2017 IEEE World Haptics Conference (WHC)
Fürstenfeldbruck (Munich), Germany
6–9 June 2017

Hardness Perception by Tapping: Effect of Dynamic Stiffness of Objects
Kosuke Higashi1 , Shogo Okamoto1 , Yoji Yamada1 , Hikaru Nagano2 , and Masashi Konyo2

Abstract— Humans can judge the hardness of an object bytapping its surface using a fingertip. The damped natural
vibration caused by tapping is a vibrotactile cue for hardness
perception. We investigated how dynamic characteristics ofan object or vibration influence the hardness perceived bytapping. Using multivariate analyses, the subjectively reported
hardness was linked with the dynamic stiffness of an object.
Dynamic stiffness, which characterizes the impulsive responseof an object, was acquired across 40–1,000 Hz for fourteen typesof material cuboid through a hammering test. These blocks
were also ranked by seven participants based on their hardness
perceived when the participants tapped them with a finger. It
was found that the dynamic stiffness did not equally affectthe hardness perception across the wide frequency range. Itssensitivity displayed a peak around 200–400 Hz and decreased
or disappeared at greater frequency bands in which humanperceptual capability is limited.

I. I NTRODUCTION
In our daily lives, we tap an object’s surface in order tojudge its hardness properties when the object is rigid andcannot be deflected by pinching or pushing. Surprisingly,
thus far, the principle of hardness perception by tapping has
yet to be intensively studied. The general agreement among
previous studies seems to be that the dominant frequencyof the transient vibration yielded by tapping is a major cue
for judging the hardness-related properties of objects [1], [2],[3], [4]. Greater frequencies lead to a greater perceived hardness. Other studies reported intriguing properties, although
they still need to be retested and endorsed under various
conditions by several studies before they can be widely
accepted. For example, using vibrotactile display devices,
artificially displaying multiple frequency components in asingle tapping event produces quality hardness stimuli [2],[5]. The viscosity or the damping properties of objects may
also determine the hardness perceived by tapping as well as
the stiffness of objects [6]. Although we focus on vibrotactile
cueing in the present study, we should mention that theforce cue, the rising steepness of the reaction force against
an impulsive force input, likely significantly influences thehardness perceived by striking an object [7], [8], [9]. In
spite of these previous attempts, few researchers have studied
the vibrational frequency characteristics associated with thehardness perceived by tapping.
In previous studies using force or vibrotactile displays
to virtually present the tapping event based on a transient
vibration, vibratory stimuli tended to be simplified to damped
This study was partly supported by JSPS Kakenhi (15H05923) and
ImPACT (Tough Robotics Challenge).
1 Department of Mechanical and Aerospace Engineering, Nagoya University, Nagoya, Japan. 2 Graduate School of Information Sciences, Tohoku
University, Sendai, Japan.
978-1-5090-1424-8/17/$31.00 ©2017 IEEE

vibrations including a single frequency component. However,in reality, the response of an object against an impulsive
tapping force includes a wide range of frequency components. In the present study, we investigate the perceptual
effects of wide-frequency characteristics of 40–1,000 Hz on
the hardness perception. To this end, we used the dynamic
stiffness of the object. Dynamic stiffness is the force perunit vibratory displacement and is defined in the frequency
domain. The transient vibration caused by tapping an object
is mathematically decoupled into an impulsive force input
by tapping and the dynamic stiffness unique to that object.
Hence, the dynamic stiffness expresses the unique frequency
characteristics associated with the hardness of the object.
We linked the dynamic stiffness and subjective hardness ofobjects by multivariate analysis and determined the contribution of the dynamic stiffness for each frequency band.
The dynamic stiffness was investigated by performing amechanical test using an impulse hammer, and the subjective
hardness was determined by subjecting the participants to a
psychophysical test.
II. H ARDNESS SPECIMENS
Fourteen types of specimens made of different materials
were used in both mechanical and subjective tests. Eachspecimen is a solid block made of one of the fourteen typesof materials listed in Table I. The blocks were comprised ofplastic, wood, metal, rubber, and stone, which are popularmaterials in our daily lives. All materials were rigid enoughnot to be deflected when their surfaces were pushed witha fingertip. It should be noted that the dynamic stiffness ofthe object depends on the material and dimensions of theblock. Hence, we did not purely investigate the differencesin terms of materials. It is also true that both the materialand structure of the object influence the softness perceived
by pinching or pushing it [10].
III. P SYCHOPHYSICAL EXPERIMENT FOR SUBJECTIVE
RATING OF HARDNESS SPECIMENS

