INERTIAL AND VIBRATIONAL CHARACTERISTICS OF SOFTBALL
AND BASEBALL BATS: RESEARCH AND DESIGN IMPLICATIONS
Kansas State University, Manhattan, KS, USA
The design of baseball and softball bats has been an
ongoing process since the inception of the game in the early 19th
century. Until the early 1970's, the only material to be used was wood and
the potential for design improvements was limited because they involved changing
the physical dimensions of the bat. Also, most of the changes were initiated
by the player and were carried out by local craftsmen in wood shops. However,
with the introduction of hollow-wall aluminum bats in the 1970's and other
metal alloys as well as composite materials since then, the potential for
applying relevant mechanical principles to improve the performance characteristics
of bats has improved dramatically. In recent years a plethora of bats with
innovative features have become available to the consumer. Some of these
innovations are a result of the research and development efforts of the bat
manufacturing industry, often in collaboration with scientists not directly
affiliated with the industry, while others are a result of flawed ideas that
are not completely thought through. Braham (1997) provides a recent, up-to-date
summary of recent innovations by the leading bat manufacturers, but an evaluation
of the claims of these products using relevant criteria is lacking. The purpose
of this paper is to provide a scientific basis and a focus for examining
and developing new bat design features. The paper will provide an overview
of the factors that are relevant to the design of baseball and softball bats,
including related theoretical and empirical studies.
Factors relevant to the design
of baseball and softball bats
When developing or evaluating a design feature of a baseball
or softball bat, the following factors must be considered: (1) the manner
in which the bat is swung and forces are transmitted to the bat during the
swing, (2) the constraints resulting from rules in the particular sport,
and (3) the relevant properties of the bat. I will review each of these
general factors in detail, making reference to related scientific literature.
Characteristics of batting relevant to bat design
Baseball and softball batting is a two-handed sidearm
striking skill with the bat held near one end and swung as a physical pendulum.
Power hitters attempt to impart maximum velocity to the impacted ball in
the desired direction by generating maximum linear velocity of the impacting
part of the bat to the ball. The motion of the bat is predominantly in the
horizontal plane and the rotation axis ranges from .15 to.20 m off the knob
end of the bat toward the hitter's body during the swing. During the swing,
the maximum linear (COP) and rotational velocity of the bat is approximately
33 m/s and 36 r/s, respectively, for college females and 38 m/s and 44 r/s,
respectively, for college males. Because the primary goal of the power
swing is to maximize bat velocity on impact, it is somewhat surprising that
maximum bat velocity has repeatedly been found at from .01 to .05 s prior
to impact (Shapiro, 1974; McIntyre & Pfautsch, 1982; Messier & Owen,
1984; Spragg, 1986). While this finding has been reported in the scientific
literature frequently, a plausible explanation of the reason has not been
found. A review of the elastic properties of the bat (appearing later in
this paper) and the characteristics of the swing may provide a tenable hypothesis.
It is possible that, at the beginning of the swing, torque applied to the
bat handle to rotate the bat toward the incoming ball and the inertia of the
barrel end of the bat cause the bat to bend, with the barrel of the bat lagging
behind. This bending mode, usually referred to as the diving board mode
, would begin to release when the rotational acceleration of the bat begins
to drop. It is conjectured that the elite hitter learns through trial and
error to adopt a bat and swing that are matched such that the velocity of
the impact point of the bat is maximized at impact. To accomplish this end,
accelerating forces would be reduced quickly at either 1/4 or 1 3/4 of the
period of oscillation of the diving board mode. For example, if the fundamental,
diving board mode of a bat is 25 Hz, then the period of oscillation is 40
ms. For the hitter to take advantage of this elastic behavior, this bending
mode would need to be "released" at approximately 10 or 70 ms prior to impact.
While this characteristic of the skilled golf swing has been empirically
verified (Cochran & Stobbs, 1986), no empirical data in support of this
hypothesis applied to softball or baseball bats have been found.
Rules most relevant to bat design
The rules regarding baseball bat characteristics are
different from those regarding softball bats. Also, rules are different
for different genders and different levels of play. For adult males, the
maximum baseball and softball bat barrel size is 2.25 and 2.75 in (.057
and .070 m), respectively. The maximum bat length is 42 and 34 inches (1.067
and .864 m) for baseball and softball, respectively, while the maximum softball
bat weight is 38 oz (10.569 N). There is no maximum baseball bat weight.
