What is the Hardness Perceived by Tapping?
Kosuke Higashi(B) , Shogo Okamoto, and Yoji Yamada
Graduate School of Engineering, Nagoya University, Nagoya, Japanhigashi.kousuke@d.mbox.nagoya-u.ac.jp, okamoto-shogo@mech.nagoya-u.ac.jp

Abstract. A human can perceive the hardness of an object by tapping its surface. We compared the ranked subjective hardness values andphysical properties of objects, including their stiﬀness, viscosity, density,and Shore hardness, and the frequencies and time constants of the natural vibrations caused by tapping. The stiﬀness, frequency, and viscosity exhibited a relatively strong positive correlation with the perceived
hardness. The results show that the viscosity inﬂuences the hardness
perceived by tapping, as well as the stiﬀness, whereas the stiﬀness andelasticity are considered to be major factors in the hardness perceived
by pinching or pushing.
Keywords: Hardness perception · Damped natural vibration · Realobject1

Introduction

Hardness is an important characteristic of an object, which humans typicallyperceive by pinching or pushing the object. Thus far, many researchers havestudied how humans perceive the hardness and softness of objects. For example,
cutaneous and proprioceptive cues were found to be integrated for the perceptionof softness [2,3,10]. Because the maximum human force is limited, hardness
perception based on the reaction force and deﬂection of an object caused bypinching or pushing is limited. Indeed, human hardness perception is saturated
and less sensitive for harder objects [4]. Similarly, hardness perception using thedeformation of a ﬁngertip when contacting an object’s surface is not eﬀective foran object that is substantially harder than the ﬁngertip. Hence, humans makeuse of tapping and the resultant vibrotactile signal to estimate the hardness ofan object, for which the perceptual strategy based on the deformation of theobject or ﬁngertip is not eﬀective.
Hardness perception by tapping has not yet been thoroughly studied.
Okamura et al. [8,9] reported that the frequency and attenuation of the damped
natural vibration caused by tapping inﬂuence the material perception. This ﬁnding has been exploited to allow force display devices with limited output forces
to deliver hardness values greater than the device is capable of expressing ina quasi-static manner [6,7]. These studies agree that higher vibratory frequencies lead to the perception of greater hardness. However, few researchers havec Springer International Publishing Switzerland 2016

F. Bello et al. (Eds.): EuroHaptics 2016, Part I, LNCS 9774, pp. 3–12, 2016.
DOI: 10.1007/978-3-319-42321-0 1

4

K. Higashi et al.

reported the exact physical properties of objects that humans perceive by tapping. In other words, a question is raised about how accurately the hardness
perceived by tapping corresponds to the physical hardness.
In a former study, we researched the eﬀects of the mass, viscosity, and stiﬀnessof virtual objects on hardness perception by tapping [5]. The result suggested
that an increase in stiﬀness and decrease in mass lead to the greater perceived
hardness. We assume that these parameters are also eﬀective for hardness perception in the case of real objects.
In this study, we investigated the correlations between the hardness perceived by tapping and representative physical parameters related to an object’s
hardness, including its stiﬀness, viscosity, density, and Shore hardness, and thefrequency and time constant of the natural vibration caused by tapping. Understanding the characteristics of hardness perception by tapping will help us todevelop materials or object structures that feel hard or soft, along with a method
to render object hardness for virtual reality systems. It can be also linked witha hardness test for industrial products by tapping or hammering.

