The experimental apparatus was designed to simulate the interaction between the rubber blade and glass surface of a wiper system, reproducing the dynamic conditions and frictional behavior observed in practical use. The experimental apparatus allows for precise control of normal load, drive velocity, and yaw angles, enabling detailed study of friction-induced vibration. Figures 1, 2 show a photograph and a schematic diagram of the experimental apparatus, which consists of drive and driven parts that simulate the wiper system, along with an optical setup for monitoring the motion of the rubber blade. A glass plate (material: borosilicate glass, diameter: 50 mm, thickness: 5 mm, and arithmetic average roughness: 4 nm) and a short piece of a commercial rubber blade (made by DENSO WIPER SYSTEMS, INC., model number: AS70GN, width: 10 mm, and arithmetic average roughness: 1 µm) were placed in the drive and driven parts respectively, forming a line contact. In the drive part, the glass plate was mounted on a motorized linear stage (KS101-30R, SURUGA SEIKI, JAPAN) via a holder and a loading mechanism to drive at a constant drive speed V. The loading mechanism employed a lever mechanism with a coil spring, including a load cell (stiffness: 2.4 kN/mm, LMB-A-50N, Kyowa Electronic Instruments, JAPAN) above the coil spring to monitor the normal load. The normal load was applied to the contact by the restoring force generated by deforming the coil spring. The coil spring could be replaced with one of three coil springs with different stiffness values (K = 4, 10, and 60 N/mm). This change would adjust the natural frequency of the loading mechanism to f _L = 18, 26, and 40 Hz, respectively. To monitor the Z-axis motion of the loading mechanism including the glass plate during the sliding test with the rubber blade, an accelerometer (frequency range: 1 Hz–10 kHz, NP-3211, Ono Sokki, JAPAN) was placed on the opposite surface of the glass plate from the contact. In the driven part, the rubber blade was mounted on a base plate supported by parallel leaf springs (stiffness k: 4 N/mm), where the natural frequency of the base structure of the rubber blade was f _B = 29 Hz. The displacement x _base of the base plate was measured by a gap sensor (resolution: 0.4 µm, EX-614V, KEYENCE, JAPAN). A manual Z-axis stage was placed right under the parallel leaf springs to adjust the height position of the rubber blade tip. A manual rotational stage was placed under the parallel leaf springs to adjust the yaw angle φ ₁ between the direction of drive velocity V and the direction of the deformation x _base of the parallel leaf springs. Another manual rotational stage was placed between the rubber blade and the base plate to adjust the yaw angle φ ₂ between the direction of drive velocity V and the direction of the deformation of the rubber blade.

Fluorescence observation of the contact between the glass plate and rubber blade was performed to measure the water film thickness and the position of the rubber blade tip using the optical setup consisting of a laser (wavelength: 520 nm), microscope, and an intensified CMOS camera (ORCA-Flash4.0 V3, HAMAMATSU PHOTONICS, JAPAN). A fluorescent solution was prepared by adding Rhodamine-B (concentration: 0.3 mol/m³) to pure water, and this solution was used as a lubricant in the contact. The laser beam was directed at the contact area lubricated with the water-based fluorescent solution. The emitted light (wavelength: 570–590 nm) was captured using the camera at a frame rate of 200 fps (exposure time: 1/frame rate) through the microscope equipped with a 5x objective lens and long-pass filter (cut-off wavelength: 550 nm) in the optical path. The resolution of the captured images was 500 × 500 px; the pixel size was 1.28 µm/px; the gradation was 65,536 levels of gray. A manual XY-axis linear stage was mounted at the bottom of the microscope to adjust the camera’s view to the contact area.

The glass plate was cleaned with acetone and hexane for 5 min each using an ultrasonic cleaner, dried with hot air from a blower, and installed into the holder of the drive part. The accelerometer was fixed to the glass plate with solid glue. The rubber blade was wiped by an ethanol-moistened paper towel and then installed onto the base plate via the blade holder. Each yaw angle (φ ₁ and φ ₂) was adjusted to be 0, 15, 30, or 45 degrees using the manual rotational stages, as shown in Figure 2. The contact between the glass plate and rubber blade was made while keeping the glass plate horizontal using the loading mechanism in the drive part and the manual Z-axis stage in the driven part. The water-based fluorescent solution of 100 mm³ was injected around the contact. After driving the glass 10 mm at V = 20 mm/s, the normal load W was adjusted to be 0.17 N while maintaining the glass plate horizontal and the rubber blade bent with static friction. The microscope position was adjusted using the manual XY-axis linear stage to observe the contact with the camera. The laser beam was then directed at the contact. When the glass was moved 20 mm at V = 5, 10, 15, and 20 mm/s, the displacement x _base of the base plate and the acceleration a _z of the glass plate were measured at sampling frequencies of 1,000 Hz and 51.2 kHz, respectively. A signal of 1 V for 0.05 s was used as a trigger to synchronize the start timing of recording x _base and a _z and capturing the fluorescence intensity images. The ambient temperature and relative humidity during the experiments were 25°C and 20%–40%, respectively.

