Inaugural International Conference of the Engineering Mechanics Institute
IMPACT OF SMALL STEEL SPHERES QUANTIFIED BY STRESS WAVE
MEASUREMENT
Gregory C. McLaskey and Steven D. Glaser
University of California, Berkeley, CA, 94709
[email protected]
Abstract
This paper presents the seismic analysis of the collision of small steel balls onto a massive steel plate. The
kinematics of the steel sphere is calculated from the timing of successive bounces while the impulse that the steel
ball imparts into the plate is quantified by the stress waves that are produced from the impact. Stress waves are
detected by high fidelity broadband NIST-Glaser transducers. Transducers of this type are typically calibrated by
analyzing the stress waves produced by breaking a glass capillary at a known force. By comparing the observed
change in momentum of the ball with the impulse measured via the stress waves, an absolutely quantitative
calibration scheme has been developed. This experimental procedure is also used to assess the assumptions on which
the Hertz theory of impact is based.
Introduction
Sensor calibration requires a comparison of the transducer to a known and independently
measurable quantity. In the case of stress wave measurement devices, this is a particularly
challenging problem since stress waves produce displacements which are often less than a
nanometer in height and change very rapidly over time. Some stress wave transduction devices,
such as capacitance or optical techniques, can be calibrated absolutely based on independently
measurable or known properties such as the theoretical capacitance of the transducer or the
wavelength of light. For very small (sub-nanometer) amplitude stress waves the piezoelectric
sensor cannot be directly compared to its optical and capacitive counterparts simply because the
piezoelectric sensors are often orders of magnitude more sensitive (Boltz and Fortunko 1995). In
this study, the stress waves produced by the collision of a small steel sphere with a massive steel
plate were chosen as a “known” quantity. The kinematics of the steel ball can be measured,
either optically with cameras or based on the timing of successive bounces. The stress waves
produced by the collision can be calculated by combining the Hertz theory of impact with wave
propagation solutions. The stress waves calculated from the marriage of theses two theories are
then compared to those measured by the piezoelectric sensor. The steps in this process are shown
in Figure 1.
Figure 1. The steps which establish the link between stress waves measured by a piezoelectric sensor
and those calculated from ball kinematics.
For this calibration scheme to be effective, both the theory of elastic wave propagation
and Hertz theory of impact must be correct. All of their inherent assumptions must be satisfied.
In this study, each step of the process is systematically analyzed.
Hertz Theory of Impact
Hertzian theory has been used by the vast majority of the theoretical work concerning the
collision of two elastic bodies. (For a review of this work see Love (1944) or Goldsmith (2001)).
The Hertz theory of impact is formulated from the stress equilibrium equation (Hunter 1957).
This quasistatic formulation completely ignores the existence of stress waves, yet despite this
limitation, the results of Hertz’s original calculations have withstood the test of time, and have
been verified experimentally (Breckenridge et al. 1990). In addition to the omission of stress
waves, Hertz’s theory assumes that the materials which comprise the ball and the plate are
completely linear elastic. As a result of these assumptions, the rebound velocity of the ball is
identically equal to the incoming velocity. This implies a restitution coefficient of 1, which is
impossible due to the second law of thermodynamics. The Hertz theory of impact neglects losses
due to stress waves and anelastic properties of the ball and plate.
Hertz gave formulations for the size of the contact area between the two bodies and the
displacements as a function of position and time. The two most relevant parameters for this study
are the contact time tc, and the maximum force between the two bodies fmax. Both of these
parameters can be described as functions of the ball radius r and incoming velocity vin
t c =k1rvin -1/5
(1)
f max =k 2 r 2 vin 6/5
(2)
where k1 and k2 are functions of the material properties and geometries of the ball and plate.
Wave Propagation
The first consideration of the stress waves produced by rapidly changing forces at the surface of
a half space was by Lamb (1904). Extension of the theory for specific geometries and loading
situations has been completed by Miller and Pursey (1954), Pekeris (1955), and many others.
Today, the types of propagating waves and their corresponding displacements produced by
forces acting on the surface of a material are well known (Aki and Richards 1980), and Green’s
functions for a point force acting on a plate have been calculated (Johnson 1974). From the
Green’s function, the solution due to arbitrary loading can be calculated via convolution, and this
solution can be compared to the output from a piezoelectric sensor.
When using Green’s functions to calculate wave propagation solutions, another set of
assumptions are implied. The material is once again assumed to be linear elastic with no
damping, and, under our formulation, the force is assumed to be a point source acting only in the
direction normal to the plane of the plate.
Experimental Procedure
Figure 2 shows the experimental configuration of the sensors and the ball drop location on the
steel plate. Sensors were located at two different orientations relative to the location of the ball
drop, both on the opposite side of the plate: one at the epicenter, and one 35 mm off epicenter (d
= 35 mm). The location and orientation of the sensors was chosen to maximize the amplitude of
the P wave at the epicentral sensor location and maximize the amplitude of both the P and S
wave at the off-epicentral location. The mild steel plate was 50 mm thick and 600 mm square. At
the maximum wave speed for steel, 6000 m/s, it takes 100 us for the stress waves to reach the
edge of the plate and return to the center of the plate where the source and sensors are located.
Therefore the first 100 us of each sensors’ record will be unpolluted by side reflections and the
plate can be treated as infinite during this time period.
Figure 2. Experimental setup for ball drop tests.
