Effect of high-frequency excitation on friction induced vibration caused by the combined action of velocity-weakening and mode-coupling

Abstract

The present article studies the effects of both tangential and normal high-frequency excitations on a two-degree-of-freedom moving-mass-on-belt which represents a minimal model incorporating both velocity-weakening instability (so-called Stribeck effect) and mode-coupling instability (so-called binary flutter). The method of direct partition of motion is employed for studying the characteristics of the system in slow time scale. Linear stability analysis is performed near the equilibrium point of the system for both with and without sinusoidal high-frequency excitation. It is observed that the instability can be suppressed by the tangential high-frequency excitation only for a specific range of strength of excitation. However, stability does not improve under normal high-frequency excitations, though amplitude of the self-excited oscillation can be controlled to some extent. Direct numerical simulations are carried out in MATLAB SIMULINK to validate the analytical predictions.

Keywords

Stribeck instability mode-coupling instability method of direct partition of motion tangential high-frequency excitation normal high-frequency excitation

1. Introduction

Friction-induced oscillation can be harmful in mechanical systems like brakes, squealing sounds from rail wheel-track interaction, lead screws, chattering in machine tools, rattling of robotic joints, bearings, stick-slip effects during oil-well drilling and many more if not suppressed properly. Extensive studies on friction-induced vibration (FIV) have been carried out for many years. The fundamental mechanisms of friction driven vibration are explained by Ibrahim (1994a). Generation of friction due to mechanical interaction and its related behavioral dynamics like stick-slip, chatter and squeals is well furnished by Ibrahim (1994b).

Four different types of self-excited oscillations occur in the friction driven mechanisms as elaborated by Hoffmann and Gaul (2008) with great details. One of the major causes of the self-sustained oscillation in friction driven mechanisms is stick-slip phenomena which is ascribed to the velocity-weakening characteristics of kinetic friction (often called as the Stribeck effect). Its geneses with a historical point of view and vast illustrations are given by Feeny et al. (1998). Denny (2004) has described different conditions for the onset of stick-slip and pure slip motion. One of the inceptive conditions for stick-slip motion is a negative rate of decrement of friction with the rise in relative velocity between the belt and oscillator surface. Difference between static and kinetic friction gives rise to a stick-slip action in a basic friction driven oscillator governed by the nonlinear friction function. Different bifurcation scenarios of stick-slip motions in one-degree-of-freedom (1-DOF) systems (Elmer, 1997; Hinrichs et al., 1998) and two-degree-of-freedom (2-DOF) systems (Awrejcewicz and Delfs, 1990) have been discussed.

Another type of self-excited FIV is mode-coupling instability, often called as the binary flutter. The condition of onset of mode-coupling instability is discussed by Hoffmann et al. (2002). Effects of structural damping and other system parameters on mode-coupling instability have been studied by Hoffmann and Gaul (2003) and Sinou and Jézéquel (2007). Mode-coupling effects may initiate squeal which is different from chatter as it generates high-pitched noise on the engagement of brakes. Various minimal models of disc brake squeal have been studied by Shin et al. (2002) and Von Wagner et al. (2007), along with linearized stability analysis.

Many researchers have discussed various methods of reducing FIV. Use of dynamic vibration absorbers for minimizing FIV is studied in Rudolph and Popp (2003) and Popp and Rudolph (2004). Zinjade and Mallik (2007) studied the effect of the impact damper in suppressing friction-induced oscillation. Nakano and Maegawa (2009) have concluded that a soft material can be used at the contact surface which is less stiff than the supporting spring to prevent stick-slip motion. Nonlinear contact stiffness can be attached at the contact interface to eliminate binary flutter (Li et al., 2016).

Many attempts have been made to employ active feedback control to reduce FIV. Heckl and Abrahams (1996) proposed an active feedback control technique to stabilize the friction-induced instability. Chatterjee (2007a) has derived a control law using Lyapunov’s second method to modulate normal load for minimizing friction driven oscillations. Nath and Chatterjee (2016, 2018) used a positive acceleration feedback controller for modulating tangential and normal force to minimize the Stribeck instabilities. Nechak (2019) adopted the nonlinear control scheme based on the feedback linearization approach to suppress the mode-coupling instabilities. Applications of time-delayed feedback control have been discussed recently (Chatterjee, 2007b; Chatterjee and Mahata, 2009a, 2009b; Das and Mallik, 2006). Adaptive fuzzy, neuro-fuzzy and data mining technique based compensation methods are proposed by Wang et al. (2011, 2012, 2013) to suppress friction-induced instabilities.