2

Experiment: Ranking of Perceived Hardness
by Tapping

Participants tapped the surfaces of objects and ranked their hardness basedon their damped natural vibrations. All of the experimental procedures wereapproved by the internal review board of the School of Engineering in Nagoya
University (#15-12).
2.1

Specimens

We performed an experiment using two types of specimens: spring-damper specimens and material specimens. All of the specimens were hard enough that we
could not judge their hardness by pushing on them with a ﬁnger. As shown in
Fig. 1, the spring-damper specimen was composed of two aluminum plates, a linear spring, and a shock absorber (EMACN1212A, B, and C, MISUMI, Japan).
Each aluminum plate was 80×80 mm in size, and the upper and lower plates were
3 mm and 5 mm thick, respectively. Three types of springs and shock absorbers
Table 1. Stiﬀness and viscosity of spring-damper structures
Sample Stiﬀness [N/m] Viscosity [Ns/m]
I

15700

0.047

II

7830

0.024

III

24700

0.047

IV

15700

0.026

V

15700

0.078

What is the Hardness Perceived by Tapping?

5

Fig. 1. Spring-damper structurewith diﬀerent strengths could be used. Thus, as listed in Table 1, there were ﬁvetypes of specimens. The details of the vibration properties of each specimen are
described in Sect. 3.
The material specimen was a block formed of a single material (60 × 60 ×
30 mm). We used eight types of materials, including wood, nitril rubber, urethane
rubber, modeling wax (ZW-200, Roland DG), ABS resin, acryl, stainless steel,and aluminum. The densities of these specimens are shown in Fig. 2. The densityof stainless steel was remarkably higher than those of the other materials. There
was a trend for the metal to have a high density and the wood to have a low
density. However, there were no signiﬁcant diﬀerences among the others.

Fig. 2. Ranks of densities of material specimens.

6

2.2

K. Higashi et al.

Tasks and Participants

The participants separately ranked the hardness of each of the ﬁve spring-damper
specimens and eight material specimens. They evaluated the hardness of a specimen by tapping its surface with the index ﬁnger of their dominant hand. It wasstressed to the participants that they should tap the specimens with the same
speed as much as possible and keep their ﬁnger touching a specimen while it
was vibrating. The participants could tap each material as many times as theywanted. They ranked the perceived hardness of each specimen, and the samerank could be assigned to multiple specimens. They wore foggy glasses andheadphones playing pink noise to block out visual and audio cues. The specimen was placed upon a buﬀer component, and its vibration was isolated fromthe ambient environment. They repeated the same task three times at intervalsof a few minutes. The participants were ﬁve naive outsiders from the researchgroup, who were males in their twenties.
2.3

Results

Figures 3 and 4 show the medians and 25th and 75th percentiles of the perceived
hardness ranks. Friedman tests indicated that these ranks signiﬁcantly variedamong the specimens at a signiﬁcance level of 0.05. In a post hoc manner, wetested the rank diﬀerences for all the pairs of specimens using Wilcoxon’s rank
sum tests, and the results are shown in the ﬁgures. As shown in Fig. 3, specimen
V, which had a greater viscosity, was ranked ﬁrst, and specimen III, which had agreater stiﬀness, was ranked second. Specimen I, which had medium stiﬀness andviscosity ranks, had a medium rank of perceived hardness. As shown in Fig. 4,
the aluminum and stainless steel were ranked ﬁrst, and the nitril rubber was
ranked last among the material specimens. There were no signiﬁcant diﬀerencesamong the ABS resin, acryl, urethane, modeling wax, and wood.

Fig. 3. Perceived hardness ranks of spring-damper specimens, where * and + indicate
p < 0.05 and p < 0.10, respectively.

What is the Hardness Perceived by Tapping?

7

Fig. 4. Perceived hardness ranks of material specimens, where * indicates p < 0.05.

3

Physical Characteristics of Hardness Specimens

3.1

Characteristics of Natural Vibration

We tapped the spring-damper specimens and measured their damped natural
vibration. An accelerometer (ADXL335, ANALOG DEVICES) was attachedto the center part of the upper surface of a specimen. The measurement was
repeated 10 times for each specimen with a sampling rate of 10 kHz. The specimen was placed on a buﬀer component, and its vibration was isolated from theambient environment, including the table and ﬂoor. We speciﬁed the main frequency of the damped natural vibration by computing the fast Fourier transform(FFT) of the acceleration signals. The vibration waveform was then band-passﬁltered so as to include the natural frequency to determine the time constant.
The time constant was obtained as the time until the amplitude of the ﬁlteredwaveform was damped to 37 % of the maximum amplitude.
We assumed that the damped natural vibration of a spring-damper specimen was the impulse response of the one-degree-of-freedom spring-mass-damper
system. In this assumption, the time constant τ and natural frequency f of thevibration are described as follows:
2m
c

