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Fig: 7. Maximum tip displacement for measurement at frequency f,.

Figure 9 7. Maximum tip displacement for measurement at frequency f,.

Related Figures (9)

In order to recover ao(t), and, hence, y(r), with a low- pass filter (LPF), it is necessary that we investigate the bandwidth (BW) occupied by the terms within the brack- ets in the expression for v,(t). In the frequency domain, v,(t) has the Fourier Transform
In order to recover ao(t), and, hence, y(r), with a low- pass filter (LPF), it is necessary that we investigate the bandwidth (BW) occupied by the terms within the brack- ets in the expression for v,(t). In the frequency domain, v,(t) has the Fourier Transform
Referring to Fig. 4 and using wy = 2w,, a conservative upper bound on the cutoff frequency of the low pass filter is given by
Referring to Fig. 4 and using wy = 2w,, a conservative upper bound on the cutoff frequency of the low pass filter is given by
Fig. 5. (a) Signal proportional to tip displacement, (b) spectrum of tip dis- placement. Euler-Bernoulli partial differential equation whose ana- lytic solution can be written as an infinite summation of modes. In practice it is customary to consider a truncated approximation that includes the first N dominant modes of vibration. The beam’s tip displacement can therefore be expressed as
Fig. 5. (a) Signal proportional to tip displacement, (b) spectrum of tip dis- placement. Euler-Bernoulli partial differential equation whose ana- lytic solution can be written as an infinite summation of modes. In practice it is customary to consider a truncated approximation that includes the first N dominant modes of vibration. The beam’s tip displacement can therefore be expressed as
where L is the length of the beam, ¢; is the ith mode shape of cantilevered beam, w,; is the ith frequency, ET is the bending stiffness, p is the linear mass density, and Yy is an upper bound amplitude defined as one-half the wave- length of the ultrasonic signal after frequency division. This bound is introduced to keep the vibration amplitudes within the range of this particular phase detection system. The set of frequencies {¢,,/ = 1,2, --- , N} is arranged in ascending order.
where L is the length of the beam, ¢; is the ith mode shape of cantilevered beam, w,; is the ith frequency, ET is the bending stiffness, p is the linear mass density, and Yy is an upper bound amplitude defined as one-half the wave- length of the ultrasonic signal after frequency division. This bound is introduced to keep the vibration amplitudes within the range of this particular phase detection system. The set of frequencies {¢,,/ = 1,2, --- , N} is arranged in ascending order.
filter in the phase detection circuit, the following terms are substituted in (7):
filter in the phase detection circuit, the following terms are substituted in (7):
For the beam parameters given in Table I, Fig. 6 and 7 show Yj, and the low pass filter cutoff frequency as a func- tion of ultrasonic frequency (after frequency division by 16). In the experiment, the ultrasonic frequency after di- vision by 16 (effective acoustic frequency) is f, = 5 kHz. Therefore, the required cutoff frequency is approximately 937 Hz, and the maximum measurable tip displacement is approximately 3.5 cm. Notice that for acoustic fre- quencies below 315 Hz the filter cutoff frequency be- comes negative, and the filter is therefore physically un- realizable.
For the beam parameters given in Table I, Fig. 6 and 7 show Yj, and the low pass filter cutoff frequency as a func- tion of ultrasonic frequency (after frequency division by 16). In the experiment, the ultrasonic frequency after di- vision by 16 (effective acoustic frequency) is f, = 5 kHz. Therefore, the required cutoff frequency is approximately 937 Hz, and the maximum measurable tip displacement is approximately 3.5 cm. Notice that for acoustic fre- quencies below 315 Hz the filter cutoff frequency be- comes negative, and the filter is therefore physically un- realizable.
Related topics:
Instrumentation and Measurement SciencePhase ModulationElectrical and Electronic EngineeringPHASE DETECTOROscillating Motion
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