Application of high-frequency excitation (HFE) (often called as ‘dither’) to the mechanical systems for quenching friction-induced oscillation is discussed by Chatterjee et al. (2004) and Thomsen (1999). They show that the Stribeck instabilities can be completely removed by increasing the strength of excitation (product of dither frequency and dither amplitude). Hoffmann et al. (2005) have shown that mode-coupling instabilities can also be successfully controlled by HFE. In the recent works (Kapelke et al., 2017; Kapelke and Seemann, 2018), effects of longitudinal high-frequency (HF) on contact compliance exerted by different dynamic friction models have been studied theoretically and experimentally. Wiper squeal and brake squeal can be suppressed using HFE, as studied in Stallaert et al. (2006) and Cunefare and Graf (2002), respectively.

In the present study, a 2-DOF mass-on-moving-belt model of FIV (as considered by Hoffmann et al. (2005)) is considered. The model is different from that of Hoffmann et al. (2005) in respect of the friction characteristics. Hoffmann et al. (2005) considered only the velocity independent model of friction; thus, their model can explain only the mode-coupling instability. However, in the present model, the velocity dependent model of friction is considered; thus, the present system can model the combined effects of Stribeck and mode-coupling instability. The major objective of the article is to study the effects of tangential HFE and normal HFE individually on such a system where the influences of both the Stribeck and mode-coupling effects are present.

2. Effect of tangential HFE on FIV

2.1. Mathematical model

The system considered in the paper is depicted in Figure 1 (Hoffmann et al., 2005). A spring-loaded mass M is slipping on a moving belt. The mass has 2-DOF, one in the tangential direction and the other in the normal direction of slipping. The motion of the vibratory system is described by 2-DOF, p₁ and p₂. The mass is supported by two linear dampers having damping coefficients c₁ and c₂ and two linear springs having stiffness k₁ and k. k is the stiffness of the supporting spring, connected to the mass at an angle of θ = 45^°. The role of the oblique spring is that it introduces asymmetry in the system’s stiffness matrix as required to model mode-coupling instability. k₂ represents the contact stiffness between the mass and the belt. A constant normal force N is applied on the mass; simultaneously, an extra HFE of frequency ω_f is also applied on the mass in the direction of slipping.

The non-dimensional forms of equations of motion are written as

\begin{array}{l} \frac{d^{2} p_{1}}{d t^{2}} + 2 ξ_{1} Ω_{1} \frac{d p_{1}}{d t} + Ω_{1}^{2} p_{1} + {1 - μ (Ω_{2}^{2} - 1)} p_{2} = q Ω_{f}^{2} \sin (Ω_{f} t), \\ \frac{d^{2} p_{2}}{d t^{2}} + 2 ξ_{2} Ω_{2} \frac{d p_{2}}{d t} + Ω_{2}^{2} p_{2} + p_{1} = 1 \end{array}

(1)

In the above non-autonomous equations, q is the non-dimensional amplitude of the HFE and Ω_f refers to the non-dimensional dither frequency, which is very large compared to the natural frequencies of the system. The other non-dimensional parameters involved in the above equations are defined as

ω_{0}^{2} = \frac{k}{2 M}, p_{0} = \frac{N}{M ω_{0}^{2}}, Ω_{i}^{2} = \frac{k_{i} + (k / 2)}{M ω_{0}^{2}}, Ω_{f} = \frac{ω_{f}}{ω_{0}}

ξ_{i} = \frac{c_{i}}{2 \sqrt{M (k_{i} + (k / 2))}}, p_{i} = \frac{p_{i}}{p_{0}}, q = \frac{a}{M p_{0}}

where i = 1, 2. In the above non-dimensional parameters, ω₀ is the reference frequency and p₀ is the characteristic length. The non-dimensional time is denoted as t = ω₀t^∗. μ is the coefficient of friction and is a nonlinear function of relative velocity. A nonlinear polynomial friction function (Thomsen, 1999) is considered here as

μ (v_{r}) = c_{0} sgn (v_{r}) + c_{1} v_{r} + c_{3} v_{r}^{3}

(2)

where the non-dimensional relative velocity of slipping is

v_{r} = v_{0} - (d p_{1} / d t)

and the non-dimensional belt velocity is written as

v_{0} = v / ω_{0} p_{0}

c_{0} = μ_{s}, c_{1} = - (3 / 2) ((μ_{s} - μ_{m}) / v_{m}) and c_{3} = (1 / 2) ((μ_{s} - μ_{m}) / v_{m}^{3})

, in which μ_s is the coefficient of static friction and μ_m is the minimum coefficient of kinetic friction. The corresponding belt velocity for the minimum coefficient of kinetic friction is denoted by v_m, and μ(v_m) = μ_m. For the sake of computational simplicity, the signum function may be regularized by a continuous and smooth hyperbolic tangent function; thus, equation (2) reads as

μ (v_{r}) = c_{0} \tanh (β v_{r}) + c_{1} v_{r} + c_{3} v_{r}^{3}

(3)

where the parameter β controls the shape of the hyperbolic tangent function; here, β is a very large quantity: β ≫ 1. For a considerable higher value β = 1000, the hyperbolic tangent function is analogous to the signum function.

