2015 IEEE International Conference on Systems, Man, and Cybernetics

Effects of Mechanical Parameters on Hardness
Experienced by Damped Natural Vibration
Stimulation
Kosuke HIGASHI∗ , Shogo OKAMOTO∗ , Hikaru NAGANO∗∗ and Yoji YAMADA∗∗ Graduate School of Engineering, Nagoya University, Nagoya, Japan∗∗ Graduate School of Information Sciences, Tohoku University, Sendai, Japan

Abstract—Humans can perceive the hardness of an object
from the damped natural vibration produced by tapping its
surface. This damped natural vibration can be used in haptic
rendering to effectively present a sense of the hardness of an
object. Although its frequency is known to inﬂuence the subjective
hardness, the effects of the viscosity or decay rate have yet to bestudied. We researched the contributions of the mass, viscosity,and stiffness to the hardness perception using a commercial
force display and psychophysical experiments, where only thevibratory displacement was presented. As a result, the stiffness
was found to have a positive effect on the subjective hardness,similar to the usual haptic rendering of an object. The mass ofthe object was negatively correlated with the subjective hardness.
These results indicated that the vibratory frequency was thedominant parameter in determining the perceived hardness,
which agreed with the previous reports on the perceptual effect of
damped natural vibration. In contrast, the perceptual effect of theviscosity depended on the individual. For approximately half theparticipants, the viscosity did not directly inﬂuence the subjective
hardness of an object. Some participants felt that a damped
natural vibration with greater viscosity was harder. However, theother participants felt that it was harder with a smaller viscosity,
which suggested that the decay rate of the mechanical system wasused as a criterion to judge the hardness of an object. Therefore,
the results indicated that the vibratory frequency is a generalparameter that can be used to specify the subjective hardness,whereas the perceptual inﬂuence of the viscosity or decay ratedepends on the individual.
Index Terms—Haptics, Hardness perception, Vibration

I. I NTRODUCTION
When a human taps an object’s surface, they perceive its
hardness based on the vibration, which consists mainly of adamped natural vibration. Okamura et al. [1], [2] demonstrated
that the damped natural vibration elicited the hardness andmaterial perception of a virtual object using a force display. The material perception based on impulsive vibrations
also functions in crinkling fragile objects by foot [3], [4].
Kuchenbecker et al. [5] showed that a more realistic damped
vibration that included multiple frequency components wasmore effective at presenting the hardness. The effective use of
a damped natural vibration realizes the rendering of an impactthat is perceived as a contact with an object harder than what
can be perceived with the force feedback methods based on
the penalty against penetration between two rigid bodies orthe deformation of an object when being pushed [5], [6]. Such
978-1-4799-8697-2/15 $31.00 © 2015 IEEE
DOI 10.1109/SMC.2015.272

vibration also provides the holder of a long object such as abeam or stick information about the point of contact [7], [8].
The damped natural vibration is characterized not only by
the stiffness of the object but also by its mass and viscosity. Therefore, these multiple types of mechanical parametersinﬂuence the hardness perception. However, previous studies
have not investigated their inﬂuence on hardness perception.
According to previous studies [1], [2], [5], [6], [9], it isevident that higher frequencies of vibration lead to greater
perceived hardness values. Meanwhile, the effect of the decayrate, or time constant, has been unclear. As mentioned later,
this depends mostly on the mass and viscosity of an object.
Thus far, the effects of the mass and viscosity of an object
on the hardness experienced as the result of a damped natural
frequency have not been clariﬁed.
This study was performed to determine the contributions of
the decay rate and vibration frequency of a damped natural
vibration and mechanical parameters, including mass, viscosity, and stiffness, to the hardness perception. We modeled
the damped natural vibration as the response of a springmass-damper system with one degree of freedom (DOF).
We then presented damped natural vibrations with different
mechanical parameters using a commercial force display andresearched the relationships between the vibration and hardness perception in two experiments. In the ﬁrst experiment,
participants reported the degree of perceived hardness basedon the magnitude estimation method. We then discussed therelationships between the subjective hardness and mechanicalparameters. In the magnitude estimation method, the deﬁnitionof perceived hardness was not necessarily shared with theparticipants. Hence, we also researched the subjective hardness
without using such a descriptor in the second experiment. We
speciﬁed sets of parameters that produced subjectively equal
perceived hardness values using the method of adjustment.
Based on the results of these two experiments, we analyzed
the contributions of the mechanical parameters to the hardness
perception.
II. DAMPED NATURAL VIBRATION
We deﬁned an object to be tapped as a 1-DOF spring-massdamper system. The damped natural vibration was modeled as
its step response including the inﬂuence of gravity. As shown
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v
m+M

M

Tapping

Force
display

m
x(t)
c

k

k

VirtualsurfacecVibratory
displacement

Fig. 2. Experimental apparatus. The subject gripped the stylus of a force
display and tapped a virtual surface with variable mechanical parameters.
Fig. 1.

