Assessment of transducer mass effects on measured FRFs in shaker modal testing

2.2.1. Assessment of mass effects on point FRF

This situation presents no complication when the point FRF is assessed because the response co-ordinate coincides with the shaking point, as shown in Fig. 3. The two-transducers masses can be considered as an integrated mass. By replacing $m_f$ with ($m_f+m_a$) in Eq. (3), we can easily obtain Eq. (6) from which the natural frequencies ${\omega }_r$ of the exact FRF $A_{pp}$ can be predicted. If natural frequencies of the measured FRFs $A_{pp}^{(p_1,p_2)}$ are denoted as ${\omega }_r^{(p_1,p_2)}$, the overall masses effects of the two transducers can be assessed by the comparison of ${\omega }_r$ and ${\omega }_r^{(p_1,p_2)}$. Note that $p_1$ and $p_2$denotes force transducer and accelerometer mass effects, respectively:

To assess the force transducer mass effects separately, we should obtain the natural frequencies (denoted as ${\omega }_r^{\left(p_1\right)}$) of point FRF $A_{pp}^{\left(p_1\right)}$ which is the “special FRF” mentioned above. This situation becomes complicated since $A_{pp}^{\left(p_1\right)}$ is not available from a straightforward measurement. To obtain ${\omega }_r^{\left(p_1\right)}$, a similar method as that of predicting the exact natural frequency ${\omega }_r$ (see Eq. (3)) is utilized again. First, $A_{pp}^{\left(p_1\right)}$ is derived by employing the mass correction method [7, 8] (removing the accelerometer mass effects from $A_{pp}^{(p_1,p_2)}$), as shown in Eq. (7):

It can be found that $A_{pp}^{\left(p_1\right)}$ has natural frequencies at ${\omega }_r^{\left(p_1\right)}$ that make the denominator of Eq. (7) approach zero. By equating the denominator of Eq. (7) to zero and then solving the resulting equation, one yields:

Then, ${\omega }_r^{\left(p_1\right)}$ can be easily obtained according to Eq. (8) by the graphical method. Thus, the mass effects of force transducer can be assessed by the comparison of ${\omega }_r^{\left(p_1\right)}$ and ${\omega }_r$.

Similarly, to assess the accelerometer mass effects individually, $A_{pp}^{\left(p_2\right)}$ can be obtained by removing the force transducer mass from $A_{pp}^{(p_1,p_2)}$:

where $A_{pp}^{\left(p_2\right)}$ is point FRF in which only accelerometer mass is involved. Likewise, $A_{pp}^{\left(p_2\right)}$ has natural frequencies at ${\omega }_r^{\left(p_2\right)}$ that make the denominator of Eq. (9) equal to zero and then solving the resulting equation, one yields:

Then, accelerometer mass effects can be assessed by the comparison of ${\omega }_r^{\left(p_2\right)}$ and ${\omega }_r$.

Fig. 3Measuring point FRF App(p1,p2)

2.2.2. Assessment of mass effects on transfer FRFs

The discussion in the previous section indicates that the mass effect on point FRFs is the same as that on transfer FRFs in case 1. In case 2, however, the mass effects are changing due to the movement of the accelerometer for different transfer FRFs measurement. Assuming the natural frequencies of $A_{lp}^{(p_1,l)}$ to be ${\omega }_r^{(p_1,l)}$, the total mass effects of the two transducers on the measured transfer FRF can be assessed by the comparison of ${\omega }_r^{(p_1,l)}$ and ${\omega }_r$ which can be readily obtained from Eq. (6).

