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Research Papers

Response of a Harmonically Forced Dry Friction Damped System Under Time-Delayed State Feedback

[+] Author and Article Information
R. K. Mitra

Department of Mechanical Engineering,
National Institute of Technology,
Durgapur 713209, West Bengal, India
e-mail: rkmitra.me@gmail.com

A. K. Banik

Department of Civil Engineering,
National Institute of Technology,
Durgapur 713209, West Bengal, India
e-mail: akbanik@gmail.com

S. Chatterjee

Department of Mechanical Engineering,
Indian Institute of Engineering
Science and Technology,
Shibpur 711103, West Bengal, India
e-mail: shychat@gmail.com

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received February 17, 2017; final manuscript received October 27, 2017; published online December 7, 2017. Assoc. Editor: Massimo Ruzzene.

J. Comput. Nonlinear Dynam 13(3), 031001 (Dec 07, 2017) (14 pages) Paper No: CND-17-1074; doi: 10.1115/1.4038445 History: Received February 17, 2017; Revised October 27, 2017

Nonlinear dynamics, control, and stability analysis of dry friction damped system under state feedback control with time delay are investigated. The dry friction damped system is harmonically excited, and the nonlinearities in the equation of motion arise due to nonlinear damping and spring force. In this paper, a frequency domain-based method, viz., incremental harmonic balance method along with arc-length continuation technique (IHBC) is first employed to identify the primary responses which may be present in such system. The IHBC is then reformulated in a manner to treat the dry friction damped system under state feedback control with time delay and is applied to obtain control of responses in an efficient and systematic way. The stability of uncontrolled responses is obtained by Floquet's theory using Hsu' scheme, and the stability of the controlled responses is obtained by applying a semidiscretization method for delay differential equation (DDE).

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References

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Figures

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Fig. 1

Single DOF dry friction damped model with time-delayed feedback control scheme

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Fig. 2

Linear stability charts under time-delayed state feedback. Solid lines represent the critical stability lines for different n (Eq. (10)) on which some of the characteristic roots are purely imaginary. The original marginally stable system can be stabilized by selecting gains and delays from the primary stability regions. (a) gv = 0, (b) gd = 0, (c) gd = 0.1, (d) gd = 0.3, (e)gd = 0.5, and (f) gd = 0.9.

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Fig. 3

Phase planes for controlled responses (system parameters: c = 0.05, k = 0.5, f = 4 and ω = 1; control parameters: gd = 0.5, gv = 0 and (a) τd=0.01, (b) τd = 0.5, (c) τd = 1.3, (d) τd = 2, (e) τd = 2.5, and (f) τd = 3). ′A′: controlled amplitude, number of “*” s: order of subharmonics, uncontrolled amplitude: 2.327 m.

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Fig. 4

Phase planes for controlled responses (system parameters: c = 0.05, k = 0.5, f = 4 and ω = 1; control parameters: gd = 0.5, gv = −0.5 and (a) τd=0.01, (b) τd = 0.4, (c) τd = 0.85, (d) τd = 1.2, (e) τd = 1.7, and (f) τd = 2). ′A′: controlled amplitude, number of “*” s: order of subharmonics, uncontrolled amplitude: 2.327 m.

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Fig. 5

(a) Approximation of the time delay (τd) term, (b) zoomed view of the encircled part shown in a, and (c) stability criteria depend on real and imaginary parts of eigenvalues of the transition matrix

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Fig. 6

Uncontrolled frequency response plots with different stiffness nonlinearity (k) (system parameters: c = 0.05, f = 4)

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Fig. 13

Controlled responses with high negative velocity gain (system parameters: c = 0.05, k = 0.3; control parameters: gd = 0.5, gv = −5, τd = 0.15)

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Fig. 14

Comparison of results obtained by IHBC and NI (system (c,k,f) and control (gd,gv,τd) parameters: (i) (0.05, 0.3, 3) and (0, 0, 0) (ii) (0.05, 0.3, 7) and (0, −1, 0.53), and (iii) (0.05, 0.3, 2) and (0.5, −1.5, 0.5))

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Fig. 7

Effect of control on frequency response plots shown in Fig. 6 (control parameters: gd = 0.1, gv = −0.75, τd = 0.61)

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Fig. 8

Effect of control on frequency response plots shown in Fig. 6 (control parameters: gd = 0.9, gv = −1.25, τd = 0.53)

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Fig. 9

Comparison of results obtained by IHBC and NI method (system (c,k,f) and control (gd,gv,τd) parameters: (i) (0.05, 0.5, 4) and (0, 0, 0), (ii) (0.05, 0.1, 4) and (0.1, −0.75, 0.61), and (iii) (0.05, 0.005, 4) and (0.9, −1.25, 0.53))

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Fig. 10

Uncontrolled frequency response plots with different forcing amplitude (f) (system parameters: c = 0.05, k = 0.3)

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Fig. 11

Effect of control on frequency response plots shown in Fig. 10 (control parameters: gd = 0, gv = −1, τd = 0.53)

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Fig. 12

Effect of control on frequency response plots shown in Fig. 10 (control parameters: gd = 0.5, gv = −1.5, τd = 0.5)

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Fig. 21

(a) Uncontrolled phase planes (system parameters: c = 0.05, k = 0.1, f = 2) and (b) controlled phase planes (control parameters: gd = 0.9, gv = −1.25, τd = 0.53)

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Fig. 15

Uncontrolled frequency response plots with different damping ratio (c) (system parameters: k = 0.25, f = 3)

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Fig. 16

Effect of control on frequency response plots shown in Fig. 15 (control parameters: gd = 0.3, gv = −0.5, τd = 0.74)

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Fig. 17

Effect of control on frequency response plots shown in Fig. 15 (control parameters: gd = 0.5, gv = −1.5, τd = 0.5)

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Fig. 18

Comparison of results obtained by IHBC and NI (system (c,k,f) and control (gd,gv,τd) parameters: (i) (0.05, 0.25, 3) and (0, 0, 0) (ii) (0.005, 0.25, 3) and (0.3, −0.5, 0.74), and (iii) (0.2, 0.25, 3) and (0.5, −1.5, 0.5))

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Fig. 19

Response amplitude (A) versus forcing amplitude (f) plots with different excitation frequency (uncontrolled) (system parameters: k = 0.1, c = 0.05)

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Fig. 22

max(Eig) versus response frequency (ω) plots (system parameters: c = 0.05, k = 0.3, f = 5; control parameters (gd, gv, τd): (i) (0, 0, 0), (ii) (0, −1, 0.53), and (iii) (0.5, −1.5, 0.5))

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Fig. 23

Closed loop control system

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Fig. 20

Effect of control on the A versus f plots shown in Fig.19 (control parameters: gd = 0.9, gv = −1.25, τd = 0.53)

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Fig. 24

Root locus plot under positive displacement feedback with time delay (τd) 0.1

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Fig. 25

Root locus plot under negative velocity feedback with time delay (τd) 0.1

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