Harris C.M., Piersol A.G. Harris Shock and vibration handbook
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Π
tra
=Π
1 → 2
= 2πfη
12
E
1
(11.77)
For a subsystem of length L
1
, δ
f
1
= c
g
1
/(2L
1
) and E
1
= L
1
E
1
′. Solving for the coupling
loss factor gives
η
12
= (11.78)
A more detailed analysis indicates that this result is valid for point connections in
a system with a modal overlap greater than 1. If the system has a constant modal
frequency spacing δ
f
, then the Nth mode will occur at f = Nδ
f
. If the damping loss
factor is η, the system modal overlap is given by M
S
=πηf/(2δ
f
). Then the
modal overlap is greater than 1 for frequencies f > 2δ
f
/(πη) or for mode numbers
N > 2/(πη). SEA is still valid below this frequency and mode number, but the vari-
ance of the model calculations (and in the measured frequency response functions)
becomes large.
For point-connected subsystems the transmission coefficient can be evaluated
from the junction impedances:
4
τ
12
= (11.79)
where R
i
is the real part of the impedance Z
i
(ratio of force to velocity at a point
excitation) at the junction attachment point of subsystem i. When more than two
subsystems are connected at a common junction, the denominator of Eq. (11.79)
must include the sum of all impedances at the junction.
For subsystems with line and area junctions the analysis of the coupling loss fac-
tor is complicated by the distribution of angles of the waves incident on the junction.
However, approximate results have been worked out for many important cases. Eq.
(11.78) can be generalized for all cases as
η
12
= (11.80)
where τ
12
(0) is the normal incidence transmission coefficient for waves traveling per-
pendicular to the junction, and I
12
contains the result of an average over all angles of
incidence.
For line-connected plates the coupling loss factor between bending modes is
found using
I
12
=
1/4
(11.81)
where L
j
is the length of the junction and k
i
= 2πf/c
Bi
is the wave number of the
modes in subsystem i.
When experimental verification of the evaluation of the coupling loss factor is
desired, measurements similar to those used for damping can be used. A decay rate
measurement of a subsystem connected to another (heavily damped) subsystem will
give a loss factor equal to the sum of the damping and coupling loss factor for the
first subsystem.Alternately, subsystem 1 can be excited alone and the spatially aver-
aged response levels of the two connected subsystems can be measured. Using
Π
12
=Π
2,diss
, the coupling loss factor is found from
k
1
4
k
2
4
k
1
4
+ k
2
4
L
j
4
I
12
τ
12
(0)
2 −τ
12
(0)
δ
f
1
πf
4R
1
R
2
|Z
1
+ Z
2
|
2
τ
12
2 −τ
12
δ
f
1
πf
STATISTICAL METHODS FOR ANALYZING VIBRATING SYSTEMS 11.27
8434_Harris_11_b.qxd 09/20/2001 11:17 AM Page 11.27
η
12
=
(11.82)
This result indicates a potential problem in determining the coupling loss factor
from measured results. If E
2
δ
f
2
E
1
δ
f
1
, then taking the difference between their val-
ues in Eq. (11.82) will greatly magnify the experimental errors in determining the
parameters used in this formula. This indicates why it is mathematically unstable to
use measured levels in a multiple subsystem model to back calculate the coupling
loss factors. However, good results can be obtained for a single junction between two
subsystems if one is excited and the other is artificially damped in order to increase
difference between E
1
δ
f
1
and E
2
δ
f
2
. Figure 11.15 shows the results of an experimen-
η
2
E
2
E
1
−δ
f
2
E
2
/δ
f
1
11.28 CHAPTER ELEVEN
FIGURE 11.15 Coupling loss factor η
12
for point connected plates;
measured data with 95 percent confidence intervals; —— calculated values
using Eqs. (11.80) and (11.81).
tal validation of Eqs. (11.80) and (11.81) for the coupling loss factor between two
plates connected at a point. The experimental error is also included, which even in
this idealized laboratory environment is more than 50 percent. While the back cal-
culation of the coupling loss factors tends to be unstable, the forward calculation in
the SEA model is relatively insensitive to errors in the coupling loss factor values,
making the model fairly robust.
MODAL EXCITATIONS
The power put into subsystem modes by the system excitations is needed in order to
use the statistical energy analysis (SEA) model for calculations of absolute response
levels. The mode counts, damping, and coupling loss factors can be used to evaluate
relative transfer functions in the system for a unit input power. However, for actual
response-level calculations the modal input power from the actual excitation
sources must be calculated.
