Cui Song,Chen Huaihai,He Xudong,Zheng Wei

State Key Laboratory of Mechanics and Control of Mechanical Structures,Nanjing University of Aeronautics and Astronautics,Nanjing 210016,China

Multi-input multi-output random vibration control using Tikhonov filter

Cui Song,Chen Huaihai*,He Xudong,Zheng Wei

State Key Laboratory of Mechanics and Control of Mechanical Structures,Nanjing University of Aeronautics and Astronautics,Nanjing 210016,China

Noises always disturb the control effect of an environment test especially in multi-input multi-output(MIMO)systems.If the frequency response function matrices are ill-conditioned,the noises in the driving forces will be amplified and the response spectral lines may awfully exceed their tolerances.Most of the major biases between the response spectra and the reference spectra are produced by the amplified noises.However,ordinary control algorithms can hardly reduce the level of noises.The influences of the noises on both the auto-and cross-power spectra are analyzed in this paper.As a conventional frequency domain method on the inverse problem,the Tikhonov filter is adopted in the environment test to suppress the exceeding spectral lines.By altering regularization parameters gradually,the auto-power spectra can be improved in a closed control loop.Instead of using the traditional way of selecting regularization parameters,we observe the coherence change to estimate noise eliminations.Incidentally,the requirement of coherence control can be realized.The errors of the phase are then studied and a phase control algorithm is introduced at the end as a supplement of cross-power spectra control.The Tikhonov filter and the proposed phase control algorithm are tested numerically and experimentally.The results show that the noises in the vicinity of lightly damped resonant peaks are more stubborn.The response spectra are able to be greatly improved by the combination of these two methods.

1.Introduction

Vibration environment tests are performed to ensure that a device can withstand the vibrations encountered during its service lifetime in its environment.Several conventional vibration tests are used for simulating an environment:random,shock,swept sine wave,or their combinations.Among these environments,random loads are the most common forces that products may endure.1,2Traditionally,a random environment test is conducted with only one exciter.However,there are circumstances which cannot be simulated properly with this manner.It is pointed out in the MIL-STD-810G that the most practical environment endured by a materiel should be reproduced with multiple exciters.Therefore,the multi-exciter-test(MET)method has been involved in the MIL-STD-810G since 2008.

In an environment test,two disturbances are able to compromise the quality of the test:inaccurate measurements offrequency response function matrices(FRFMs)and ill-condition of the FRFMs.The former errors are usually caused by lowlevel noises or lack of high enough measuring accuracy when some resonances are lightly damped.3A number of control algorithms4–6have been developed to eliminate biases between the references and the response spectra caused by this kind of errors in multi-input multi-output(MIMO)environment tests.The latter kind is also noise-related.In an MET,FRFMs are used to generate dynamic forces by an inverse operation.Once they are severely ill-conditioned at some frequencies,the noises in the driving forces will be amplified to unacceptable levels.The amplification of noises is often marked as an inverse problem.Unlike the errors brought by inaccurate FRFMs,the amplified noises in the driving forces will produce exceeding spectral lines in the response spectra,which cannot be suppressed by the above-mentioned algorithms.These high-level noise components in the response spectra may eventually result in a test failure.Many papers have studied the inverse problem in different frameworks.We can distinguish two main categories:one considering physical means and the other using mathematical tools.

Early works tend to reveal the reason of inverse difficulty through physical phenomena.Fabunmi7,8investigated the modes participating in FRFMs,and found that at a given frequency,the number of orthogonal modes had a direct relationship with the condition of the FRFMs.In the vicinity of lightly damped frequencies,only one mode dominated the FRFMs.Thus,the FRFMs are ill-conditioned severely.Lee and Park9proposed a two-step method to stabilize the inversion in the domain offorce identification.The first is to improve the condition offRFMs by a proper selection of measurement positions to avoid the smallest singular values.The second is to modify the structures by attaching dampers.Their suggestions are constructive.A damped structure has less acute resonant peaks at whichFRFMscontainthemosterrors.However,inanenvironment test,the modifications on a product’s structure violate the intention of the test.Even if the inverse problem can be solved easily,the test article is not the original one and we still have no idea of the operating condition of the unmodified product.Chiementin et al.10,11explained the relations between modal analysis and the determinant offRFMs.They proposed an experimental approach by placing sensors in well-conditioned areas.Although their achievements are thorough and contributive,the method is not developed for environment tests either.