2.2. Analysis using the method of direct partition of motion

According to the standard formalism of the method of direct partition of motion (MDPM) (Blekhman, 2000), the system response under HFE can be split into slow and fast components, where the average of the fast response tends out to be a trivial quantity. After averaging out the fast component of the response, one finally obtains a set of autonomous equations that correctly predict the slow response of the system.

To this end, equation (1) can be rewritten in the matrix form as follows

[\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}] (\begin{matrix} \frac{d^{2} p_{1}}{d t^{2}} \\ \frac{d^{2} p_{2}}{d t^{2}} \end{matrix}) + [\begin{matrix} 2 ξ_{1} Ω_{1} & 0 \\ 0 & 2 ξ_{2} Ω_{2} \end{matrix}] (\begin{matrix} \frac{d p_{1}}{d t} \\ \frac{d p_{2}}{d t} \end{matrix}) + [\begin{matrix} Ω_{1}^{2} & 1 - μ (Ω_{2}^{2} - 1) \\ 1 & Ω_{2}^{2} \end{matrix}] (\begin{matrix} p_{1} \\ p_{2} \end{matrix}) = (\begin{matrix} q Ω_{f}^{2} \sin (Ω_{f} t) \\ 1 \end{matrix})

(4)

Equation (4) can be recast in the standard form (Thomsen, 2003)

M (p, t) \frac{d^{2} p}{d t^{2}} + S (p, \frac{d p}{d t}, t) + \sum_{j = 1}^{m} (H_{j} (p, \frac{d p}{d t}, t) + Ω_{f} F_{j} (p, t)) \frac{\partial^{2}}{\partial {(Ω_{f} t)}^{2}} Θ_{j} (t, Ω_{f} t) = 0

(5)

where

Ω_{f}

= excitation frequency,

Ω_{f} ≫ Ω_{1,2}

[p] = {[\begin{matrix} p_{1} & p_{2} \end{matrix}]}^{T}

describes the positional state.

[M] = [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}]

is an inertia matrix, here an identity matrix.

[S] = [S_{0}] + [S_{11}] [p] + [S_{12}] [d p / d t]

is the 2 × 1-matrix of slow forces with

[S_{0}] = [\begin{matrix} 0 \\ - 1 \end{matrix}]

[S_{11}] = [\begin{matrix} Ω_{1}^{2} & 1 - μ (Ω_{2}^{2} - 1) \\ 1 & Ω_{2}^{2} \end{matrix}]

and

[S_{12}] = [\begin{matrix} 2 ξ_{1} Ω_{1} & 0 \\ 0 & 2 ξ_{2} Ω_{2} \end{matrix}]

[H]

: 2 × 1-matrix, here a zero matrix,

[F] = [\begin{matrix} - q Ω_{f} \\ 0 \end{matrix}]

defines the fast forces and

Θ_{1} = - \sin Ω_{f} t

. m is the total number of HF forces acting on the system. For the present system, m = 1 (tangential HF force).

According to the formalism of the MDPM, the solution of equation (5) is split into two components, namely, the slow component vector ‘ $p_{s}$ ’ and the fast component vector ‘ $p_{f}$ ’. Thus, one writes

p (t) = p_{s} (t) + Ω_{f}^{- 1} p_{f} (t, Ω_{f} t)

(6)

where

Ω_{f}^{- 1}

is a small quantity,

Ω_{f}^{- 1} ≪ 1

(as Ω_f is large). The slow vector component ‘

p_{s}

’ represents the slow-motion operating on the slow natural time scale ‘t’ of the system, whereas the fast motion ‘

p_{f}

’ takes place on the fast time scale

Ω_{f} t = τ

. One can write

〈 p_{f} (t, τ) 〉 = \frac{1}{2 π} \int_{0}^{2 π} p_{f} (t, τ) d τ = 0

(7)

where ‘

〈 \cdot 〉

’ is the averaging operator.

Differentiating equation (6) with respect to slow time ‘t’ one obtains

\begin{array}{l} \frac{d p}{d t} = \frac{d p_{s}}{d t} + \frac{\partial p_{f}}{\partial τ} + Ω_{f}^{- 1} \frac{\partial p_{f}}{\partial t} \\ \frac{d^{2} p}{d t^{2}} = Ω_{f} \frac{\partial^{2} p_{f}}{\partial τ^{2}} + \frac{d^{2} p_{s}}{d t^{2}} + 2 \frac{\partial^{2} p_{f}}{\partial t \partial τ} + Ω_{f}^{- 1} \frac{\partial^{2} p_{f}}{\partial t^{2}} \end{array}

(8)

Spatial derivatives of [F] and [M] are zero as these are independent of position variables. Inserting equation (8) into (4) one obtains

