21.7 Multiple Modes of Vibration

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21.7.1 The System

In engineering, multimode oscillations are common. Many, if not most, cases have mode mixing features

even though they may in some cases be too small to be readily observed. The importance of nonlinearity

to these problems is not widely appreciated, so a case to illustrate salient features is provided here. Freedecay

records were obtained with an oscillator in the form of a vertically oriented (hanging) tungsten

wire, of length 24 cm and diameter 0.31 mm. It was clamped at the top end, and at the bottom a

rectangular plate was attached that was 11.3 cm long, 1.3 cm wide, and 0.8 mm thick. The plate was cut

from double-sided copper circuit board. The board was positioned between the stationary plates of a

capacitive sensor, as shown in Figure 21.28.

5.4

0

20

40

60

80

Sensor Output (mV)

100

120

140

160

5.5 5.6

5.46 Hz

5.46 Hz

5.83 Hz

Resonance’ response of Oscillator with multiple nonlinearities

6.01 Hz

6.20 Hz

5.7 5.8

Drive Frequency (Hz)

5.9 6 6.1 6.2

’

FIGURE 21.27 Resonance response (steady state) of the driven oscillator.

FIGURE 21.28 Photograph of the detector used to monitor the multimode oscillator.

21-24 Vibration and Shock Handbook

© 2005 by Taylor & Francis Group, LLC

For the picture, the apparatus was disassembled and the plate allowed to rest on the top of the bottom

electrode set. Operationally, the plate was positioned midway between the upper and lower static

electrode sets (separation distance of 4 mm); and there was no mechanical contact during oscillation. As

can be seen, the top of the circuit board containing the upper electrode set contains more than a dozen

electronic components; these are of the surface mount technology type. The detector is of the SDC type

and this particular embodiment is manufactured in Poland for TEL-Atomic Inc., Jackson, MI, for use in

the Computerized Cavendish Balance.

As can be seen in the picture, the wire was rather kinked instead of straight, which is expected to be a

significant source of nonlinearity. For this reason, not to mention that it is very difficult to make larger

diameter tungsten wires reasonably straight, no serious attempt was undertaken to remove the kinks.

21.7.2 Some Experimental Results

An example decay record generated with this apparatus is illustrated in Figure 21.29.

21.7.3 Short Time Fourier Transform

When multiple modes are present in a decay, as in Figure 21.29, it is not possible to readily estimate Q for

all of the various modes using time data. The decays can be estimated using the FFT, in a technique called

the short time Fourier transform, which is built-in to various software packages related to acquisition

systems, such as LabVIEW (see Appendix 15A). With the versatile software supplied with the Dataq A/D

converter, it is straightforward to employ an equivalent manual technique. Using the number of points to

define the FFT a value (always a power of 2 total) that is substantially smaller than the number of points

in the record, a manual scan is performed in which one simply increments from start to finish, calculating

a separate FFT at each position in time along the way. As an illustration of this powerful tool, Figure 21.30

shows spectra corresponding to the start and the finish of the data in Figure 21.29.

All the modes decay in time, and the rate of decay is especially large for those modes that correspond to

sum and difference frequencies of the primary modes at 1.19 and 2.19 Hz. Table 21.2 gives the spectral

intensities in dB for the two times considered. Where the rows are blank for the end of record case, the

values were insignificantly small.

FIGURE 21.29 Example free-decay of a multimode wire oscillator.

Experimental Techniques in Damping 21-25

© 2005 by Taylor & Francis Group, LLC

The decibel values in the table are referenced to the bit-size (16 corresponding to 65536) of the ADC.

In terms of the sensor output voltage, V ; it is defined by Dataq as:

dB ¼ 20 log10ð32; 768 £ V =FSÞ ð21:6Þ

where FS is the full-scale voltage as determined by the gain setting.

Elsewhere in this chapter, the decibel is calculated with a different reference. For example, for an FFT

spectral line having real and imaginary components R and I; respectively (voltage based), the intensity in

dB is calculated using

dB ¼ 20 log10

ffiffiffiffiffiffiffiffiffi

R2 þ I2

p 􀀝

n

2

􀀒 􀀏 􀀐􀀓

ð21:7Þ

where n is the number of points in the FFT. This is convenient for determining noise levels. For example,

from later graphs showing electronics noise, the floor of the SDC sensor is found to be of the order of

2 120 dB, corresponding to a microvolt. The position resolution defined by this noise level is about

500 nm, i.e., the wavelength of visible light.