The most widely adopted mathematical tools on the inverse problem are frequency domain methods.Ill-conditioned singular values are altered by different algorithms in the frequency domain.The first kind tends to cancel some singular values.Powell and Seering12suggested to reject the lowest singular values which were considered as noises by them.The method proposed by Romano and Lopez13was not so radical.Their idea was to abandon the singular values lower than 10%of the highest singular values.Thite and Thompson14rejected singular values according to the errors in both FRFMs and responses.The second kind consists of regularizing these singular values.Nelson and Yoon15,16adopted singular values discarding and the Tikhonove filter as well within the framework offorce identif ication.The threshold selection criterion is also presented in their papers.Hansen17proposed the well-known L-curve principle,which was very helpful to choose the optimal regularization parameters.Recently,Jia etal.18performed an experimental study based on a weighted regularization method.Although a lot of methods and principles have been developed,few of them have been developed for environment tests.

In an environment test,the inverse problem has been noticed since Smallwood et al.5,19–24built the foundation of this domain.Smallwood was inclined to correct FRFMs before a test.The earliest method of generating pseudoinverse of ill-conditioned FRFMs by Smallwood20is to remove the offending row and column from FRFMs,invert the reduced matrix,and expand and insert a row and a column or zero back.In 1993,Smallwood and Paez23suggested using singular value decomposition(SVD)to eliminate meaningless small singular values.When Smallwood proposed his random control algorithm in 1999,5a further emphasis was made that great care must be taken at frequencies where FRFMs were illconditioned.However,no concrete method or related experiment was described in that paper.Generally,his idea was nothing different from singular values discarding.12–14In fact,a simple rejection of singular values has a number of incidental defects no matter how accurate the thresholds are.Afterwards,Underwood et al.25,26also made some progresses on this problem.Underwood and Keller25managed to conduct a closedloop control which updated FRFMs every loop rather than only correcting the FRFMs at the beginning of a test to alleviate ill conditioning.In 2002,Underwood26summarized the singular problem of MIMO random environment tests.A control algorithm,which is known as adaptive control algorithm,was published in the meantime.The algorithm is alleged to be capable of updating FRFMs to reduce noises and achieve the goal of control.Details of this algorithm are covered by patented rights.Thus,developing an algorithm which can give consideration to both ill conditioning and control necessity is still a good option.

The following section provides a brief overview of the theory on MIMO random vibration tests and the inverse problem in environment tests.The details can be obtained in works of Smallwood et al.19–24

2.Theory on MIMO environment tests and inverse problem

2.1.Driving signal generation

The configuration of an environment test is shown in Fig.1.The frequency response function matrix G of the under-test system is initially measured,and the original L is calculated from the reference spectrum R.The driving spectrum D is obtained with a random phase matrix P.The driving signal x in the time domain is then generated by inverse fast Fourier transformation(IFFT)and time domain randomization.The response signal y is measured and the power spectrum matrix Syyis calculated.Syyis compared with the reference spectrum R and a corrected new Lnewis obtained by the control algo-rithm.After that,the driving spectrum D is updated and a new driving signal x is generated.The process is continued until the end of the test.

Fig.1 Block diagram of an MIMO random vibration test.

According to Fig.1,the control target of an n-input noutput random vibration environment test is to keep the power spectrum of the response be identical to the reference spectrum R,i.e.,

where the diagonal elements of Syyand R are auto-power spectra and the off-diagonal elements are cross-power spectra.

In order to achieve Eq.(1),the corresponding drive spectrum should be23

where A is the Moore-Penrose pseudo-inverse of G

L in Eq.(2)is the lower triangular matrix of the Cholesky decomposition of the reference matrix

where the superscript ‘H” denotes the operation of conjugate transpose.P is an accessional random phase matrix whose offdiagonal elements are zeros and diagonal elements are exp(iθi)(i=1,2,...,n),in which θidenotes the phase angle with a uniform distribution in the interval[-π，π]and i equals to

Theoretically,the response of the system in the frequency domain can be expressed as

Then the response spectrum matrix should be

Since the matrix A is the inverse of the system G,the response Y is

Then Syycan be simplified to be

Unfortunately,the matrix A could never be the real inverse of the system G,which is obvious because of noise and some other errors.It can be written as

where E is an error matrix.

SubstitutingEq.(9)intoEq.(6),therealresponsespectrumis

That is why one needs a control algorithm.Most of the available control algorithms,such as Smallwood’s algorithm5and the matrix power control algorithm,6are devoted to eliminate this kind of errors.However,there exist other errors which are more complicated and difficult to be controlled by these kinds of algorithms.