The situation becomes more complicated when the two-transducers mass effects on the measured transfer FRFs are separately assessed. To assess accelerometer mass effects, we should first get the natural frequencies ${\omega }_r^{\left(l\right)}$ of transfer FRF $A_{lp}^{\left(l\right)}$ in which only accelerometer mass is involved. Since $A_{lp}^{\left(l\right)}$ is also not available from the straightforward measurement, the mass correction method is utilized again. $A_{lp}^{\left(l\right)}$ can be obtained by removing the force transducer mass effects from $A_{lp}^{(p_1,l)}$, as shown in Eq. (11):

Unfortunately, $A_{lp}^{\left(l\right)}$ is still unachievable as $A_{pp}^{(p_1,l)}$ in Eq. (11) cannot be measured directly. Correction method is utilized again, and $A_{pp}^{(p_1,l)}$ can be obtained by removing the accelerometer mass effects from $A_{pp}^{(p_1,p_2,l)}$, as it can be seen in Eq. (12):

It should be noted that $A_{pp}^{(p_1,p_2,l)}$ is measured with a dummy mass $m_d$ (equals the mass of the accelerometer $m_a$) attached to point $l$, as shown in Fig. 5.

Fig. 4Measurement of transfer FRF Alp(p1,l)

Fig. 5Measurement of point FRF App(p1,p2,l)

Substituting Eq. (12) into Eq. (11), one yields:

Thus, $A_{lp}^{\left(l\right)}$ has natural frequencies at ${\omega }_r^{\left(l\right)}$ that make the denominator of Eq. (13) approach zero and then solving the resulting equation, one yields:

Thus, ${\omega }_r^{\left(l\right)}$ can be obtained from Eq. (14). Then, the accelerometer mass effects on the measured transfer FRF can be assessed by the comparison of ${\omega }_r^{\left(l\right)}$ and ${\omega }_r$ which can be readily obtained from Eq. (6). So far, the quality of accelerometer mass effects on the measured transfer FRF has been assessed.

To assess force transducer mass effects separately, we should first get the natural frequencies ${\omega }_r^{\left(p_1\right)}$ of transfer FRF $A_{lp}^{\left(p_1\right)}$ in which only force transducer mass gets involved. The situation of measuring $A_{lp}^{\left(p_1\right)}$ is contradictory since the accelerometer mass should not be attached (at point l), while, the measurement cannot be performed without the accelerometer. Fortunately, the transfer FRF $A_{lp}^{\left(p_1\right)}$ have the same natural frequencies ${\omega }_r^{\left(p_1\right)}$ as those of the point FRF $A_{pp}^{\left(p_1\right)}$ which are already available from Eq. (8). Thus, the natural frequencies ${\omega }_r^{\left(p_1\right)}$ of $A_{lp}^{\left(p_1\right)}$ can also be predicted using Eq. (8).

Besides, some researchers have investigated the mass correction methods which can also be employed here to obtain $A_{lp}^{\left(p_1\right)}$ by removing the accelerometer mass from the measured transfer FRF $A_{lp}^{(p_1,l)}$. Two different methods have been discussed by Carkar [8] and Ashory [11], respectively. According to the measurement scheme in Fig. 6 given in [8], $A_{lp}^{\left(p_1\right)}$ can be acquired by removing the accelerometer mass effects from $A_{lp}^{(p_1,l)}$ using Eq. (15):

The detailed measurement process is as follows:

• Measuring point FRF $A_{pp}^{(p_1,p_2)}$ according to Fig. 3,

• Measuring transfer FRF $A_{lp}^{(p_1,p_2,l)}$ according to Fig. 6(a),

• Measuring point FRF $A_{pp}^{(p_1,p_2,l)}$ according to Fig. 6(b).

It should be noted that the chosen dummy mass $m_d$ equals the accelerometer mass $m_a$in Fig. 6.

The accelerometer mass can then be removed from the measured transfer FRF $A_{lp}^{(p_1,l)}$ according to Eq. (15).

Fig. 6Measurements using accelerometers and dummy mass

a) Measurement of $A_{lp}^{(p_1,p_2,l)}$

b) Measurement of $A_{pp}^{(p_1,p_2,l)}$

In Eq. (15), $A_{lp}^{\left(p_1\right)}$ has natural frequencies at ${\omega }_r^{\left(p_1\right)}$ that make the denominator of Eq. (15) approach zero and then solving the resulting equation, one yields:

Thus, the natural frequencies ${\omega }_r^{\left(p_1\right)}$ of $A_{lp}^{\left(p_1\right)}$ can be predicted from Eq. (16). It is not surprising that the Eq. (16) coincides with Eq. (8) which is previously mentioned for predicting the natural frequencies of $A_{lp}^{\left(p_1\right)}$.

