Influence of fluid induced tangential forces,nonlinear stiffness,and internal damping on the vibration response of a Jeffcott Rotor System

Abstract

This study investigates horizontal and vertical oscillations of a horizontally supported Jeffcott Rotor System (JRS), emphasizing the impact of key dynamic parameters. These include the eccentricity ratio, fluid induced tangential forces, nonlinear restoring forces due to large shaft deformations, and internal shaft damping. Under simultaneous resonance, the nonlinear amplitude–phase modulation equations are formulated using the analytical technique of the Method of Multiple Scales (MMS). A detailed analysis of both localized and non-localized oscillations is performed. Numerical simulations comprising time responses, phase portraits, amplitude frequency responses, backbone curves, and phase angle diagrams demonstrate complex dynamic phenomena such as the jump phenomenon, multi-valued solutions, and multiple loops. The analytical results are compared through numerical simulations, and the stability of the steady-state solutions is assessed using linear stability analysis (LSA). Furthermore, analytical expressions are derived to identify critical parameter thresholds corresponding to initiating turning points in localized oscillations. The findings provide valuable insights into the nonlinear dynamics and stability characteristics of rotor systems operating under multi-parameter excitation. The findings demonstrate that key parameters related to lubricant viscosity, geometry, and material properties of the rotor play a crucial role in control or mitigating the vibration amplitudes and unwanted jump phenomena.

Keywords

Fluid induced tangential force multi-jumps rotor dynamics simultaneous resonance

Introduction

Rotating components, ranging from toys to aerospace machinery such as jet engines, are crucial. It is essential to ensure the dynamic stability of these components, as excessive vibrations in rotating machinery can lead to degradation and catastrophic failures. As a result, rotor dynamics has evolved as a significant field of research, with a focus on mitigating vibration in rotor systems. A comprehensive understanding of the nature of vibrations of the rotating systems is vital for developing and implementing effective control strategies to achieve the aforementioned objective. Among the various theoretical models utilized for rotor dynamics, the JRS is a fundamental model for analysing the key dynamic characteristics of flexible rotor systems. The JRS is a lumped parameter model consisting of a rigid disc attached to a massless, flexible shaft supported at both ends.

A primary source of instability in the JRS arises from centrifugal forces generated due to eccentricity between the center of gravity and the center of mass of the disc. As operating speeds rise, the influence of other factors such as nonlinear restoring forces, fluid induced tangential force, internal material damping, etc., increases, subsequently resulting in complicated behaviours. Understanding these effects is necessary to predict and mitigate dynamic instabilities in practical rotor systems accurately.

There is considerable literature on JRS investigations and the factors influencing system vibrations. When oscillations in a rotor system have a large amplitude, geometric nonlinearities are induced. The dominant nonlinearities in the governing equations of motion are cubic in nature, arising either from large transverse deflections of the shaft^1–3 or from nonlinear restoring forces associated with bearing clearance.⁴ As proposed by Saito,⁴ the critical unbalance in the rotating system consequently rises as radial clearance increases. Several nonlinear dynamic behaviors have been reported in the JRS with nonlinear restoring forces due to large deformation of shaft due to bending or bearing clearence. These include period doubling phenomenon,^5–8 multivalued solutions,^3,5,9,10 pitchfork bifurcation,⁶ Hopf bifurcation,^8,10–12 jump phenomenon,^5–7,10 flip bifurcation,⁸ saddle node bifurcation,^6,8,10,11 subharmonic whirling motion,^8,12–14 quasi periodic whirling motion,^6,8,10,12 fold bifurcation,⁷ sensitivity to initial conditions,^6–10,15 non-synchronous vibration,¹⁶ forward whirling motions,¹⁷ backward whirling motions,¹⁷ etc. Ganesan^18,19 observed the mitigation of larger amplitudes in the rotor system using non-symmetric bearing clearance. Yabuno et al.¹² analytically derived both localized and nonlocalized nonlinear normal modes of the JRS, including nonlinear restoring forces.

Another critical aspect that governs the vibration response of the JRS is the internal damping.^20–31 The analysis of a JRS with internal damping led to some results, like subharmonic motion,³² multivalued solutions,^{23,27,29–31,33} forward whirling motion,²⁵ chaotic behavior,²² quasiperiodic motion,^22,32 jump phenomena,^{22,23,27,29–31} Hopf bifurcation,^{23,27,29–31} multiple loops,^{23,27,29–31} spring hardening,^23,28–31 critical speeds,²⁵ backward whirling motion,²⁵ etc.

An additional key parameter affecting the oscillatory behaviour of the JRS is fluid induced rotating force, which has been studied for its role in energy dissipation and its effect on system stability.^32,34–39 Fluid induced forces are pivotal in rotor systems' dynamic behavior and stability, particularly in cases involving hydrodynamic bearings and seals. The fluid-induced instability in journal-bearing-supported rotors, revealing that nonlinear fluid forces can lead to lower and higher order harmonic,^32,36 forward and backward whirling motion,³⁴ multivalued solutions,^32,36 oil whirl and oil whip,^32,35–37 rotor stability,^34,39 subharmonic oscillations,³² quasiperiodic behavior,³² chaotic motion,³² complex vibration patterns,³⁶ swirling flows,³⁷ destabilizing influence in high-speed operations,^37,39 critical speeds,³⁹ etc. These studies underscore the need to incorporate accurate fluid force models to predict and mitigate fluid-induced instabilities in rotating machinery.

The significant contribution of this study lies in the comprehensive combination of multiple significant factors such as eccentricity ratio, fluid-induced rotational forces, nonlinear restoring force due to large deformation, and internal damping into a Jeffcott Rotor System. Localized and non-localized oscillation responses are investigated, revealing complex dynamic phenomena such as the jump phenomenon, multiple loops, sensitive to initial conditions, multivalued solutions, and quasiperiodic motions. Under simultaneous resonance conditions, the MMS is utilized to find analytical formulations for the amplitude and phase modulation equations. Furthermore, critical parameter thresholds associated with imitating limit points in localized oscillations are determined analytically. The selected parameters are explored with practical relevance, ensuring that the insights derived apply to real-world rotor systems and inform the design of more robust vibration control strategies.

This study investigates the impact of various nonlinearities and dynamic effects on the vibration behavior of the Jeffcott Rotor System, incorporating the effects of nonlinear restoring forces from large shaft deformations, fluid-induced rotating forces, and internal damping within the model. The structure of the paper is organized as follows: Mathematical modeling section presents the derivation of the governing mathematical model, incorporating all relevant dynamic forces and nonlinearities. The amplitude phase modulation equations are derived using the Method of Multiple Scales (MMS) section. The steady-state solutions of these equations, followed by a linear stability analysis (LSA) of the steady-state responses in analytical section. Analytical expressions for the critical system parameters associated with the initiation of turning points in localized oscillations are provided at end of analytical section. A comprehensive discussion of the numerical and analytical findings, and finally concludes the study with key observations and insights.

Mathematical modeling

The ideal Jeffcott rotor model comprises a disc positioned at the midspan of a flexible shaft, with the shaft supported at both ends, as illustrated in Figure 1. A small eccentricity e, defined as the offset between the disc’s center of gravity (G) and the geometric center of the shaft axis (M), as depicted in Figure 2, is typically inevitable due to manufacturing imperfections. The exponential growth of an unbalanced force in the Jeffcott rotor system is caused by eccentricity. The intersection of the disc plane and the bearing’s center line is where the origin (O) in Figure 1 is located.

Figure 1.

Schematic diagram of Jeffcott rotor system.

Figure 2.

Fluid film radial damping and stiffness.

In Figure 2, the centrifugal force generated by the disk rotation is shown acting along ‘MG’. As the rotating speed (ω) rises, so does the centrifugal force. The centrifugal force causes the JRS to vibrate in both horizontal and vertical directions. Primary, internal, and simultaneous resonance are the three types of resonance conditions. The present work investigates simultaneous resonance, characterized by the coincidence of the rotational speed with the natural frequencies along both horizontal and vertical axes.

The governing equation of motion of a horizontally supported JRS can be expressed in complex form

m \ddot{z} + F_{d} + F_{r} + F_{h} + F_{g} = m e ω^{2} e^{i ω τ} - i m g,

(1)

In equation (1),

z

represents the rotor center complex displacement at point M, expressed as

z = x + i y

and

x

and

y

indicate the rotor centre displacements (M) in the both horizontal and vertical directions, respectively.

The external damping force acting on the Jeffcott rotor, as described in Ref. 3,15, is linearly proportional to the velocity of the rotor centre ( $\dot{z}$ ) and can be expressed using complex notation accordingly.

F_{d} = c_{z} \dot{z},

(2)

where

c_{z} = c_{x} + i c_{y}

is the damping coefficient

The nonlinear restoring force arising from large shaft deformation due to bending, expressed in complex form, can be represented as shown in Ref. 3,10,11,15,17.

F_{r} = k_{1} z + k_{2} z (z \bar{z}),

(3)

Where,

\bar{z}

is the complex conjugate of

z

The internal damping of the shaft can be written as^{27,29–31,33}

F_{i} = ξ (\dot{z} + i ω z),

(4)

The fluid induced tangential force can be written as^3,15

F_{t} = K z - i D λ ω z,

(5)

where

D

is the coefficient of the fluid rotating damping,

K

(= \frac{1}{(\frac{1}{K_{b}}) + (\frac{1}{K_{1}})} + K_{o})

represents the lateral stiffness of the overall system.

