Abstract

This article investigates the nonminimum phase (NMP) zeros in the transfer function, between actuated load input and measured displacement output, of a multi-degrees-of-freedom (DoF) flexible system in the presence of proportional viscous damping. NMP zeros have a negative impact on the dynamics and control of flexible systems and therefore are generally undesirable. Viscous damping is one potential means to guarantee that no NMP zeros exist in the system. However, the impact of viscous damping on NMP zeros of multi-DoF flexible systems is not adequately studied or understood in the literature. To address this gap, a change of variable method is used to first establish a simple mathematical relationship between the zeros of a multi-DoF undamped flexible system and its proportionally damped counterpart. The “proportional” viscous damping model is used due to its practical amenability, conceptual simplicity, and ease of application. This mathematical relationship (between zeros of an undamped system and its damped counterpart) is used to derive the necessary and sufficient condition for the absence of NMP zeros in proportionally damped flexible systems. A graphical analysis of this necessary and sufficient condition is provided, which leads to the formulation of simple proportional damping strategies. A case study of a 4DoF flexible system is presented to demonstrate how a proportional viscous damping strategy can be used to simultaneously guarantee the absence of NMP zeros in multiple single-input single-output (SISO) transfer functions of a multi-DoF flexible system.

References

1.
Chen
,
J.
,
Qiu
,
L.
, and
Toker
,
O.
,
2000
, “
Limitations on Maximal Tracking Accuracy
,”
IEEE Trans. Autom. Control
,
45
(
2
), pp.
326
331
.
2.
Freudenberg
,
J.
, and
Looze
,
D.
,
1985
, “
Right Half Plane Poles and Zeros and Design Tradeoffs in Feedback Systems
,”
IEEE Trans. Autom. Control
,
30
(
6
), pp.
555
565
.
3.
Middleton
,
R. H.
,
1991
, “
Trade-Offs in Linear Control System Design
,”
Automatica
,
27
(
2
), pp.
281
292
.
4.
Loix
,
N.
,
Kozanek
,
J.
, and
Foltete
,
E.
,
1996
, “
On the Complex Zeros of Non-Colocated Systems
,”
J. Struct. Control
,
3
(
1–2
), pp.
79
87
.
5.
Cannon
,
R. H.
, and
Schmitz
,
E.
,
1984
, “
Initial Experiments on the End-Point Control of a Flexible One-Link Robot
,”
Int. J. Rob. Res.
,
3
(
3
), pp.
62
75
.
6.
Di Gennaro
,
S.
,
2003
, “
Output Stabilization of Flexible Spacecraft With Active Vibration Suppression
,”
IEEE Trans. Aerosp. Electron. Syst.
,
39
(
3
), pp.
747
759
.
7.
Hu
,
Q.
,
2008
, “
Sliding Mode Maneuvering Control and Active Vibration Damping of Three-Axis Stabilized Flexible Spacecraft With Actuator Dynamics
,”
Nonlinear Dyn.
,
52
(
3
), pp.
227
248
.
8.
Hughes
,
P. C.
,
1974
, “
Dynamics of Flexible Space Vehicles With Active Attitude Control
,”
Celestial Mech.
,
9
(
1
), pp.
21
39
.
9.
Friedmann
,
P. P.
, and
Millott
,
T. A.
,
1995
, “
Vibration Reduction in Rotorcraft Using Active Control—A Comparison of Various Approaches
,”
J. Guid. Control Dyn.
,
18
(
4
), pp.
664
673
.
10.
Uddin
,
M. M.
,
Sarker
,
P.
,
Theodore
,
C. R.
, and
Chakravarty
,
U. K.
,
2018
, “
Active Vibration Control of a Helicopter Rotor Blade by Using a Linear Quadratic Regulator
,”
ASME International Mechanical Engineering Congress and Exposition
, Vol.
52002
,
Pittsburgh, PA
,
Nov 9–15
, p.
V001T03A014
.
11.
Chang
,
J. Y.
,
2007
, “
Hard Disk Drive Seek-Arrival Vibration Reduction With Parametric Damped Flexible Printed Circuits
,”
Microsyst. Technol.
,
13
(
8
), pp.
1103
1106
.
12.
Feng
,
G.
,
Fook Fah
,
Y.
, and
Ying
,
Y.
,
2005
, “
Modeling of Hard Disk Drives for Vibration Analysis Using a Flexible Multibody Dynamics Formulation
,”
IEEE Trans. Magn.
,
41
(
2
), pp.
744
749
.
13.