For a point force excitation F(t) the average power put into a system is
Π
in
=
F
2
G
(11.83)
8434_Harris_11_b.qxd 09/20/2001 11:17 AM Page 11.28
where G
is the average real part of the mobility at the excitation point. For a pre-
scribed point velocity source ˙y(t) the average power put into a system is
Π
in
= ˙
y
2
R
(11.84)
where R
is the average real part of the impedance at the excitation point.
The normalized variance in the input power due to variations in the mode shapes
and frequency response function of the system is approximated by
= (11.85)
where ∆f is the bandwidth of the excitation. For more complicated excitations the
input power can be estimated by measuring the response of a system to the excita-
tion and using the SEA model to back calculate the input power. Alternatively, the
measured response levels of the excited subsystem can be used as “source” levels,
and the power flow into the rest of the system can be evaluated using the SEA
model.
SYSTEM RESPONSE DISTRIBUTION
To solve for the distribution of vibrational energy in a system it is convenient to
rewrite Eq. (11.61) in symmetric form:
[B]{Φ} + [I]
= {Π
in
} (11.86)
where [I] is the identity matrix, {Φ} = 2π{E/δ
f
} is the vector of modal power poten-
tial, and [B] is the symmetric matrix of coupling and damping terms with off-
diagonal terms B
ij
=−f η
ij
/δ
f
i
and diagonal terms B
ii
= ( f/δ
f
i
)(η
i
+Σ
j
η
ij
). This system
of equations can be solved using standard numerical methods. Solving for the values
of E gives a mean value estimate of the energy distribution.
The variance in E is more difficult to evaluate because it depends on the evalua-
tion of the inverse matrix [B]
−1
. If the variance of each term in [B] is small compared
to its mean-square value, then the variances in [B]
−1
can be approximated by
[σ
B
−1
2
] [(B
ij
−1
)
2
][σ
B
2
][(B
ij
−1
)
2
] (11.87)
where the notation [(B
ij
−1
)
2
] refers to a matrix with the squares of the elements in
[B]
−1
, term for term.
The subsystem energy values can be converted to dynamic response quantities
using the relation E = m˙y
2
. For a narrow-band vibration at frequency f
c
(which could
be a single one-third octave band response in a broad-band analysis) the displace-
ment response is
y
2
˙y
2
(2πf
c
)
2
and the acceleration response is
ÿ
2
(2πf
c
)
2
˙y
2
.The
relation between the vibration velocity response and the maximum dynamic strain
depends on the type of motion involved. For longitudinal motion the mean-square
strain is
2
= ˙y
2
/c
L
2
. For bending motion of a uniform beam or plate the maximum
strain is
m
a
x
2
= 3˙y
2
/c
L
2
.
dE
dt
3δ
f
πfη+∆f
σ
Π
in
2
Π
in
2
STATISTICAL METHODS FOR ANALYZING VIBRATING SYSTEMS 11.29
8434_Harris_11_b.qxd 09/20/2001 11:17 AM Page 11.29
When the response values in a complex system are plotted on a logarithmic scale,
a surprising result occurs. The log-values are distributed with an approximately
Gaussian distribution over frequency. This is illustrated in Fig. 11.16 for a beam net-
work. The system frequency response function is computed numerically using a
transfer impedance model including bending and longitudinal and torsional motions
11.30 CHAPTER ELEVEN
FIGURE 11.16 Numerical calculation of the vibration response of a
four-beam network. (A) Normalized frequency response function for
a point on beam 4. (B) Probability density function of log-levels,
Numerical data histogram, — Normal distribution.
in each of the four beam segments.A histogram of the computed response values on
the decibel scale compares very well with a Gaussian distribution. This result can be
explained by noting that the response value at any particular frequency results from
the product of a large number of quantities. Then the logarithm of the response
value will be the sum of a large number of terms. If the complexity in the system
causes the responses at different frequencies to be independent, then by the central
limit theorem the log-values will tend to have a Gaussian distribution. This means
that the mean-square response values will have a log-normal distribution.