\frac{\partial^{2} p_{f}}{\partial τ^{2}} = - M^{- 1} \sum F \frac{\partial^{2} Θ}{\partial τ^{2}} - Ω_{f}^{- 1} (\frac{d^{2} p_{s}}{d t^{2}} + 2 \frac{\partial^{2} p_{f}}{\partial t \partial τ}) - Ω_{f}^{- 1} M^{- 1} S + O (Ω_{f}^{- 2})

(9)

where

M = M (p_{s}, t), F = F (p_{s}, t), Θ = Θ (t, τ) and S = S (p_{s}, \frac{d p_{s}}{d t} + \frac{\partial p_{f}}{\partial τ}, t)

Integrating equation (9) twice with respect to fast time ‘τ’ one obtains

p_{f} (t, τ) = - M^{- 1} \sum F Θ + O (Ω_{f}^{- 1}) = [\begin{matrix} - q Ω_{f} \sin (Ω_{f} t) \\ 0 \end{matrix}] + O (Ω_{f}^{- 1})

(10)

To find out the slow equation, in equation (9), the fast components have to be averaged out. Using equation (7) and equation (10), one obtains

\frac{d^{2} p_{s}}{d t^{2}} + S (p_{s}, \frac{d p_{s}}{d t}, t) + V (p_{s}, \frac{d p_{s}}{d t}, t) = 0

(11)

The term $V = V (p_{s}, (d p_{s} / d t), t)$ is called the vibrational force (Thomsen, 2003). For the present system, the vibrational force consists of averaged slow forces, which is a nonlinear function of relative velocity (that is inherently present in the dry friction function defined in equation (2)). The generic expression of vibrational forces for the present model reads

V (p_{s}, \frac{d p_{s}}{d t}, t) = 〈 S (p_{s}, \frac{d p_{s}}{d t} + \frac{\partial {\hat{p}}_{f}}{\partial τ}, t) - S (p_{s}, \frac{d p_{s}}{d t}, t) 〉

(12)

where ‘

{\hat{P}}_{f}

’ is the zero order approximation of the fast component, and thus

{\hat{p}}_{f} = - [M] \sum [F] {Θ}

(13)

Following the above procedure, one obtains the autonomous equation describing the effect of HFE on the non-autonomous equation as

[\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}] (\begin{matrix} \frac{d^{2} p_{1 s}}{d t^{2}} \\ \frac{d^{2} p_{2 s}}{d t^{2}} \end{matrix}) + [\begin{matrix} 2 ξ_{1} Ω_{1} & 0 \\ 0 & 2 ξ_{2} Ω_{2} \end{matrix}] (\begin{matrix} \frac{d p_{1 s}}{d t} \\ \frac{d p_{2 s}}{d t} \end{matrix}) + [\begin{matrix} Ω_{1}^{2} & 1 - μ^{*} (Ω_{2}^{2} - 1) \\ 1 & Ω_{2}^{2} \end{matrix}] (\begin{matrix} p_{1 s} \\ p_{2 s} \end{matrix}) = (\begin{matrix} 0 \\ 1 \end{matrix})

(14)

The exclusive effect of HF on the friction-induced oscillation is underpinned by the effective friction function

μ^{*} = 〈 \begin{array}{l} c_{0} sgn (v_{0} - \frac{d p_{1 s}}{d t} + q Ω_{f} \cos (Ω_{f} t)) \\ + c_{1} (v_{0} - \frac{d p_{1 s}}{d t} + q Ω_{f} \cos (Ω_{f} t)) \\ + c_{3} {(v_{0} - \frac{d p_{1 s}}{d t} + q Ω_{f} \cos (Ω_{f} t))}^{3} \end{array} 〉

(15)

From equation (15), one obtains the expression of the effective friction function as

μ^{*} = μ + \frac{3}{2} c_{3} {(q Ω_{f})}^{2} v_{r} for | v_{r} | \geq q Ω_{f}

(16a)

μ^{*} = c_{0} (1 - \frac{2}{π} \cos^{- 1} (\frac{v_{r}}{q Ω_{f}})) + (c_{1} + \frac{3}{2} c_{3} {(q Ω_{f})}^{2}) v_{r} + c_{3} v_{r}^{3} for | v_{r} | < q Ω_{f}

(16b)

2.3. Stability analysis

In this section, local stability analysis is performed for both unexcited and excited systems to make the effects of dither distinguishable. As signum function is not differentiable at zero, a hyperbolic tangent function is being substituted for the signum function (which is easily differentiable at zero) as described in equation (3).

