Fig. 1(a) shows a pictorial view of a typical belt grinding procss of a workpiece (golf club head) held by two hands on a belt grinding machine observed at a workplace (Chen et al., 2017). The possible vibration sources and their hypothesized transmission pathways are illustrated in Fig. 1(b). The basic model of the entire machine-workpiece-hand-arm system used in the current study is illustrated in Fig. 1(c), which was developed in a previous study (Dong et al., 2018). Because additional components must be considered to analyze some of the engineering intervention methods, the basic model was extended to include these components. The specific extensions, together with their associated intervention methods, are described in Section 2.3. For the purpose of this study, six test treatments were considered in the basic modeling analyses, which are the combinations of bare-hand operations with two feed forces (15 N and 30 N) on three different grinding interfaces (R45, R55, and R65). The model parameters for each of the six treatments are listed in Table 1. Different from the reported model parameters that were calibrated using experimental data measured on assumed laboratory grinding interfaces (Xu et al., 2018; Dong et al., 2018), the parameters listed in Table 1 were calibrated in the current study using a set of reported human subject experimental data measured using three real interface sections cut from the drive wheels of belt grinding machines at a workplace (Dong et al., 2018). The parameter calibration method used in the current study was the same as that developed and used in the previously-reported study (Dong et al., 2018).

The equation of motion for the system model of each intervention method was generally expressed as follows: (1)[M]{Z..}+[C]{Z.}+[K]{Z}={F}

This equation was used to calculate the system displacement responses (z) to each of the two force excitations (F): F_W = F_W exp^jωt and F_C = F_C exp^jωt, in which j=−1, ω is angular frequency, and t is time.

When the machine vibration excitation (F_W) was considered, the calculated responses were used to determine the conventional workpiece vibration transfer function (T_W) using the workpiece response (z_c) and drive wheel response (z_W): (2)TW=zc/zw=AC−W/AW, where A_W ( = −ω²z_w) is the drive wheel vibration acceleration, and A_C-W (= −ω²z_c) is the workpiece vibration acceleration resulting from the machine vibration acting on the drive wheel.

When the interface vibration force (F_C) was used as the input, the resulting acceleration response (A_C-C) was used to define a new transfer function (T_C) as follows: (3)TC=AC−C/(FC/mc)

The force-mass ratio (F_C/m_c) in this equation is the acceleration of the workpiece that is free or not constrained in any system but subjected to the same vibration force (F_C) as that generating the system responses. Therefore, this transfer function represents the normalization of the vibration acceleration of the workpiece constrained in the system with respect to the free workpiece acceleration resulting from the same excitation force. It provides a dimensionless measure of the efficiency of the force-motion transformation in the frequency domain; it is reasonable to term this transfer function as force-motion transfer function.

As the phase angles of these transfer functions were not of concern in this study, only the magnitudes of the two transfer functions (T_W = |T_W|; T_C = |T_C|) were quantified and used in this study, which were referred to as transmissibility in the following presentations. For the same reason, only the magnitude of the system vibration was considered in this study.

We hypothesize that these two vibration transmissibility spectra can be used as a basis to evaluate the engineering intervention methods for controlling the workpiece vibration and its transmission to the human hands. The hypothesis was tested in this study, together with the test of the hypothesized vibration transmission pathway shown in Fig. 1(b), through the development and test of an approximation method for estimating workpiece vibration from grinding machine vibration. According to the flowchart shown in Fig. 1(b), the vibration on the drive wheel should be used to represent the machine vibration. Unfortunately, it is very difficult to measure the drive wheel vibration. Alternatively, the vibration on the grinding machine body near the grinding drive wheel was measured in the previous study (Chen et al., 2017). It was used in the current study to represent the machine vibration. Because the peak frequencies of the machine vibration were correlated with those measured on the workpieces in the frequency range of 20–125 Hz (Chen et al., 2017), it is reasonable to assume that the machine body vibration (A_MB) is correlated with the drive wheel vibration (A_W) in this frequency range, as the drive wheel is located between the machine body and workpiece. Then, their linear regression relationship can be expressed as follows: (4)AW≈χAMB, where χ is the regression slope/coefficient. According to Eq. (2), the workpiece vibration magnitude (A_C-W) resulting from the drive wheel vibration can be estimated as follows: (5)AC−W≈TWAW≈TW(χAMB)