The calculated mean values and variances in the SEA model can be converted to
the decibel scale as follows. If the mean-square velocity ˙y
2
has a log-normal distribu-
tion with variance σ
˙y
2
2
, then the velocity level L
˙y
10 log
10
( ˙y
2
/˙y
ref
2
) has a normal dis-
tribution with a mean value and variance given by
8434_Harris_11_b.qxd 09/20/2001 11:17 AM Page 11.30
L
˙y
= 10 log
10
− 5 log
10
1 +
σ
L˙
y
2
= 43 log
10
1 +
(11.88)
Note that the mean of the decibel levels is not equal to the decibel level of the mean-
square value.
TRANSIENT (SHOCK) RESPONSE USING SEA
The statistical energy analysis (SEA) model can solve for the transient response of a
system using Eq. (11.61). The numerical solution methods for equations of this form
can be illustrated using the finite difference method. Given an initial energy state
E(0), the energy state at a short time later is approximated by
E(∆t) E(0) +∆t (11.89)
where dE/dt =Π
in
−Π
out
.This new energy distribution is then used to project forward
to the next time step, etc. The accuracy of the solution depends on the size of ∆t rel-
ative to the energy flow time constants in the system, (2πfη)
−1
. For the finite differ-
ence solution, using ∆t ≤ (6πfη)
−1
usually provides accurate results.
dE
dt
σ
˙y
2
2
(
˙y
2
)
2
σ
˙y
2
2
(
˙y
2
)
2
˙y
2
˙y
ref
2
STATISTICAL METHODS FOR ANALYZING VIBRATING SYSTEMS 11.31
FIGURE 11.17 Transient response of an equipment shelf. (A) Experi-
mental structure showing the locations of the impact F and the accelera-
tion response a. (B) Comparison of the transient response of the structure;
— Measured data, ---SEA model.
8434_Harris_11_b.qxd 09/20/2001 11:17 AM Page 11.31
An example of a transient analysis using SEA is shown in Fig. 11.17. The meas-
ured acceleration response of a shelf on an equipment rack for an impact at the leg
is shown along with the corresponding transient SEA solution of Eq. (11.61). The
energy level of the shelf builds up for the first 0.01 sec before beginning to decay.
Modeling the transient mean-square response with Eq. (11.39), the undamped
shock spectrum for this response signal can be estimated using Eq. (11.42). Alter-
nately, if a shock excitation is modeled as a time-dependent power input to the SEA
model, then the peak response spectrum of the system components can be estimated
directly from the maximum mean-square values in the transient SEA solution.
REFERENCES
1. Crandall, S. H., and W. D. Mark: “Random Vibration in Mechanical Systems,” Academic
Press, New York, 1963.
2. Lyon, R. H., and R. G. DeJong: “Theory and Application of Statistical Energy Analysis,” 2d
ed., Butterworth-Heinemann, Boston, Mass., 1995.
3. Caughey, T. K.: “Nonstationary Random Inputs and Responses,” chap. 3, in S. H. Crandall
(ed.), Random Vibration, vol. 2, M.I.T. Press, Cambridge, Mass., 1963.
4. Cremer, L., M. Heckl, and E. E. Ungar: “Structure-Borne Sound,” 2d ed., Springer-Verlag,
Berlin, 1988.
11.32 CHAPTER ELEVEN
8434_Harris_11_b.qxd 09/20/2001 11:17 AM Page 11.32
CHAPTER 12
VIBRATION TRANSDUCERS
Anthony S. Chu
INTRODUCTION
This chapter on vibration transducers is the first in a group of seven chapters on the
measurement of shock and vibration. Chapter 13 describes typical instrumentation
used in making measurements with such devices; Chap. 15 covers the mounting of
vibration transducers and how they may be calibrated under field conditions; more
precise calibration under laboratory conditions is described in detail in Chap. 18.
The selection of vibration transducers is treated in Chaps. 15 and 16. This chapter
defines the terms and describes the general principles of piezoelectric and piezoresis-
tive transducers; it also sets forth the mathematical basis for the use of shock and vibra-
tion transducers and includes a brief description of piezoelectric accelerometers,
piezoresistive accelerometers, piezoelectric force and impedance gages, and piezoelec-
tric drivers, along with a review of their performance and characteristics. Finally, the
following various special types of transducers are considered: optical-electronic trans-
ducers, including laser Doppler vibrometers, displacement measurement systems,
fiber-optic reflective displacement sensors, electrodynamic (velocity coil) pickups, dif-
ferential-transformer pickups, servo accelerometers, and capacitance-type transducers.
