Around resonance, the force transmitted by an exciter into a structure will be smaller or greater than a reference force generated by its coils due to electromechanical interaction. A simple analysis is presented which reveals how this phenomenon of force drop-off is controlled by three factors. The first factor, called Armature Mass Factor, describes a purely mechanical interaction between the structure and the exciter. The electromechanical energy conversion and its interaction with the structure yields two additional factors, called Electrical Resistance and Electrical Inductance Factors. They describe the effects of coil resistance, inductance and magnetic field strength relative to structural damping and stiffness. Present analysis indicates that, under proper circumstances, more than 90 percent of the force drop-off can be eliminated if armature-to-structure mass ratio is smaller or equal to half of modal loss factor

Rao, Dantam K.

English

NASA Technical Reports Server

N88- 13624
On the 'Glitches' in the Force Transmitted by an
Elcctrodynamic Exciter to a Structure
Dantam K. Rao
Around resonance, the force transmitted by an exciter into a
structure will be smaller or greater than a reference force
generated by its coils due to electromechanical interaction.
I simple analysis presented herein reveals how this
phenomenon of force drop-off is controlled by three factors.
The first factor, called Armature Mass Factor, describes a
purely mechanical interaction between the structure and
exciter. It signifies the value of armature-of-structure
mass ratio relative to the modal loss factor. The
electromechanical energy conversionand its interaction with
the structure yields two additional factors, called
Electrical Resistance and Electrical Inductance Factors.
They describe the effects of coil resistance, inductance and
magnetic field strength relative to structural damping and
stiffness. Present analysis indicates that, under proper
circumstances, more than DOg of the force drop-off can be
eliminated if armature-to-structure mass ratio is smaller or
equal to half of modal loss factor.
I_rI_RODUCTION
Traditionally, in a typical measurement set-up, the force needed to
vibrate a grounded structure is generated by an attached electrodynamic
exciter. We usually assume that, in such set-up, a constant force is
transmitted into the structure if a uniform sine voltage is inputted into the
exiter via a power amplifier as shown in Fig. I. But in reality, the
amplitude and phase of transmitted force is substantially different from the
force generated in the coils (around the resonance frequency) due to
electromechanical interaction between exciter and structure, even if input
voltage is constant. A Force Glitch describes these local differences in the
force transmitted into a structure around its resonance frequency. (In
contrast, a Xotion Glitch describes local variations in the table base-motion
excitation of a free structure. We do not intend to study them here). These
glitches can be smoothened by a compressor loop, but we assume that our
measurement setup does not have such a loop.
As shown in Fig. 2, a glitch consists of a Peak and a Notch in the plot
of transmitted force vs frequency around the structural resonance. At the
Notch frequency, this force drops to the lowest level, while at the Peak it
rises to its highest value. The Notch frequency equals the resonance of the
entire vibrating system.
PRI_CEDINO PAGE BLANK NOT FILMED
245
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Our major interest lies in analyzing the factors causing Force [ExDp-off viz. drop
in level of force transmitted (from that generated in the ooils) to the Notch value.
We review below some (but not all) literature dealing with the foroe drop-off.
Historically, many '_cal" models have been used to explain the force drop-
off. They account only for the mechanical parts; they also presume that armature
coil generates a constant-amplitude force. Ewins [i] used a 1-degree model to
explain how the transmitted force becomes small at the stru_ resonance
frequency. Earlier, Granick and Stern [2] analyzed a 2-degree model to show that
the Notch frequency equals the stru_ resonance, while Bangs [3] analyzed the
effect of structural nonlinearity. Rao [4] described a 3-degree model to include a
force transducer.
A few researchers have also employed an "el_cal" model. This model
accounts for all vibrating parts, including electrical and electromechanical
conversions; they presume that the armature coil generates a force proportional to
current flow. An earlier review by Rao [5] recorded some pertinent literature on
equations for exciters; these equations are identical to Crandall et al [6].
