Roughness-Induced Vibration
Caused by a Tangential
Oscillating Mass on a Plate
Joachim Feldmann
Technische Universität Berlin,
Einsteinufer 25,
D-10587 Berlin, Germany
e-mail: [email protected]
This work examines the dynamic behavior of a system consisting of a mass-block on the
rough surface of a simply supported plate, harmonically excited in the tangential direction. The vertical excitation emerges from roughness, tracked by the mass-block. Low-frequency sliding results in high-frequency vertical excitation up to the ultrasonic range.
The conditions of the elastic contact between the two bodies are modeled in the form of
vertical contact stiffness. A specific friction law with a behavior similar to an elastically
coupled coulomb damper represents the tangential direction. The model allows for the
study of the interaction between the tangential friction behavior and the vertical
roughness-induced vibrations. Parameters of interest are friction velocity, mass-block
weight, surface roughness, and contact material. Because of nonlinearities, the theoretical model must be solved within the time domain. The theoretical results are verified
through experimental results of a corresponding setup. The subject combines material
science, contact mechanics, and structural dynamics. [DOI: 10.1115/1.4005828]
Keywords: sliding friction, rough contact, oscillating mass-plate system, high-frequency
excitation
1
Introduction
This investigation was inspired by the well-known fact that rolling contact of a wheel on a rail generates frequencies in the ultrasonic range [1]. Contact low-pass filter effects [2] prevent direct
displacement generation of ultrasonic frequencies through surface
roughness. The hypothesis is that the excitation of such high frequencies originates from sliding friction components in the contact patch between wheel and rail in conjunction with well-known
sinusoidal tracing. Consequently, if this hypothesis holds true,
there must be feedback to the friction conditions, because the
influence of high-frequency vibrations on friction and vice versa
is well known in the industrial manufacture of metal drawing
products [3,4]. Vibration can decrease the static friction and
increase the potential for sliding, impacting stability.
The other cause of ultra-high frequencies could be impulse excitation during rolling. This assumption, for instance, offers various approaches from literature [5–7]. The author’s investigation
[8] could not clearly verify impulse excitation as a main vibration
mechanism in purely rolling contacts, especially with respect to
the amount of vibration generated.
To ensure a more controlled situation for this investigation, the
assumed effects were completely separated from any actual engineering system. A basic laboratory experiment consisting of a
low-frequency, externally excited gross sliding mass-block on a
statistically rough plate was conducted. This configuration was
examined both in theory and practice. The parameters were sliding velocity, load, surface roughness, and contact material.
Numerous approaches from different points of view document
the attempt to resolve the issue of roughness-induced-vibration.
The issue concerns many technical problems. It originates in questions pertaining to friction, wear, and lubrication. Many aspects
were investigated to obtain important parameters in connection
with topography of roughness or surface interaction. Most problems are quasi-static. With respect to noise generation, however,
Contributed by the Design Engineering Division of ASME for publication in the
JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received November 5, 2010; final
manuscript received December 6, 2011; published online May 29, 2012. Assoc.
Editor: Philip Bayly.
Journal of Vibration and Acoustics
the impact of system dynamics along with the coupling conditions
of contacting bodies cannot be overlooked. An excellent survey of
the entire topic can be found in Ref. [9]. Actual aspects tangent to
the subject, which also delve into questions down to the atomic
scale of roughness can be found in Ref. [10].
Engineering often treats friction force in a global sense, as a
function of the relative velocity between the sliding bodies. On a
smooth, plain contact, this approach is primarily a purely tangential issue. This fact was investigated in detail from various angles;
for example, in Refs. [11–13]. Important aspects pertaining to the
vertical behavior of a rough frictional contact can be found in
Refs. [14] and [15]. The model described herein tries to connect
three subjects: the vertical system dynamic, the tangential frictional aspects, and a free displacement input based on an elastic
rough surface contact.
2
Model Description
2.1 Governing Equations. The model consists of a flat, rigid
mass that slides on a rough, flat rectangular plate, Fig. 1. An external harmonic force excites the mass in the tangential direction.
The model has both a vertical and tangential system dynamic,
along with the contact conditions between the two bodies. The
two objects are metal.
