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Research Papers

Two-Dimensional Input Shaping for One-Dimensional Continua

[+] Author and Article Information
Amir Lotfi-Gaskarimahalle

Mechatronics Research Lab, The Pennsylvania State University, University Park, PA 16802lotfi@psu.edu

Christopher D. Rahn

Mechatronics Research Lab, The Pennsylvania State University, University Park, PA 16802cdrahn@psu.edu

J. Vib. Acoust 130(2), 021004 (Jan 30, 2008) (8 pages) doi:10.1115/1.2827982 History: Received October 03, 2006; Revised September 24, 2007; Published January 30, 2008

This paper extends input shaping control to one-dimensional continua. Unlike discrete systems where the input is shaped only in the temporal domain, temporal and spatial input shaping can produce zero residual vibration in setpoint position control of distributed strings and beams. For collocated and noncollocated boundary control of strings and domain control of strings and pinned beams, the response to step inputs is solved in closed form using delays. For a clamped beam model, a closed form infinite modal series is used. The boundary controlled string can be setpoint regulated using two-pulse zero vibration (ZV) and three-pulse zero vibration and derivative (ZVD) shapers but ZVD is not more robust to parameter variations than ZV, a unique characteristic of second-order partial differential equations systems. Noncollocated ZV and ZVD boundary control enables rigid body translation of a string with zero residual vibration. Domain controlled strings and pinned beams with spatial input distributions that satisfy certain orthogonality conditions (e.g., midspan point load or uniformly distributed load) can be setpoint regulated with shaped inputs. For the cantilevered beam, modal shaping of the input distribution and ZV or ZVD temporal shaping drives the tip to the desired position with zero residual vibration.

FIGURES IN THIS ARTICLE
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Copyright © 2008 by American Society of Mechanical Engineers
Topics: String , Vibration
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Figures

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Figure 5

Noncollocated boundary controlled string: input u0(t) (dotted), acceleration (u0)tt (dashed), and response u(1,t) (solid). Inset shows displacement distribution u(x,ti) in response to the ZVD input at times ti=0,2,3,4,6.

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Figure 1

Four typical string applications: (a) boundary controlled string, (b) noncollocated boundary controlled string, (c) domain controlled string, and (d) noncollocated boundary controlled string with mass

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Figure 2

Sawtooth function in Eq. 6 with τ=0

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Figure 3

Boundary controlled string response to ZV (p(t)=dotted and u(1,t)=solid) and ZVD (p(t)=dash dotted and u(1,t)=dashed) shapers. Inset shows displacement distribution response u(x,ti) to the ZV input at times ti=i∕2.

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Figure 4

Boundary controlled string robustness (a) response to ZV (solid) and ZVD (dotted) shapers with a 10% error in the natural frequencies and (b) sensitivity curve for the first mode under ZV (solid) and ZVD (dashed) and the second mode under ZV (dotted) and ZVD (dash dotted) shapers. (c) Sensitivity curve for the string under ZV and ZVD shapers (same curve).

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Figure 6

Noncollocated boundary controlled string with mass: input F(t) (dotted) and response u(1,t) (solid). Inset shows displacement distribution u(x,ti) at times ti=i.

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Figure 7

ZV (p(t)=dotted and u(12,t)=solid) and ZVD (p(t)=dash dotted and u(12,t)=dashed) shaped domain controlled string responses to (a) point force at x=12 and (b) distributed force. Insets show displacement distribution u(x,ti) at times ti=i∕4.

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Figure 8

Two typical beam applications: (a) pinned-pinned beam and (b) cantilevered beam

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Figure 9

ZV (p(t)=dotted and u(12,t)=solid) and ZVD (p(t)=dash dotted and u(12,t)=dashed) domain control pinned beam response to a midspan point load. Inset shows displacement response to ZV inputs u(x,ti) at times ti=i∕4π.

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Figure 10

Pinned-pinned beam robustness (a) response to ZV (solid) and ZVD (dotted) shapers with 10% error in the first natural frequency and (b) sensitivity curves for ZV (solid) and ZVD (dotted) shapers.

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Figure 11

Cantilevered beam response to ZV (p(t)=dotted and u(1,t)=solid) and ZVD (p(t)=dash dotted and u(1,t)=dashed) shapers. Inset shows the displacement distribution u(x,ti) at times ti=i∕4.

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