All bats used in professional baseball must be made of wood. The most recent
rule, which places an upper limit on the coefficient of restitution (Bat Performance
Factor) for bats at different levels and types of play, is having a tremendous
impact on the direction of bat design activity. Bat performance factor will
be discussed in greater detail later in this paper.
Inertial and vibrational properties relevant to
Several inertial and vibrational properties of the bat
are relevant to its effective use: (1) mass, (2) moment of inertia, (3) coefficient
of restitution, (4) location of node of the fundamental vibration node, and
(5) center of percussion location.
Mass and moment of inertia determine the amount of effort
required to swing the bat. There is an inverse relationship between bat
linear and angular acceleration and mass and moment of inertia, respectively,
for a given linear and angular impulse (integral of force/torque and time).
Thus, the more mass and/or moment of inertia, the more impulse required
to produce a given change in bat speed or direction. In other words, greater
mass and moment of inertia compromise the hitter's ability to control the
path of the bat as it moves toward the ball as well as to generate bat velocity
during the swing. Theoretical models of the relationship between mass and
impact parameters indicate that lighter bats than have been used by most
skilled players would be more effective (Kirkpatrick, 1963; Adair, 1990).
In a study which sought to empirically determine an individual's ideal bat,
this concept was supported (Bahill & Karnavas, 1991). Furthermore,
bats now used by elite softball and baseball players are much lighter than
they were 10 years ago. Bat manufacturers and retailers do not provide moment
of inertia measurements with their products; however, moment of inertia is
a critical design parameter and is also used to develop bat selection guidelines.
When the ball and bat are impacted, during impact the
bat behaves in some respects as a physical pendulum and in some respects
as an elastic body. Taking both rigid-body and elastic properties into consideration,
the best part of the bat on which to hit the ball is called the "sweet spot".
The "sweet spot" is a general, nonscientific term, that means that the best
overall results are obtained from impacts on this point. In other words,
impacts on the sweet spot feel best to the hitter and results in imparting
velocity to the ball are best. Or, in more precise terms, the sweet spot
is the impact location where the transfer of energy from the bat to the ball
is maximal while the transfer of energy to the hands is minimal. On closer
examination, four parameters have been identified as having an effect on
the "sweetness" (liveliness) and location of the sweet spot: (1) center of
percussion (COP), (2) node of the fundamental vibrational mode, (3) coefficient
of restitution, and (4) the maximum "power" point.
Center of percussion. When the ball
hits the bat at the center of percussion (COP), there is no reaction impulse
(shock) at the axis. The impact axis for bats has been shown to be the point
under the first knuckle of the top hand (Plagenhoef, 1971; Noble, 1983).
Thus, COP impacts are more comfortable than at other locations because there
is no painful impact "shock" that is experienced during impacts at other
locations. The distance from the impact axis to the COP of a bat can be
found from the following equation:
COPdist = T2g/42 = .2483877*T2
(units in meters)
where T is the period of one oscillation when the bat
is suspended from the axis, and g is the gravitational constant in meters.
The COP has been demonstrated to be the impact location producing the greatest
post-impact velocity with stationary bats (Weyrich, Messier, Ruhmann, &
Berry, 1989). Another empirical study involving 18 elite slow-pitch softball
hitters reported a correlation of .58 between the perceived location of the
sweet spot and the COP. Thus, COP location explained 33% of the variability
in perceived sweet spot impact location (Noble, 1983). Brody developed a
theoretical construct for determining the impact location of a swinging bat
with a pitched ball that would result in greatest postimpact ball velocity
(Brody, 1986). This location was not on the COP, but was a function of the
relative velocity and mass of the ball and bat as well as the inertial properties
of the bat.