By separating the real and imaginary components of equation (6), the governing equations of motion for the JRS along both directions are obtained as follows:

m \ddot{z} + c_{z} \dot{z} + k_{1} z + k_{2} z (z \bar{z}) + ξ (\dot{z} + i ω z) + K z - i D λ ω z = m e ω^{2} e^{i ω τ} - i m g .

(6)

Separating the real and imaginary parts of equation (6), the governing equations of motion of the JRS along the horizontal and vertical directions, respectively are written as

m \ddot{x} + c_{x} \dot{x} + k_{1} x + k_{2} x^{3} + k_{2} y^{2} x + D λ ω y + ξ ω y = m e ω^{2} \cos (ω τ),

(7a)

m \ddot{y} + c_{y} \dot{y} + k_{1} y + k_{2} y^{3} + k_{2} x^{2} y - D λ ω x - ξ ω x = m e ω^{2} \sin (ω τ) - m g,

(7b)

For static equilibrium $\dot{x} = \ddot{x} = \dot{y} = \ddot{y} = ω = 0$ , and $y = V_{s t}$ . Substituting in equations (7a) and (7b), hence the conditions for static equilibrium in the x-direction and y-direction are described as

x_{s t} = 0,

(8a)

k_{1} V_{s t} + k_{2} V_{s t}^{3} + k_{o} V_{s t} = - m g .

(8b)

Shifting the coordinates to the static equilibrium states as $x = x^{*}$ , $y = V_{s t} + y^{*}$ are presented as

m {\ddot{x}}^{*} + c_{x} {\dot{x}}^{*} + (k_{1} + k_{2} V_{s t}^{2}) x^{*} + 2 k_{2} V_{s t} x^{*} y^{*} + k_{2} ({x^{*}}^{2} + y^{*}^{2}) x^{*} + D λ ω (V_{s t} + y^{*}) + ξ ω (V_{s t} + y^{*}) = m e ω^{2} \cos (ω τ),

(9a)

m {\ddot{y}}^{*} + c_{y} {\dot{y}}^{*} + (k_{1} + 3 k_{2} V_{s t}^{2}) y^{*} + k_{2} V_{s t} ({x^{*}}^{2} + 3 y^{*}^{2}) + k_{2} ({x^{*}}^{2} + y^{*}^{2}) y^{*} - λ ω x^{*} + ξ ω x^{*} = m e ω^{2} \sin (ω τ) .

(9b)

The dimensionless parameters are introduced.

t = ω_{0} τ, ω_{0} = \sqrt{\frac{k_{1}}{m}}, u = \frac{x^{*}}{V_{s t}}, v = \frac{y^{*}}{V_{s t}}, f = \frac{e}{V_{s t}}, μ_{x} = \frac{c_{x}}{\sqrt{k_{1} m}}, μ_{y} = \frac{c_{y}}{\sqrt{k_{1} m}}, α = \frac{k_{2} {V_{s t}}^{2}}{k_{1}}, γ = \frac{k_{o}}{k_{1}},

ω_{1}^{2} = 1 + α + γ, ω_{2}^{2} = 1 + 3 α + γ, Ω = \frac{ω}{ω_{0}}, r_{i} = \frac{D λ ω}{k_{1}}, c_{i} = \frac{ξ ω}{k_{1}},

The dimensionless equations governing the horizontal and vertical modes of the horizontally mounted Jeffcott Rotor System (JRS) are expressed as follows:

u^{″} + μ_{1} u^{'} + ω_{1}^{2} u + 2 λ u v + λ (u^{2} + v^{2}) u + (r_{i} + c_{i}) (1 + v) = f Ω^{2} \cos (Ω t),

(10a)

v^{″} + μ_{2} v^{'} + ω_{2}^{2} v + λ (u^{2} + 3 v^{2}) + λ (u^{2} + v^{2}) v - (r_{i} + c_{i}) u = f Ω^{2} \sin (Ω t),

(10b)

Here, primes denote differentiation concerning the dimensionless time ‘t’. Equations (10a) and (10b) represent the horizontal and vertical oscillations of the JRS, respectively. In the upcoming section, the governing equations of motion (10a) and (10b) are analyzed analytically using perturbation techniques.

Analytical studies

In this section, the nonlinear amplitude and phase equations describing the vibrations of the Jeffcott Rotor System (JRS) are formulated using the Method of Multiple Scales (MMS). The corresponding steady-state responses are then determined, followed by a linear stability analysis to assess the stability of these solutions.

Method of multiple scales

In this section, the amplitude and phase equations for the rotor’s horizontal and vertical oscillations are derived using the method of multiple scales (MMS).^10,23,30 The analysis is performed for simultaneous resonance condition

Ω = ω_{1} + σ_{1}, σ_{2} {= ω}_{2} - ω_{1},

(11)

The series solutions for displacements in both horizontal and vertical directions are defined as:

u (f_{k}) = \sum_{i = 1}^{n} ε^{i} u_{i} (f_{k}),

(12a)

v (f_{k}) = \sum_{i = 1}^{n} ε^{i} v_{i} (f_{k}) .

(12b)

Where,

f_{k} = (T_{0}, T_{1}, T_{2})

. The time derivatives are expressed in terms of fast and slow time scales as follows:

\frac{d}{d t} = \frac{\partial}{\partial T_{o}} + ε \frac{\partial}{\partial T_{1}} + ε^{2} \frac{\partial}{\partial T_{2}}, \frac{d^{2}}{d t^{2}} = \frac{\partial^{2}}{\partial {T_{o}}^{2}} + 2 ε \frac{\partial^{2}}{\partial T_{o} \partial T_{1}} + ε^{2} (\frac{\partial^{2}}{\partial {T_{1}}^{2}} + 2 \frac{\partial^{2}}{\partial T_{o} \partial T_{2}}) .

(13)

Scaled system parameters are written as:

μ_{x} = ε^{2} μ_{h x}, μ_{y} = ε^{2} μ_{h y}, f = ε^{2} f_{h}, r_{i} = ε^{2} r_{i h}, c_{i} = ε^{2} c_{i h} .

(14)

Equations (12a)–(14) substituted into equations (10a) and (10b), segregating coefficients of

ε

O (ε) : \frac{\partial^{2}}{\partial {T_{0}}^{2}} u_{1} + {ω_{1}}^{2} u_{1} = 0,

(15a)

\frac{\partial^{2}}{\partial {T_{0}}^{2}} v_{1} + {ω_{2}}^{2} v_{1} = 0 .

(15b)

O (ε^{2}) : \frac{\partial^{2}}{\partial {T_{0}}^{2}} u_{2} + {ω_{1}}^{2} u_{2} = f_{h} Ω^{2} \cos (Ω T_{0}) - 2 λ v_{1} u_{1} - 2 (\frac{\partial^{2}}{\partial T_{1} \partial T_{0}} u_{1}) - r_{i h} - c_{i h},

(16a)

\frac{\partial^{2}}{\partial {T_{0}}^{2}} v_{2} + {ω_{2}}^{2} v_{2} = - λ {u_{1}}^{2} - 2 (\frac{\partial^{2}}{\partial T_{1} \partial T_{0}} v_{1}) + f_{h} Ω^{2} \sin (Ω T_{0}) - 3 λ {v_{1}}^{2} .

(16b)

O (ε^{3}) : \frac{\partial^{2}}{\partial {T_{0}}^{2}} u_{3} + {ω_{1}}^{2} u_{3} = - 2 λ v_{1} u_{2} - 2 (\frac{\partial^{2}}{\partial T_{2} \partial T_{0}} u_{1}) - 2 λ v_{2} u_{1} - λ {v_{1}}^{2} u_{1} - μ_{h 1} (\frac{\partial}{\partial T_{0}} u_{1}) - \frac{\partial^{2}}{\partial {T_{1}}^{2}} u_{1} - c_{i h} v_{1} - 2 (\frac{\partial^{2}}{\partial T_{1} \partial T_{0}} u_{2}) - λ {u_{1}}^{3} - r_{i h} v_{1} = 0,

(17a)

\frac{\partial^{2}}{\partial {T_{0}}^{2}} v_{3} + {ω_{2}}^{2} v_{3} = - 6 λ v_{1} v_{2} - 2 (\frac{\partial^{2}}{\partial T_{1} \partial T_{0}} v_{2}) - 2 λ u_{1} u_{2} - λ {u_{1}}^{2} v_{1} - μ_{h 2} (\frac{\partial}{\partial T_{0}} v_{1}) - \frac{\partial^{2}}{\partial {T_{1}}^{2}} v_{1} + c_{i h} u_{1} - 2 (\frac{\partial^{2}}{\partial T_{2} \partial T_{0}} v_{1}) - λ {v_{1}}^{3} + r_{i h} u_{1} = = 0 .

(17b)

The solutions of equations (15a) and (15b) are expressed as:

u_{1} (f_{k}) = A (f_{d}) e^{i ω_{1} T_{0}} + \bar{A} (f_{d}) e^{- i ω_{1} T_{0}},

(18a)

v_{1} (f_{k}) = B (f_{d}) e^{i ω_{2} T_{0}} + \bar{B} (f_{d}) e^{- i ω_{2} T_{0}} .