1
k
1
k
2
f=
(1 − ζ ) 
2π
m
2π m
τ=

(1)
(2)

where m, c, k, and ζ represent the mass, viscosity, stiﬀness of the system, andthe damping ratio of the vibration, respectively. The stiﬀness k was deﬁned

8

K. Higashi et al.

Fig. 5. Time constants and frequencies of vibrations of spring-damper specimens.

as the sum of the spring constants of the spring and damper composing thespecimen. The viscosity c listed in Table 2 was obtained by substituting τ and f
into Eqs. (1) and (2), respectively.
The time constants and natural frequencies of the damped natural vibrationsof the specimens are shown in Fig. 5. The deviations of these values in repeatedmeasurements were negligibly small. Specimens with greater spring constants
exhibited greater natural frequencies. In addition, specimens with greater damping ratios had smaller time constants.
We also analyzed the vibration waveform of the material specimens by thesame procedure. However, the waveforms contained multiple signiﬁcant frequency components, which deviated from the damped natural vibration containing a single natural frequency component. Hence, we did not specify thevibration characteristics of the material specimens.
3.2

Shore Hardness

We measured the hardness of each specimen using a method based on the Shore
C hardness test. In the original Shore C hardness test, a diamond hammer isdropped on an object, and its rebound height is measured. The Shore hardness
is evaluated based on the ratio of the rebound height to the drop height. The
Shore hardness may align with a human’s hardness perception because both areaﬀected by the physical properties of the object in a momentary contact state.
In this experiment, we used a small ball made of hard rubber instead of the
diamond hammer to avoid any possible damage to the specimens.
We ﬁxed the specimen in a vise and dropped a rubber ball with a diameterof 22 mm from right above it. The position and initial height of the rubber ballwere controlled using a guide frame. We tested each specimen ﬁve times, andrecorded the drop and rebound heights using a high-speed camera (RX10II,
SONY, Japan). We then derived the average rebound ratio for each specimen.
The ranks of the rebound ratios are shown in Fig. 6. Specimen III, which
had a greater spring constant, exhibited the highest rebound ratio. Moreover,a specimen with a smaller viscosity had a greater rebound height. Among thematerial specimens, urethane was ranked ﬁrst, and nitril rubber was ranked

What is the Hardness Perceived by Tapping?

9

Fig. 6. Percentage of rebound height against falling height. Error bars indicate standard deviations. Left: spring-damper specimens. Right: material specimens.

last. There were no signiﬁcant diﬀerences in the rebound heights of the wood,
aluminum, stainless steel, acryl, ABS resin, and modeling wax.

4

Discussion: Comparison Between Perceived Hardness
and Physical Characteristics Related to Hardness

We researched the relationships between the hardness perception by tapping andthe representative parameters related to hardness. We calculated Spearman’s
correlation coeﬃcient for the perceived hardness ranks and physical characteristics. The average perceived hardness rank among the participants was usedin the calculation. Specimens with physical characteristics with no signiﬁcantdiﬀerences or diﬀerences that were too small to discriminate were ranked as thesame level. Tables 2 and 3 show the ranks of the physical characteristics andtheir correlation coeﬃcients with the perceived hardness.
It was found that the stiﬀness and viscosity of an object, and the frequency ofthe damped natural vibration, exhibited relatively strong correlations with theperceived hardness. Following these parameters, the density of a material andTable 2. Ranks of physical characteristics of spring-damper specimens and Spearman’s
correlation with perceived hardness
ParameterI

