0
Research Papers

On Averaging and Vibrational Control of Mechanical Systems With Multifrequency Inputs

[+] Author and Article Information
Sevak Tahmasian

Department of Biomedical Engineering
and Mechanics,
Virginia Tech,
Blacksburg, VA 24061
e-mail: sevakt@vt.edu

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT,AND CONTROL. Manuscript received March 19, 2017; final manuscript received May 8, 2018; published online June 18, 2018. Assoc. Editor: Douglas Bristow.

J. Dyn. Sys., Meas., Control 140(11), 111007 (Jun 18, 2018) (10 pages) Paper No: DS-17-1158; doi: 10.1115/1.4040296 History: Received March 19, 2017; Revised May 08, 2018

This paper discusses the averaging, control authority, and vibrational control of mechanical control-affine systems with high-frequency, high-amplitude inputs. The inputs have different frequencies of the same order. This work is an extension of the existing averaging method for high-frequency mechanical systems with single-frequency inputs. Vibrational control authority of mechanical control-affine systems is introduced, and the effects of inputs' waveform and frequency on vibrational control authority are investigated. The results show that, in general, using multifrequency inputs may result in lower control authority of mechanical systems compared to single-frequency inputs, especially when using harmonic inputs. The results on vibrational control authority of the systems with multifrequency inputs are demonstrated using vibrational control of a horizontal pendulum with two inputs. This paper also discusses the averaging of multiple-time-scale control systems.

FIGURES IN THIS ARTICLE
<>
Copyright © 2018 by ASME
Your Session has timed out. Please sign back in to continue.

References

Bogoliubov, N. N. , and Mitropolsky, Y. A. , 1961, Asymptotic Methods in the Theory of Non-Linear Oscillations, Hindustan Publishing Corporation, Delhi, India.
Mitropolsky, Y. A. , 1967, “ Averaging Method in Non-Linear Mechanics,” Int. J. Nonlinear Mech., 2(1), pp. 69–96. [CrossRef]
Guckenheimer, J. , and Holmes, P. , 1983, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Applied Mathematical Sciences, Springer-Verlag, New York. [CrossRef]
Sanders, J. A. , and Verhulst, F. , 1985, Averaging Methods in Nonlinear Dynamical Systems. Applied Mathematical Sciences, Springer-Verlag, New York. [CrossRef]
Meerkov, S. M. , 1980, “ Principle of Vibrational Control: Theory and Applications,” IEEE Trans. Autom. Control, AC, 25(4), pp. 755–762. [CrossRef]
Bellman, R. E. , Bentsman, J. , and Meerkov, S. M. , 1986, “ Vibrational Control of Nonlinear Systems: Vibrational Stabilizability,” IEEE Trans. Autom. Control, 31(8), pp. 710–716. [CrossRef]
Bellman, R. E. , Bentsman, J. , and Meerkov, S. M. , 1986, “ Vibrational Control of Nonlinear Systems: Vibrational Controllability and Transient Behavior,” IEEE Trans. Autom. Control, 31(8), pp. 717–724. [CrossRef]
Kapitza, P. L. , 1965, “ Dynamical Stability of a Pendulum When Its Point of Suspension Vibrates,” Collected Papers of P. L. Kapitza, Vol. 2, P. L. Kapitza , and D. ter Haar , ed., Pergamon, Oxford, UK, pp. 714–725.
Hong, K. S. , 2002, “ An Open-Loop Control for Underactuated Manipulators Using Oscillatory Inputs: Steering Capability of an Unactuated Joint,” IEEE Trans. Control Syst. Technol., 10(3), pp. 469–480. [CrossRef]
Bentsman, J. , and Hong, K. S. , 1991, “ Vibrational Stabilization of Nonlinear Parabolic Systems With Neumann Boundary Conditions,” IEEE Trans. Autom. Control, 36(4), pp. 501–507. [CrossRef]
Bentsman, J. , Hong, K. S. , and Fakhfakh, J. , 1991, “ Vibrational Control of Nonlinear Time Lag Systems: Vibrational Stabilization and Transient Behavior,” Automatica, 27(3), pp. 491–500. [CrossRef]
Bentsman, J. , and Hong, K. S. , 1993, “ Transient Behavior Analysis of Vibrationally Controlled Nonlinear Parabolic Systems With Neumann Boundary Conditions,” IEEE Trans. Autom. Control, 38(10), pp. 1603–1607. [CrossRef]
Tahmasian, S. , and Woolsey, C. A. , 2015, “ A Control Design Method for Underactuated Mechanical Systems Using High Frequency Inputs,” ASME J. Dyn. Syst., Meas., Control, 137(7), p. 071004.
Tahmasian, S. , and Woolsey, C. A. , 2016, “ Flight Control of Biomimetic Air Vehicles Using Vibrational Control and Averaging,” J. Nonlinear Sci., 27(4), pp. 1193–1214.
Bullo, F. , 2002, “ Averaging and Vibrational Control of Mechanical Systems,” SIAM J. Control Optim., 41(2), pp. 542–562. [CrossRef]
Blekhman, I. I. , 2000, Vibrational Mechanics, World Scientific Publishing Co, Singapore. [CrossRef]
Crouch, P. E. , 1981, “ Geometric Structures in Systems Theory,” IEE Proc. D (Control Theory Appl.), 128(5), pp. 242–252. [CrossRef]
Bullo, F. , and Lewis, A. D. , 2005, Geometric Control of Mechanical Systems. Texts in Applied Mathematics, Springer, New York. [CrossRef]
Weibel, S. , Baillieul, J. , and Kaper, T. J. , 1995, “ Small-Amplitude Periodic Motions of Rapidly Forced Mechanical Systems,” Conference on Decision and Control (CDC), New Orleans, LA, Dec. 13–15, pp. 533–539.
Fidlin, A. , and Thomsen, J. J. , 2008, “ Nontrivial Effects of High-Frequency Excitation for Strongly Damped Mechanical Systems,” Int. J. Non-Linear Mech., 43(7), pp. 569–578. [CrossRef]
Khalil, H. K. , 1996, Nonlinear Systems, Prentice Hall, Upper Saddle River, NJ.
Agrachev, A. A. , and Gamkrelidze, R. V. , 1979, “ The Exponential Representation of Flows and the Chronological Calculus,” Math. USSR: Sb., 35(6), pp. 727–785. [CrossRef]
Tahmasian, S. , Allen, D. W. , and Woolsey, C. A. , 2016, “ On Averaging and Input Optimization of High-Frequency Mechanical Control Systems,” J. Vib. Control, 24(5), pp. 937–955.
Tahmasian, S. , Jafari, F. , and Woolsey, C. A. , 2016, “ On Vibrational Stabilization of a Horizontal Pendulum,” ASME Paper No. DSCC2016-9675.
Tsakalis, K. S. , and Ioannou, P. A. , 1993, Linear Time-Varying Systems, Prentice Hall, Upper Saddle River, NJ.