As also illustrated in Fig. 1(b), besides directly producing the workpiece vibration, the vibration transmitted to the grinding interface can also serve as a geometric input to cause the vibration of the interface cutting force (Hahn and Lindsay, 1970; Oliveira et al., 2008; Aurich et al., 2009). While the cutting vibration force (F_C-W) resulting from the transmitted vibration may be complex, it is reasonable to hypothesize that the resulting cutting vibration force is correlated with the transmitted vibration, or their regression relationship can be expressed as follows: (6)FC−W≈ζAC−W≈ζTWχAMB where ζ is their regression slope/coefficient. According to Eq. (3), the resulting additional workpiece vibration (A_C-CW) can be written as follows: (7)AC−CW≈TC(FC−W/mc)≈(ζχ/mc)TCTWAMB

Then, the total workpiece vibration (A_C-M) resulting from the machine vibration is the sum of these two types of vibrations or it can be expressed as follows: (8)AC−M≈AC−W+AC−CW≈χTWAMB[1+(ζmc)TC]

This approximation method was tested using the experimental data measured in a previous study (Chen et al., 2017). A series of analyses were performed to examine the correlations among the measured machine body vibration, the estimated vibrations at different stages, and the measured workpiece vibration (A_C-Measured). The results are presented in section 3.3.

The results presented in Section 3.3 support the hypothesis that the two types of transmissibility (T_W and T_C) can be used to represent the frequency responses of the workpiece to the two excitations for assessing the effectiveness of any intervention method for controlling the vibration transmission. Therefore, they were calculated for each of the analyzed vibration control methods and used as a basis for assessing their effectiveness. If not specified, the basic model shown in Fig. 1(c) was used in the analyses.

The machine vibration transmissibility of the handheld workpiece was predicted from Eq. (2) using the system responses yielded from Eq. (1) with F_W (F_W = 1.0) as the sole excitation and the model parameters listed in Table 1. The predicted transmissibility spectra for the six test treatments are illustrated in Fig. 4, together with their corresponding experimental data measured in a previous study (Dong et al., 2018). The modeling results agree well with experimental data for each of the six test treatments in the major frequency range (20–1000 Hz) of concern. It should also be noted that the agreement was not imposed in the model calibration process, as the measured transmissibility spectra of the workpiece was not used in the model calibration (Dong et al., 2018). These observations suggest that the system model is valid for the prediction of the basic vibration response of the handheld workpiece.

The force-motion transmissibility of the handheld workpiece was calculated from Eq. (3) using the modeling responses yielded from Eq. (1) with F_C (F_C = 1.0) as the sole excitation and the model parameters listed in Table 1. The calculated transmissibility spectra for the six test treatments are illustrated in Fig. 5. Different from the response to the machine vibration shown in Fig. 4, the response to the cutting vibration force is small for each test treatment in the low-frequency range (<25 Hz). This is because low-frequency vibration can be transmitted to the entire system; the interface vibration force must drive the mass of the entire system with fixed boundaries that restrict the system motion. With the reduction of the vibration transmission distance from its source, the response gradually increases with the increase in frequency before reaching a peak value for each test treatment. The peak value occurs at a frequency slightly higher than the undamped fundamental resonant frequency of the workpiece listed in Table 1, which was estimated from the workpiece mass (m_c) and the interface contact stiffness (K₁). Naturally, the increase in the interface contact stiffness shifts the resonance to a higher frequency range. There is a minor second peak in the force-motion transmissibility shown in Fig. 5. This peak is related to the major resonance of the fingers on the workpiece (Xu et al., 2018; Dong et al., 2018). At higher frequencies, the responses for all the test treatments decrease with the increase in frequency, but they all converge to its limit or unity (1.0), regardless of the grinding interface conditions. This is because the cutting vibration at high frequencies cannot be effectively transmitted beyond the workpiece; the resulting workpiece acceleration should become increasingly close to its free mass acceleration (F_C/m_c) with the increase in vibration frequency.