According to literature [11], the mass effects of accelerometer can be eliminated from the $A_{lp}^{(p_1,l)}$ using Eq. (17), see Fig. 7:

where $A_{lp}^{(p_1,l)}$ and ${\stackrel{-}{A}}_{lp}^{(p_1,l)}$ are measured according to Fig. 7(a) and (b), respectively. $A_{lp}^{\left(p_1\right)}$ is corrected transfer FRF after removing the mass effects of accelerometer.

Fig. 7Repeated measurement of transfer FRF using two accelerometers with different masses

a)

b)

Thus, $A_{lp}^{\left(p_1\right)}$ has natural frequencies at ${\omega }_r^{\left(p_1\right)}$ that make the denominator of Eq. (17) approach zero and then solving the resulting equation, one yields:

Thus, ${\omega }_r^{\left(p_1\right)}$ can also be obtained from Eq. (18). And it can be seen that both Eq. (16) (or Eq. (8)) and (18) can be employed to obtain the natural frequencies ${\omega }_r^{\left(p_1\right)}$ of $A_{lp}^{\left(p_1\right)}$.

So far, assessments of transducer mass effects on both point and transfer FRFs have been investigated. Besides, the overall as well as individual mass effects of two transducers have been assessed, respectively. The conclusions are given in Table 1.

3.2.1. Assessing the transducer mass effects on point accelerance ${\mathit{A}}_{22}$

According to the proposed method, only the measurement of point accelerance $A_{22}^{(m_a,m_f)}$ is required to predict the natural frequencies of $A_{22}^{\left(m_a\right)}$, $A_{22}^{\left(m_f\right)}$, and $A_{22}$, hence assessing the force accelerometer, transducer and their overall mass effects on the measured point accelerance $A_{22}^{(m_a,m_f)}$, respectively. The overall mass effects of the two transducers can be assessed by comparison of the natural frequencies of the “measured” accelerance $A_{22}^{(m_a,m_f)}$ and those of the exact accelerance $A_{22}$. According to Eq. (6), the natural frequencies of $A_{22}$ can be acquired by searching the intersection points of $A_{22}^{(m_a,m_f)}$ and 1$/(m_f+m_a)$. As is shown in Fig. 14, the curves of $A_{22}^{(m_a,m_f)}$ and 1$/(m_f+m_a)$ intersect at points $C_1$, $C_2$ and $D_1$, $D_2$. However, $D_1$ and $D_2$ are excluded from the solutions by the common sense that the natural frequencies of $A_{22}$ (without transducers attached) should be higher than those of $A_{22}^{(m_a,m_f)}$ (with transducers attached). Thus, the remaining points $C_1$ (25.7 Hz) and C2 (52.1 Hz) correspond with the first and second natural frequency of $A_{22}$, respectively. For comparison purpose, the exact accelerance $A_{22}$ is also numerically generated in Fig. 14. And it can be found that the resonances frequencies $A_1$ (25.7 Hz) and $A_2$ (52.1 Hz) of $A_{22}$ are in a quite good agreement with the predicted ones $C_1$ (25.7 Hz) and $C_2$(52.1 Hz), which demonstrate the effectiveness of the proposed method.

Similar observations can also be made for the assessment of the individual mass effects of accelerometer and force transducer illustrated in Fig. 15 and Fig. 16. In Fig. 15, the natural frequencies of $A_{22}^{\left(m_a\right)}$can be found at the intersection points ($C_1$ (21.7 Hz) and $C_2$ (46 Hz)) of $A_{22}^{(m_a,m_f)}$ and 1$/m_f$. And the natural frequencies of $A_{22}^{\left(m_f\right)}$can be found at the intersection points ($C_1$ (22.6 Hz) and $C_2$ (46.9 Hz)) of $A_{22}^{(m_a,m_f)}$ and 1$/m_a$, as it can be seen in Fig. 16.

