Interfacial damping in assembled structures is difficult to predict and control since it depends on numerous system parameters such as elastic mismatch, roughness, contact geometry, and loading profiles. Most recently, phase difference between normal and tangential force oscillations has been shown to have a significant effect on interfacial damping. In this study, we conduct microscale (asperity-scale) experiments to investigate the influence of magnitude and phase difference of normal and tangential force oscillations on the energy dissipation in presliding spherical contacts. Our results show that energy dissipation increases with increasing normal preload fluctuations and phase difference. This increase is more prominent for higher tangential force fluctuations, thanks to larger frictional slip along the contact interface. We also show that the energy dissipation and tangential fluctuations are related through a power law. The power exponents we identify from the experiments reveal that contacts deliver a nonlinear damping for all normal preload fluctuation amplitudes and phase differences investigated. This is in line with the damping uncertainties and nonlinearities observed in structural dynamics community.

## Introduction

Interfacial damping arises from frictional shear and slip at the mating surfaces. Frictional energy dissipation is difficult to predict because frictional interactions exhibit high degree of variability, nonlinearity, and uncertainty. Lack of adequate understanding of interfacial damping affects the predictive modeling of assembled structures subjected to vibrations.

This power-law relation holds for both material and interfacial damping [3,6]. Power-law exponents closer to 2 suggest linear damping because in a structure with linear compliance, maximum strain energy scales as the square of the applied stress, and since the damping can be regarded as the ratio of the energy dissipation to the maximum strain energy; the damping scales as $\sigma n\u22122$, and thus becomes independent of loading if $n=2.$ Therefore, the deviation of $n$ from two signals the degree of nonlinearity in damping.

Mindlin et al. in 1952 [1] proposed the first analytical solution for the interfacial damping. This study showed that under a constant normal load and small amplitude of tangential oscillations, the energy dissipated by frictional slip scaled with the cube of the maximum tangential force, i.e., power-law exponent, $n=3.$ The same analytical solution predicts power-law exponents higher than 3 for larger oscillations. Starting with Johnson’s [2], and Goodman and Brown’s [3] experiments, theoretical estimation of power-law exponents 3 or higher were never observed in measurements of interfacial damping. Experiments with various sample geometries, materials, and instruments the power-law exponents consistently ranged between 2.2 and 2.8 [4,7–9].

A solution to frictional dissipation in spherical contacts under in-phase normal and tangential loading was first proposed by Mindlin and Deresiewicz [10]. The power-law exponent for lower amplitude of cyclic loading was again cubic. The importance of phase difference between normal preload and tangential force in controlling frictional dissipation was first observed by Griffin and Menq [11]. They proposed a two degrees-of-freedom dynamical system with Coulomb friction to study interfacial damping and found that 90 deg phase difference between the normal and tangential motions resulted in 36% reduction in resonance amplitude when compared to in-phase motion. Menq et al. [12] obtained similar conclusions for an elliptical motion with 45 deg phase difference. Those modeling efforts hold for sliding contacts, but not for presliding contacts that are more prevalent in assembled structures. This gap in modeling was addressed recently by several research groups [13–17]. In particular, Putignano et al. [15] used the Ciavarella [18,19] and Jager’s [20] superposition approach to evaluate the frictional energy dissipated during out-of-phase normal and tangential loading of spherical contacts under presliding conditions. In their study, they assumed that the contacting bodies are half-planes with Poisson’s ratio set to zero and the contact is elastic with no elastic mismatch. Those studies showed that frictional dissipation increased with the oscillations in normal load and their phase difference with the tangential oscillations. In addition, power-law exponents as low as 2.2 were obtained for particular loading configurations and phase differences (90 deg out-of-phase).

