Acoustic loss at substantial ultrasonic strain
in
6Al6V2Sn and
sintered
6Al4V titanium
David Wuchinich[1]
ã David
Wuchinich 2004
Keywords: Q, attenuation, dissipation, fatigue, titanium,
ultrasonic, sintered, powdered metal, stress
PACS code:
74.75.Ld
The mechanical Q of the high
strength 6Al6V2Sn titanium alloy and of two alloys of sintered 6Al4V is
measured and compared with values obtained for standard cast and rolled 6Al4V
at 20 kHz cyclic strains ranging for 0.18 to 0.42 percent. 662 was found to exhibit comparable loss
to the standard samples while the sintered specimens revealed much lower
dissipation. Annealing of one of the
sintered samples reduced the Q to those found typical for cast and rolled
material. The 662 material survived
continuous operation at strains in the neighborhood of 0.4 percent. It appears that sintering may provide an
economical process for making ultrasonic horns and that intensities greater
than can typically be produced by use of 64 Ti may be available using 662Ti.
Mechanical dissipation
resulting from hysteresis in the stressstain relationship in many metals has
been studied extensively in relation to fatigue and the operational life of
components subjected to cyclic stress at ultrasonic frequencies because this
loss limits the maximum excursion that can reliably be produced[2]. As the alloy 6Al4V titanium is used
extensively in ultrasonic welding and surgical horns, and is often subjected to
strains in the range of 0.25 percent during operation, Mason’s exemplary method
and comprehensive studies have provided a guide for fatigue safe design for
this alloy[3].
A complete treatise of his methods has also been published[4]. Additional studies of similar alloys have
produced comparable results[5]: The mechanical Q is constant for strains up
to 0.25 percent, thereafter decreasing rapidly. In this respect, this alloy of titanium is distinguished among
other metals employed as horns, such as steel and aluminum, both of which exhibit
progressively greater loss with increasing strain[6]. A strain of 0.25 percent corresponds to a
stress of 276 Mpa (40 kpsi) and it is this value that has been adopted as the
practical endurance limit for titanium horns operating at ultrasonic frequencies.
Horns and ultrasonic
surgical tips have commonly been made by machining rolled stock taken from
castings. Recently, sintered versions
of the alloy have become available[7]
and offer substantial economies in the manufacture of large quantities of
identical items as the process results in horns having near to net finished
shape from molds. Sintering, however,
may result in porosity which, if occurring in regions subject to substantial
strain, can cause stress concentrations that precipitate failure. Hence, an acoustic evaluation of this
material’s tolerance of large strains is of interest.
Also of interest in this
study is the performance of the titanium alloy 6Al6V2Sn, known for having
higher strength than 6Al4V and which has become available to users in common
rolled bar form in less than mill run quantities.
The half wavelength
resonator utilized by Mason and which closely resembles in action the classical
model of simple harmonic vibration utilizing two opposing and equal masses
connected by a massless spring is a simple candidate for evaluation of
acoustic loss. A schematic
representation of the structure is shown in Figure 1.
Figure 1 Test specimen geometry
This model is chosen for two
principal reasons: (1) stress is
confined to the spindle portion of the resonator and is substantially uniform
in this region and (2) the abrupt transistion in cross sectional area between
the mass and spring elements permit the production of large cyclic stress for
modest and easily produced free end amplitude excursions. As elastic loss is often a function of the
peak stress level, confinement and uniformity of stress permits a reliable
measurement of loss. Production of
stress levels likely to reveal substantial losses without resort to excitation
by high intensity transducer and horn amplifiers is also an advantage.
Exact and finite element
analysis reveal that indeed, for the first extensional resonant mode, the end
sections with the larger diameter move uniformly in opposite directions and
that the slender center section is subjected to substantially uniform strain
with a frequency approximately given by
_{} (1)
with E representing
Young’s modulus.
The stored energy of
vibration, E_{s}, can be computed as:
_{} (2)
where s is the peak stress and V the volume. As the stress is principally located within
the slender stem of length l whose cross section, A, is constant, Eq.
(2) becomes:
_{} .
(3)
V = Al, the
volume of the spindle. The exact value
can be computed from finite element modal analysis or from solutions for the
wave equation for a model consisting of connected and appropriately sized
prismatic steps[8] , but Eq.
(3) is sufficiently accurate for computations within fifteen percent of the
true value if the geometry is chosen so that the stress is predominantly
present in the stem.