Figure 3 illustrates the calibration of film thickness measurements for the water-based fluorescent solution using the images captured by the camera. To calibrate the intensity of the captured images to the water film thickness, a shim film (material: stainless steel) with known thickness h _s of 5, 10, and 15 µm was placed between the glass plate and rubber blade, creating three distinct contact situations: contact via the shim film, contact via the fluorescent water, and direct contact of the glass plate and rubber blade, as shown in Figure 3A. In Figure 3B, green, red, and blue plots represent the fluorescence intensity values measured with shim films of different thicknesses (5, 10, and 15 μm, respectively). Each color plot showed that the fluorescence intensity was maximum at the end of the shim film (position: 0 µm) and decreased with increasing distance from the shim film. The water film thickness at the position where the fluorescence intensity was the maximum was assumed to be equal to the thickness of the shim film. The minimum value of the fluorescence intensity was assumed to indicate the water film thickness of zero. The relationship between the water film thickness and the fluorescence intensity measured at the position of zero using each shim is shown in Figure 3C. The solid back line represents a fitting line with the equation: h = a (I–b), where h is the water film thickness, I is the intensity measured by fluorescence observation, and a and b are fitting parameters. The fitting line shown in Figure 3C has parameters a = 1.15 and b = 102.5 (determined by the least squares method, R ²: 0.95).

Figure 4 illustrates a typical result of fluorescence observation for the sliding contact of the glass and rubber blade at K = 10 N/mm, φ ₁ = φ ₂ = 0 deg, and V = 5 mm/s, where t = 0.065 s in Figure 5. In the fluorescence intensity image with a grayscale range of 0–500 shown in Figure 4A, the intensity decreased as it approached X = 2.04 mm but did not change significantly along the Y-axis direction. The black line in Figure 4B represents the mean values of the film thickness h of water, calculated from the intensity averaged from Y = −0.32 to 0.32 mm. The red and blue lines represent the mean values of h plus and minus standard deviation (STD), respectively, also calculated from Y = −0.32 to 0.32 mm. The variation of h with X shows the tip-corner shape of the rubber blade and the minimum value h _min of the water film thickness, which is 2.3 μm at X = 2.04 mm. The position where the minimum film thickness h _min was observed is defined as x _rubber. Considering the STD value shown by the blue line, the value of h at X = 2.04 mm was almost zero, implying that the lubrication regime in the experiment was mixed lubrication.

Figure 5 illustrates temporal changes in fluorescence intensity, x _rubber, h _min, x _base, and a _z as a typical experimental result at K = 10 N/mm, φ ₁ = φ ₂ = 0 deg, and V = 5 mm/s, where the values from the last one second of the four-second sliding time in the test were extracted. A low-pass filter (cut-off frequency: 200 Hz) was applied to the temporal change in a _z . In addition, the spectra shown in the right-hand side column of Figure 5 were obtained using the fast Fourier transform (FFT) method applied to the temporal changes in x _rubber, h _min, x _base, and a _z shown in the left-hand side column of Figure 5. The values of x _rubber and h _min were obtained from the fluorescence image at the top of the left-hand side column in Figure 5, which was made by combining the intensity profiles averaged in the Y-axis direction from each snapshot. x _base is the displacement of the base, which was connected to parallel leaf springs. a _z is the acceleration of the glass plate with the lever in the loading mechanism. x _rubber, x _base, and a _z showed steady waveforms of 29 Hz, which matched the natural frequency of the base (f _B = 29 Hz), as shown in the spectra, where their amplitudes were 30 μm, 32 μm, 0.13 m/s², respectively. On the other hand, the waveform of h _min contained two components of f ₁ = 29 Hz and f ₂ = 57 Hz. The first-order frequency f ₁ was the same as f _B = 29 Hz, and the second-order frequency f ₂ was approximately twice f _B. The amplitude A _{h_min} at f ₁ was 0.63 µm and that at f ₂ was 0.88 µm.