A. Participants
The participants were seven male university students who
agreed and provided informed consent. All of the participants
were in their 20s and all were right-handed.
B. Procedures: Ranking task
Fourteen types of hardness specimens were randomly
placed on a desk. The participants compared and ranked themby tapping their surfaces. The participants were instructed totap the center of the largest area of each specimen using theindex finger of their writing hand. To avoid judgment based

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TABLE I
M ATERIAL AND SIZE OF HARDNESS SPECIMENS

The dynamic stiffness of an object is defined by theratio of the force acting on the object f (t) and its surface
displacement x(t) in the frequency domain. Given that weused an accelerometer to measure the surface vibration, thedynamic stiffness of the object is determined by

Urethane (S)H(ω) =Cork

Wood

A. Theory and analysis

NitrileGraniteConcrete

2

Brick

Nylon

Acryl

Aluminum

Modeling wax-2

Polycarbonate0
Stainless steel

Normalized rank ofperceived hardness

STIFFNESS

Size [mm]60 × 60 × 30
60 × 60 × 30
60 × 60 × 30
60 × 60 × 30
60 × 60 × 30
60 × 60 × 30
60 × 60 × 30
60 × 60 × 30
60 × 60 × 30
60 × 60 × 30
100 × 100 × 30
100 × 100 × 60
200 × 100 × 60
200 × 100 × 60

Urethane (H)Material
Acrylic resin
Polycarbonate resin
Nylon resin
Nitrile rubber
Urethane rubber (Soft)
Urethane rubber (Hard)
Wood
Wax
Stainless steel
Aluminum
GraniteBrickConcreteCork

IV. H AMMERING TEST FOR DETERMINING DYNAMIC

Fig. 1. Perceived hardness of specimens. Mean and standard deviations ofnormalized ranks among the participants.

on the surface texture or thermal properties, placing a finger
on the specimen or pushing was not allowed. The numberof taps was not specified and a participant could assign thesame ranks to multiple specimens (i.e., multiple specimens
could be perceived to have the same hardness). During theexperiment, the participants wore headphones that played apink noise to shutout the sounds of tapping. Furthermore,since sunglasses with opaque films were worn by the participants, judgment based on the appearance of the specimen
was impossible. Note that the specimen blocks could barely
be seen through the glasses. Most of the participants wereable to finish the ranking task within 10 min.
C. Analysis of ranking data
For a better application of multivariate analyses described
below, we converted the rank of each specimen assigned
by individual participants into an interval scale using anormalized ranking method [11].
D. Results
Fig. 1 shows the mean and standard deviation of thenormalized ranking scores for each specimen. The metal
blocks were ranked highest, followed by plastic, stone, wood,and rubber. The hardness ranks were largely consistent withintuitive understanding, and irregularity was not observed. In
the latter analysis, the mean scores were used as representative subjective hardness values.

38

F[f (t)]F[f (t)]=F[x(t)]
F[ẍ(t)]/(jω)2(1)

where F and ω denote Fourier transform and the angularfrequency, respectively. As described below, ẍ(t) and f (t)were measured by an accelerometer fixed to the specimen
surface and a load cell installed on an impulse hammer,
respectively. Both types of values were then transformed into
the values in the frequency domain using Fourier transform.
Finally, we computed the absolute value of the ratio of theforce to the displacement as the dynamic stiffness. For eachspecimen, the dynamic stiffness was averaged from the dataacquired from ten hammer strikes. Taking into account thecapabilities of the accelerometer that we used and humanperception, we used the stiffness across 40–1,000 Hz forlater analysis.
B. Experimental setup
Fig. 2 shows the measurement apparatus of the hammering
test. An impulse hammer (GK-3100, Ono Sokki Co. Ltd.,
Japan) and a force amplifier (480M96A, Ono Sokki Co. Ltd.)
intended for the load cell embedded in the hammer wereused. A high-precision piezo accelerometer (2302B, Showa
Sokki Co. Ltd., Japan, valid over 20 Hz) was fixed near thecenter of the surface that was to be struck by the hammer,
and its output was acquired through an amplifier (4035-50,
Showa Sokki Co. Ltd., Japan). The force and acceleration
data were sampled using an oscilloscope at 10 kHz. Eachspecimen was fixed on a large metal plate (800 × 800 × 100mm) for the hammering test.
The frequency response of the end tip of the hammer
should ideally be similar to that of the human fingertip.
Few end tips are commercially available for the impulse
hammer we used. Among them, we chose the one whose
frequency characteristics are most similar to those of thehuman fingertip. Fig. 3 shows the frequency spectrum of theimpulsive force when the end tip or fingertip was used. Both
responses agree fairly well with each other.
C. Procedure of hammering test
Each specimen was struck by the hammer at the centerof its largest surface. Furthermore, each of the specimens
was tested multiple times such that ten valid data values
were acquired. Most common invalid trials were due to nonimpulsive strikes with double hammering or a contact periodthat was too long. These invalid trials were easily detectedby checking the force records. Fig. 4 shows an example ofthe force and acceleration acquired during a valid strike. The
contact period of valid trials was generally 2–3 ms.

