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Abstract

In this study, a model of two nonlinear spring masses subjected to a smooth friction-velocity curve is examined. The second order governing system of motion is obtained in view of the system friction forces. This system is transformed to another suitable one of first order to investigate the points of equilibrium using Hurwitz’s theory. The excitation and critical stick-slip (SS) speeds are determined in accordance with the characteristic equation for the Jacobian matrix of the system. The stability and behavior of the system motion, along with the behavior of the SS movement, are examined. The effect of various parameters, such as excitation speed, damping and fraction coefficients, linear and nonlinear stiffness of springs, and masses that affect the motion and stability of the system is analyzed and studied. This research has numerous practical applications in a variety of industries, including airports, modern car brakes, well drilling, explaining and understanding seismic events, mechanical transportation, friction in bridges, and civil systems, which all exhibit impacted friction in dynamics and stick-slip phenomena.

Keywords

nonlinear dynamics vibration of stick-slip stability of dry friction phase plane friction-induced vibration

Introduction

As a result of the resistance of friction during operation, most mechanical devices in use today generate undesired noise and vibration on their own. These vibrations, which are of the friction-self-excited variety,¹ are common in machines and tools, such as those used to slow down moving objects mechanically. Scientifically speaking, vibration is a well-known phenomenon with significant implications for dynamical systems,² the place where their structures can deteriorate and break down due to vibrations induced by friction surfaces. In worst-case scenarios, it might even lead to the breakdown of mechanical parts. Approximately 80% of mechanical component failures can be traced back to friction-induced vibration.³

Over the past few years, numerous vibrational models have been studied due to their significance in several engineering applications, for example, Refs. 4–12. In Refs. 4–8 and Refs. 9–12, respectively, various dynamical movements of damped linear/nonlinear springs and rigid body pendulums are examined. The regulating equations of motion (EOM) were obtained in accordance with Lagrange’s equations of the second kind, and approximate solutions of these equations were obtained via a multiple scales approach (MSA).¹³ For several classes of resonances, the necessary conditions for solvability and the corresponding modulation equations were established. In addition, the stable and unstable zones are plotted and discussed in Refs. 7, 8, 11, and 12 via the nonlinear criteria of Routh–Hurwitz.¹⁴

The vibrating planar motion of rigid body has been examined in Refs. 15–17. The movement of a connected rigid body with a linear spring, as a pendulum, is investigated in Ref. 15 in which the authors considered its pivot point to be fixed. The generalization of this problem has been studied in Ref. 16 when the pendulum has two connected rigid bodies with a nonlinear spring. The authors restricted the motion of its suspension point to being on a trajectory of Lissajous curve. Recently, the analytic approximate solutions of the EOM of a triple rigid body with a stationary fixed point up to the third approximation using the MSA are examined in Ref. 17. The arising resonance cases are categorized, in which three of them are studied simultaneously.

However, many academics have become interested in the use of absorbers in the configuration of various dynamical model structures, for example, Refs. 18–23. The reason is that they are used in engineering industries to control vibration. The behavior of a three-degrees-of-freedom (DOF) nonlinear spring pendulum was examined in Ref. 18 to shed light on how a longitudinal absorber might stabilize and control the vibration of a ship roll motion. Whereas the behavior of a 2DOF tuned absorber’s in an elliptic trajectory of its suspension point was studied in Ref. 19. The stability criteria were established and the steady-state solutions around the specified resonance circumstances were examined. In Ref. 20, it is discussed how the behavior of the considered model in Ref. 19 is affected by the existence of a damped nonlinear elastic pendulum. In Ref. 21, the problem of analyzing the vibrations of a coupled inverted pendulum using a passive mass as a spring absorber was addressed. In Ref. 22, the authors showed how to determine the phase between the primary structure’s vibration and the absorber’s vibration in order to automatically adjust the rotating speed of a 2DOF pendulum absorber. The authors of Ref. 23 concentrated on the investigation of a few non-conservative oscillator cases. The procedure for the solutions of conservative oscillators has been greatly simplified by the modification of the homotopy perturbation method (HPM). Additionally, this alteration is thought to be exclusive to HPM and is not reflected in the other perturbation approaches. In Ref. 24, the fractal Toda oscillator is established using the theory of fractal variation and the proper analytical solution is obtained using the non-perturbative approach.