Extensive work by Tcmlinson [7,8] showed that the transmitted force can be distorted
if the table vibrations are so large that nonlinear solenoid effects come into play.
Recently Olsen [9] established that a "smaller" armature-to-structural mass
ratio, viz., lighter armature, is required to reduce the force drop-off. (Research
prior to 60's showed [i0] that the motion glitch can be smoothened by selecting a
heavier armature, i.e., a larger armature-to-structural mass ratio. )
Thus we know that a "smaller" armature-to-structural mass ratio reduoes force
drop-off. But, a question of practical interest to the experimenter is, how "small"
should this ratio be? Should it be i/i00 or 1-in-million? This paper attempts to
quantify this ratio. Another major aim of this paper is to identify and investigate
the effect of any "el_ical" factors that reduce the foroe drop-off (in
addition to mechanical factors).
/ y
\ r • • "4% ENERGY
PNViRONMENT (m) Ix ii (m) LOSS
FORCE v_ MOTION
S_PPLY _EXCITEROWER OU_,trr
I b i
INPUT_ POWER AMP IVOLTAGE TYPE" (m) - MOUNT MEANS
PEAK
-- _ 'GLITCH'
REF - "----T'-
/_ NOTCHN
FREGUENCY Nz
(oJ 'GLITCH' IN THE FORCE (I)1 GLITCH CHARACTERISTICS
SPECTRUM
force
drop -off
Fig. 1 Typical Setup for Measuring
Frequency Response
Fig. 2 Force Glitch Consisting of
a Peak and a Notch.
246
NOTATION
e _-
i =
f¢ =
fd =
fo ----
j --
k_ =
k =
k =
L =
m =
m 8 =
R
X ----
3 --
&0 =
% =
d t =
subo=
input sine voltage
current flowing in the coils
ampl. of force ,, ,,
force drop-off
ampl. of force transmitted
/-q--
force-to-current ratio
the complex modal stiffness
_+J _p_) of modal stiffness
self-inductance of the coil
the modal mass of structure +
seismic part of force transducer
the eff. mass of armature +
part of force _dcr + stinger
resistance of coil + source
displacement of str. + armature
the structural modal loss factor
frequency of excitation
i3atllz-dlfrequency of structure
time derivative d( )/dt
amplitude (real or complex)
Factors Controlling the Foroe Drop-off
M = Armature Mass Factor (eq. 4)
C = Electrical Resistance Factor (eq. 9)
K = Electrical Inductance Factor (eq.10)
STRUCTURE
/I///F// / /
< .
ARMATURE
FORCE
GENERATED
BY COILS, fc exp(juJt)
Fig. 3 '_echanical" Model
'_ECHANICAL" MODEL OF STRUCIURE ATEAf_ED TO AN EXCITER
Formula for Force Drop-off as a Function of Armature Mass
The equation of motion of a grounded structure attached rigidly to the armature
of an exciter is (see Fig. 3 and Ref. [5] for assumptions)
(m + ma)x + k'x = fc exp (j_t) (i)
We rewrite this equation in the standard form m x + k*x = fo exp(j_t)
where f o , denoting the c_mplex a_plitude of foroe transmitted into the structure,
is given by the differenoe [i] between the force generated in the coils and the
inertial force needed to vibrate the armature,
fo(@)) = fc + _02 ma Xo
k*- &_2m
k* - 6o2 (m+ma)
fc (2)
247
whez_the complex amplitude of displacement x o is obtained by solving (i). As shown
in Fig. 2, we define the "force drop-off" f_ as the difference between the
amplitude of reference force generated in the colls at zero-frequ_, fc , and the
amplitude of the force transmitted into the structure at the natural frequenc_ &_N
= _ We use (2) to express the force drop-off in terms of a nondimensional
factor M as given below:
fd = fo (0) - fo(_N )
1
= [ Z-- --]
/(i + M 2}
fc (3)
where the Armature Mass Factor M controlling force drop off is
m a /m armature-to-structure mass ratio
M = = (4)
str. loss factor
Effect ofArmatureMassonAmplitudeandPhaseofTransmittedForoe
We display in Fig. 4 how armature mass influ_ transmitted force. This
figure confirms the well known fact that a lighter armaturebeneficially reduces the
force drop-off; but this also detrimentally reduoes the frequency range between the
Peak andNotch.
