21.6 Oscillator with Multiple Nonlinearities

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An oscillator can have significant nonlinearities of both the elastic and damping types. An example is the

mechanical system pictured in Figure 21.21.

The instrument is a modified extensometer that was sold by TEL-Atomic and which was designed

around the SDC sensor to measure Young’s modulus and thermal expansion coefficients. The wire

sample normally used with the instrument (along with a hollow power resistor that fits in the black

clamp) has been removed, and two rare earth magnets have been employed. One magnet is

superglued to the bottom of the pan where the weights are normally placed, as shown in Figure

21.22; and the other magnet is attached to the bottom of the inductor that is sitting on the top of the

oscillator (cased instrument, Pasco) used for drive. The pair of magnets are positioned in close

proximity so as to repel each other, thus supporting the mass of the moveable arm of the

extensometer.

The study of nonlinear systems requires a linear sensor; i.e., any nonlinear contributions from the

detector must be negligible. Figure 21.23 shows the calibration results for the instrument and its

linear response for the range of amplitudes used in the study.

The potential energy of this oscillator was assumed to have the following form

UðxÞ ¼

b

xn þ cx ð21:5Þ

and the parameters were estimated by measuring x as small masses were placed on the pan. A linear

regression fit to a log – log plot (using the sensor calibration constant of 550 V/m) yielded

b ¼ 1:02 £ 1025; n ¼ 1:526; and c ¼ 0:304 (system international units). Anharmonicity of the

FIGURE 21.21 Mechanical oscillator with multiple nonlinearities.

Experimental Techniques in Damping 21-21

© 2005 by Taylor & Francis Group, LLC

potential is readily apparent in the plot shown

in Figure 21.24, with the force of restoration

being greater in compression (x decreasing) than

it is in extension. This feature is reminiscent of

interatomic potentials, with anharmonicity being

responsible for thermal expansion.

Because of the elastic nonlinearity, the mean

position depends on the amplitude of the

oscillation, as is evident in the free-decay curve

in Figure 21.25.

The damping of this oscillator was also found to

be nonlinear, as seen in Figure 21.26.

The oscillator exhibits hysteresis when driven at

larger amplitudes, as shown in Figure 21.27, where

it can be seen that the location of an amplitude

jump depends on which way the oscillator is

adjusted, either up or down in frequency. Such

jumps (well known with oscillators with nonlinear

elasticity) stand in stark contrast with the behavior

of a linear oscillator, as can be seen by comparing

Figure 21.27 with the screen picture in Figure

21.20.

A surprise from this study involves the frequency

of oscillation. In general, the oscillator did not

entrain to the drive. Moreover, the preferred frequencies

were not necessarily the same as the freedecay

frequency of 6.01 Hz. Some of the frequencies

(measured with power spectra) are indicated in

FIGURE 21.22 Closeup picture of the oscillator in Figure 21.21 (nonoperational configuration), showing

placement of the rare earth magnets.

1.5

1

0.5

0

−3 −2 −1−0.5 0

Position (mm)

SDC calibration data

Sensor output (V)

y = 0.55x + 0.0039

R2 = 0.9999

1 2 3

−1

−1.5

FIGURE 21.23 Calibration data for the sensor used

with the oscillator having multiple nonlinearities.

−0.01 0

0.002

Potential Energy, oscillator with multiple nonlinearities

U(x)−U(0.02)

0

0.004

0.01

x−0.02 (m)

0.02 0.03

FIGURE 21.24 Plot of the potential energy function of

the oscillator.

21-22 Vibration and Shock Handbook

© 2005 by Taylor & Francis Group, LLC

Figure 21.27. The 3% frequency jumps observed in Figure 21.27 (in going from 5.46 Hz to 6.20 Hz) are

not real but rather artifacts of the finite resolution of the 1024 point transforms that were employed.

Figure 21.27 demonstrates why nonlinear damping measurements should be done in free-decay.

Figure 21.25 demonstrates how exponential fits make no sense for some oscillators. Fortunately, the

STFT can be used to determine the amplitude dependence of the Q:

FIGURE 21.25 Free-decay curve showing the mean position shift as a function of oscillator amplitude. (Decreasing

sensor voltage corresponds to increasing x:) The frequency of oscillation is 6.01 Hz.

2

dB

0

0

−5

−10

−15

−20

−25

−30

−35

−40

−45

4

Shift time (s)

STFT data, oscillator with multiple nonlinearities

y = −0.1058x2 − 0.7069x − 19.989

R2 = 0.9985

6 8 10 12

0 2

0

50

100

Q

150

200

250

4

Time (s)

Time dependence of the damping

6 8 10 12

FIGURE 21.26 Free-decay character as determined using the short time Fourier transform.

Experimental Techniques in Damping 21-23

© 2005 by Taylor & Francis Group, LLC

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