The static equilibrium of the system with tangential HFE is obtained from equation (14) as

\begin{array}{l} p_{1 s e} = 1 - Ω_{2}^{2} p_{2 s e}, \\ p_{2 s e} = \frac{Ω_{1}^{2}}{Ω_{1}^{2} Ω_{2}^{2} + (Ω_{2}^{2} - 1) (μ_{0}^{*}) - 1} \end{array}

(17)

The Jacobian of the system at the static solutions is given by

[J^{*}] = [\begin{matrix} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ - Ω_{1}^{2} & J_{32}^{*} & J_{33}^{*} & 0 \\ - 1 & - Ω_{2}^{2} & 0 & - 2 ξ_{2} Ω_{2} \end{matrix}]

(18)

For low strength of the excitation regime, i.e. $| v_{0} | \geq q Ω_{f}$

\begin{array}{l} J_{32}^{*} = (Ω_{2}^{2} - 1) μ_{0}^{*} - 1 \\ J_{33}^{*} = - 2 ξ_{1} Ω_{1} + {(μ_{0}^{*})}^{'} (Ω_{2}^{2} - 1) p_{2 s e} \end{array}

where

\begin{array}{l} μ^{*} (v_{0}) = μ_{0}^{*} = μ_{0} + \frac{3}{2} c_{3} {(q Ω_{f})}^{2} v_{0} \\ {(μ_{0}^{*})}^{'} = {\frac{d μ^{*} (v_{r})}{d v_{r}} |}_{v_{r} = v_{0}} = - c_{0} β \sec h^{2} (β v_{0}) - (c_{1} + \frac{3}{2} c_{3} {(q Ω_{f})}^{2}) - 3 c_{3} v_{0}^{2} \end{array}

(19)

For high strength of the excitation regime, i.e. $| v_{0} | < q Ω_{f}$ , the Jacobian matrix is modified due to change in both $μ_{0}^{*}$ and ${(μ_{0}^{*})}^{'}$ , so expressions for $μ_{0}^{*}$ and ${(μ_{0}^{*})}^{'}$ in such cases are written as

\begin{array}{l} μ^{*} (v_{0}) = μ_{0}^{*} = c_{0} (1 - \frac{2}{π} \cos^{- 1} (\frac{v_{0}}{q Ω_{f}})) + (c_{1} + \frac{3}{2} c_{3} {(q Ω_{f})}^{2}) v_{0} + c_{3} v_{0}^{3} \\ {(μ_{0}^{*})}^{'} = {\frac{d μ^{*} (v_{r})}{d v_{r}} |}_{v_{r} = v_{0}} = \frac{- \frac{2}{π} c_{0}}{(q Ω_{f}) (\sqrt{1 - {(\frac{v_{0}}{q Ω_{f}})}^{2}})} - (c_{1} + \frac{3}{2} c_{3} {(q Ω_{f})}^{2}) - 3 c_{3} v_{0}^{2} \end{array}

(20)

The stability for the equilibrium is ascertained by the eigenvalues of the Jacobian matrix. The eigenvalues must have a negative real part for stability. Stability plots for the system without HFE can be performed using equations (18) and (19) by substituting qΩ_f = 0. Thus, the stability plots are obtained and shown in Figure 2(a)–(d) for different values of μ_m. One can easily observe the regions of Stribeck and mode-coupling instability from Figure 2(a). It is evident from the stability charts that with the increasing value of μ_m, instability spreads across the whole range of the modal frequency range.

Using the Jacobian given in equation (18), the local stability plots are constructed in the plane of two modal frequencies, Ω₁ and Ω₂, for the system with HFE and are depicted in Figure 3. From these figures, one can observe, for small excitation (qΩ_f = 0.1), the instability region is unaltered. However, with the gradual increment in the strength of excitation, the region of stability widens, and beyond a critical value of the strength of excitation (say at qΩ_f = 0.5 as shown in Figure 3(b) and (e)), the region of instability disappears completely (at least in the range of modal frequencies chosen for the study) signifying stable static equilibrium. However, regions of instabilities reappear at some higher strength of excitation.

2.4. Numerical results and discussion

2.4.1. Pure Stribeck instability

Numerical simulations of the system both with and without HFE are performed in MATLAB Simulink (using the Dormand–Prince algorithm). From Figure 2(a), it is observed that the Stribeck instability occurs when the natural frequency in the tangential direction (Ω₁) is much lower than the natural frequency in the normal direction (Ω₂). One set of natural frequencies (Ω₁ = 1.1 and Ω₂ = 4) is picked from Figure 2(a) in order to simulate pure Stribeck instability. Figure 4 illustrates the stability region in the plane of the belt velocity versus. the strength of excitation for constant damping ratios ξ₁ = ξ₂ = 0.01 and varying minimum coefficient of kinetic friction μ_m. From Figure 4, it is observed that the Stribeck instability region grows with the increasing difference between μ_s and μ_m. Stability boundaries are validated by the direct numerical simulation. It is also observed that beyond a critical value of the strength of HFE, the Stribeck instability is completely removed.

