Пресс-релиз популярных книг

Авторы: 111 А Б В Г Д Е Ж З И Й К Л М Н О П Р С Т У Ф Х Ц Ч Ш Щ Э Ю Я

Книги: 164 А Б В Г Д Е Ж З И Й К Л М Н О П Р С Т У Ф Х Ц Ч Ш Щ Э Ю Я

На сайте 111 авторов, 92 книг, 72 статей, 5913 глав.

21.4 Damping Examples

Back

21.4.1 Case 1: Vibrating Bar — Linear with Significant Noise

The simplest means to measure the Q of an oscillator whose frequency is in the range of the human ear is

to use a microphone connected to a digital oscilloscope. In this case, the microphone was an inexpensive

dynamic type and the oscilloscope was a Tektronix TDS 3054. A better choice, had it been available,

would be an electret microphone. The ring-down of a xylophone bar, following a strong (sharp) hammer

strike, is shown in Figure 21.5.

The voltage versus time of the microphone output was saved to memory in the oscilloscope, from

which the digital record was output to a floppy disk, using the CSV format. Data from the disk were read

into columns A and B of an Excel spreadsheet using “Open file.” An envelope fit was then performed on

the turning points by placement of trial and error data into column C, using “autofill.” A separate graph

was generated for each value of the constant b in the expression “¼ 0:04 p expð2b p A1Þ” typed into Cell

C1. (The lower turning points were obtained by typing “¼ 2b1” into Cell D1 and using autofill.

21-8 Vibration and Shock Handbook

(Additional details concerning the use of Excel in this manner will be provided in the discussion of

seismometer damping that follows.)

Although optimizing algorithms could be generated to perform such a fit (with a probable slight

increase in accuracy), this visual technique is preferred here as it is more understandable, user-friendly,

and its performance is proven. The total time required in Excel to generate Figure 21.5 using a 2K record

(2048 points) is typically only a few minutes with a modern Pentium computer.p

Once a satisfactory fit was obtained (b ¼ 1:6 in Figure 21.5), the Q was estimated using

Q ¼

D ¼

b ð21:1Þ

There is a fair amount of electronic noise in Figure 21.5 because the microphone was connected

directly to the oscilloscope. The smallest bandwidth of the oscilloscope, at 20 MHz, causes a large

amount of Johnson (white) noise. Narrowing the bandpass by means of a preamplifier would improve

the quality of the data dramatically. Such is typically true of signal to noise ratio (SNR) improvement

by tailoring the electronics to the need.

21.4.2 Case 2: Vibrating Reed — Example of Nonlinear Damping

To illustrate another sensing technique, the system shown in Figure 21.6 was used.

A 908 twist was given to a hacksaw blade after heating with a torch and quenching. One end was

clamped to the vertical post shown and a piece of cardboard was taped to the other end. An incandescent

lamp is placed above the cardboard, which vibrates horizontally, and a solar panel below the cardboard is

used as a sensor. The solar panel in this case is a commercial unit that comes with a cigarette lighter plug

for charging automobile batteries. The output from the panel goes to the Tektronix digital scope also in

the picture.

Unlike other sensing schemes described in this chapter, the solar panel output is not bipolar but

instead has a constant voltage offset corresponding to the equilibrium position of the reed.

The frequency of oscillation is too low to operate the oscilloscope with a.c. coupling. Thus, it is

important to make the d.c. offset as small as possible. This was accomplished by shielding nonactive parts

pA disclaimer is in order at this point. Present comments by the author should not be interpreted as an endorsement of

Microsoft products in general. Although QuickBasic and Excel have both proven unusually beneficial to the work described

in this chapter, they are the only software packages marketed by the company to have received a strong endorsement from the

author.

FIGURE 21.5 Free-decay record of a vibrating bar.

