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Research Papers

Energy-Neutral Transfer of Vibration Energy Across Modes by Using Active Nonlinear Stiffness Variation of Impulsive Type

[+] Author and Article Information
Thomas Pumhössel

Institute of Mechatronic Design and Production,
Johannes Kepler University Linz,
Linz 4040, Austria
e-mail: thomas.pumhoessel@jku.at

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received August 10, 2015; final manuscript received July 7, 2016; published online September 1, 2016. Assoc. Editor: Bogdan I. Epureanu.

J. Comput. Nonlinear Dynam 12(1), 011001 (Sep 01, 2016) (11 pages) Paper No: CND-15-1239; doi: 10.1115/1.4034264 History: Received August 10, 2015; Revised July 07, 2016

The effect of impulsive stiffness variation to the modal energy content of dynamical systems is investigated in this contribution. Therefore, the overall number of modes of vibration is divided into a set of lower and a set of higher modes. It is shown analytically that impulsive stiffness variation, applied in a state-dependent, nonlinear manner allows a targeted transfer of discrete amounts of energy across mode sets. Analytical conditions are presented, holding for a transfer from the lower to the higher mode set or vice versa. The existence of transfer cases where no energy crosses the system boundary, i.e., the energy-neutral case, is investigated in a comprehensive manner. Some numerical investigations underline that shifting vibration energy to higher modes causes a faster decay of vibration amplitudes, as the damping properties of a mechanical system can be utilized more effectively. Moreover, it is demonstrated that the proposed approach allows to eliminate vibration frequencies from the frequency spectrum of mechanical systems.

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Figures

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Fig. 1

Possible variations of mode set energies ΔEA,k and ΔEB,k of mode sets A and B

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Fig. 2

Dependency of variations of mode set energies from strength scaling factor εk, for εA,k>0 and εA,k<εB,k (a), and 0<εB,k<εA,k (b)

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Fig. 3

Dependency of variations of mode set energies from strength scaling factor εk, for εA,k>0 and εB,k<0, where (a) depicts the case χB,k<−χA,k and (b) χB,k>−χA,k

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Fig. 4

Schematic of the investigated mechanical system

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Fig. 5

Occurrence of transfer cases (A→B)= and (B→A)=, where A={1} and B={2,3} (a), and relative variation ΔEA,k(εS,k)/EA,0 of mode set A (b), without damping and stiffness variation. Detail view of possible variations of mode set energies for different values of εk (c).

Grahic Jump Location
Fig. 6

Impulsive strength factor εk and modal coordinates yi, i=1…3, for transferring energy from mode set A={1} to B={2,3}, where no energy crosses the system boundary (a). Corresponding time series of mode set energies EA and EB, modal energies E1, E2, and E3, and total energy Etot of the mechanical system (b). No structural damping present.

Grahic Jump Location
Fig. 7

Occurrence of transfer cases (A→B)= and (B→A)=, where A={1} and B={2,3}, in the presence of impulsive stiffness variation (a), and relative variation ΔEA,k(εS,k)/EA,0 of mode set A (b), without damping. Detail view of possible variations of mode set energies for different values of εk where almost all energy is contained within mode set B (c).

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Fig. 8

Occurrence of transfer cases (A→B)= and (B→A)=, where A={1,2} and B={3} (a), and relative variation ΔEA,k(εS,k)/EA,0 of mode set A (b), without damping and stiffness variation. Detail view of possible variations of mode set energies for different values of εk (c).

Grahic Jump Location
Fig. 9

Impulsive strength factor εk and modal coordinates yi, i=1…3, for transferring energy from mode set A={1,2} to B={3}, where no energy crosses the system boundary (a). Corresponding time series of mode set energies EA and EB, modal energies E1, E2, and E3, and total energy Etot of the mechanical system (b). No structural damping present.

Grahic Jump Location
Fig. 10

Occurrence of transfer cases (A→B)= and (B→A)=, where A={1,2} and B={3}, in the presence of impulsive stiffness variation (a), and relative variation ΔEA,k(εS,k)/EA,0 of mode set A (b), without damping. Detail view of possible variations of mode set energies for different values of εk where almost all energy is contained within mode set B (c).

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Fig. 11

Comparison of total relative energy Etot,r of the mechanical system for different values of maximum allowable impulsive strength factor εmax, total relative energy Etot,r,wo,IPE if no stiffness variation is present, and relative work Wp,r of the impulsive elastic forces, for the transfer case (A→B)=, where A={1} and B={2,3} (a), and A={1,2} and B={3} (b)

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Fig. 12

Comparison of relative mode set energies EA,r and EB,r of mode sets A and B, total relative energy Etot,r, and total relative energy Etot,wo,IPE if no stiffness variation is present, for the transfer case (A→B)=, where A={1} and B={2,3} (a), and A={1,2} and B={3} (b). εmax is set to 0.4.

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Fig. 13

Modal coordinates yi, i=1…3 with and without (gray colored lines) stiffness variation, and rectangular stiffness function p for the transfer case (A→B)=, where A={1} and B={2,3} (a), and A={1,2} and B={3} (b). εmax is set to 0.4.

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Fig. 14

Comparison of physical coordinates xi, i=1…3 with and without (gray colored lines) stiffness variation, for the transfer case (A→B)=, where A={1} and B={2,3} (a), and A={1,2} and B={3} (b). εmax is set to 0.4.

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