More significant is the additional ph_n of pP_%se-drop revealed by this
Figure. The phase of the force signal (relative to that of foroe in the coils)
drops to its iciest value at the Notch frequency and rises beyond it. This results
in oonsiderable fluctuations in the phase around the resonance frequency.
,,, 20
iF
m <
-'It) 10
2is .oils o
\ ' pA-'-__
o-
- \ v:o,
-_ n='O01 I
I
.975 1.000 1.025
Fig. 4
maim
.10
.950 .975 .1_0
Effect ofArmatureMass on the Force Transmitted
18o-
go e
0 _
-90"
-180"
1.025
(._ EXCITATION FREQUENCY
_N NATURAL FREQUENCY
248
For lighter armatures, this figure shows that the phase can fluctuate by as muchas
two full out-of-phase 180 deg. turns over a very narrow frequency range. The rate of
rise in the Phase beyond Notch frequency appears, hc_ever, to be independent of
armature mass. Hence although a lighter armature reduces the force drop-off and
phase drop, we still need to use adequate frequency resolution to follc_ the sharp
rise in the phase beyond the Notch frequency.
We display in Fig. 5 how the loss factor affects the force transmitted. This
figure shows that heavier damping reduoes the force drop off and widens the
frequency range between the Peak and Notch. It also has the beneficial effect of
reducing the _Pk3se drop; further the phase changes at a slower pace around the
2010
-I°
w>
1.0
"1°z w<UE _g
_" 0 .10
II
01"° .01
.950
' A ° ' IA.P,,Tuoe
- .005 -
[ .01 -
.02 -
, V -
__VN - -
FORCE DROP-OFF
I I
• 975 1.000 1.025 .950
I I PHASE ,
,7= ,02 --
.01 --
.005
.001
I
F
N
.975 1.000
180"
90 °
0 •
-90"
-180"
1.025
= EXCITATION FREQUENCY
_N NATURAL FREQUENCY
Fig. 5 Effect of Structural Loss Factor on Force Transmitted
Fig. 6 shows how the force drop-off is controlled by the Armature Mass Factor.
From this figure, we conclude that 90% of coil-generated force can be transmitted
into structure by choosing an armature-to-structure mass ratio that is less than
half of the structural loss factor. This leads to a thumb rule, herein called the
Half-Loss Factor Rule. Briefly stated, it recommends use of a light armature whose
weight obeys the rule:
armature-to-structure mass ratio < half-of-loss factor (5)
Then it is possible to transmit 90% of generated force into the structure at the
frequency of resonance. For example, a structure with a modal mass = i0 kg and
modal loss factor = 1/50 will require an armatur weighing 0.I kg for the force drop-
off to be 10%.
249
_CAL MODELOFSTRUCIIYREATTA(_EDTOANEXCITER
Formula for Force Drop-off Including El_ical Factors
Exciters work the principle of el_cal conversion, an idealized version
of which is shown in Fig. 7 as an conversion box. Ideal lossless electrical
ir_sutted into it outputs mechanical force on a mass-less, frictionless push-rod.