The sensors used for these experiments were Glaser-NIST-type broad-band, piezoelectric
displacement transducers, made in our laboratory, and modeled after the NBS conical transducer
(Proctor 1982) and described in Glaser et al. (1998). The sensing element is PZT-5a (leadzirconate-titanate) truncated cone with an aperture diameter of 1.5mm. These sensors have a
virtually flat frequency response (+/- 2 dB) from 12 kHz to 960 kHz and with extremely
unidirectional sensitivity in the direction of the axis of the truncated cone (normal to the
specimen surface). The signals were recorded with a 14 bit High-TechniquesTM digitizer. One
sensor was designated the trigger sensor, and when the ouput voltage from this sensor exceeded a
1 mV threshold about two million (221) samples of data were recorded at a sampling rate of 5
MHz. In this way, a total of 0.4194 seconds of displacement time history was recorded for each
sensor, corresponding to 205 us before the time of triggering and about 419 ms after the time of
triggering.
A typical time record from the off-epicentral sensor is shown in Figure 3. In the 0.4 seconds
shown (Figure 3 (a)), this displacement time record contains the displacement time histories from
stress waves due to four different collisions between the same ball and the steel plate. These four
peaks correspond to four successive bounces of the steel ball on the steel plate. In between each
bounce the stress waves reflect off of the edges of the plate and the amplitude of these reflections
decays in a roughly exponential fashion due to the anelasticity of the steel (i.e. the stress waves
are eventually converted into heat or another form of energy).
Figure 3. Displacement time history collected from the off-epicentral sensor at three different time
magnifications.
Figure 3 (b) shows the first 4 ms after the second bounce at a greater time magnification. The
features of the first waves to arrive at the location of the off-epicentral sensor are shown in detail
in Figure 3 (c). The Green’s function for the off-epicentral sensor is very nearly equal to two
delta functions, one corresponding to the P-wave and one corresponding to the later arriving Swave. When this Green’s function is convolved with a force time function of finite width as
described by the Hertz theory of impact, the resulting theoretical displacements will contain two
pulses of finite width. These pulses (due to the P-wave followed by the larger S-wave) can be
seen in the Figure 3 (c). The width of each pulse corresponds to the total contact time tc (Eq. (1))
of the ball on the plate during the collision, and the height of the pulses is a proportional to the
maximum force between the plate and the steel, as described by the Hertz theory of impact (Eq
(2)). At this distance of 61 mm from the source, the P- and S-waves traveling at 6000 m/s and
3200 m/s, respectively, have separated by 9 µs. Based on this separation, the P wave will remain
distinct from the S wave for a contact times of less than 9 µs (as shown in the figure).
Relevant parameters of the initial P-wave pulse such as initial pulse width, height, and shape can
be measured from the sensor’s output for each bounce. By combining the theories of impact and
wave propagation, these same parameters are simultaneously defined as functions of the ball
kinematics such as the incoming velocity and radius of the colliding sphere which are
independently measured via the timing of the successive bounces of the ball. Each successive
bounce is numbered and all parameters associated with the resulting collision are labeled with
that number. For example, the incoming velocity from the second bounce vin2 can be calculated
by the “air time” between the first and second bounces t2-(t1+tc1) where tc1 is the time of contact
of the collision associated with the first bounce. Likewise, the rebound velocity of the second
bounce vre2 can be calculated by the air time between the second and third bounces t3-(t2+tc2). In
this way the kinematics of the ball can be calculated and compared to the contact parameters
found from characteristics of the initial P-wave pulse. These parameters were independently
calculated for each sensor, and produce identical results.
Preliminary results
By comparing the relationship between the ball kinematics and the contact properties
measured from the P-wave pulse it was found that theory and experiment match to first order but
that some modifications are needed. It was found that the contact time roughly follows the v-1/5
relationship and pulse height roughly follows the v6/5 relationship, but for any bounce, the
rebound velocity is always less than the incoming velocity due to losses from the anelasticity of
the steel, and the energy contained in the stress waves which radiate away from the impact
location. Instead of attempting to measure contact time and maximum force, it is more effieicent
to measure the total impulse delivered by the ball into the plate. The impulse P can be defined as:
P = ∫ ∫ σ(x,t) dA dt = ∫ f(t) dt
t A
(3)
t
where σ is the stress as a function of space and time (Goldsmith 2001). This can be related, via
Hertz theory of contact and the Green’s functions, to the area under the initial P-wave pulse.
Alternatively, from the impulse momentum laws (Goldsmith 2001), the impulse
P = m∆v = m(v re -vin )
can be calculated from the kinematics of the ball, which are measured and calculated
independently of the stress wave pulse characteristics.
(4)
The change in momentum calculated from measurements of ball kinematics should always equal
the impulse P derived from stress wave pulse measurements from the piezoelectric transducer
regardless of the size of the ball and incoming velocity. Therefore, when the change in
momentum calculated in (4) is scaled by the Green’s function (in nm/N) and plotted versus the
impulse calculated from (3) for a variety of ball sizes and velocities, the resulting trend should be
linear and pass through the origin, and the slope of the line will be equal to the sensitivity of the
sensor in volts/nm.
Unfortunately, the stress wave measurements observed with the piezoelectric transducer
rarely match those calculated via theory. Differences between experiment and theory could be
due to one or more of the following reasons: (1) inaccurate measurements of the kinematics of
the ball, (2) limitations inherent in the Hertz theory of impact (3) limitations inherent in the
theory of elastic wave propagations, and (4) the inability of the sensor to accurately describe the
stress waves. A systematic series of experiments is being performed to identify the sources of
discrepancy.
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