II

III IVV

Correlation

Stiﬀness

3

5

1

3

3

0.77

Viscosity

2.5

5

2.5 4

1

0.82

Frequency

3

5

1

3

3

0.77

Time constant

4

2.5

2.5 1

5

0.08

Rebound height

3

4.5

1

4.5

0.45

2

Perceived hardness 3.23 3.77 2.2 3.73 2.13 -10

K. Higashi et al.

Table 3. Ranks of physical characteristics of material specimens and Spearman’s correlation with perceived hardness
Parameter

Alum. SST∗ ABS Acryl Ure. Wax Wood Nitril Correl.

Density

2

1

6.5

4.5

4.5

4.5

Per. hardness
1.73
2.07 4.2∗
SST indicates stainless steel

Rebound height 4.5

4.5

6.5

8

4.5

1

4.5

4.5

4.57

5.37 4.87 5.73

3

0.58

8

0.50

7.47

-

its rebound hardness were moderately correlated with the perceived hardness(ρ > 0.5). Here, we discuss the interpretation of these results, although any correlations in the results were insigniﬁcant because of the small number of samples.
Stiﬀness and Frequency. The perceived hardness ranks had a strong correlation with those of the stiﬀness and the frequency of the damped natural vibrationof the specimens. It was experimentally proved that a human has the ability todiscriminate the stiﬀness of an object by tapping. To the best of our knowledge, very few studies have attempted to examine this perceptual mechanism. It
was assumed that the eﬀect of the frequency was similar to that of the stiﬀness
because they are related by the vibration property, as shown in Eq. (2). The
result, the positive eﬀect of the frequency on the perceived hardness, matches
past conclusions on presenting the hardness of a virtual object through vibration
stimuli [5,7–9].
Viscosity. The viscosity was found to be correlated with the perceived hardness
as strongly as the stiﬀness, indicating that the viscosity was not a secondary factor for the perceived hardness. Note that the stiﬀness and viscosity were fairly
independent of each other in our setup, with the correlation between these two
types of values being 0.35. The viscosity has a role as important as the stiﬀness in restraining the amplitude of the vibration caused by tapping. Although
the viscosity inﬂuences the hardness perception by pushing a compliant virtual
object [1], the eﬀect of the viscosity on the hardness perceived by tapping real
objects was ﬁrst conﬁrmed in this study to the best of our knowledge. Because
these two types of hardness perception are based on diﬀerent perceptual mechanisms, we should not simply translate the eﬀect of the viscosity between thehardness perception by pushing and that by tapping.
Density. The density of a material specimen had a moderate correlation withthe perceived hardness. Note that the density and mechanical hardness of amaterial do not necessarily correspond to each other. For example, the hardnessof a certain kind of urethane rubber can be changed, while its volume and mass
are maintained by changing the ratio of the curing agent. We assumed that thiscorrelation was caused by a change in the vibration properties as a result of achange in the density. The density of an object aﬀects its vibration properties,

What is the Hardness Perceived by Tapping?

11

including the amplitude, time constant, and frequency. The cause of the correlation would become clear by examining the eﬀect of the density on the hardness
perception with changes in the vibration properties.
Time Constant. There was no signiﬁcant correlation between the time constant of the damped natural vibration and the hardness perception. As shownin Eq. (1), the time constant has a relationship with the viscosity. As previously
mentioned, the viscosity exhibited a positive correlation with hardness perception. Thus, it seems natural that the time constant may have a negative one.
Nonetheless, we are still uncertain about the contribution of the time constantto the perceived hardness, because in another study [5], the time constant ofa simulated damped natural vibration inﬂuenced the perceived hardness. The
eﬀect of the time constant should be examined considering its interaction withother physical parameters and the human ability to discriminate the time changeof vibration stimuli in future works.
Rebound Ratio. The rebound ratio was also found to be a secondary parameter aﬀecting the hardness perception. According to the deﬁnition of Shore
hardness, a harder object exhibits a higher rebound ratio as a result of thelower energy dissipation in the elastic deformation. An object with a higher stiﬀness whose energy dissipation is lower exhibits a higher rebound ratio. On theother hand, an object with a higher viscosity whose energy dissipation is higher
exhibits a lower rebound ratio. Therefore, the hardness based on the reboundratio has positive and negative correlations with the stiﬀness and viscosity of theobject, respectively. However, as previously mentioned, the hardness perception
has positive correlations with both the stiﬀness and viscosity. It is speculated
that the rebound ratio is inferior to the individual use of the stiﬀness or viscosity
as an index of perceived hardness by tapping.

