................................... 61 A.3 Discrete Fourier Transform and FFT .................................................................................. 62 A.4 Auto- and cross spectrum calculation................................................................................. 62 A.5 Coherence ....................................................................................................................................... 63 A.6 FRF calculation ............................................................................................................................. 63
Appendix C. The Moore-Penrose pseudoinverse ................................................. 65
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1 Introduction
1.1 Background To meet the demand of low fuel consumption and low emissions, car manufacturers have to develop new technologies in order to increase the fuel efficiency of their vehicles. One way to accomplish this is to use an automatic engine start/stop system. This system automatically stops the engine when certain driving conditions are fulfilled, e.g. vehicle comes to a standstill together with a moderate or high battery charge level. The engine may be programmed to start again at clutch engage. This study focuses on determining the character of the transient vibration excitation of a start/stop of the powertrain and the vibration transfer from source to receiver. The goal is to characterize the vibrations with a suitable measure and to predict the perceived human comfort sensation. This work is performed at Saab Automobile AB in Trollhättan.
1.2 Problem formulation The concept of a start/stop engine system cannot be directly transferred to a vehicle with a conventional system because of the high transient vibration levels this setup yields. The powertrain mounting system or the powertrain control strategy must in some way be altered to fit this new approach. This is part of future work. The goal of this study is to characterize the engine start up and shut down vibrations of a vehicle with a suitable measure and investigate how these vibrations are transferred and perceived as discomfort at the driver’s seat. A second purpose is to supply Saab with suitable computer scripts to support further investigation on transient powertrain vibrations.
1.3 Limitations This study focuses on the rotational degree of freedom (DOF) of the powertrain in the pitch direction since the engine is transversally mounted in the vehicle. Figure 1 displays a transversally mounted powertrain.
Figure 1. A transversally mounted engine and its 6 degree of freedom coordinate system.
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2 Theory
2.1 Human body vibration Vibration is often characterized with acceleration. There are three categories of human exposure to vibrations [1]:
Whole-body vibration where the body is supported on a vibrating surface. Motion sickness, less than 1 Hz. Hand transmitted vibration.
2.1.1 Frequency weighting This study focuses on whole body vibration since the seat supports the driver in a sitting posture, which is relevant in this case. The human body is a complex mechanical system with inner organs that respond differently to excitation depending on vibration frequency, amplitude and direction. The amount of discomfort experienced by a person depends on vibration level, frequency, direction and time duration [1]. However, there is no evidence on how specific time dependences affects perceived discomfort [2]. To reflect this, weighting functions should be applied to the measured data. The weighting functions in lateral, longitudinal and vertical direction according to ISO 2631-1 are presented in figure 2. The human sensitivity curves (HSC) in [3] have very similar characteristics as the ISO 2631-1 weighting curves.
Figure 2. Weighting functions in the lateral directions X and Y and the vertical direction Z.
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The ISO 2631-1 weighting functions were applied to all measured time data records before any further analysis was performed. An example of the frequency weighting in both the longitudinal and vertical direction is demonstrated in figure 3.
Figure 3. Frequency weighted seat rail acceleration displayed together with the unmodified measured data. Start up event for a Saab 9-5 in x- and z-direction. Note the similar amplitudes in both directions after the weighting functions have been applied.
The perception threshold for Wk weighted vertical vibrations in the frequency range 1 – 100 Hz is roughly 0.015 m/s2, with quartile ranges that extend from 0.01 m/s2 to 0.02 m/s2 [2]. A magnitude of 0.1 m/s2 RMS is easily noticeable, 1 m/s2 RMS is considered as uncomfortable, and 10 m/s2 RMS is usually dangerous [1].
2.1.2 Measurable quantities ISO 2631-1 also recommends a number of measures that are appropriate to use when investigating human vibration excitation. These are:
Root mean square value (RMS value) Maximum transient vibration value (MTVV) Vibration dose value (VDV)
Another widely accepted statistical measure is the
Peak-to-peak value (PTP value)
X Z
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The RMS value is a measure of the total magnitude of a periodic or stationary random time signal but is often used when analyzing transient vibration transferred to the human body, see equation 1 and figure 4:
aw 1
Taw
2 (t)dt0
T
12
(1)
where
aw is the weighted acceleration and T is measurement time.
Figure 4. Weighted time data with corresponding RMS value. Start up event for a Saab 9-3 in longitudinal x-, lateral y- and vertical z-direction.
According to [1] however, this may not be an accurate measure of acceleration severity if the vibration is transient or intermittent, contains shocks or varies in magnitude. In these cases the RMS value may underestimate the vibration severity and the VDV should be calculated. The VDV increases with increasing measurement time, so in order to maintain a VDV when increasing the excitation time, e.g. number of start up events, is to lower the acceleration level. VDV has the dimension m/s1.75 [2], see Appendix A. The VDV is defined as:
VDV aw (t) 4
dt0
T
14
.
(2)
X Y Z
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Figure 5. Weighted time data with corresponding running VDV. The VDV of the signal is the last and highest value over the entire measurement. Start up event for a Saab 9-3 in longitudinal x-, lateral y- and vertical z-direction.
The MTVV is defined as the maximum running RMS value with an integration time τ, see equation 3 and figure 6. In [4] there were efforts made to correlate subjectively graded vibration levels of discomfort to measurable quantities and it was concluded that the MTVV yielded the smallest deviations from a curve fitted relationship between these two domains.
aw(t0) 1
aw (t)
2
dtt0
t0
12
(3)
MTVV max aw(t0) (4)
The integration time,
, is set to 0.25 s.
X Y Z
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Figure 6. Weighted time data with corresponding running RMS and MTVV. The signal MTVV is defined as the maximum running RMS in the specified time interval. The integration time,
, is 0.25 s. Start up event for a Saab 9-3 in longitudinal x-, lateral y- and vertical z-direction.
The peak-to-peak value (PTP) is defined as the maximum difference between the two local extreme values in a confined interval running along the entire length of the signal at a fixed overlap, see figure 7. An interval overlap of 75% was used in this study.
X Y Z
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Figure 7. Weighted time data with corresponding running PTP. The signal PTP value is defined as the maximum running PTP in the specified time interval. Start up event for a Saab 9-3 in longitudinal x-, lateral y- and vertical z-direction.