Roy
,
N. K.
, and
Cullinan
,
M. A.
,
2018
, “
Design and Characterization of a Two-Axis, Flexure-Based Nanopositioning Stage With 50 mm Travel and Reduced Higher Order Modes
,”
Precis. Eng.
,
53
, pp.
236
247
.
14.
Xu
,
Q.
,
2013
, “
Design and Development of a Compact Flexure-Based XY Precision Positioning System With Centimeter Range
,”
IEEE Trans. Ind. Electron.
,
61
(
2
), pp.
893
903
.
15.
Yong
,
Y. K.
,
Aphale
,
S. S.
, and
Moheimani
,
S. R.
,
2008
, “
Design, Identification, and Control of a Flexure-Based XY Stage for Fast Nanoscale Positioning
,”
IEEE Trans. Nanotechnol.
,
8
(
1
), pp.
46
54
.
16.
Awtar
,
S.
, and
Parmar
,
G.
,
2013
, “
Design of a Large Range XY Nanopositioning System
,”
ASME J. Mech. Rob.
,
5
(
2
), p.
021008
.
17.
Choi
,
K.-B.
, and
Lee
,
J. J.
,
2005
, “
Passive Compliant Wafer Stage for Single-Step Nano-Imprint Lithography
,”
Rev. Sci. Instrum.
,
76
(
7
), p.
075106
.
18.
Rath
,
S.
,
Cui
,
L.
, and
Awtar
,
S.
,
2021
, “
On the Zeros of an Undamped Three-DOF Flexible System
,”
ASME Lett. Dyn. Syst. Control
,
1
(
4
), p.
041010
.
19.
Gevarter
,
W. B.
,
1970
, “
Basic Relations for Control of Flexible Vehicles
,”
AIAA J.
,
8
(
4
), pp.
666
672
.
20.
Martin
,
G. D.
,
1978
, “
On the Control of Flexible Mechanical Systems
,”
Ph.D. dissertation
,
Stanford University
,
Stanford, CA
.
21.
Coelingh
,
E.
,
de Vries
,
T. J.
, and
Koster
,
R.
,
2002
, “
Assessment of Mechatronic System Performance at an Early Design Stage
,”
IEEE/ASME Trans. Mech.
,
7
(
3
), pp.
269
279
.
22.
Cui
,
L.
,
Okwudire
,
C.
, and
Awtar
,
S.
,
2017
, “
Modeling Complex Nonminimum Phase Zeros in Flexure Mechanisms
,”
ASME J. Dyn. Syst. Meas. Control
,
139
(
10
), p.
101001
.
23.
Goodman
,
L.
,
1976
, “
Material Damping and Slip Damping
,”
Shock Vib. Handb.
,
36
, pp.
1
28
.
24.
Bowden
,
M.
, and
Dugundji
,
J.
,
1990
, “
Joint Damping and Nonlinearity in Dynamics of Space Structures
,”
AIAA J.
,
28
(
4
), pp.
740
749
.
25.
Li
,
P.
, and
Hu
,
R.
,
2007
, “
On the Air Damping of Flexible Microbeam in Free Space at the Free-Molecule Regime
,”
Microfluid. Nanofluid.
,
3
(
6
), pp.
715
721
.
26.
Varanasi
,
K. K.
, and
Nayfeh
,
S. A.
,
2006
, “
Damping of Flexural Vibration Using Low-Density, Low-Wave-Speed Media
,”
J. Sound Vib.
,
292
(
1
), pp.
402
414
.
27.
Greene
,
M.
,
1987
, “
Robustness of Active Modal Damping of Large Flexible Structures
,”
Int. J. Control
,
46
(
3
), pp.
1009
1018
.
28.
Rath
,
S.
, and
Awtar
,
S.
,
2023
, “
On the Zeros of Three-DoF Damped Flexible Systems
,”
J. Sound Vib.
560
,
117698
.
29.
Pang
,
S. T.
,
Tsao
,
T.
, and
Bergman
,
L. A.
,
1993
, “
Active and Passive Damping of Euler-Bernoulli Beams and Their Interactions
,”
ASME J. Dyn. Syst. Meas. Control
,
115
(
3
), pp.
379
384
.
30.
Lin
,
J. L.
, and
Juang
,
J. N.
,
1995
, “
Sufficient Conditions for Minimum-Phase Second-Order Linear Systems
,”
J. Vib. Control
,
1
(
2
), pp.
183
199
.
31.