V. L INKAGE OF DYNAMIC STIFFNESS AND PERCEIVED
HARDNESS BY MULTIVARIATE ANALYSIS

A. Multivariate model and strategy of analysis
Based on the fact that the frequency of the damped
vibration at the tapping event influences the perceived hardness [1], [2], [3], [5] and the perception of vibrotactile
stimulus highly depends on its frequency [12], we linked
the dynamic stiffness values si ∈ Rn×1 of n representative
frequencies for an object i and its normalized hardness rank
hi byhiFig. 2.

Measurement apparatus for the hammering test

= sTi w + cn∑=sij wj + c.

(2)jForce [dB]0where sij and wj were the stiffness value at the jth representative frequency for object i and its perceptual weight,
respectively. For m samples of dynamic stiffness, the regression model is
h = Sw + c.
(3)

HammerFinger

-10
-20
-30
-40
0

Fig. 3.

200

400
600
Frequency [Hz]

800

1000

Spectra of force impulses using a hammer and fingertip

D. Results: Dynamic stiffness of hardness specimens

Force [N]

Accel. [m/s2]

Fig. 5 shows the dynamic stiffness computed for each of
the fourteen specimens. The dynamic stiffness values agreewith our intuitive understanding of hardness. The specimens
composed of materials that we typically perceive as hardincluding, aluminum, stainless steel, concrete, and polycarbonate, displayed a relatively large stiffness value across thewide frequency range. In contrast, those composed of thesoft materials, including urethane rubber, nitrile rubber, cork,
and wood, were found to display less stiffness. The peak
stiffness values differed for each of specimens, especially forsofter materials. However, hard materials displayed similar
dynamic stiffness values in the low frequency range.

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where S = [s1 , s2 , ..., sm ]T , h = [h1 , h2 , ..., hm ]T , and c
are the matrix of dynamic stiffness, the vector of hardness
scores, and a constant value that represents the interceptof hardness scores, respectively. Because of the collinearity
among the explanatory variables, (3) cannot be directly
solved by multiple regression analysis using the generalized
inverse matrix of S. To avoid this issue, using principal
component analysis, we factorized S into a few independentvectors before resolving (3) using those vectors.
B. Principal component analysis of dynamic stiffness
We computed the principal components of the dynamic
stiffness of the fourteen types of specimens. Matrix S ∈Rm×n was composed of the stiffness values at n representative frequencies for m hammering test trials. Twenty-six
points within 40 to 1,000 Hz were used as representative
frequencies with the equal interval of 40 Hz: n = 26. The
hammering test included 140 (14 types of specimens × 10
trials) trials: m = 140. With a varimax rotation applied, S′
was decomposed into the component scores A ∈ Rm×n ofn′ ×nall trials and the principal component vectors B ∈ R
by
S = AB(4)
where n′ < n. The number of principal components wasdetermined to be three such that the cumulative contribution
ratio reached 95%: n′ = 3.

0-40

C. Multiple regression analysis between hardness scoresand dynamic stiffness using the reduced set of explanatory
variables

100
0
0

10
Time [ms]20

Fig. 4. Example of acceleration and force data acquired by performing ahammering test (Hard urethane)

We applied a multiple regression analysis with the principal component scores and subjective hardness scores being
explanatory and objective variables, respectively. For thisanalysis, the subjective hardness scores h was modeled to

39

ǆ 10

7

Stainless
Aluminum
Polycarbonate
Acryl
Wax
Nylon
Brick
Concrete
GraniteUrethane (H)Wood
NitrileCork
Urethane (S)Dynamic Stiffness [N/m]

8
6
4
2
0
0

200

400
600
Frequency [Hz]Fig. 5.

0.5

Regression coefficient [m/N]

Factor loading
0
200

400
600
Frequency [Hz]

800

1000

Dynamic stiffness of the material specimens

Component 1
Component 2

0

800

1000

Fig. 6. Loading vectors of principal components. Loadings are scaled byusing their eigenvalues.

ǆ 10
4

-9

3
2
1
0
0

200

400
600
Frequency [Hz]

800

1000

Fig. 7. Frequency characteristics of the weight function for the subjective
hardness and dynamic stiffness′

be a product of A and regression coefficients z ∈ Rn ×1
and determined by
h = Az + c.