Time acceleration data in the three main directions can be reduced to a single value
av (kx2awx
2 ky2awy
2 kz2awz
2 )12 (5)
where awx, awy, awz are weighted accelerations in the x-, y- and z-direction respectively with the multiplying factors kwx = kwy = kwz = 1 when considering seated people as done in this study. This is denoted the vibration total value or component ride value and should be used for health and safety evaluation if no dominant axis of vibration exists [2]. Perception of discomfort is not only dependent on source vibration but also on the human response to it. The authors in [1] claims that resonances can be found at about 5 Hz and sometimes also at 7 – 12 Hz in the vertical direction. This corresponds quite well with data in [5], see figure 8. This means that it may not be sufficient to replace the body with a single mass in physical measurements or computer models when predicting the vibration transmission through the seat. In [6] it was concluded that it is hard to get reproducible measurements of the seat transmission using human subjects due to differences in weight, length and load distribution. Mannequins could be used to achieve identical loading conditions throughout an entire measurement series in order to minimize these bias errors. This method has the downside that it may not represent the human body dynamics realistically.
X Y Z
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Figure 8. The human body is a complex mechanical system, but from approximately 2-100 Hz it can be approximated by a system of particles [5]. These are the approximate resonances for the internal organs that can cause discomfort by amplifying input vibrations. Abdomen and diaphragm resonances (4-12 Hz) are in the same frequency range as powertrain vibration modes (6-18 Hz), see section 2.3.
Whole body vibrations are transmitted through the seat cushion and backrest. The foam inside the cushion that supports the person is a highly non-linear material and it is characterized by high stiffness at both large and small amplitude excitation, but low stiffness at medium amplitudes [6]. This leads to large discrepancies in frequency responses when using different human subjects that load the seat with different levels of compression. In addition to this, body posture also influences seat stiffness. The seat effective amplitude transmissibility value (SEAT) has been introduced to characterize seat influence on vibration transmission to the human body. It is defined as:
SEATRMS vibration RMS value on seat
vibration RMS value on floor 100%
(6)
The SEAT value can also be calculated by using the VDV if the vibration time domain signal contains shocks. Note that the seat-body system actually can amplify the vibration input from the seat rail, giving SEAT values larger than 100 %, see figure 9. Usually, during idling conditions in cars, the SEAT value is approximately 60-80 % [1].
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Figure 9. The acceleration severity is often higher at the seat cushion compared to the seat rail in the vertical direction. The measurement was made in a Saab 9-3. SEATVDV = 225.1%.
2.2 Powertrain mounting system designs The vibration source for start up and shut down vibrations is the powertrain. It is mounted to the car chassis with the powertrain mounting system (PMS), which has four functions [7]. They are to:
support the weight of the powertrain. (weight support) control the displacement of center of gravity (C.G.) of the powertrain.
(motion control) control the powertrain vibrations. (vibration control) isolate the powertrain vibrations from the vehicle structure. (vibration
isolation) The PMS is therefore an important subsystem in the transmission from powertrain acceleration to seat rail acceleration. The challenge in developing a PMS is to determine the mount properties like stiffness and damping due to conflicting design parameters requirements. The mounts need to have low dynamic stiffness and damping in order to isolate the powertrain vibrations from the chassis, but at the same time they are required to be sufficiently stiff to ensure that powertrain motion is within range.
2.2.1 Neutral torque axis (NTA) The neutral torque axis (NTA), also called torque roll axis (TRA), is the axis around which the engine rotates when torque is applied. The location and direction of this axis has to be taken into account when designing a layout for a PMS. To reduce mount reactions forces due to powertrain rotation it is
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recommended to place the weight carrying mounts aligned along the NTA together with one or more torque struts, see sections below. Different powertrain configurations have slightly different NTA.
2.2.2 Standard 3 point All three mounts provide weight support. One mount is located at the right-hand side (RHM) on the NTA, and the other two are located on the left-hand side at front and rear positions (LHFM, LHRM) of the engine. The LHFM and LHRM also support torque forces and are positioned at an inclined angle to match the elastic centre to the NTA.
Figure 10. A standard 3 point PMS [7].
2.2.3 NTA 3 point Two mounts aligned along the NTA on either side of the engine (LHM, RHM) carry the weight of the engine. A torque strut supports the output torque in the direction of expansion because of instability effects in compression. The bounce and pitch modes are uncoupled. Another advantage is that the PMS is relatively easy to tune for road input as well as for idle conditions. A disadvantage is that the layout is space demanding. A more compact PMS is built by placing the two mounts beneath the engine (low 3 point PMS), alternatively moving one mount away from the NTA (balanced 3 point PMS).
Figure 11. From left: A 3 point NTA PMS, a balanced 3 point NTA PMS, a low 3 point NTA PMS [7].
2.2.4 NTA 4 point This strategy uses two side mounts as in section 2.2.3 for weight support, while engine torque is supported by two mounts at the fore and aft positions of the engine. According to [7], this setup can be somewhat problematic since pre-stresses can occur in the fore/aft mounts due to creep in the weight supporting mounts.
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Figure 12. A 4 point NTA PMS.
2.3 Mounts There are two measures to characterize a mount in a PMS, the static Force vs. Displacement (F-D) relation and the Dynamic Stiffness Loss Angle (DSLA). The F-D relation of a mount is designed by using the requirements of natural frequencies of the PMS, usually in the range of 6 – 18 Hz [7], see figure 13.
Figure 13. Important vehicle frequency ranges [7]. The PMS rigid modes are located within 6 - 18 Hz.
The DSLA of a mount under low frequency and large amplitude is developed based on reduction of displacement of the powertrain C.G. and the forces transmitted to the car body. Minimization of cabin noise and the vibration of steering wheel and seat rail determine the suitable DSLA of a mount under high frequency and small amplitude excitations.