TABLE IPARAMETERS OF STIMULI USED IN EXPERIMENT 1

Tapping of object modeled as 1-DOF spring-mass-damper system.

in Fig. 1, in this system, a hand and stylus with a total massof M collides with an object that is characterized by massm, viscosity c, and stiffness k at velocity v(t). The mass andstylus then collectively vibrate as one object with a mass ofm (= m+M ). By superimposing the impulse response causedby the momentum of the contact object M v(t) and the step
response caused by the gravity load M g, the displacement of
the vibration at time t, with t = 0 being the moment of impact,
is represented byx(t) =Mg−t+ Aexp( )sin(ωt − φ)kτv 2 k − vgc + g 2 mA = 2Mk(4m k − c2 )Test Stimulus

(1)
(2)



2m
c
√
4m k − c2
ω=

√ 2m

g 4m k − c2
φ = tan−1
2vk − gc
τ=

Reference Stimulus

(3)
(4)
(5)

where A, τ , ω, and φ are the amplitude, time constant,vibration frequency, and phase offset, respectively. Because ω
and τ rely on m or c as in (3) and (4), the perceptual qualityof the damped natural vibration is characterized not only by
the stiffness of the object but also by its mass and viscosity.
III. E XPERIMENTAL SETUP AND CONDITIONS
A participant manipulated a commercial force display,
Phantom Omni (SensAble), as shown in Fig. 2 and tapped avirtual surface that was located parallel to the table. When thesurface was tapped, the force display, with the stylus controlled
by a PI controller, presented the displacement of the damped
natural vibration represented by (1). The mass, viscosity, andstiffness of the virtual object were variable. The contact speed,which affects the amplitude of vibration was limited to control
the experimental condition, which allowed us to focus on thecontribution of mechanical parameters. If a participant tappedthe virtual object at a speed that was outside the range of
0.15–0.35 m/s, no vibration was presented to them, in orderm [kg]
0.07
0.07
0.01
0.04
0.10
0.13
0.07
0.07
0.07
0.07
0.07
0.07
0.07
0.07

c [Ns/m]
0.60
0.60
0.60
0.60
0.60
0.60
0.30
0.45
0.75
0.90
0.60
0.60
0.60
0.60

k [N/m]
5000
5000
5000
5000
5000
5000
5000
5000
5000
5000
2000
3500
6500
8000

to prevent a large variation in the contact speed among theparticipants and trials. The mass of the contact object (M )was approximated to be 0.1 kg by measuring the weight ofthe stylus held by a participant that relaxed with his elbow
on the table. In the experiments of this study, each participant
held a stylus in their writing hand and placed their elbow on adesk with a cushion between them. A reference stimulus waspresented on the left half of the force display’s work space,and a test stimulus was presented on the right half. Participants
were allowed to freely experience the reference and test stimuli
during a trial. They wore headphones playing pink noise toblock out any auditory cues.
An important point of the experimental design was that theforce display delivered the vibratory displacement deﬁned by(1), which did not include the reaction force of the virtual
plane. No force feedbacks were presented based on the deformation of the plane or the penalty against the penetration of thestylus through the plane. In this way, we could investigate theeffects of the damped natural vibration while decoupling thequasi-static force feedback related to the mechanical propertiesof the virtual object.
IV. E XPERIMENT
A. Experiment 1: Speciﬁcation of perceived hardness by magnitude estimation method
1) Participants and tasks: The experiment involved ﬁvenaive participants that signed a written informed consent. Each1540

