20.6 Measurements of Damping

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20.6.1 Sensor Considerations

The challenge to any measurement is to accomplish the task without significantly altering the system

under study (see Chapter 15). For measurements on mechanical oscillators of the type described in

Box 20.1

DAMPING CHARACTERISTICS

Type Equation of Motion Damping Capacity Q

Viscous €x þ 2b_x þv20

x ¼0 2pbvA2 m

v

2b

Hysteretic

(linear approximation)

x€ þ

h

mv

x_ þv2x ¼ 0 phA2 mv2

h

Hysteretic

(modified Coulomb)

x€ þ chA sgnðx_Þ þv2x ¼ 0 4chA2m pv2

4ch

Coulomb x€ þ

f

m

sgnðx_Þ þv2x ¼ 0 4fA pmv2 A

4f

Amplitude dependent x€ þ cf A2 sgnðx_Þ þv2x ¼ 0 4cf A3m pv2

4cf A

Damping Theory 20-23

© 2005 by Taylor & Francis Group, LLC

this document, two types of sensor are generally superior to every other kind: (i) optical and

(ii) capacitive. Optical sensors are probably the least perturbative but they do not readily yield

themselves to large dynamic range with good linearity (and small quantization errors for digital type).

Inductive sensors, such as the linear variable differential transformer (LVDT), are known from

seismology to be inherently more noisy (up to 100 times) because of ferromagnetic granularity.

Additionally, transformers are not amenable to miniaturization, and the components are inherently

less stable. It is therefore a mystery why the widespread use of the fully differential inductive sensor

(LVDT) continues when we have available the superior fully differential capacitive sensor, which is

electrically equivalent (apart from its reactance type) and capable of miniaturization to the MEMS

level. The challenge with really small capacitive sensors is the increase in output reactance of the device

as they approach femtoFarad levels of individual capacitors.

All measurements reported in this document were taken with the fully differential unit whose patent

name is “symmetric differential capacitive” (see Peters, 1993a, 1993b). It is especially useful for studying

mechanical oscillators of macroscopic size and, morphed to various forms, it recently has found

application in MEMS. It is capable of great sensitivity when configured in the form of an array, as shown

in Figure 20.4.

Various lines in Figure 20.4 correspond to narrow insulator strips, such as the single vertical line in the

set that connects to the amplifier. In the cross-connected static set, the plates labeled “1” are electrically

distinct from the others labeled “2”. The total-plate arrangement constitutes a symmetric AC bridge, and

the central position of the moving set (x ¼ 0 as shown in the figure) corresponds to bridge balance with

V0 ¼ 0: Displacement away from balance gives a voltage output that is linear between 2w=4 and w=4; as

illustrated in the graph at the bottom of Figure 20.4.

The oscillator frequency is typically tens of kHz, and the amplifier is of instrumentation type

(Horowitz and Hill, 1989). Unlike a bridge null detector, the linear response through x ¼ 0 is realized

cross-connected

static 4 equipotential pairs

1

w

x

2

2

2

2

2

Oscillator

response using

synchronous

detection

moving electrode set

static pair of electrodes

Amplifier

V0

V0

−w/4 w/4x

1

1

1

1

−x

FIGURE 20.4 Illustration of a fully differential capacitive transducer array. For clarity, the three electrode-sets are

shown separated from their operating positions (parallel with a small separation gap, with the moving electrodes in

the middle).

20-24 Vibration and Shock Handbook

© 2005 by Taylor & Francis Group, LLC

when synchronous detection is employed. This can be accomplished with a lock-in amplifier, but the

most recent Cavendish balance to employ the sensor uses diodes (Tel-Atomic Inc., online at http://www.

telatomic.com/sdct1.html).

A tutorial (“detailed explanation”) of the SDC sensor using diodes can also be found at this website.

In Figure 20.4, four individual SDC units have been shown connected in parallel. The total number, N;

of individual units in an array depends on the characteristic width, w; for which the total range of

detectable motion is w=2: If the requirement on range is small, then N can in principle be made very large,

which is desirable for the following reason. The sensitivity of this position sensor is inversely

proportional to w if output capacitance of the device is not a factor. As w is reduced, however, the

degrading influence of increased output reactance (capacitive) is more significant than the improved

sensitivity that would result if the sensor could be connected to an amplifier with infinite input

impedance. Since the instrumentation amplifier’s input capacitance is not negligible, shrinking w is

beneficial only if the output reactance can by some means be kept low. This is accomplished with the

array of individual units. In principle, the output capacitance could be held reasonably constant as N

approaches 100, by using photolithographic techniques and small spacing between the parallel

electrodes. The concept has been deemed feasible because of existing technologies as well as the following:

although not in the form of an array, Auburn University has fabricated a mesoscale accelerometer

around the SDC sensor. The prototype was built on printed wire board (PWB) under U.S. Army contract

(Dean, 2002).

No doubt the popular silicon-based MEMS accelerometers marketed by Analog Devices utilize the

impedance advantages of an array, employing a large number of “fingers” in a force-feedback

arrangement. Although employed mostly otherwise, the first case of a fully differential capacitive

transducer using force-feedback was one based on simultaneous action of actuator and sensor functions

in a single unit of nonlinear type (Peters et al., 1991).

