Figure 1.
Schematic diagram of (a) the ball bearing system, and (b) its two degrees-of-freedom spring model.
Figure 1.
Schematic diagram of (a) the ball bearing system, and (b) its two degrees-of-freedom spring model.
Figure 2.
For δ0 = 1.5 μm, stable (solid) and unstable (dashed) varying compliance (VC) periodic peak-to-peak frequency–response curves in x (red line) and y (black line) directions.
Figure 2.
For δ0 = 1.5 μm, stable (solid) and unstable (dashed) varying compliance (VC) periodic peak-to-peak frequency–response curves in x (red line) and y (black line) directions.
Figure 3.
For δ0 = 1.5 μm, frequency–response curves of 1/2-order subharmonic resonances of the two degrees-of-freedom of the system, (a) in the horizontal and (b) vertical directions.
Figure 3.
For δ0 = 1.5 μm, frequency–response curves of 1/2-order subharmonic resonances of the two degrees-of-freedom of the system, (a) in the horizontal and (b) vertical directions.
Figure 4.
For δ0 = 1.5 μm, response characteristics of the VC period-1 motion when Ω = 310 rad/s, (a) orbit (black line) and its Poincare mapping (red dots), (b) time series and their frequency spectra.
Figure 4.
For δ0 = 1.5 μm, response characteristics of the VC period-1 motion when Ω = 310 rad/s, (a) orbit (black line) and its Poincare mapping (red dots), (b) time series and their frequency spectra.
Figure 5.
For δ0 = 1.5 μm, response characteristics of the VC period-2 motion when Ω = 330 rad/s, (a) orbit (black line) and its Poincare mapping (red dots), (b) time series and their frequency spectra.
Figure 5.
For δ0 = 1.5 μm, response characteristics of the VC period-2 motion when Ω = 330 rad/s, (a) orbit (black line) and its Poincare mapping (red dots), (b) time series and their frequency spectra.
Figure 6.
For δ0 = 1.5 μm, numerical bifurcation diagrams of local maxima (a) y(τ) and (b) x(τ) when Ω sweeps up (black dots) and down (red dots).
Figure 6.
For δ0 = 1.5 μm, numerical bifurcation diagrams of local maxima (a) y(τ) and (b) x(τ) when Ω sweeps up (black dots) and down (red dots).
Figure 7.
For δ0 = 1.5 μm, VC quasi-period response characteristics, (a) orbit (black line) and its Poincare mapping (red dots), (b) frequency spectrum when Ω = 400 rad/s.
Figure 7.
For δ0 = 1.5 μm, VC quasi-period response characteristics, (a) orbit (black line) and its Poincare mapping (red dots), (b) frequency spectrum when Ω = 400 rad/s.
Figure 8.
For δ0 = −1.5 μm, stable (solid) and unstable (dashed) VC periodic peak-to-peak frequency–response curves in x (red line) and y (black line) directions.
Figure 8.
For δ0 = −1.5 μm, stable (solid) and unstable (dashed) VC periodic peak-to-peak frequency–response curves in x (red line) and y (black line) directions.
Figure 9.
For δ0 = −1.5 μm, frequency–response curves of (a) primary resonances, and (b) 1/2-order subharmonic resonances of the two degrees-of-freedom of the system.
Figure 9.
For δ0 = −1.5 μm, frequency–response curves of (a) primary resonances, and (b) 1/2-order subharmonic resonances of the two degrees-of-freedom of the system.
Figure 10.
For δ0 = −1.5 μm, numerical bifurcation diagrams of local-maxima (a) y(τ) and (b) x(τ) when Ω is sweeping up (black dots) and down (red dots).
Figure 10.
For δ0 = −1.5 μm, numerical bifurcation diagrams of local-maxima (a) y(τ) and (b) x(τ) when Ω is sweeping up (black dots) and down (red dots).
Figure 11.
For δ0 = −1.5 μm, response characteristics of the VC superharmonic motion when Ω = 131.6 rad/s, (a) orbit (black line) and its Poincare mapping (red dots), (b) time series and their frequency spectra.
Figure 11.
For δ0 = −1.5 μm, response characteristics of the VC superharmonic motion when Ω = 131.6 rad/s, (a) orbit (black line) and its Poincare mapping (red dots), (b) time series and their frequency spectra.
