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Journal of Vibration and Acoustics | Volume 138 | Issue 6 | research-article
Research Papers

Experimental and Numerical Validation of Digital, Electromechanical, Parametrically Excited Amplifiers

[+] Author and Article Information
Amit Dolev

Dynamics Laboratory,
Mechanical Engineering,
Technion,
Technion City, Haifa 3200003, Israel
e-mail: amitdtechnion@gmail.com

Izhak Bucher

Dynamics Laboratory,
Mechanical Engineering,
Technion,
Technion City, Haifa 3200003, Israel
e-mail: bucher@technion.ac.il

1Corresponding author.

Contributed by the Technical Committee on Vibration and Sound of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received November 11, 2015; final manuscript received June 5, 2016; published online July 19, 2016. Assoc. Editor: Jeffrey F. Rhoads.

J. Vib. Acoust 138(6), 061001 (Jul 19, 2016) (14 pages) Paper No: VIB-15-1470; doi: 10.1115/1.4033897 History: Received November 11, 2015; Revised June 05, 2016
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A parametric amplifier having a tunable, dual-frequency pumping signal and a controlled cubic stiffness term is realized and investigated experimentally. This device can be tuned to amplify a desired, single frequency weak signal, well below resonance. The transition between a previously described theoretical model and a working prototype requires an additional effort in several areas: modeling, design, calibration, identification, verification, and adjustment of the theoretical model. The present paper describes these necessary steps and analyzes the results. Tunability is achieved here by adding a digitally controlled feedback, driving a linear mechanical oscillator with an electromechanical actuator. The main advantage of the present approach stems from the separation of the controlled parametric and nonlinear feedback terms which are linked to the resonating element. This separation allows for the realization of feedback in an electronic form where a digital implementation adds further advantages as the feedback coefficients can be tuned in situ. This arrangement benefits from the mechanical resonance of a structure and from the ability to set the parametric excitation such that it accommodates sinusoidal input signals over a wide range of frequencies. The importance of an in situ identification phase is made clear in this work, as the precise setting of model and feedback parameters was shown to be crucial for successful application of the amplifier. A detailed model-identification effort is described throughout this paper. It has been shown through identification that the approach is robust despite some modeling uncertainties and imperfections.
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References

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Figures

Grahic Jump Location
Fig. 1

A damped linear single degree-of-freedom (SDOF) oscillator with nonlinear and time dependent stiffness subjected to parametric excitation and direct excitation

+A damped linear single degree-of-freedom (SDOF) oscillator with nonlinear and time dependent stiffness subjected to parametric excitation and direct excitation
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A damped linear single degree-of-freedom (SDOF) oscillator with nonlinear and time dependent stiffness subjected to parametric excitation and direct excitation
Grahic Jump Location
Fig. 2

Nonlinear frequency response, analytical solution versus numerical simulation when the parametric amplifier is subjected to different direct excitation frequencies, and accordingly different pumping frequencies. The subfigures are divided into two domains, (I) contains a single stable solution branch and (II) contains two stable solution branches and a single U.S. solution branch. (a) Response amplitude near the natural frequency ωa/2≈ωn. (b) Response phase near the natural frequency. Analytical stable (S) branches are shown by continuous lines, analytical U.S. branches are shown by dashed lines, numerical results are shown by data markers.

+Nonlinear frequency response, analytical solution versus numerical simulation when the parametric amplifier is subjected to different direct excitation frequencies, and accordingly different pumping frequencies. The subfigures are divided into two domains, (I) contains a single stable solution branch and (II) contains two stable solution branches and a single U.S. solution branch. (a) Response amplitude near the natural frequency ωa/2≈ωn. (b) Response phase near the natural frequency. Analytical stable (S) branches are shown by continuous lines, analytical U.S. branches are shown by dashed lines, numerical results are shown by data markers.
View Large  |  Save Figure  |  Download Slide (.ppt)
Nonlinear frequency response, analytical solution versus numerical simulation when the parametric amplifier is subjected to different direct excitation frequencies, and accordingly different pumping frequencies. The subfigures are divided into two domains, (I) contains a single stable solution branch and (II) contains two stable solution branches and a single U.S. solution branch. (a) Response amplitude near the natural frequency ωa/2≈ωn. (b) Response phase near the natural frequency. Analytical stable (S) branches are shown by continuous lines, analytical U.S. branches are shown by dashed lines, numerical results are shown by data markers.
Grahic Jump Location
Fig. 3

The SDOF experimental system consists of a linear voice-coil actuator, modular mass, displacement sensor, and leaf springs. The modular mass vibrates in the direction of y(t).