Fig. 6 shows direct comparisons of the two types of vibration transmissibility spectra (T_W and T_C) and their summations. Interestingly, the two types of transmissibility spectra are almost symmetrical with respect to the undamped resonant frequency of the workpiece at the interface. Both types of responses exhibit their large responses in the resonant frequency range. As a result, the summed response has a large peak in the resonant frequency range.

Fig. 7(a) illustrates the one-third octave band acceleration spectra of a large belt grinding machine (A_MB) reported in a previous study (Chen et al., 2017). They were measured on the machine body close to the grinding drive wheel operating at three different speeds (1,200, 1,800, 2400 rpm). The major peak frequency for each speed is associated with the operation speed of the drive wheel, which suggests that these peaks resulted primarily from the unbalanced mass of the drive wheel. Fig. 7(e) illustrates the reported vibration acceleration spectra of handheld workpieces (golf club heads) (A_C-Measured) measured during their grinding on the same machine as that for the measurement of the machine body vibration (Chen et al., 2017). The peak frequencies at 20, 31.5, and 40 Hz for three speeds are the same as those observed in Fig. 7(a). There is also a common peak at about 125 Hz for each operating speed in both the machine body acceleration spectra (A_MB) and the measured workpiece acceleration spectra (A_C-Measured). However, their peak magnitudes are different. The overall trends of these two types of spectra are also different. As a result, the overall correlation between A_C-Measured and A_MB in the frequency range of 20–125 Hz is not very good at some speeds, as shown in Fig. 8(a) and as reflected by the correlation coefficients listed in Table 2 (1st row of R-values). At higher frequencies (160–1600 Hz), the correlation between these two types of vibration spectra is poor for 1200 rpm and 1800 rpm but good for 2400 rpm, as indicated by the R-values also listed in Table 2.

The workpiece vibration spectrum resulting from the machine vibration at each speed was estimated from the machine body vibration shown in Fig. 7(a) using the approximate method described in Section 2.2. Each of the six pairs of the predicted T_W and T_C shown in Figs. 4 and 5 were tested in the estimations. Their corresponding R-values for each prediction stage in the two frequency ranges are listed in Table 2. As examples, Fig. 8 illustrates the regression curves of the correlations for Interface R55 under 30 N feed force and Fig. 7(b) illustrates the acceleration spectra (A_C-W) estimated using Eq. (5) with T_W for this test treatment, which represents the first part of the workpiece vibration resulting from the transmitted machine vibration in Eq. (8). The change of the regression slope (χ) does not affect the results of the correlation analyses, but its value (χ = 3) was selected such that the estimated peak accelerations at 20, 30, and 40 Hz (related to the machine operation speeds) were comparable with those measured on the workpieces (Fig. 7(e)). As reflected by the R-values listed in Table 2 and the regression relationships shown in Fig. 8(a and b), the correlation between A_C-Measured and A_C-W for Interfaces R55 and R65 in the selected frequency range (20–125 Hz) is better than that between A_C-Measured and A_MB. The improved correlation confirms that the machine vibration transmissibility (T_W) plays a certain role in determining the workpiece vibration. However, as also shown in Table 2, the correlation between A_C-Measured and A_C-W for Interface R45 generally became worse than that between A_C-Measured and A_MB. This may be because R45 is a brand-new interface and its dynamic properties may not be sufficiently representative of those used in the measurement of the workpiece vibration in the reported study (Chen et al., 2017).