participant compared the hardness perceived by the reference
and test stimuli and reported the degree of perceived hardnessof each test stimulus based on the method of magnitude
estimation with the reference stimulus being a modulus. The
thirteen test stimuli were presented in random order in a single
set. A total of three sets were performed for each participant
with a 5-min break between sets.
2) Parameters of stimuli: Table I lists the mechanicalparameters used for the reference and test stimuli. The parameters for the reference stimulus were set to m = 0.07 kg,
c = 0.6 Ns/m and k = 5000 N/m. A value of 10 was
designated as the magnitude of its hardness as a modulus.
The parameters of the test stimuli were varied using rangesof 0.01–0.13 kg, 0.3–0.9 Ns/m, and 2000–8000 N/m for m,c, and k, respectively. The test stimuli included one whoseparameters were the same as those of the reference stimulus.
These parameters of stimuli were determined as to be within
the stably controllable range of the force display. As a resultof these parameter sets, the vibratory frequency, time constant,and amplitude varied within 15–35 Hz, 0.3–0.9 s, 0.8–1.6 mm
respectively. In terms of the frequency, some of the earlier
studies used two–three times higher than our value [1], [2], [3].
Especially, Ikeda and Hasegawa [7] used a 300 Hz vibration
using a high performance haptic display.
3) Data analysis: Trials that ended in a remarkable discrepancy between the measured and set displacements ofthe stylus were excluded from the statistics. The allowablemaximum error was set to 0.5 mm at RMS. Such vibratory
displacement errors might have occurred as a result of theirregular dynamism of a human hand. The reported values ofhardness were geometrically averaged for individuals, and themechanical parameters were normalized to z-scores. Using aleast-squares method, these values were then ﬁtted to a curved
surface H(m, c, k), which presented the subjective hardness
as a function of the mechanical parameters. The function
H(m, c, k) was denoted by
H(m, c, k) = p1 (m − p2 )2 + p3 (c − p4 )2+ p5 (k − p6 )2 + p7

(6)

where pi were regression coefﬁcients. The data acquired
through experiment 1 were well ﬁtted with the sufﬁcient
coefﬁcients of determination (R2 ): 0.83, 0.98, 0.93, 0.98, and0.89 for participants A–E, respectively.
B. Experiment 2: Investigation of subjectively equal hardness
using method of adjustment
1) Participants and tasks: This experiment involved ﬁvenaive participants, two of which also participated in experiment 1. Each participant adjusted the k value of the test
stimulus so that the perceived hardness was equal to thatperceived using the reference stimulus. The k values were
adjusted using a keyboard within a range of 100–10000 N/m,for which the variable pitch was 100 N/m. The eight types oftest stimuli were presented in random order in a single set. A
total of three sets were performed for each participant, with a
5-min break between sets.

TABLE IIPARAMETERS OF STIMULI USED IN EXPERIMENT 2
m [kg]c [Ns/m]
0.030
0.50
0.015
0.50
0.020
0.39
0.020
0.61
Test Stimulus
0.030
0.35
0.030
0.65
0.041
0.39
0.041
0.61
0.045
0.50
Adj.: Adjusted by participants.

Reference Stimulusk [N/m]
5000
Adj.
Adj.
Adj.
Adj.
Adj.
Adj.
Adj.
Adj.

TABLE IIIGRADIENT OF H IN m, c, AND k DIRECTIONS∂H∂H∂HParticipant
∂m
∂c
∂k
A
−1.49
0.49
1.35∗
B
−1.07∗
0.71∗
1.02∗
C
−0.94∗−0.82∗
0.80∗
D−2.07∗
1.66∗
1.92∗
E
−4.12∗−0.43
2.30∗* means signiﬁcant slope at p < 0.05.

2) Parameters of stimuli: The parameters for the reference
stimulus were set to m = 0.03 kg, c = 0.5 Ns/m, and k =5000 N/m. The parameters of the eight types of test stimuli
were set in the vicinity of those for the reference stimulus, as
listed in Table II. For the test stimuli, the m and c values werevaried in the ranges of 0.015–0.045 kg and 0.35–0.65 Ns/m,
respectively. These parameters of stimuli were determined as
to be within the stably controllable range of the force display.
3) Data analysis: As was done for experiment 1, trials that
ended in a remarkable position error between the measured
and set displacements of the stylus were excluded from thestatistics. The sets of parameters acquired from each participantthat were considered to provide subjectively equal hardness
values were normalized to z-scores. These values were thenﬁtted to a curved surface on the m-c-k space using a leastsquares method. The equation of the surface was deﬁned bya quadratic polynomial function of m, c, and k. This function
described the acquired data with coefﬁcients of determination(R2 ) of 0.85, 0.62, 0.94, 0.59, and 0.81 for participants D–H,
respectively.
V. R ESULTS
A. Gradients of H along each mechanical parameter fromexperiment 1
In order to analyze the effects of the mechanical parameterson the subjective hardness, we computed the gradients of H
in the m, c, and k directions in the vicinity of the reference
stimulus for each participant. Table III lists the gradientsof H in the directions of m, c, and k for each participant
in experiment 1. Statistic signiﬁcance in the table is thesigniﬁcance of regression coefﬁcients. In addition, Fig. 3 shows
the curved surface of H established from one participant.
For all the participants, the gradients of H along k werepositive, which indicated that the subjective hardness increased
as the stiffness of the object increased. In this study, the force