The stress can be computed
from optical measurements of the displacement of the end mass using, for
example, a microscope with a calibrated reticule as:
_{} (4)
where d is the excursion (peak to peak motion of either end
mass) and l is the length of stem having a constant diameter.
Computation of the acoustic
loss at any given level of vibration can be made by measurement of the
temperature rise in the stem over a prescribed duration of vibration. The rate of heat generation in the spindle
can be written as:
_{} (5)
where Q is the actual
heat contained, Q_{m}_{ } is the heat generated by mechanical loss and Q_{l}
is the heat lost by conduction and radiation.
Measurement has shown that Q_{l}
can be accurately represented as
_{} (6)
where K is the heat
loss coefficient and T is the difference between the spindle and
ambient temperature. K can be
found by measuring the fall in temperature once vibration ceases (dQ_{m}/dt
= 0). Since
_{}
with r the material density, C the specific heat
capacity and V the spindle volume, it follows that:
_{}
from
which
_{} (7)
where
the temperature falls from T_{1} to T_{2}_{ } over an interval of time t_{d}.
Knowing
K, equation 5 may be integrated and solved for Q_{m:}
_{} (8)
Actual
measurement has shown that the rise in temperature during vibration is closely
proportional to the time so that T may be approximated as;
_{}
where
t_{b} –t_{a} is the interval of time over which
the temperature rises from T_{a} to T_{b} . Equation 8 can then be written as:
_{} (8a)
The energy dissipated per cycle of vibration, E_{l},
is then
_{} (9)
_{}
where ¦ is the
frequency of vibration
Q
is defined as the ratio of the energy stored to that lost per cycle multiplied
by 2p, and becomes, using the
expressions for E_{s} given by equation 3 and E_{l} by equation 9 as:
_{} (10)
Two samples of sintered
6Al4V titanium were obtained in the form of rough cylindrical blanks. Sample (#1) contained 10 percent titanium
carbide. The second sample (#2) had the
conventional formulation of the cast material.
. Figure 2 is a photograph of
the samples as received.
Figure 2  Sintered sample
blanks as received
The extensional sound
velocity, c, was found from the sample dimensions and by measuring the
frequency of the first freefree flexural mode of vibration[9]
and the density obtained by measurement of displaced weight of water under
suspended immersion. Knowing both the
density and sound velocity permitted computation of Young’s modulus:
_{} (1)
where E is Young’s
modulus, r the density and c the
extensional sound velocity. For both
samples the density was found to be 4400 Kg/m^{3 } (0.16 lbf/in^{3}). The sound velocity for sample 1 was computed
from the flexural frequency to be 5100 m/s (200,000 in/s), while that for
sample 2 was 5300 m/s (210,000 in/s).
Hence the moduli:
Density and Elastic
moduli for sintered samples
Sample 
Density, Kg/m^{3}
(lbsf/in^{3}) 
Modulus, GPa (Mpsi) 
1 (10 percent TiC) 
4400 (0.16) 
117 (17) 
2 (64) 
4400 (0.16) 
122 (18) 
To determine the mechanical
Q of these materials test specimens of Mason’s massspringmass resonators[10],
also known as spool or dumb bell resonators, were machined from each. Figure 3 is a drawing of the powdered metal
specimens designed to resonate at 20 kHz.
Figure 3 – 20 kHz half
wavelength acoustic test specimen
The specimens were attached
to a 20 kHz transducer driven stepped horn as shown in Figure 4 using a
cyanoacrylate adhesive. The temperature
rise in the center of the stem, which is a motional node, was made using a
noncontacting infrared thermometer and confirmed by measurement using a Type K
thermocouple. The test duration varied
with the level of applied vibration[DW1].
Table 2 provides a
representative sample of the raw data.
The resonant frequency for all tests and both specimens was measured to
be 19,990 Hz. The temperature measurements
shown are averages of at least three separate runs.
Figure 4  Testing apparatus
arrangement
Table 2
20 kHz Typical Thermal Measurement Data
Specimen 
d,
microns (.001in) 
Duration, s 
DT_{av}
,^{o}C (^{o}F) 
Computed
stress[11],
MPa (kpsi) 
Strain
% 
Comment 
2 
64 (2.5) 
10 
8.33 (15.0) 
260 (37) 
0.21 

2 
76 (3.0) 
10 
11.9 (21.5) 
310 (44) 
0.24 

Figure 5 is a representative
oscilloscope trace of the spindle temperature rise and fall during a single
test.