2.2.Inverse problem in environment test

Considering all kinds of noises,we write the real response spectrum Syyas

where Snis the power spectrum produced by all kinds of noises and the noises in the environment test always take effect in the form of a spectrum.Assuming that Lycomes from the Cholesky decomposition of Syy,it can be expressed as

where Lnis also the noise component.Substituting Eq.(12)into Eq.(2),the real driving spectrum will be

In Eq.(13),Drealis the real driving spectrum applied to the test specimen and Dncomes from the noises.Generally,the level of Dnis too low to contribute remarkable responses.However,when the system G is ill-conditioned at some frequencies,it will be different.Let the SVD of G be

where σiis the ith singular value(i=1,2,...,n),and viand uiare the column vectors of V and U,respectively.Then,Eq.(13)can be written as

It is more obvious that when any singular value of G approaches zero,the components caused by the noises in the driving spectrum will become unexpectedly large.

2.3.Tikhonov filter

The Tikhonov filter is a typical method on regularizing singular values and has been widely adopted in force identification.The function of the filter is

where Fαis the regularized driving loads and Yαis the measured responses.In this equation,the regularized singular values are

Choosing proper regularization parameters α,the singular values can be decreased and the noises contained in Yαwill be reduced.We assume that

The curve of the function with k changing is shown in Fig.2.It can be obtained that when k is less than 10-3,the filter is invalid,and when k is over 103,the corresponding singular values are totally eliminated.

A lot of principles have been developed for regularization parameters selection.The most widely used principle is to evaluate the errors brought by this regularization.According to Vogel’s work27,the representation of Morozov’s discrepancy principle is

where Y0is the non-regularized responses.Along with the change of α,the error εαreaches its minimum and the regularization parameters are the best ones.More details can be obtained in Ref.27.

Fig.2 Curve of Tikhonov filter function wα（σ2）as k changes.

3.Algorithms for MIMO random control

Under most circumstances,amplified noises caused by ill conditioning are the main reason that the identified forces do not match their origins in the domain of input estimation.This inverse difficulty in force identification is actually an ill-posed problem in which there are less equations than unknown variables.The Tikhonov filter is one of the widely used methods which can stabilize the ill-posed problem.However,some differences are noteworthy when we decide to adopt the filter in MIMO power-spectra regeneration.Firstly,the Tikhonov filter involved in least square schemes28for force reconstruction requires corresponding methods,such as the L-curve principle,to determine the optimal regularization parameters.In MIMO vibration control,we have our own thresholds,namely±3 dB tolerances,to estimate the quality of response spectra.Sometimes we don’t have to find the optimal regularization parameters if the response spectra have been adjusted properly in terms of the MIL-STD-810G.Secondly,cross properties are seldom considered as a problem in force identification while we monitor coherence and phase in the MIMO environment test to evaluate the condition of cross-power spectra.Moreover,crosspower spectra can hardly be controlled only using singular value modification methods.

We adopt the Tikhonov filter in this section to eliminate noises in auto-power spectra and propose a criteria to select regularization parameters based on environment control.Afterwards,a phase control algorithm is recommended to reduce the errors brought by inaccurate measurement of the phase offRFMs.

3.1.Auto-power spectra control

The regularization parameters’selection described in Section 2.3 is developed for force identification and the idea of adopting the Tikhonov filter has not been used in environment tests before.

The noises in the response spectra originate from two sources.One is the native noises in the measurement channels.Their levels are low and can hardly be reduced.The other is the components produced by the amplified noises in the driving forces.These noise components in the response spectra have a relatively high level and the corresponding response spectra may exceed their tolerances because of these high-level noises.Therefore,the selection of ill-conditioned frequencies is in terms of auto-power spectra tolerances.Assuming that the system has n inputs and n outputs.The method is

where Aαis the regularized impedance matrix,Syy,iiis the ith auto-power spectrum,and toliis the upper tolerance of the ith auto-power spectrum.

The initial parameter α0is related to the smallest singular values σn.Usually,the smallest singular values take the responsibility of amplifying the noises.The regularization should start with σn.It has been mentioned in Section 2.3 that the initial multiple of the filter function is 0.001,and thus we set α0as below:

In addition,the multiple b in Eq.(23)should be given.The initial parameter is 10-3σ2n.At the beginning,only the smallest singular value σnis altered.When α is overis going to be changed.If the auto-power spectra do not drop below the upper tolerances,the rest of singular values will be altered in this manner.As a result,for n-input and n-output systems,the multiple b should be

where ε is the expected times of the control loop.For a simpler situation,in which there are only two inputs and two outputs,the multiple b should be

where κ（G）denotes the condition number of the FRFMs.When the system is ill-conditioned,the multiple b is relatively large.