## Problem Statement

In this study, we investigate the effects of normal and tangential fluctuations, and the phase difference between those fluctuations on frictional dissipation in presliding contacts. In particular, we conduct experiments on a contact configuration featuring a rigid sphere-on-deformable half space. As shown in Fig. 1, variable normal $(P)$ and tangential loads $(Q)$ are imposed on this contact. Mean normal load causes tip to penetrate into the sample to a depth of $d,$ and forms a circular contact area with radius, $a.$ Normal load fluctuations following the mean preloading are chosen smaller than 30% of the mean preload to mimic sustained structural vibrations without significant loss of preload. The tangential loading is imposed by tangential displacements fluctuating around zero mean. Fluctuations in tangential displacement are chosen to ensure presliding contact conditions. Mechanical energy losses as measured by the load–displacement response of the contact are used to monitor frictional dissipation. The details of the test samples, experimental setup, loading parameters, and processing of raw data are given in Sec. 3. The results and discussions are presented in Sec. 4. The paper ends with a list of findings and conclusions in Sec. 5.

## Experiments

### Tested Samples.

The surfaces tested are cut from a 0.5 mm-thick polyimide film and used as received from the supplier (UPILEX). Polyimides are used safely in extreme thermomechanical and chemical conditions, thanks to their high glass transition temperature and low creep properties [21]. Practical uses of polyimide films include electronic circuits, wire insulations, and composite components in aero-engines, turbine blades, and launch vehicles [22–25]. We elect to test polyimide samples for several reasons. First of all, polyimide films show linear isotropic response in elastic loading cases [26,27]. Also, we will demonstrate that polyimide exhibit negligible viscoelasticity at the loading rates used in our experiments. Those properties enable us to compare the experimental observations to known analytical solutions of elasticity. Second, polyimide is inert to rigid diamond probes, and thus adhesion is negligible compared to deformation forces. Therefore, frictional slip dominates the interactions between the sample surfaces and probe. Lastly, polyimide delivers more compliant response compared to metals, and thus tangential displacements in presliding regime are comparable to contact radii (∼a few micrometer). Our presliding experiments can resolve and control the tangential displacements on the order of microns with significantly better resolution than nanoscale slips observed at metallic contacts.

Prior to presliding experiments, ultrasonic cleaning with acetone bath is applied to all polyimide samples. After cleaning, the samples are glued to metallic pucks and mounted on the triboindenter (TI950 by Hysitron, Inc., Minneapolis, MN), and roughness of the test surfaces is measured by scanning probe microscopy (SPM). The SPM imaging load, size, and frequency are 2 *μ*N, 10 *μ*m × 10 *μ*m, and 1 Hz, respectively.

Figure 2 shows a representative SPM image of the surface heights of a tested region. The root-mean-square (RMS) and peak-to-valley roughness measured in tested regions vary around 15 and 100 nm, respectively. As will be presented, the mean preload used in the experiments causes penetration depths on the order of 100 nm, and contact radii on the order of 2 *μ*m. The effect of roughness on the indentation response of a spherical contact is investigated elsewhere [28] and the main parameter dictating the amount of deviation is found to be the ratio of RMS roughness to penetration depth. Based on the RMS roughness and penetration depth values, this ratio is around 0.15 in our experiments. In terms of maximum contact pressure, this leads to 12–17% of deviation from the Hertzian theory, whereas the deviation is around 10% for contact radius. In addition, the difference between the surface heights of the contact zone (a 5 *μ*m × 5 *μ*m area in the center) measured before and after experiments (Fig. 2(c)) shows that the change in surface heights range between 4.5 nm and −6.4 nm, which is comparable to the asperity size. This suggests that the loading conditions used in the experiments cause predominantly elastic deformations and the extent of plasticity, if there is any, did not go beyond the asperity scale. Negative values for the height difference between the two images can be attributed to thermal drift, the difference between the two contact detection points preceding SPM imaging sequences or nanoscale debris that is swept out of the contact zone. Spatial matching of the coordinate systems of the before and after images was carried out by offsetting the latter along the *x* and *y* directions. Therefore, the calculated differences are also inevitably subjected to some error from rotational mismatch.