(18b)

Where,

f_{d} = (T_{1}, T_{2})

. The solutions equations (18a) and (18b), along with equation (11), are substituted into equations (16a) and (16b), and eliminating secular terms by setting the coefficients of

e^{i ω_{1} T_{0}} = 0

. The following equations are solved to yield:

\frac{\partial A (f_{d})}{\partial T_{1}} = - \frac{i e^{i T_{1}} σ_{h 1} {f_{h} Ω}^{2}}{4 ω_{2}},

(19a)

\frac{\partial B (f_{d})}{\partial T_{1}} = - \frac{e^{i T_{1} (σ_{h 1} - σ_{h 2})} {f_{h} Ω}^{2}}{4 ω_{2}} .

(19b)

The expressions for $\frac{\partial A (f_{d})}{\partial T_{2}}$ and $\frac{\partial B (f_{d})}{\partial T_{2}}$ are obtained by and eliminating secular terms by setting the coefficients of $e^{i ω_{1} T_{0}} = 0$ , and are given in Appendix A. The solutions of $u_{1}$ , $v_{1}$ , $u_{2}$ and $v_{2}$ are substituted in equations (17a) and (17b). The time derivative relations between $A$ and $B$ are follows:

\dot{A} = \frac{d A}{d t} = ε \frac{\partial A (f_{d})}{\partial T_{1}} + ε^{2} \frac{\partial A (f_{d})}{\partial T_{2}},

(20a)

\dot{B} = \frac{d B}{d t} = ε \frac{\partial B (f_{d})}{\partial T_{1}} + ε^{2} \frac{\partial B (f_{d})}{\partial T_{2}} .

(20b)

The explicit expressions for $\dot{A}$ and $\dot{B}$ are lengthy and therefore omitted. Instead, the polar representations of $A$ and $B$ are expressed as given in Ref. 10.

A (f_{d}) = \frac{1}{2} {{\hat{a}}_{1} e}^{i δ_{1}} = \frac{1}{2 ε} {a_{1} e}^{i δ_{1}},

(21a)

B (f_{d}) = \frac{1}{2} {\hat{a}}_{2} e^{i δ_{2}} = \frac{1}{2 ε} a_{2} e^{i δ_{2}},

(21b)

Here,

a_{1}

and

δ_{1}

represent the steady-state amplitude and phase in the horizontal direction, while

a_{2}

and

δ_{2}

correspond to those in the vertical direction of the JRS. The corresponding amplitude and phase modulating equations are derived from equations (21a) and (21b).

{\dot{a}}_{1} = \frac{d a_{1}}{d t} = 2 ε . Re (\dot{A}) e^{- i δ_{1}}, {\dot{δ}}_{1} = \frac{d δ_{1}}{d t} = \frac{2 ε}{a_{1}} . Im (\dot{A}) e^{- i δ_{1}},

(22a)

{\dot{a}}_{2} = \frac{d a_{2}}{d t} = 2 ε . Re (\dot{B}) e^{- i δ_{2}}, {\dot{δ}}_{2} = \frac{d δ_{2}}{d t} = \frac{2 ε}{a_{2}} . Im (\dot{B}) e^{- i δ_{2}},

(22b)

Substituting equations (20a) and (20b) into equations (22a) and (22b) to obtain the expressions for amplitude and phase modulating equations in both directions is written as follows

{\dot{a}}_{1} = - \frac{1}{2} μ_{1} a_{1} - \frac{λ}{8 ω_{1}} a_{2}^{2} a_{1} \sin (2 ϕ_{1} - 2 ϕ_{2}) + (8 λ - 16 λ ω_{1} + 8 ω_{1}^{3} - 4 ω_{2} ω_{1}^{2}) {(c_{i} + r_{i}) a_{2} ω}_{2}^{2} \sin (ϕ_{1} - ϕ_{2}) - \frac{λ^{2}}{4 ω_{1} (2 ω_{1} - ω_{2}) ω_{2}^{2}} a_{2}^{2} a_{1} \sin (2 ϕ_{1} - 2 ϕ_{2}) - \frac{1}{4} \frac{(- 3 ω_{1} + Ω) f Ω^{2} \sin ϕ_{1}}{{ω_{1}}^{2}},

(23a)

{\dot{ϕ}}_{1} = σ_{1} - \frac{(3 λ {a_{1}}^{2} + 2 λ {a_{2}}^{2})}{8 ω_{1}} - \frac{λ}{8 ω_{1}} {a_{2}}^{2} \cos ({2 ϕ}_{1} - 2 ϕ_{2}) - \frac{(- 3 ω_{2} + Ω) f Ω^{2} \cos ϕ_{1}}{4 a_{1} {ω_{1}}^{2}} - \frac{λ^{2} {a_{2}}^{2} \cos (2 ϕ_{1} - {2 ϕ}_{2})}{4 (- {ω_{2}}^{2} + 4 {ω_{1}}^{2}) ω_{1} {ω_{2}}^{2}} (2 ω_{1} - 3 ω_{2}) + (16 ω_{1}^{4} - 32 λ ω_{1}^{2} - 4 ω_{2}^{2} ω_{1}^{2} - 8 λ ω_{2}^{2}) (c_{i} + r_{i}) a_{2} ω_{2}^{2} \cos (ϕ_{1} - ϕ_{2}) + \frac{λ^{2}}{4} \frac{(24 {ω_{1}}^{2} {a_{2}}^{2} - 2 {ω_{2}}^{2} {a_{2}}^{2} - 3 {ω_{2}}^{2} {a_{1}}^{2} + 8 {ω_{1}}^{2} {a_{1}}^{2})}{(- {ω_{2}}^{2} + 4 {ω_{1}}^{2}) ω_{1} {ω_{2}}^{2}},

(23b)

{\dot{a}}_{2} = - \frac{1}{2} μ_{2} a_{2} + \frac{1}{4} \frac{(- 3 ω_{2} + Ω) f Ω^{2} \cos ϕ_{2}}{{ω_{2}}^{2}} + (32 λ ω_{1}^{2} - 4 ω_{2}^{2} ω_{1}^{2} + 16 ω_{1}^{4} - 8 λ ω_{2}^{2}) (c_{i} + r_{i}) a_{1} ω_{2}^{2} \sin (ϕ_{1} - ϕ_{2}) - \frac{λ^{2}}{4 (4 {ω_{1}}^{2} - {ω_{2}}^{2}) {ω_{2}}^{2}} {a_{1}}^{2} a_{2} \sin (2 ϕ_{1} - 2 ϕ_{2}) + \frac{λ}{8 ω_{2}} {a_{1}}^{2} a_{2} \sin (2 ϕ_{1} - 2 ϕ_{2}),

(23c)

{\dot{ϕ}}_{2} = σ_{1} - σ_{2} - \frac{(2 λ {a_{1}}^{2} + 3 λ {a_{2}}^{2})}{8 ω_{2}} - \frac{λ}{8 ω_{2}} {a_{1}}^{2} \cos ({2 ϕ}_{1} - 2 ϕ_{2}) - \frac{1}{4} \frac{(- 3 ω_{2} + Ω) f Ω^{2} \sin ϕ_{2}}{a_{2} {ω_{2}}^{2}} - \frac{λ^{2} {a_{1}}^{2} \cos (2 ϕ_{1} - {2 ϕ}_{2})}{4 (- {ω_{2}}^{2} + 4 {ω_{1}}^{2}) {ω_{2}}^{2}} (4 ω_{1} - ω_{2}) + \frac{1}{4} \frac{(60 {ω_{1}}^{2} {a_{2}}^{2} - 15 {ω_{2}}^{2} {a_{2}}^{2} - 2 {ω_{2}}^{2} {a_{1}}^{2} + 24 {ω_{1}}^{2} {a_{1}}^{2}) λ^{2}}{(- {ω_{2}}^{2} + 4 {ω_{1}}^{2}) {ω_{2}}^{3}} + (16 ω_{1}^{4} - 4 ω_{1}^{2} + 32 λ ω_{1}^{2} - 8 λ ω_{2}^{2}) (c_{i} + r_{i}) a_{1} ω_{2}^{2} \cos (ϕ_{1} - ϕ_{2}) .

(23d)

Here, ${\dot{ϕ}}_{1} = σ_{1} - \dot{δ_{1}}$ , and ${\dot{ϕ}}_{2} = σ_{1} - σ_{2} - \dot{δ_{2}}$ . These autonomous equations describe the evolution of amplitudes and phases corresponding to the vertical and horizontal vibrations of the JRS. Numerical simulations of these equations are carried out using MATLAB’s built-in ODE solvers to track the amplitude and phase variations. Additionally, Matcont, a numerical bifurcation analysis toolbox within MATLAB, is employed to generate the amplitude frequency response curves. A detailed discussion of the results is presented in Section 4.

Steady-state oscillations

The steady-state response is characterized by the following conditions¹⁵

{\dot{a}}_{1} = {\dot{ϕ}}_{1} = {\dot{a}}_{2} = {\dot{ϕ}}_{2} = 0 .