3.2.Coherence control

The tolerances of cross-power spectra are defined in terms of coherenceandphase.Theordinary coherenceequation between two noise-free signals is

where E（·）denotes the expected value,Syy,ijdenotes the magnitude of the cross-power spectrum between the ith and jth signals,and Syy,iiand Syy,jjare the auto-power spectrum of the ith and jth signals,respectively.We can obtain from Eq.(26)that the coherence control is highly related with the control of the auto-power spectra.We assume that Yiand Yjare the true response signals in the frequency domain,and Niand Njare the related noises mixed in the responses.These noises are usually independent on response signals and each other.Then,

Substituting the above three equations into Eq.(26),the noisy coherence is

From Eqs.(28)and(29),we can conclude that the autopower spectra of any control point are able to be polluted by independent noises,thus making the acquired auto-power spectra abnormal.This noisy auto-power spectra can eventually influence the coherence.Eq.(30)indicates that a noisy coherence will be smaller than the true coherence because of the enlarged denominator in the equation compared with Eq.(26).Especially,the level of Snin Eq.(30)can be extreme if the corresponding FRFMs are severely ill-conditioned,which makes the acquired coherence worse.It is worth mentioning that the auto-power spectra may exceed their tolerances for reasons not involving the noise problem.We assume that the true response signals Yiand Yjare turned to be tiYiand tjYj,where tiand tjare multiples caused by some other disturbances such as inaccurate measurement.Then,

Eq.(34)shows that disturbances without noise involved can hardly affect the accurate identification of the coherence.The conclusion tells that we could use the coherence as a noise indicator to observe the noise level in the auto-power spectra.Equally,we could also suppress the abnormal auto-power spectra to control the coherence.If the noises in the autopower spectra are filtered out,the coherence will be highly improved.The level of Snremained after filtering is related to the quality of coherence control and we certainly hope it will be as small as possible.

In Section 3.1,the thresholds of the Tikhonov filter are chosen to be the upper tolerances for the auto-power spectra in Eq.(22).According to the MIL-STD-810G,the tolerances for the auto-power spectra are±3 dB for f≤500 Hz and±6 dB for f＞ 500 Hz while the tolerances for an ordinary coherence in the range of 0.707≤γ＜1 are±0.1.The tolerances for the auto-power spectra are preliminarily set to be±3 dB in the whole control bandwidth in this method.Meanwhile,the Tikhonov filter will be applied until the auto-power spectra drop below the+3 dB tolerances.However,the coherence might not be controlled when the abnormal auto-power spectra are just suppressed below±3 dB.We assume that an auto-power spectrum near+3 dB at an ill-conditioned frequency is

Fig.3 Diagram of simulation of cantilever beam.

Therefore,the noisy coherence will be

If the coherence is in the range of 0.707≤γ＜1,the+3 dB tolerance for the auto-power spectra cannot make the coherence inside the±0.1 boundaries.

For a better control on the coherence,the tolerances for auto-power spectra should be lowered down gradually to filter more noises as far as the coherence is controlled properly.

3.3.Phase control

With the application of the Tikhonov filter,the auto-power spectra and coherence are able to be controlled.The regularization on the singular values is to reduce the magnitude of the noises.This noise magnitude decreasing will help reduce the noise influence on the phase.We can obviously observe a phase improvement after the application of the Tikhonov filter in the simulation and experiment section.

The major bias in the phase comes from noise disturbances andinaccuratemeasurementofFRFMs.Ifthecoherenceiscontrolled inside its tolerances,we believe that the noises in the response signals have been reduced to their minimum.Under most conditions,the phase may not stay inside its tolerances afterthecoherenceiscontrolled.Westillneedtodevelopaphase control algorithm to finish the cross-power spectra control.

We assume that φiand φjare the phase biases of the response signals Yiand Yjbrought by inaccurate measurement offRFMs.The response signals polluted by the phase biases are

It can be seen from Eq.(38)that the difference between φiand φjcan influence the phase of the cross-power spectra.Meanwhile,the auto-power spectra and coherence will not be affected.Thus,we develop a phase control algorithm as below:

Table 1 Parameters of cantilever beam example.

Fig.4 Auto-power spectral densities(APSDs)of reference at control points 1 and 2.

Fig.5 Definitions of coherence and phase.

Fig.6 Singular values offrequency response function matrices(FRFMs).

Fig.7 Condition numbers offRFMs.

Fig.8 Auto-and cross-power spectral densitiesof cantilever beam.

where φ1ncan be determined by the difference between the phase and its reference of Syy,1n.It is worth mentioning that φ11is zero.Thus,the correction on the phase of Syy,1nwill be

where φ1nis the difference between the real phase and its reference of Syy,1n.When Δ is obtained by this manner,a more general phase of Syy,ijcan be corrected consequently as

Moreover,Eq.(39)can be simplified.With the application of Cholesky decomposition,Eq.(39)can be transformed to

Fig.9 Controlled auto-power spectra,coherence,and phase of cantilever beam using Smallwood’s algorithm.