Experimental Techniques in Damping 21-9

of the solar cell from the lamp. Although the d.c.

offset could be removed with a voltage bucking

battery, this was found to introduce unacceptable

noise spikes. With the offset, the gain of the

electronics was limited by the amount of vertical

position shift allowed by the scope. The results of

this study are illustrated in Figure 21.7.

The nonzero value of b (0.018) in Figure 21.7

indicates the presence of amplitude-dependent

damping (refer to Equation 20.61, Chapter 20).

The nonlinear damping in this case probably

derives from the air, rather than internal friction of

the hacksaw blade. Its presence causes the Q of the

system to increase with time.

The Q of the system is calculated from the expression

Q ¼

ðb þ ayÞt ð21:2Þ

where t is the period of oscillation. At the start of the record ðy ¼ 0:12Þ the Q is 390 and it approaches

2700 as the amplitude approaches zero.

21.4.3 Case 3: Seismometer

Since the ability of a seismometer to detect tremors is proportional to the square of the period of the

instrument, they require a good low-frequency sensor. The most common sensor for the latest generation

commercial instruments is a half-bridge (differential) capacitive type. Because of the greater sensitivity and

linearity of the full-bridge symmetric differential capacitive (SDC) sensor mentioned in Chapter 20, it is

well suited to these applications, being easy to employ. (The full-bridge character is described in a TELAtomic

tutorial (Peters, 2002).) A significant advantage of the SDC symmetry (equivalent electrically to

the inductive LVDT) is its relative insensitivity to construction imperfections, such as roughness of surface

and nonparallelism of electrodes. Thus, construction can be done crudely without serious degradation of

performance. For example, electrodes of the first prototype of the SDC sensor were fabricated from sheet

FIGURE 21.7 Vibrating reed decay with amplitudedependent

damping.

FIGURE 21.6 Setup for measurement of vibrating reed free-decay.

21-10 Vibration and Shock Handbook

copper that was cut with shears and subsequently flattened by hammer on a hard plane surface. This stands

in stark contrast to the optically polished surfaces necessary for the use of some sensors.

The Sprengnether vertical seismometer discussed at several points in Chapter 20 was studied in a

configuration for which Q ¼ 4:9; as determined by an external 990 V resistor to provide induced-current

damping. The resistor was connected across the coil that is part of the original equipment and which

moves (with the mass of the instrument) in the field of a stationary magnet, i.e., a Faraday’s law (velocity)

detector. Excitation to initiate the free-decay study was accomplished by applying an alternating (square

wave) current to the coil, reversing the direction of the current at each turning point of the motion of the

mass. The fundamental (Fourier series) of a square-drive generated this way is shifted 908 from the mass

motion, corresponding therefore to resonance. After cessation of the drive, data as shown in Figure 21.8

were collected with a Dataq DI-700 ADC (16-bit).

The graph in Figure 21.8 was generated with Excel after the Dataq record was saved to floppy disc as

an p .dat (CSV) file. It was imported to Excel using “open file” with “comma delimiter.” Once in Excel,

these data were shifted one place to the right (from the default A column to the B column) to

accommodate computer generation of a time-data column. The column of time values was generated

according to the sample rate, the value of which is by default saved to the data file. To generate the time

column, a 0 was placed in the first row, n corresponding to the start of data. Dropping down one row in

the A column, “¼ An þ 1=ðsample rateÞ” was typed, to increment the time. Then the lower right hand

corner “small solid square” of the box containing this time was grabbed and held with the left button of

the mouse to autofill all the way to the last time point of the data. The computer-generated exponentials,

which correspond to the turning points, were obtained by generating two additional columns. These

were obtained by placing the cursor at a row corresponding to the time An (in column C) and then typing

“¼ A0 p expð2ðomega=2=QÞ p AnÞ:” The value of A0 is obvious from the data and a first estimate for Q can

be quickly obtained from about a dozen turning points (read with the Dataq software before the data are

ever saved).