Fig. 8 shc_s how, in practical situations, the ideal electrical input is modified by
the electrical resistance R and self-_ L of the ooil and the
,. uJ "z
.. ,,-
0 0
,, ,, .al 
I_ 01
u. • 99%
.1 1.0 10
M - ARMATURE-TO-STRUCTURE MASS RATIO
STRUCTURAL LOSS FACTOR
6 Combined effect of Armature Mass and Str. Loss Factor on Force Drop-Off.
mechanical output by the mass of armature and structural properties. The equation
of motion (i) thus modifies to (see [5] for details assumptions and derivation)
(m + ma)x + k*x - ksi = 0
k B x + (R+Ldt)i = e o exp(j_t) (6)
We rewrite first of this equation in the standard form m x + k*x = f0 exp(j _t)
where f0 , denoting the (x_plex amplitude of force transmitted into the structure,
is given by the diff_ between the force generated in the ooils (that is now
proportional to the current) and the inertial force needed to vibrate the armature,
fo(&_) = ksi o
[k* -_2(m+ma)][R+j_L] + j_k8 2
fc (7)
250
/7777
FIELD DUE TO
CURRENT I
AWAY FROM
OBSERVER
BACK emf eb=-kB_
FORCE GENERATED
BY COILS fc ---- kB I
COIL
Fc = kBi-- X
_FRICTIONLESS
_" PUSH-ROD
eb__kB__ kB
(a) ELECTROMECHANICAL TRANSDUCTION (b) IDEAL MODEL
Fig. 7 Ideal Electrom_chanical Transducer
STRUCTURE
FORCE
TRANBMI'I-rED
Fig. 8 Electromechanical Model of a Structure+Exciter
where x o and i o denote amplitudes of displacement and current that are obtained by
solving (6) and f c now denotes the force kee_/R transmitted into the structure at
the zero frequency. The force drop-off now depends on two factors since
fd = fo(0) - fo(&JN)
1
= [ i - ] f_ (8)
V'{ (I_c/K_)2 + (mc/_)2 )
251
Here C, called the Electrical Resistance Factor, quantifies nondimensionalized
el_ical damping whereas K, called the Electrical Inductance Factor, quantifies
nondimensionalized electrical stiffness, both expressed relative to structural loss
factor, and are defined by
(k, 2/R)I /km elec. damping-to-str, crit. damping
c = = (9)
str. loss factor
(kB2/L)/k elec. stiffness-to-str, stiffness
K = = (10)
str. loss factor
Effect of El_cal Factors
Fig. 9 shows how the Electrical Resistance Factor C affects the force
transmitted. It reveals that lower resistance can reduce the transmitted; it can
also introduce unacceptable violent fluctuations in the phase. For example, for the
parameters illustrated, the phase shows a drop-rise-drop-rise pattern over -180 ° to
+180 ° between Peak and Notch. This is in contrast to the drop-rise pattern exhibited
by the mechanical model as shown by Figs. 4 and 5.
Fig. i0 exhibits how electrical inductance factor K influences the force drop-
off. This figure shows that a i_ inductance can reduoe the force transmitted
and introduce unacceptable _rise-drop-rise fluctuations in the _hase. These two
figures re-emphasize the need for adequate frequency resolution to measure the phase
of the force signal.
ii
10
PO
zu_
a 1
<o
_- 0 0.1
116
II
0.1
1.0
5.0--
10.
--- o.1 1
- 1.0
- 5.0
180"
I PHASE
_--1.0
-- 5.0
L--IO.
90 °
0 o
°l°
" "" .01 50 .975 1.00 1.025 .950 .975 1.000 1.025
-90"
-180"
(_J = EXCITATION FREQUENCY
(.d N NATURAL FREQUENCY
Fig. 9 Effect of Electrical Resi_ Factor C on Force Transmitted
252
2.0| K T T I ] I I ] 180"
/ l Pm./.,=.O'/
.1o K .1o
g _ .os " .50 .05 .
.0 .,o
O il _ ".
| -11_m
• 950 .975 1 ._ 1.025 .950 .975 1 ._ 1.025
(ON
EXCITATION FREQUENCY
NATURAL FREQUENCY
Fig. i0 Effect of Electrical Inductance Factor K on Force Transmitted.
The effect of Resistance Factor C on the force drop-off is revealed in Fig ii.
This figure shows how a reduction in C value (i.e., _ in resistance) can
eliminate the force drop-off. Similar effect can be obtained by increasing the K
value (i.e., reducing the inductance) as shown in Fig. 12.
