# Correction of Transducers Mass Effects from the Measured FRFs in Hammer Impact Testing

Authors: Jun Ren; Jun Wang; Shusheng Bi
Journal: Shock and Vibration
 (2017)

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## Abstract

In modal testing, the measured frequency response functions (FRFs) are often affected by transducer mass effects. Especially when multiple sensors are employed, transducers mass effects are more significant. This paper deals with removing sensors mass effects from the measured FRFs in hammer impact testing. The proposed method offers some advantages over the available techniques in that extra FRFs measurements with different configurations are not required during the correction process. Firstly, correction method for single accelerometer case is discussed. Further, general correction formulation for multiple accelerometers case is derived. Then, validations of the proposed method are demonstrated using simulated and experimental data. It shows good effectiveness in experimental validation for single accelerometer case. Numerical simulation also demonstrates a good performance for multiple accelerometers case. However, further simulations with noisy-polluted data show that the accuracy of correction results will be affected to some extent by the noise. It is suggested that the measured FRFs be preprocessed using the curve-fitting procedure or noise reduction processing before applying the proposed method.


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## Body

## 1. Introduction

Frequency response functions (FRFs) are widely utilized in structural dynamics analysis and modification. However, accuracies of the measured FRFs are usually affected due to a variety of factors, one of which is transducer mass effect [1, 2]. For large-scale test structures, mass effect of the sensor is relatively small and it is usually artificially ignored in the test. But for lightweight or small-scale structure, mass effect introduced by the sensor is notable [3, 4]. Especially when multiple sensors are employed, transducers masses loading effects are significant. The measured FRFs may contain larger errors. It is thus necessary to correct the measured FRFs before they are utilized for further analysis.In early, a mass effects correction method for driving point FRF was introduced by Ewins [1]. Afterwards, some publications focused on removing such effects from transfer FRFs. Decker and Witfeid [5] presented a method to correct the transducer mass effects from transfer FRFs based on substructuring techniques. Ashory [6] proposed a method in which transfer FRFs can be corrected if the measurement is repeated with an accelerometer of different mass. Cakar and Sanliturk [7] presented a correction method by employing dummy mass. Ren et al. [8, 9] proposed a correction method for eliminating accelerometer and force transducer mass loading effects from measured FRFs in shaker modal testing. These researches mainly focus on removing mass effects of individual sensor. And extra FRFs measurements with different configurations are needed in the process of correcting the measured transfer FRFs, such as moving the sensor to different response locations, employing accelerometers of different masses or adding extra accelerometer (or dummy mass) on the structure. Zamani et al. [10] presented a strategy for correcting accelerometer mass effect using sensitivity analysis and the effect of accelerometer’s mass can be neglected using this strategy. Yin [11] presented a method for eliminating transducer additional mass effects in time-domain responses.In practical modal test, it usually required to install multiple sensors for measuring some certain FRFs. The current authors presented a method for correcting multiple transducers mass effects from the measured FRFs in an earlier work [12]. This article expands on that work with differences on the following: (1) correction method for both single accelerometer and multiple accelerometers cases are discussed; (2) experiments validation, analysis, and discussions are supplemented except for numerical case study. Firstly, correction method for single accelerometer case is discussed. Further, general correction formulation for multiple accelerometers case is derived. Then, validations of the proposed method are demonstrated using simulated as well as experimental data. The proposed method offers some advantages over the available techniques in that extra FRFs measurements with different configurations are not required in the process of correcting the measured FRFs. “Different configurations” here means that test structure is changed due to movement of sensors to different response locations or addition of extra sensors (or dummy masses) on the structure.


