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Vibrations attenuation of a system excited by unbalance and the ground movement by an impact element

Marek Lampart

Marek Lampart

Department of Applied Mathematics & IT4Innovations, VŠB - Technical University of OstravaOstrava, Czech Republic
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Lampart, Marek
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Jaroslav Zapoměl

Jaroslav Zapoměl

  • Department of Applied Mathematics, VŠB - Technical University of OstravaOstrava, Czech Republic
  • Department of Dynamics and Vibrations, Institute of ThermomechanicsPrague, Czech Republic
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Zapoměl, Jaroslav
  
Dec 08, 2016
Vibrations attenuation of a system excited by unbalance and the ground movement by an impact element's Cover Image
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Published Online: Dec 08, 2016
Page range: 603 - 616
Received: Sep 11, 2016
Accepted: Dec 08, 2016
DOI: https://doi.org/10.21042/AMNS.2016.2.00046
Keywords
© 2016 Marek Lampart, Jaroslav Zapoměl, published by Sciendo
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Fig. 1

Model of vibrating system
Model of vibrating system

Fig. 2

Vibration of the baseplate given by function (6).
Vibration of the baseplate given by function (6).

Fig. 3

Model of vibrating system, damping device without damper.
Model of vibrating system, damping device without damper.

Fig. 4

Peak-to-peak vibration of the rotor frame dependent on the clearance c1 = c2 and the mass of the impact element mt for the excitation frequency (left) ω = 100 rad s−1 and (right) ω = 115 rad s−1.
Peak-to-peak vibration of the rotor frame dependent on the clearance c1 = c2 and the mass of the impact element mt for the excitation frequency (left) ω = 100 rad s−1 and (right) ω = 115 rad s−1.

Fig. 5

Contour plot of the peak-to-peak vibration of the rotor frame dependent on the clearance c1 = c2 and the mass of the impact element mt for the excitation frequency (left) ω = 100 rad s−1 and (right) ω = 115 rad s−1.
Contour plot of the peak-to-peak vibration of the rotor frame dependent on the clearance c1 = c2 and the mass of the impact element mt for the excitation frequency (left) ω = 100 rad s−1 and (right) ω = 115 rad s−1.

Fig. 6

Impact occurrence dependent on the clearance c1 = c2 and the mass of the impact element mt for the excitation frequency (left) ω = 100 rad s−1 and (right) ω = 115 rad s−1. Here, red box = impact occurs, green dot = no impacts.
Impact occurrence dependent on the clearance c1 = c2 and the mass of the impact element mt for the excitation frequency (left) ω = 100 rad s−1 and (right) ω = 115 rad s−1. Here, red box = impact occurs, green dot = no impacts.

Fig. 7

Peak-to-peak vibrations amplitudes of the rotor frame dependent on the excitation frequency for the mass of the impact element mt from 7 kg to 9 kg. There are no impacts in all these cases.
Peak-to-peak vibrations amplitudes of the rotor frame dependent on the excitation frequency for the mass of the impact element mt from 7 kg to 9 kg. There are no impacts in all these cases.

Fig. 8

Peak-to-peak vibrations amplitudes of the rotor frame dependent on the excitation frequency for clearances c1 = c2 from 2 mm to 4 mm where (left) corresponds to mt = 22 kg and (right) corresponds to mt = 25 kg. In all these cases impacts occur.
Peak-to-peak vibrations amplitudes of the rotor frame dependent on the excitation frequency for clearances c1 = c2 from 2 mm to 4 mm where (left) corresponds to mt = 22 kg and (right) corresponds to mt = 25 kg. In all these cases impacts occur.

Fig. 9

Black and blue: the dependence of the peak-to-peak vibration amplitude of the rotor frame and impact element, respectively, on angular frequency of excitation of the baseplate for unlimited movement of the impact body; red: the dependence of the peak-to-peak vibration amplitude of the rotor frame on angular frequency of excitation of the baseplate for the movement of the impact body limited by clearances; for parameters mt = 25 kg and bt = 0 N s m−1.
Black and blue: the dependence of the peak-to-peak vibration amplitude of the rotor frame and impact element, respectively, on angular frequency of excitation of the baseplate for unlimited movement of the impact body; red: the dependence of the peak-to-peak vibration amplitude of the rotor frame on angular frequency of excitation of the baseplate for the movement of the impact body limited by clearances; for parameters mt = 25 kg and bt = 0 N s m−1.

Fig. 10

Black and blue: the dependence of the peak-to-peak vibration amplitude of the rotor frame and impact element, respectively, on angular frequency of excitation of the baseplate for unlimited movement of the impact body; red: the dependence of the peak-to-peak vibration amplitude of the rotor frame on angular frequency of excitation of the baseplate for the movement of the impact body limited by clearances; for parameters mt = 25 kg and bt = 500 N s m−1.
Black and blue: the dependence of the peak-to-peak vibration amplitude of the rotor frame and impact element, respectively, on angular frequency of excitation of the baseplate for unlimited movement of the impact body; red: the dependence of the peak-to-peak vibration amplitude of the rotor frame on angular frequency of excitation of the baseplate for the movement of the impact body limited by clearances; for parameters mt = 25 kg and bt = 500 N s m−1.