4.Numerical applications

We use the finite element method in this simulation.As we can see from Fig.3,the model of a cantilever beam consists of 5 plane beam elements whose node has one translational freedom and one rotational freedom.Two control points are located at positions where the beam is excited.Parameters for the example are listed in Table 1.The references for auto-and cross-power spectral densities are defined respectively in Figs.4 and 5.The alarm boundary tolerances of the auto-power spectral density are±3 dB and the abort boundary tolerances are±6 dB.The tolerances for the coherence is±0.1.The phase boundaries are ±20°.

Fig.10 Controlled auto-power spectra,coherence,and phase of cantilever beam using matrix power algorithm.

Table 2 Multiple b at ill-conditioned frequencies.

Here we use the noise model as

Fig.11 Auto-power spectra,coherence,and phase of cantilever beam when tolerances of noise filtering are+3 dB boundaries.

where yi（ω）refers to the simulated frequency domain acceleration of the ith channel,ynioisy（ω）is the noised yi（ω）,and Δyi（ω）is the magnitude Gaussian noise with a mean equal to unity and a standard deviation equal to 0.05.In addition,Δφiis a phase Gaussian noise with a mean equal to zero and a standard deviation equal to 5°.

As shown in Fig.6,the second singular values offRFMs are extremely small compared with the first singular values in the frequency bands around 30 Hz and 50 Hz.We can also see from Fig.7 that the FRFMs are ill-conditioned at these frequencies.It is very likely that exceeding spectral lines will occur at these points.Fig.8 displays auto-and cross-power spectral densities.Clearly,both auto-and cross-power spectra have some abnormal spectral lines at the frequencies where illconditioning occurs.

Table 3 Finial regularization parameters of Tikhonov filter in numerical example.

Fig.12 Auto-power spectra,coherence,and phase of cantilever beam when tolerances of noise filtering are 0.8tol+3dB.

Fig.13 Auto-power spectra,coherence,and phase of cantilever beam after application of phase control algorithm.

Fig.14 Under-test beam.

4.1.Control process using traditional control algorithms

Fig.15 Other test equipment.

Fig.16 APSDs of reference at control point 1 and point 2 in experiment.

Traditionally,we tend to use control algorithms to suppress abnormal spectral lines of responses.Most of biases between response spectra and reference spectra are able to be effectively reduced to a satisfied level except those caused by illconditioning.We take advantage of two typical control algorithms in this part to demonstrate the application effects.

Smallwood’s algorithm5is carried out at first.Fig.9 shows the response spectra controlled by this algorithm at the second control loop.Once the algorithm is applied,both auto-and cross-power spectra are quickly divergent.

Then we use another algorithm to test its reaction on illconditioned spectra lines.In fact,Cui et al.6has already noticed the instability of Smallwood’s algorithm and developed a matrix power algorithm.We choose the power of the algorithm to be 0.8 in this simulation.Fig.10 depicts the response spectra at the tenth control loop.Compared with those in Fig.8,the response spectra have been improved a lot.However,the control process is ineffective on speed and it might take another ten more control loops to achieve a satisfactory result.

Fig.17 Coherence and phase definitions in experiment.

Fig.18 Condition number of under-test beam.

4.2.Control process using the proposed procedure

We set the expected number of control loops to be ten and take the preliminary tolerances of the auto-power spectra to be+3 dB boundary.The multiple b is calculated in terms of Eq.(25)as listed in Table 2.

Fig.11 demonstrates the response spectra after 10 control loops.We can see from the figure that the auto-power spectra have been suppressed below the+3 dB boundary.According to the MIL-STD-810G,the auto-power spectra are satisfactory.Although the coherence and phase have been improved through this filtering,they still exceed their lower tolerances at some frequencies.The tolerances for the auto-power spectra are going to be lowered down as

Fig.19 Auto-power spectra,coherence,and phase of cantilever beam in experiment.

Then,we update the tolerances in Eq.(44).When conducting a further noise filtering,the number of control loops should be set as large as we can.Thus the multiple b can be as small as possible so as to prevent excessive singular value regularization,in which the spectral lines may exceed their-3 dB tolerances and the coherence may lose control again.

The coherence can be seen as the noise indicator.The±0.1 boundaries for the coherence are quite strict tolerances.As a result,when the coherence is controlled inside the±0.1 boundaries,we think that the noises have been totally filtered out.The final regularization parameters are given in Table 3.

Fig.20 Controlled auto-power spectra,coherence,and phase of cantilever beam using matrix power algorithm in experiment.