5

Conclusion

We investigated the physical parameters that aﬀect hardness perception based on
the tapping of real objects. Through psychophysical experiments and mechanicaltests, we compared the hardness perception ranks and six physical properties,
including the stiﬀness, viscosity, density, and Shore hardness values of objects,
and the frequencies and time constants of the natural vibrations caused by tapping. Our results suggested that the stiﬀness and viscosity of an object and thefrequency of its damped natural vibration had correlations with the perceived
hardness. In contrast with the quasi-static hardness perception by pinching or
pushing, the viscosity was also found to be an eﬀective factor, indicating that weshould not assume that the perceived hardness is entirely related to the actual
hardness of an object. The result suggests that viscosity can compensate for theperceived hardness of objects instead of increasing the stiﬀness. The ﬁndings
can be applied to, for example, rendering a hard virtual object by using thepowerless displays or the material design of industrial products.

12

K. Higashi et al.

Acknowledgment. This study was partly supported by ImPACT (Tough Robotics
Challenge) and MEXT Kakenhi (15H05923).

References
1. van Beek, F.E., Heck, D.J., Nijmeijer, H., Bergmann Tiest, W.M., Kappers,
A.M.L.: The eﬀect of damping on the perception of hardness. Proceedings of IEEE
World Haptics Conference pp. 82–87 (2015)
2. Tiest, B.: W.M., Kappers, A.M.L.: Cues for haptic perception of compliance. IEEE
Trans. Haptics 2(4), 189–199 (2009)
3. Bicchi, A., Schilingo, E.P., De Rossi, D.: Haptic discrimination of softness in teleoperation: the role of the contact area spread rate. IEEE Transactions on Robotics& Automation 16(5), 496–504 (2000)
4. Harper, R., Stevens, S.S.: Subjective hardness of compliant materials. Quarterly
Journal of Experimental Psychology 16(3), 204–215 (1964)
5. Higashi, K., Okamoto, S., Nagano, H., Yamada, Y.: Eﬀects of mechanical parameters on hardness experienced by damped natural vibration stimulation. Proceedings of IEEE International Conference on Systems, Man, and Cybernetics pp.
1539–1544 (2015)
6. Ikeda, Y., Hasegawa, S.: Characteristics of perception of stiﬀness by varied tapping
velocity and penetration in using event-based haptic. Proceedings of 15th Joint
Virtual Reality Eurographics Conference on Virtual Environments pp. 113–116(2009)
7. Kuchenbecker, K.J., Fiene, J., Niemeyer, G.: Improving contact realism throughevent-based haptic feedback. IEEE Trans. Visual Comput. Graphics 12(2),219–230 (2006)
8. Okamura, A.M., Hage, M.W., Cutkosky, M.R., Dennerlein, J.T.: Improving realitybased models for vibration feedback. Proceedings of ASME Dynamic Systems &Control Conference 69(2), 1117–1124 (2000)
9. Okamura, A.M., Cutkosky, M.R., Dennerlein, J.T.: Reality-based models for vibration feedback in virtual environments. IEEE/ASME Trans. Mechatron. 6(3),245–252 (2001)
10. Srinivasan, M.A., LaMotte, R.H.: Tactual discrimination of softness. J.
Neurophysiol. 73(1), 88–101 (1995)