The following conditions are introduced: (i) the dry surfaces of
the contacting bodies are broken-in, i.e., contact deformations in
the plastic regime can be neglected; (ii) the resulting roughness of
both surfaces is concentrated on the plate as an arbitrary input
quantity into the system, a special process reduced the primary 3D
contact in two dimensions; (iii) the tangential dry frictional contact is described by a law similar to a type of elastically connected
coulomb damper; and (iv) the elastic vertical contact is modeled
through separate and equal springs. The model was further
reviewed for the following aspects: (v) extension in both the input
momentum mobility and tangential input mobility of the plate;
(vi) development of the vertical plate mobility with respect to the
overall size of the nominal contact patch; and (vii) consideration
of actual multipoint contact. These additional extensions do not
C 2012 by ASME
Copyright V
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Fig. 1
Scheme of the investigated theoretical model: i 5 actual contact point
show a real benefit compared with the experimental results, and
are not documented in this paper.
The time-dependent system equations are (one dot on the displacement denotes a derivative with respect to time d/dt , and two
dots denote d2 /dt2 ):
X
Fstat m n€mass r1 n_mass Fi ¼ 0
(1)
i
Equation (1) defines the balance of all vertical forces, with the
static load Fstat ¼ m g (g acceleration of gravity), and the vertical
single spring contact force.
Fi ¼ si Dni HðDni Þ
(2)
Dni is the vertical displacement difference between mass and
plate.
Dni ¼ nmass nplate;i nrough;i
(3)
n is the time-dependent displacement in the x-direction.
The symbol R denotes the possibility of considering more than
one contact point, depending on the number of mean contact
points in the nominal contact surface, information which is
derived from a subprogram referred to “contact module” (mentioned below). In this case, si ¼ stotal =i simply refers to a portion
of the total contact stiffness, where i is the number of contact
points, which should be considered.
HðDnÞ is the well-known Heaviside function, which formally
prevents the mass from being lifted in the event the difference Dn
becomes negative. This condition results in a nonlinearity of the
system. The vertical displacement of the plate is calculated by
convolution of its impulse response (also known as Green’s function) with the corresponding contact force
nplate;i ¼ Fi gplate;i
(4)
The impulse response of a simply supported plate is applied [16]
4
a b m00
1 X
1
X
sin an x1 sin bm y1 sin an x2 sin bm y2
xnm
n
m
with its natural frequencies
041002-2 / Vol. 134, AUGUST 2012
and
an ¼
np
;
a
bm ¼
mp
E h3
; B00 ¼
b
12ð1 t2 Þ
x1 ; y1 ; x2 ; y2 are the coordinates of excitation and response, respectively; a; b are the length and width, respectively; m00 being the
weight per unit area; B00 being the bending stiffness per unit area;
h is the thickness; E is the elasticity modulus; t being the Poisson
ratio; and dp denotes the loss factor. For multi-contact, i, a set of
impulse responses has to be calculated according to both the corresponding contact point and response coordinates.
Figure 2 provides an example of the impulse response applied,
showing both time dependency and corresponding frequency
transform (spectrum represents the relation acceleration to force).
Here the response is related to the entire nominal contact area of
the mass-block, which results in a decrease of the high-frequency
amplitudes.
The tangential force balance of the system is
Texc X
Ti þ m g€mass þ r2 g_ mass ¼ 0
(5)
(6)
i
with the tangential excitation force
Texc ¼ T0 cosðxtÞ
(7)
and the single friction force related to the corresponding vertical
contact force
Ti ¼ lðDg_ i Þ Fi
(8)
The tangential velocity difference between mass and plate is
Dg_ i ¼ g_ mass g_ plate;i ffi g_ mass
gðx1 ; y1 ; x2 ; y2 ; tÞi ¼
sin xnm t edp xt
xnm
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
B00
¼ ða2n þ b2m Þ2 00
m
(9)
g is the time-dependent displacement in the y direction.
Compared with the mass, the mobility of the plate in the
purely tangential direction is quite low so its impact was
disregarded.