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20

1
H 10

k 0
-1

0

1

-2
m

0
2 -2

0
k

2

Fig. 3.
Subjective hardness H as function of normalized m, k, and c.
A sample of participant A in experiment 1 when the c value was ﬁxed at
0.60 Ns/m. H increased as k increased and m decreased.
TABLE IVGRADIENT OF k IN m AND c DIRECTIONS
Experiment 1
Experiment 2
∂k
∂k
∂k
∂k
ParticipantParticipant
∂m
∂c
∂m
∂c
A
0.73∗
−0.17
F
0.94∗
0.14
∗
∗
B
0.97
−0.69
G
0.64
0.07
C
1.08∗
0.92∗
H
0.66∗
0.57∗
∗
∗
D
1.36
−0.74
D
0.43
−0.80∗
E
1.04
0.19
E
0.64
0.10
* means signiﬁcant slope at p < 0.05.

was not fed back based on the deformation of the virtual
plane. Only the damped natural vibration was delivered to theparticipant. Even under this condition, k was found to be thedominant parameter when determining the subjective hardness.
Furthermore, the gradients of H along m were negative forall the participants. As discussed later, these negative gradients
indicated the dependence of the hardness perception on thefrequency of the damped natural vibration.
In contrast, the gradients of H along c varied among
the participants. For two of the ﬁve participants, statistically
positive gradients were observed, whereas one participant hada negative gradient. For the other two participants, not statistically valid gradients along c were exhibited. Such individual
characteristics of c for the hardness experienced as a result of adamped natural vibration are different from those experienced
by pushing a deformable object. In the case of the hardness
perceived by pushing a deformable object, an increase inc simply leads to the increase in the perceived hardness.
However, the effect of c is completely different in the caseof vibration-based hardness perception. To the authors’ best
knowledge, this aspect of human hardness perception has notbeen previously reported.
B. Gradients of k along m and c
Fig. 4 shows the values reported by one participant (participant F) in experiment 2, and the surface of subjectively equal
hardness that was established by the reported values. For thisparticipant, k and m were highly dependent on each other. As
k increased, m also increased. In contrast, c was insigniﬁcantly

0
c

-1

-1

0
m

1

Fig. 4. Surface of subjectively equal hardness in normalized m-c-k space.
The results for participant F in experiment 2.

associated with the changes in the k value.
Given that k is an indicator of physical and perceived
hardness, we computed the gradients of k along m and c in
the vicinity of the reference stimulus for each experiment. For
this computation, the k values of samples in experiment 1 wereﬁtted to a polynominal quadratic function of m, c, and H. The
gradients were then computed around the m, c, H values of thereference stimulus used in experiment 2. Table IV lists these
gradients. Positive gradients of k along m were observed for
all the participants. In other words, to maintain the subjective
hardness, k and m varied in the same direction, which isin line with the idea that the hardness perceived by damped
natural frequencies is inﬂuenced by its vibratory frequency,
as discussed later. In contrast, the signs of the gradients ofk along c varied among the participants. The gradient wassigniﬁcantly positive for two participants, but it was negative
for two participants for both types of experiments. The otherparticipants did not exhibit any statistically signiﬁcant gradients. Such an individuality for the effect of c was consistentin the two experiments.
VI. D ISCUSSION
As listed in Table IV, the gradients from participants D
and E were consistent between the two types of experiments.
Thus, we could reach similar conclusions in both types ofexperiments.
A. Signiﬁcant effect of vibratory frequency on hardness perception
A positive gradient of k along m indicated the dependenceof the hardness perception on the frequency of the damped natural vibration. As shown in (4), the frequency is substantially1
proportional to (k/m) 2 . The frequency remains constant whenincreasing m and k simultaneously. Such a hardness perception
based on the vibration frequency is consistent with the resultsof previous research [1], [2], [5], [6], [9], where a higher
frequency led to a greater subjective hardness. Furthermore,this effect of the frequency was rigorous for all the participants,
because the positive gradients were statistically signiﬁcant formost of the participants.