Thus, by a judicious choice of M,C and K values, we can oontrol the force drop-
off observed at the resonance frequenc_ of the structure.
It
3
'" w
o o
-" a:
O0
M. 14.
O
IJ,.
.i
'u
1.0
.01 I I I [ ,oL,,,,_, t I
.1 1
ARMATURE-TO-STRUCTURE MASS RATIO
0%
2O
6O
Z
3 --
MJ
II o
3 "0
90 _ _
0 uJ
0 --
u.
M _
STRUCTURAL LOSS FACTOR
99%
10
Fig. ii Effect of Electrical Resistance Factor on Force Drop-off
253
.I
i
w_
II
._ .01
.1
I
k.<
/ 1'/2 - L __1
I
I I I I L I I I I I
1 10
ARMATURE-TO-STRUCTURE MASS RATIO
M=
STRUCTURAL LOSS FACTOR
Fig. 12 Effect of Electrical Inductance Factor on Force Drop-off
6O
99%
o
OONCLUSIONS
The present paper identified three factors that affect the force transmitted by
an electrcdynamic exciter into a structure around the resonanoe frequency. This
force transmitted is shown to depend on three factors. A purely mechanical factor,
called Armature Mass Factor, describes the armature mass-to-structural mass ratio
relative to the structural loss factor; it should be less than 1/2 to transmit more
than 90% of force generated in the ooils. The remaining two factors, called
Electrical Resistance Factor C and Electrical Irductance Factor K describe the
effect of coil resistance, inductance and magnetic field strength relative to
structural damping and stiffness. Present analysis also revealed the phezKmnenon of
phase-drop (in addition to the well-known ph_n of force drop-off) that occurs
around the resonance frequency. It also shows that the Electrical Resistance Factor
should be decreased while Inductance Factor should be increased in order to reduce
the force drop-off.
REFERENCES
i. D.J. Ewins, Modal Testinq Theory and Practice. John Wiley Sons, 1984.
2. N. Granick and J.E. Stern, '_Material Damping of Aluminum by a Resonant Dwell
Technique," NASA TN D-2893, 1965.
3. W.F. Bangs, "Sinusoidal Vibration Testing of Nonlinear Spacecraft Structures,"
Shock, Vibration and Associated Environments, Bulletin No. 33, Part III, Mar.
1964, pp. 195-201.
254
. D.K. Rao and D.I.G. Jones, "Investigation of Modes, Frequencies and Forced
Response of a High Frequency Test System," Shock and Vibration Bulletin, No. 55,
Part 2, pp. 27-38, June 1985.
5. D.K. Rao, "Electrodynamic Interaction Between a Resonating Structure and an
Exciter," Proc. 5th Int. Conf. Modal Analysis, Apr. 1987, London.
6. S.H. Crandall, Dynamics of Mechanical and El_ical System_,McGraw-Hill
Book Co., 1968, p. 308.
o G.R. Tomlinson, "A Simple Theoretical and Experimental Study for the Force
Characteristics from El_c Exciters on Linear and Nonlinear Systems",
Proc. 5th Int. Modal Anal. Conf., London, Vol. 2, pp. 1479-1486, Apr. 1987.
. G.R. Tcmlinson, "Force Distortion in Resonanoe Testing of Structures with
electro-dyr_%T_c vibration Exciters", J. Sound and Vib., Vol. 63, No. 3, pp. 337-
350, 1979.
. N.L. Olsen, 'K/sing and Understanding Electrodynamic Shakers in Modal
Applications," Proc. 4th Int. M,d. Anal. Conf., Ins Angeles, Vol. 2, pp. 1160-
1167, Jan. 1986.
I0. K. Unholtz, '_Vibration Testing Machines", in Shock and Vibration Handbook, ed.
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255


On the glitches in the force transmitted by an electrodynamic exciter to a structure

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