2.3.1 Weight carrying mounts There are several types of powertrain mounts available:
Elastomeric mounts are commonly used in powertrain mount systems. The dynamic stiffness at high frequencies and small displacements are higher than at low frequency, large amplitude stiffness [8]. They are often designed from a single block of rubber into complex shapes in order to give them different stiffness components in the three different axes. Using bush-type mounts is a space efficient strategy, but since they have relatively low damping they cannot limit the displacement of the powertrain C.G. in the low-frequency range according to [7] and at the same time satisfying the vibration isolation demand. A HEM uses elastic rubber for vibration isolation and a hydraulic mechanism to provide a large amount of damping at a specific frequency. This is useful when trying to minimize displacement of the powertrain C.G. close to certain PMS resonances. The development cycle is in the third generation with a design based on an inertia track (1st gen.), a decoupler (2nd gen.) and a plunger (3rd gen.), see figure 14.
Figure 14. Mount development history [7].
When the mount is compressed, the pressure in the upper chamber increases. This causes the viscous fluid to flow through the narrow inertia track into the
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chamber below that expands to contain the fluid. According to [7] the fluid in the inertia track cannot follow high frequency – small amplitude vibrations, which gives a high dynamic stiffness. This is not positive for the vibration isolation characteristics of the mount, and therefore a decoupler is inserted to allow vibrations at high frequencies and small displacements to bypass the inertia track. The travel of the decoupler is mechanically limited to small amplitude vibrations to force the inertia track activation when the mount is subject to large amplitude excitation. To further reduce transmission of high frequency vibrations a ring shaped feature, plunger, is fastened inside the upper chamber. A principal dynamic stiffness is presented in figure 15.
Figure 15. Frequency characteristics for mounts from three generations [7].
Semi-active mounts are able to change the stiffness and damping characteristics in a short period of time (ms scale). At engine idle conditions a low stiffness is required in order to minimize the vibration transmission to the car body. There are several techniques that accomplish this:
Vacuum switchable mounts features a valve that allows air to flow into a chamber between the rubber spring and the hydraulic chamber, decoupling their relative motions. This is positive when low stiffness is required e.g. in idling.
Orifice controlled mounts have an orifice placed in parallel with the inertia track. The orifice is opened to allow the fluid to bypass the inertia track when a low stiffness is required.
Inertia track controlled mounts have in idle conditions two inertia tracks working in parallel. This moves the stiffness shift higher up in frequency, see figure 16. One of those is closed in driving conditions to increase damping at the shake frequency.
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Figure 16. Frequency characteristics for mounts with one or two inertia tracks [7].
An active mount assembly consists of a passive mount, vibration sensor, a force generating device and an electronic feedback control system. It minimizes transmitted vibrations by exciting the structure with a 180° phase shifted copy of the measured vibration. These systems require external power, are expensive, increase cost, packaging space and weight of the PMS compared to regular systems.
2.3.2 Torque struts Torque struts are often used in assemblies of transversally mounted powertrains to provide torque counteracting the powertrain pitch movement. This setup is common in front wheel-driven vehicles. The strut consists of two bushings connected by a rod. The bushings are usually either both elastomeric or one elastomeric and one hydraulic. It is important to design the strut so that the natural frequency of the mass spring system bushing-rod-bushing is well above the fundamental engine firing frequency [7]. Also, as the stiffness often increases when a longitudinal force affects the mount, it is ideal to physically separate the load-bearing mount from the strut. Torque struts should also be located so that they are loaded in tension as they become unstable in compression.
2.4 The start up and shut down event During a start up event a diesel engine spins from zero rpm up to approximately 1100 rpm, then settling at an idle mean value about 850 rpm. It passes the PMS resonances at 5 – 20 Hz (corresponding to 150 – 600 rpm for engine order 2), with the special interest in the pitch mode resonance that is typically located around 11 Hz (330 rpm for order 2) for transversally mounted engines. The rotational motion oscillates with the frequency of engine order 2, i.e. for a 4-
cylinder engine
f rpm
60 2. The rotation decelerates when air is compressed
inside the combustion chamber, and accelerates when the same air is expanded or even ignited, some 50 ms later. The first combustion occurred at 0.55 s in figure 17, in which the fine resolution rpm signal was sampled at 112/rotation and the standard rpm signal at 1/rotation.
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Figure 17. A start up event in a Saab 9-5. Seat rail acceleration with both fine and standard resolution rpm signals.
The shut down is similar to the start up event in that the engine rotational velocity passes the PMS resonances, but the system initial condition is different. Also, the magnitude of the rotational speed variation is less compared to a start up. The peak acceleration magnitude is approximately 1/5 of the peak magnitude in a start up event, see figure 18.
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Figure 18. A shut down event in a Saab 9-5. Seat rail acceleration with both fine and standard resolution rpm signals. This one out of the 10 measurements made showed a rotational acceleration increase after the rotational speed had slowed to zero rpm. This effect is probably due to an inertia reaction combined with air compression and expansion in one or more cylinders.
2.4 Signal acquisition and analysis
2.4.1 Data recording A multichannel FFT analyzer system was used to record the time data at a sampling frequency of 256 Hz. The measurement acquisition procedure in a dual channel FFT analyzer is presented in figure 19. These steps are described in further detail in appendix B.
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Figure 19. Measurement chain of a dual channel analyser [9]. A double-sided spectrum is denoted G.
2.4.2 FRF estimation A frequency response function (FRF) describes the linear relationship of magnitude and phase over frequency between two signals, where one of them is the input and the other is the output of a linear time invariant system (LTI-system). The H1 estimator in [10] was used to calculate frequency response functions since the dominant noise source was expected to mostly disturb the output channels, e.g. the accelerometers glued to the seat rail, see figure 20.
Figure 20. Sketch of signal path [10].
The cross correlation between the input
x(t) and the output
y(t) of a system is
defined as:
Ryx () E x(t) y(t ) E x(t) y(t ) h()d
(7)
Ryx () E x(t) y(t ) h()d
(8)
Ryx () Ryx ( ) h()d
(9)
The cross spectrum is defined as:
Syx ( f ) F Ryx () (10)
The estimated FRF was calculated with:
H1 GaT ()
GTT ()
(11)
where
GaT () is the double sided cross spectrum between the acceleration
signal and the torque excitation signal and
GTT () is the autospectrum of the
acceleration time signal. This FRF is a transfer point torque accelerance, which indicates that the torque input point was not the same as the accelerometer point in the global coordinate system.
