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Analytical investigation on nonlinear vibration behavior of an unbalanced asymmetric rotor using the method of multiple scales

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Abstract

In this paper, the vibration behavior of a rotor with an asymmetric shaft subjected to unbalanced forces is analyzed theoretically. The model is a rotor composed of a rigid disk and a flexible shaft. The shaft is considered to be a beam with a rectangular cross section. The general equations of motion were first derived by considering the effect of high order large deformation in bending. In this process, a continuous shaft, gyroscopic effects, and rotor mass unbalance are taken into account to study the rotor’s nonlinear vibratory behavior near the main resonances. The equations are discretized using the Rayleigh–Ritz method. The obtained equations are nonlinear coupled differential equations which are solved using the multiple-scales method. It can be concluded from the results that nonlinearity due to an asymmetric shaft severely affects the resonance behavior of the rotor.

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Abbreviations

a 1 :

Amplitude at the equilibrium position (m) in x direction

a 2 :

Amplitude at the equilibrium position (m) in z direction

\(\varOmega\) :

Angular Velocity of the rotor (rad sec−1)

\(\omega_{1}\), \(\omega_{2}\) :

First and second critical speed of the rotor (rad sec−1)

I :

Average area moment of inertia of shaft (m4)

c :

Coefficient of damping (N s m−1)

R 1 :

Cross-sectional radius of shaft/internal radius of disk (m2)

A :

Cross-sectional area of shaft (m2)

\(\rho\) :

Density of material (kg m−3)

\(\sigma\) :

Detuning parameter (rad sec−1)

u(y, t):

Displacement along x-axis of rotor (m)

w(y, t):

Displacement along z-axis of rotor (m)

T S :

Kinetic energy of shaft (N m)

T D :

Kinetic energy of disk (N m)

T u :

Kinetic energy of mass unbalance (N m)

L :

Length of shaft (m)

I D x :

Mass moment of inertia of disk in direction x (kg m2)

I D y :

Mass moment of inertia of disk in direction y (kg m2)

M D :

Mass of disk (kg)

d :

Position of mass unbalance from geometric center of shaft (m)

L 1 :

Position of disk on shaft (m)

U s :

Strain (deformation) energy of shaft (N m)

T R :

Total kinetic energy of rotor (N m)

U R :

Total strain (deformation) energy of rotor (N m)

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Acknowledgements

The authors would like to thank Professor Roozbeh Dargazany of University of Michigan University for his support and corrections on an earlier version of the manuscript. The authors are also grateful to the anonymous reviewers for their valuable comments.

Author information

Authors and Affiliations

  1. Department of Mechanical Engineering, KNT University, Tehran, Iran

    P. Jamshidi

  2. Faculty Member of Department of Mechanical Engineering, KNT University, Tehran, Iran

    A. A. Jafari

Authors
  1. P. Jamshidi
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  2. A. A. Jafari
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Correspondence to P. Jamshidi.

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Technical Editor: José Roberto de França Arruda.

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Appendices

Appendices

1.1 Linear analysis

The rotor is analyzed as a free linear system without damping to determine natural frequencies, and in Fig. 7a, Campbell diagram is plotted to determine the rotor critical speeds. Two critical speeds are found to be 1442.1 rpm and 1462 rpm. The mass unbalance response is also shown in Fig. 7b.

Fig. 7
figure 7

a Campbell Diagram. b Mass unbalance response

Full size image

1.2 Numerical data

Geometric and material properties

$$\begin{aligned} & \rho = 7800\,{\text{kgm}}^{ - 3} ,\;E = 200 \times 10^{9} \,{\text{Nm}}^{ - 2} ,\;c = 0.001 \\ & {\text{Shaft:}}\;L_{1} = 92\,{\text{mm}},\;L = 24\,{\text{mm,}}\;L_{1} = 80\,{\text{m}},\;Sx = 4\,{\text{mm}},\;Sz = 6\,{\text{mm}}\;(\text{rectangular}\,{\text{cross}}\,{\text{section}}\,4 \times 6) \\ & {\text{Disk:}}\;M_{\text{D}} = 1.561\,{\text{Kg}},\;R_{1} = 19\,{\text{mm}},\;R_{2} = 150\,{\text{mm}},\;h = 10\,{\text{mm}},\;I_{{{\text{D}}x}} = M_{\text{D}} (3R_{1}^{2} + 3R_{2}^{2} + h^{2} ) /12 \\ & I_{{{\text{D}}y}} = M_{\text{D}} (R_{1}^{2} + R_{2}^{2} ) /2,\;m_{\text{u}} = 1\,{\text{g}} \\ \end{aligned}$$