## 2. Correction of Mass Effects of Single Accelerometer from the Measured FRFs

Hammer impact test of a cantilever beam is shown in Figure1. The beam is excited at point p by a hammer. And an accelerometer with a mass of ma is mounted at point l to measure the response. Then dynamics of the beam are changed due to accelerometer mass effects.Figure 1
Hammer impact testing of cantilever beam.To facilitate the analysis, accelerometer is separated from the original system, as shown in Figure2. If excitation force and reaction force of the added masses at point j are Fp and Fl, displacement of point l can be expressed as(1)Xl=HlpFp+HllFl.Displacement of accelerometer can also be established by using the receptance and reaction force F′:(2)Xl′=-1maω2Fl′.Constraint equations of forces and displacements at l are(3)Fl=-Fl′Xl=-Xl′.By substituting (2) and (3) into (1), ones yields(4)Hlpl=XlFp=Hlp1+maω2Hll,where Hlp(l) is measured FRF (with mass effects). And Hlp and Hll are exact FRFs (without mass effects). FRFs in (4) are receptances. In practical test, however, accelerances are utilized more frequently. This poses no difficulty since receptances and accelerances are just two different forms of presenting the same FRF. Multiplying both sides by -ω2 of (4), one yields(5)Alpl=Alp1+maAll,where A denotes accelerance. If excitation point is moved from p to l, accelerance of point l can be easily obtained as(6)Alll=All1+maAll.By rearranging (5) and (6), correction formulas of measured accelerances can be obtained as follows: (7)All=Alll1-maAlll(8)Alp=Alpl1-maAlll.Figure 2
Free body diagram of the system.It can be seen from (7) and (8) that the measured accelerances can be corrected if both point and transfer accelerances are measured.


## 3. Theoretical Development of the Method for Multiple Accelerometers Case

AnN dof mass-spring system is presented in Figure 3(a). To measure the responses, N accelerometers with masses Δm1,Δm2,…,Δmn are installed on point 1,2,…,N, respectively. Then the system is changed due to additional masses, as it can be seen in Figure 3(b).Figure 3
Ann dof mass-spring system.
(a)
Original system (corresponding to FRFα)
(b)
Modified system by adding the extra masses (corresponding to FRFα′).To facilitate the analysis, accelerometers are separated from the original system, as shown in Figure4. The system is excited at point k and the excitation force is Fk. Then displacements of point 1,2,…,N can be expressed as(9)x1=α1kFk+∑j=1Nα1jRjx2=α2kFk+∑j=1Nα2jRj⋮xN=αNkFk+∑j=1NαNjRj,where x1,x2,…,xN are displacements of points 1, 2, and N, respectively. αik is receptance (relating to excitation point k and response point i) of original system. And R1, R2, and RN are reaction forces of the added masses at point 1,2,…,N, respectively. The constraint equations are(10)x1=x1′,x1=x1′⋮xN=xN′R1=-R1′,R1=-R2′⋮RN=-RN′.The governing equations for the added masses Δm1,Δm2,…,Δmn are(11)H11′=x1′R1′=-1Δm1ω2H22′=x2′R2′=-1Δm2ω2⋮HNN′=xN′RN′=-1ΔmNω2,where H11′, H22′, and HNN′ are receptances of the added masses. Then the reaction forces of the added masses are(12)R1=x1Δm1ω2R2=x2Δm2ω2⋮RN=xNΔmNω2Substituting (12) into (9) yields(13)x1=α1kFk+∑j=1Nα1jxjΔmjω2x2=α2kFk+∑j=1Nα2jxjΔmjω2⋮xN=αNkFk+∑j=1NαNjxjΔmjω2.Dividing Fk by both sides of (13) yields (14)α1k′=α1k+∑j=1Nα1jαjk′Δmjω2α2k′=α2k+∑j=1Nα2jαjk′Δmjω2⋮αNk′=αNk+∑j=1NαNjαjk′Δmjω2,where αs are receptances of original system and α′s are receptances of modified system (with transducer masses attached). Rearranging (14) by separating α′s and αs yields (15)1-α11Δm1ω2⋯-α1NΔmNω2⋮⋱-αN1Δm1ω21-αNNΔmNω2α1k′⋮αNk′=α1k⋮αNk.Figure 4
