Contact Impact Forces at Discontinuous 2-DOF Vibroimpact

Bazhenov, V.A.; Pogorelova, O.S.; Postnikova, T.G.

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Contact Impact Forces at Discontinuous 2-DOF Vibroimpact

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Mar 14, 2016

Applied Mathematics and Nonlinear Sciences

Volume 1 (2016): Issue 1 (January 2016)

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Published Online: Mar 14, 2016

Page range: 183 - 196

Received: Dec 03, 2015

Accepted: Mar 10, 2016

DOI: https://doi.org/10.21042/AMNS.2016.1.00014

Keywords
vibroimpact system, discontinuous, contact force, Hertz’s law, bifurcation, multiplier, nonlinear, stability

© 2016 V.A. Bazhenov, O.S. Pogorelova, T.G. Postnikova, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Global and partial views of contact force depending on exitation amplitude (ω=7.23 rad·s-1 )

Contact forces, external load, and phase portraits for: (a) (5,2)-regime; (b) quasiperiodic; (c) (1,1)-regime; (d) (2,3)-regime (ω=7.23 rad·s-1 )

Contact forces during impact for quasiperiodic regime (ω=7.23 rad·s-1 )

Floquet multipliers jump (ω=7.23 rad·s-1 )

Global view of contact force depending on exitation frequency (P=500 N)

Contact forces during impact at (1,1)-regime for different stiffness k2 (P=500 N, ω=7.23 rad·s-1 )

Contact forces and phase portrait for chatter (P=500 N, ω=7.23 rad·s-1 )

"Stick" & "bell" for rigid and soft impacts (P=500 N, ω=7.0 rad·s-1 )

Global and partial views of contact force depending on exitation amplitude for soft impact (ω=7.23 rad·s-1 )

Global and partial views of contact force depending on exitation frequency for soft impact (P=500 N)

j_AMNS_2016_1_00014_tab_001_w2aab3b7e1030b1b6b1ab1b2b4b9aAa

P	Re(μ₁)	Im(μ₁)	\|μ₁\|	Re(μ₂)	Im(μ₂)	\|μ₁\|
98.48	-0.25	0.83	0.87	-0.25	0.83	0.87
98.98	-0.25	0.83	0.87	-0.25	0.83	0.87
99.48	3.57	0	3.57	1.12	0	1.12

j_AMNS_2016_1_00014_tab_005_w2aab3b7e1030b1b6b1ab1b2b6c10aAa

R₁=R₂, m	A=B, m^-1	F_con,N·10⁵
1	1	1.49
2	0.5	1.71
5	0.2	2.05
10	0.1	2.36

Parameters of vibroimpact system

Bodies’ characteristic	Rigid impact		Soft impact
Bodies’ characteristic	Main body	Attached body	Main body	Attached body
Mass m_i, kg	1000	100	1000	310
Partial vibration frequency ω_i,rad·s^-1	6.283	5.646	6.283	3.606
Young’s modulus E_i, N·m^-2	2.1 · 10¹¹	2.1 · 10¹¹	2.44 · 10⁵	2.1 · 10¹¹
Contact surface radius R_i, m	2	2	1	0.5
Coefficients A,B, m^-1	A = 0.5 B = 0.5		A = 1.5 B = 1.5
Poisson’s ratio v_i	0.3	0.3	0.3	0.3
Damper coefficient ξ_i	0.036	0.036	0.036	0.036
Initial distance between bodies D, m	0.05		0.05
Impact duration T_con, s	7.82 · 10^-4		0.19
Coefficient of impact duration k_con, %	0.09		20.9

j_AMNS_2016_1_00014_tab_003_w2aab3b7e1030b1b6b1ab1b2b5b5aAa

ω_i, rads.s^-1	8.03	8.04	8.05	8.06	8.07	8.10	8.16
Re(μ₁)	-0.595	-0.593	151.4	90.1	65.9	37.4	19.7
IM(μ₁)	0.654	0.652	0	0	0	0	0
\|μ₁\|	0.8828	0.8829	151.4	90.1	65.9	37.4	19.7

j_AMNS_2016_1_00014_tab_002_w2aab3b7e1030b1b6b1ab1b2b4c12aAa

P	Re(μ₁)	Im(μ₁)	\|μ₁\|	Re(μ₂)	Im(μ₂)	\|μ₁\|
392.5	0.197	0.977	0.997	0.197	-0.977	0.997
392.0	0.197	0.981	1.001	0.197	-0.981	1.001
391.0	0.197	0.983	1.003	0.197	-0.983	1.003

j_AMNS_2016_1_00014_tab_004_w2aab3b7e1030b1b6b1ab1b2b6b7aAa

Material	Steel	Copper	Aluminium	Rubber
Young’s modulus, N m^-2	2.10	1.11	0.69	0.00008
Maximum F_con,N·10⁵	1.71	1.33	1.10	0.0309

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Contact Impact Forces at Discontinuous 2-DOF Vibroimpact

Published Online: Mar 14, 2016

Page range: 183 - 196

Received: Dec 03, 2015

Accepted: Mar 10, 2016

DOI: https://doi.org/10.21042/AMNS.2016.1.00014

Keywords
vibroimpact system, discontinuous, contact force, Hertz’s law, bifurcation, multiplier, nonlinear, stability

© 2016 V.A. Bazhenov, O.S. Pogorelova, T.G. Postnikova, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

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j_AMNS_2016_1_00014_tab_001_w2aab3b7e1030b1b6b1ab1b2b4b9aAa

j_AMNS_2016_1_00014_tab_005_w2aab3b7e1030b1b6b1ab1b2b6c10aAa

Parameters of vibroimpact system

j_AMNS_2016_1_00014_tab_003_w2aab3b7e1030b1b6b1ab1b2b5b5aAa

j_AMNS_2016_1_00014_tab_002_w2aab3b7e1030b1b6b1ab1b2b4c12aAa

j_AMNS_2016_1_00014_tab_004_w2aab3b7e1030b1b6b1ab1b2b6b7aAa

Contact Impact Forces at Discontinuous 2-DOF Vibroimpact

V.A. Bazhenov

O.S. Pogorelova

T.G. Postnikova

Published Online: Mar 14, 2016

Page range: 183 - 196

Received: Dec 03, 2015

Accepted: Mar 10, 2016

DOI: https://doi.org/10.21042/AMNS.2016.1.00014

Keywordsvibroimpact system, discontinuous, contact force, Hertz’s law, bifurcation, multiplier, nonlinear, stability

© 2016 V.A. Bazhenov, O.S. Pogorelova, T.G. Postnikova, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Fig. 1

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Fig. 11

j_AMNS_2016_1_00014_tab_001_w2aab3b7e1030b1b6b1ab1b2b4b9aAa

j_AMNS_2016_1_00014_tab_005_w2aab3b7e1030b1b6b1ab1b2b6c10aAa

Parameters of vibroimpact system

j_AMNS_2016_1_00014_tab_003_w2aab3b7e1030b1b6b1ab1b2b5b5aAa

j_AMNS_2016_1_00014_tab_002_w2aab3b7e1030b1b6b1ab1b2b4c12aAa

j_AMNS_2016_1_00014_tab_004_w2aab3b7e1030b1b6b1ab1b2b6b7aAa

Keywords
vibroimpact system, discontinuous, contact force, Hertz’s law, bifurcation, multiplier, nonlinear, stability