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

Nader Jalili

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

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: 0.4e0.5.

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Fig. 1

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

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Fig. 2

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

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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)

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Fig. 4

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

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Fig. 5

Effect of e on mode shapes 1–8

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Fig. 6

Fractional modal strain energy versus e for the first eight modes

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Fig. 7

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

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Fig. 8

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

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Fig. 9

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

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Fig. 10

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





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