Fig. 11

Optimal choice of the clearance c1 = c2 for the maximal attenuation of vibrations of the rotor body corresponding to the red curve in Fig. 9, for parameters mt = 25 kg and bt = 0 N s m−1.
Optimal choice of the clearance c1 = c2 for the maximal attenuation of vibrations of the rotor body corresponding to the red curve in Fig. 9, for parameters mt = 25 kg and bt = 0 N s m−1.

Fig. 12

Optimal choice of the clearance c1 = c2 for the maximal attenuation of vibrations of the rotor body corresponding to the red curve in Fig. 9, for parameters mt = 25 kg and bt = 500 N s m−1.
Optimal choice of the clearance c1 = c2 for the maximal attenuation of vibrations of the rotor body corresponding to the red curve in Fig. 9, for parameters mt = 25 kg and bt = 500 N s m−1.

Fig. 13

Bifurcation diagram with respect to the angular frequency ω for parameters mt = 25 kg and bt = 500 N s m−1.
Bifurcation diagram with respect to the angular frequency ω for parameters mt = 25 kg and bt = 500 N s m−1.

Fig. 14

Bifurcation diagram with respect to the angular frequency ω for parameters mt = 25 kg and bt = 0 N s m−1.
Bifurcation diagram with respect to the angular frequency ω for parameters mt = 25 kg and bt = 0 N s m−1.

Fig. 15

Phase portrait (left), Y − Yt versus Y.−Y.t$\begin{array}{} \displaystyle {\bf{\dot Y}} - {{\bf{\dot Y}}_t} \end{array}$ and Fourier spectra (right) for the angular frequency ω = 100 rad s−1 and the gaps between the rotor frame and the impact element c1 = c2 = 6 mm.
Phase portrait (left), Y − Yt versus Y.−Y.t$\begin{array}{} \displaystyle {\bf{\dot Y}} - {{\bf{\dot Y}}_t} \end{array}$ and Fourier spectra (right) for the angular frequency ω = 100 rad s−1 and the gaps between the rotor frame and the impact element c1 = c2 = 6 mm.

Fig. 16

Phase portrait (left), Y − Yt versus Y.−Y.t$\begin{array}{} \displaystyle {\bf{\dot Y}} - {{\bf{\dot Y}}_t} \end{array}$ and Fourier spectra (right) for the angular frequency ω = 150 rad s−1 and the gaps between the rotor frame and the impact element c1 = c2 = 1 mm.
Phase portrait (left), Y − Yt versus Y.−Y.t$\begin{array}{} \displaystyle {\bf{\dot Y}} - {{\bf{\dot Y}}_t} \end{array}$ and Fourier spectra (right) for the angular frequency ω = 150 rad s−1 and the gaps between the rotor frame and the impact element c1 = c2 = 1 mm.

Fig. 17

Phase portrait (left), Y − Yt versus Y.−Y.t$\begin{array}{} \displaystyle {\bf{\dot Y}} - {{\bf{\dot Y}}_t} \end{array}$ and Fourier spectra (right) for the angular frequency ω = 110 rad s−1 and the gaps between the rotor frame and the impact element c1 = c2 = 5 mm.
Phase portrait (left), Y − Yt versus Y.−Y.t$\begin{array}{} \displaystyle {\bf{\dot Y}} - {{\bf{\dot Y}}_t} \end{array}$ and Fourier spectra (right) for the angular frequency ω = 110 rad s−1 and the gaps between the rotor frame and the impact element c1 = c2 = 5 mm.

Fig. 18

Phase portrait (left), Y − Yt versus Y.−Y.t$\begin{array}{} \displaystyle {\bf{\dot Y}} - {{\bf{\dot Y}}_t} \end{array}$ and Fourier spectra (right) for the angular frequency ω = 185 rad s−1 and the gaps between the rotor frame and the impact element c1 = c2 = 1 mm.
Phase portrait (left), Y − Yt versus Y.−Y.t$\begin{array}{} \displaystyle {\bf{\dot Y}} - {{\bf{\dot Y}}_t} \end{array}$ and Fourier spectra (right) for the angular frequency ω = 185 rad s−1 and the gaps between the rotor frame and the impact element c1 = c2 = 1 mm.

Parameters of the system (1)_

valuequantityformatdescription
m100kgmass of the damping body
mR40kgmass of the rotor
mt kgmass of the impact element
k11.5 ×105N m−1linear stiffness coefficient
JRT5kg m2moment of inertia of the rotor
b1.5 × 103N s m−1damping coefficient of the suspension
kt8 × 104N m−1coupling stiffness of the impact element
bt500N s m−1damping coefficient of the impact element
eT2mmeccentricity of the rotor center of gravity
Φ radrotation angle of the rotor
MZ100N mstarting moment
kM8N m s rad−1negative of the motor characteristic slope
α1s−1parameter of the baseplate excitation
ω rad s−1baseplate excitation frequency
A1mmground vibration amplitude
kc4 × 107N m−1contact stiffness
bc3 × 103N s m−1coefficient of contact damping
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