### Presliding Experiments.

*μ*m is first pressed on the polyimide samples. After the initial preloading, normal loading and tangential displacement fluctuations with various amplitudes and phase differences are imposed on the contact. The normal loads and tangential displacements used are expressed as

where $P0$ is the normal preload, $P1$ and $x1$ are the fluctuations in the normal load and tangential displacement, $w$ is the fluctuation frequency, and $\varphi $ is the phase difference. As shown in Fig. 3, the fluctuations $P1$ and $x1$ are chosen as triangular waves with $w=\pi /10.$ The mean normal load, $P0$ is set to 1 mN, and the fluctuation amplitudes, $P1$ are adjusted as the 0%, 10%, 20%, and 30% of the mean preload. Note that the first of those fluctuation conditions corresponds to the constant normal load case, which is heavily studied in the literature. For each normal loading condition, the tangential displacement fluctuations are increased from 70 to 120 nm in four equal increments. To ensure presliding conditions, we determined the displacements at the sliding inception by unidirectional sliding experiments on polyimide under a normal load of 1000 *μ*N. As shown in Fig. 4, the tangential displacement (*δ _{T}*) of 60 nm is significantly lower than the displacement at the onset of macrosliding.

The phase difference between the normal loads and tangential displacements is adjusted to yield low (5.8 deg ± 2.3 deg) and high (93.6 deg ± 2.7 deg) phase differences between the normal and tangential loads. Note that normal loads and tangential displacements are the boundary conditions we control throughout all the presliding experiments, and we measure the corresponding normal displacements and tangential forces. Four trials are conducted at each loading scenario to check repeatability. The triangular fluctuations are imposed for a total of eight periods as shown in Fig. 3(a). Previously, three cycles were found sufficiently long enough for transient slip process to shake down and reach steady-state in cyclically loaded contacts [29]. To ensure steady-state contact conditions, we will use the eighth cycle while estimating the frictional energy dissipation. Figure 3(c) shows a representative tangential displacement versus force response (also known as hysteresis or friction loop). The area inside the hysteresis loop quantifies the energy dissipation during one loading–unloading cycle.

### Cyclic Indentation Tests.

The mechanical energy loss measured by the hysteresis (friction) loops can be attributed to frictional slip, interface failure, viscoelastic losses, plasticity, and/or surface damage (fracture, wear, etc.) of the polyimide samples. The roughness maps of the tested surfaces before and after presliding tests suggest no noticeable plasticity or damage (see Figs. 2(a) and 2(b)). To identify the relative contribution of viscoelastic losses, we conduct cyclic indentation tests using the triboindenter at the fluctuation frequency used in the presliding experiments (0.05 Hz). Normal loading imposed in these tests is identical to the profile depicted in Fig. 3(a). The phase differences between the normal load and displacement responses are used to monitor the degree of viscous response, and viscoelasticity-born losses. Note that the elastic mismatch between the probe and the samples also causes frictional slip during normal load fluctuations. However, the mismatch-induced slip is expected to be very limited compared to the slip caused by tangential loading. Therefore, phase difference obtained by the cyclic indentation tests can reveal the relative importance of viscoelasticity when compared to the phase differences observed in the hysteresis (friction) loops.

### Adhesion Tests.

If significantly high, adhesive interactions can suppress frictional slip, and thus, sliding at the interface can only occur through atomic attrition, cleavage, fracture, and crack propagation. All of those dissipative processes can contribute to the mechanical losses in the presliding tests. We conduct pull-off experiments to quantify the adhesive strength of the contact. The triboindenter and the same probe used in presliding experiments are employed in the pull-off experiments. Loading in normal direction is imposed by controlling the probe displacement in loading, unloading, retracting, and resetting phases as shown in Fig. 5. In the retracting stage, the maximum tensile force needed for complete loss of contact is reported as the pull-off force. This force quantifies the importance of adhesive interactions relative to the loads used in the presliding experiments. Note that the energy loss due to adhesive interactions scales with the pull-off force generated at the continuum length scales [30]. Therefore, pull-off forces provide a direct measure of the influence of adhesion.