Table 3 shows the shifts of natural frequencies of point accelerance $A_{22}$ due to the transducer mass effects. In this table, the second, third and forth columns correspond with natural frequencies shifts caused by the accelerometer, force transducer and their overall mass loading effects, respectively. The natural frequencies shift in the fourth column indicate the quantity of the transducers masses effects in shaker $+$ accelerometer modal testing (the mass of both two transducers is involved). And the second column can be considered as the accelerometer mass effects in hammer + accelerometer testing (only accelerometer mass is involved). The third column can be considered as force transducer mass effects in shaker + Laser Doppler vibrometer modal testing (only force transducer mass is involved). It can be predicted from the second and third column that if this system is respectively tested in hammer + accelerometer and shaker + Laser Doppler vibrometer cases, the mass effects of accelerometer in the former are larger than that of force transducer in the later. This is due to that the mass of the acceleration sensor is larger than that of the force sensor, and two sensors are installed in the same location.

Fig. 14Prediction of natural frequencies of A22 for assessing overall mass effects of two transducers

Fig. 15Prediction of natural frequencies of A22(ma) for assessing accelerometer mass effects

Fig. 16Prediction of natural frequencies of A22(mf) for assessing force transducer mass effects

3.2.2. Assessment of transducer mass effects on transfer accelerance ${\mathit{A}}_{12}$

The overall mass effects of the two transducers on the transfer accelerance can be assessed by a comparison of the natural frequencies of the “measured” accelerance $A_{12}^{(m_a,m_f)}$ and exact accelerance $A_{12}$. This presents no complication since the natural frequencies of $A_{12}$ are the same as those of $A_{22}$ which are readily acquired in the previous discussion. Thus, the natural frequencies of $A_{12}$ can also be found at the intersection points ($C_1$ (25.7 Hz) and $C_2$ (52.1 Hz)) of $A_{22}^{(m_a,m_f)}$ and 1$/(m_f+m_a)$, as shown in Fig. 17.

To assess the individual mass effects of accelerometer, an extra measurement of point FRF $A_{22}^{(m_a,m_f,m_d)}$ is required to predict the natural frequencies of $A_{12}^{\left(m_a\right)}$ according to Eq. (14). As shown in Fig. 18, $A_{22}^{(m_a,m_f,m_d)}$ and 1$/(m_f+m_a)$ intersect at points $C_1$ (24 Hz) and $C_2$ (49.7 Hz) which correspond with the first and second natural frequency of $A_{12}^{\left(m_a\right)}$, respectively. The force transducer mass effect on the transfer accelerance can be assessed by a comparison of the natural frequencies of the “measured” accelerance $A_{12}^{\left(m_f\right)}$ and those of the exact accelerance $A_{12}$. The natural frequencies of A₁₂ are identical with those of A₁₁ which are readily obtained before. Since $A_{12}^{\left(m_f\right)}$ has the same natural frequencies as $A_{22}^{\left(m_f\right)}$, the natural frequencies of $A_{12}^{\left(m_f\right)}$can also be found at the intersection points ($C_1$ (22.6 Hz) and $C_2$ (46.9 Hz)) of $A_{22}^{(m_a,m_f)}$ and 1$/m_a$, as it can be seen from Fig. 19. It is not surprising that the same results (intersection points $C_1$ (22.6 Hz) and $C_2$ (46.9 Hz) in Fig. 20) can also be obtained by employing another method (using two accelerometers with different masses, see Eq. (18)). For the comparison purpose, $A_{12}$, $A_{12}^{\left(m_a\right)}$ and $A_{12}^{\left(m_f\right)}$ are numerically calculated and also shown in Figs. 17-20, respectively. It can be seen that the predicted natural frequencies are in a quite good agreement with the calculated ones.