(24)

Substituting equations (23a)–(23d) into equation (24), we obtain

- \frac{1}{2} μ_{1} a_{1} - \frac{λ}{8 ω_{1}} a_{2}^{2} a_{1} \sin (2 ϕ_{1} - 2 ϕ_{2}) + (8 λ - 16 λ ω_{1} + 8 ω_{1}^{3} - 4 ω_{2} ω_{1}^{2}) {(c_{i} + r_{i}) a_{2} ω}_{2}^{2} \sin (ϕ_{1} - ϕ_{2}) - \frac{λ^{2}}{4 ω_{1} (2 ω_{1} - ω_{2}) ω_{2}^{2}} a_{2}^{2} a_{1} \sin (2 ϕ_{1} - 2 ϕ_{2}) - \frac{1}{4} \frac{(- 3 ω_{1} + Ω) f Ω^{2} \sin ϕ_{1}}{{ω_{1}}^{2}} = 0,

(25a)

σ_{1} - \frac{(3 λ {a_{1}}^{2} + 2 λ {a_{2}}^{2})}{8 ω_{1}} - \frac{λ}{8 ω_{1}} {a_{2}}^{2} \cos ({2 ϕ}_{1} - 2 ϕ_{2}) - \frac{(- 3 ω_{2} + Ω) f Ω^{2} \cos ϕ_{1}}{4 a_{1} {ω_{1}}^{2}} - \frac{λ^{2} {a_{2}}^{2} \cos (2 ϕ_{1} - {2 ϕ}_{2})}{4 (- {ω_{2}}^{2} + 4 {ω_{1}}^{2}) ω_{1} {ω_{2}}^{2}} (2 ω_{1} - 3 ω_{2}) + (16 ω_{1}^{4} - 32 λ ω_{1}^{2} - 4 ω_{2}^{2} ω_{1}^{2} - 8 λ ω_{2}^{2}) (c_{i} + r_{i}) a_{2} ω_{2}^{2} \cos (ϕ_{1} - ϕ_{2}) + \frac{λ^{2}}{4} \frac{(24 {ω_{1}}^{2} {a_{2}}^{2} - 2 {ω_{2}}^{2} {a_{2}}^{2} - 3 {ω_{2}}^{2} {a_{1}}^{2} + 8 {ω_{1}}^{2} {a_{1}}^{2})}{(- {ω_{2}}^{2} + 4 {ω_{1}}^{2}) ω_{1} {ω_{2}}^{2}} = 0,

(25b)

- \frac{1}{2} μ_{2} a_{2} + \frac{1}{4} \frac{(- 3 ω_{2} + Ω) f Ω^{2} \cos ϕ_{2}}{{ω_{2}}^{2}} + (32 λ ω_{1}^{2} - 4 ω_{2}^{2} ω_{1}^{2} + 16 ω_{1}^{4} - 8 λ ω_{2}^{2}) (c_{i} + r_{i}) a_{1} ω_{2} \sin (ϕ_{1} - ϕ_{2}) - \frac{λ^{2}}{4 (4 {ω_{1}}^{2} - {ω_{2}}^{2}) {ω_{2}}^{2}} {a_{1}}^{2} a_{2} \sin (2 ϕ_{1} - 2 ϕ_{2}) + \frac{λ}{8 ω_{2}} {a_{1}}^{2} a_{2} \sin (2 ϕ_{1} - 2 ϕ_{2}) = 0,

(25c)

σ_{1} - σ_{2} - \frac{(2 λ {a_{1}}^{2} + 3 λ {a_{2}}^{2})}{8 ω_{2}} - \frac{λ}{8 ω_{2}} {a_{1}}^{2} \cos ({2 ϕ}_{1} - 2 ϕ_{2}) - \frac{1}{4} \frac{(- 3 ω_{2} + Ω) f Ω^{2} \sin ϕ_{2}}{a_{2} {ω_{2}}^{2}} - \frac{λ^{2} {a_{1}}^{2} \cos (2 ϕ_{1} - {2 ϕ}_{2})}{4 (- {ω_{2}}^{2} + 4 {ω_{1}}^{2}) {ω_{2}}^{2}} (4 ω_{1} - ω_{2}) + \frac{1}{4} \frac{(60 {ω_{1}}^{2} {a_{2}}^{2} - 15 {ω_{2}}^{2} {a_{2}}^{2} - 2 {ω_{2}}^{2} {a_{1}}^{2} + 24 {ω_{1}}^{2} {a_{1}}^{2}) λ^{2}}{(- {ω_{2}}^{2} + 4 {ω_{1}}^{2}) {ω_{2}}^{3}} + (16 ω_{1}^{4} - 4 ω_{1}^{2} + 32 λ ω_{1}^{2} - 8 λ ω_{2}^{2}) (c_{i} + r_{i}) a_{1} ω_{2}^{2} \cos (ϕ_{1} - ϕ_{2}) = 0 .

(25d)

The steady-state amplitudes and corresponding phases of both horizontal and vertical oscillations of the JRS can be determined numerically by solving equations (25a)–(25d).

Linear stability analysis (LSA)

To determine the stability of the steady-state solutions obtained via equations (23a)–(23d), the LSA is performed. Let { $stst = (a_{10}, ϕ_{10}, a_{20}, ϕ_{20}$ )} be a collection of steady-state (stst) solutions. ( $a_{11}$ , $ϕ_{11}$ , $a_{12}$ , $ϕ_{12}$ ) is intended to be an infinitesimal addition to this collection

a_{1} = a_{10} + a_{11}, ϕ_{1} = ϕ_{10} + ϕ_{11}, a_{2} = a_{20} + a_{12}, ϕ_{2} = ϕ_{20} + ϕ_{12} .

(26)

The linearized equations can be expressed in matrix form by substituting equation (26) into equations (23a)–(23d)

[\begin{array}{l} {\dot{a}}_{11} \\ {\dot{ϕ}}_{11} \\ {\dot{a}}_{21} \\ {\dot{ϕ}}_{21} \end{array}] = {[\begin{array}{l} \frac{\partial {\dot{a}}_{1}}{\partial a_{1}} & \frac{\partial {\dot{a}}_{1}}{\partial ϕ_{1}} & \frac{\partial {\dot{a}}_{1}}{\partial a_{2}} & \frac{\partial {\dot{a}}_{1}}{\partial ϕ_{2}} \\ \frac{\partial {\dot{ϕ}}_{1}}{\partial a_{1}} & \frac{\partial {\dot{ϕ}}_{1}}{\partial ϕ_{1}} & \frac{\partial {\dot{ϕ}}_{1}}{\partial a_{2}} & \frac{\partial {\dot{ϕ}}_{1}}{\partial ϕ_{2}} \\ \frac{\partial {\dot{a}}_{2}}{\partial a_{1}} & \frac{\partial {\dot{a}}_{2}}{\partial ϕ_{1}} & \frac{\partial {\dot{a}}_{2}}{\partial a_{2}} & \frac{\partial {\dot{a}}_{2}}{\partial ϕ_{2}} \\ \frac{\partial {\dot{ϕ}}_{2}}{\partial a_{1}} & \frac{\partial {\dot{ϕ}}_{2}}{\partial ϕ_{1}} & \frac{\partial {\dot{ϕ}}_{2}}{\partial a_{2}} & \frac{\partial {\dot{ϕ}}_{2}}{\partial ϕ_{2}} \end{array}]}_{s t s t} [\begin{array}{l} a_{11} \\ ϕ_{11} \\ a_{21} \\ ϕ_{21} \end{array}] = {[B]}_{s t s t} [\begin{array}{l} a_{11} \\ ϕ_{11} \\ a_{21} \\ ϕ_{21} \end{array}],

(27)

where the square matrix

B = [\begin{array}{l} B_{11} & B_{12} & B_{13} & B_{14} \\ B_{21} & B_{22} & B_{23} & B_{24} \\ B_{31} & B_{32} & B_{33} & B_{34} \\ B_{41} & B_{42} & B_{43} & B_{44} \end{array}]

is the Jacobian matrix. Appendix B provides the Jacobian matrix coefficients. The Jacobian matrix eigenvalues are determined as

\det [B ‐ η I] = | \begin{array}{c} B_{11} - η & B_{12} & B_{13} & B_{14} \\ B_{21} & B_{22} - η & B_{23} & B_{24} \\ B_{31} & B_{32} & B_{33} - η & B_{34} \\ B_{41} & B_{42} & B_{43} & B_{44} - η \end{array} | = 0,

(28)

where

η

are the Jacobian matrix eigenvalues and I is an identity matrix. The determinant of equation (28) is evaluated, yields

η^{4} + α_{1} η^{3} + α_{2} η^{2} + α_{3} η + α_{4} = 0 .

(29)

The coefficients $Ν_{1}, Ν_{2}, Ν_{3}$ and $Ν_{4}$ in equation (29) are presented in Appendix C. Routh–Hurwitz criterion are as follows.

{R H}_{1} = Ν_{1} > 0, {R H}_{2} = Ν_{1} Ν_{2} - Ν_{3} > 0, {R H}_{3} = Ν_{3} (Ν_{1} Ν_{2} - Ν_{3}) - {Ν_{1}}^{2} Ν_{4} > 0, {R H}_{4} = Ν_{4} > 0 .

(30)

In the subsequent section, these conditions are utilized to evaluate the stability of steady-state solutions for a specified set of system parameters.

Critical values for the initiation of limit points for localized oscillations

Localized oscillations describe system responses that are restricted to either the horizontal or the vertical direction of motion, with the other direction neglected. Specifically, horizontal localization corresponds to $a_{1} \neq 0$ , $ϕ_{1} \neq 0$ and $a_{2} = 0$ , $ϕ_{2} = 0$ , while vertical localization corresponds to $a_{2} \neq 0$ , $ϕ_{2} \neq 0$ and $a_{1} = 0$ , $ϕ_{1} = 0$ . In these cases, the governing equations become uncoupled. In contrast, nonlocalized oscillations involve simultaneous motion in both directions ( $a_{1} \neq 0$ , $ϕ_{1} \neq 0$ , $a_{2} \neq 0$ , and $ϕ_{2} \neq 0$ ).