Certain solid-state materials are electrically responsive to mechanical force; they
often are used as the mechanical-to-electrical transduction elements in shock and
vibration transducers. Generally exhibiting high elastic stiffness, these materials can
be divided into two categories: the self-generating type, in which electric charge is
generated as a direct result of applied force, and the passive-circuit type, in which
applied force causes a change in the electrical characteristics of the material.
A piezoelectric material is one which produces an electric charge proportional to
the stress applied to it, within its linear elastic range. Piezoelectric materials are of
the self-generating type. A piezoresistive material is one whose electrical resistance
depends upon applied force. Piezoresistive materials are of the passive-circuit type.
A transducer (sometimes called a pickup or sensor) is a device which converts
shock or vibratory motion into an optical, a mechanical, or, most commonly, an elec-
trical signal that is proportional to a parameter of the experienced motion.
A transducing element is the part of the transducer that accomplishes the conver-
sion of motion into the signal.
A measuring instrument or measuring system converts shock and vibratory
motion into an observable form that is directly proportional to a parameter of the
12.1
8434_Harris_12_b.qxd 09/20/2001 11:15 AM Page 12.1
experienced motion. It may consist of a transducer with transducing element, signal-
conditioning equipment, and device for displaying the signal. An instrument con-
tains all of these elements in one package, while a system utilizes separate packages.
An accelerometer is a transducer whose output is proportional to the acceleration
input. The output of a force gage is proportional to the force input; an impedance
gage contains both an accelerometer and a force gage.
CLASSIFICATION OF MOTION TRANSDUCERS
In principle, shock and vibration motions are measured with reference to a point
fixed in space by either of two fundamentally different types of transducers:
1. Fixed-reference transducer. One terminal of the transducer is attached to a
point that is fixed in space; the other terminal is attached (e.g., mechanically, elec-
trically, optically) to the point whose motion is to be measured.
2. Mass-spring transducer (seismic transducer). The only terminal is the base of a
mass-spring system; this base is attached at the point where the shock or vibra-
tion is to be measured.The motion at the point is inferred from the motion of the
mass relative to the base.
MASS-SPRING TRANSDUCERS (SEISMIC TRANSDUCERS)
In many applications, such as moving vehicles or missiles, it is impossible to establish
a fixed reference for shock and vibration measurements. Therefore, many transduc-
ers use the response of a mass-spring system to measure shock and vibration. A
mass-spring transducer is shown schematically in Fig. 12.1; it consists of a mass m
suspended from the transducer case a by a spring of stiffness k. The motion of the
mass within the case may be damped by
a viscous fluid or electric current, sym-
bolized by a dashpot with damping coef-
ficient c. It is desired to measure the
motion of the moving part whose dis-
placement with respect to fixed space is
indicated by u. When the transducer
case is attached to the moving part, the
transducer may be used to measure
displacement, velocity, or acceleration,
depending on the portion of the fre-
quency range which is utilized and
whether the relative displacement or
relative velocity dδ/dt is sensed by the
transducing element. The typical re-
sponse of the mass-spring system is ana-
lyzed in the following paragraphs and
applied to the interpretation of trans-
ducer output.
Consider a transducer whose case
experiences a displacement motion u,
12.2 CHAPTER TWELVE
FIGURE 12.1 Mass-spring type of vibration-
measuring instrument consisting of a mass m
supported by spring k and viscous damper c. The
case a of the instrument is attached to the mov-
ing part whose vibratory motion u is to be meas-
ured. The motion u is inferred from the relative
motion δ between the mass m and the case a.
1
8434_Harris_12_b.qxd 09/20/2001 11:15 AM Page 12.2
and let the relative displacement between the mass and the case be δ. Then the
motion of the mass with respect to a reference fixed in space is δ+u, and the force
causing its acceleration is m[d
2
(δ+u)/dt
2
].Thus, the force applied by the mass to the
spring and dashpot assembly is −m[d
2
(δ+u)/dt
2
]. The force applied by the spring is
−kδ, and the force applied by the damper is −c(dδ/dt), where c is the damping coeffi-
cient. Adding all force terms and equating the sum to zero,
−m − c − kδ=0 (12.1)
Equation (12.1) may be rearranged:
m + c + kδ=−m (12.2)
Assume that the motion u is sinusoidal, u = u
0
cos ωt, where ω=2πf is the angular
frequency in radians per second and f is expressed in cycles per second. Neglecting
transient terms, the response of the instrument is defined by δ=δ
0
cos (ωt −θ); then
the solution of Eq. (12.2) is
ω
2
=
−ω
2
2
+
ω
2
(12.3)
ω
θ=tan
−1
−ω
2
(12.4)
The undamped natural frequency f
n
of the instrument is the frequency at which
=∞
when the damping is zero (c = 0), or the frequency at which θ=90°. From Eqs. (12.3)
and (12.4), this occurs when the denominators are zero:
ω
n
= 2πf
n
=
rad/sec (12.5)
Thus, a stiff spring and/or light mass produces an instrument with a high natural fre-
quency. A heavy mass and/or compliant spring produces an instrument with a low
natural frequency.