The applied friction law [17], comparable to that of an elastically connected coulomb damper [11], has the form
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Fig. 2 Time dependence (a), and corresponding spectrum (b) of the plate’s applied impulse response with boundary conditions “simply supported”
_ g_ hyst Þ
_ g_ hyst Þ
_ ¼ lslip þ ðlstick lslip Þ ea1 ðDgþ
lðDgÞ
lstick ea2 ðDgþ
for ðDg_ þ g_ hyst Þ > 0;
_ g_ hyst Þ
_ g_ hyst Þ
_ ¼ lslip ðlstick lslip Þ ea1 ðDgþ
þ lstick ea2 ðDgþ
lðDgÞ
for ðDg_ þ g_ hyst Þ < 0
(10)
lslip , lstick being the well-known quasi-static friction coefficients
of slip and stick, respectively, a1 and a2 are corresponding constants, and g_ hyst is a consistent amount of hysteresis. In the calculation scheme, one must distinguish between the two cases
Dg_ 0 and Dg_ < 0. This law has the advantage of being wellsuited with respect to handling the solution procedure. For example, Fig. 3 shows the friction law of Eq. (10) with a friction velocity of 10 mm/s rms.
The wanted unknown quantities are vertical contact force, vertical displacement of both mass and plate, friction force, and tangential velocity of mass. Quantities with controlling character are
the actual roughness inputs at any contact point. The necessary
five equations for each unknown can be extracted from the above
Eqs. (1)–(10). After discretizing time, the system was solved on
the basis of the well-known Newton-Raphson method including
the corresponding derivatives of each equation to each unknown.
The resultant matrix system was solved according to Lorentz unit
(LU)-decompostion. The selected time increment was 105 s,
which corresponds to the maximum frequency range of 50 kHz. In
Fig. 3 Friction versus friction velocity, exemplary for 10 mm/s
rms, one complete cycle
Journal of Vibration and Acoustics
conformance with the measurement situation, the results are illustrated either in time-dependence or as frequency spectrums. Vertical amplitudes of both mass and plate represent acceleration instead
of displacement. The simulated excitation frequency was 7 Hz.
2.2 Roughness Input. The free vertical displacement input
of the system ensues from the resulting roughness of both the
plate surface and the oscillating mass-block surface on it. In the
actual case, both surfaces have roughly the same statistical properties so that the entire roughness can formally be focused on one
surface with a corresponding larger standard deviation [18]. To
obtain near actual data for the model input, the roughness of the
plate used in experiment was measured and specially treated. The
plate had a dimension 0.3 m 0.26 m. The wavelength range of
interest spanned from 50 106 m to 0.04 m, the latter measure
being that of the maximum mass-block dimensions. In principle,
the entire contact could be treated completely threedimensionally, for an example, see Ref. [19]. But for this application, it was quite sufficient to treat the contact two-dimensionally.
To this end, parallel traces were recorded across a width corresponding
to the quadratic mass-block with spacing according to
p
ffiffiffi
3
2-law. The length of every trace was 0.16 m, and the radius of
detection was 5 106 m. This discrete 3D recording was followed by reducing to 2D profile according to Ref. [20], also refer
to Ref. [21]. In this approach, originally developed through the
aerodynamics of wings, the cross correlating mean summit height
of all parallel traces over sections perpendicular to the profile length
are calculated by span, here at a resolution interval of 10 106 m.
The described procedure was verified experimentally. As it turns
out, compared to an original reference roughness profile, the reduction of one dimension shows an amplitude decrease of roughly 20
dB without an essential change in frequency content. Additionally,
the obtained 2D profile was functionally filtered based on the length
dimension of the mass. In principle, the basis of such filtering is a
Si-function [18]. Section 4 (Fig. 5) shows the spectrum of the
applied input roughness profile.
The main calculation program required a correction through an
interpolation procedure between the two different increments pertaining to the interface of mass-block position g_ mass Dt (here time
resolution Dt ¼ 105 s) and roughness profile (here space resolution Dx ¼ 10 106 m). This measure has a significant impact on
the high-frequency level of the acceleration spectrums of both
plate and mass.
2.3 Vertical Contact Stiffness. Because of rough contact, a
subprogram was used to calculate the vertical contact stiffness
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as simply supported. The quadratic mass-block measured
0.04 m 0.04 m, and two weights, 0.221 kg and 0.543 kg, were
used. For both objects, the contact material used was regular steel.
The contacting surfaces had statistically equal roughness properties. They were first ground through a manufacturing process to
ensure flatness of the contact and allowing a very precise parallel
sliding condition without momenta. A second step produced a
defined, statistically distributed roughness through sandblasting.
