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Research Papers

The Method of Receptances for Continuous Rods

[+] Author and Article Information
Amit Maha

Department of Mechanical Engineering,
Louisiana State University,
Baton Rouge, LA 70803
e-mail: amaha@tigers.lsu.edu

Y. M. Ram

Department of Mechanical Engineering,
Louisiana State University,
Baton Rouge, LA 70803
e-mail: ram@me.lsu.edu

We used the first three numbers generated by MATLAB pseudo random number generator after invoking the software.

Manuscript received January 7, 2014; final manuscript received April 3, 2014; accepted manuscript posted April 9, 2014; published online April 21, 2014. Assoc. Editor: Alexander F. Vakakis.

J. Appl. Mech 81(7), 071009 (Apr 21, 2014) (7 pages) Paper No: JAM-14-1020; doi: 10.1115/1.4027370 History: Received January 07, 2014; Revised April 03, 2014; Accepted April 09, 2014

It is shown that it is possible in general to assign 2n poles of an axially vibrating nonuniform rod by sensing the state at n points along the rod and by using an actuator which applies the control force. Moreover, the assignment of poles may be done without continuous or discrete analytical modeling. In particular, the rigidity, the density, and the variable cross-sectional area of the rod may be considered unknown. The control gains are determined by the receptances between the point of actuation and the points of sensing, which may be measured experimentally. In the analytical arena, where the receptances are exact, the assignment of the desired poles is also exact and suffers from no discretization or model reduction errors. However, the control force affects in general the distribution of the poles which are not intended to be desirably placed or to remain unchanged. Although the analysis is carried out for simplicity on an axially vibrating rod, the above mentioned results are generally applicable to other linear elastic structures of higher dimensions.

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References

Ram, Y. M., and Mottershead, J. E., 2007, “The Receptance Method in Active Vibration Control,” AIAA J., 45(3), pp. 562–567. [CrossRef]
Ram, Y. M., and Mottershead, J. E., 2013, “Multiple-Input Active Vibration Control by Partial Pole Placement Using the Method of Receptances,” Mech. Syst. Signal Process., 40(2), pp. 727–735. [CrossRef]
Mottershead, J. E., Tehrani, M. G., James, S., and Ram, Y. M., 2008, “Active Vibration Suppression by Pole-Zero Placement Using Measured Receptances,” J. Sound Vib., 311(3-5), pp. 1391–1408. [CrossRef]
Mottershead, J. E., Tehrani, M. G., James, S., and Court, P., 2012, “Active Vibration Control Experiments on an AgustaWestland W30 Helicopter Airframe,” Proc. Inst. Mech. Eng., Part C, 226(6), pp. 1504–1516. [CrossRef]
Papatheou, E., Wei, X., Jiffri, S., Tehrani, M. G., Bode, S., Singh, K. V., Mottershead, J. E., and Cooper, J. E., 2012, “Flutter Control Using Vibration Test Data: Theory, Rig Design and Preliminary Results,” International Conference on Noise and Vibration Engineering, Leuven, Belgium, September 17–19, Paper No. 424.
Papatheou, E., Tantaroudas, N. D., Da Ronch, A., Cooper, J. E., and Mottershead, J. E., 2013, “Active Control for Flutter Suppression: An Experimental Investigation,” International Forum on Aeroelasticity and Structural Dynamics, Royal Aeronautical Society, Bristol, UK, June 24–26, Paper No. S8D.
Tehrani, M. G., Elliott, R. N. R., and Mottershead, J. E., 2010, “Partial Pole Placement in Structures by the Method of Receptances: Theory and Experiments,” J. Sound Vib., 329(24), pp. 5017–5035. [CrossRef]
Bai, Z. J., Chen, M. X., and Datta, B. N., 2013, “Minimum Norm Partial Quadratic Eigenvalue Assignment With Time Delay in Vibrating Structures Using the Receptance and the System Matrices,” J. Sound Vib., 332(4), pp. 780–794. [CrossRef]
Ram, Y. M., Mottershead, J. E., and Tehrani, M. G., 2011, “Partial Pole Placement With Time Delay in Structures Using the Receptance and the System Matrices,” Linear Algebra Appl., 434(7), pp. 1689–1696. [CrossRef]
Ram, Y. M., Singh, A., and Mottershead, J. E., 2009, “State Feedback Control With Time Delay,” Mech. Syst. Signal Process., 23(6), pp. 1940–1945. [CrossRef]
Singh, K. V., and Ouyang, H., 2013, “Pole Assignment Using State Feedback With Time Delay in Friction-Induced Vibration Problems” Acta Mech., 224(3), pp. 645–656. [CrossRef]
Tehrani, M. G., Wilmshurst, L., and Elliott, S. J., 2013, “Receptance Method for Active Vibration Control of a Nonlinear System,” J. Sound Vib., 332(19), pp. 4440–4449. [CrossRef]
Ouyang, H., 2011, “A Hybrid Control Approach for Pole Assignment to Second-Order Asymmetric Systems,” Mech. Syst. Signal Process., 25(1), pp. 123–132. [CrossRef]
Ouyang, H., 2010, “Pole Assignment of Friction-Induced Vibration for Stabilisation Through State-Feedback Control,” J. Sound Vib., 329(11), pp. 1985–1991. [CrossRef]
Ouyang, H., Richiedei, D., and Trevisani, A., 2013, “Pole Assignment for Control of Flexible Link Mechanisms,” J. Sound Vib., 332(12), pp. 2884–2899. [CrossRef]
Tehrani, M. G., and Ouyang, H., 2012, “Receptance-Based Partial Pole Assignment for Asymmetric Systems Using State-Feedback,” Shock Vib., 19(5), pp. 1135–1142. [CrossRef]
Tehrani, M. G., Mottershead, J. E., Shenton, A. T., and Ram, Y. M., 2011, “Robust Pole Placement in Structures by the Method of Receptances,” Mech. Syst. Signal Process., 25(1), pp. 112–122. [CrossRef]
Mottershead, J. E., Tehrani, M. G., and Ram, Y. M., 2009, “Assignment of Eigenvalue Sensitivities From Receptance Measurements,” Mech. Syst. Signal Process., 23(6), pp. 1931–1939. [CrossRef]

Figures

Grahic Jump Location
Fig. 1

Closed loop system

Grahic Jump Location
Fig. 2

Exponential excitation

Grahic Jump Location
Fig. 4

Frequency function of the closed loop uniform rod

Grahic Jump Location
Fig. 5

Frequency function of the closed loop exponential rod

Grahic Jump Location
Fig. 6

Frequency functions of the open and closed loops exponential rod

Grahic Jump Location
Fig. 7

Rod with multiple actuators

Grahic Jump Location
Fig. 8

(a) Transversely vibrating beam and (b) vibrating plate

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