Figure 5  Spindle temperature
variation with time
Figure 6 plots the Q, computed
using the measured values of temperature rise and stress according to Eq. (6b),
versus the strain. The specific heat, C,
of titanium was taken as 565 J/^{o}K/kg (0.135 BTU/lb/^{o}F =
1260 in.lbs/lb^{0}F). It is
noted that Q’s computed without correction for heat loss were only ten percent
larger than the values shown.
Figure
6
The
specimen containing 10 percent TiC fractured from repeated operation at 310 MPa
cyclic stress. The 64 specimen
fractured when testing was attempted at 450 MPa.
Similar
measurements were also made on a test specimen made of the alloy 6Al6V2Sn
which was first annealed for one hour at 700 C (1300 F), followed by air
cooling. The pertinent material
properties, again obtained by measurement of the flexural vibration frequency
and density are tabulated below.
Table
3
Density and Elastic
moduli for 6Al6V2Sn sample
Density, Kg/m^{3}
(lbsf/in^{3}) 
Modulus, GPa (Mpsi) 
4510 (0.163) 
99.3 (14.4) 
A plot
of the Q of this sample versus operating cyclic stress is provided in Figure 7.
Figure 7
Although
this material exhibits a substantially lower Q than either of the sintered
6Al4V samples, it was also tested for endurance at 414 Mpa (60 kpsi) operating
cyclic stress with the spindle cooled by a water spray as shown in Figure
8. The sample survived repeated 6
millioncycle operation at this stress level.
Figure 8 – Endurance
test of 662 Ti alloy
Test specimens
were also made for two samples made from 6Al4V round bar stock, revealing a Q
of approximately 5000 at strains up to 0.33 percent. Annealing of these samples did not affect the Q.
A sample
of the 64 sintered material was also annealed at 730 C for one hour and
allowed to air cool. A specimen made
from this sample was also tested.
Annealing reduced the Q from the values shown in Figure 6 to those found
for standard rolled round stock.
As a
check upon the thermal method applied here, samples of 662 and 64 specimens
were cemented to an identically resonant transducerbooster combination whose
quiescent power consumption at a measured vibration amplitude was known. The additional power consumed at the same
amplitude with the specimen attached is due to losses in the specimen and the Q
can be computed by dividing the energy of vibration by the power lost per
cycle. The results obtained agreed with
those listed here within ten percent.
Discussion
While the values of Q measured
here are substantially below those found by Mason ^{3}, all alloys evaluated appear comparable or superior
in performance to the venerable 6Al4V titanium alloy.
The Q of sintered 6Al4V Ti
appears to be substantially greater than that measured for cast and rolled
material, and indicates that the material is a promising substitute for use in
highly stressed ultrasonic horns as its acoustic losses begin to mount at the
same stress threshold of approximately 275 Mpa (40 kpsi) exhibited by the
conventional product. The alloy
containing 10 percent Titanium carbide and recommended for its ease in
machining, while exhibiting more modest performance and progressive degradation
at substantially lower dynamic stress, appears worthy of evaluation for use in
applications where the maximum dynamic stress does not exceed 200 Mpa (30
kpsi).
The 662 alloy exhibits about
the same losses as that shown in the samples made of common rolled 64 but it
appears capable of sustaining cyclic strains in the neighborhood of 0.4 percent
at 20 kHz in applications where cooling can be provided.
Acknowledgements
The author thanks Stanley
Abkowitz of Dynamet Technology for discussion of sintered products’ properties
and for supplying the samples and Professor Daniel Bershers of Columbia
University for advice regarding his own and W.P. Mason’s dynamic strain studies.
[1] Modal Mechanics, 431 Hawthorne Avenue, Yonkers, NY 10705
[2] Neppiras, Mechanical transformers for producing very large motion, Letters to the editor, Acustica 13, 369370, 1963.
[3]Mason, W.P. and J. Wehr, Internal friction and ultrasonic yield stress of the alloy 90Ti6Al4V, J. Phys. Chem. Solids, 31:19251933, 1970.
[4] Mason, W.P., Low and high amplitude internal friction measurements in solids and their relation to imperfection motions, Advances in Material Research, Vol. 2, (C. J. McMahon, ed.), pp. 287364, Interscience 1968.