Figure 5 depicts the simulated time-history plots to show the effect of different strength of HFE. It is observed from Figure 5(a) and (b) that the normal vibration is small compared to the tangential vibration. However, even when the strength of excitation is chosen from the unstable region qΩ_f = 0.1 (the unstable region in Figure 4), both the normal and tangential vibration amplitude decrease. For higher strength of excitation, qΩ_f = 0.5 (the stable region in Figure 4), the average vibration in both the directions is completely suppressed. In Figure 5(a) and 5(b), different response curves of the tangential displacement and the normal displacement for different values of the strength of excitation are plotted and compared by simulating the non-autonomous equation (4). It is generally observed the amplitude of vibration decreases with the increasing strength of excitation. In Figure 5(c), response of the tangential displacement obtained by simulating both the non-autonomous equation (4) and the averaged autonomous equation (14) is plotted. It is evident that the small HFE exists around the static slow and averaged response.

2.4.2. Pure mode-coupling instability

From Figure 2(a), it is observed that the mode-coupling instability occurs when the natural frequency in the tangential direction (Ω₁) is close to the natural frequency in the normal direction (Ω₂). Under such circumstances, Figure 6 illustrates the stability region in the plane of the belt velocity versus. the strength of excitation. Stability boundaries are validated by the direct numerical simulation. As observed from Figure 6(a), the mode-coupling instability is stabilized beyond the critical values of strength of excitation for some parameter values. However, it is quite evident from Figure 6(b) that for some other parameter values, the mode-coupling instabilities are stabilized only for a definite range of strength of excitation.

Numerically simulated time-history plots are depicted in Figures 7 and 8 to show the effect of different strength of HFE on pure mode-coupling instability. It is observed from Figures 7 and 8 that both the tangential vibration and the normal vibration are prominent. It is visualized from Figure 7(a) and (b) that both displacement in the tangential direction and the normal direction increase for the strength of excitation qΩ_f = 0.3 (the unstable region in Figure 6(a)). Although for the strength of excitation qΩ_f = 0.5 (the stable region vicinity to the stability boundary in Figure 6(a)) the low-frequency vibration is completely eliminated, only the HF component is present in the system response in the tangential direction.

Low-frequency self-excited vibration in both the tangential direction and the normal direction as shown in Figure 8(a) and (b) is quenched for the strength of excitation qΩ_f = 1 (the stable region in Figure 6(b)). It is also observed from Figure 8(a) and (b) that the friction-induced self-excited vibration reappears for the strength of excitation qΩ_f = 1.5 (the unstable region in Figure 6(b)).

2.4.3. Combined action of both Stribeck instability and mode-coupling instability

Both Stribeck instability and mode-coupling instability coalesce at Ω₁ = 1.75 and Ω₂ = 2.2 as shown in Figure 2(c) and (d). The corresponding stability chart in the belt velocity versus strength of the excitation plane is plotted in Figure 9. The stability boundary is validated by the direct numerical simulation. It is also noted from Figure 9 that the HFE stabilizes the combined action of both the instabilities only for a limited range of strength of excitation.

Time-history plots of the response of the system are depicted in Figure 10(a) and (b). It is observed that the self-excited vibration can be quenched only for a certain range of strength of excitation. Very low and very high strength of excitation is not beneficial.

3. Effect of normal HFE on FIV

In the present section, the effect of HFE applied normal to the sliding direction is investigated. A mathematical model in this regard is constructed, and numerical simulations are performed to explore the steady state behavior of the system

3.1. Mathematical model

The mathematical model of the primary system under investigation is similar to that which has been discussed in Section 2. The only exception here is that instead of tangential HFE an actuator is placed to provide HFE in a direction normal to the velocity of sliding. The model is depicted in Figure 11.

The non-dimensional equations of motion of the above system are written as

\begin{array}{l} \frac{d^{2} p_{1}}{d t^{2}} + 2 ξ_{1} Ω_{1} \frac{d p_{1}}{d t} + Ω_{1}^{2} p_{1} + {1 - μ (Ω_{2}^{2} - 1)} p_{2} = μ q Ω_{f}^{2} \sin (Ω_{f} t) \\ \frac{d^{2} p_{2}}{d t^{2}} + 2 ξ_{2} Ω_{2} \frac{d p_{2}}{d t} + Ω_{2}^{2} p_{2} + p_{1} = 1 - q Ω_{f}^{2} \sin (Ω_{f} t) \end{array}

(21)

The angle θ is 45^°. All the non-dimensional parameters involved in the above equations are defined earlier in Section 2.1.