In an early study (Bryant, Bryant, Chen, & Krahenbuhl,
1977) comparing the dynamic and performance characteristics of aluminum and
wood bats, data were reported showing an impact area of several cm in length
on hollow-wall aluminum bats where there was zero-order reaction impulse
while reaction impulse on wood bats was a direct linear of function of distance
from the COP. However, a later study by Noble and Eck (Noble & Eck,
1986) presented a theoretical construct and empirical data demonstrating
that, assuming the bat is rigid during impact, reaction impulse is a direct
linear function of the distance of the impact from the COP (Figure 1). Also,
the slope of the regression line of impact reaction impulse on impact-COP
distance is a direct linear function of the distance of the COP from the
impact axis (Figure 2). In other words, the greater the distance of the
COP from the hands, the smaller the reaction impulse resulting from an impact
of a given distance from the COP. This study also demonstrated that the
distance of the COP from the axis was:
COP = I/Mr
where I = moment of inertia about the axis, M = the total
bat mass, and r = distance from the axis to the center of mass. Strategies
were later presented for systematically displacing the location of the COP
(Noble & Eck, 1985) by placing mass at various locations along the longitudinal
axis of the bat. While these strategies were effective
in displacing the COP to a more distal location on the
bat, this "improvement" did not enjoy wide acceptance by the players because
of the greater excitation of the fundamental vibrational mode resulting from
COP impacts (Noble & Walker, 1994b).
Figure 1. Mechanical system
describing center of percussion (COP) location as a function of impact location.
(Legend: P = impact impulse, P1 = impact reaction impulse,
A = impact location, ap = distance from COM to COP, O = impact
axis, M = mass, S = axis-COM distance, Io = moment of inertia
about impact axis). Note. From "Effects of Selected Softball Bat Loading
Strategies on Impact Reaction Impulse" by L. Noble and J. Eck, 1986, Medicine
and Science in Sports and Exercise, 18, p. 51. Copyright 1986 by the
American College of Sports Medicine. Adapted with permission.
Figure 2. Theoretical relationship between impact reaction
impulse and impact location. (Legend: P = impact impulse, P1
= impact reaction impulse, ap = COP location, a = distance from
impact axis to impact, M = mass, S = axis-COM distance, Io = moment
of inertia about impact axis). Note. From "Effects of Selected Softball
Bat Loading Strategies on Impact Reaction Impulse" by L. Noble and J. Eck,
1986, Medicine and Science in Sports and Exercise, 18, p. 51. Copyright
1986 by the American College of Sports Medicine. Adapted with permission.
Vibrational properties. The above discussion
relates to the rigid body behavior of the bat during impact. The bat also
exhibits important and relevant elastic properties during impact as well
as during the swing because the bat is not completely rigid. The vibrational
behavior of a bat approximates that of a uniform beam, described in detail
in most engineering textbooks on vibrations (Thompson, 1993). If we assume,
for simplicity, that a bat can be represented by a uniform rod rigidly suspended
at the point of contact with the hands, then the various normal vibrational
modes are only those for a rigid clamped rod. Figure 3a illustrates the
approximate length (modeled after a uniform rod), node locations, and relative
amplitude of these modes. The lowest frequency mode, commonly called the
diving board mode, corresponds to a vibration where the axis is at a nodal
point and the barrel end is at a maximum.. This node has a wavelength (0)of
approximately 4H, where H is the distance from the axis to the barrel end
of the bat. This fundamental mode only has one node and it is located at
the clamped point. The next highest frequency mode has a node at the handle
and another at 3/4H. If the ball strikes the bat at a node of a given vibrational
mode, then that mode will not be excited. Since all modes have an anti-node
unclamped, or barrel end of the bat, all modes of vibration
can be excited when striking the bat at the barrel end. The frequency of
the diving board mode has been
Figure 3. Vibrational modes of bat as (a) a uniformed
rod clamped at the axis, and (b) as a free-free rod. (i
= approximate wavelength of ith harmonic modeled as a uniform rod).
Note: Part a from "Effects of Selected Softball Bat Loading Strategies
on Impact Reaction Impulse" by L. Noble and J. Eck, 1986, Medicine and
Science in Sports and Exercise, 18, p. 51. Copyright 1986 by the American
College of Sports Medicine. Part b from "Baseball Bat Inertial and vibrational
Characteristics and Discomfort Following Ball-Bat Impacts" by L. Noble and
H. Walker, 1994, Journal of Applied biomechanics, 10, p.
134. Copyright 1994 by Human Kinetics, Inc. Adapted with permission.
reported to be 27 Hz for an aluminum softball bat and
18 Hz for a wood softball bat (Brody, 1990) and that of the first overtone
mode was 317 and 209 Hz, respectively. Both of these bats were 34 inches
(.864 m) in length. Shorter bats and bats with greater strength/mass ratios
will have higher fundamental frequencies. It is likely that this mode is
excited during the swing, as has been demonstrated during the swinging of
golf clubs(Cochran & Stobbs, 1986); however, this low-frequency mode
is not excited by the impacting ball (Brody, 1990; Noble & Walker, 1994a;
Noble & Walker, 1994b).