From Table 3,we could find that the regularization parameters from 28.75 Hz to 33.75 Hz are smaller than the 10-3σ21values while these parameters at the other frequencies are more close to the 10-3σ21values.The values of 10-3σ21are important thresholds of singular values regularization.When parameters are equal or greater than the thresholds,the first singular values are changed.In fact,the FRFMs of the simulation model in this section are representative.The ill conditioning of the FRFMs around 30 Hz are caused by extremely small second singular values.On the contrary,the FRFMs around 45 Hz are ill-conditioned because of extraordinary great first singular values.We can see from Fig.6 about the corresponding valley and peak of singular values.The ill conditioning only related to small singular values is easy to be handled.The exceeding spectral lines induced by this category can be rapidly suppressed,but it takes time to stabilize the ill-conditioned problems related to extremely great first singular values,which are often in the vicinity of lightly damped resonant peaks.A lot of errors exist around lightly damped resonances.These errors bring control difficulties frequently.In this simulation,the exceeding spectral lines around 45 Hz are stubborn and are the last to be suppressed.Fig.12 shows the result of the second-stage noise filtering.Compared with those in Fig.11,the auto-power spectra at ill-conditioned frequencies are further lowered down and the coherence is controlled properly.However,the phase is not obviously improved at this time.This phenomenon is common especially in a field test.Actually,the phase is more affected by the inaccurate measurement offRFMs.Through noise filtering,the influence of noises on the phase can be reduced to their minimum but the phase cannot be totally controlled.

Fig.21 Auto-power spectra,coherence,and phase of cantilever beam when tolerances of noise filtering are+3 dB boundaries in experiment.

Adopting the newly developed phase control algorithm,the phase is able to be adjusted properly as shown in Fig.13.It is worth stressing that the algorithm is better to be applied after noise filtering.Noise components can hardly be controlled by a regular control algorithm.

Fig.22 Auto-power spectra,coherence,and phase of cantilever beam when tolerancesofnoise filtering are 0.7tol+3dB in experiment.

Table 4 Final regularization parameters of experiment.

Fig.23 Singular values of under-test item.

Fig.24 Auto-power spectra,coherence,and phase of cantilever beam after application of phase control algorithm in experiment.

5.Experimental application

5.1.Parameter setting

We can see from Fig.14 that a typical MET with two exciters and two control points was conducted on a cantilever beam.Two exciters vibrate vertically and two sensors are located on the beam to catch the accelerations.A digital signal analyzer VXI,two power amplifiers,and a personal computer are also used in the system as shown in Fig.15.The control software is written in MATLAB.The sampling frequency is 5120 Hz,the control frequency band is 20–2000 Hz,and the frequency resolution is 2.5 Hz.

The two auto-power spectral densities of the reference are indicated in Fig.16 and its cross-power spectral densities are defined in Fig.17.We set±3 dB as the alarm boundaries and±6 dB as the abort boundaries for the auto-power spectra.The tolerances for the coherence is±0.1.The phase boundaries are ±20°.

Fig.18 shows the curve of the condition number of the under-test beam.We can see that ill conditioning happens in the frequencies around 1400 Hz and 1700 Hz.The levels of condition numbers around these two frequencies are extremely high.Fig.19 demonstrates the original response spectra.Unlike the response spectra in the numerical study,the noise distribution in the field test is more complicated and the spectral lines may exceed the+3 dB limits at some relatively wellconditioned frequencies.The most major bias between the reference and the response spectra occurs in the frequencies around 1700 Hz and the corresponding condition number is almost over 1500.

5.2.Control process using matrix power algorithm

Since Smallwood’s algorithm is potentially unstable,we do not adopt the method in the experiment for safety.The response spectra are controlled with the matrix power algorithm.In the field test,we find that the abnormal spectral lines are much more stubborn than those in the simulation.The over-range response spectra around 1700 Hz can hardly be lowered down any more at the point shown in Fig.20,while the exceeding spectral lines at other frequencies are controlled properly.The uncontrollable issue happens when the ill conditioning at some frequencies is severe.We can find from Fig.18 that the condition numbers around 1700 Hz have already been over 1000 which are extreme compared with those at the other frequencies.

5.3.Noise filtering and phase control

Then we set+3 dB as the preliminary tolerances for noise filtering.Fig.21 shows the result of the first-stage filtering.The former exceeding spectral lines of the auto-power spectra are near but below the+3 dB boundaries.The coherence and phase are improved,more or less.