Fig. 7(c) illustrates the second part of the workpiece acceleration (A_C-CW) resulting from the transmitted machine vibration, which was estimated using Eq. (7) with T_W and T_C for Interface R55 under 30 N feed force. The transformation with T_C greatly improved the correlation (A_C-Measured vs. A_C-CW) in the first frequency range (20–125 Hz) for every case, as shown in Table 2 and Fig. 8(c). In fact, a direct transformation with T_C alone (T_C*A_MB) also resulted in a very strong correlation for every case, as also shown in Table 2. This is because the T_C-transformation made the overall slope of the estimated spectra consistent with that of the measured workpiece vibration spectra in the selected frequency range. These observations confirm that the machine vibration can play a very important role in determining the cutting vibration force at the grinding interface (Oryński and Pawłowski, 1999; Hahn and Lindsay, 1970; Oliveira et al., 2008; Aurich et al., 2009), and that it is essential to consider the force-motion transmissibility in the analysis of the workpiece vibration. However, as shown in Fig. 8(c), the T_C-transformation also skewed the correlation relationship: the data points in the low vibration range are all located at one side of the regression line for each case; the predicted vibrations at the frequencies (20, 31.5 and 40 Hz) related to the operation speeds shown in Fig. 7(c) are not comparable with the measured data shown in Fig. 7(e). This suggests that A_C-CW alone is insufficient to fully represent the workpiece vibration resulting from the transmitted machine vibration, although the high R-values suggest that A_C-CW is more important than A_C-W in the cases examined in this study. Then, it is necessary to use Eq. (8) to estimate the total workpiece vibration resulting from the machine vibration.

The summed workpiece vibration (A_C-M) calculated using Eq. (8) for each operation speed is illustrated in Fig. 7(d). The constant (ζ/m_c) in Eq. (8) actually represents the relative weighting of the interface cutting vibration (A_C-CW) resulting from the machine vibration in its combination with the directly-transmitted machine vibration (A_C-W). Because A_C-CW is more important than (A_C-W), A_C-CW should have larger weighting in the combination to form the estimated final workpiece vibration. Therefore, The constant (ζ/m_c = 2.5) was determined such that the A_C-M spectra shown in Fig. 7(d) for all the three operation speeds were comparable with those measured on the workpieces (Fig. 7(e)), while the correlations between A_C-M and A_C-Measured spectra in the selected frequency range remained at a high level (Table 2). As shown in Fig. 8(d), the regression relationship for the correlation between A_C-W and A_C-Measured at each operation speed is much less skewed than that shown in Fig. 8(c).

As found in exploring analyses, the use of different sets of interface stiffness (K₁) and damping (C₁) values listed in Table 1 in the described analyses do not change the basic trends of the effects of intervention methods on workpiece response functions. As examples, the modeling results of the intervention methods obtained with the K₁ and C₁ values listed in Table 1 for Interface R55 under a 30 N feed force as the basis in the analyzes were presented in this section. The T_w and T_C spectra calculated with the K₁ and C₁ values from the original model shown in Fig. 1(c) were used as the baseline references for the comparisons to clearly identify the potential effects of each analyzed intervention method.

As shown in Fig. 9, proportionally reducing the interface stiffness and damping values can generally reduce the fundamental resonant frequency of the workpiece on the grinding interface and its vibration in the entire frequency range of concern.

As shown in Fig. 10(a), adding a solid adapter or mass to the workpiece without changing the grinding interface conditions or feed force can reduce the fundamental resonant frequency of the workpiece. The added mass can also proportionally reduce the force-motion transmissibility in the frequency range above the resulting resonant frequency of the workpiece, as shown in Fig. 10(b). However, results shown in Fig. 10(a) also indicate that the added mass can substantially increase the major resonant peak.

As shown in Fig. 11, in addition to the solid adapter, installing a vibration absorber on the adapter can further reduce the workpiece vibration in the original resonant frequency range. Its effectiveness, however, depends on the combination of the mass, stiffness, and damping values of the vibration absorber.

As predicted with the model shown in Fig. 2(b), if some cushioning materials are inserted between an adapter and the workpiece, and the adapter is held by the hands of a grinding operator, the vibration on the adapter at frequencies above the fundamental resonant frequency of the workpiece can be substantially reduced, especially in the very high*frequency range, as shown in Fig. 12. This is because the cushioning function generally increases with the increase in vibration frequency. The effectiveness of this intervention method also increases with the reduction of the cushion stiffness. This cushioning method, however, may increase the adapter vibration at frequencies below the original resonant frequency of the workpiece, as also shown in Fig. 12.