During impact, the vibration behavior of the bat corresponds
to that of a free, non-supported bat whether irregardless of the firmness
of the grip. Figure 3b illustrates the lowest (fundamental) and first harmonic
modes, approximate wavelengths and node locations of the free, unsupported
bat. The approximate locations of the two nodes for the first mode are
29% of the bat length from each end. The next highest mode has three nodes
with approximate locations as depicted in Figure 3b. The number of nodes
for each successively higher mode increases by one at each step. Also, the
amplitude associated with each mode decreases as the frequency increases
and increases as the distance from the node increases.
The node locations and frequency of the fundamental and
first harmonic modes can be measured by, first, supporting the bat in a horizontal
position by threads attached to the ceiling. A vibration exciter and velocity
sensor are horizontally oriented and placed as indicated in Figure 4. Specific
placement of the exciter and sensor is not critical as long as they are not
placed on one of the nodes. A resister is put in series with the exciter
coil, and the voltage displayed on the horizontal axis of an oscilloscope.
This voltage is proportional to the current through the resister and exciter
coil which is proportional to the force applied by the exciter. The output
of the velocity sensor is displayed on the vertical axis of the oscilloscope.
This display is generally an ellipse whose axes are oriented at an angle
determined by the gain setting of the oscilloscope, the phase relationship
of the two signals, and the location of the velocity sensor. The input is
gradually increased in frequency until resonance is achieved. At resonance,
the exciting force and velocity are in phase and the ellipse is reduced to
a straight line. The straight line changes slope as the velocity sensor
4. Vibration measurement system. From "Baseball Bat Inertial and
vibrational Characteristics and Discomfort Following Ball-Bat Impacts" by
L. Noble and H. Walker, 1994, Journal of Applied biomechanics, 10,
p. 134. Copyright 1994 by Human Kinetics, Inc. Adapted with permission.
is moved along the long axis of the bat. The slope changes
from positive to negative as the sensor passes over a node and is horizontal
when the sensor is located at a node.
The fundamental frequency (free condition) of 34-inch
(.864m) aluminum softball bats is usually within the range of 180-360 Hz
and can be varied throughout this range by changing the shape of the bat
and strategic redistribution of mass (Noble & Walker, 1993). The easiest
way to change the fundamental frequency is to modify the diameter of the
bat in the taper region, which is near the antinode of the fundamental mode.
Node locations of the fundamental mode can also be strategically displaced
as much as .06 mm using similar strategies. The distal node of the fundamental
mode has been identified as one of the determinants of the sweet spot. Impacts
on the node do not excite the low-frequency, fundamental mode, resulting
in a higher-frequency sound and smaller, higher frequency vibrations of the
bat handle. Thus, it is reasonable to expect that nodal impacts would be
more comfortable for the hitter. Fortunately, in most bats the location
of this node and the COP are less than .02 m apart, approximately 25 per
cent of the bat length from the barrel end. When using bats with the node
and COP close together, impacts at both locations are more comfortable than
at other locations (Noble & Walker, 1994a) with no significant difference
between the two. However, strategies developed to strategically displace
the COP toward the barrel end were found to significantly increase the distance
(.06 m) between the node and COP. Impacts at the node on these bats were
found to be more comfortable than impacts at any other location, including
the COP (Noble & Walker, 1994b). This is consistent with results from
a study investigating the sweet spot location of tennis racquets indicating
the node as the predominant predictor of sweet spot location (Hatze, 1994). Furthermore,
the node-COP distance has been found to be one of the most powerful predictors
of player's perception of bat performance and preference (Noble & Dzewaltowski,
No empirical studies have been found investigating the
effects of impact location relative to the location of the distal node of
the fundamental mode on post-impact ball velocity. However, an excellent
theoretical presentation of this effect has been developed, but unfortunately
it remains unpublished (Van Zandt, 1991). This paper applied the theory
of the elastic behavior of an irregularly shaped, cylindrically symmetric
object to a wood baseball bat (frequency of diving board mode and first harmonic
- 27 Hz and 137 Hz, respectively). The node-COP distance was .01 m. A set
of equations was developed and used to find the first 20 normal modes of
vibration and calculate each mode's effect on the recoil of the ball. Figure
5 illustrates estimates of the post-impact ball velocity as a function of
impact point along the length of the bat. The impact ball and bat velocities
were 40 m/sec and 16 m/sec, respectively. The solid curve shows the expected
result, the dashed curve shows the effect of suppressing all modes above
the fundamental for the free condition (137 Hz), and the starred curve shows
the performance expected for an infinitely rigid club having otherwise identical
properties. Two observations relative to the effect of impact location and
post-impact ball velocity are notable: (1) post-impact velocity is significantly
lower for impacts not on the node, especially as the impact location moves
toward the handle of the bat; and (2) the higher-frequency modes serve to
increase post-impact ball velocity. For a collision only .10 m toward the
bat handle from the node, post-impact ball velocity is expected to decrease
by 5%. The effect of this loss in ball velocity would cost a distance in
ball flight of 10%, or 42 ft (12.8 m). The higher elastic modes play an
important role in bat performance, restoring approximately 50% of the loss
in ball velocity. This degradation in bat performance would be less with
stiffer bats with higher natural frequencies, such as those made from composites,
aluminum, and other metal alloys. The loss in bat performance
Figure 5. Theoretical estimates of the degradation of post-impact
ball velocity at different impact locations due to bat vibrations.
