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Research Papers

Vibration Instability in a Large Motion Bistable Compliant Mechanism Due to Stribeck Friction

[+] Author and Article Information
Alborz Niknam

Department of Mechanical Engineering and
Energy Processes,
Southern Illinois University Carbondale,
Carbondale, IL 62901
e-mail: alborz@siu.edu

Kambiz Farhang

Department of Mechanical Engineering and
Energy Processes,
Southern Illinois University Carbondale,
Carbondale, IL 62901
e-mail: farhang@siu.edu

1Corresponding author.

Contributed by the Technical Committee on Vibration and Sound of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received April 16, 2018; final manuscript received May 31, 2018; published online July 5, 2018. Assoc. Editor: Stefano Lenci.

J. Vib. Acoust 140(6), 061017 (Jul 05, 2018) (9 pages) Paper No: VIB-18-1168; doi: 10.1115/1.4040513 History: Received April 16, 2018; Revised May 31, 2018

The present paper investigates friction-induced self-excited vibration of a bistable compliant mechanism. A pseudo-rigid-body representation of the mechanism is used containing a hardening nonlinear spring and a viscous damper. The mass is suspended from above with the spring-damper combination leading to the addition of geometric nonlinearity in the equation of motion and position- and velocity-dependent normal contact force. Friction input provided by a moving belt in contact with the mass. An exponentially decaying function of sliding velocity describes the friction coefficient and, thereby, incorporates Stribeck effect of friction. Eigenvalue analysis is employed to investigate the local stability of the steady-state fixed points. It is observed that the oscillator experiences pitchfork and Hopf bifurcations. The effects of the spring nonlinearity and precompression, viscous damping, belt velocity, and the applied normal force on the number, position, and stability of the equilibrium points are investigated. Global system behavior is studied by establishing trajectory maps of the system. Critical belt speed is derived analytically and shown to be only the result of Stribeck effect of friction. It is found that one equilibrium point dominates the steady-state response for very low damping and negligible spring nonlinearity. The presence of damping and/or spring nonlinearity tends to diminish this dominance.

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Figures

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Fig. 1

Mechanical model of the oscillator: (a) bistable compliant mechanism with a flexible link on a moving belt of constant velocity vb and (b) pesudo-rigid-body representation of the mechanism

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Fig. 2

An illustration of a bistable mechanism with double-well potential energy: for three values of Γ and α=3

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Fig. 3

Loci of equilibrium points for V=0.5 and α=3: (a) location of equilibrium points versus normal force and (b) location of equilibrium points versus Γ. Dashed and solid lines display saddle and spiral, respectively.

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Fig. 10

Global response of sustained oscillation with several initial conditions (dot). Solid lines show trajectories. Dashed lines divide the phase plane: (a) The system possesses two spirals (+) and one saddle (×) and (b) the right spiral undergoes a Hopf bifurcation; V=0.22, n=1, α=3, Γ=0, ζ=0.1.

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Fig. 11

Global response of sustained oscillation with several initial conditions (dot). The system possesses two spiral sinks (+) and one saddle (×). Both spirals undergo Hopf bifurcation. Solid lines show trajectories. Dashed lines divide the phase plane. V=0.5, n=1, α=3, Γ=0, ζ=0.1.

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Fig. 12

Freebody diagram of the mass

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Fig. 13

Critical dimensionless applied normal load versus spring nonlinearity: 0.01<V<0.5,α=3

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Fig. 9

Global response of sustained oscillation with several initial conditions (dot). The system possesses two spiral sources (+) and one saddle (×). Solid lines show trajectories. Dashed lines divide the phase plane; V=0.1, n=1, α=3, Γ=0, ζ=0.1.

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Fig. 8

(a) Global responses for two damped systems, ζ=0.01 (dashed line) and ζ=0.1 (solid line) from identical initial conditions (dot): The former experiences sustain oscillation around the right spiral, the latter oscillates around the left spiral. Line segment labeled A shows where fne>0. V=0.1, n=1, α=3, Γ=0 and (b) overlay of trajectory for ζ=0.1 and the second fnet>0 region near the left spiral.

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Fig. 7

fnet>0 region boundaries in (V,u1) and (ζ,u1) planes for V = 0.1, α=3, n=1: (a) fnet>0 regions shown as hashed regions for a linear spring, (b) zoomed in view of fnet>0 region near the left spiral and mass displacement as vertical dashed line, (c) fnet>0 regions in which the arrows indicate the region, and (d)fnet>0 regions in the (V,u1) plane for Γ=0

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Fig. 6

Schematic diagram showing the definition of net tangential force, fnet=fk+fd−fs, on the slider

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Fig. 5

Illustration of local stability of the linearized system for α=3, n=1: belt velocities correspond to critical Hopf bifurcation parameters versus damping ratio

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Fig. 4

Loci of equilibrium points for V=0.5 and n=1: (a) location of equilibrium points versus α and (b) location of equilibrium points versus Γ. Dashed and solid lines display saddle and spiral, respectively.

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