## Results and Discussion

### Viscoelastic and Adhesive Losses.

In this section, we first study viscoelasticity of polyimide at the loading rate used in the presliding experiments. Figure 6(a) shows the normal load imposed on four different locations on the sample, and Fig. 6(b) shows the corresponding normal displacement responses. Note that no tangential loading is imposed in the cyclic indentation tests. The displacement responses exhibit cyclic behavior as the loading imposed, with an average phase difference of 0.3 deg. As will be discussed in Sec. 4.3, this phase difference is significantly smaller than the phase differences observed in the hysteresis loops (∼2.2 deg). Therefore, we conclude that the contribution of the viscoelastic losses in the total mechanical loss is negligible. This result is not surprising given the very low loading rates used in our experiments. Previous studies also report loss tangents lower than 0.01 from the dynamic mechanical testing of polyimide films with 1 Hz or lower loading rates [31]. In fact, master curves obtained from the dynamic mechanical analyses of polyimide films exhibit very low loss moduli for loading rates below 100 Hz. Therefore, the response of polyimide samples in both cyclic indentation and presliding tests with loading rates of 0.05 Hz can be safely assumed purely elastic. A counterintuitive observation from the normal displacement curves shown in Fig. 6(b) is the negative creep response. The mean normal penetration in to the sample surface decreases by 10 nm at the end of 180 s test period, yielding a linear creep rate of around −0.056 nm/s. The triboindenter manual lists the ideal drift rates as 0.05 nm/s, which is nearly the same as the measured negative creep, and thus we conclude that this is an artifact of the inevitable thermal drift in the triboindenter.

Next, we quantify the adhesive strength between the probe and the polyimide samples. Figure 7 shows a representative load versus displacement response in adhesion tests. Note that zero penetration depth corresponds to the undeformed sample surface. In adhesion tests, the probe first approaches to the sample surface, and then penetrates up to 140 nm. This penetration depth is chosen to yield similar normal loads (870–950 *μ*N) as in presliding experiments. Unloading begins immediately after the maximum penetration depth is reached.

As evident from Fig. 7, the unloading response follows the loading curve with a minute hysteresis. As discussed in cyclic indentation tests, this is thanks to predominantly elastic loading. At the retracting phase, the forces become tensile, and reach a maximum value referred as pull-off force, after which the contact between the probe and the sample is lost momentarily. The pull-off forces obtained in those tests are on the order of 1 *μ*N. (inset figure in Fig. 7). This range is negligibly small compared to the mean preload, and so are the energy losses due to adhesive interactions.

where $Fpull-off$ is the pull-off force, $\Delta \gamma $ is the surface energy, and $R$ is the combined contact radius. This equation returns a surface energy value of 4.24 mJ/m^{2} when we use 1 *μ*N (from Fig. 7) and 50 *μ*m for $Fpull-off$ and $R$, respectively. This value is significantly small compared to 40 mJ/m^{2}, the surface energy value reported for polyimide [35]. This significant difference confirms that the effect of roughness in our adhesion tests is influential. Regardless of the mechanisms leading to low adhesive strength, we can eliminate adhesive interactions from the list of possible contributors to the mechanical losses, leaving only frictional slip as the dominant contributor.