We set the tolerance of second-stage noise filtering to be 0.7tol+3dB.The result is illustrated in Fig.22.In Fig.22(b),the coherence has been inside its tolerances.In this experiment,there are over 30 frequency points at which the FRFMs are corrected using the Tikhonov filter.Part of typical regularization parameters of the noise filtering are listed in Table 4.Combining with Fig.23,we may figure that the singular values around 1450 Hz and 1740 Hz are different from the ill-conditioned singular values in the simulation.The minimum second singular values are located near the maximum first singular values.The corresponding regularization parameters in Table 4 show that the first singular values are changed largely around 1740 Hz,where there is a high resonant peak.

After noise filtering,the phase control algorithm is applied and the result is demonstrated in Fig.24.The exceeding phase has been controlled as expected.The auto-power spectra and coherence are not affected by this algorithm.The phase control algorithm can be combined with any other control algorithms to enhance their ability of controlling the phase.

6.Conclusions

In this work,the Tikhonov filter was studied and used in an MIMO environment test to eliminate noise components of the response spectra.The influences of the noises on both the auto-and cross-power spectra are introduced.We can conclude from the results that the control of the coherence is highly related with the noises in the auto-power spectra.The traditional tolerances for the auto-power spectra cannot satisfy the demand of coherence control.Based on that,a further noise filtering is suggested by lowering down the tolerances of the autopower spectra.After the coherence is controlled properly,a phase control algorithm is put forward to finish the crosspower spectra control.Both the numerical study and the experimental application verify the whole procedure can solve the noise disturbance and control the response spectra.

From the results of the simulation and the experiment,we can see that the exceeding spectral lines of the auto-power spectra in the lightly damped resonances are more stubborn.The first singular values are fixed to suppress these spectral lines.Although a number of works has concentrated on the relations between damping and ill conditioning,the problem is worth a further study in the framework of an MIMO environment test.

Acknowledgements

The study was supported by the Fundamental Research Funds for the Central Universities(No.NS2015008)and the corresponding work was performed in the State Key Laboratory of Mechanics and Control of Mechanical Structures.

1.Wang Y,Wang Z,Sun J,Zhang J,Zissimos M.Gray bootstrap method for estimating frequency-varying random vibration signals with small samples.Chin J Aeronaut 2014;27(2):383–9.

2.Wang Z,Wang Y,Wang Q,Zhang J.Fuzzy norm method for evaluating random vibration of airborne platform from limited PSD data.Chin J Aeronaut 2014;27(6):1442–50.

3.Phillips AW,Allemang RJ.An overview of MIMO-FRF excitation/averaging/processing techniques.J Sound Vib 2003;262(3):651–75.

4.Karshenas A,Dunnigan M,Williams B,Goodfollow J.Multiresolution processing for random vibration control.J IES 1997;40(1):35–40.

5.Smallwood DO.Multiple shaker random vibration control—An update.Proceedings of the institute of environmental sciences;1999 May 1–3;Ontario(CA).Mount Prospect(IL):Institute of Environmental Sciences;1999.p.212–21.

6.Cui X,Chen H,He X,Jiang S.Matrix power control algorithm for multi-input multi-output random vibration test.Chin J Aeronaut 2011;24(6):741–8.

7.Fabunmi JA.Modal constraints on structural dynamic force determination.J Am Helicopter Soc 1985;30(4):48–54.

8.Fabunmi JA.Effects of structural modes on vibratory force determination by the pseudoinverse technique.AIAA J 1986;24(3):504–9.

9.Lee JK,Park YS.Response selection and dynamic damper application to improve the identification of multiple input forces of narrow frequency band.Mech Syst Sig Proc 1994;8(6):649–64.

10.Chiementin X,Bolaers F,Dron JP,Rasolofondraibe L.Inverse approach to the reconstruction and quantification of vibratory sources.J Vib Control 2007;13(8):1169–90.

11.Chiementin X,Bolaers F,Rasolofondraibe L,Dron JP.Localization and quantification of vibratory sources:Application to the predictive maintenance of rolling bearings.J Sound Vib 2008;316(1–5):331–47.

12.Powell R,Seering W.Multichannel structural inverse filtering.J Vib Acoust 1984;106(1):22–8.

13.Romano J,Lopez J.Practical application of transfer path analysis to resolve structure-borne noise problems in vehicle design.Proceedings of ISMA;1996 Feb 12–15;Dearborn(MI).Bethel(CT):Society for Experimental Mechanics;1996.p.527–36.

14.Thite AN,Thompson DJ.The quantification of structure-borne transmission paths by inverse methods.Part 1:Improved singular value rejection methods.J Sound Vib 2003;264(2):411–31.

15.Nelson PA,Yoon SH.Estimation of acoustic source strength by inverse methods:Part I,Conditioning of the inverse problem.J Sound Vib 2000;233(4):639–64.