3.2. Analysis using the MDPM

Equation (21) is recast in the form of equation (5) as

\begin{array}{l} [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}] (\begin{matrix} \frac{d^{2} p_{1}}{d t^{2}} \\ \frac{d^{2} p_{2}}{d t^{2}} \end{matrix}) + [\begin{matrix} 2 ξ_{1} Ω_{1} & 0 \\ 0 & 2 ξ_{2} Ω_{2} \end{matrix}] (\begin{matrix} \frac{d p_{1}}{d t} \\ \frac{d p_{2}}{d t} \end{matrix}) + [\begin{matrix} Ω_{1}^{2} & 1 - μ (Ω_{2}^{2} - 1) \\ 1 & Ω_{2}^{2} \end{matrix}] (\begin{matrix} p_{1} \\ p_{2} \end{matrix}) - [\begin{matrix} 0 \\ 1 \end{matrix}] + \\ {[\begin{matrix} q Ω_{f}^{2} (\begin{array}{l} - c_{0} sgn (v_{0} - \frac{d p_{1}}{d t}) + (- c_{1} + 3 v_{0}^{2} c_{3}) \frac{d p_{1}}{d t} \\ - 3 c_{3} v_{0} {(\frac{d p_{1}}{d t})}^{2} + 3 c_{3} {(\frac{d p_{1}}{d t})}^{3} \end{array}) \\ 0 \end{matrix}] + [\begin{matrix} Ω_{f} (- μ_{k} q Ω_{f}) \\ Ω_{f} (q Ω_{f}) \end{matrix}]} \frac{\partial^{2} (- \sin (Ω_{f} t))}{\partial {(Ω_{f} t)}^{2}} = 0 \end{array}

(22)

where

μ_{k} = c_{1} v_{0} + c_{3} v_{0}^{3}

Ω_{f}

= excitation frequency,

Ω_{f} ≫ Ω_{1,2}

[p] = {[\begin{matrix} p_{1} & p_{2} \end{matrix}]}^{T}

, describes the positional state.

[M] = [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}]

is an inertia matrix, here an identity matrix.

[S] = [S_{0}] + [S_{11}] [p] + [S_{12}] [d p / d t]

is the 2 × 1-matrix of slow forces with

[S_{0}] = [\begin{matrix} 0 \\ - 1 \end{matrix}]

[S_{11}] = [\begin{matrix} Ω_{1}^{2} & 1 - μ (Ω_{2}^{2} - 1) \\ 1 & Ω_{2}^{2} \end{matrix}]

and

[S_{12}] = [\begin{matrix} 2 ξ_{1} Ω_{1} & 0 \\ 0 & 2 ξ_{2} Ω_{2} \end{matrix}]

The rapidly oscillating excitation components are $[H] = [\begin{matrix} h_{1} \\ 0 \end{matrix}]$ and $[F] = [\begin{matrix} - μ_{k} q Ω_{f} \\ q Ω_{f} \end{matrix}]$ , where

h_{1} = q Ω_{f}^{2} (- c_{0} s g n (v_{0} - \frac{d p_{1}}{d t}) + (- c_{1} + 3 v_{0}^{2} c_{3}) \frac{d p_{1}}{d t} - 3 c_{3} v_{0} {(\frac{d p_{1}}{d t})}^{2} + 3 c_{3} {(\frac{d p_{1}}{d t})}^{3})

The corresponding scalar time function is $Θ = - \sin (Ω_{f} t)$ . For the present discussion, one takes $v_{0} < v_{m}$ so that both the Stribeck and mode-coupling instabilities can be addressed simultaneously.

Considering both definitions and assumptions of the MDPM (Blekhman, 2000), the expression for the fast component vector reads as

p_{f} (t, τ) = - M^{- 1} \sum F Θ + O (Ω_{f}^{- 1}) = [\begin{matrix} - μ_{k} q Ω_{f} \sin (Ω_{f} t) \\ q Ω_{f} \sin (Ω_{f} t) \end{matrix}] + O (Ω_{f}^{- 1})

(23)

The general solution of equation (22) is computed as

p = [\begin{matrix} p_{1 s} - μ_{k} q \sin (Ω_{f} t) \\ p_{2 s} + q \sin (Ω_{f} t) \end{matrix}]

(24)

The first derivative of the general solution w.r.t slow time is

\frac{d p}{d t} = [\begin{matrix} \frac{d p_{1 s}}{d t} - μ_{k} q Ω_{f} \cos (Ω_{f} t) \\ \frac{d p_{2 s}}{d t} + q Ω_{f} \cos (Ω_{f} t) \end{matrix}]

(25)

Similarly, the vibrational forces are formulated using equation (25) as follows

V (p_{s}, \frac{d p_{s}}{d t}, t) = [\begin{matrix} μ^{*} (Ω_{2}^{2} - 1) p_{2 s} + 〈 h 〉 〈 h_{1} \frac{\partial^{2}}{\partial {(Ω_{f} t)}^{2}} (Θ) 〉 \\ 0 \end{matrix}]

(26)

where

μ^{*} = μ + (3 / 2) c_{3} {(μ_{k} q Ω_{f})}^{2} v_{r}

; the calculation of the effective friction function is obtained from equation (15), where