If a rest support is connected to the machine body, as simulated in the model shown in Fig. 3(a), the workpiece vibration in the original resonant frequency range can be substantially reduced, as shown in Fig. 13. However, the workpiece vibration at some lower frequencies may be largely increased, depending on the dynamic properties of the machine body and rest support, as also shown in Fig. 13.

Fig. 14 illustrates direct comparisons of the vibration transmissibility spectra of the machine body, rest support, and workpiece for four different sets of machine connection properties. Obviously, the new resonant peak frequency of the workpiece on the rest support is approximately the same as that of the machine body and rest support. Both the machine body and rest support resonances can significantly affect the workpiece response peak frequency and magnitude.

Fig. 15 illustrates the effects of an independent rest support on the workpiece responses. The physical separation of the rest support from the machine body had little influence on the force-motion transmissibility if the dynamic properties of the rest support remain unchanged, as shown in the right columns of Figs. 13 and 15. However, the separation greatly reduced the machine vibration transmissibility of the workpiece, as shown in the left columns of Figs. 13 and 15. Increasing the stiffness, damping, and effective mass values of the independent rest support further reduced the machine vibration transmissibility, as shown in Fig. 15(c1).

The characteristics of the two response functions shown in Figs. 4–6 theoretically prove that the workpiece vibrations in the low- and low/middle-frequency ranges on many belt grinding machines are likely to be associated primarily with the grinding machine vibration. This is further confirmed by the combined modeling and experimental results illustrated in Fig. 7. The current standard method for assessing the risk of hand-transmitted vibration exposures emphasizes this frequency range (ISO 5349–1, 2001). Therefore, the control of the machine vibration should be at the top of the list of strategies for minimizing grinding operator hand-transmitted vibration exposures. This can be achieved by selecting or developing low-vibration grinding machines, regularly monitoring the machine vibration, and maintaining the machines in good condition. Some standards and technologies have been established or developed to help achieve these objectives (ISO ISO 20816–1, 2016; ISO 21940–1, 2019). Alternatively, reducing the stiffness of grinding interface and/or using a solid seat support can also effectively reduce the workpiece vibration in the low and middle frequency ranges, as shown in Figs. 9 and 15.

As shown in Fig. 6, both types of vibration sources can result in large workpiece responses in the resonant frequency range. The resonant responses can be attenuated by using a workpiece adapter equipped with a vibration absorber, as shown in Fig. 11. The useful frequency range of the adapter can be effectively extended by using the nonlinear technique reported in a recent study (Lindell et al., 2015). The use of a seat support can also effectively reduce the resonant responses, as shown in Fig. 15.

At higher frequencies, the force-motion transmissibility (T_C) plays the dominant role in determining the overall response, as also shown in Fig. 6. Several studies found that high-frequency vibration exposures to fingers could also be harmful (Starck et al., 1990; Dandanell and Engstrom, 1986). It is always desired to reduce the exposure as much as feasible. Because some high-frequency cutting forces may be useful to maintain the grinding quality and efficiency (Tönshoff and Degenhardt, 1982; Oryński and Pawłowski, 1999; Hahn and Lindsay, 1970; Oliveira et al., 2008; Aurich et al., 2009), it is neither desired nor feasible to eliminate them in the workpiece grinding process. Fortunately, while it is difficult to find or develop a practically useable vibration-reducing (VR) glove to effectively reduce the low-frequency vibrations transmitted to the fingers, VR gloves can effectively attenuate high-frequency vibrations and reduce sharp peaks, especially for vibration frequencies above 1000 Hz (Welcome et al., 2012; Xu et al., 2019). As shown in Figs. 10–12, some intervention methods can also be used to effectively reduce the vibrations that could result from high-frequency cutting forces without affecting the grinding efficiency.

As anticipated, there are some differences between the estimated workpiece responses and the measured vibration data, especially at frequencies higher than 125 Hz, as shown in Fig. 7. They result from the following reasons/sources:

The actual transfer function between the drive wheel and machine body was not available because it is extremely difficult to reliably measure the vibrations on the rotating drive wheel. As the first degree of approximation, a linear transformation was actually used to represent the transfer function, as expressed in Eq. (4). This is reasonable in the low- and middle-frequency ranges, because the connection between the drive wheel and the machine body must be stiff. This, however, can introduce a large error in the estimated high-frequency vibrations.