Constants

$$\lambda_{1} = 0.017,\quad \lambda_{2} = 2.0354 \times 10^{ - 6} ,\quad \beta_{1} = 2.325 \times 10^{4} ,\quad \beta_{2} = 8.9424 \times 10^{3} ,\quad \beta_{3} = 8.0482 \times 10^{9}$$

1.3 Functions H 1 and H 2 used in Eqs. (30) and (31) are given as

$$\begin{aligned} H_{1} & = - 0.5\beta_{3} [(1 + \varLambda_{1}^{2} )A_{1}^{3} \exp [3i\omega_{1} T_{0} ] \\ & \quad + \,(3 + \varLambda_{2}^{2} + 2\varLambda_{1} \varLambda_{2} )A_{1} A_{2}^{2} \exp [i(\omega_{1} + 2\omega_{2} )T_{0} ] + (3 + \varLambda_{1}^{2} + 2\varLambda_{1} \varLambda_{2} )A_{1}^{2} A_{2} \exp [i(2\omega_{1} + \omega_{2} )T_{0} ] \\ & \quad + \,[ - 2i\omega_{1} D_{1} A_{1} (T_{1} ) + \lambda_{1} \varOmega \varLambda_{1} D_{1} A_{1} + (3 + \varLambda_{1} \bar{\varLambda }_{2} + \varLambda_{1} \bar{\varLambda }_{1} + \varLambda_{1}^{2} )A_{1}^{2} \bar{A}_{1} + (6 + 2\varLambda_{2} \bar{\varLambda }_{2} + 2\varLambda_{1} \bar{\varLambda }_{2} + \varLambda_{1} \varLambda_{2} )A_{1} A_{2} \bar{A}_{2} - ic\omega_{1} A_{1} ]\exp [i\omega_{1} T_{0} ] \\ & \quad + \,[ - 2i\omega_{2} D_{1} A_{2} (T_{1} ) + \varLambda_{2} \lambda_{1} \varOmega D_{1} A_{2} + (6 + 2\bar{\varLambda }_{1} \varLambda_{2} + 2\varLambda_{1} \bar{\varLambda }_{1} + 2\varLambda_{1} \varLambda_{2} )A_{1} \bar{A}_{1} A_{2} + (3 + 2\varLambda_{2} \bar{\varLambda }_{2} + \varLambda_{2}^{2} )A_{2}^{2} \bar{A}_{2} - ic\omega_{2} A_{2} ]\exp [i\omega_{2} T_{0} ] \\ & \quad + \,(1 + \bar{\varLambda }_{2}^{2} )A_{1} \bar{A}_{2}^{2} \exp [i(\omega_{1} - 2\omega_{2} )T_{0} ] + (2 + 2\varLambda_{1} \bar{\varLambda }_{2} )A_{1}^{2} \bar{A}_{2} \exp [i(2\omega_{1} - \omega_{2} )T_{0} ] \\ & \quad + \,(1 + \varLambda_{2}^{2} )A_{2}^{3} \exp [3i\omega_{2} T_{0} ] + (2 + 2\bar{\varLambda }_{1} \varLambda_{2} )\bar{A}_{1} A_{2}^{2} \exp [i(2\omega_{2} - \omega_{1} )T_{0} ] \\ & \quad + \,(1 + \bar{\varLambda }_{1}^{2} )\bar{A}_{1}^{2} A_{2} [i(\omega_{2} - 2\omega_{1} )T_{0} ] + cc] - 0.5im_{e} \varOmega^{2} df(L_{1} )\exp (i\varOmega T_{0} ) + cc \\ & \quad \quad 0.5\,[ - \lambda_{2} \omega_{1}^{2} - \lambda_{2} \omega_{1} (2\varOmega ) - \lambda_{2} i\omega_{1}^{2} \varLambda_{1} - \lambda_{2} \omega_{1} \varLambda_{1} i(2\varOmega ) + \beta_{2} + \beta_{2} i\varLambda_{1} ]A_{1} \exp [i(2\varOmega + \omega_{1} )] \\ & \quad \quad 0.5\,[ - \lambda_{2} \omega_{1}^{2} + \lambda_{2} \omega_{1} (2\varOmega ) + \lambda_{2} i\omega_{1}^{2} \varLambda_{1} - \lambda_{2} \omega_{1} \varLambda_{1} i(2\varOmega ) + \beta_{2} - \beta_{2} i\varLambda_{1} ]\bar{A}_{1} \exp [i(2\varOmega - \omega_{1} )] \\ & \quad \quad 0.5\,[ - \lambda_{2} \omega_{2}^{2} - \lambda_{2} \omega_{2} (2\varOmega ) - \lambda_{2} i\omega_{2}^{2} \varLambda_{2} - \lambda_{2} \omega_{2} \varLambda_{2} i(2\varOmega ) + \beta_{2} + \beta_{2} i\varLambda_{2} ]A_{2} \exp [i(2\varOmega + \omega_{2} )] \\ & \quad \quad 0.5\,[ - \lambda_{2} \omega_{2}^{2} + \lambda_{2} \omega_{2} (2\varOmega ) + \lambda_{2} i\omega_{2}^{2} \varLambda_{2} - \lambda_{2} \omega_{2} \varLambda_{2} i(2\varOmega ) + \beta_{2} - \beta_{2} i\varLambda_{2} ]\bar{A}_{2} \exp [i(2\varOmega - \omega_{2} )] \\ \end{aligned}$$
$$\begin{aligned} H_{2} & = - 0.5\beta_{3} [(\varLambda_{1} + \varLambda_{1}^{3} )A_{1}^{3} \exp [3i\omega_{1} T_{0} ] + (\varLambda_{1} + 2\varLambda_{2} + 3\varLambda_{1} \varLambda_{2}^{2} )A_{1} A_{2}^{2} \exp [i(\omega_{1} + 2\omega_{2} )T_{0} ] \\ & \quad + \,(2\varLambda_{1} + \varLambda_{2} + 3\varLambda_{1}^{2} \varLambda_{2} )A_{1}^{2} A_{2} \exp [i(2\omega_{1} + \omega_{2} )T_{0} ] \\ & \quad + \,[ - 2\varLambda_{1} i\omega_{1} D_{1} A_{1} - \lambda_{1} \varOmega D_{1} A_{1} (T_{1} ) + (2\varLambda_{1} + \bar{\varLambda }_{1} + 3\varLambda_{1}^{2} \bar{\varLambda }_{1} )A_{1}^{2} \bar{A}_{1} + (2\varLambda_{1} + 2\varLambda_{2} + 2\bar{\varLambda }_{2} + 6\varLambda_{1} \varLambda_{2} \bar{\varLambda }_{2} )A_{1} A_{2} \bar{A}_{2} - ic\omega_{1} \varLambda_{1} A_{1} ]\exp [i\omega_{1} T_{0} ] \\ & \quad + \,[ - 