+The SDOF experimental system consists of a linear voice-coil actuator, modular mass, displacement sensor, and leaf springs. The modular mass vibrates in the direction of y(t).
View Large  |  Save Figure  |  Download Slide (.ppt)
The SDOF experimental system consists of a linear voice-coil actuator, modular mass, displacement sensor, and leaf springs. The modular mass vibrates in the direction of y(t).
Grahic Jump Location
Fig. 4

Experimental system flowchart, including the peripheral devices, control and data-flow schematics. The control loop is programed in matlab using Simulink and realized by the digital signal processor, also serving as a data acquisition device. The control-loop feedback is the mass position, y(t).

+Experimental system flowchart, including the peripheral devices, control and data-flow schematics. The control loop is programed in matlab using Simulink and realized by the digital signal processor, also serving as a data acquisition device. The control-loop feedback is the mass position, y(t).
View Large  |  Save Figure  |  Download Slide (.ppt)
Experimental system flowchart, including the peripheral devices, control and data-flow schematics. The control loop is programed in matlab using Simulink and realized by the digital signal processor, also serving as a data acquisition device. The control-loop feedback is the mass position, y(t).
Grahic Jump Location
Fig. 5

An example of a measured and curve-fitted step response. Measurements—data points, fitted model—continuous line and error (model deviation)—data points.

+An example of a measured and curve-fitted step response. Measurements—data points, fitted model—continuous line and error (model deviation)—data points.
View Large  |  Save Figure  |  Download Slide (.ppt)
An example of a measured and curve-fitted step response. Measurements—data points, fitted model—continuous line and error (model deviation)—data points.
Grahic Jump Location
Fig. 6

Measured leaf springs force versus the mass deflection, measurements—data points and fitted models—continuous lines. (a) Linear spring model and (b) nonlinear, hardening (Duffing) spring model.

+Measured leaf springs force versus the mass deflection, measurements—data points and fitted models—continuous lines. (a) Linear spring model and (b) nonlinear, hardening (Duffing) spring model.
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Measured leaf springs force versus the mass deflection, measurements—data points and fitted models—continuous lines. (a) Linear spring model and (b) nonlinear, hardening (Duffing) spring model.
Grahic Jump Location
Fig. 7

Example of a measured and fitted step response. Measurements—data points and fitted model—continuous line.

+Example of a measured and fitted step response. Measurements—data points and fitted model—continuous line.
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Example of a measured and fitted step response. Measurements—data points and fitted model—continuous line.
Grahic Jump Location
Fig. 8

Steady-state amplitude under principal parametric excitation versus the relative offset of the pumping magnitude. Measured—data points, theoretical—continuous line, additional two analytical solutions for slightly higher pumping frequencies, for ωa=2(1.01ωn), and for ωa=2(1.005ωn).

+Steady-state amplitude under principal parametric excitation versus the relative offset of the pumping magnitude. Measured—data points, theoretical—continuous line, additional two analytical solutions for slightly higher pumping frequencies, for ωa=2(1.01ωn), and for ωa=2(1.005ωn).
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Steady-state amplitude under principal parametric excitation versus the relative offset of the pumping magnitude. Measured—data points, theoretical—continuous line, additional two analytical solutions for slightly higher pumping frequencies, for ωa=2(1.01ωn), and for ωa=2(1.005ωn).
Grahic Jump Location
Fig. 9

Nonlinear frequency response. (a) Response amplitude near the natural frequency ωa/2≈ωn. (b) Response amplitude at the direct excitation frequency ωr. Analytical stable (S) branches are shown by continuous lines, analytical U.S. branch is shown by a dashed line, numerical and experimental results are shown by data markers.

+Nonlinear frequency response. (a) Response amplitude near the natural frequency ωa/2≈ωn. (b) Response amplitude at the direct excitation frequency ωr. Analytical stable (S) branches are shown by continuous lines, analytical U.S. branch is shown by a dashed line, numerical and experimental results are shown by data markers.
View Large  |  Save Figure  |  Download Slide (.ppt)
Nonlinear frequency response. (a) Response amplitude near the natural frequency ωa/2≈ωn. (b) Response amplitude at the direct excitation frequency ωr. Analytical stable (S) branches are shown by continuous lines, analytical U.S. branch is shown by a dashed line, numerical and experimental results are shown by data markers.
Grahic Jump Location
Fig. 10

The nonlinear frequency response gain. Analytically computed gain of the theoretical model is shown by a line, continuous (S)—stable branch, dashed (U.S.)—U.S. branch. Numerically computed gain from the measurements is shown by data markers.

+The nonlinear frequency response gain. Analytically computed gain of the theoretical model is shown by a line, continuous (S)—stable branch, dashed (U.S.)—U.S. branch. Numerically computed gain from the measurements is shown by data markers.
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The nonlinear frequency response gain. Analytically computed gain of the theoretical model is shown by a line, continuous (S)—stable branch, dashed (U.S.)—U.S. branch. Numerically computed gain from the measurements is shown by data markers.
Grahic Jump Location
Fig. 11

(a) The amplifier's amplitude near the natural frequency versus the direct excitation phase. (b) The amplifier's amplitude at the direct excitation frequency (ωr) versus the direct excitation phase. Analytical results are shown by continuous lines, experimental and numerical results—are shown by data markers.