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2.4.3 Transient signal frequency content visualization The energy spectral density (ESD) spectrum in [10] is used to display the frequency content of energy signals, i.e. signals with limited power such as the measured transient acceleration signals in this study. It is defined as:
ESD 2 T DFT x(n)
2
f
(12)
where the factor 2 converts the double sided spectrum level to match a single sided spectrum,
T is the measurement time, [s],
f is the frequency resolution, [Hz] and
DFT x(n) is the frequency components from the discrete Fourier transform of
the time signal x. The area beneath the ESD represents the energy in the transient signal within a certain frequency interval, see figure 20. A rectangular time weighting window is used in order to achieve better amplitude estimation than a Hanning window would give. The already low-pass filtered transient time signal is periodically repeatable without the effect of energy leakage.
Figure 20. ESD spectrum of the frequency weighted seat rail acceleration during a start up event in the vertical z-direction in a SAAB 9-3 with the 1.9 liter diesel engine with 6 speed manual transmission.
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2.5 Torque sensitivity analysis The torque sensitivity analysis is an important method because it shows how much of the powertrain vibration due to engine torque that is transferred to the seat rail position. The result is a FRF that can be used to set engine vibration limits, e.g. during a start up event. In standard torque sensitivity analysis, the powertrain is excited with a torque excitation, e.g. two mirrored and counteracting (within certain limits, see [11] or section 3.2.2) sine or random force excitations, and calculates the FRF corresponding to some other point of interest on the vehicle such as the seat rail. The FRF indicates the vibration transmission efficiency through a certain frequency range, determined by the sampling frequency and excitation frequency range. By choosing seat rail acceleration limits it is then possible to define engine torque restrictions versus frequency. The process can then be validated by comparing a measured seat rail acceleration time record to the calculated, convoluted acceleration;
asr(t) Top(t)*h(t) (13)
where
asr(t) is the modeled seat rail acceleration,
Top(t) is the calculated
operating input engine torque. The operation becomes a multiplication in the frequency domain:
Asr( f ) Top( f )H( f ) (14)
The input torque is defined as:
Top( f ) op HT1
(15)
where
op is the operating powertrain rotational angle measured in a start up or
shut down event and then projected onto the NTA,
HT is the torque impedance
calculated from [11].
2.6 NTA estimation The total deformation at a point i,
ui , is described by:
ui uirb ui
fb (16)
where
uirb is the rigid body deformation and
uifb is the flexible body deformation
[19]. The transversally mounted powertrain can typically be approximated as a rigid body at low frequencies up to at least 200 Hz [12]. Since the upper frequency limit of the measurement is
fs2 128Hz, equation 11 is simplified to;
ui uirb
(17)
with;
uirb(x1) ucg
rb r(x1,xcg ) (18)
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where
uirb(x1) is the local displacement in the point
x1, [m],
ucgrb is the displacement of the powertrain center of gravity, [m],
r(x1,xcg ) is the distance vector from
xcg to
x1, [m] and
is the angle of rotation of the rigid body [1]. The model can be written as:
u T (19)
where the vector
u contains the translational acceleration time data from all sensors,
T is a geometric transformation matrix that contains the positions of the sensors relative to the powertrain C.G. and
is a vector containing all six DOF of the powertrain, three translational and three rotational. Each
accelerometer contributes to the system in each sample with
um and
Tm as:
um
uxm
uym
uzm
,
Tm
1 0 0 0 rzm ry
m
0 1 0 rzm 0 rx
m
0 0 1 rym rx
m 0
,
ucg xrb
ucg yrb
ucg zrb
x
y
z
(20)
where
m 1, 2, 3, 4, 5 is a sensor index and
rxm, ry
m, rzm are distances from the origin
to the sensor in x-, y-, and z-direction in a Cartesian coordinate system defined in the CAD-model describing the vehicle.
Figure 21. The distances from the PMS C.G. to mount m is defined in the T-matrix by using this geometry.
As all five accelerometers are added to the system to decrease measurement error (at least two three-axial accelerometers are needed to ensure a working model [19]),
u becomes a 15x1 vector:
rm
rzm
rxm
rym
z
x
y
uxm
uzm
uym
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u
u1
u2
u3
u4
u5
(21)
When describing the distance matrix as:
Rm
0 rzm ry
m
rzm 0 rx
m
rym rx
m 0
(22)
and including all five accelerometers the transformation matrix becomes a 15x6 matrix;
T
I R1
I R2
I R3
I R4
I R5
,
(23)
where
I is a 3x3 diagonal matrix with ones occupying the main diagonal. This system, equation 14 combined with equations 16, 17 and 18, is solved by using the Moore-Penrose pseudoinverse technique from [13] for every measured time sample.
2.6.1 Estimating the NTA by using Moore-Penrose pseudoinverse This method yields a solution with the smallest possible 2-norm when compared to other matrix inversion techniques [14]. The solution is;
Tu (24)
where
T is the Moore-Penrose pseudoinverse of
T , see appendix C. A pseudoinverse is possible to calculate even for non-square matrices. This method uses singular value decomposition (SVD) to create this pseudoinverse. Using SVD it is possible to decompose any complex matrix
ACmn as;
A UV H , diag 1,,...,r . (25)
where
UCmm is a unitary matrix that contain the eigenvectors for
AHA,
Rmn is a pseudo-diagonal or diagonal matrix that contain the positive square roots to the positive eigenvalues for
AHA and
AAH and
V Cnn is a unitary matrix that contain the eigenvectors for
AAH .
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The pseudoinverse is then calculated by:
A VU H , diag1
1,
,...,1
r,
.
(26)
2.7 Error consideration Two types of errors that can affect the result of a measurement are bias and random errors [14]. Bias errors are systematic errors and often linked to the measurement equipment or procedures. Random errors originate from measurement noise, computational noise and other unmeasured sources of energy that contribute to signal input or output in an unpredictable or uncounted for way. The influence of random errors can be reduced by averaging the measurement data [9], e.g. the averaged autospectrum of a number of time records gAi, i = 1, 2, 3, …, Nav;
GAA () E GA ()GA () limNav
GAi ()G*Ai ()i1
Nav
Nav,
(27)
and the averaged cross spectrum of two unique time signals gAi and gBi;
GAB () E GA ()GB () limNav
GAi ()G*Bi ()i1
Nav
Nav.