2\varLambda_{2} i\omega_{1} D_{1} A_{2} - \lambda_{1} \varOmega D_{1} A_{2} (T_{1} ) + (2\varLambda_{1} + 2\varLambda_{2} + 2\bar{\varLambda }_{1} + 6\varLambda_{1} \bar{\varLambda }_{1} \varLambda_{2} )A_{1} \bar{A}_{1} A_{2} + (2\varLambda_{2} + \bar{\varLambda }_{2} + 3\varLambda_{2}^{2} \bar{\varLambda }_{2} )A_{2}^{2} \bar{A}_{2} - ic\omega_{2} \varLambda_{2} A_{2} ]\exp [i\omega_{2} T_{0} ] \\ & \quad + \,(\varLambda_{1} + 2\bar{\varLambda }_{2} + 3\varLambda_{1} \bar{\varLambda }_{2}^{2} )A_{1} \bar{A}_{2}^{2} \exp [i(\omega_{1} - 2\omega_{2} )T_{0} ] + (2\varLambda_{1} + \bar{\varLambda }_{2} + 3\varLambda_{1}^{2} \bar{\varLambda }_{2} )A_{1}^{2} \bar{A}_{2} \exp [i(2\omega_{1} - \omega_{2} )T_{0} ] \\ & \quad + \,(\varLambda_{2} + \varLambda_{2}^{3} )A_{2}^{3} \exp [3i\omega_{2} T_{0} ] + cc] + 0.5m_{e} \varOmega^{2} df(L_{1} )\exp (i\varOmega T_{0} ) + cc \\ & \quad \quad 0.5\,[\lambda_{2} \omega_{1}^{2} \varLambda_{1} + \lambda_{2} \omega_{1} \varLambda_{1} (2\varOmega ) - \lambda_{2} i\omega_{1}^{2} - \lambda_{2} i\omega_{1} (2\varOmega ) + \beta_{2} i - \beta_{2} \varLambda_{1} ]A_{1} \exp [i(2\varOmega + \omega_{1} )] \\ & \quad \quad 0.5\,[ - \lambda_{2} \omega_{1}^{2} \varLambda_{1} + \lambda_{2} \omega_{1} \varLambda_{1} (2\varOmega ) - \lambda_{2} i\omega_{1}^{2} + \lambda_{2} i\omega_{1} (2\varOmega ) + \beta_{2} i + \beta_{2} \varLambda_{1} ]\bar{A}_{1} \exp [i(2\varOmega - \omega_{1} )] \\ & \quad \quad 0.5\,[\lambda_{2} \omega_{2}^{2} \varLambda_{2} + \lambda_{2} \omega_{2} \varLambda_{2} (2\varOmega ) - \lambda_{2} i\omega_{2}^{2} + \lambda_{2} i\omega_{2} (2\varOmega ) + \beta_{2} i - \beta_{2} \varLambda_{2} ]A_{2} \exp [i(2\varOmega + \omega_{2} )] \\ & \quad \quad 0.5\,[ - \lambda_{2} \omega_{2}^{2} \varLambda_{2} + \lambda_{2} \omega_{2} \varLambda_{2} (2\varOmega ) - \lambda_{2} i\omega_{2}^{2} + \lambda_{2} i\omega_{2} (2\varOmega ) + \beta_{2} i + \beta_{2} \varLambda_{2} ]\bar{A}_{2} \exp [i(2\varOmega - \omega_{2} )] \\ \end{aligned}$$