16.Yoon SH,Nelson PA.Estimation of acoustic source strength by inverse methods:Part II,Experimental investigation of methods for choosing regularization parameters.J Sound Vib 2000;233(4):665–701.

17.Hansen P.Analysis of discrete ill-posed problems by means of the L-curve.SIAM Rev 1992;34(4):561–80.

18.Jia Y,Yang Z,Song Q.Experimental study of random dynamic loads identification based on weighted regularization method.J Sound Vib 2015;342:113–23.

19.Smallwood DO.Multiple shaker random control with cross coupling.Proceedings of the institute of environmental sciences;1978 Apr 18–20;Fort Worth(TX).Mount Prospect(IL):Institute of Environmental Sciences;1978.p.341–7.

20.Smallwood DO.Random vibration testing of a single test item with a multiple input control system.Proceedings of the institute of environmental sciences;1982 Apr 21–23;Atlanta(GA).Mount Prospect(IL):Institute of Environmental Sciences;1982.p.42–9.

21.Smallwood DO,Woodall T,Buksa EJ.Minimum drive requirements for a multiple input multiple output linear system.Proceedings of the institute of environmental sciences;1986 Apr18–20;Dallas(TX).Mount Prospect(IL):Institute of Environmental Sciences;1986.p.295–301.

22.Paez T,Smallwood D,Buska E.Random control at n2 points using n shakers.Proceedings of the institute of environmental sciences;1987 May 5–7;San Jose(CA).Mount Prospect(IL):Institute of Environmental Sciences;1987.p.271–5.

23.Smallwood DO,Paez TL.A frequency domain method for the generation of partially coherent normal stationary time domain signals.Shock Vibrat 1993;1(1):45–53.

24.Smallwood DO.Generation of stationary non-Gaussian time histories with a specified cross-spectral density.Shock Vibrat 1997;4(5–6):361–77.

25.Underwood MA,Keller T.Recent system developments for multiactuator vibration control.Sound Vibrat 2001;35(10):16–23.

26.Underwood MA.Multi-exciter testing applications:theory and practice.Proceedings-institute of environmental sciences and technology;2002 Apr 17–19;San Marcos(CA).San Jose(CA):Institute of Environmental Sciences and Technology;2002.p.12–5.

27.Vogel C.Computational methods for inverse problems.Philadelphia(PA):SIAM;2002.p.2–6.

28.Liu Y,Shepard Jr WS.Dynamic force identification based on enhanced least squares and total least-squares schemes in the frequency domain.J Sound Vib 2005;282(1–2):37–60.

Cui Song received his B.S.and M.S.degrees in engineering mechanics from Nanjing University of Aeronautics and Astronautics in 2010 and 2013,respectively,and then became a Ph.D.student there.His main research interests are multi-exciter vibrational environment test.

Chen Huaihai is a professor and Ph.D.advisor in the Institute of Vibration Engineering Research at Nanjing University of Aeronautics and Astronautics,Nanjing,China.He received his Ph.D.degree from Dalian University of Technology.His current research interests are multi-exciter vibrational environment test and vibration fatigue.

He Xudong is an associate professor in the Institute of Vibration Engineering Research at Nanjing University of Aeronautics and Astronautics,where he received his Ph.D.degree in 2007.His main research interest is multi-exciter vibrational environment test.

Zheng Wei is a Ph.D.student in the Institute of Vibration Engineering Research at Nanjing University of Aeronautics and Astronautics.His main research interest is sweep sine vibration test in multi-axis.

21 September 2015;revised 22 June 2016;accepted 25 August 2016

Available online 7 November 2016

Coherence;

Environmental testing;

Multi-input multi-output(MIMO);

Noise;

Phase control;

Tikhonov filter

Ⓒ2016 Chinese Society of Aeronautics and Astronautics.Production and hosting by Elsevier Ltd.This is anopenaccessarticleundertheCCBY-NC-NDlicense(http://creativecommons.org/licenses/by-nc-nd/4.0/).

*Corresponding author.Te.:+86 25 84893082.

E-mail addresses:cuisong@nuaa.edu.cn(S.Cui),chhnuaa@nuaa.edu.cn(H.Chen),hexudong@nuaa.edu.cn(X.He),kantslife@qq.com(W.Zheng).

Peer review under responsibility of Editorial Committee of CJA.

Production and hosting by Elsevier

http://dx.doi.org/10.1016/j.cja.2016.10.016

1000-9361Ⓒ2016 Chinese Society of Aeronautics and Astronautics.Production and hosting by Elsevier Ltd.

This is an open access article under the CC BY-NC-ND license(http://creativecommons.org/licenses/by-nc-nd/4.0/).