Lin
,
J. L.
,
1999
, “
On Transmission Zeros of Mass-Dashpot-Spring Systems
,”
J. Dyn. Syst. Meas. Control
,
121
(
2
), pp.
179
183
.
32.
Williams
,
T.
,
1989
, “
Transmission-Zero Bounds for Large Space Structures, With Applications
,”
J. Guid. Control Dyn.
,
12
(
1
), pp.
33
38
.
33.
Rath
,
S.
, and
Awtar
,
S.
,
2021
, “
Non-Minimum Phase Zeros of Two-DoF Damped Flexible Systems
,”
IFAC-PapersOnLine
,
54
(
20
), pp.
579
585
.
34.
Trombetti
,
T.
, and
Silvestri
,
S.
,
2004
, “
Added Viscous Dampers in Shear-Type Structures: The Effectiveness of Mass Proportional Damping
,”
J. Earthquake Eng.
,
08
(
02
), pp.
275
313
.
35.
Schwarz
,
B.
, and
Richardson
,
M.
,
2014
, “
Proportional Damping From Experimental Data
,”
Topics in Modal Analysis, Volume 7: Proceedings of the 31st IMAC, A Conference on Structural Dynamics, 2013
,
New York
,
June 6
,
Springer
, pp.
179
186
.
36.
Adhikari
,
S.
,
2006
, “
Damping Modelling Using Generalized Proportional Damping
,”
J. Sound Vib.
,
293
(
1
), pp.
156
170
.
37.
Caughey
,
T. K.
, and
O’Kelly
,
M. E. J.
,
1965
, “
Classical Normal Modes in Damped Linear Dynamic Systems
,”
ASME J. Appl. Mech.
,
32
(
3
), pp.
583
588
.
38.
Caughey
,
T. K.
,
1960
, “
Classical Normal Modes in Damped Linear Dynamic Systems
,”
ASME J. Appl. Mech.
,
27
(
2
), pp.
269
271
.
39.
Rath
,
S.
,
2024
, “
On Zeros of Flexible Systems
,”
Ph.D. dissertation
,
University of Michigan
,
Ann Arbor, MI
.
40.
Hoagg
,
J. B.
,
Chandrasekar
,
J.
, and
Bernstein
,
D. S.
,
2006
, “
On the Zeros, Initial Undershoot, and Relative Degree of Collinear Lumped-Parameter Structures
,”
ASME J. Dyn. Syst. Meas. Control
,
129
(
4
), pp.
493
502
.
41.
Wu
,
W. H.
, and
Chen
,
C. Y.
,
2001
, “
Simple Lumped-Parameter Models of Foundation Using Mass-Spring-Dashpot Oscillators
,”
J. Chin. Inst. Eng.
,
24
(
6
), pp.
681
697
.
42.
Ritto
,
T.
,
Aguiar
,
R.
, and
Hbaieb
,
S.
,
2017
, “
Validation of a Drill String Dynamical Model and Torsional Stability
,”
Meccanica
,
52
(
11
), pp.
2959
2967
.
43.
Tang
,
X.
,
Hu
,
X.
,
Yang
,
W.
, and
Yu
,
H.
,
2017
, “
Novel Torsional Vibration Modeling and Assessment of a Power-Split Hybrid Electric Vehicle Equipped With a Dual-Mass Flywheel
,”
IEEE Trans. Veh. Technol.
,
67
(
3
), pp.
1990
2000
.
44.
Reichl
,
K. K.
, and
Inman
,
D.
,
2017
, “
Lumped Mass Model of a 1D Metastructure for Vibration Suppression With No Additional Mass
,”
J. Sound Vib.
,
403
, pp.
75
89
.
45.
Li
,
D.
,
Ikago
,
K.
,
Yin
,
A. J. M. S.
, and
Processing
,
S.
,
2023
, “
Structural Dynamic Vibration Absorber Using a Tuned Inerter Eddy Current Damper
,”
Mech. Syst. Signal Process
,
186
, p.
109915
.
46.
Wang
,
Z.
,
Chen
,
Z.
,
Wang
,
J. J. E. E.
, and
Vibration
,
E.
,
2012
, “
Feasibility Study of a Large-Scale Tuned Mass Damper With Eddy Current Damping Mechanism
,”
Earthquake Eng. Eng. Vib.
,
11
(
3
), pp.
391
401
.
47.
Sodano
,
H. A.
,
Bae
,
J.-S. J. S.
, and
Digest
,
V.
,
2004
, “
Eddy Current Damping in Structures
,”
Shock Vib. Dig.
,
36
(
6
), p.
469
.
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