Figure 12.
Power spectra of response amplitude around the bearing clearance-free operations.
Figure 12.
Power spectra of response amplitude around the bearing clearance-free operations.
Figure 13.
The locations of primary resonances (blue lines), 1/2-order subharmonic resonances (black lines), 2-order superharmonic resonances (green lines), and combination resonances (red lines) predicted by the equivalent resonant frequencies ωx0 and ωy0.
Figure 13.
The locations of primary resonances (blue lines), 1/2-order subharmonic resonances (black lines), 2-order superharmonic resonances (green lines), and combination resonances (red lines) predicted by the equivalent resonant frequencies ωx0 and ωy0.
Figure 14.
For δ0 = 4.0 μm, stable (solid) and unstable (dashed) VC periodic peak-to-peak frequency–response curves in x (red line) and y (black line) directions.
Figure 14.
For δ0 = 4.0 μm, stable (solid) and unstable (dashed) VC periodic peak-to-peak frequency–response curves in x (red line) and y (black line) directions.
Figure 15.
For δ0 = 4.0 μm, numerical bifurcation diagrams of local maxima (a) x(τ) and (b) y(τ) when Ω sweeps up (black dots) and down (red dots) around 345–375 rad/s.
Figure 15.
For δ0 = 4.0 μm, numerical bifurcation diagrams of local maxima (a) x(τ) and (b) y(τ) when Ω sweeps up (black dots) and down (red dots) around 345–375 rad/s.
Figure 16.
For δ0 = 4.0 μm, response characteristics of the VC quasi-period motion when Ω = 368 rad/s, (a) orbit (black line) and its Poincare mapping (red dots), (b) frequency spectrum.
Figure 16.
For δ0 = 4.0 μm, response characteristics of the VC quasi-period motion when Ω = 368 rad/s, (a) orbit (black line) and its Poincare mapping (red dots), (b) frequency spectrum.
Figure 17.
For δ0 = 4.0 μm, response characteristics of the VC quasi-period motion (the tori-doubling case) when Ω = 363 rad/s, (a) orbit (black line) and its Poincare mapping (red dots), (b) frequency spectrum.
Figure 17.
For δ0 = 4.0 μm, response characteristics of the VC quasi-period motion (the tori-doubling case) when Ω = 363 rad/s, (a) orbit (black line) and its Poincare mapping (red dots), (b) frequency spectrum.
Figure 18.
For δ0 = 4.0 μm, response characteristics of the VC chaotic motion when Ω = 353 rad/s, (a) orbit (black line) and its Poincare mapping (red dots), (b) frequency spectrum.
Figure 18.
For δ0 = 4.0 μm, response characteristics of the VC chaotic motion when Ω = 353 rad/s, (a) orbit (black line) and its Poincare mapping (red dots), (b) frequency spectrum.
Figure 19.
For δ0 = 4.0 μm, response characteristics of the VC period-35 motion when Ω = 357 rad/s, (a) orbit (black line) and its Poincare mapping (red dots), (b) frequency spectrum.
Figure 19.
For δ0 = 4.0 μm, response characteristics of the VC period-35 motion when Ω = 357 rad/s, (a) orbit (black line) and its Poincare mapping (red dots), (b) frequency spectrum.
Figure 20.
For δ0 = 4.0 μm, response characteristics of the VC period-8 motion when Ω = 355 rad/s, (a) orbit (black line) and its Poincare mapping (red dots), (b) frequency spectrum.
Figure 20.
For δ0 = 4.0 μm, response characteristics of the VC period-8 motion when Ω = 355 rad/s, (a) orbit (black line) and its Poincare mapping (red dots), (b) frequency spectrum.
Figure 21.
For δ0 = 4.0 μm, numerical bifurcation diagrams of local maxima (a) x(τ) and (b) y(τ) when Ω sweeps up (black dots) and down (red dots) around 100–200 rad/s.
Figure 21.
For δ0 = 4.0 μm, numerical bifurcation diagrams of local maxima (a) x(τ) and (b) y(τ) when Ω sweeps up (black dots) and down (red dots) around 100–200 rad/s.
Figure 22.
For δ0 = 4.0 μm, response characteristics of the VC quasi-period motion when Ω = 181.2 rad/s, (a) orbit (black line) and its Poincare mapping (red dots), (b) frequency spectrum.