(28)
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3 Measurement setup Four different measurements were conducted. The goals of them are listed below. The sampling frequency was set to 256 Hz in accordance with Saab standards and to allow for some headroom from estimated natural frequencies of the steering wheel assembly at 40 Hz and powertrain modes at 6-18 Hz for future analysis.
1. Transient start/stop measurement with focus on the seat rail, seat cushion and steering wheel vibrations in a Saab 9-3, with the purpose to acquire statistical vibration data.
2. Transient start/stop measurement with focus on seat rail vibrations in a Saab 9-5, with the purpose to acquire time domain vibration data for future analysis. A fine resolution rpm signal was also recorded in order to obtain engine rotational irregularity for future analysis.
3. Measurement with focus on NTA estimation in a Saab 9-5 using a stationary random torque excitation, with the purpose to calculate transfer functions between engine rotational acceleration to seat rail and steering wheel acceleration.
4. Engine torque sensitivity with focus on seat rail acceleration in a Saab 9-5 normalized to the estimated NTA.
3.1 Equipment The signal acquisition system used was Lms® [16], which also was used to calculate FRF and coherence functions from the torque input to seat rail and steering wheel acceleration. Signal analysis was also performed in Matlab® [17]. The equipment used is presented in table 1. Table 1. Measurement equipment. 1. - 3 3-axial low-sensitivity accelerometers - Tachometer 1/ rotation - LMS multichannel analyzer - Computer
3.2 Test setup The measurements were conducted to investigate the vibration problem and provide operating data input to models. Details of the measurement setup are listed below.
3.2.1 Transient start/stop vibrations in a Saab 9-3 10 measurements on start up/ shut down transient vibrations were made according to GMW 14384 and GMW 8477. 3-axial accelerometers were placed at standardized positions at the seat rail, seat cushion and also the steering wheel
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for which the 12 o’clock position was used. Each start up- and shut down event was recorded within a time window of 2 seconds. This is a suitable time length for future frequency analysis due to the FFT demand of block sizes of powers of 2, in this case:
nfft fs t 256 2 512 samples. (29)
A tachometer was used to record the one-rotation average engine speed in rpm. It was powered by external power in order to provide a reliable rpm signal throughout the entire measurement.
3.2.2 Transient start/stop vibrations in a Saab 9-5 with a fine resolution rpm sensor Also 10 measurements on start up/ shut down transient vibrations were made for a Saab 9-5. As with the Saab 9-3 measurement, 3-axial accelerometers were placed at the seat rail. Each start up- and shut down event was recorded within a time window of 2 seconds.
3.2.3 NTA estimation and FRF estimation using a known torque excitation The engine torque during a start up event may be difficult to measure and therefore it was suggested that the powertrain rotational acceleration data projected onto the NTA should be used as input to the model instead of torque data. The NTA was estimated from an arithmetic average of 40 seconds of operating data, where each NTA sample was estimated by solving the linear system of equations described in section 2.6. Five accelerometers was located on the surface of the engine block, see figure 21 and 22. The corresponding accelerometer coordinates were imported from a CAD-model describing the engine. One accelerometer was placed at the 12 o’clock steering wheel position and one at the seat rail position.
Figure 21. Five high sensitivity accelerometers were mounted onto the engine surface. Four of them on the upper side of the engine block.
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Two electrodynamic shakers were vertically fastened at fore and aft positions to the engine block from underneath the vehicle using shaker rods to reach these positions, see figure 22. The shakers were also firmly grounded.
Figure 22. Two shakers with impedance heads and one of five high sensitivity accelerometers were mounted to the lower engine surface.
The excitation was a stationary random torque excitation calibrated by using [13]. With the first 8 seconds of time data removed, to ensure that transient start up effects had dissipated, six FRF in total were estimated. One FRF was estimated between the rotational powertrain acceleration, projected onto the estimated NTA, and each of the translational directions, x, y and z, for both the seat rail position and the steering wheel 12 o’clock position. The block size was;
samples that resulted in a total of 99 averages with an overlap of 50% between these blocks. The measurement frequency resolution was:
18s 0.125 Hz (31)
3.2.4 Torque sensitivity measurement using a known torque excitation The torque sensitivity measurement was made according to GMW 14902, utilizing two shakers 180 ± 2 degrees out of phase exciting the powertrain with a pitch torque (the two forces relate as F1/F2 = 1.00 ± 0.02 in the frequency interval 10-70 Hz according to [11]). The excitation was an 8 second long linearly swept sine (0.125-100 Hz). A random force input signal (0-100 Hz) was also used for comparison.
nfft fs t 256 Hz 8s 2048 (30)
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4 Results
4.1 Transient start/stop vibrations in a Saab 9-3 Several different vibration quantities (RMS, VDV, MTVV and PTP) illustrate the variation of vibration severity from a sequence of 10 unique start up events and 10 unique shut down events in a Saab 9-3. The boxes extend from the 25th percentile q1 to the 75th percentile q3. The median is marked with a line inside the box. Points are drawn as outliers if they are larger than
q3 w(q3 q1) or
smaller than
q1 w(q3 q1) , where the maximum whisker length
w is set to 1.5.
This corresponds to a range of approximately ±2.7σ and 99.3% coverage if the data are normally distributed. The plotted whisker extends to the value that is the most extreme but not an outlier. Things to be noted in these results are that the magnitude of the vertical vibrations at the seat cushion is far greater than at the seat rail. This is an effect of a resonance at about 10 Hz of the mass-spring system formed by the human body and the seat. Vibrations in the lateral direction are very low due to the low source excitation in this direction. What also is interesting is that vertical seat rail vibrations show larger spread in start up event compared to shut down events. A possible cause for this is that the engine start sequence is not synchronized, e.g. the number of revolutions before first ignition differs from event to event and that this affects the start up more than the shut down. The acceleration RMS variation for the start up sequence is presented in figure 23.
Figure 23. Acceleration RMS value variation based on 10 start up events in a Saab 9-3.
X Y Z X Y Z Seat rail Seat cushion
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The MTVV variation for a start up sequence is presented in figure 24.
Figure 24. MTVV variation based on acceleration data from 10 start up events in a Saab 9-3.