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B. Individual effect of c or decay rate on hardness perception
The gradients of k along c varied among the participants.
Hence, we could not reach a general conclusion about therelationship between k and c or the effect of c on the perceived
hardness.
For two and three of the ﬁve participants in experiments 1
and 2, respectively, the gradients were insigniﬁcant, which indicated that c was not signiﬁcantly related to k. This indicatedthat for some participants, the subjective hardness based on thedamped natural vibration did not rely on the viscosity of theobject, or the hardness originating from k qualitatively differedfrom that originating from c.
For one participant in each type of experiment, the gradientof k along c was signiﬁcantly positive. For these participants,
the decay rate might have become a cue for hardness perception because the decay rate of 1-DOF spring-mass-damper1system is affected by the ratio of c to k 2 as follows:
c
.
(7)
ζ= √
2 mk
For the other participants, the gradient of k along c wasnegative, suggesting that an increase in c led to a greater
subjective hardness. This was because the negative gradientmeant that the increase in c was needed to compensate for thedecrease in k to maintain the subjective hardness. This appears
to be in line with the hardness perception of a deformableobject by pushing, where an increase in c also increases
the perceived hardness. For these participants, the damped
natural vibration might have been linked with the mechanical
impedance of the object, and the mechanical impedance played
a greater role in the hardness perception rather than the decayrate.
In summary, although the reasons remain to be studied, achange in the value of c inﬂuenced individuals differently.
This indicates that we should not expect the decay rate tohave similar effects on individuals in terms of the perceived
hardness.

of the stiffness and mass of the virtual object were prominentand shared among the participants. An increase in the stiffness
resulted in an increase in the perceived hardness. An increasein the mass led to a decrease in the perceived hardness. These
results support the conclusion that the vibratory frequency
strongly inﬂuences the hardness perceptions, which is consistent with the results of previous studies.
In contrast, individual differences were observed in theeffects of the viscosity of the virtual object. For approximately
half the participants, changes in the viscosity were not accompanied by changes in the perceived hardness, suggesting an
insigniﬁcant effect of the viscosity on the perceived hardness.
For some participants, an increase in the viscosity decreased
the subjective hardness. Such an effect makes sense, considering that the decay rate plays a role as a perceptual cue.
For the rest of the participants, an increase in the viscosity
increased the subjective hardness, which is in line with theperceptual phenomenon held for deformation- or penetrationbased object rendering. Therefore, the effects of the viscosity
could substantially vary across individuals.
Because of the limited abilities of the commercial force
display used in this study, these results were based on a narrow
range of vibratory stimuli. For more general aspects, further
studies with wider range of vibratory stimuli and more participants are necessary. Nonetheless, to summarize the resultsof this study, the hardness presented by a damped natural
vibration could be adjusted by changing the frequency of thevibration because its perceptual effects were prominent forthe majority of those who experienced the stimuli. However,
the effect of the decay rate or viscosity on the hardness
perception may vary among people, whereas its effect onmaterial perception is not denied.
ACKNOWLEDGMENT
This study was in part supported by KAKENHI Shitsukan
25135717 and 15H05923.

VII. C ONCLUSIONSR EFERENCES

A hardness presentation based on a damped natural vibration effectively delivers the sense of hardness. Furthermore,this method is applicable not only for force displays but also
for vibrotactile displays. Hence, it is expected to become ageneral technique and be widely used for mobile devices.
In general, the vibratory frequency is known to regulate thehardness experienced as a result of a damped natural vibration.
However, the effects of other types of mechanical parameters
have yet to be determined. We investigated their contributions
using a commercial haptic display. In the experiments, thedamped natural vibration caused by the contact between twobodies was presented, and no force feedback related to thepenetration of the tip of a stylus to the virtual object or
its deformation was conveyed. In this way, we could purely
investigate the perceptual inﬂuence of the vibration.
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[7] R. Okazaki and H. Kajimoto, “Altering distance perception from hittingwith a stick by superimposing vibration to holding hand,” Haptics:
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