Figure 22.
For δ0 = 4.0 μm, response characteristics of the VC quasi-period motion when Ω = 181.2 rad/s, (a) orbit (black line) and its Poincare mapping (red dots), (b) frequency spectrum.
Figure 23.
For δ0 = 4.0 μm, response characteristics of the VC period-4 motion when Ω = 180 rad/s, (a) orbit (black line) and its Poincare mapping (red dots), (b) frequency spectrum.
Figure 23.
For δ0 = 4.0 μm, response characteristics of the VC period-4 motion when Ω = 180 rad/s, (a) orbit (black line) and its Poincare mapping (red dots), (b) frequency spectrum.
Figure 24.
For δ0 = 4.0 μm, the orbits (black line) and their Poincare mappings (red dots) of the VC quasi-period motion when (a) Ω = 126 rad/s, and (b) Ω = 154.6 rad/s.
Figure 24.
For δ0 = 4.0 μm, the orbits (black line) and their Poincare mappings (red dots) of the VC quasi-period motion when (a) Ω = 126 rad/s, and (b) Ω = 154.6 rad/s.
Figure 25.
For δ0 = 4.0 μm, response characteristics of the VC chaotic motion when Ω = 154.5 rad/s, (a) orbit (black line) and its Poincare mapping (red dots), (b) time series and their frequency spectra.
Figure 25.
For δ0 = 4.0 μm, response characteristics of the VC chaotic motion when Ω = 154.5 rad/s, (a) orbit (black line) and its Poincare mapping (red dots), (b) time series and their frequency spectra.
Figure 26.
For δ0 = −1.1 μm, stable (solid) and unstable (dashed) VC periodic peak-to-peak frequency–response curves in x (red line) and y (black line) directions.
Figure 26.
For δ0 = −1.1 μm, stable (solid) and unstable (dashed) VC periodic peak-to-peak frequency–response curves in x (red line) and y (black line) directions.
Figure 27.
For δ0 = −1.1 μm, the response characteristics in 1/2-order subharmonic resonance, (a) the frequency–response curves in x (red line) and y (black line) directions, and (b) numerical bifurcation diagrams of local maxima of y(τ) when Ω sweeps up (black dots) and down (red dots).
Figure 27.
For δ0 = −1.1 μm, the response characteristics in 1/2-order subharmonic resonance, (a) the frequency–response curves in x (red line) and y (black line) directions, and (b) numerical bifurcation diagrams of local maxima of y(τ) when Ω sweeps up (black dots) and down (red dots).
Figure 28.
Stable (solid) and unstable (dashed) VC periodic peak-to-peak frequency–response curves in x (red line) and y (black line) directions.
Figure 28.
Stable (solid) and unstable (dashed) VC periodic peak-to-peak frequency–response curves in x (red line) and y (black line) directions.
Figure 29.
Response characteristics of the VC period-2 motion when Ω = 88 rad/s, (a) orbit (black line) and its Poincare mapping (red dots), (b) time series and their frequency spectra.
Figure 29.
Response characteristics of the VC period-2 motion when Ω = 88 rad/s, (a) orbit (black line) and its Poincare mapping (red dots), (b) time series and their frequency spectra.
Figure 30.
The peak-to-peak frequency–response curves of literature [
14], in (
a)
x and (
b)
y directions. Herein, the blue and red lines are the period-1 and period-2 solution branches of
Figure 28, respectively, obtained by our work.
Figure 30.
The peak-to-peak frequency–response curves of literature [
14], in (
a)
x and (
b)
y directions. Herein, the blue and red lines are the period-1 and period-2 solution branches of
Figure 28, respectively, obtained by our work.
Figure 31.
Response characteristics of the VC quasi-period motion when
Ω = 242.2277 rad/s (i.e., in literature [
14],
ωs takes 6000 rpm), (
a) orbit (black line) and its Poincare mapping (red dots), (
b) frequency spectrum.
Figure 31.
Response characteristics of the VC quasi-period motion when
Ω = 242.2277 rad/s (i.e., in literature [
14],
ωs takes 6000 rpm), (
a) orbit (black line) and its Poincare mapping (red dots), (
b) frequency spectrum.
Figure 32.