The VDV variation for a start up sequence is presented in figure 25.
Figure 25. VDV variation based on acceleration data from 10 start up events in a Saab 9-3.
X Y Z X Y Z Seat rail Seat cushion
X Y Z X Y Z Seat rail Seat cushion
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The PTP variation for a start up sequence is presented in figure 26.
Figure 26. PTP value variation based on acceleration data from 10 start up events in a Saab 9-3.
The acceleration RMS variation for the shut down sequence is presented in figure 27.
Figure 27. Acceleration RMS value variation based on 10 shut down events in a Saab 9-3.
X Y Z X Y Z Seat rail Seat cushion
X Y Z X Y Z Seat rail Seat cushion
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The MTVV variation for a shut down sequence is presented in figure 28.
Figure 28. MTVV variation based on acceleration data from 10 shut down events in a Saab 9-3.
The VDV variation for a shut down sequence is presented in figure 29.
Figure 29. VDV variation based on acceleration data from 10 shut down events in a Saab 9-3.
X Y Z X Y Z Seat rail Seat cushion
X Y Z X Y Z Seat rail Seat cushion
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The PTP variation for a shut down sequence is presented in figure 30.
Figure 30. PTP value variation based on acceleration data from 10 shut down events in a Saab 9-3.
4.1.1 SEAT value Results based on 10 measurements showed that the seat isolated vibrations in the lateral and longitudinal direction well while amplifying vibrations in the vertical direction. This indicates that the magnitude of the vertical vibrations can increase above the longitudinal even if they are measured to be lower at the seat rail. Remember that these values were obtained by loading the seat with only one specific weight that affects the nonlinear properties of the foam seat. Figures below show that the SEAT value calculated from the measurements also is highly dependent on excitation. The SEATRMS variation for a start up sequence is presented in figure 31.
X Y Z X Y Z Seat rail Seat cushion
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Figure 31. SEATRMS value variation based on acceleration data from 10 start up events in a Saab 9-3.
The SEATMTVV variation for a start up sequence is presented in figure 32.
Figure 32. SEATMTVV value variation based on acceleration data from 10 start up events in a Saab 9-3.
X Y Z Seat
X Y Z Seat
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The SEATVDV variation for a start up sequence is presented in figure 33.
Figure 33. SEATVDV value variation based on acceleration data from 10 start up events in a Saab 9-3.
The SEATPTP variation for a start up sequence is presented in figure 34.
Figure 34. SEATPTP value variation based on acceleration data from 10 start up events in a Saab 9-3.
X Y Z Seat
X Y Z Seat
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The SEATRMS variation for a shut down sequence is presented in figure 35.
Figure 35. SEATRMS value variation based on acceleration data from 10 shut down events in a Saab 9-3.
The SEATMTVV variation for a shut down sequence is presented in figure 36.
Figure 36. SEATMTVV value variation based on acceleration data from 10 shut down events in a Saab 9-3.
X Y Z Seat
X Y Z Seat
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The SEATVDV variation for a shut down sequence is presented in figure 37.
Figure 37. SEATVDV value variation based on acceleration data from 10 shut down events in a Saab 9-3.
The SEATPTP variation for a shut down sequence is presented in figure 38.
Figure 38. SEATPTP value variation based on acceleration data from 10 shut down events in a Saab 9-3.
X Y Z Seat
X Y Z Seat
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4.2 Transient start/stop vibrations in a Saab 9-5 with a fine resolution engine speed sensor The fine resolution rpm sensor shows dips at every compression (2 EO), see figure 39. This speed variation leads to a vibration excitation into the vehicle structure and should be as low as possible in magnitude and as high as possible in frequency to increase start up comfort. A pitch mode resonance at 11 Hz corresponds to an engine speed (2 EO) of 330 rpm. Excitation should therefore be avoided at and below this engine speed if possible. The vibration increases dramatically when the engine passes this engine speed, see figures 39 and 40.
Figure 39. Seat rail acceleration RMS value variation based on 10 shut down events.
The engine speed variation is however lower in the shut down event, which is followed by a smaller vibration response at the seat rail, see figure 40.
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Figure 40. Acceleration RMS value variation based on 10 shut down events.
The acceleration RMS variation for a start up sequence is presented in figure 41.
Figure 41. Acceleration RMS value variation based on 10 start up events in a Saab 9-5.
The acceleration RMS variation for a shut down sequence is presented in figure 42.
X Y Z Seat rail
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Figure 42. Acceleration RMS value variation based on 10 shut down events in a Saab 9-5.
The VDV variation for a start up sequence is presented in figure 43.
Figure 44. VDV variation based on acceleration data from 10 start up events in a Saab 9-5.
X Y Z Seat rail
X Y Z Seat rail
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The VDV variation for a shut down sequence is presented in figure 45.
Figure 45. VDV variation based on acceleration data from 10 shut down events in a Saab 9-5.
The MTVV variation for a start up sequence is presented in figure 48.
Figure 48. MTVV variation based on acceleration data from 10 start up events in a Saab 9-5.
X Y Z Seat rail
X Y Z Seat rail
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The MTVV variation for a shut down sequence is presented in figure 49.
Figure 49. MTVV variation based on acceleration data from 10 shut down events in a Saab 9-3.
The PTP variation for a start up sequence is presented in figure 50.
Figure 50. PTP value variation based on acceleration data from 10 start up events in a Saab 9-5.
X Y Z Seat rail
X Y Z Seat rail
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The PTP variation for a shut down sequence is presented in figure 51.
Figure 51. PTP value variation based on acceleration data from 10 shut down events in a Saab 9-5.
4.3 NTA estimation and FRF estimation using a torque excitation
4.3.1 NTA estimation The estimated NTA is compared to the reference NTA from CAD in order to verify the model, see figure 52 to 54.
X Y Z Seat rail
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Figure 52. The estimated NTA, X-Y-view.
Figure 53. The estimated NTA, X-Z-view.
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Figure 54. The estimated NTA, Y-Z-view.
4.3.2 Transfer between powertrain rotational acceleration and seat rail acceleration The relationship between powertrain rotational acceleration and seat rail acceleration in x-, y- and z-direction is presented in figure 55. Excitation was a stationary random torque. Peaks between 3-7 Hz originate from primary ride rigid body modes, the coherence is lower in this region indicating a lower than optimal linear relationship. Peaks between 11-13 Hz originate from powertrain rigid body modes.