Fig. 17Prediction of natural frequencies of A12 for assessing overall mass effects of two transducers

Fig. 18Prediction of natural frequencies of A12(ma) for assessing accelerometer mass effects

Fig. 19Prediction of natural frequencies of A12(mf) for assessing force transducer mass effects

Fig. 20Prediction of natural frequencies of A12(mf) (using two accelerometers with different masses) for assessing force transducer mass effects

Table 4 shows the shifts of natural frequencies of transfer accelerance $A_{12}$ due to the transducer mass effects. It is not surprising to find that individual mass effects of force transducer on the transfer accelerance $A_{12}$ are identical to that on the point accelerance $A_{22}$, as discussed before (see Table 1). However, compared with Table 3, a significant difference in Table 4 is that the individual mass effects of accelerometer are less than that of force transducer although the mass of accelerometer is larger than that of force transducer. This is primarily due to the fact that installation positions of the two sensors are different. It is further illustrated that the quality of transducer mass effects is not only related to magnitude of transducer mass, but also to installation location of transducer.

3.2.3. Discussion of methods with noisy data

It can be seen from the above discussion that the natural frequencies of $A_{22}^{\left(m_a\right)}$, $A_{22}^{\left(m_f\right)}$, $A_{22}$ and $A_{12}^{\left(m_a\right)}$, $A_{12}^{\left(m_f\right)}$, $A_{12}$ are needed for assessing the transducer mass effects on point and transfer accelerance, respectively. And a total of only two point FRFs $\left(A_{22}^{(m_a,m_f)}\right.$ and $\left.A_{22}^{(m_a,m_f,m_d)}\right)$ measurements are required to predict these natural frequencies. It means that the accuracy of the proposed assessment method depends largely on the measurement accuracy of $A_{22}^{(m_a,m_f)}$ and $A_{22}^{(m_a,m_f,m_d)}$. The following simulations are presented to verify the effectiveness of the methods with noise-contaminated FRFs. The same system given in Fig. 12 is utilized here. The difference is that $A_{22}^{(m_a,m_f)}$ and $A_{22}^{(m_a,m_f,m_d)}$ are numerically incorporated with the white Gaussian noise (zero mean and variance ${\delta }_n^2$). The noise level $\gamma$ is defined as:

where ${\left|H_{ij}\left(\omega \right)\right|}_{\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{n}}$ is the mean absolute value of $H_{ij}\left(\omega \right)$ in the frequency range of interest. The noise levels in this example are set to be 0.5 %, 1 %, respectively. Note that the FRF is a complex value; the random noise should be added to the real and imaginary parts of the FRF, respectively. The prediction results corresponding to 0.5 % noise are presented in Fig. 21.

Fig. 21Prediction of natural frequencies of A22 (0.5 % noise)

Fig. 22Prediction of natural frequencies of A22 (1 % noise)

It is seen that the results ($C_1$ and $C_2$) are less affected by the noise. This is because $A_{22}^{(m_a,m_f)}$ and 1$/(m_f+m_a)$ intersects at the points ($C_1$ and $C_2$) corresponding to relatively large amplitudes which are contaminated less than the small amplitudes. With the noise level increasing to 1 %, the affected $A_{22}^{(m_a,m_f)}$ exhibits a lot of glitches. And several intersections points of $A_{22}^{(m_a,m_f)}$ and 1$/(m_f+m_a)$ being found in the frequency ranges $\left[f_1,f_2\right]$ and $\left[f_3,f_4\right]$ make it difficult to determine the accurate results (see Fig. 22). In this case, it is suggested that the measured FRFs be preprocessed using the curve-fitting procedure before the proposed assessment method is applied. The method may work well with the curve fitted data. Similar observations can also be made for predicting the natural frequencies of $A_{22}^{\left(m_a\right)}$, $A_{22}^{\left(m_f\right)}$ and $A_{12}^{\left(m_a\right)}$, $A_{12}^{\left(m_f\right)}$. However, they are not presented here for sake of brevity.