Figures

Grahic Jump Location
Fig. 1

The 2π-periodic functions sin(t) (full), S2π(t) (dashed), and T2π(t) (dash dotted)

Grahic Jump Location
Fig. 2

Time histories of the original (full) and averaged (dashed) dynamics

Grahic Jump Location
Fig. 4

The horizontal pendulum system and the path ellipse

Grahic Jump Location
Fig. 3

The horizontal pendulum system

Grahic Jump Location
Fig. 5

Time histories of the original and averaged dynamics for θe = 35 deg

Grahic Jump Location
Fig. 6

The path ellipse (full) with its major axis (dashed) along the desired orientation θe = 35 deg

Grahic Jump Location
Fig. 7

Time histories of the original and averaged dynamics of multi-frequency horizontal pendulum

Grahic Jump Location
Fig. 8

The path tracked by the endpoint A of the pendulum with ω1 = 30 rad/s, ω2 = 40 rad/s, and ϕ = 30 deg

Grahic Jump Location
Fig. 10

Time history of θ using inputs (36) with θe = 40 deg and r2 = 3

Grahic Jump Location
Fig. 11

The trajectory tracked by the endpoint A of the pendulum using inputs (36) with θe = 40 deg and r2 = 3

Grahic Jump Location
Fig. 12

Time histories of the original system (26) (full), slowly time-varying (STV) system (43) (dashed) and time-invariant (TI) system (44) (dash-dotted)

Grahic Jump Location
Fig. 9

The 2π-periodic functions w1(t) and w2(t) and their second derivatives over one period

Tables

Errata

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

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In