FIGURE 21.30 Beginning and end spectra corresponding to the temporal data from Figure 21.29. Ordinate values

are spectral intensity in dB, abscissa values are frequency in Hz (linear scale).

TABLE 21.2 Spectral Intensities for Some of the Lines Shown in Figure 21.30

Frequency (Hz) Start of Record (dB) End of Record (dB)

2.19 78.3 63.0

1.19 68.1 55.6

1.00 44.6

0.19 40.8

3.38 35.0 6.7

4.34 26.7

4.53 22.4

6.53 22.9

5.53 17.8

21-26 Vibration and Shock Handbook

© 2005 by Taylor & Francis Group, LLC

Of the two primary modes of this kinked-wire

case study, the higher frequency (2.19 Hz) is the

twisting mode and the lower frequency (1.19 Hz)

is the swinging mode. The swinging mode is a

little higher frequency than that which would

result if the wire were completely flexible,

yielding a near simple pendulum (1.02 Hz for

24 cm length). The swinging mode is two

dimensional (pendulum equivalent called conical),

but the sensor only responds (first order) to

motion perpendicular to the long axis of the

electrodes. It should also be noted that this

motion is attenuated, relative to the twisting

response, because of the mechanical commonmode

rejection feature discussed in Chapter 20.

The manual STFT was used on the data that

generated Figure 21.29 to estimate the decay history of three different modes — both of the primary

ones (twist and swing) and also the mode whose frequency is the difference between the frequencies

of the primaries, i.e., 1 Hz. Figure 21.31 shows the results, where a Hanning window was used, and

the total number of points in the record permitted five equally time-spaced FFTs, when working with

a 1024 point transform.

Although the decay of the twisting mode is seen to be reasonably exponential, there was large beating

between the modes (readily observed in Figure 21.29). Beating alone would not yield a mix signal whose

frequency is 1.0 Hz. However, beating in a linear system can cause amplitude variations in the weaker

swinging mode.

21.7.4 Nonlinear Effects — Mode Mixing

At least two signals in the spectra are the result of nonlinearity, i.e., the lines corresponding to the sum

and difference of the frequencies of the primary pair — at 3.38 and 1.00 Hz, respectively. If the system of

oscillator and detector were completely linear, then no such sum and difference cases would be possible.

It is also to be noted that these mixtures are not the result of sensor nonlinearity, which as noted

previously one must be careful to avoid.

The amplitude of a mix signal was expected to approximately obey the following relation:

Am / A1A2 ð21:8Þ

To test this premise, the STFT was used to estimate the amplitudes of each of the three components

indicated in Equation 21.8. The amplitudes were all normalized, relative to the starting value for each

case, and the results used to generate the graphs in Figure 21.32.

The amplitude of oscillation for a given mode, at the time of the transform, is found by using the peak

value in dB of the intensity of the spectral line for that mode, according to

A / 10dB=20 ð21:9Þ

where the factor of 20 is used since the spectral intensities were calculated in terms of voltages. Although

calibration constants (in V/m and V/rad) could be used to express the amplitude in meters or in radians,

corresponding to the mode, nothing is gained by doing so for the present purposes.

The mixing index for these cases is defined by the expression

index ¼

ffiAffiffimffiffiffiffi

A1A2 p ð21:10Þ

0

0.0

0.2

0.4

normalized amplitude

0.6

0.8

1

1.2

0.2 0.4

Time (fraction of record length)

0.6 0.8 1.0

swinging

twisting

mix (difference freq.)

Decay of modes, wire oscillator

FIGURE 21.31 Decay of modes of the wire oscillator,

determined by the manual STFT.

Experimental Techniques in Damping 21-27

© 2005 by Taylor & Francis Group, LLC

which is similar to expressions encountered in optics. It can be seen that the sum and difference

frequencies are approximated reasonably well by theoretical expectation.

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