The technique is illustrated in Table 21.1. For example, in the case of Figure 21.8, Q ¼ 4:8 from the 13

turning points. Thus the argument of the exponential was set to 0.0385. Using autofill, the columnar

(upper) exponential was then quickly produced. Then a second (adjacent) column D was generated in

similar manner, by taking the negative of the last point and then autofilling to the top row. (When one

autofills downward, the rate with which Excel traverses the rows increases exponentially after the last row

of data has been passed; thus, it is much easier to fill upwards rather than downwards.)

Once the pair of exponentials being fitted to the data have been graphed, along with the data, it is

simple to adjust the curves by varying the argument (in this case, small changes around 0.0385) until a

0.50

0.40

0.30

0.20

0.10

0.00

0 50 100 Time (s)

(Q = 4.9 from fit, 4.8 from turning pts)

(Period = 17 s)

Sensor Output (V)

−0.10

−0.20

−0.30

−0.40

−0.50

Seismometer Free-Decay

FIGURE 21.8 Free-decay record of the Sprengnether vertical seismometer, period 17 sec.

Experimental Techniques in Damping 21-11

good fit is obtained (using autofill each time). The fit is rapid and accurate when there are not too many

parameters to vary, since the eye is well suited to this operation. Upon obtaining the best-fit by this

means, the preliminary value of Q ¼ 4:8 was altered to the final value of Q ¼ 4:9:

Note that in Table 21.1 a new Excel worksheet was employed (column A no longer the time as in the

discussion above). The voltages corresponding to the turning points (maximum and minimum) were

each read by placing the cursor at an extremum and manually recording the value displayed in turn by

the Dataq software. These values were then typed into Excel, as opposed to the “file open” method for

importing large datasets, i.e., as used to generate Figure 21.7.

21.4.4 Case 4: Rod Pendulum with Photogate Sensor

One of the simplest ways to measure damping at larger levels is to use a photogate of the type common to

general education physics laboratories. The infrared beam of the photogate is tripped by a “flag” attached

TABLE 21.1 Estimation of Q from the Turning Points

Use of Excel to estimate logarithmic decrement from turning points of the motion

Free-decay of Cavendish balance

Free-decay of Seismometer

1234567

−0.252

0.247

−0.185

−0.194

−0.133

−0.155

−0.095

−0.062

−0.037

0.107

0.088

0.024

0.292266

0.25322

0.272743

0.122

−2*ln(1−(A1−A13)/(A1−A2+A3−A4+A5−A6+A7−A8+A9−A10+A11−A12))

−2*ln(1−(A2−A12)/(A2−A3+A4−A5+A6−A7+A8−A9+A10−A11))

B15 = −2*ln(1−(A3−A15)/(A3−A4+A5−A6+A7−A8+A9−A10+A11−A12+A13−A14))

(A15+A16)/2 = mean

(final estimate from computerized fit = 0.271)

(final estimate from computerized fit = 4.9)

average yields Q = 4.8 (p/0.653)

Turning Pt (V)

B C

1st set

2nd set

3 0.3643

−0.2607

−0.1343

−0.0687

−0.0371

−0.0194

−0.0109

0.0071 0.6554 0.651

0.6532

0.0149

0.0298

0.0549

0.1913

0.1029

4567

15 C15 = −2*ln(1−(A4−A14)/(A4−A5+A6−A7+A8−A9+A10−A11+A12−A13))

21-12 Vibration and Shock Handbook

to the oscillator. The rod pendulum pictured in Figure 21.9 was studied by this means. Figure 21.10

shows a closeup of the flag which is attached to the top of the pendulum and which passes through the

photogate during oscillation.

As seen in the figures, various parts of the pendulum are clamped to a vertical steel rod. Both upper

and lower masses are made of lead, each with a pair of holes drilled in it — one to pass the brass rod of the

pendulum through and the other (after tapping) to hold a thumbscrew for securing the mass at different

vertical positions on the rod.