From The dynamical theory of the baseball bat
, by L.L. Van Zandt,
1991. Unpublished manuscript. Purdue University, W. Lafayette, Indiana, USA..
would be eliminated if the fundamental frequency of the
bat was "tuned" to the ball-bat contact, or dwell, time. To obtain impact
frequency tuning, the fundamental frequency should equal the reciprocal of
twice the dwell time. For example, the dwell time of a softball and bat collision
(velocity = 31 m/s) has been measured at .0035 sec (Plagenhoef, 1971). For
this case, the frequency-matched bat would have a fundamental frequency of
143 Hz. This procedure would be difficult to effectively apply with
precision because the dwell time is a function of collision velocity, which
is not entirely under the control of the hitter, as well as the hardness
of the ball.
Coefficient of restitution. The coefficient
of restitution (COR) of two colliding objects, such as the ball and bat,
is the ratio of the difference between their velocities immediately after
impact compared to the difference between their velocities prior to impact.
This ratio has been shown to be a function of collision velocity as well
as temperature. For simplicity, the COR of balls and bats are evaluated
separately. Ball COR is usually determined by impacting the balls with a
wooden wall backed by concrete. . Rules for different levels of play for
softball and baseball have been established setting an upper limit for the
COR, usually determined by impacting the balls with a steel plate at 60 mph
(26.4 m/s). The COR is now often stamped on the cover of the ball. For
most adult softball competition, the maximum COR is .55. The COR of bats
has been shown to be a weak function of impact location along the barrel
of the bat where the diameter is constant. Thus, it does not play a significant
role in determining the location of the sweet spot. If a given ball impacts
with a ball with these conditions held constant, the bat with the higher
coefficient of restitution will produce the greatest post-impact ball velocity.
Improving the COR of bats has been the primary focus of the research and
development efforts of the major bat manufacturers during the past decade.
The COR has been significantly improved through the use of materials of
higher strength/mass ratios and strategic manipulation of the wall thickness
of the barrel of the bat. These dramatic increases in COR have caused great
concern on the part of coaches and officials associated with all levels of
play. This concern relates to the safety of the participants and to the
changes in the nature and integrity of the games that those associated with
the game identify with. An outgrowth of this concern is the adoption of
a standard for evaluating the "liveliness" (COR) of bats and implementing
rules placing an upper limit on this aspect of bat performance. A standard
method of testing to measure the COR of bats has recently been adopted by
most of the baseball and rules committees in the USA (ASTM, 1995). This
procedure involves impacting a ball with a known COR with a stationary bat
with a fixed axis (free to rotate) at the bat's COP at a ball speed of 88
ft/sec (26.8 m/s). The COR is calculated by the standard method of comparing
the difference between the velocity of the bat and ball before impact to
that following impact. A Bat Performance Factor (BPF) is then calculated
from the ratio of bat and ball COR to the ball COR. For example, if the
predetermined ball COR is .5 and the measured COR of the ball-bat collision
is .55, then the BPF of the bat would be 1.1. The BPF of most wood bats
is between .9 and 1.0 while that of the latest aluminum alloy bats is typically
above 1.1. Maximum bat performance standards have now been adopted by most
levels of play for both baseball and softball. For example, the maximum
BPF allowed for most adult slow pitch softball is 1.2 and that established
for collegiate baseball is 1.15. The adoption of this standard has changed
the recent trend in bat design from focusing on increasing COR to improving
other performance characteristics that comply with the rules.