(5)

Using the stepwise method, we selected the explanatory
variables that significantly affect the objective variables. As
a result, two of the three principal components that are
shown in Fig. 6 were found to influence the hardness scores.
Hence, finally, using the scores for these two components,
the regression equation was determined to be
h = A12 z 12 + c(6)where A12 ∈ Rm×2 and z 12 ∈ R2×1 are a submatrix of
A and corresponding regression coefficients, respectively.
Above equation is identical toh = SB −1
12 z 12 + c(7)where B 12 ∈ R2×n included the two principal component
vectors that significantly affected the hardness scores. Here,w = B −1
12 z 12 is regarded as perceptual weights against the
stiffness values for n representative frequencies.
D. Results
Fig. 6 shows the two principal components of dynamic
stiffness involved in the regression analysis. One of the component displayed greater stiffness values at middle frequency

40

range with the peak value being around 300 Hz. The othercomponent included larger stiffness values at high frequency
range around 800 Hz.
Fig. 7 shows the perceptual weights (w) on the dynamic
stiffness values from 40–1,000 Hz. Except for 1,000 Hz,
the weights were positive, which indicates that the greater
stiffness values led to the greater hardness perception. The
weight values differed across the frequency, suggesting thatthe perceptual effects of dynamic stiffness depend on thefrequency. The stiffness values around 300 Hz were mostweighted, and those at high frequencies decreased as thefrequency increased. The constant value c in the regression
equation was c = −1.20.
Fig. 8 compares the hardness scores observed by theparticipants and those estimated by the dynamic stiffnessof objects using the regression model. The correlation coefficient between the observed and estimated scores was
0.93, and the regression model effectively represented theperceived hardness scores.
VI. D ISCUSSION
From Fig. 8, the dynamic stiffness of the object was foundto be an effective predictor for the hardness perceived bytapping. In other words, the hardness perception is likely to
be based on the relationships between the force acted on theRanks estimated by dynamic stiffness2

their reasonable relationships, whereas further studies arenecessary before being conclusive.

Nylon

VII. C ONCLUSION

Aluminum
Polycarbonate

Concrete
Granite0-1

Stainless

Acryl

1

Urethane
(S)

Wax
Brick
Urethane (H)Nitrile
Wood
Cork

-2
-2

-1

0

1

Normalized ranks by observation

2

Fig. 8. Correlation between the observed and estimated hardness ranks.
r = 0.93.

object surface and its surface vibration. In earlier studies
using force or vibrotactile display devices, the reaction
stimuli toward tapping were produced based on the contact
speed or momentum of the virtual hand or tool [1], [2],[3], [4], [13], which was physically correct and perceptuallyeffective in terms of the realism of virtual stimuli. Given
that the momentum change of the virtual hand is identical tothe impulse of the reaction force, it is natural for the humanperception of contact events to be linked with the dynamic
stiffness values that are defined by the ratio of the tapping
force to reactive vibration amplitudes.
The weights on the stiffness values shown in Fig. 7
suggested that the effects of dynamic stiffness depends onthe frequency. This may be due to the human vibrotactile
characteristics that is also frequency-dependent. We shouldnote that although multivariate analysis mathematically finds
the best linkage between the human perception and thephysical properties of the objects, their results do not always correctly reflect the characteristics of human perceptual system. Nonetheless, unequal contribution of dynamic
stiffness across the wide frequency range is reasonable.
The human vibrotactile sensitivity is highest around 250 Hz
and deteriorates above this frequency [12]. At even higher
range, the perception of vibrotactile stimuli is difficult.
Such sensitivity largely agrees with the weights on dynamic
stiffness shown in Fig. 7. The weights were maximum atapproximately 300 Hz near which the vibrotactile perception
is most sensitive. Above 900 Hz in which the vibrotactile
perception is unlikely to operate, the weights were nearlyzero and suggested that the stiffness values have no impact on
hardness perception. These consistency between the characteristics of human vibrotactile perception and the weights ofdynamic stiffness computed in the present study corroborates

To seek for the relationships between the wide-range
dynamic characteristics of an object and the hardness perception by tapping, we linked their dynamic stiffness values
and subjective hardness scores. We found that dynamic stiffness across 40–1,000 Hz effectively estimates the hardness
perception. Their frequency-dependent contributions to theperception can be reasonably interpreted based on the vibrotactile characteristics of humans. These findings may leadto the design of frequency-filters or frequency-weightingfunctions for vibrotactile rendering methods using display
devices whose frequency responses are limited.
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