Free body diagram of then dof system.It shows that the measured FRFsαik′(i=1⋯N) can be obtained according to (15) if exact FRFs αij(i=1⋯N,j=1⋯N) are known. However, it is the opposite in the problem of correcting the transducer mass effects that exact FRFs αij are to be obtained and FRFs αik′ are known (by measurement). Obviously, exact FRF αik cannot be directly obtained according to (15). This paper proposed a method by adding virtual negative masses to solve this problem, as it can be seen in Figure 5. Modification of the structure by adding extra masses is shown in Figure 5(a) which corresponds with (15). In Figure 5(b), however, the modified system  (a3) is considered as “original system” (b1). And then the “modified system” can be generated by adding virtual negative masses -Δmi (i=1⋯N) to point 1⋯N, respectively. The new obtained “modified system” (b3) is equivalent to original system (a1).Figure 5
Modification of the system by adding extra masses.
(a)
Modification of the system by adding extra mass Δm
(b)
Modification of modified system by adding –ΔmThe process of modification of the system in Figure5(a) can be described by employing (15). To describe Figure 5(b), it only requires to replace αik′(i=1⋯N), αij(i=1⋯N,j=1⋯N), and Δmi(i=1⋯N) with αik(i=1⋯N), αij′(i=1⋯N,j=1⋯N), and -Δmi (i=1⋯N) in (15). The result is shown in the following:(16)1+α11′Δm1ω2⋯α1N′ΔmNω2⋮⋱αN1′Δm1ω21+αNN′ΔmNω2α1k⋮αNk=α1k′⋮αNk′.It can be seen from (16) that exact FRF αik (i=1⋯N) of original system can be obtained given that transducer masses Δmi (i=1⋯N) and FRFs αij′ of modified system are known. By extending (8), the genera formula for correcting FRFs αij′ (i=1⋯N, j=1⋯N) can be easily deduced in(17)1+α11′Δm1ω2⋯α1N′ΔmNω2⋮⋱⋮αN1′Δm1ω2⋯1+αNN′ΔmNω2α11⋯α1N⋮⋱⋮αN1⋯αNN=α11′⋯α1N′⋮⋱⋮αN1′⋯αNN′.By rearranging (17), exact FRF αij (i=1⋯N, j=1⋯N) can be solved by(18)α11⋯α1N⋮⋱⋮αN1⋯αNN=1+α11′Δm1ω2⋯α1N′ΔmNω2⋮⋱⋮αN1′Δm1ω2⋯1+αNN′ΔmNω2-1α11′⋯α1N′⋮⋱⋮αN1′⋯αNN′.Discussion above refers to receptancesα. In practical test, accelerances A are utilized more frequently. Multiplying both sides by -ω2 of (18), one yields(19)A11⋯A1N⋮⋱⋮AN1⋯ANN=1-A11′Δm1⋯-A1N′ΔmN⋮⋱⋮-AN1′Δm1⋯1-ANN′ΔmN-1A11′⋯A1N′⋮⋱⋮AN1′⋯ANN′,where As are accelerances of original system and A′s are accelerances of modified system (with mass effects). It is not surprising that (19) will be coincident with (7) and (8) when only one accelerometer is used.As it can be seen from (16)~(19), all the measured FRFs associated with the response points are required for FRFs corrections. In the case of n response points, a symmetric matrix consisting of n2 FRFs needs to be measured. Since FRF Aij′ equals Aji′, only N(N+1)/2 FRFs are required to be measured in fact.