1.4 Constants

1.4.1 (a) Constants used in Eqs. (39) and (40) are given as

$$\begin{aligned} c_{2} & = - \frac{{C_{2} }}{{C_{1} }},\quad c_{3} = - \frac{{C_{3} }}{{C_{1} }},\quad c_{4} = - \frac{{C_{4} }}{{C_{1} }},\quad c_{5} = - \frac{{C_{5} }}{{C_{1} }},\quad c_{6} = - \frac{{C_{6} }}{{C_{1} }} \\ d_{2} & = - \frac{{D_{2} }}{{D_{1} }},\quad d_{3} = - \frac{{D_{3} }}{{D_{1} }},\quad d_{4} = - \frac{{D_{4} }}{{D_{1} }} \\ \end{aligned}$$

where

$$\begin{aligned} C_{1} & = (\beta_{1} - \omega_{1}^{2} )( + 2\varLambda_{1} i\omega_{1} + \lambda_{1} \varOmega ) + i\omega_{1}^{2} \lambda_{1} ( - 2i\omega_{1} + \lambda_{1} \varOmega \varLambda_{1} ) \\ C_{2} & = - (\beta_{1} - \omega_{1}^{2} )\frac{{\beta_{3} }}{2}(2\varLambda_{1} + \bar{\varLambda }_{1} + 3\varLambda_{1}^{2} \bar{\varLambda }_{1} ) + i\omega_{1}^{2} \lambda_{1} \frac{{\beta_{3} }}{2}(3 + 2\varLambda_{1} \bar{\varLambda }_{1} + \varLambda_{1}^{2} ) \\ C_{3} & = - (\beta_{1} - \omega_{1}^{2} )\frac{{\beta_{3} }}{2}(2\varLambda_{1} + 2\varLambda_{2} + 2\bar{\varLambda }_{2} + 6\varLambda_{1} \varLambda_{2} \bar{\varLambda }_{2} ) + i\omega_{1}^{2} \lambda_{1} \frac{{\beta_{3} }}{2}(6 + 2\varLambda_{2} \bar{\varLambda }_{2} + 2\varLambda_{1} \bar{\varLambda }_{2} + 2\varLambda_{1} \varLambda_{2} ) \\ C_{4} & = \frac{1}{2}(\beta_{1} - \omega_{1}^{2} )m_{e} \omega_{1}^{2} df(L_{1} ) - \frac{1}{2}\omega_{1}^{2} \lambda_{1} m_{e} \omega_{1}^{2} {\text{d}}f(L_{1} ) \\ C_{5} & = - (\beta_{1} - \omega_{1}^{2} )ic\omega_{1} \varLambda_{1} - \omega_{1}^{3} \lambda_{1} c \\ C_{6} & = \frac{1}{2}[(\beta_{1} - \omega_{1}^{2} )( - \lambda_{2} \omega_{1}^{2} \varLambda_{1} + \lambda_{2} \omega_{1} \varLambda_{1} (2\varOmega ) - \lambda_{2} i\omega_{1}^{2} + \lambda_{2} i\omega_{1} (2\varOmega ) + \beta_{2} i + \beta_{2} \varLambda_{1} ) \\ & \quad - \,i\omega_{1}^{2} \lambda_{1} ( - \lambda_{2} \omega_{1}^{2} + \lambda_{2} \omega_{1} (2\varOmega ) + \lambda_{2} i\omega_{1}^{2} \varLambda_{1} - \lambda_{2} \omega_{1} \varLambda_{1} i(2\varOmega ) + \beta_{2} - \beta_{2} i\varLambda_{1} )] \\ D_{1} & = (\beta_{1} - \omega_{2}^{2} )(2\varLambda_{2} i\omega_{1} + \lambda_{1} \varOmega ) + i\omega_{1} \omega_{2} \lambda_{1} (2i\omega_{2} - \varLambda_{2} \lambda_{1} \varOmega ) \\ D_{2} & = - (\beta_{1} - \omega_{2}^{2} )\frac{{\beta_{3} }}{2}(2\varLambda_{2} + \bar{\varLambda }_{2} + 3\varLambda_{2}^{2} \bar{\varLambda }_{2} ) - i\omega_{1} \omega_{2} \lambda_{1} \frac{{\beta_{3} }}{2}(3 + 2\varLambda_{2} \bar{\varLambda }_{2} + \varLambda_{2}^{2} ) \\ D_{3} & = - (\beta_{1} - \omega_{2}^{2} )\frac{{\beta_{3} }}{2}(2\varLambda_{1} + 2\varLambda_{2} + 2\bar{\varLambda }_{1} + 6\varLambda_{1} \bar{\varLambda }_{1} \varLambda_{2} ) - i\omega_{1} \omega_{2} \lambda_{1} \frac{{\beta_{3} }}{2}(6 + 2\bar{\varLambda }_{1} \varLambda_{2} + 2\varLambda_{1} \bar{\varLambda }_{1} + 2\varLambda_{1} \varLambda_{2} ) \\ D_{4} & = - (\beta_{1} - \omega_{2}^{2} )ic\omega_{2} \varLambda_{2} + \omega_{1} \omega_{2}^{2} \lambda_{1} c) \\ \end{aligned}$$