Figure 6 shows the cross-sections of h _min, x _base, and z versus x _rubber at K = 10 N/mm, φ ₁ = φ ₂ = 0 deg, and V = 5 mm/s, where the values were extracted at t = 0.065–0.100 s as shown in Figure 5. z is the Z-axis displacement of the glass plate, calculated by dividing a _z by – (2πf)², assuming a _z to be a harmonic vibration, where f = 29 Hz as shown in Figure 5. The open plot represents the start point of the period (t = 0.065 s). Regarding the cross-section of h _min versus x _rubber in Figure 6A, the values of h _min at the top and bottom end points of x _rubber were the minima of 2 µm in one stroke. The value of h _min at the center of the backward stroke of x _rubber was 6 μm, larger than that at the center of the forward stroke, which was 4 µm. This result suggests that the in-plane motion of the rubber blade tip generated a hydrodynamic effect, such as the wedge effect, increasing the film thickness of water in the middle of the forward and backward strokes where the velocity of the rubber blade tip reached two local maxima in one stroke, causing the 2f _B component in h _min. On the other hand, the cross-section of x _base versus x _rubber shown in Figure 6B displayed a positively inclined elliptical orbit. As the value of x _rubber increased, the value of x _base also increased, but with a lag relative to the change in x _rubber. This result suggests that the motion of the rubber blade tip is the origin of the vibration. In addition, the cross-sections of z versus x _rubber in Figure 6C showed a horizontal elliptical orbit. The difference between the cross-sections of z versus x _rubber and h _min versus x _rubber indicates that the change in h _min was in the Z-axis direction but did not affect the Z-axis motion of the lever in the loading system.

Figure 7 illustrates the effect of drive speed on the amplitude and frequency of x _rubber, h _min, x _base, and a _z at K = 10 N/mm and φ ₁ = φ ₂ = 0 deg. The amplitude and frequency were extracted at a predominant frequency in each spectrum, calculated using the FFT method for the temporal change in x _rubber, h _min, x _base, and a _z for 1.0 s, as shown in Figure 5. A _{x_rubber}, A _{h_min}, A _{x_base}, and A _{a_ z} represent the amplitude of x _rubber, h _min, x _base, and a _z , respectively. The diameter of circles in Figure 7 represents the amplitude. The thresholds for extracting the predominant components in the spectrum were A _{x_rubber} = 5 μm, A _{h_min} = 0.25 µm, A _{x_base} = 5 μm, and A _{a_ z} = 0.025 m/s². For x _rubber, x _base, and a _z , the frequency of the vibration was consistent with f _B = 29 Hz, as shown in Figure 5, and did not change with increasing drive speed V. On the other hand, the values of A _{x_rubber}, A _{x_base}, and A _{a_ z} increased with increasing V. The tendencies for the vibration frequency to correspond to the natural frequency derived from the supporting structure, rather than the natural frequency of the rubber blade alone, and for the amplitude to increase with drive speed are consistent with the results of Suzuki et al.‘s research (Suzuki and Yasuda, 1998). Regarding h _min, there are three predominant frequencies: f ₁, f ₂, and f ₃. The first-order frequency f ₁ = 29 Hz was the same as f _B = 29 Hz, the second-order frequency f ₂ = 57 Hz, as also shown in Figure 5, was approximately 2 × f _B, and the third-order frequency f ₃ = 85 Hz was close to 3 × f _B. The values of f ₁, f ₂, and f ₃ did not change with increasing V. In contrast, the values of A _{h_min} at f ₂ = 57 Hz increased with increasing V, but those at f ₁ = 29 Hz and f ₃ = 85 Hz did not increase significantly with increasing V. Note that the second-order frequency was not the natural frequency of the system’s second mode, but could have been caused by the speed-dependent hydrodynamic effect.

Figure 8 illustrates the comparison between the maximum speed of the rubber blade tip and the drive speed. The maximum speed of the rubber blade tip was calculated with A _{x_rubber} · ω, where ω was obtained from 2π f ₁. The plot of A _{x_rubber} · ω at each V was along the dashed line, which shows A _{x_rubber} · ω = V. This result implies that the amplitude A _{x_rubber} of vibration was saturated when the maximum speed A _{x_rubber} · ω of the rubber blade tip reached the drive speed V.