This mechanical oscillator is a compound pendulum; the period can be made long as the knife-edge

(also clamped to the rod) approaches the center of mass. At long periods, the instrument is not very

responsive to external accelerations of the supporting frame; but it is sensitive to internal structural

changes. Low-frequency instability is encountered as the upper parts of the instrument experience creep,

particularly in the materials just above the knife-edge. The upper pendulum has similarities to an

inverted pendulum, except that it is rigidly connected to the lower pendulum, and causes the oscillator to

eventually exhibit double-well (Duffing) characteristics. This happens at larger amplitudes as the period

is increased toward really long times. The tendency toward mesoanelastic complexity depends on the

dimensions of the rod. As expected because of the well-known engineering properties of rods and tubes, a

large diameter, thin-wall tube will behave differently from a solid rod made from the same amount of

material (same total mass). This will be true if the tube does not experience localized (sharp) deformation

prone to creasing.

Damping measurements with a photogate require that the time required for the flag to pass through the

beam be fairly small — thus larger amplitudes of motion are required than with other sensors. Of course

really large motion would result in a period increase, consistent with long-understood pendulum

dynamics. For amplitudes within the acceptable range (which in practice is not overly restrictive), the

velocity of the pendulum as it passes through the equilibrium position is inversely proportional to the time

FIGURE 21.9 Rod pendulum in which damping measurements are made with a photogate.

Experimental Techniques in Damping 21-13

interval between interrupts of the photogate beam by the two vertical arms of the flag. If the period does not

change with amplitude, then there is also an inverse relationship between the gate time and the amplitude.

A plot of the inverse of these times versus cycle number is, for the constraints indicated, a reasonable

approximation of the turning points of the free-decay.

In this case the single gate time interval measurements were made by a Pasco Smart Timer. It is a

user-friendly instrument that also permits the period of the pendulum to be accurately measured by

the flag (by using two different lengths of the flag arms). For experiments of this type, it may prove

more convenient to measure the period with a stopwatch (infrequently as compared to the velocity).

The sequentially increasing time intervals are read manually from the Smart Timer and recorded by

hand, once per cycle. Of course, to do so requires that the period be long enough to permit these

operations. The recorded values are conveniently analyzed by typing to a spreadsheet, which is then

used to graph damping curves such as shown in

Figure 21.11.

A pure exponential fit is not appropriate to the

decay of Figure 21.11, in which the upper mass had

been removed and a business card taped to the

bottom of the pendulum to cause turbulent air

damping (period near 1 sec). The fit shown,

however, involving both linear and quadratic

dampings, is seen to be quite reasonable. As in

the case of the vibrating reed discussed earlier, this

system is adequately described by the nonlinear

damping equation 20.61 given in Chapter 20. For

the data of Figure 21.11, Q ¼ 25 initially and

increases to 70 at the end of the record.

FIGURE 21.10 Top of the pendulum showing the upper mass and “flag” for tripping the photogate.

peak velocity (m/s)

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

5 10

cycle number

exponential

(0.08)

Free-decay, brass rod with sail’

(a = 0.2, b = 0.045)

15 20 25

’

FIGURE 21.11 Free-decay of a pendulum as determined

by photogate measurements.

21-14 Vibration and Shock Handbook

Without the business card, and with the upper mass in place, the decay of this pendulum was found

with the photogate measurement technique to be exponential, as expected for viscous damping with

periods of about 5 sec. At periods in excess of about 10 sec, however, internal friction of the rod becomes

more important than air damping. Although the decay is then still exponential at larger levels, the

frequency dependence is not the same as required by linear air damping.

21.4.5 Case 5: Rod Pendulum Influenced by Material under the Knife-Edge

The data in Figure 21.11 were collected with the

knife-edges resting on hard ceramic alumina

flats. When supported by other materials, the

damping of a rod pendulum can be influenced

by anelastic flexure other than that of the rod.

Hardness of the material does not guarantee

low damping, as will be seen in the following

examples. The data that follows were

collected with a different pendulum, depicted

in Figure 21.12.