Section 4.2 presents a parametric study showing that, under localized conditions, one steady-state solution occurs across the entire frequency range for small values of a certain parameter. As this parameter exceeds a critical threshold, three solutions appear within a narrow frequency range, with two stable and one unstable solution, forming limit points and the onset of jump phenomena. This section outlines the methodology for determining the critical parameter values that give rise to these multivalued solutions and dynamic instabilities.

Substituting $a_{2} = 0$ , and $ϕ_{2} = 0$ in equations (23a)–(23d) yields the equations governing localized oscillations in the horizontal direction:

{\dot{a}}_{1} = - \frac{1}{2} μ_{1} a_{1} - \frac{1}{4} \frac{(- 3 ω_{1} + Ω) Ω^{2} \sin ϕ_{1}}{ω_{1}^{2}},

(31a)

{\dot{ϕ}}_{1} = σ_{1} - \frac{3 λ a_{1}^{2}}{8 ω_{1}} + \frac{1}{4} \frac{(- 3 ω_{2}^{2} + 8 ω_{1}^{2}) a_{1}^{2}}{(ω_{2}^{2} + 4 ω_{1}^{2}) ω_{1}^{2} ω_{2}^{2}} - \frac{1}{4} \frac{(- 3 ω_{1} + Ω) f Ω^{2} \sin ϕ_{1}}{a_{1} ω_{1}^{2}} .

(31b)

By setting

{\dot{a}}_{1} = 0

and

{\dot{ϕ}}_{1} = 0

in equations (31a) and (31b) to obtain the steady-state solutions, and subsequently eliminating

ϕ_{1}

, the resulting expression yields:

q_{6} a_{1}^{6} + (q_{41} Ω + q_{40}) a_{1}^{4} + (q_{22} Ω^{2} + q_{21} Ω + q_{20}) a_{1}^{2} +

(32)

(q_{06} Ω^{6} + q_{05} Ω^{5} + q_{04} Ω^{4}) f^{2} = 0,

The coefficients

q_{6}

q_{41}

q_{40}

, etc., are listed in Appendix D.

\frac{d a_{1}}{d Ω} = - \frac{q_{41} a_{1}^{4} + (2 q_{22} Ω + q_{21}) a_{1}^{2} + (6 q_{06} Ω^{5} + 5 q_{05} Ω^{5} + 4 q_{04} Ω^{3}) f^{2}}{{6 q}_{6} a_{1}^{5} + 4 (q_{41} Ω + q_{40}) a_{1}^{3} + 2 (q_{22} Ω^{2} + q_{21} Ω + q_{20}) a_{1}},

(33)

At the limit points, $\frac{d a_{1}}{d Ω} \to \infty$ . By equating the denominator of equation (33) to zero, a quadratic equation in $a_{1}$ is obtained.

3 q_{6} a_{1}^{4} + 2 (q_{41} Ω + q_{40}) a_{1}^{2} + ({q_{22} Ω}^{2} + q_{21} Ω + q_{20}) = 0 .

(34)

The roots of equation (34) are written as

a_{1}^{2} = - \frac{2 (q_{41} Ω + q_{40}) \pm \sqrt{4 {(q_{41} Ω + q_{40})}^{2} - 12 q_{6} ({q_{22} Ω}^{2} + q_{21} Ω + q_{20})}}{6 q_{6}} .

(35)

The quadratic equation of $Ω$ is obtained by setting the discriminant in equation (35) to zero

{α_{1} Ω}^{2} + β_{1} Ω + γ_{1} = 0,

(36)

where

α_{1} = (q_{41}^{2} - 3 q_{6} q_{22})

β_{1} = (2 q_{40} q_{41} - 3 q_{6} q_{21})

, and

γ_{1} = (q_{40}^{2} - 3 q_{6} q_{20})

. The roots of equation (36) are

Ω_{1, 2} = \frac{- β_{1} \pm \sqrt{β_{1}^{2} - 4 α_{1} γ_{1}}}{2 α_{1}} .

(37)

The value of $a_{1}$ at the critical point is calculated by equating the discriminant in equation (35) to zero.

a_{1 c} = \sqrt{- \frac{2 (q_{41} Ω_{c} + q_{40})}{6 q_{6}}},

(38)

As the coefficients

q_{6}

q_{41}

q_{40}

, etc., are independent of the eccentricity ratio

f

, the expression for the critical value

{(f_{c})}_{h}

in horizontal direction is written as

{(f_{c})}_{h} = \sqrt{- \frac{q_{6} a_{1 c}^{6} + (q_{41} Ω_{c} + q_{40}) a_{1 c}^{4} + (q_{22} Ω_{c}^{4} + q_{21} Ω_{c} + q_{20}) a_{1}^{2}}{(q_{06} Ω_{c}^{6} + q_{05} Ω_{c}^{5} + q_{02} Ω_{c}^{4})}} .

(39)

{(f_{c})}_{h}

denotes the critical value of eccentricity in horizontal direction, while the corresponding critical value for vertical oscillations is represented by

{(f_{c})}_{v}

{(f_{c})}_{v} = \sqrt{- \frac{p_{6} a_{2 c}^{6} + (p_{41} Ω_{c} + p_{40}) a_{2 c}^{4} + (p_{22} Ω_{c}^{4} + p_{21} Ω_{c} + p_{20}) a_{2}^{2}}{(p_{06} Ω_{c}^{6} + p_{05} Ω_{c}^{5} + p_{02} Ω_{c}^{4})}},

(40)

The following quadratic equation in $a_{2}$ is expressed as:

3 p_{6} a_{2}^{4} + 2 (p_{41} Ω + p_{40}) a_{2}^{2} + ({p_{22} Ω}^{2} + p_{21} Ω + p_{20}) = 0 .

(41)

The coefficients

p_{6}

p_{41}

p_{40}

, etc., are listed in Appendix D.

Results and discussions

This section discusses most of the results obtained from the analysis.

To investigate the impact of the eccentricity ratio $(f)$ , fluid induced tangential coefficient $(r_{i})$ , stiffness ratio $(α)$ , internal damping coefficient ( $c_{i}$ ) and fluid radial stiffness ratio $(γ)$ , on the nature of vibration of JRS, a detailed parametric analysis is performed in subsequent section. Equations (23a)–(23d) numerically simulated to obtained, the time response plots, phase plane curves, phase angle plots and frequency response graphs in “Matcont”. The amplitude and phase modulation equations (equations (23a)–(23d)) are typically used to evaluate the JRS characteristic.

Numerical and analytical results comparison

Figure 3 presents a comparison between the results obtained from the Method of Multiple Scales (MMS) and the time-domain and phase-plane responses for horizontal vibrations of the JRS, based on the governing equations (equations (10a) and (10b)). The specific set of JRS parameter values employed for this analysis is provided in Table 1. Figure 3 shows that the MMS results and numerical simulation results of the governing equations (10a) and (10b), respectively, correspond reasonably well. The MMS results are utilized to predict the system dynamics in upcoming analyses assuming that the qualitative behaviour of system remains uniform. Most JRS oscillations in this study are single-periodic for the selected parameter values, though quasiperiodic motions occur within a limited parameter range.

Figure 3.

(a) Time responses and (b) corresponding phase plane plots of the JRS in horizontal direction. The system parameters are listed in Table 1, with initial conditions set as (0.01, 0, 0.01, 0).

Table 1.

JRS parameters and its values.

Parameter	$μ_{x}$	$μ_{y}$	$c_{i}$	$f$	$α$	$γ$	$r_{i}$
Value	0.0154	0.0247	0.00085	0.045	0.05	0.05	0.0005

In upcoming amplitude frequency responses curves (a) depicts the horizontal oscillations, while (b) illustrates the vertical oscillations. Figure 4 illustrates the amplitude frequency response curves for different values of the eccentricity ratio $f$ . In these plots, the numerical solutions of the original governing equations (equations (10a) and (10b)) are indicated by discrete points, while the continuous solid lines represent the analytical solutions derived using the Method of Multiple Scales (equations (23a)–(23d)). A close comparison reveals that the MMS solutions capture the qualitative dynamics of the system with good fidelity relative to the numerical simulations. Given the computational efficiency and compatibility with bifurcation analysis tools like Matcont, the subsequent amplitude–frequency response analyses are based on the MMS derived equations.

Figure 4.

Frequency response curves of the JRS for various values of the eccentricity ratio $f$ , with system parameters as listed in Table 1 and $α \neq 0$ . (a) horizonatal oscillations (b) vertical oscillations.

Localized oscillations

As described in Section 3.4, localized oscillations along the horizontal direction are analyzed independently by imposing the constraints $a_{2} = 0$ , and $ϕ_{2} = 0$ in equations (23a) and (23b). Similarly, localized oscillations in the vertical direction are examined by setting $a_{1} = 0$ , and $ϕ_{1} = 0$ in equations (23c) and (23d). Notably, upon implementing these constraints either in equations (10a) and (10b) or (23a)–(23d) the internal damping coefficient ( $c_{i}$ ) and fluid induced tangential coefficient ( $r_{i}$ ) is eliminated from the reduced set of equations. This phenomenon occurs because the velocity-dependent terms in one direction appear in the governing equation of motion for the perpendicular direction. Consequently, the localized oscillatory behavior is decoupled from these effects, rendering it independent of the parameter internal damping coefficient ( $c_{i}$ ) and fluid induced tangential coefficient ( $r_{i}$ ).

