0
Research Papers

Effective Placement of a Cantilever Resonator on Flexible Primary Structure for Vibration Control Applications—Part 1: Mathematical Modeling and Analysis

[+] Author and Article Information
Troy Lundstrom

Department of Mechanical and
Industrial Engineering,
Piezoactive Systems Laboratory,
Northeastern University,
Boston, MA 02115
e-mail: lundstrom.t@husky.neu.edu

Professor
Fellow ASME
Department of Mechanical and
Industrial Engineering,
Piezoactive Systems Laboratory,
Northeastern University,
Boston, MA 02115
e-mail: n.jalili@northeastern.edu

1Corresponding author.

Contributed by the Technical Committee on Vibration and Sound of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received February 15, 2017; final manuscript received February 9, 2018; published online April 17, 2018. Assoc. Editor: Mohammed Daqaq.

J. Vib. Acoust 140(5), 051003 (Apr 17, 2018) (15 pages) Paper No: VIB-17-1066; doi: 10.1115/1.4039531 History: Received February 15, 2017; Revised February 09, 2018

Abstract

In this Part 1 of a two-part series, the theoretical modeling and optimization are presented. More specifically, the effect of attachment location on the dynamics of a flexible beam system is studied using a theoretical model. Typically, passive/active resonators for vibration suppression of flexible systems are uniaxial and can only affect structure response in the direction of the applied force. The application of piezoelectric bender actuators as active resonators may prove to be advantageous over typical, uniaxial actuators as they can dynamically apply both a localized moment and translational force to the base structure attachment point. Assuming unit impulse force disturbance, potential actuator/sensor performance for the secondary beam can be quantified by looking at fractional root-mean-square (RMS) strain energy in the actuator relative to the total system, and normalized RMS strain energy in the actuator over a frequency band of interest with respect to both disturbance force and actuator beam mount locations. Similarly, by energizing the actuator beam piezoelectric surface with a unit impulse, one can observe RMS base beam tip velocity as a function of actuator beam position. Through such analyses, one can balance both sensor/actuator performance and make conclusions about optimally mounting the actuator beam sensor/actuator. Accounting for both sensing and actuation requirements, the actuator beam should be mounted in the following nondimensionalized region: $0.4≤e≤0.5$.