## 4. Verification of the Method

### 4.1. Experimental Study for Single Accelerometer Case: Impact Testing of a Rectangle Plate

Free-free impact modal testing of a rectangle plate shown in Figure6 was chosen for the verification of the proposed method. The rectangular plate is suspended by elastic rope, to simulate free-free boundary conditions. The size and weight of the plate are 250×148×6 mm3 and 0.61 Kg, respectively. The plate is equipped with 15 measuring points; the acceleration sensor (approximately 8 g) is installed at point 2. Hammer impact is moving sequentially from point 1 to point 15 for measuring accelerances A21(2),A22(2),…,A2_15(2). Obviously all the measured FRFs are affected by acceleration sensor mass loading effects. However, only a point FRF A22(2) and a transfer FRF A2_14(2) are chosen to be corrected here for the sake of brevity.Figure 6
Free-free impact modal testing of a rectangle plate.The process of removing accelerometer mass is shown in Figure7(a). Effectiveness of the proposed method can be assessed by comparing the corrected FRF with the exact FRF (Figure 7(a3)). However, exact value is unavailable which makes it difficult to assess. To solve this difficulty, a comparative experiment is designed, as shown in Figure 7(b). An additional mass mext is attached point l when accelerance is measured (Figure 7(b1)). And corrected accelerance can be obtained by removing additional mass mext from the measured value. Then, the method can be assessed by comparing corrected FRF with target FRF (Figure 7(b3)) which can be measured directly.Figure 7
Comparative experiments.
(a)
Elimination of transducer massma
(b)
Elimination of additional massmextAccording to the idea of comparative experiments, Target FRFsA22 and A2_14 (without additional mass mext) are firstly measured. Then A22(2) and A2_14(2) (with additional mass mext 37 g) are measured. Then corrected FRFs A22∗ and A2_14∗ can be calculated by employing (7) and (8). Effectiveness of the correction method can thus be assessed by comparing A22 and A2_14 with A22∗ and A2_14∗, respectively.As shown in Figure8, resonance frequencies of the measured accelerance A22(2) are lower than those of the target accelerance A22 due to extra mass effects. After correction of extra mass mext, as expected, the frequencies of corrected accelerance A22∗ as a whole are in quite good agreement with those of the target accelerance A22. However, it is not very satisfying that amplitudes at resonance peaks of A22∗ are slightly inconsistent with those of A22, for example, the first resonant peak shown in Figure 9. This is because additional damping effects are unavoidably involved when extra mass is installed on the structure and the proposed method only considers removing the mass effects rather than damping effects. Similar observation can also be found in correction of transfer FRF A2_14(2) and point FRF A22(2); see Figures 10 and 11, respectively. It can be seen that the proposed method shows a good effectiveness in resonance frequencies correction. Comparison of frequencies of exact, “measured,” and corrected FRFs is given in Table 1. It is shown that frequencies errors of 1st and 2nd order are obviously reduced after correction, from 5.9%, 7.1% to 0.2%, and 0.6%, respectively, while frequencies errors of 3rd and 4th order are slightly reduced after correction. In particular, for 3rd order frequency, the error reduction is merely 0.1% (from 0.8% to 0.7%) after correction. This is due to the fact that the additional mass effects on different orders of modal are not the same. In this test, accelerometer installation position (point 2) coincides with modal node of 3rd order. Then additional mass has little effect on frequency of this order.Table 1
Comparison of frequencies of target, “measured,” and corrected FRFs.
1st order/Hz
2nd order/Hz
3rd order/Hz
4th order/Hz
Target
486.3
1341
1480
1824
Measured
457.5
1246
1468
1789
Corrected
487.5
1333
1490
1845
Errors before correction
5.9%
7.1%
0.8%
1.9%
Errors after correction
0.2%
0.6%
0.7%
1.2%Figure 8
Comparison of “measured,” corrected, and target FRFA22.Figure 9
Comparison of “measured,” corrected, and target FRFA22 (350–650 Hz).Figure 10
Comparison of “measured,” corrected, and target FRFA2_14.Figure 11
Comparison of “measured,” corrected, and target FRFA2_14 (350–650 Hz).