The damping in a transducer is specified as a fraction of critical damping. Critical
damping c
c
is the minimum level of damping that prevents a mass-spring transducer
from oscillating when excited by a step function or other transient. It is defined by
c
c
= 2 k
m
(12.6)
Thus, the fraction of critical damping ζ is
ζ= = (12.7)
It is convenient to define the excitation frequency ω for a transducer in terms of
the undamped natural frequency ω
n
by using the dimensionless frequency ratio
c
2k
m
c
c
c
k
m
δ
0
u
0
k
m
c
m
c
m
k
m
δ
0
u
0
d
2
u
dt
2
dδ
dt
d
2
δ
dt
2
dδ
dt
d
2
(δ+u)
dt
2
VIBRATION TRANSDUCERS 12.3
8434_Harris_12_b.qxd 09/20/2001 11:15 AM Page 12.3
ω/ω
n
. Substituting this ratio and the relation defined by Eq. (12.7), Eqs. (12.3) and
(12.4) may be written
=
2
(12.8)
1 −
2
2
+
2ζ
2
θ=tan
−1
2ζ
(12.9)
1 −
2
The response of the mass-spring transducer given by Eq. (12.8) may be expressed
in terms of the acceleration ü of the moving part by substituting ü
0
=−u
0
ω
2
.Then the
ratio of the relative displacement amplitude δ
0
between the mass m and transducer
case a to the impressed acceleration amplitude ü
0
is
1
=−
1 −
2
2
+
2ζ
2
(12.10)
The relation between δ
0
/u
0
and the frequency ratio ω/ω
n
is shown graphically in
Fig. 12.2 for several values of the fraction of critical damping ζ. Corresponding
curves for δ
0
/ü
0
are shown in Fig. 12.3. The phase angle θ defined by Eq. (12.9) is
shown graphically in Fig. 12.4, using the scale at the left side of the figure. Corre-
sponding phase angles between the relative displacement δ and the velocity ˙u and
acceleration ü are indicated by the scales at the right side of the figure.
ACCELERATION-MEASURING TRANSDUCERS
As indicated in Fig. 12.3, the relative displacement amplitude δ
0
is directly propor-
tional to the acceleration amplitude ü
0
=−u
0
ω
2
of the sinusoidal vibration being
measured, at small values of the frequency ratio ω/ω
n
. Thus, when the natural fre-
quency ω
n
of the transducer is high, the transducer is an accelerometer. If the trans-
ducer is undamped, the response curve of Fig. 12.3 is substantially flat when ω/ω
n
<
0.2, approximately. Consequently, an undamped accelerometer can be used for the
measurement of acceleration when the vibration frequency does not exceed approx-
imately 20 percent of the natural frequency of the accelerometer.The range of meas-
urable frequency increases as the damping of the accelerometer is increased, up to
an optimum value of damping. When the fraction of critical damping is approxi-
mately 0.65, an accelerometer gives accurate results in the measurement of vibration
at frequencies as great as approximately 60 percent of the natural frequency of the
accelerometer.
As indicated in Fig. 12.3, the useful frequency range of an accelerometer
increases as its natural frequency ω
n
increases. However, the deflection of the spring
in an accelerometer is inversely proportional to the square of the natural frequency;
i.e., for a given value of ü
0
, the relative displacement is directly proportional to 1/ω
n
2
[see Eq. (12.10)]. As a consequence, the electrical signal from the transducing ele-
ment may be very small, thereby requiring a large amplification to increase the sig-
nal to a level at which recording is feasible. For this reason, a compromise usually is
ω
ω
n
ω
ω
n
1
ω
n
2
δ
0
ü
0
ω
ω
n
ω
ω
n
ω
ω
n
ω
ω
n
ω
ω
n
δ
0
u
0
12.4 CHAPTER TWELVE
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