The effects of various system parameters on localized oscillations are demonstrated through the frequency response plots as shown in Figures 5–9. In upcoming figures, the Limit Point (LP) denotes a bifurcation where the nature of the system’s response transitions between stable and unstable regions. Figure 5 illustrates the frequency responses curves for various values of the $f$ (eccentricity ratio), in the absence of nonlinear effects ( $α = γ = 0$ ). The peak amplitude increases significantly with increasing eccentricity ratio. For lower values, such as $f$ equals 0.008, a single-valued response indicates linear JRS behavior. However, as $f$ increases ( $f = 0.045$ ), the resonance peak becomes sharper and more pronounced. The amplitude response in the horizontal direction is notably more sensitive to changes in $f$ compared to the vertical direction, which can be attributed to the lower damping ratio assumed in the horizontal direction. This sensitivity may increase the chance to large-amplitude oscillations or instability in practical scenarios.

Figure 5.

Frequency response curves of the JRS for various values of the eccentricity ratio $f$ , with system parameters as listed in Table 1 and $α = γ = 0$ . (a) horizonatal oscillations (b) vertical oscillations.

Figure 6.

Figure 7.

Frequency response curves of the JRS are presented for various values of $α$ , with parameters listed in Table 1 and $γ = c_{i} = r_{i} = 0$ . (a) horizonatal oscillations (b) vertical oscillations.

Figure 8.

Frequency response curves of the JRS for varying values of $μ_{y}$ , with system parameters as listed in Table 1: (a) $λ = 0$ and (b) $λ \neq 0$ .

Figure 9.

Frequency response curves of the JRS for varying values of $γ$ , with system parameters as listed in Table 1. (a) horizonatal oscillations (b) vertical oscillations.

Figure 6 illustrates the amplitude-frequency responses of JRS for various values of the $f$ , under the condition of a nonzero nonlinear stiffness parameter $α$ not equal to zero, with system parameters adopted from Table 1. It is evident from Figure 6 that the oscillation amplitude increases with increasing eccentricity ratio. For small values of $f$ ( $f = 0.008$ ), the system exhibits a unique, stable solution across the entire frequency range. However, multi valued solutions and jump phenomena are observed as f increases after a particular critical threshold. The emergence of multiple Limit Points (LP), signifies the transition between stable and unstable branches of the solution manifold. As $f$ increases, the system exhibits pronounced softening-type nonlinearity, leading to amplitude jumps and multivalued response curves.

The trajectories of the limit points (LP) corresponding to varying values of $f$ are represented by the curves $B_{1}$ and $B_{2}$ , as illustrated in Figure 6. The critical curve $B_{1}$ is often referred to as the backbone curve. The area enclosed between these critical curves defines the unstable zone, which contains all unstable steady-state solutions of the Jeffcott Rotor System (JRS) across different values of $f$ . Following the methodology outlined in Section 3.4, the critical eccentricity ratios for the onset of localized oscillations are determined as ${(f_{c})}_{h} = 0.01221$ for the horizontal direction and ${(f_{c})}_{v} = 0.03121$ for the vertical direction. For small amplitude oscillations, the nonlinear restoring force components are negligible due to higher-order terms involving the stiffness ratio $α$ . However, when the eccentricity ratio exceeds the critical value ${(f_{c})}_{h}$ , the resulting large-amplitude horizontal oscillations significantly enhance the nonlinear contribution to the restoring force. This leads to a rightward bending of the frequency response curve, characteristic of a spring-hardening effect. As $f$ increases further, this bending becomes more pronounced along the horizontal direction. This enhanced nonlinearity is attributed to the relatively lower damping coefficient in the horizontal direction ( $μ_{1} = 0.0154$ ) compared to vertical direction ( $μ_{1} = 0.0247$ ).

Figure 7 illustrates the effect of the stiffness ratio $α$ on system dynamics in the absence of other nonlinear parameters ( $γ = c_{i} = r_{i} = 0$ ). As inferred from the governing equations (equations (10a) and (10b)), the system exhibits linear behavior when both $α$ and $γ$ are zero, resulting in a single stable steady-state response across the frequency range, with a clear resonance peak. When nonlinear stiffness components are introduced ( $α = 0.02$ , $γ = 0.05$ and $α = γ = 0.05$ ), the system demonstrates nonlinear phenomena such as multivalued responses and jump phenomenon. These effects are especially prominent in the horizontal direction, where the amplitude–frequency curves show a distinct rightward bending due to the lower horizontal damping coefficient ( $μ_{x} = 0.0154$ ) compared to the vertical direction ( $μ_{y} = 0.0247$ ). Additionally, the gap between the stable and unstable solution branches narrows near the upper limit points, making it difficult to maintain stable operation in these regions. Even small perturbations can force the system to jump to a lower amplitude branch, potentially leading to undesirable dynamic transitions. Another important observation from Figure 7 is that increasing the stiffness ratio $α$ leads to both an upward shift in the resonant frequency and an amplification of the peak vibration amplitude. It is worth noting that the stiffness ratio $α$ is intrinsically linked to the material properties and geometry of the shaft and operating conditions.

Furthermore, each bent frequency response curve in Figure 7 is observed to contain two LPs: one near the peak amplitude (denoted as P₁) and the other at a lower amplitude (denoted as P₂). The segment of the response curve between these two LPs corresponds to unstable steady-state solutions. The unstable solutions between the two limit points, LP1 and LP2, give rise to a characteristic jump phenomenon in the system response. The direction and nature of the jumps are indicated by arrows in Figure 7. To illustrate this behaviour, the frequency response curve for an eccentricity ratio $f = 0.045$ and nonlinear stiffness ratio $α = 0.05$ . As the excitation frequency is gradually increased from a lower level, the system response tracks the upper stable branch of solutions until it approaches the limit point LP₂. Beyond this point, any further increase in excitation frequency results in a sudden transition to point P₁ on the lower branch, accompanied by an abrupt drop in oscillation amplitude. Conversely, when the excitation frequency is reduced from this state, the system response gradually traces the lower branch until the lower limit point LP₁ is encountered. At this stage, a further decrease in excitation frequency causes the solution to abruptly jump back to point P₁ on the upper branch, thereby completing a hysteresis loop defined by the sequence LP₂– P₁– LP₁–P₂– LP₂. This loop indicates a strong nonlinear response, wherein the amplitude undergoes sudden and significant changes. Such jump phenomena are highly undesirable in practical systems, as they are often associated with large, severe oscillations that can lead to instability or catastrophic failure. Therefore, mitigating these jumps is critical for ensuring the reliable operation of rotor-bearing systems.

Figure 8 presents the frequency response curves of the JRS for various values of the dimensionless mass unbalance parameter $μ_{y}$ , illustrating the dynamic characteristics in both horizontal and vertical directions. In Figure 8(a), the system exhibits a symmetric resonance curve for horizontal oscillations with $λ = 0$ , indicating linear behavior and the absence of nonlinear restoring forces. The peak amplitude remains largely invariant across different $μ_{y}$ values, reflecting a negligible effect of mass unbalance in the horizontal direction under linear stiffness conditions. Conversely, Figure 8(b) reveals a significantly nonlinear response in the vertical direction with $λ \neq 0$ , where the effects of increasing $μ_{y}$ are more pronounced. As $μ_{y}$ increases, the amplitude frequency curves show marked shifts, leading to the emergence of multiple limit points (LP), signifying regions of bistability and jump phenomena. For lower values of $μ_{y}$ , the response curve contains a sharp bend followed by a sudden drop, defined by two LPs. This characteristic behaviour indicates the presence of a nonlinear hardening effect, which becomes more dominant as $μ_{y}$ increases. The coexistence of stable and unstable branches, highlights the systems sensitivity to initial conditions and excitation amplitude, emphasizing the need for careful parameter selection in rotor-bearing system design.

The effect of the fluid radial stiffness ratio $γ$ on the amplitude frequency response of the JRS, showing both horizontal and vertical oscillations, is presented in Figure 9. As observed in Figure 9(a), the nonlinear response in the horizontal direction becomes increasingly complex with decreasing $γ$ . For $γ = 0.025$ , multiple limit points (LP) emerge, indicating the existence of bistable regions where the system can exhibit two coexisting stable solutions. This nonlinear behaviour is further highlighted by the sharp fold in the response curve, resulting in jump phenomena. The zoomed-in inset clearly shows how the stable and unstable branches with variation in $γ$ , underlining the sensitivity of the system to fluid radial stiffness ratio. In contrast, Figure 9(b) shows that the vertical oscillations are relatively less sensitive to changes in fluid radial stiffness ratio $(γ)$ , with only slight variations in peak amplitude and resonance frequency. However, the inset magnifies the subtle differences near the peak, confirming the presence of $γ$ modifications in the response curve. As $γ$ increases, the system transitions toward a more linear and single-valued response, suppressing the bistable behaviour and eliminating the limit points.