Figure 9 illustrates the effect of the yaw support angles (φ ₁ and/or φ ₂) and stiffness K of the loading system at V = 5 mm/s on the amplitude and frequency of the rubber blade’s vibration x _rubber, which were extracted with thresholds of A _{x_rubber} = 5 µm. The diameter of circles in Figure 9 represents the amplitude. Figures 9B, E, H show the effect of the yaw support angles (φ ₁ and/or φ ₂) at K = 10 N/mm, where the value of A _{x_rubber} at φ ₁ = φ ₂ = 0 was extracted from that shown at V = 5 mm/s in Figure 7. Increasing only φ ₁ (φ ₂ = 0) or φ ₂ (φ ₁ = 0) at K = 10 N/mm as shown in Figures 9B, E, the values of A _{x_rubber} around 29 Hz did not significantly change. On the other hand, increasing the value of φ ₁ = φ ₂ at K = 10 N/mm as shown in Figure 9H, the value of A _{x_rubber} decreased and at φ ₁ = φ ₂ = 45 deg was lower than the threshold of A _{x_rubber} = 5 µm. When the stiffness K of the loading system was 4 N/mm, which made the situation f _B > f _L, the frequency of the vibration shifted from f _B to around f _L with increasing only φ ₁ (Figure 9A) or both φ ₁ and φ ₂ (Figure 9G). When K = 60 N/mm, which made the situation of f _B < f _L in Figures 9C, F, I, the frequency of the vibration did not change with increasing φ ₁ and/or φ ₂. A _{x_rubber} decreased with increasing the value of only φ ₁ (Figure 9C) or both φ ₁ and φ ₂ (Figure 9I), and disappeared at φ ₁ = φ ₂ > 30 deg.

Figure 10 illustrates the effect of the yaw support angles (φ ₁ and/or φ ₂), the stiffness K of the loading system, and the drive speed V on the maximum amplitude A _{x_r_max} of the rubber blade’s vibration x _rubber, which included the maximum amplitude at a different frequency as shown in Figures 9A, G. Increasing the value of φ ₁ = φ ₂ in Figures 10G–I, the value of A _{x_r_max} decreased under any drive speed and stiffness conditions. When the stiffness of the loading system was relatively high (K = 60 N/mm), the value of A _{x_r_max} decreased with increasing only φ ₁ (φ ₂ = 0) in Figure 10C. These results suggest that the yaw support angles in a wiper system could work as the yaw angular misalignment effect (Kado et al., 2013; Nakano et al., 2013-01; Tadokoro et al., 2021), which provides a positive damping effect generated by rotating the friction force vector and suppress the friction-induced vibration. For an actual wiper system in which the wiping motion is in an arc, a parallel misalignment between the drive rotational axis and the driven torsional axis (Tadokoro et al., 2018) could be effective in adding the yaw angular misalignment effect.

Figure 11 illustrates the effect of drive speed V on the friction coefficient µ and average value h _{min_mean} of h _min. Black and red plots represent the values at φ ₁ = φ ₂ = 0 and 45 deg, respectively. The friction coefficient in Figure 11A was calculated using the following Equation 1 (Kado et al., 2013): μ = k x base _ mean W ⁡ cos ⁡ φ 1 (1) where x _{base_mean} is the time-averaged value of x _base. The friction coefficient at φ ₁ = φ ₂ = 0 and 45 deg showed the velocity-weakening property, with black and red lines representing the fitting lines with the following Equation 2: μ = α μ V + β μ (2)

The black line corresponds to α _µ = −0.001 and β _µ = 0.88 (R ²: 0.92), and the red line corresponds to α _µ = −0.003 and β _µ = 0.92 (R ²: 0.99). The slope of the fitting lines at φ ₁ = φ ₂ = 0 deg was slightly lower than that at φ ₁ = φ ₂ = 45 deg. In Figure 11B, h _{min_mean} is the time-averaged value of h _min. At φ ₁ = φ ₂ = 0 deg, h _{min_mean} increased with increasing the drive speed V, where the fitting line was adjusted with the following Equation 3: h min ⁡ _ mean = α h V + β h (3)

The black line corresponds to α _h = 0.6 and β _h = 1.5 (R ²: 0.42). On the other hand, at φ ₁ = φ ₂ = 45 deg, h _{min_mean} slightly decreased with increasing V; the red line corresponds to α _h = −0.02 and β _h = 2.0 (R ²: 0.58). These results show that the time-averaging value of h _min did not significantly affect the change in the friction coefficient. Also, note that friction-induced vibration causes measurement errors in the friction coefficient (Kado et al., 2013). Therefore, to fully understand the details of the friction-induced vibration in the wiper system, further verifications through numerical approaches are also needed.