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Figure 55. FRF from PMS rotational acceleration around the NTA to seat rail acceleration.
The torque sensitivity FRF is presented in figure 56. It represents the relationship between input force and acceleration at the seat rail.
Figure 56. FRF from a random pitch torque excitation to seat rail acceleration.
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The torque sensitivity FRF acquired with a sine torque excitation is presented in figure 57.
Figure 57. FRF from a sine pitch torque excitation to seat rail acceleration.
If inverted, this sensitivity describes the vibration transfer from a target point of view, see figure 58.
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Figure 58. Inverse of the torque sensitivity, from a sine pitch torque excitation to seat rail acceleration.
4.3.3 Transfer between powertrain rotational acceleration and steering wheel acceleration The relationship between powertrain rotational acceleration and steering wheel acceleration in x-, y- and z-direction is presented in figure 59. Excitation was a stationary random torque.
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Figure 59. FRF from PMS rotational acceleration around the NTA to steering wheel acceleration.
The torque sensitivity FRF is presented in figure 60.
Figure 60. FRF from a random pitch torque excitation to steering wheel acceleration.
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The torque sensitivity FRF is presented in figure 61. The excitation was a sine torque excitation.
Figure 61. FRF from a sine pitch torque excitation to steering wheel acceleration in.
4.3.4 Input accelerance The input accelerance at the reference shaker (torque) input point shows resonant behavior at 11 Hz, see figures 62-65. Though the swept sine produced a higher coherence, especially in the range 1-30 Hz, see figure 14, the measured phase was smoother in the random measurement.
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Figure 62. FRF, swept sine torque excitation, 0.125-100 Hz.
Figure 63. Coherence, swept sine torque excitation 0.125-100 Hz.
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Figure 64. FRF, stationary random torque excitation, 0-100 Hz.
Figure 65. Coherence, random torque excitation 0-100 Hz.
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5 Conclusions The powertrain average NTA was estimated. A transfer function between rotational acceleration around the NTA and seat rail acceleration was found. Also, a transfer function between shaker force and seat rail acceleration was found. This function is called torque sensitivity, which if inverted describes a frequency dependent target value. The seat rail is a more convenient measurement position to use in order to achieve reproducible results than the seat cushion. This, since the acceleration at this location also depends on several other factors such as person size, weight and body posture. Vibration severity can be characterized using a number of different measures, RMS, MTVV, peak-to-peak and VDV. The severity variation is high, especially in the vertical z-direction. This variation is presumably mainly due to the fact that the engine starts from different positions each start up event. The standard ISO 2631-1 utilizes weighting functions to reflect the sensitivities of the human body. The body-seat system is important to consider and is not corrected for by the standard. The SEAT-value is a measure of this effect and was found to be approximately 85% in the lateral directions (x, y) and 200% in the vertical direction (z), for start up events. Seat values at idling are approximately 70% in all directions.
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6 Future work Expand the model to include input operating torque data. Make the NTA estimation model independent on CAD data input on the location of the powertrain center of gravity. The goal of continuing work should be to develop a seamless transition in both start and stop events.
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References [1] M. J. Crocker. Handbook of noise and vibration control. ISBN 978-0-471-39599-7, 2007. [2] ISO 2631-1:1997. Vibration and shock. Evaluation of human exposure to whole-body vibration – Part 1: General requirements, 1997. [3] GMW14384. Hybrid Start-up and Shutdown Vibration Evaluation Procedure, 2006. [4] P. Jönsson, Ö. Johansson. Prediction of vehicle discomfort from transient vibrations, Journal of sound and vibration, Vol. 282 p.1043-1064, 2005. [5] H. Bodén, U. Carlsson, R Glav, H. P. Wallin, M Åbom. Ljud och vibrationer. ISBN 91-7170-434-5, 2001. [6] S.K. Kim, S.W. White, A.K. Bajaj, P. Davies. Simplified models of the vibration of mannequins in car seats, Journal of sound and vibration, Vol. 264 p.49-90, 2003. [7] Shangguan, W-B. Engine mounts and powertrain mounting systems: a review, Int. J. Vehicle Design, Vol. 49, No. 4, pp. 237-258, 2009. [8] Evert Andersson, Mats Berg, Sebastian Stichel. Rail Vehicle Dynamics. KTH, 2007. [9] U. Carlsson. Experimental Structure Dynamics. MWL, KTH, 2008. [10] H. Bodén, K. Ahlin, U. Carlsson. Signaler och Mekaniska system, MWL, KTH, 2007. [11] GMW 14902. Vehicle Sensitivity to Powertrain Crank Axis Torque Input Evaluation Procedure, 2006. [12] Personal communication with P-O Sturesson, Saab Automobile AB. [13] Lars Eldén. A weighted pseudoinverse, generalized singular values, and constrained least squared problems. Linköping University, 1982. [14] Matlab help. This method yields a solution with the smallest possible norm when solving over determined systems of equations. [15] T. Karlsson, T. Ågren. Requirements of Measurements in Transfer Path Analysis using Matrix Inversion. M.sc. Thesis, KTH, 2005. [16] S. Spanne. Matrix Theory. Department of Mathematics, Lund Institute of Technology, 1995
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[17] Lms® user’s manual. [18] Matlab® user’s manual. [19] P-O. Sturesson, F. Wrang. Teoretisk Analys av Ivägkörning samt Lågvarvsbrum - Saab 640 - Delrapport Statisk Analys - stelkroppsanalys, nr TKV-97-0093
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Appendix A. Dimension of the VDV-value The Vibration Dose Value is calculated from acceleration data and has the dimension m/s1.75, see equation A.1.
m
s2
4
s
14
m4
s8
s
14
m4
s7
14
m
s74
m
s1.75
(A.1)
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Appendix B. Signal acquisition The measurement chain is presented below.
Figure B1. Measurement chain of a dual channel analyser [10]. The double-sided spectra are denoted G.