### Identification of Elastic Properties.

where $E*=[(1\u2212\upsilon 12)/E1+(1\u2212\upsilon 22)/E2]\u22121$ and $R=(1/R1+1/R2)\u22121$ are the equivalent Young’s modulus and radius of the contact pair, and $E1,2$, $\upsilon 1,2$, and $R1,2$ are the Young’s modulus, Poisson’s ratio, and radius of contacting materials 1 and 2, respectively. When the material properties and geometry of the spheroconical diamond probe given in Table 1 are inserted in Eq. (5), the only unknowns are the elastic properties of the polyimide film. The literature reports the Poisson’s ratio of 0.34 for polyimides. With those values, we estimate the Young’s modulus of the polyimide films by the best fits to the normal load versus penetration depth graphs shown in Fig. 8. This procedure is applied to the initial loading stage of all the presliding experiments ($n=112$), and the Young’s modulus is found to range from 1.6 to 2.9 GPa, with a mean of 2.4 GPa. The Young’s moduli range we identify agrees reasonably well with the values reported in the literature [36–38]. The analytical predictions of Eq. (5) with the minimum, maximum, and mean values of Young’s moduli are also overlaid on the experimental data in Fig. 8. The match between the analytical curves and the experimental data is quite significant beyond ∼20 nm of penetration depth, whereas below that level, experimental data exhibit a slightly smaller slope than 3/2. This is in support of the previously presented argument that once the penetration depth exceeds the RMS roughness of the surface, the indentation curves gradually converge to the Hertzian response. This also agrees with the use of the normal loading portion of the tests in estimating Young’s modulus as an alternative to the Oliver–Pharr method.

For each experiment, we determined the residual normal displacement upon complete unloading. Then, we calculated the total thermal drift by multiplying the thermal drift rate measured and total test duration. The mean and standard deviation values were calculated to be 5.84 nm and 5.4 nm for residual normal displacements and 26.98 nm and 11.39 nm for thermal drift values. This shows that experimentally observed residual normal displacements lie within the uncertainty of thermal drifts. Hence, we conclude that the extent of plasticity is minimal.

We will use the elastic properties identified experimentally in the analytical interpretation of the presliding experiments in Sec. 4.3.

### Presliding Experiments and Energy Dissipation.

We conduct a total of 112 presliding experiments to study the effect of loading fluctuations and phase differences on the frictional dissipation. Those experiments can be grouped into three major categories according the loading conditions: (i) no normal load fluctuation (Fig. 9(a)); (ii) normal and tangential load fluctuations with low phase (LP) difference (Fig. 9(b)); and (iii) normal and tangential load fluctuations with high phase (HP) difference (Fig. 9(c)). Loading condition in Fig. 9(a) is extensively studied in the literature both experimentally and analytically. We will apply this loading condition on the polyimide samples to validate the experiments against analytical solutions. Loading conditions in Figs. 9(b) and 9(c) differ from each other by the phase difference between the normal and tangential fluctuations. As evident from the loading curves, Fig. 9(b) approximates in-phase linear loading (i.e., oblique loading), whereas the loading in Fig. 9(c) causes an elliptical loading in $P\u2212Q$ plane. Those extreme cases provide sufficiently large contrast between loading conditions, and thus enable us to study the effect of phase difference on frictional energy losses.

Figure 10 shows the hysteresis loops obtained from the eighth cycle of loading for the constant normal load case under four different maximum tangential fluctuations. The areas inside those loops will be studied as the frictional losses at the steady state.

Figure 11 shows the frictional dissipation as a function of maximum tangential load for all testing configurations. Starting with the constant normal load case, we observe that higher tangential loads cause larger frictional dissipation. This is expected since higher tangential loads lead to larger frictional slip. Frictional slip and associated energy dissipation in spherical contacts under cyclic loading conditions are worked out by Mindlin and Deresiewicz [10], yielding the following expression for the energy loss:

where $G*=(2\u2212\upsilon 1)/G1$ and $G1$ and $\upsilon 1$ are the shear modulus and Poisson’s ratio of the test material, respectively; $a$ is the contact radius, and $\mu $ is the coefficient of friction of the test surface.