Response characteristics of the VC quasi-period motion when
Ω = 211.9492 rad/s (i.e., in literature [
14],
ωs takes 5250 rpm), (
a) orbit (black line) and its Poincare mapping (red dots), (
b) frequency spectrum.
Figure 32.
Response characteristics of the VC quasi-period motion when
Ω = 211.9492 rad/s (i.e., in literature [
14],
ωs takes 5250 rpm), (
a) orbit (black line) and its Poincare mapping (red dots), (
b) frequency spectrum.
Figure 33.
Numerical bifurcation diagrams of local maxima of y(τ) when Ω sweeps up (black dots) and down (red dots).
Figure 33.
Numerical bifurcation diagrams of local maxima of y(τ) when Ω sweeps up (black dots) and down (red dots).
Table 1.
Specifications of JIS6306 ball bearing [
22].
Table 1.
Specifications of JIS6306 ball bearing [
22].
Item | Value |
---|
Contact stiffness Cb (N/m3/2) | 1.334 × 1010 |
Ball diameter Db (mm) | 11.9062 |
Pitch diameter Dh (mm) | 52.0 |
Number of balls Nb | 8 |
Equivalent mass m (kg) | 20 |
Damping factor c (Ns/m) | 200 |
Radial load W (N) | 196 |
Table 2.
Period-1 motion Floquet multipliers λm around A1.
Table 2.
Period-1 motion Floquet multipliers λm around A1.
Ω | 316.3645 | 316.4645 | 316.5645 | 316.6645 |
---|
λm | 0.021 + 0.987i | 0.019 + 0.987i | 0.018 + 0.988i | 0.017 + 0.988i |
0.021 − 0.987i | 0.019 − 0.987i | 0.018 − 0.988i | 0.017 − 0.988i |
−0.988 + 0.009i | −0.995 | −1.002 | −1.006 |
−0.988 − 0.009i | −0.980 | −0.974 | −0.969 |
Table 3.
Period-2 motion Floquet multipliers
λm around
A4 (counterclockwise, see
Figure 3a).
Table 3.
Period-2 motion Floquet multipliers
λm around
A4 (counterclockwise, see
Figure 3a).
Ω | 341.7310 | 341.7316 | 341.7316 | 341.7315 |
---|
λm | −0.536 + 0.817i | −0.535 + 0.818i | −0.535 + 0.818i | −0.535 + 0.818i |
−0.536 − 0.817i | −0.535 − 0.818i | −0.535 − 0.818i | −0.535 − 0.818i |
0.987 | 0.998 | 1.001 | 1.002 |
0.968 | 0.957 | 0.954 | 0.953 |
Table 4.
Period-1 motion Floquet multipliers λm around B7.
Table 4.
Period-1 motion Floquet multipliers λm around B7.
Ω | 406.9645 | 407.0645 | 407.1645 | 407.2645 |
---|
λm | −0.018 + 1.031i | −0.019 + 1.006i | −0.019 + 0.971i | −0.066 + 0.923i |
−0.018 − 1.031i | −0.019 − 1.006i | −0.019 − 0.971i | −0.066 − 0.923i |
−0.014 + 0.831i | −0.016 + 0.851i | −0.017 + 0.883i | 0.027 + 0.925i |
−0.014 − 0.831i | −0.016 − 0.851i | −0.017 − 0.883i | 0.027 − 0.925i |
Table 5.
The equivalent resonant frequencies of the system predicted from Equation (23).
Table 5.
The equivalent resonant frequencies of the system predicted from Equation (23).
Ω | 353 | 355 | 357 | 363 | 368 |
---|
ωx0 (rad/s) | 1767.54 | 1775.00 | 1795.20 | 1833.00 | 1867.97 |
ωy0 (rad/s) | 1056.46 | 1065.00 | 1060.80 | 1071.00 | 1076.03 |
Table 6.
Specifications of the ball bearing in [
14].
Table 6.
Specifications of the ball bearing in [
14].
Item | Value |
---|
Contact stiffness Cb (N/m3/2) | 1.334 × 1010 |
Ball diameter Db (mm) | 11.9062 |
Pitch diameter Dh (mm) | 52.0 |
Number of balls Nb | 8 |
Bearing clearance δ0 (μm) | 20 |
Equivalent mass m (kg) | 16 |
Damping factor c (Ns/m) | 2940 |
Radial load W (N) | 58.8 |