1.4.2 (b) Constants used in Eq. (43) are given as

$$c_{2}^{{\prime }} = - \frac{{C_{2}^{{\prime }} }}{{C_{1}^{{\prime }} }},\quad c_{3} = - \frac{{C_{3}^{{\prime }} }}{{C_{1}^{{\prime }} }},\quad c_{4} = - \frac{{C_{4}^{{\prime }} }}{{C_{1}^{{\prime }} }},\quad c_{5} = - \frac{{C_{5}^{{\prime }} }}{{C_{1}^{{\prime }} }},\quad c_{6} = - \frac{{C_{6}^{{\prime }} }}{{C_{1}^{{\prime }} }}$$

where

$$\begin{aligned} C_{1}^{{\prime }} & = (\beta_{1} \, - \omega_{2}^{2} )(2\varLambda_{2} i\omega_{1} + \lambda_{1} \omega_{2} ) - i\omega_{2}^{2} \lambda_{1} ( - 2i\omega_{2} + \varLambda_{2} \lambda_{1} \omega_{2} ) \\ C_{2}^{{\prime }} & = (\beta_{1} \, - \omega_{2}^{2} ) - \frac{{\beta_{3} }}{2}(2\varLambda_{2} + \bar{\varLambda }_{2} + 3\varLambda_{2}^{2} \bar{\varLambda }_{2} ) + i\omega_{2}^{2} \lambda_{1} - \frac{{\beta_{3} }}{2}(3 + 2\varLambda_{2} \bar{\varLambda }_{2} + \varLambda_{2}^{2} ) \\ C_{3}^{{\prime }} & = (\beta_{1} \, - \omega_{2}^{2} ) - \frac{{\beta_{3} }}{2}(2\varLambda_{1} + 2\varLambda_{2} + 2\bar{\varLambda }_{1} + 6\varLambda_{1} \bar{\varLambda }_{1} \varLambda_{2} ) + i\omega_{2}^{2} \lambda_{1} - \frac{{\beta_{3} }}{2}(6 + 2\bar{\varLambda }_{1} \varLambda_{2} + 2\varLambda_{1} \bar{\varLambda }_{1} + 2\varLambda_{1} \varLambda_{2} ) \\ C_{4}^{{\prime }} & = + (\beta_{1} \, - \omega_{2}^{2} )\frac{1}{2}m_{e} \omega_{2}^{2} {\text{d}}f(L_{1} ) + \omega_{2}^{2} \lambda_{1} \frac{1}{2}m_{e} \omega_{2}^{2} {\text{d}}f(L_{1} ) \\ C_{5}^{{\prime }} & = (\beta_{1} \, - \omega_{2}^{2} )( - ic\omega_{2} \varLambda_{2} ) + \omega_{2}^{3} \lambda_{1} c \\ C_{6}^{{\prime }} & = \frac{1}{2}\,(\beta_{1} \, - \omega_{2}^{2} )( - \lambda_{2} \omega_{2}^{2} \varLambda_{2} + \lambda_{2} \omega_{2} \varLambda_{2} (2\,\omega_{2} ) - \lambda_{2} i\omega_{2}^{2} + \lambda_{2} i\omega_{2} (2\,\omega_{2} ) + \beta_{2} i + \,\beta_{2} \varLambda_{2} ) \\ & \quad + \,\frac{1}{2}\,i\omega_{2}^{2} \lambda_{1} ( - \lambda_{2} \omega_{2}^{2} + \lambda_{2} \omega_{2} (2\,\omega_{2} ) + \,\lambda_{2} i\omega_{2}^{2} \varLambda_{2} - \lambda_{2} \omega_{2} \varLambda_{2} i(2\,\omega_{2} ) + \beta_{2} - \beta_{2} i\varLambda_{2} ). \\ \end{aligned}$$

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Jamshidi, P., Jafari, A.A. Analytical investigation on nonlinear vibration behavior of an unbalanced asymmetric rotor using the method of multiple scales. J Braz. Soc. Mech. Sci. Eng. 41, 456 (2019). https://doi.org/10.1007/s40430-019-1960-z

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