+(a) The amplifier's amplitude near the natural frequency versus the direct excitation phase. (b) The amplifier's amplitude at the direct excitation frequency (ωr) versus the direct excitation phase. Analytical results are shown by continuous lines, experimental and numerical results—are shown by data markers.
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(a) The amplifier's amplitude near the natural frequency versus the direct excitation phase. (b) The amplifier's amplitude at the direct excitation frequency (ωr) versus the direct excitation phase. Analytical results are shown by continuous lines, experimental and numerical results—are shown by data markers.
Grahic Jump Location
Fig. 12

(a) Amplitude near the natural frequency versus the direct excitation amplitude. (b) Amplitude at the direct excitation frequency versus the direct excitation amplitude. Analytical results are shown by continuous lines, numerical and experimental results are shown by data markers, and a fitted curve of the experimental results is shown by a continuous line.

+(a) Amplitude near the natural frequency versus the direct excitation amplitude. (b) Amplitude at the direct excitation frequency versus the direct excitation amplitude. Analytical results are shown by continuous lines, numerical and experimental results are shown by data markers, and a fitted curve of the experimental results is shown by a continuous line.
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(a) Amplitude near the natural frequency versus the direct excitation amplitude. (b) Amplitude at the direct excitation frequency versus the direct excitation amplitude. Analytical results are shown by continuous lines, numerical and experimental results are shown by data markers, and a fitted curve of the experimental results is shown by a continuous line.
Grahic Jump Location
Fig. 13

Experimentally computed gain

+Experimentally computed gain
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Experimentally computed gain
Grahic Jump Location
Fig. 14

The amplitude sensitivity near the natural frequency to variations in the direct excitation amplitude versus the direct excitation amplitude. Analytical sensitivity is shown by a continuous line, numerical sensitivity is shown by data markers, and the numerically computed sensitivity from the measurements using a fitted line is shown by a continuous line.

+The amplitude sensitivity near the natural frequency to variations in the direct excitation amplitude versus the direct excitation amplitude. Analytical sensitivity is shown by a continuous line, numerical sensitivity is shown by data markers, and the numerically computed sensitivity from the measurements using a fitted line is shown by a continuous line.
View Large  |  Save Figure  |  Download Slide (.ppt)
The amplitude sensitivity near the natural frequency to variations in the direct excitation amplitude versus the direct excitation amplitude. Analytical sensitivity is shown by a continuous line, numerical sensitivity is shown by data markers, and the numerically computed sensitivity from the measurements using a fitted line is shown by a continuous line.
Grahic Jump Location
Fig. 15

The effect of sensitivity tuning (a) Amplitude near the natural frequency versus the direct excitation amplitude. (b) Amplitude at the direct excitation frequency versus the direct excitation amplitude. Analytical results are shown by continuous lines, numerical and experimental results are shown by data markers, and a fitted curve of the experimental results is shown by a continuous line.

+The effect of sensitivity tuning (a) Amplitude near the natural frequency versus the direct excitation amplitude. (b) Amplitude at the direct excitation frequency versus the direct excitation amplitude. Analytical results are shown by continuous lines, numerical and experimental results are shown by data markers, and a fitted curve of the experimental results is shown by a continuous line.
View Large  |  Save Figure  |  Download Slide (.ppt)
The effect of sensitivity tuning (a) Amplitude near the natural frequency versus the direct excitation amplitude. (b) Amplitude at the direct excitation frequency versus the direct excitation amplitude. Analytical results are shown by continuous lines, numerical and experimental results are shown by data markers, and a fitted curve of the experimental results is shown by a continuous line.
Grahic Jump Location
Fig. 16

The amplitude sensitivity near the natural frequency to variations in the direct excitation amplitude versus the direct excitation amplitude. Analytical sensitivity is shown by a continuous line, numerical sensitivity is shown by data markers, the numerically computed sensitivity from the measurements is shown by data markers, and the numerically computed sensitivity from the measurements using a fitted line is shown by a continuous line.

+The amplitude sensitivity near the natural frequency to variations in the direct excitation amplitude versus the direct excitation amplitude. Analytical sensitivity is shown by a continuous line, numerical sensitivity is shown by data markers, the numerically computed sensitivity from the measurements is shown by data markers, and the numerically computed sensitivity from the measurements using a fitted line is shown by a continuous line.
View Large  |  Save Figure  |  Download Slide (.ppt)
The amplitude sensitivity near the natural frequency to variations in the direct excitation amplitude versus the direct excitation amplitude. Analytical sensitivity is shown by a continuous line, numerical sensitivity is shown by data markers, the numerically computed sensitivity from the measurements is shown by data markers, and the numerically computed sensitivity from the measurements using a fitted line is shown by a continuous line.

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