For these reasons, the method expressed in Eq. (8) can only be used to estimate the workpiece vibrations below the fundamental resonant frequency of the workpiece and resulted from the vibrations measured at a point on the machine body close to the drive wheel. Furthermore, the regression coefficients (χ = 3, ζ = 2.5m_c) could also vary with different machines and workpieces. However, these limitations or deficiencies do not affect the validity of the methodology proposed and used in this study to evaluate the effectiveness of the engineering control methods based on the two transfer functions (T_W, T_C); the crude estimation method was only designed and used to test the hypothesis on the two vibration transmission pathways from the drive wheel to the workpiece and to demonstrate that the two transfer functions (T_W, T_C) play an essential role in determining the workpiece vibrations.

The modeling results illustrated in Fig. 9 suggest that the workpiece vibration in the entire frequency range of concern generally decreases with the reduction of the grinding interface hardness if the other grinding conditions remain unchanged. These results suggest that reducing the feed force, and especially the material stiffness, can reduce the vibration exposure. As shown in Fig. 4, the use of interface R45 (a brand-new wheel rubber tread) is obviously better than the other two interfaces (R55 and R65 were used and partially worn). This suggests that the wheel rubber tread should be replaced with a new one when it is worn to a certain extent.

The results shown in Figs. 5 and 9 also suggest that this intervention method is especially effective in the original resonant frequency range and at higher frequencies, but it may not be effective at the heavily-weighted lower frequencies. However, the interface hardness must be substantially reduced (e.g., >50% of the reference value (K₁ = 96.828 kN/m)). It may be difficult to further reduce the interface stiffness by simply replacing the rubber tread on the belt grinding machine with a much softer rubber tread; such a replacement may increase not only the machine noise but also the unbalanced mass of the drive wheel which may result in higher machine vibrations (Tönshoff and Degenhardt, 1982). A grinding machine designed to operate a flexible grinding disc may be selected to realize a substantial reduction in the grinding interface hardness (https://www.google.com/se, 1120). Alternatively, a free belt grinding machine can also be considered (https://www.google.com/se, 1120). Unfortunately, the reduction of the interface stiffness is likely to reduce the grinding efficiency or productivity (Tönshoff and Degenhardt, 1982). In addition to other factors (e.g., the geometric requirement of the ground surfaces and the desired surface quality), the grinding machine with a soft interface is usually selected to conduct the final fine grinding or polishing of workpiece surfaces. Then, the workpiece vibration in a polishing process should be generally less than that in the other grinding processes. This theoretical prediction is consistent with field observation (Chen et al., 2017).

The modeling responses illustrated in Fig. 10 suggest that increasing the mass of a workpiece may reduce the vibration exposure in the high-frequency range, but such a change may reduce or increase the vibration resulting from the machine vibration, primarily depending on the dominant frequency range of the machine vibration. For example, because the dominant weighted machine vibrations shown in Fig. 7(a) are in the range of 20–40 Hz, increasing the workpiece mass should increase the frequency-weighted acceleration of the workpiece. This is consistent with that observed in the previous study (Chen et al., 2017): the ground steel alloy golf club heads weighed more than the ground titanium club heads; the weighted acceleration of the alloy club heads for every machine operating speed on each model of grinding machines (large and small) was larger than that of the titanium club heads. However, it should be noted that the change of workpiece mass is usually accompanied with the changes of workpiece material, grinding surface geometries, hand coupling conditions, etc.; these changes may also influence the workpiece vibration. As a result, it may not be reliable to predict the workpiece vibration solely based on the weight or mass of a workpiece.