### 4.2. Numerical Simulation for Multiple Accelerometers Case: Impact Testing of Cantilever Beam

Simulated hammer impact test of a cantilever beam model is shown in Figure12 and the physical parameters are given in Table 2. Six points are evenly arranged along the longitudinal direction of the beam. Acceleration sensors with masses of 0.38 Kg, 0.42 Kg, and 0.4 Kg are installed at points 2, 4, and 5, respectively. And the excitation point is chosen to be point 2. Obviously, the measured FRFs A22′, A42′, and A52′ are affected by the mass effects of accelerometers (m2, m4, and m5) and need to be corrected.Table 2
Parameters of the cantilever beam.
Parameters
l/m
b/m
h/m
ρ/(Kg⋅m−3)
E/MPa
Values
1.5
0.05
0.012
7547
2.07
×
10
5Figure 12
Cantilever beam model.According to (19), additional FRFs A44′, A55′, and A45′ are required to be “measured” for correction of FRFs A22′, A42′, and A52′ (note that A45′ equals A54′), as it can be seen in (20). The “measured” accelerances are numerically generated in this simulation. For comparison purpose, exact FRFs A22∗, A42∗, and A52∗ corresponding with original structure (without attaching transducers) are also numerical calculated.(20)A22A42A52=1-A22′Δm2-A24′Δm4-A25′Δm5-A42′Δm21-A44′Δm4-A45′Δm2-A52′Δm2-A54′Δm41-A55′Δm5-1A22′A42′A52′.The correction results and comparisons are shown in Figures13, 14, and 15. As shown in these figures, resonance frequencies of the “measured” FRFs are lower than those of exact FRFs due to the mass effects of transducers mass effects. And the magnitudes of resonance frequency reduction of each order are different. It demonstrates that transducers bring different degrees of mass effects on different orders [13–15]. In this study, decreases of low-order resonances frequencies are much smaller than those of high-order resonances frequencies. As a result, transducers mass effects can be ignored if only the first order is concerned. However, elimination of transducers mass effects should be taken into consideration if high-order (e.g., the third order) is of interest. After correction, as expected, corrected FRFs A22, A42, and A52 are in quite good agreement with exact FRFs A22∗, A42∗, and A52∗ which demonstrates a good performance of the proposed method. Although the method has been confirmed to be quite promising in a simulated test, some aspects of the method need further consideration before it can readily be used in practical modal test. The measured FRFs in a practical test are inevitably polluted by noise to some extent. Further simulations presented here examine the performance of the method in the case of noisy data. The same system given in Figure 12 is utilized again. The difference is that A22′, A42′, A52′, A44′, A55′, and A45′ are numerically incorporated with commonly used white Gaussian noise (zero mean and variance δn2) [15]. The noise level γ is defined as(21)γ2=δn2Hijωmean2,where Hij(ω)mean is the mean absolute value of Hij(ω) in the frequency range of interest. The noise levels in present study are set to be 0.5% and 1%, respectively. Since FRFs are complex values, the random noise should be added to real and imaginary parts of the FRFs, respectively.Figure 13
Comparison of “measured,” exact, and corrected FRFA22.Figure 14
Comparison of “measured,” exact, and corrected FRFA42.Figure 15
Comparison of “measured,” exact, and corrected FRFA52.Comparisons of measured, exact, and corrected FRFs are shown in Figures8–13. It can be seen from Figures 16, 17, 19, and 20 (corrections of A22 and A42) that frequencies bands (15–25 Hz, 45–70 Hz) near second and third resonant peak have been effectively corrected, while others including frequencies bands near first resonant peak exhibit poor correction results. In Figures 18 and 21 (correction of A52), however, the frequencies bands near all the resonant peaks show desirable correction results. This is mainly due to the fact that the frequencies bands of the “measured” FRFs with relatively low amplitudes are greatly polluted by the noise, thus affecting the correction results. Therefore, the proposed correction method may be sensitive to a certain extent to the noise in practical test. In this case, it is suggested that the measured FRFs be preprocessed using the curve-fitting procedure before the proposed correction method is applied. Another possible solution to this problem is to eliminate the noise from the measured FRFs before they are utilized for calculations.Figure 16
Comparison of “measured,” exact, and corrected FRFA22 (0.5% noise).Figure 17
Comparison of “measured,” exact, and corrected FRFA42 (0.5% noise).Figure 18
Comparison of “measured,” exact, and corrected FRFA52 (0.5% noise).Figure 19
Comparison of “measured,” exact, and corrected FRFA22 (1% noise).Figure 20
Comparison of “measured,” exact, and corrected FRFA42 (1% noise).Figure 21
Comparison of “measured,” exact, and corrected FRFA52 (1% noise).


