The Dynamic Response of Tuned Impact Absorbers for Rotating Flexible Structures

[+] Author and Article Information
Steven W. Shaw1

Department of Mechanical Engineering,  Michigan State University, East Lansing, MI 48824-1226shawsw@egr.msu.edu

Christophe Pierre

College of Engineering,  McGill University, Montreal, Quebec, H3A 2K6 Canadachristophe.pierre@mcgill.ca

It should be noted that in (3) spherical balls are used for the absorber mass, and these presumably roll without slipping, in which case their effective mass will include rotational effects; see (8) for a similar analysis involving the rollers of bifilar torsional absorbers.

This system is used for reference since it includes the inertia of the absorber as fixed to the primary system, but not its dynamics relative to the primary system.

Grazing refers to a zero-velocity impact. As will be demonstrated in the simulations below, another type of grazing bifurcation occurs in this system, one that leads directly to chaos.

This is a consequence of the phase relations dictated by the zero free flight damping.

This grazing is quite different in nature than the one wherein the symmetric impact motion merges with the linear nonimpacting resoponse. See (25) for a discussion of the classification of grazing bifurcations.


Address all correspondence to this author.

J. Comput. Nonlinear Dynam 1(1), 13-24 (May 18, 2005) (12 pages) doi:10.1115/1.1991872 History: Received April 14, 2005; Revised May 18, 2005

This paper describes an analytical investigation of the dynamic response and performance of impact vibration absorbers fitted to flexible structures that are attached to a rotating hub. This work was motivated by experimental studies at NASA, which demonstrated the effectiveness of these types of absorbers for reducing resonant transverse vibrations in periodically excited rotating plates. Here we show how an idealized model can be used to describe the essential dynamics of these systems, and used to predict absorber performance. The absorbers use centrifugally induced restoring forces so that their nonimpacting dynamics are tuned to a given order of rotation, whereas their large amplitude dynamics involve impacts with the primary flexible system. The linearized, nonimpacting dynamics are first explored in detail, and it is shown that the response of the system has some rather unique features as the hub rotor speed is varied. A class of symmetric impacting motions is also analyzed and used to predict the effectiveness of the absorber when operating in its impacting mode. It is observed that two different types of grazing bifurcations take place as the rotor speed is varied through resonance, and their influence on absorber performance is described. The analytical results for the symmetric impacting motions are also used to generate curves that show how important absorber design parameters—including mass, coefficient of restitution, and tuning—affect the system response. These results provide a method for quickly evaluating and comparing proposed absorber designs.

Copyright © 2006 by American Society of Mechanical Engineers
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Figure 1

(a) Schematic diagram of the pendulum-type blade model with attached pendulum absorber. (b) System attached to the rotating hub.

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Figure 2

Normalized linear natural frequencies versus normalized rotation rate σ. The dashed line is the primary system frequency with the absorber locked, ω11, showing centrifugal stiffening; the dashed-dotted line is the absorber frequency with the primary system locked, ω22=ñσ; the solid and dotted lines represent the in-phase and out-of-phase system natural frequencies, ω1 and ω2, respectively. Note the veering that occurs for small absorber mass. Parameter values: μ=0.024, δ=0.67, α=0.84, ñ=3.007.

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Figure 3

Modal amplitude ratios as a function of σ; line types and parameters are the same as those in Fig. 2

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Figure 4

Campbell diagram. Primary system frequency (with no absorber) versus σ along with order excitation lines for μ=0, δ=0.67, and orders n=1, 2, 3, and 4.

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Figure 5

System resonance frequency versus ñ. Parameter values: μ=0.024, δ=0.67, α=0.84, n=3.

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Figure 6

Excitation force amplitude thresholds above which impacts must occur, depicted as f versus σ, for absorber tuning values ñ=2.993 (dashed line), ñ=3 (solid line), and ñ=3.007 (dotted line). The nonimpacting steady-state response exists below these curves, and impacting motions must occur above them. Note that impacting and nonimpacting steady-state responses can co-exist below these curves. Parameter values: μ=0.024, δ=0.67, α=0.84, n=3.

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Figure 7

A typical symmetric periodic impacting response. (a) ẋ versus x. (b) ẏ versus y. Parameter values: μ=0.024, δ=0.67, α=0.84, ñ=3.007, n=3, e=0.5, f=0.01, σ=0.17, and modal damping ratios of 1.0%.

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Figure 8

Symmetric impacting response versus rotor speed near primary system resonance, shown as the amplitudes of dynamic variables at impact. Solid lines represent the stable impacting response and dashed lines represent the unstable impacting response. Parameter values: δ=0.67α=0.84, e=0.8, μ=0.024, ñ=3.007, f=0.005, n=3. (a) Primary system amplitude log10∣x∣, where the dotted line is the reference linear primary system response with the absorber locked, showing resonance. (b) Primary system velocity ẋ. (c) Absorber velocity ẏ. (d) Excitation phase t0.

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Figure 9

Representative absorber phase planes, ẏ versus y, showing steady-state responses. Parameter values: δ=0.67, α=0.84, e=0.8, μ=0.024, ñ=3.007, f=0.005, n=3. (a) Well below resonance, σ=0.17; (b) just below resonance, σ=0.34; (c) close to the first near-resonance grazing bifurcation, σ=0.3415; (d) chaos, immediately beyond the grazing bifurcation, σ=0.3416; (e) close to the second near-resonance grazing bifurcation, σ=0.34348; (f) above resonance, near the grazing coincidence with the linear response, σ=0.35.

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Figure 10

Some details of a typical chaotic response near resonance. Parameter values: δ=0.67, α=0.84, e=0.8, μ=0.024, ñ=3.007, σ=0.343, f=0.005, n=3. (a) x(t) versus t; (b) ẋ(t) versus x(t) sampled at impact times; (c) ẏ(t) versus the forcing phase, tomod(2π∕ω), sampled at impact times.

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Figure 11

Peak response of the primary system at resonance versus force amplitude f, for δ=0.67, α=0.84, and n=3: (a) μ=0.024, ñ=3.007, and e=0.9,0.7,0.5,0.3; (b) e=0.8, ñ=3.007, and μ=0.024,0.018,0.0120,0.006; (c) μ=0.024, e=0.8, and ñ=2.81,3.007,3.28.




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