B.1 Anti-aliased A/D conversion (ADC) The analogue signal from the sensor is converted into a discrete signal in both time and magnitude due to the fact that signal analysis is performed within digital computer systems. The relationships in the time domain are
fs 1
t Nf
T Nt N
f s
f 1
Tf sN
(B.1)
where
fs is the sampling frequency [Hz], fny is the Nyquist frequency [Hz], fmax is the highest frequency of interest [Hz], dt is the sample time [s], T is the record time [s]. Analogue signals with frequencies higher than fs/2 fold around fs/2 back towards 0 Hz and can thereby not be separated from signals with frequencies originally below fs/2. An analogue low-pass filter, with cut-off frequency of fs/2.56, is therefore applied on the signal before the digital sampling is made to reduce these aliasing effects. The factor is slightly larger than 2 because to compensate
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for the non-ideal filter decay rate. It is necessary to use a filter with high damping at frequencies above fmax to be able to achieve a high dynamic range. The dynamic range is defined as the ratio between the highest and lowest signal level the measurement system can handle simultaneously [10]. The amplitude of a signal converted by a n-bit ADC (Analogue Digital Converted) is determined in 2n levels , e.g. 216= 65536 levels. Input voltage levels must be adjusted properly to take advantage of this range. If input voltage is ± 10 V, each step from a 216 bit ADC is Δ = (10-(-10))/216 ≈ 0.3 mV. This process introduces a quantization error that is assumed to be independent in each sample and evenly distributed from −Δ/2 to Δ/2. The mean square error is then:
2 p(x) x2dx 1
x2dx
2
12 / 2
/ 2
(B.2)
where
σ2 is the mean square error [V2], p is a distribution function [1/V], x is an analogue signal value [V], Δ is the interval length [1]. The dynamic range is often presented as:
rms of largest sine
rms of quantisation noise
V
2
12
12 V
2 V 6
V2V2
(B.3)
6V2V2
1.5 2n (B.4)
or in decibel relative 1 with
20 log10( 1.5 2n ) 1.86n dB (B.5)
where V is the range [V].
B.2 Time window weighting The time record is assumed to be periodic by the DFT, see B.3. This may result in sharp jumps at the beginning and end of the time record, especially if random excitation is used. A problem with this is that it introduces power at additional frequencies into the spectrum of the analyzed signal. A time window is usually applied to the time record to reduce this power leakage. The Hanning window forces the time record to zero at both ends with a smooth transition the center of the time interval, making the signal periodic when repeated. This specific window is a good compromise between frequency and amplitude resolution and is often used when analyzing random signals or signals with unknown origin or characteristics.
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B.3 Discrete Fourier Transform and FFT The Discrete Fourier Transform (DFT) is used to transform the time record to the frequency domain [10]. Frequency coefficients in the discrete time signal
x(n) are calculated with
X(k) 1
Nx(n) e
i 2nkN
n0
N 1
(B.6)
where k corresponds to frequency. The inverse Fourier Transform is calculated with
x(n) X(k) ei 2nkN
k0
N 1
. (B.7)
The Fast Fourier Transform (FFT) is a widely used and efficient way to calculate the DFT. It requires a time series length (block size) of N = 2n points where n is a positive integer. The reason for this is that the signal can be divided into two equally long parts. These parts are then further divided by two. The DFT of each part is then calculated, and after that, for the whole signal by combining all the DFT. The number of calculations required becomes N*log2N instead of N2. The calculation time using FFT compared to not using FFT if N = 1024 is less than one percent: (N*log2N)/N2 = 0.0098.
B.4 Auto- and cross spectrum calculation Linear frequency response functions (FRF) were used to characterize the mechanical system. The calculations needed to estimate a FRF is printed in further detail below. The auto correlation of the time domain signal
x(t) is defined as
Rxx () limT
1
Tx(t)x(t )dt
T 2
T 2
(B.8)
where The cross correlation between two unique time domain signals indicates is defined as
Rxy () limT
1
Tx(t)y(t )dt
T 2
T 2
(B.9)
Auto correlation is defined as
Rxx () x(t)x(t )dt
(B.10)
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cross correlation is defined as
Rxy () x(t)y(t )dt
(B.11)
The autospectrum is defined as
Sxx ( f ) Rxx () ei2fd
(B.12)
The cross spectrum from the time domain data is calculated as
Syx ( f ) Ryx () ei2fd
(B.13)
B.5 Coherence Coherence is a measure of linear dependence between two signals and is defined as
yx2
Syx ( f )2
Sxx ( f ) Syy ( f ) .
(B.14)
A value of 1 indicates a perfectly linear relationship between the two channels at a particular frequency. Coherence values below 1 can indicate several things:
The relationship between input and output signals is non-linear, e.g. at resonances.
The input, output or both signals are contaminated with uncorrelated noise, e.g. too low input force amplitude at these frequencies.
Frequency resolution is insufficient. A time delay is so large that the response signal is truncated due to too
short record length.
B.6 FRF calculation
The FRF was obtained by using the
Hxy
1 estimator in [10], see equation B.15. This
estimation is the optimum choice in a least square sense and reduces the influence of noise applied to the accelerometer channels.
Hxy1 ( f )
Syx ( f )
Sxx ( f )
(B.15)
This is an underestimate compared to using H2, which is used if the noise source mainly affects the input channel, since
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Hxy
2 ( f ) Syy ( f )
Sxy ( f )Syy ( f )
Syx* ( f )
Hxy
1 ( f ) Hxy
2 ( f ) 2( f )
.
(B.16)
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Appendix C. The Moore-Penrose pseudoinverse A pseudoinverse is the matrix equivalent to the inverse for non-invertible and therefore also non-quadratic matrices [16]. The matrices A and X are pseudoinverses to each other if
i) AXA A
ii) XAX X, A is m n X is n m
(C.1)
A more specific definition of the Moore-Penrose pseudoinverse
A
to the matrix
A is the pseudoinverse
X A that has the following properties
iii) AX H AX
iv) XA H XA
.
(C.2)
Writing
A UV H and assuming
X VYU H, where Y is to be determined leads
to
AX UV HVYU H
AXA U Y V H
(C.3)
from which the following result can be deducted:
i) Y
ii) YY Y
iii) Y H Y
iv) Y H
Y
(C.4)
The matrix Y is then a Moore-Penrose pseudoinverse to