The pivotal amplitudes were within a range of 1–10 lm. Two
characteristics comparable with sandpaper grit of 80 and 40 were
used as per ISO 6344.
Using a very stiff yet light carbon rod, the mass-block was
externally excited by an electro-dynamical shaker with a frequency of 7 Hz. The entire setup was arranged on a flat, heavy optical table made of granite so the experiments could be carried out
under highly controlled conditions, Fig. 4.
Fig. 4 Test setup: mass-block on a rough steel plate, external
shaker excitation
and both the number and location of mean contact points in the
nominal contact patch. Refer to [8] for the primary theoretical
aspects. Calculations are based on the well-known Hertzian nonlinear contact theory (for example, refer to Ref. [22]) along with a
statistical contact approach developed primarily by Greenwood
and Tripp [23]. The calculated contact stiffness is referred to as
incremental as it exhibits the gradient of the contact force and the
elastic deformation with respect to a specific working point by the
static load. In principle, the contact stiffness depends on: the static
load, the Young’s modulus of the contacting material, the contact
geometry, the standard deviation of the probability distribution of
summit heights, and the radius of the mean spherical summit caps.
The property of the stiffness is that of a sum of single, statistically
equal springs, whose stiffness is summarized. So the total stiffness
is proportional to the number of contacts, a property termed
mechanically parallel. That means the smoother the contact, the
higher the number of contacts and the stiffer the contact.
3
Experimental Setup
As mentioned above, the test setup consisted of a steel plate
measuring 0.3 m 0.26 m with a thickness of 0.003 m. With
respect to the lengthwise dimension, both sides were supported
elastically, and the other two sides were unsupported. In the calculation model, the boundary conditions of the plate were treated
4 Theoretical and Experimental Results: Comparison
and Discussion
4.1 Actual Roughness Input. Dependent on the displacement stroke in tangential oscillation amplitude, a contact point
only ”sees” a corresponding section of the overall surface roughness profile. This means additional nonlinear wavelength filtering depending on the tangential frictional stroke (velocity
divided by the oscillation angular frequency). This is a highly
significant effect, which research thus far had not yet focused on.
For a certain time, every contact point has its own share in the
roughness profile, all parts are completely superimposed with
respect to the system input, the same as the four wheels in vehicular excitation on an uneven road. Because this sum is determined by the contact points on the highest profile maximums,
the contribution of the other contact points can consistently
decrease. This could be the reason why a one-point contact, at
the highest maximum of the profile, is more or less representative. The difference from the example of a forward-moving car
is that in this case, the oscillation repeats the corresponding
inputs. In the model, the ten highest profile maximums are used
for ten corresponding contact points. The locations of these maximums are generated by the sub-module and transferred into the
main program, as mentioned above. The number of contacts to
be considered depends on the macro-structure of the roughness
profile. Figure 5 shows wave-number spectrums of the original
2D-reduced functional filtered global input profile (thin line)
along with the actual input at one identical contact point. The
Fig. 5 Wavenumber spectrums of mean original roughness input (thin line), and actual roughness input (thick line). (a) Oscillation stroke 6 0.227 mm rms, (b) oscillation stroke 6 2.27 mm rms. Values result from friction velocity 10 mm/s and 100 mm/s,
respectively, divided by the excitation angular frequency 6.28 3 7 Hz. Results based on measured and processed 3D
roughness.
041002-4 / Vol. 134, AUGUST 2012
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Fig. 6
Tangential velocity (10 mm/s rms) of mass-block versus time: (a) calculated, (b) measured
Fig. 7
Tangential velocity (100 mm/s rms) of mass-block versus time: (a) calculated, (b) measured
two applied friction velocities are (i) 10 mm/s, and (ii) 100 mm/s
rms. As shown, the actual excitation input clearly depends on the
stroke of the oscillating mass, whereas the original input roughness is not impacted.
4.2 Tangential Friction Velocity. The tangential velocity
measured on the mass-block is roughly identical to the relative
friction velocity provided there are no considerable amplitudes in
tangential direction in the plate, compare Eq. (9). According to
the respective friction property, as shown in Fig. 3, the tangential
velocity shows typical slip and stick regions where the stick areas
diminishing more with increasing tangential excitation amplitude,
cf [11]. Both Fig. 6 and Fig. 7 are exemplary illustrations of the
behavior: theoretically for the two velocity amplitudes 10 mm/s
rms and 100 mm/s rms (Fig. 6(a) and Fig. 7(a)), versus the corresponding measured results (Fig. 6(b) and Fig. 7(b)).