Nonlocalized oscillations

The horizontal and vertical motions associated with nonlocalized oscillations are inherently coupled. To systematically assess the influence of various system parameters on these oscillations, the following subsection is organized into several focused sub-sections.

Effect of eccentricity ratio f

Figure 10(a) and (b) present the frequency response curves for various $f$ , in the absence of the radial fluid damping coefficient ( $γ$ ), internal damping coefficient ( $c_{i}$ ) and fluid induced tangential coefficient ( $r_{i}$ ) effects. For a lower value of $f$ ( $f = 0.0015$ ), the system exhibits a single stable solution for horizontal and vertical oscillations. However, a more complex nonlinear behaviour is observed as $f$ increases. For $f = 0.02$ , the jump phenomenon exhibits in both directions, it occurs at frequencies significantly lower than the resonance. Additionally, the magnitude of the jump is significantly smaller in the vertical direction.

Figure 10.

Frequency responses of the JRS are presented for various values of $f$ , with system parameters specified in Table 1 and $γ = c_{i} = r_{i} = 0$ . Subfigures show: (a) horizontal oscillations, (b) vertical oscillations, and (c) and (d) phase plane plots in the horizontal and vertical directions under different initial conditions.

For a larger eccentricity ratio (

f = 0.045

), the system displays multi-valued solutions, multiple loops and multiple jump phenomena, as evident in both Figure 10(a) and (b). The frequency response curves exhibit self-intersections, forming closed loops that correspond to seven distinct steady-state solutions at (

σ_{1} = 0.095

). The stability of solutions is assessed using the Routh–Hurwitz criteria (equation (30)), as detailed in Section 3.3. According to the criterion, all four coefficients

{R H}_{1}

through

{R H}_{4}

must be positive for a solution to be stable. Table 2 summarizes the stability verification for the parameters

c_{i} = r_{i} = 0

, and

σ_{1} = 0.095

. The results show that three of the seven steady-state solutions are stable while the remaining four are unstable.

Table 2.

Validation of the Routh–Hurwitz stability criterion with parameters from Table 1, where $c_{i} = r_{i} = 0$ , and $σ_{1} = 0.095$ .

SL. no	Steady-state (stst) solutions				Criterion				Stability of solution
SL. no	$a_{10}$	$ϕ_{10}$	$a_{20}$	$ϕ_{20}$	${R H}_{1}$	${R H}_{2}$	${R H}_{3}$	${R H}_{4}$
1	0.295	103.59	0.554	21.391	>0	>0	>0	>0	Stable
2	0.716	103.62	0.682	21.613	>0	>0	>0	<0	Unstable
3	1.132	103.15	1.480	21.430	>0	>0	>0	>0	Stable
4	1.882	103.15	1.532	22.093	>0	>0	>0	<0	Unstable
5	2.156	102.96	1.970	22.659	>0	>0	>0	>0	Unstable
6	2.177	102.54	1.970	22.852	>0	>0	<0	>0	Unstable
7	2.288	101.43	2.090	23.295	>0	>0	>0	>0	Stable

To further validate these findings, phase-plane diagrams obtained from direct numerical simulations of the full governing equations (equations (10a) and (10b)) are shown in Figure 10(c) and (d). As expected, only the three stable solutions manifest in the phase portraits, consistent with the Routh–Hurwitz analysis. The corresponding sets of initial conditions leading to these distinct stable solutions are provided in Table 3.

Table 3.

Initial values.

$IC$	0.15, 0.2, 0.25, 0.3	1.5, 1.3, 1.3, 1.2	3.15, 0.2, 3.25, 0.3
$stst$	$u_{s t s t 1}, v_{s t s t 1}$	$u_{s t s t 2}, v_{s t s t 2}$	$u_{s t s t 3}, v_{s t s t 3}$

The limit points (LP) identified in Figure 10 correspond to the turning points of the frequency response curves, where stable and unstable solution branches meet. At these points, the slope of the frequency response curve becomes vertical to the $σ_{1}$ axis, signifying the occurrence of a fold bifurcation. In the diagrams, stable steady-state responses of the JRS are depicted with solid lines, while unstable responses are shown with dashed lines. The markers labelled ‘H’ indicate bifurcations to neutral saddle-type solutions, which are inherently unstable. The segments of the solution curves connecting these ‘H’ points are confirmed to be unstable, as reported in earlier studies.^{10,23,27,29,30} The stability of the steady-state solutions is further corroborated through numerical phase-plane analysis and by applying the Routh–Hurwitz stability criterion, as detailed in Section 3.3.

A significant observation from Section 4.2 about localized oscillations is that the resonance peak amplitude remains consistently smaller in the vertical direction than in the horizontal direction. This behavior is primarily due to the higher damping coefficient applied vertically ( $μ_{y} = 0.0247$ ) compared to the horizontal damping coefficient ( $μ_{x} = 0.0154$ ). This trend is also evident in non-localized oscillations for lower eccentricity ratios ( $f = 0.0015$ and $f = 0.02$ ), as illustrated in Figure 10(a) and (b). However, for $f = 0.045$ , this trend reverses the peak amplitude in the vertical direction exceeds that of the horizontal direction. This reversal is due to the stronger dynamic coupling along the vertical axis induced by the higher vibration amplitudes associated with the large eccentricity. In contrast, the oscillation amplitudes are insufficient for the smaller eccentricity values to activate significant coupling in the vertical direction. It is important to note that, the governing equations along the horizontal and vertical directions are effectively decoupled in the case of localized oscillations.

Figure 11 presents the frequency response curves for values eccentricity ratios $f$ , in the presence of radial fluid damping coefficient ( $γ$ ), internal damping coefficient ( $c_{i}$ ) and fluid induced tangential coefficient ( $r_{i}$ ) effects. For $f = 0.045$ , a noticeable reduction in the number of solution branches and loops is observed in Figure 11 when compared to the more detailed amplitude frequency structure as shown in Figure 10. This simplification can be attributed to the combined influence of system-specific parameter values. However, despite this simplification multiple jump phenomena are observed.

Figure 11.

Frequency response curves of the JRS are presented for various values of f using the parameters listed in Table 1. (a) horizonatal oscillations (b) vertical oscillations.

The direction and nature of the jumps are indicated by arrows in Figure 11. To illustrate this behavior, the frequency response curve for an eccentricity ratio $f = 0.045$ . An increase in excitation frequency from lower levels leads the system to follow the upper stable solution path until it reaches the limit point LP₁. Beyond this point, any further increase in excitation frequency results in a sudden transition to point P₁ on the lower branch, accompanied by an abrupt drop in oscillation amplitude. Conversely, when the excitation frequency is decreased from this state, the system response follows the lower branch until it reaches the second limit point (LP₂). Beyond this point, any further reduction in frequency leads to a sudden transition back to point P₂ on the upper branch, completing a hysteresis loop characterized by the sequence LP₁– P₁– LP₂–P₂– LP₁. This loop indicates a strong nonlinear response, wherein the amplitude undergoes sudden and significant changes. Such highly undesirable jump phenomena in practical systems may lead to instability or catastrophic failure.

Influence of stiffness ratio $α$

The effect of the stiffness ratio $α$ on the dynamic behaviour of the JRS in horizontal and vertical direction is illustrated in Figure 12. Initially, the analysis is conducted in the absence of stiffness ratio ( $α = 0$ ), for a clearer interpretation of the influence of nonlinear shaft stiffness. Under these conditions, the governing equations of motion (equations (10a) and (10b)) reduce to a linear system. The corresponding frequency response curves presented in Figure 12 clearly reflect this linear behaviour, which display a single and stable solution across the entire frequency range. As the stiffness ratio $α$ increased ( $α = 0.02$ ), the system transitions into a nonlinear region, where characteristic features of nonlinear dynamics become prominent. Specifically, the amplitude frequency responses exhibit multi-valued solutions, multiple limit points, and closed loops, revealing multi-jump phenomena. Moreover, increasing $α (α = 0.02)$ , results in a noticeable resonant frequency shift towards higher values, reflecting an effective system stiffening. This is further supported by the rightward bending of the frequency response curves, a classical signature of the spring hardening effect in nonlinear oscillatory systems.

Figure 12.

Frequency response curves of the JRS are presented for various values of $α$ (Table 1). (a) horizonatal oscillations (b) vertical oscillations.

Influence of fluid radial stiffness ratio $γ$

Figure 13 presents the amplitude frequency responses of the JRS in horizontal and vertical motions for different values of the fluid radial stiffness ratio $γ$ . When $γ = 0$ , both horizontal and vertical responses exhibit nonlinear characteristics including multiple limit points (LPs) and Hopf bifurcations (H). In particular, the system shows multi-jump phenomena, where sudden transitions between branches of stable and unstable solutions occur. As $γ$ is increased to 0.025, the nonlinear interactions between the horizontal and vertical modes become more evident. The presence of $γ$ alters the topology of the frequency response curves, with previously isolated branches now connecting through additional LPs and H points. Notably, the closed-loop behaviour becomes more confined, especially in the vertical direction.

Figure 13.

Frequency response curves of the JRS are presented for various values of $γ$ (Table 1). (a) horizonatal oscillations (b) vertical oscillations.