Maximum "power" point. The impact point
along the longitudinal axis of the bat that will result in the greatest post-impact
ball velocity, or maximum power point, is another important performance characteristic
of the bat. Brody (1986) developed a theoretical framework and estimates
of the maximum power point for a bat that is swung and impacts with a pitched
ball, behaving as a free-free body during impact. Estimates of the location
of the distance from the maximum power point to the bat COM (PPTDIST) were
found to increase as the mass and moment of inertia of the bat increases and
as the ratio of bat/ball mass increases. Furthermore, PPTDIST increases as
the ratio of bat angular velocity to ball linear velocity increases. Thus,
PPTDIST was estimated to be less for baseball than for fastpitch softball.
PPTDIST was calculated for an aluminum softball bat and ball with varying
bat/ball velocity and mass ratios and found the location of the maximum power
point to be located between the COM and COP. No empirical data relative
to these estimates was reported. Weyrich, Messier, Ruhmann (1989) provided
empirical data on the effects of impact location on postimpact ball velocity.
Bats were not swung, but held stationary by either clamps or strings mounted
to the ceiling while balls were propelled against the bat at 27 m/s. Impacts
on the COP produced greater postimpact ball velocities than impacts on the
COM and impacts near the barrel end. While these results are apparently contradictory
to the theoretical work of Brody(1986), these bats were not swung, as Brody's
estimates assume, and no impacts were studied in the area between the COM
and COP. The empirical data provided by Noble (1983) in an earlier study
does, however, appear to conflict with Brody's estimates. In this study,
the location of 52 self-perceived "sweet spot" impacts of 12 elite male softball
players and 5 highly skilled college male baseball player was examined using
cinematographic methods (200 f/s). The "sweet spot" location, the point
where the results were best, coincided very closely with the COP location.
Furthermore, sweet spot location relative to the COP did not appear to be
different for softball and baseball impacts. If it can be assumed that these
elite players' perception of sweet spot is synonymous with the maximum power
point, then these results do not seem to support Brody's 1986) theoretical
estimates of maximum power point location. However, because COP location
only accounted for 33 per cent of the variability in sweet spot location
in the empirical study (Noble, 1983), other characteristics of the bat, ball,
hitter-bat interface and of the swing probably affected these results.
Further empirical research is necessary to clarify this issue.
Several important factors relevant to the design of baseball
and softball bats were identified.: (1) how is the bat swung and how forces
are transmitted to the bat during the swing, (2) what are the constraints
resulting from rules in the particular sport, and (3) what mechanical properties
are relevant. The most important mechanical properties of the bat are: (1)
mass, (2) moment of inertia, (3) coefficient of restitution (COR), (4) vibrational
properties, (5) center of percussion location (COP), and (6) maximum power
point location. While mass and moment of inertia determine the magnitude
of the efforts to swing and control the bat during the swing, COR, COP, and
vibrational properties largely determine the behavior of the bat during impact.
Obviously, all of these factors play a vital role in the overall effectiveness
of the bat in imparting maximum energy to the ball while imparting minimal
energy to the hitter. Correlating studies related to the swing characteristics
and vibrational properties of the bat provides a hypothesis for the surprising
and recurring finding of maximum bat velocity occurring prior to contact,
rather than at contact. Bat design objectives during the past few years
has involved: (1) interior loading strategies to displace the COP toward
the barrel end, (2) changing the weight distribution and shape of the bat
to displace the COP toward the barrel end while keeping the distance between
the distal node of the fundamental vibrational mode and COP small, and (3)
increasing the COR. Implementing these strategies has involved the adoption
of improved metal alloys and composite materials having a greater strength/mass
ratio. Because of recent rule changes setting an upper limit on COR, research
and design activity will undoubtedly change direction and focus. One of
the potentially fruitful areas of research and design activity appears to
develop bats that are "frequency tuned". A frequency tuned bat would be
designed to take advantage of the elastic properties of the bat during
the swing as well as during the impact.
Swing tuning involves coupling the diving board mode (clamped-free
condition)with characteristics of the swing so this mode is excited and released
at the most opportune time. Impact tuning involves coupling
the mode of the fundamental node to the ball-bat dwell time.
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