20.6.2 Common-Mode Rejection

In attempts to measure damping, one can be confronted with difficulties of mode mixing. For example,

the historical Cavendish experiment, using optical detection, has been traditionally difficult unless the

instrument is placed in a very quiet location to avoid pendulous swinging of the boom. The highfrequency

pendulous motion (of the order of 1 Hz) as a “noise” becomes superposed on the lowfrequency

torsional signal. The computerized Cavendish balance sold by Tel-Atomic overcomes this

problem by means of a mechanical common-mode rejection feature. An SDC sensor placed near one

boom end is connected in electrical phase opposition to a second SDC sensor placed near the other end

of the boom. The boom itself serves as the moving electrode for both sensors. Neither sensor has a

first-order response to boom motion parallel to its long axis. Pendulous motion perpendicular to the

boom orientation is largely canceled.

20.6.3 Example of Viscous Damping

The aforementioned Sprengnether– LaCoste spring seismometer is well-suited to the demonstration of

viscous damping, when damping is imposed in the following manner: the instrument was built with a

Faraday Law (velocity) detector; i.e., a coil that moves with the mass of the instrument, in the field of a

stationary magnet. As originally employed, the coil was connected to the amplifier of a recorder. In the

present configuration, however, the velocity detector is not employed, since its sensitivity is severely

limited at low frequencies. Instead, an SDC array of the type shown in Figure 20.4 is used to measure the

position of the mass (a pair of lead weights, total mass 11 kg). If the instrument is operated with the coil

open-circuit, there is no induced current. By connecting a resistor across the coil (through very fine

copper wires that go to terminals on the case), mass motion induces a current. The induced current

opposes the motion through Lenz’s Law, resulting in damping. The damping depends on the size of the

Damping Theory 20-25

© 2005 by Taylor & Francis Group, LLC

current and is thus an inverse function of the resistor’s magnitude through Ohm’s Law. The phenomenon

is illustrated in Figure 20.5.

As compared with the “undamped” instrument, whose Q is approximately 80 at a period of 17 sec, it is

seen that the addition of a 990-ohm resistor lowered the Q by more than an order of magnitude to 4.9.

A 330-ohm resistor reduced it even further to 3.1. The amount of damping is also governed by the

resistance of the coil winding, which is 480 ohm.

The envelopes that have been fitted to the decay curves were the basis for estimating the Q: The decay

data were imported to Excel by first outputting the Dataq DI-154RS A/D generated record as a p.dat

(CSV) file. The fits were produced by trial and error using the drag and autofill functions. Notice that the

990-ohm resistor (first) case is not as pure an exponential decay as the other case because of creep.

The rate of creep is greater at large initial amplitudes of the motion.

20.6.4 Another Way to Measure Damping

Curve fitting (full nonlinear, in general) is the best way to estimate damping parameters, especially

if the decay is not exponential. For more routine cases, simpler methods can be used. Among the

host of ways that have been defined to specify the damping of an oscillator, one of the most

common uses the logarithmic decrement. The solution to Equation 20.1 with zero right-hand side is

given by

xðtÞ ¼ x0 e2bt cosðvt þ fÞ: ð20:16Þ

The full-cycle turning points, xN ¼ x0 e2bNT ; with N ¼ 0; 1; 2; … can be used to compute the

logarithmic decrement through

bT ¼

1

N

ln

x0

xN ð20:17Þ

Unfortunately, an estimate based on Equation 20.17 can be difficult due to the presence of either or

both of two problems: (i) mean position offset in the decay record or (ii) asymmetry of the decay,

where the turning points on one side of equilibrium decay at a different rate than those on the

other side. Case (ii) occurs more often than one might expect; it is frequently a consequence of

material complexity and not the result of nonlinearity in the electronics of the detector. It is

important, however, to be sure that the detector is either linear or that corrections for the

nonlinearity be utilized before estimating the damping.

1.5

1

0.5

0

0 20 40

creep rate = 0.8 mV/s

Time (s)

Sensor calibration constant = 2000 V/m

Time (s)

Q = 4.9

Period = 17 s

Sensor output (V)

(R = 990 Ω)

Q = 3.1

(R = 330 Ω)

60 80

−0.5

−1

−1.5

Sensor output (V)

1

0.5

0

0 20 40 60

−0.5

−1

FIGURE 20.5 Examples of induced current damping of a vertical seismometer using two different resistors.

20-26 Vibration and Shock Handbook

© 2005 by Taylor & Francis Group, LLC

A method to provide partial compensation uses half-cycle turning points n ¼ 2N; and works with a

minimum of three such points.

bT ¼ 22 ln½1 2 ðxn21 2 xnþ1Þ=ðxn21 2 xnÞ􀀉 ð20:18Þ

Advantage is taken of random error reduction by using Equation 20.18 on a set of turning points

(optimal number sometimes being about a dozen). The calculations are straightforward in a spreadsheet

such as Excel by means of the autofill function.

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