The sensor in this case was an SDC unit,

connected to the computer through the Dataq DI-

700 A/D converter. The upper and lower masses

are each approximately 1 kg and their separation

distance on the aluminum hunting arrow from

which the pendulum was fabricated was about

70 cm.

21.4.5.1 Lithium Fluoride Samples

The samples used to collect the data in Figure 21.13 were identical pairs, except that one pair had been

irradiated with a huge dose of gamma rays. The resulting changes to the structure of the crystal are

responsible not only for color centers as noted in the photograph in Figure 21.14, but also a dramatic

change in the internal friction. It is clear from Figure 21.13 that internal friction in the LiF is the

dominant source of damping of the rod pendulum that was used (Peters, 2003a, 2003b, 2003c, 2003d).

Lithium fluoride is used in thermoluminescent film badges (radiation monitors). When exposed to

energetic radiation, atoms are “knocked” from their crystal lattice sites into metastable states

corresponding to interstitial positions of the lattice. Upon ramping the temperature of the sample in an

oven fitted with a photomultiplier tube, jumps from the metastable state are accompanied by the release

of photons. The amount of light so generated is a measure of the dose that was received by the crystal.

Because light flashes are observed with rather small changes in the temperature, it is reasonable to expect

that mechanical strains might also cause a significant change to the defect state of such crystals. This

postulate is confirmed by the data in Figure 21.13, which show a dramatic difference in the decay

character of the pure (clear) crystals (bottom figure) and those which were extensively damaged by

gammas (top figure).

In both of the decays in Figure 21.13 there is significant nonlinear damping, as evidenced in the

early portions of each of the two records. The top case is nearly pure Coulombic, and the bottom

case is partially amplitude dependent. This is revealed from estimates of the Q; shown in Figure 21.15.

The Q values in Figure 21.15 were computed from successive triplet-values of the turning points of the

motion, read directly from the decay pattern displayed on the monitor by the Dataq software. The

equation used is

Q ¼

2 2 ln½1 2 ðun 2 unþ2Þ=ðun 2 unþ1Þ􀀉

; n ¼ 0; 1; 2; … ð21:3Þ

FIGURE 21.12 Long period rod pendulum used to

study the influence of different materials under the

knife-edge.

Experimental Techniques in Damping 21-15

21.4.6 Hard Materials with Low Q

It is commonly (and mistakenly) thought that

hard materials must necessarily also have low

damping. The following two examples show that

this is not necessarily so. Even though cast iron is

very hard, it is also quite dissipative, which makes

it an ideal material for engine blocks. Figure 21.16

shows a decay curve for the steel knife-edges of

the pendulum resting on cast iron samples.

At the start of the record the damping with

cast iron is nearly twice as great as that of steelon-

sapphire or steel-on-silicon, where the Q was

found to be of the order of 80. This large

damping measurement is consistent with the

known excellent properties of cast iron for use in

engine blocks, although the frequencies for such

applications are much higher.

Figure 21.17 is another very hard material which has large damping — the ceramic piezoelectric

wafer formed from lead, zirconium, and titanium (PZT), which by means of a mechanical impulse

is commonly used to generate an electric spark to ignite a gas grill. The secular decline of Q based

on the short temporal record indicates Coulomb damping. It is consistent with the nearly straight-line

turning points for the early part of the long-term record, also shown. The long-term record

FIGURE 21.13 Illustration of damping difference according to specimen type under the knife-edge.

FIGURE 21.14 Photograph of LiF single crystals used

to obtain the data in Figure 21.13.

21-16 Vibration and Shock Handbook

is labeled as anomalous because it does not appear to be consistent with several simultaneously acting

dissipation mechanisms. Instead, the strong Coulomb damping seen early on seems to disappear later,

once the amplitude has dropped below a particular level. This suggests activation processes of a quantal

type. It would be interesting to study the PZT wafers in a different pendulum configuration, and not

operating “open-circuit” as in the present case, but rather with different resistors connected between the

top and bottom of the wafers.