<>

References

Frahm, H. , 1911, “Device for Damping Vibrations of Bodies,” U.S. Patent No. US989958A.
Page, C. , Avitabile, P. , and Niezrecki, C. , 2012, “Passive Noise Reduction Using the Modally Enhanced Dynamic Absorber,” Topics in Modal Analysis II, Vol. 6, Springer, New York, pp. 637–648.
Dayou, J. , and Brennan, M. , 2002, “Global Control of Structural Vibration Using Multiple-Tuned Tunable Vibration Neutralizers,” J. Sound Vib., 258(2), pp. 345–357.
Fang, J. , Li, Q. , and Jeary, A. , 2003, “Modified Independent Modal Space Control of MDOF Systems,” J. Sound Vib., 261(3), pp. 421–441.
Dadfarnia, M. , Jalili, N. , Liu, Z. , and Dawson, D. M. , 2004, “An Observer-Based Piezoelectric Control of Flexible Cartesian Robot Arms: Theory and Experiment,” Control Eng. Pract., 12(8), pp. 1041–1053.
Gu, H. , and Song, G. , 2004, “Adaptive Robust Sliding-Mode Control of a Flexible Beam Using PZT Sensor and Actuator,” IEEE International Symposium on Intelligent Control, Taipei, Taiwan, Sept. 2–4, pp. 78–83.
Olgac, N. , and Holm-Hansen, B. , 1994, “A Novel Active Vibration Absorption Technique: Delayed Resonator,” J. Sound Vib., 176(1), pp. 93–104.
Olgac, N. , and Holm-Hansen, B. , 1995, “Tunable Active Vibration Absorber: The Delayed Resonator,” ASME J. Dyn. Syst. Meas. Control, 117(4), pp. 513–519.
Bashash, S. , Salehi-Khojin, A. , and Jalili, N. , 2008, “Forced Vibration Analysis of Flexible Euler-Bernoulli Beams With Geometrical Discontinuities,” American Control Conference, Seattle, WA, June 11–13, pp. 4029–4034.
Rao, S. S. , 2007, Vibration of Continuous Systems, Wiley, Hoboken, NJ.
Xu, J. W. , Liu, Y. B. , Shao, W. W. , and Feng, Z. , 2012, “Optimization of a Right-Angle Piezoelectric Cantilever Using Auxiliary Beams With Different Stiffness Levels for Vibration Energy Harvesting,” Smart Mater. Struct., 21(6), p. 065017.
Erturk, A. , Renno, J. M. , and Inman, D. J. , 2009, “Modeling of Piezoelectric Energy Harvesting From an L-Shaped Beam-Mass Structure With an Application to UAVs,” J. Intell. Mater. Syst. Struct., 20(5), pp. 529–544.
Gurjar, M. , and Jalili, N. , 2007, “Toward Ultrasmall Mass Detection Using Adaptive Self-Sensing Piezoelectrically Driven Microcantilevers,” IEEE/ASME Trans. Mechatronics, 12(6), pp. 680–688.
da Rocha, T. L. , da Silva, S. , and Lopes, V., Jr. , 2004, “Optimal Location of Piezoelectric Sensor and Actuator for Flexible Structures,” 11th International Congress on Sound and Vibration, St. Petersburg, Russia, July 5–8, pp. 1807–1814.
Lee, J. , Ghasemi, A. H. , Okwudire, C. E. , and Scruggs, J. , 2017, “A Linear Feedback Control Framework for Optimally Locating Passive Vibration Isolators With Known Stiffness and Damping Parameters,” ASME J. Vib. Acoust., 139(1), p. 011006.
Daraji, A. H. , Hale, J. M. , and Ye, J. , 2018, “New Methodology for Optimal Placement of Piezoelectric Sensor/Actuator Pairs for Active Vibration Control of Flexible Structures,” ASME J. Vib. Acoust., 140(1), p. 011015.
Jalili, N. , 2009, Piezoelectric-Based Vibration Control: From Macro to Micro/Nano Scale Systems, Springer Science & Business Media, New York.
Allemang, R. J. , and Brown, D. L. , 1982, “A Correlation Coefficient for Modal Vector Analysis,” First International Modal Analysis Conference, Union College Press, Orlando, FL, Nov. 8–10, pp. 110–116.
Thomson, W. , and Dahleh, M. , 1998, Theory of Vibration With Applications, Prentice Hall, Upper Saddle River, NJ, pp. 301–303.

Figures

Fig. 1

(a) Physical two-beam system, its schematic and (b) diagram of two-beam system

Fig. 2

Natural frequencies of nondimensionalized cantilever beam (dips in DetJβ̃b indicate natural frequencies)

Fig. 3

(a) λ versus γ for choosing geometry (length and thickness) of secondary beam (ha neglected) and (b) υ versus χ for actuator beam tuning (ha not neglected)

Fig. 4

(a) DetJ(β̃b) surface and (b) variation of first four natural frequencies versus e (actuator beam tuned to mode 1)

Fig. 10

Normalized RMS base beam tip velocity versus e for different combinations of modes

Fig. 6

Fractional modal strain energy versus e for the first eight modes

Fig. 7

Fractional RMS strain energy versus d and e for modes 1–8

Fig. 8

Fractional actuator RMS strain energy versus d and e for different combinations of modes

Fig. 9

Normalized RMS strain energy in actuator versus d and e for different combinations of modes

Fig. 5

Effect of e on mode shapes 1–8

Related

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related Proceedings Articles
Related eBook Content
Topic Collections