At a higher coupling value of $γ = 0.05$ , a marked transformation in the system dynamic behavior is observed. Although multi-valued solutions are still present, the number of branches is reduced, suggesting a weakening of the amplitude frequency response structure due to the influencing parameter. While limit points and Hopf bifurcations remain, the corresponding loops are reshaped, suggesting a redistribution of instability regions due to an increase in $γ$ value. Interestingly, the nonlinear resonance peaks shift toward higher frequencies in both x and y directions, indicating a spring hardening effect.

Influence of internal damping coefficient $c_{i}$

Figure 14 illustrates the amplitude frequency response of the JRS in both directions, respectively for different values of the internal damping coefficient $c_{i}$ . When $c_{i} = 0$ , system exhibits strong nonlinear behaviour, including multiple steady-state solutions, Hopf bifurcations (H), and limit points (LP). These features lead to multi-jump phenomena, where sudden transitions between solution branches occur as the detuning parameter $σ_{1}$ is varied. As a small amount of damping is introduced ( $c_{i} = 0.065$ ), a significant reduction in loops and unstable branches is observed. The multi-jump behaviour becomes less noticeable, though multivalued responses still continue. The nonlinear features are almost eliminated for higher damping values ( $c_{i} = 0.075$ ). The amplitude frequency curves reveal a single and stable solution across the entire range of $σ_{1}$ , for both horizontal and vertical motions. The disappearance of multi-valued branches indicates that the system response becomes effectively linearized due to the internal damping coefficient.

Figure 14.

Amplitude frequency response curves of the Jeffcott Rotor System (JRS) are presented for various values of $c_{i}$ (Table 1). (a) horizonatal oscillations (b) vertical oscillations.

Influence of damping coefficient ratio $r_{i}$

Figure 15 presents the amplitude frequency response curves of the JRS for different values of the external damping parameter $μ_{y}$ , which represents the viscous damping in the vertical direction. Figure 15(a) shows that the value of $μ_{y}$ strongly influences the horizontal amplitude response. As the damping coefficient increases from $μ_{y} = 0.0154$ to $μ_{y} = 0.0355$ , the peak amplitude of oscillation decreases significantly. The sharp resonance peak observed for low damping with increasing $μ_{y}$ , indicates the classical damping effect that dissipates energy and suppresses resonance amplification. Additionally, the response curves are linear and single valued across the entire frequency range, implying the absence of nonlinearities ( $γ = α = 0$ ). In Figure 15(b), the vertical amplitude response shows a similar trend. Although the variations in amplitude are less pronounced than in the horizontal direction, a consistent reduction in peak amplitude is observed with increasing damping. Thus, $μ_{y}$ serves as a stabilizing parameter that reduces oscillatory magnitudes and mitigates the risk of instabilities.

Figure 15.

Amplitude frequency response curves of the Jeffcott Rotor System (JRS) are presented for various values of $μ_{y}$ (Table 1). (a) horizonatal oscillations (b) vertical oscillations.

Influence of fluid induced tangential coefficient $r_{i}$

Figure 16 illustrates the frequency responses curves of the JRS for various values of the $r_{i}$ in horizontal and vertical oscillations, respectively. When $r_{i} = 0$ , the response curves in both directions resemble those for $c_{i} = 0$ , exhibiting pronounced nonlinear characteristics, including multiple limit points (LP), Hopf bifurcations (H), and the occurrence of multi-jump phenomena. These features indicate strong coupling between horizontal and vertical oscillations and nonlinear resonance effects. As $r_{i}$ is increased to 0.065, a notable reduction in complexity of amplitude frequency response is observed, similar to the behaviour seen with increased $c_{i}$ . While some multivalued responses and Hopf bifurcations still exist, the loops shrink and the number of unstable branches is reduced. For a slightly higher value $r_{i} = 0.075$ , the system undergoes further stabilization, the multivalued loops nearly vanish and the response approaches a single-valued stable solution.

Figure 16.

Amplitude frequency response curves of the Jeffcott Rotor System (JRS) are presented for various values of $r_{i}$ (Table 1). (a) horizonatal oscillations (b) vertical oscillations.

Conclusion

This study investigates the influence of key nonlinear dynamic parameters on the vibration behavior of a horizontally supported Jeffcott Rotor System (JRS). The parameters considered include the nonlinear restoring force arising from large shaft deformations (Stiffness ratio, $α$ ), eccentricity ratio ( $f$ ), fluid induced tangential force (Coefficient, $r_{i}$ ), fluid radial stiffness coefficient ( $γ$ ) and internal damping (Coefficient, $c_{i}$ ). In the Method of Multiple Scales (MMS), autonomous amplitude phase modulation equations are analytically derived in both directions. The results of analytical solutions (MMS) are validated through numerical integration of the governing equations of motion. The linear stability of steady-state responses is assessed via perturbation analysis of the MMS solutions. Both localized and nonlocalized oscillatory modes are explored in detail. Furthermore, numerical bifurcation analysis is performed using “Matcont”, a MATLAB-based continuation and bifurcation toolbox, to extract amplitude frequency response curves. The effects of various system parameters on the vibration behavior of the JRS are examined using these response curves, with the key findings summarized below.

(i) An increase in the eccentricity ratio $f$ leads to higher centrifugal forces, resulting in amplified resonance peaks, multi-valued solutions, and jump phenomena in the frequency response. Such abrupt amplitude jumps can cause a risk of sudden system failure and must be carefully controlled. The nonlinear restoring force, governed by the stiffness ratio $λ$ , contributes to spring hardening behavior and associated jump phenomena. The stiffness ratio can be modulated through appropriate material selection and thermal management, as elevated temperatures during continuous operation may alter shaft stiffness and amplify nonlinear effects.

(ii) The jump phenomenon can be mitigated through external damping, characterized by damping coefficients $μ_{x}$ (horizontal) and $μ_{y}$ (vertical). For localized oscillations (where the horizontal and vertical motions are uncoupled), an increased damping along the vertical direction ( $μ_{y}$ > $μ_{x}$ ) leads to a notable reduction in peak amplitude and effective mitigation of the jump phenomenon. The critical parameter values triggering the onset of jump phenomena are determined analytically. In the case of nonlocalized oscillations, coupling effects become more prominent, especially in the vertical direction, resulting in lower peak amplitudes than in the horizontal direction at larger values of $f$ . However, for small-amplitude oscillations, the coupling remains weak.

(iii) The jump phenomenon can also be effectively mitigated by enhancing the coefficient associated with fluid-induced tangential forces ( $r_{i}$ ) and internal damping ( $c_{i}$ ) through the use of high-viscosity lubricants. While this enhances the damping effect and suppresses oscillation amplitude and jump behavior, it may also elevate bearing temperatures, necessitating improved thermal management. The fluid induced tangential force and internal damping model employed in this study exhibits strong damping characteristics advantageous to stability. Further work will explore different modelling frameworks to extend the present findings.

(iv) The multi-valued solutions and associated loops in the frequency response, which become prominent at higher eccentricity ratios $f$ , can be effectively mitigated by reducing the fluid radial stiffness ratio. A lower fluid stiffness weakens the system’s nonlinear response, smoothing the system characteristics and minimizing the occurrence of jump phenomena and dynamic instabilities. This control strategy highlights the importance of fluid structure interaction parameters in tuning the system’s vibratory behavior. Adjusting the radial stiffness through appropriate lubricant selection provides a practical approach to enhancing rotor-bearing systems' operational stability, especially under high-speed and high-eccentricity conditions.

Footnotes

Acknowledgments

The authors would like to thank the anonymous reviewers for their careful reading, constructive comments, and valuable suggestions, which improved the quality and clarity of this manuscript.

CRediT authorship contribution statement

Amit Malgol: Writing – original draft, Visualization, Validation, Software, Methodology, Investigation, Formal analysis.

Javed Sikandar Shaikh: Writing – original draft, Methodology, Investigation, Formal analysis, Data curation, Conceptualization.

Vinayak S Uppin: Writing – review & editing, Formal analysis, Data curation, Project administration.

Aqleem Siddiqui: Writing – review & editing, Supervision, Validation.

Vishal G Salunkhe: Writing – review & editing, Visualization, Software.

Tejaswi Ram: Writing – original draft, Formal analysis, Data curation.

Surya Marimuthu: Visualization, Validation, Project administration, Resources, Methodology.

Uday Aswalekar: Writing – review & editing, Supervision, Validation, Software.

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The authors received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Amit Malgol

Data Availability Statement

No data was used for the research described in the article.*

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Influence of fluid induced tangential forces,nonlinear stiffness,and internal damping on the vibration response of a Jeffcott Rotor System

Abstract

Keywords

Introduction

Mathematical modeling

Analytical studies

Method of multiple scales

Steady-state oscillations

Linear stability analysis (LSA)

Critical values for the initiation of limit points for localized oscillations

Results and discussions

Numerical and analytical results comparison

Localized oscillations

Nonlocalized oscillations

Effect of eccentricity ratio f

Influence of stiffness ratio α

Influence of fluid radial stiffness ratio γ

Influence of internal damping coefficient c i

Influence of damping coefficient ratio r i

Influence of fluid induced tangential coefficient r i

Conclusion

Footnotes

Acknowledgments

CRediT authorship contribution statement

Declaration of conflicting interests

Funding

ORCID iD

Data Availability Statement

Appendix

References

Influence of stiffness ratio $α$

Influence of fluid radial stiffness ratio $γ$

Influence of internal damping coefficient $c_{i}$

Influence of damping coefficient ratio $r_{i}$

Influence of fluid induced tangential coefficient $r_{i}$