To estimate the energy dissipation using Eq. (6), we need the material properties, contact radius, loading conditions, and the coefficient of friction, all of which we discussed in Secs. 3.2 and 4.2 except for the last one. We conduct sliding friction tests using the same experimental procedure as in presliding tests but with larger tangential displacements to obtain the coefficient of friction of polyimide samples. Upon five trials, we find the mean value of coefficient of friction of 0.2, and substituting the elastic properties listed in Table 1 and contact radius estimated from the experimental data using the Hertzian assumption in to Eq. (6), we estimate the energy dissipation predictions. We calculated the shear modulus of polyimide ($G1$) using $G=E/2(1+\upsilon )$. Since the experimental Young’s moduli of polyimide varied across a range of values between 1.6 GPa and 2.9 GPa, we calculated minimum and maximum shear moduli corresponding to the same measured range, and plotted Eq. (6) for these two values separately. The band formed by these two theoretical curves as presented in Fig. 11 correlates reasonably well with the measured frictional dissipations, confirming the frictional slip as the major dissipation mechanism observed in the experiments. Among all the cases tested, high preload fluctuation with HP difference between tangential loads yields the maximum frictional dissipation. As the preload fluctuations decrease, so does the energy dissipation, and thus, constant normal load case yields the minimum frictional dissipation. For a fixed maximum tangential load, one can alter the frictional dissipation by a factor of two by adjusting the fluctuation amplitudes and phase difference. Albeit being subtler, we also find that the HP difference cases consistently dissipate more energy than the LP difference cases. Those observations are in line with the analytical solutions of Putignano et al. [15] and Patil and Eriten’s numerical results [39] stating that maximum frictional dissipation occurs when the normal and tangential components are approximately 90 deg out of phase. Similar conclusions are also made earlier for contacts under more complicated loading conditions that could result in sliding and separation [13]. The influence of normal fluctuation amplitudes and phase difference on the frictional dissipation vanishes as the maximum tangential fluctuation amplitude decreases, and thus, all the dissipation values cluster closely as shown around $Qmax=90\mu N.$ However, when the fluctuations are high, the differences are more pronounced, and controlling dissipation with both fluctuation amplitudes and phase difference is possible. Remarkably, frictional dissipations can be tuned over an order of magnitude from low to high amplitude vibrations when phase difference is increased from low to high; i.e., the triangles at around $Qmax=90\mu N$ to circles at around $Qmax=150\mu N.$

We conducted further analyses to investigate the effect of loading in the normal direction on energy dissipation. The numerical estimation of the area enclosed by the indentation curve in Fig. 7 yielded an energy dissipation of 4.44 pJ, which is comparable to the maximum energy dissipation per lap values displayed in Fig. 11. Note that this dissipation is due to entire normal loading cycle rising from 0 to 1000 *μ*N and then decreasing to zero upon unloading. The normal load fluctuations in our experiments on the other hand represent only 30% of this load at maximum. In a linear elastic material, total strain energy would scale with the square of normal load; i.e., *P*^{2}; and thus normal load fluctuations of 0.3*P* cause 10% change in total strain energy when compared to the contributions due to the mean normal load. Viscoelastic losses under constant loading rate and temperature can be related to the total strain energy by a constant loss factor, and hence, the total losses for a normal fluctuation cycle would be on the order of 0.44 pJ.

Furthermore, we estimated the dissipation due to normal loading over the last fluctuation cycle of 20 s (i.e., starting from *P*_{0} to *P*_{1} rising up to *P*_{0}* + P*_{1} and falling back to *P*_{0}* − P*_{1}). The mean and standard deviation values of the residual normal displacements from this mini normal loading/unloading cycle are 1.49 nm and 0.4 nm, respectively, whereas the same parameters are calculated to be 3.34 nm and 1.55 nm for thermal drift that occurs over the same cycle. Since the residual displacements are lower than thermal drift, the energy dissipation due to normal load fluctuations around the mean preload of *P*_{0} is unquantifiable. Our presliding experiments, however, are conducted under tangential displacements that are an order of magnitude higher than thermal drift. Therefore, frictional energy loss is expected to be quantifiably larger than drift and viscoelasticity-related losses as presented in Figs. 10 and 11.

Another possible source of energy dissipation in normal loading is elastic mismatch between the probe and the polyimide samples. Due to elastic mismatch, normal and tangential tractions are coupled; i.e., normal loading causes shear tractions around edges of a spherical contact, and shear tractions influence normal displacements and thus contact area and normal tractions. Munisamy et al. [40] showed that when the product of elastic mismatch parameter (Dundurs constant) and friction coefficient is less than 0.1, the effects of mismatch on contact parameters and thus dissipation can be neglected. In our experiments, that product attains a value of 0.048 (Dundurs constant ∼0.24 × friction coefficient ∼ 0.2). Therefore, the contribution of mismatch on the dissipation observed in the normal loading cycles is expected to be negligible.