[5] Kuz’menko, V.A., Fatique strength of structural materials at sonic and ultrasonic loading frequencies, Ultrasonics 13:1, 2130, 1975.
[6] Mason, W.P., Physical Acoustics and the Properties of Solids, p. 176, D. Van Nostrand, 1958
Puskar, A., Cyclic stressstrain curves and internal friction of steel at ultrasonic frequencies, Ultrasonics, May 1982.
[7] Dynamet Technology, Eight A Street, Burlington, MA 01803
[8] Merkulov, L.G. and A. V. Kharitonov, Theory and design of sectional concentrators, Soviet Physics – Acoustics, Vol. 5, 183190, (1959).
[9] Roberts, M.H. and J. Nortcliffe, Measurement of Young’s Modulus at High Temperatures, J. Iron and Steel Inst., Nov. 1947, pp. 345348.
[10] To avoid inadvertent stress concentration the reduction to the stem is radiussed, unlike that shown for Mason’s resonators. See Methods of Experimental Physics, Vol. 19, Acoustics, p. 346, Academic Press, 1981.
[11] Computation using Eq.4 is within ten percent of these values obtained by both finite element modal analysis and boundary value solution of the wave equation applied to a stepwise model of the geometry. Computation of Q using the simplified analysis is within fifteen percent of values tabulated.
[DW1] To view that formulae used in this note, open the note for editing.
To check the thermal measurements used to determine Q, the specimen was attached to a transducer/booster combination tuned to 20,000 Hz whose power dissipation at 3.0 mils excursion was measured (20 kHz xdcr&3to1 booster@3mils*.wfd). The average dissipation from ten independent measurements, obtained using power/t analysis in waveform manager, was found to be 5.14 watts with a standard deviation of 0.2 watts. The specimen was then attached to the booster, using cyanoacrylate adhesive applied to lapped mating faces, and the power again measured ten times at the same excursion (45,000 psi peak cyclic stress). The average power was found to be 7.5 watts with a standard deviation of 0.5 watts.
The average dissipation per cycle is then the difference between these averages divided by the frequency (20200 Hz when the specimen was attached, assuming that the dissipation of the booster/transducer combination does not change when the frequency shifts slightly from 20.000 to 20.2 kHz).
The energy equivalent mass, Me, which is the mass that if moved with the specimen's free end velocity produces the same kinetic energy as that stored in the specimen itself, was computed by adding a small mass, dm, to the specimen and computing, using FEA modal analysis, the new resonant frequency. Me then can be found by evaluating_{}, where f’' is the resonant frequency with mass dm added and fo is the frequency without the mass. Knowing Me, the stored energy of vibration can be computed as _{} where Vi is 2pf x, where f is the actual measured frequency of vibration and x is the optically measured excursion.
The mass added (SPOOL2') was 0.5 inches in diameter and 0.005 inches in length which, using a density of 0.163 lbsf/cu. in., gives 4.59E5 slugs. The frequency with the mass was measured to be 20010 Hz and with it, 19919 Hz. Hence Me was found to be 4.59E5 slugs and the stored energy of vibration computed to be 0.83 in lbs or 0.069 ft lbs or 0.093 J.
The Q is 2pE/dissipated energy per cycle. As 2.35 additional watts were consumed with the specimen attached and the frequency measured was 20200 Hz, the energy dissipated per cycle is 1.16E4 J, so that Q ~ 5,000.
Similar measurements were made of 6Al4V at 3 mils excursion (44,000 psi peak cyclic stress) where the average power was found to be 6.81 watts with a standard deviation of 0.11 watts. Using the frequency data from SPOOL and SPOOL' where the same mass was added to compute the energy equivalent mass, it is found that Me = 4.59E5 slugs. As the measurement was again made at a 3 mil excursion, E = 0.81 in lbs or .067 ft lbs = 0.091 J. As the net dissipation is 6.815.14 = 1.67 watts or 8.43E5 J/cycle (using a frequency of 19800 Hz), the Q is 6800.
Note: Power measurements were made with the static capacitance of the transducer tuned with a series inductance of 10.1 mH, with the voltage measured between common and the inductor input and thus include any dissipation present in the inductor. However, a measurement was also made directly across the transducer at 3 mils excursion. The power measured, 6.95 watts average, was within the range of measured values.