4.3 Friction Relations and Properties. Aside from the wellknown material dependencies, the examined system friction is
impacted by the vertical surface roughness input, and the vertical
system response, respectively. In Ref. [24], the questions were
investigated based on the relation between the dynamic ratios of
tangential and normal contact force, a relation that yields the
dynamic portion of the friction coefficient. An interesting theoretical statistical approach was developed in Ref. [25], where friction
is determined based on the oblique micro-contact patches of
Journal of Vibration and Acoustics
Fig. 8 Calculated relation friction force versus friction velocity,
exemplary of 10 mm/s rms
matching roughness asperities in contact, where the coupling
between the two directions ensued automatically. Such
approaches are not compatible with this more globally aimed
model.
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Fig. 9 (a) Friction force versus tangential displacement. (b) Derivation of the curve in (a), result is equivalent to the global
tangential stiffness of the contact. Example of a sliding velocity of 10 mm/s rms over 2.5 periods, compare Fig. 6.
Fig. 10 Calculated vertical acceleration amplitudes versus time. (a) Result for mass-block, and (b) result for plate. Friction
velocity is 10 mm/s rms.
The interaction between friction force and normal force leads to
a corresponding dynamic part in the friction law, as illustrated in
Fig. 8, exemplary of a friction velocity of 10 mm/s rms. This
dynamic clearly appears during the sliding state, not while
sticking.
From the relation friction force versus tangential displacement,
Fig. 9(a), one also can deduce a sort of global tangential contact
stiffness by derivation, Fig. 9(b). As shown in each stick phase,
the stiffness jumps to extremely high values, whereas during the
sliding state the stiffness tends toward zero, superposed by
dynamic deviations caused by rough contact interaction (it should
be noted that in Fig. 9(b), the value at the coordinate scale is cut
out to show the high-frequency deviations at the zero line). In
practice, the time-dependent impulse-type stiffness steps could be
the cause of damped resonance vibrations in the tangential
response; determined by the mass and the resulting tangential
stiffness, such vibrations diminish during the slip state. Potential
contact interactions with the quasi-static parameters of the applied
friction law are not considered. In an earlier paper [26], it was
found that in an oscillating case, the kinetic coefficient of friction
can be roughly 40% higher than its quasi-static value, regardless
of the sliding velocity. The significance of such behavior could
not be observed in this experiment. It appears the friction law
itself not directly being sensitive with respect to contact condi041002-6 / Vol. 134, AUGUST 2012
tions is not a shortfall of the model, with the exception of the typical material dependencies. Admittedly, Ref. [27] (see Ref. [28])
provides a strong and constant relationship of the tangential stiffness to the vertical stiffness determined only by the Poisson number; but such a relation only holds true in connection with microslip in a contact patch, not for sizable sliding where the tangential
stiffness completely breaks down in contrast to the corresponding
vertical stiffness.
4.4 Vertical System Responses. Figure 10 and Fig. 11 are
exemplary of calculated time plots of both mass and plate vertical
accelerations of the two selected friction velocities 10 mm/s rms
and 100 mm/s rms. The slip and stick regions are clearly noticeable, the latter leading to less vibration as no relative tracking
occurs between the rough surfaces. Notice that the ordinates are
scaled differently. The random character of the signals are caused
by the roughness excitation. Both theoretical and measured results
broadly concur as shown in a comparison with the corresponding
Fig. 12 and Fig. 13. The absolute amplitudes show moderate differences based on higher frequency amplitudes in the theoretical
results. In a level presentation in dB, these differences will not be
as significant, refer to the spectrums illustration below. The vertical system responses indicate no instabilities such as squealing or
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Fig. 11 Calculated vertical acceleration amplitudes versus time. (a) Result for mass-block, and (b) result for plate. Friction
velocity is 100 mm/s rms.
Fig. 12 Measured vertical acceleration amplitudes versus time. (a) Result for mass-block, and (b) result for plate. Friction
velocity is 10 mm/s rms.