## 4.1. Experimental Study for Single Accelerometer Case: Impact Testing of a Rectangle Plate

Free-free impact modal testing of a rectangle plate shown in Figure6 was chosen for the verification of the proposed method. The rectangular plate is suspended by elastic rope, to simulate free-free boundary conditions. The size and weight of the plate are 250×148×6 mm3 and 0.61 Kg, respectively. The plate is equipped with 15 measuring points; the acceleration sensor (approximately 8 g) is installed at point 2. Hammer impact is moving sequentially from point 1 to point 15 for measuring accelerances A21(2),A22(2),…,A2_15(2). Obviously all the measured FRFs are affected by acceleration sensor mass loading effects. However, only a point FRF A22(2) and a transfer FRF A2_14(2) are chosen to be corrected here for the sake of brevity.Figure 6
Free-free impact modal testing of a rectangle plate.The process of removing accelerometer mass is shown in Figure7(a). Effectiveness of the proposed method can be assessed by comparing the corrected FRF with the exact FRF (Figure 7(a3)). However, exact value is unavailable which makes it difficult to assess. To solve this difficulty, a comparative experiment is designed, as shown in Figure 7(b). An additional mass mext is attached point l when accelerance is measured (Figure 7(b1)). And corrected accelerance can be obtained by removing additional mass mext from the measured value. Then, the method can be assessed by comparing corrected FRF with target FRF (Figure 7(b3)) which can be measured directly.Figure 7
Comparative experiments.
(a)
Elimination of transducer massma
(b)
Elimination of additional massmextAccording to the idea of comparative experiments, Target FRFsA22 and A2_14 (without additional mass mext) are firstly measured. Then A22(2) and A2_14(2) (with additional mass mext 37 g) are measured. Then corrected FRFs A22∗ and A2_14∗ can be calculated by employing (7) and (8). Effectiveness of the correction method can thus be assessed by comparing A22 and A2_14 with A22∗ and A2_14∗, respectively.As shown in Figure8, resonance frequencies of the measured accelerance A22(2) are lower than those of the target accelerance A22 due to extra mass effects. After correction of extra mass mext, as expected, the frequencies of corrected accelerance A22∗ as a whole are in quite good agreement with those of the target accelerance A22. However, it is not very satisfying that amplitudes at resonance peaks of A22∗ are slightly inconsistent with those of A22, for example, the first resonant peak shown in Figure 9. This is because additional damping effects are unavoidably involved when extra mass is installed on the structure and the proposed method only considers removing the mass effects rather than damping effects. Similar observation can also be found in correction of transfer FRF A2_14(2) and point FRF A22(2); see Figures 10 and 11, respectively. It can be seen that the proposed method shows a good effectiveness in resonance frequencies correction. Comparison of frequencies of exact, “measured,” and corrected FRFs is given in Table 1. It is shown that frequencies errors of 1st and 2nd order are obviously reduced after correction, from 5.9%, 7.1% to 0.2%, and 0.6%, respectively, while frequencies errors of 3rd and 4th order are slightly reduced after correction. In particular, for 3rd order frequency, the error reduction is merely 0.1% (from 0.8% to 0.7%) after correction. This is due to the fact that the additional mass effects on different orders of modal are not the same. In this test, accelerometer installation position (point 2) coincides with modal node of 3rd order. Then additional mass has little effect on frequency of this order.Table 1
Comparison of frequencies of target, “measured,” and corrected FRFs.
1st order/Hz
2nd order/Hz
3rd order/Hz
4th order/Hz
Target
486.3
1341
1480
1824
Measured
457.5
1246
1468
1789
Corrected
487.5
1333
1490
1845
Errors before correction
5.9%
7.1%
0.8%
1.9%
Errors after correction
0.2%
0.6%
0.7%
1.2%Figure 8
Comparison of “measured,” corrected, and target FRFA22.Figure 9
Comparison of “measured,” corrected, and target FRFA22 (350–650 Hz).Figure 10
Comparison of “measured,” corrected, and target FRFA2_14.Figure 11
Comparison of “measured,” corrected, and target FRFA2_14 (350–650 Hz).