21.4.7 Anisotropic Internal Friction

With Polaroid material (H sheet) placed under the knife-edges it was found that the damping depends on

the direction of the long-chain polymeric molecules. The direction of the molecules in a sample is readily

determined by looking through the Polaroid at reflected light from a polished floor. When the reflection

occurs close to the Brewster angle, only the horizontal component of the electric field is significant in the

reflected light for unpolarized incident light. The direction of the molecules is thus determined by

rotating the sample until the minimum of level of light is found. When this occurs the molecular chains

are situated horizontally.

Time variable Damping-influence of Defects

0 5 10 15 20 25

Color–centered LIF

half-cycle number

Quality factor, Q

0 10 20 30 40 50

Quality factor, Q

half-cycle number

Clear LiF

FIGURE 21.15 Temporal dependence of the Q; LiF crystal experiments.

0 10 20 30 40

half-cycle number, n

Quality Factor vs time

1.00

0.50

0.00

− 0.50

−1.50

Sensor output (V)

200 400 600 800

Time (s)

Cast Iron Damping in Free-decay

(T =12 s, a = 0.011, b = 0.0035)

Temporal record

FIGURE 21.16 Data collected using cast iron samples.

Experimental Techniques in Damping 21-17

It was reasoned that the molecular properties of Polaroid might result in mechanical as well as optical

anisotropies. This postulate proved to be true, as shown in Figure 21.18.

When oscillating on silicon at a period of 10 sec, previous studies have found that the instrument

decays consistently with a Q of 80 (uncertainty 3%). In the present study, half a dozen free-decay records

were obtained for (i) edges parallel to the chains and (ii) edges perpendicular to the chains. The average

Q of oscillation was estimated at 50 for the parallel case and 43 for the perpendicular case.

Reproducibility proved slightly better for the parallel case (4%) as compared to the perpendicular case

(5%). Additional details are documented elsewhere (Peters, 2003a, 2003b, 2003c, 2003d).

− 0.60

− 0.40

− 0.20

0.00

0.20

0.40

0.60

Sensor output (V)

0 100 200 300 400

Time (s)

Long time record showing

anomalous damping

Time (8 s /div)

0 5 10 15 20

half cycle number

Q est. from turning

point triplets

PZT free-decay

y = − 0.4101x + 20.528

R2 = 0.312

Time record (T = 12 s )

Sensor output (100 mV/div)

FIGURE 21.17 Data from an experiment involving PZT ceramic wafers.

1.00

0.80

0.60

0.40

0.20

0.00

− 0.20

− 0.40

− 0.60

− 0.80

− 1.00

Sensor output (V)

0 100 200

Time (s)

Axis perpendicular to chains

(Q = 43) 1.00

1.50

0.00

−0.50

−1.00

Axis parallel to chains

(Q = 50)

Time (s)

Sensor output (V) 0

50 100 150 200

FIGURE 21.18 Free-decay curves showing anisotropy of the internal friction in polaroid material.

21-18 Vibration and Shock Handbook

21.4.7.1 Summary, Free-Decay Q Estimation

All of the techniques so far described are methods based on free-decay, which is especially important

for nonlinear systems. With linear systems it is also possible to use steady-state methods, as noted in

Box 21.2 (last column; de Silva, 2000). Box 21.2 summarizes the techniques used in the present chapter to

estimate the logarithmic decrement, bT; from which Q ¼ p=ðbTÞ:

The best method is to use a full nonlinear fit; the worst is to measure the time to 1=e: The expression in

Box 21.2 for the logarithmic decrement, using the STFT, is equivalent to

Q ¼ 27:29

􀀈 􀀈 􀀈 􀀈

ð21:4Þ

where f is the frequency in Hz and the STFT slope is specified in dB per s.

Пресс-релиз популярных книг

21.4 Damping Examples

Популярные книги

Популярные статьи

Навигация