Fig. 13 Measured vertical acceleration amplitudes versus time. (a) Result for mass-block, and (b) result for plate. Friction
velocity is 100 mm/s rms.
chaotic behavior. The present frequencies are mainly determined
by both the contact resonance and the resonance frequencies of the
plate, Fig. 14. The fact that there are no instabilities confirms prior
investigations on the suppression of such effects through interacting
Journal of Vibration and Acoustics
contact vibrations [29,30]. Further, it is known that instabilities are
forced if such a system is excited near the contact resonance, externally or in coincidence with a natural frequency of a substructure.
Those conditions are not met in the investigated system.
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Fig. 14 Calculated vertical acceleration spectrums of mass (thin line) and plate (thick line). (a) Results for a friction velocity of
10 mm/s rms, and (b) results for a friction velocity of 100 mm/s rms. At high frequencies, the responses of the mass are limited
by a simulated electrical noise of the corresponding measurement device, compare Fig. 15.
Fig. 15 Measured vertical acceleration spectrums of mass (thin line) and plate (thick line). (a) Results for a friction velocity of
10 mm/s rms, and (b) results for a friction velocity of 100 mm/s rms
The contact resonance is a function of the vertical contact stiffness in conjunction with the resulting masses of block and plate.
The corresponding frequency results in a global maximum in the
acceleration spectrums. Below the contact resonance, both mass
and plate behave roughly the same, they are coupled. Above the
contact resonance, vibration amplitudes on the mass diminish
gradually. In the reading, the responses are limited by the electrical noise of the measurement device. The plate response, however, shows its typical resonances over the entire frequency
range, limited only by the increasing impact of the mechanical
damping. Also refer to the spectrums illustration below. In
theory, the vertical relative vibration behavior between mass and
plate remains largely constant and is only determined by the
relation of the corresponding individual mobilities of the two
contacting bodies, except for some influences of friction velocity. Compared with the vibration amplitudes of the mass, the latter effect results in higher vibration amplitudes in the plate based
on increasing decoupling.
The vertical contact stiffness tends to decrease with increasing
friction velocity, a fact that can be observed by the contact resonance shifting toward lower frequencies at higher sliding velocities. Such behavior can be explained with increasingly tracking
of the mass-block on the peaks of surface roughness brought on
by a sort of elevation effect. The number of contacts decreases;
041002-8 / Vol. 134, AUGUST 2012
thus, the stiffness diminishes. The observed dependency results in
a factor between 1 and 0.33 for a sliding velocity range between
10 mm/s and 100 mm/s, respectively. Figure 15 shows the measurement results. The correlation between theory and experiment
is quite well; deviations are found in the upper frequency range
where the theoretical model overestimates the amplitudes. This
effect could be reduced by introducing a more suitable impulse
response for the plate. Additional effort in modeling the actual
contact conditions can also yield greater concurrence. But the
results suffice with respect to the objective of investigating the
principal behavior of the selected system.
5
Conclusions
Overall, the theoretical model developed with its simplification
is an astoundingly good representation of the experimental reality.
Therefore, the model is a useful tool for parameter studies. The
model shows the significance of roughness input, which is responsible for exciting the vertical system response with its resonance
frequencies. The model shows that low-frequency frictional contact in the tangential direction can result in vertical ultrasonicfrequency system vibration, depending on the density of natural
frequencies. It also illustrates the interaction of the vertical
dynamic with the tangential system behavior.
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The investigated parameters such as friction velocity, massblock weight, or surface roughness yield the following global
results: (i) the velocity exponent for the total vertical acceleration
levels of both plate and mass is between 3 and 3.5, roughly a
9-dB- to 12-dB-level increase for each time the friction velocity is
doubled under constant remaining parameters; (ii) increasing vertical load results in both an increase of the number of contacts
with an increase of the vertical contact stiffness, and an increase
of vibration amplitudes of mass and plate; and (iii) an increase of
the rms value of the surface roughness amplitudes results in a
decrease in the number of contact spots and therefore in a softer
vertical spring and higher vertical response amplitudes. Detailed
results of an extended parameter study shall be published later on.
Acknowledgment
The paper was written in memory of Professor Manfred Heckl,
teacher and principal to the author. The author also thanks Professor Wolfgang Kropp of Chalmers in Göteborg for his many ideas
and his outstanding support over an extended period.
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