### Power-Law Exponents.

Next, we estimate the power-law exponents giving the dependence of energy dissipation on the maximum tangential load; i.e., the exponent n in $\Delta W\u223cQmaxn$. Note that for a given loading condition, we find the best fits to 16 sets of energy loss and $Qmax.$ Table 2 lists the estimated power-law exponents for seven loading conditions tested. The power-law exponent is 2.5 for the constant normal load case. Note that this case was treated analytically by Mindlin and Deresiewicz [10], and power-law exponents of 3 were found at the small vibration approximation. For larger vibrations, the power-law exponents were predicted even higher than 3. However, experimental studies featuring constant normal loading reported power-law exponents ranging from 2.2 to 2.8 [2,8]. Our results fall into this observed range. Spatial, temporal, and load-dependent variations in coefficient of friction, plasticity, and roughness are possible reasons for the mismatch between the analytical and experimental power-law exponents. In our study, we control the loading conditions, contact geometry, and surface roughness to suppress the effects of plasticity and roughness. However, the spatial and temporal variations in frictional behavior, which links normal and tangential tractions over the contact area, might still be influential. For the fluctuating normal load cases, the power-law exponents vary between 2.3 and 2.7 with higher phase differences consistently leading to lower power-law exponents. This is in close agreement with the analytical [15] and numerical [39] observations made in the literature. Both works state that power-law exponents closer to 2 are attainable with 90 deg out-of-phase normal and tangential fluctuations. Besides, Kim and Jang investigated an axisymmetric Hertzian contact problem under cyclic loading and obtained a power-law exponent of 2 [41]. Although the trends in our experiments correlate well with the theoretical and numerical predictions, we could not obtain power-law exponents close to 2 with the loading configurations we tested. Note that any deviation from two introduces nonlinear damping into a dynamic system. Therefore, our experiments assert that nonlinear damping is inevitable in an assembled structure containing frictional contacts even when the loading conditions, contact geometry, and roughness are controlled within a reasonable degree of accuracy.

## Conclusions

The relationship between frictional energy dissipation in spherical contacts and two main loading parameters has been investigated: magnitude of and phase difference between fluctuations in normal and tangential loads. 0%, 10%, 20%, and 30% of normal load fluctuations around a mean preload of 1000 *μ*N and four different maximum tangential loads covering the range between ∼90 *μ*N and ∼150 *μ*N were used in two different phase difference scenarios, i.e., ∼0 deg and ∼90 deg. Pull-off and cyclic normal loading experiments conducted prior to the presliding experiments verified that energy losses observed in the presliding experiments can be fully attributed to frictional dissipation. Major findings of the study include:

Energy dissipation have been shown to increase with increasing tangential force fluctuations primarily because of larger frictional slip;

The maximum dissipation values were observed in the large normal and tangential load fluctuations and HP difference, whereas the minimum dissipation was obtained under constant normal load and small tangential fluctuations;

For the same tangential loading and phase difference, energy dissipation increases as normal load fluctuations increase;

As the maximum tangential load increases, the range of energy dissipation values observed for different normal loading and phase difference scenarios widens making it possible to tune energy dissipation over a wide range;

In contrast to the theoretical predictions of 3, the power-law exponents linking the tangential fluctuation magnitudes to energy dissipations were found to be 2.49 for the constant normal loading case. Additionally, none of the loading cases studied delivered power-law exponents close to 2; i.e., nonlinear damping prevailed for all the contact cases investigated.

For the same normal and tangential loading parameters, energy dissipation increased and power-law exponents decreased with increasing phase difference.

## Acknowledgment

This work is supported by the U.S. National Science Foundation (NSF) under Grant No. NSF CMMI-1462870.