As above-mentioned, a set of locking pliers or a handle may be firmly attached to a small workpiece to help safely conduct the workpiece grinding (https://www.familyhandyma, 2020; https://www.youtube.com/w, 2020). Some special adapters may also be designed to achieve the same purpose. Their effects on the workpiece vibration responses are the same as if the workpiece mass is substantially increased. As above-discussed, these adapter methods are unlikely to be effective in the low/middle-frequency range, but these techniques should be very effective for reducing high-frequency vibration exposures. Furthermore, the use of these adapter methods has the following potential ergonomic benefits: (i) the adapter can be designed to change the finger pinch grip on a workpiece to a power grip on the adapter, which can increase the skin contact surface area, reduce hand and arm muscle forces, and minimize their fatigue; and (ii) if the adapter/workpiece is suspended or supported, there may be a further reduction in the required grip force to perform the task. The suspension system or adapter support may also reduce vibration transmission; these potential benefits are further discussed in the following paragraphs and in section 4.5.

As illustrated in Fig. 11, the workpiece vibration can be further reduced in the original resonant frequency range if a vibration absorber is added to the adapter. Compared with the simple adapter method also shown in this figure (η = +∞ or m_ad = 200 g), the added vibration absorber can also significantly expand the effective frequency range of the adapter. This intervention method may be useful to reduce the workpiece vibration in the major resonant frequency range, for example, the vibration peaks in the range of 80–160 Hz shown in Fig. 7(e). A unique feature of the vibration absorber method is that it can be designed to effectively target the vibration peak at a specific frequency. For example, as shown in Fig. 11(a), the vibration at 80 Hz can be greatly reduced (>50%) using an adapter/absorber with η = 0.25 or with an absorber alone (m_ad = 0) but with m_ab = 200 g and η = 0.5. If the workpiece has a hollow space, simply filling the space with sand or clay may substantially reduce the vibrations in the resonant frequency range and at higher frequencies, as the added sand or clay is likely to function similarly to a vibration absorber.

The suspended adapter method is better than the simple mass adapter method, especially in the high-frequency range, as shown in Fig. 12. However, similar to the other adapter methods, it may amplify the vibration at frequencies below the original resonant frequency. To extend the effective frequency range to below 20 Hz, the suspension stiffness must be greatly reduced. This may make it impossible for the operator to safely and effectively control the workpiece during grinding. Alternatively, the adapter mass can be greatly increased to extend the effective frequency range. This, however, may substantially increase the burden of the hand-arm systems and can reduce productivity. A suspension system or mechanical arm may be used to support the adapter and/or workpiece, similar to that used to support a handheld grinder (McDowell et al., 2016). Then, the intervention becomes similar to the rest support method discussed in the next section.

As illustrated in Figs. 13 and 15, a rest support can greatly reduce the workpiece vibration in the original resonant frequency range, and this method is generally more effective than any of the other investigated intervention methods. However, these support systems may not reduce much workpiece vibration at very high frequencies, as shown in the right columns of these figures. Fortunately, as above-discussed, the very high-frequency vibration can be effectively attenuated using other methods.

The results shown in Fig. 14 indicate that the machine body vibration can significantly affect the vibration on the rest support connected to the machine body, and that the machine body resonance, rest support resonance, or both could result in high workpiece vibration. The physical separation of the rest support from the machine body can effectively reduce the workpiece vibration in the low- and middle-frequency range, especially when the rest support has a rigid structure and is firmly mounted on the ground, as shown in Fig. 15(c1). These modeling predictions are consistent with those observed at workplaces (HSE, 2005). These modeling predictions further suggest that if feasible and practical, the rest support should not be connected to the machine body, but it should have a rigid structure and be firmly mounted to the floor, especially for machines extensively used to perform grinding daily.

Theoretically, a metal to metal contact has the largest contact stiffness and promotes the highest rest support effectiveness. Practically, however, this may not be the best choice if the workpiece has a contact resonance on the metal surface of the rest surface with little damping. Such a resonance was observed in our experimental study during the measurement of the impedance of the workpiece-hand-arm system on a stiff interface (Xu et al., 2018). The modeling results shown in Fig. 15 suggest that it is not necessary to maintain the metal-to-metal contact to maximize the effectiveness of the rest support. We hypothesize that attaching a thin layer of non-metal material such as rubber on the rest support surface may avoid any sharp resonance on the rest support while maintaining the benefits of the rest support. This hypothesis should be tested in further studies.