## 4.2. Numerical Simulation for Multiple Accelerometers Case: Impact Testing of Cantilever Beam

Simulated hammer impact test of a cantilever beam model is shown in Figure12 and the physical parameters are given in Table 2. Six points are evenly arranged along the longitudinal direction of the beam. Acceleration sensors with masses of 0.38 Kg, 0.42 Kg, and 0.4 Kg are installed at points 2, 4, and 5, respectively. And the excitation point is chosen to be point 2. Obviously, the measured FRFs A22′, A42′, and A52′ are affected by the mass effects of accelerometers (m2, m4, and m5) and need to be corrected.Table 2
Parameters of the cantilever beam.
Parameters
l/m
b/m
h/m
ρ/(Kg⋅m−3)
E/MPa
Values
1.5
0.05
0.012
7547
2.07
×
10
5Figure 12
Cantilever beam model.According to (19), additional FRFs A44′, A55′, and A45′ are required to be “measured” for correction of FRFs A22′, A42′, and A52′ (note that A45′ equals A54′), as it can be seen in (20). The “measured” accelerances are numerically generated in this simulation. For comparison purpose, exact FRFs A22∗, A42∗, and A52∗ corresponding with original structure (without attaching transducers) are also numerical calculated.(20)A22A42A52=1-A22′Δm2-A24′Δm4-A25′Δm5-A42′Δm21-A44′Δm4-A45′Δm2-A52′Δm2-A54′Δm41-A55′Δm5-1A22′A42′A52′.The correction results and comparisons are shown in Figures13, 14, and 15. As shown in these figures, resonance frequencies of the “measured” FRFs are lower than those of exact FRFs due to the mass effects of transducers mass effects. And the magnitudes of resonance frequency reduction of each order are different. It demonstrates that transducers bring different degrees of mass effects on different orders [13–15]. In this study, decreases of low-order resonances frequencies are much smaller than those of high-order resonances frequencies. As a result, transducers mass effects can be ignored if only the first order is concerned. However, elimination of transducers mass effects should be taken into consideration if high-order (e.g., the third order) is of interest. After correction, as expected, corrected FRFs A22, A42, and A52 are in quite good agreement with exact FRFs A22∗, A42∗, and A52∗ which demonstrates a good performance of the proposed method. Although the method has been confirmed to be quite promising in a simulated test, some aspects of the method need further consideration before it can readily be used in practical modal test. The measured FRFs in a practical test are inevitably polluted by noise to some extent. Further simulations presented here examine the performance of the method in the case of noisy data. The same system given in Figure 12 is utilized again. The difference is that A22′, A42′, A52′, A44′, A55′, and A45′ are numerically incorporated with commonly used white Gaussian noise (zero mean and variance δn2) [15]. The noise level γ is defined as(21)γ2=δn2Hijωmean2,where Hij(ω)mean is the mean absolute value of Hij(ω) in the frequency range of interest. The noise levels in present study are set to be 0.5% and 1%, respectively. Since FRFs are complex values, the random noise should be added to real and imaginary parts of the FRFs, respectively.Figure 13
Comparison of “measured,” exact, and corrected FRFA22.Figure 14
Comparison of “measured,” exact, and corrected FRFA42.Figure 15
Comparison of “measured,” exact, and corrected FRFA52.Comparisons of measured, exact, and corrected FRFs are shown in Figures8–13. It can be seen from Figures 16, 17, 19, and 20 (corrections of A22 and A42) that frequencies bands (15–25 Hz, 45–70 Hz) near second and third resonant peak have been effectively corrected, while others including frequencies bands near first resonant peak exhibit poor correction results. In Figures 18 and 21 (correction of A52), however, the frequencies bands near all the resonant peaks show desirable correction results. This is mainly due to the fact that the frequencies bands of the “measured” FRFs with relatively low amplitudes are greatly polluted by the noise, thus affecting the correction results. Therefore, the proposed correction method may be sensitive to a certain extent to the noise in practical test. In this case, it is suggested that the measured FRFs be preprocessed using the curve-fitting procedure before the proposed correction method is applied. Another possible solution to this problem is to eliminate the noise from the measured FRFs before they are utilized for calculations.Figure 16
Comparison of “measured,” exact, and corrected FRFA22 (0.5% noise).Figure 17
Comparison of “measured,” exact, and corrected FRFA42 (0.5% noise).Figure 18
Comparison of “measured,” exact, and corrected FRFA52 (0.5% noise).Figure 19
Comparison of “measured,” exact, and corrected FRFA22 (1% noise).Figure 20
Comparison of “measured,” exact, and corrected FRFA42 (1% noise).Figure 21
Comparison of “measured,” exact, and corrected FRFA52 (1% noise).


## 5. Conclusion

In modal testing, the measured FRFs are often affected by transducer mass effects. Especially when multiple sensors are employed, transducers mass effects are more significant. This paper deals with removing accelerometers mass effects from the measured FRFs in hammer impacting testing. The research indicates the following:(1)
WhenN accelerometers are attached at points 1,2,…,N, respectively, at least N(N+1)/2 FRFs related to all the response points are required to measure for eliminating accelerometers mass effects from the measured FRFs A1p′,A2p′,…,ANp′ (subscript p denotes excitation point).(2)
Advantage of this method is that additional configurations of the structure for FRFs measurements (used for correction) are not required, such as moving the sensor to different response locations, adding extra sensors or dummy masses on the structure.The proposed method shows good effectiveness in experimental validation for single accelerometer case. A simple numerical example presented also demonstrates a good performance in the numerical simulation for multiple accelerometers case. However, further simulations with noisy-polluted data show that the accuracy of correction results will be affected to some extent by the noise. It is suggested that the measured FRFs be preprocessed using the curve-fitting procedure or noise reduction processing before applying this method. Since the FRFs measurements required for corrections refer to different excitation points, moving the exciter to various locations is required. This is usually easy to achieve in hammer impact testing. However, moving the exciter is usually inconvenient (if not impossible) in shaker modal testing because the excitation point is usually fixed in such case. Therefore, the proposed method is more applicable for hammer impact test. Further work is necessary to extend this proposed method so as to make it also applicable for shaker modal test case. This does not mean “to make the shaker to be movable” but rather attempts to obtain the FRFs with different excitation points by employing the FRFs with same excitation points. Thus, all the FRFs required for correction can then be obtained by using the FRFs with same excitation points which avoids troublesome movements of the shaker. However, it will probably involve additional FRFs measurements with extra dummy masses or accelerometers attached.



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*Source: 6857326-2017-11-13.xml*