μ

is defined with equation (2) and

〈 h_{1} \frac{\partial^{2}}{\partial {(Ω_{f} t)}^{2}} (Θ) 〉 = 0

The final autonomous equation set can be written in the form as follows

M (p_{s}, t) \frac{d^{2} p_{s}}{d t^{2}} + S (p_{s}, \frac{d p_{s}}{d t}, t) + V (p_{s}, \frac{d p_{s}}{d t}, t) = 0

So, the final slow flow equations of friction induced oscillations under normal HFE yield

[\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}] (\begin{matrix} \frac{d^{2} p_{1 s}}{d t^{2}} \\ \frac{d^{2} p_{2 s}}{d t^{2}} \end{matrix}) + [\begin{matrix} 2 ξ_{1} Ω_{1} & 0 \\ 0 & 2 ξ_{2} Ω_{2} \end{matrix}] (\begin{matrix} \frac{d p_{1 s}}{d t} \\ \frac{d p_{2 s}}{d t} \end{matrix}) + [\begin{matrix} Ω_{1}^{2} & 1 - μ^{*} (Ω_{2}^{2} - 1) \\ 1 & Ω_{2}^{2} \end{matrix}] (\begin{matrix} p_{1 s} \\ p_{2 s} \end{matrix}) = (\begin{matrix} 0 \\ 1 \end{matrix})

(27)

3.3. Stability analysis

The static equilibrium of the system with normal HFE is obtained from equation (27) as

\begin{array}{l} p_{1 s e} = 1 - Ω_{2}^{2} p_{2 s e} \\ p_{2 s e} = \frac{Ω_{1}^{2}}{Ω_{1}^{2} Ω_{2}^{2} + (Ω_{2}^{2} - 1) (μ_{0}^{*}) - 1} \end{array}

(28)

The Jacobian of the system at the static equilibrium solutions is given by

[J^{*}] = [\begin{matrix} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ - Ω_{1}^{2} & J_{32}^{*} & J_{33}^{*} & 0 \\ - 1 & - Ω_{2}^{2} & 0 & - 2 ξ_{2} Ω_{2} \end{matrix}]

(29)

where

\begin{array}{l} J_{32}^{*} = (Ω_{2}^{2} - 1) μ_{0}^{*} - 1 \\ J_{33}^{*} = - 2 ξ_{1} Ω_{1} + {(μ_{0}^{*})}^{'} (Ω_{2}^{2} - 1) p_{2 s e} \end{array}

\begin{array}{l} μ^{*} (v_{0}) = μ_{0}^{*} = μ_{0} + \frac{3}{2} c_{3} {(μ_{k} q Ω_{f})}^{2} v_{0} \\ {(μ_{0}^{*})}^{'} = {\frac{d μ^{*} (v_{r})}{d v_{r}} |}_{v_{r} = v_{0}} = - c_{0} β \sec h^{2} (β v_{0}) - (c_{1} + \frac{3}{2} c_{3} {(μ_{k} q Ω_{f})}^{2}) - 3 c_{3} v_{0}^{2} \end{array}

(30)

The stability of the equilibrium is ascertained by the eigenvalues of the Jacobian matrix. The eigenvalues must have a negative real part for the case of stable solution. Stability plots for the system are depicted in Figure 12 for qΩ_f = 0 (without excitation), 0.5 and 1. From Figure 12, it is quite evident that the stability plot remains unchanged for both with and without normal excitation.

3.4. Numerical simulations

Although the stability is not altered by the normal excitation, it is interesting to study the effect of normal HFE by direct numerical simulations. To this end, a point (I) is selected in the parameter plane as shown in Figure 12. The point (I) corresponds to the modal frequencies Ω₁ = 1.75 and Ω₂ = 2.2. The time–history plots are shown in Figure 13 for various strengths of excitation. For the strength of excitation 0.5, the amplitude of the response decreases in both tangential and normal directions. However, as observed from Figure 14, for higher strength of excitation, the amplitude can increase with the strength of excitation.

4. Conclusions

In the present paper, a 2-DOF mass-spring-damper system resting on a moving belt is considered as the basic model of FIV where the instability is ascribed to both the Stribeck effect and the mode-coupling effect. The central focus of the study is to investigate the effect of HFE on the dynamics and stability of the system. Two different problems are considered. In the first problem, the HFE is applied in the direction of slipping (tangential direction). In the second problem, the effect of HFE in the normal direction is investigated. The MDPM is employed to convert the non-autonomous equations of motion into autonomous equations describing the system dynamics in slow time scale. Detailed analysis has shown that the system can be stabilized for a range of strength of tangential excitation, beyond which the stability is not restored. However, normal excitation cannot improve the stability of the original system although the amplitude of the self-excited oscillation can be suppressed to a large extent by applying proper normal excitation.

Footnotes

Declaration of conflicting interests

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

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Shyamal Chatterjee

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