The challenges of friction-induced vibrations have been investigated in the past using a wide range of numerical calculation and experimental analytic techniques.^25–27 To the contrary, friction-induced vibrations are affected by a wide range of variables, such as the component’s stiffness, the frictional interface’s material properties, the damping process, the surface topography, and the working environment. Therefore, it is very challenging to investigate friction-induced self-excited vibrations.²⁸

Over the course of several decades of scientific inquiry, many researchers have studied different aspects of identifying the critical variables that cause the formation of vibrations induced by friction,^29,30 in which the frictional vibration mechanism is highly relevant to both applied and theoretical fields of study.

Customers using vehicles or other machines that experience vibration due to friction find it as an annoying phenomenon. To better understand friction-induced vibration and the resulting nonlinear motions, spring mass models have been developed and analyzed. In particular, friction-induced chaotic motions were investigated in Refs. 31–34. Stick-slip (SS) oscillations’ mechanisms and frequency patterns were also discussed in Ref. 35. If the oscillation is chaotic, then unexpected phenomena, like a spectrum of frequencies that was not anticipated, may emerge.

Lyapunov exponents are a useful tool for finding the chaotic attractor. Chaotic behavior is characterized by a high degree of sensitivity to the model’s initial conditions. The nonlinear motion displays the weird attractor if the greatest Lyapunov exponent is non-negative, in which it can be computed as in Refs. 36 and 37.

In many different studies on dynamic systems, smoothing methods are used to model the friction curve in the engaged model. The discontinuous friction curve is changed to a differentiable one via the smoothing technique. The friction curve has been modeled using smoothing techniques in the friction-engaged model.^38–40 Wolf’s algorithm³⁷ can be used to calculate Lyapunov exponents in a smooth frictional system since the smoothing method converts the discontinuous friction curve into a differentiable one.

The coefficient of friction changes depending on a number of factors, including the material of the two rubbing surfaces, the amount of debris on the surface, the speed at which they are moving, and so on. When a system is put through the rigors of frictional contact, the friction curve will shift significantly, which might cause a variation in the forms of nonlinear motion.^41,42 For nonlinear motion, this could mean a transition between different forms. The authors looked at how the friction curve affects the emergence of the chaotic attractor, in which the velocity-dependent friction curve was used. The parameters of the main system were determined by adjusting the slope of the curve and the friction-velocity magnitude.⁴³

However, complicated mechanical characteristics, such as chaos and bifurcation, are frequently present in nonlinear vibration systems, making it challenging to compute and analyze them analytically. As a result, approximate analytical algorithms are frequently used. In Ref. 44, the authors studied the planar excitation-driven suspension of elastic cables with roughly comparable natural frequencies. According to the analysis of first order, there is a relationship between cubic nonlinearity, both jump phenomena and saturation. The approaches of multiple scales and Runge–Kutta are used in Ref. 45 to investigate the nonlinear vibration properties of multilayered cylindrical composite shells with axial movement. The achieved outcomes demonstrate several nonlinear characteristics, like the phenomena of internal resonance, which highlight the influence of damping, velocity, and excitation amplitude on the internal resonance range, amplitude response, and soft characteristic of the system. In Ref. 46, the piecewise nonlinear characteristics of the viscoelastic shock absorber and the relationship between amplitude-frequency characteristics and system parameters are investigated. Using the same methodology, Ref. 47 investigates whether delay feedback control of the gyroscope system under stress is feasible. Using a shooting approach, the vibration phenomena, with one DOF magnetically levitated system, is examined taking into account the influence of the electromagnet’s nonlinearity. An analytical technique for assessing the periodic behavior of steady-state for a nonlinear system was proposed in Refs. 48 and 49. This technique is based on the approach of multiple scales and the substructure synthesis formulation.

The primary objective of this work is to examine the SS vibration of a 2DOF dynamical model. This model consists of two masses connected with nonlinear damping springs. In light of the system friction forces, the motion’s governing system of second order is obtained. The points of equilibrium are explored using Hurwitz’s theory, in which the controlling system of motion is converted into another suitable one of first order. According to the Jacobian matrix of the investigated system, the excitation and critical SS speeds are calculated. It is looked at for how the SS movement behaves as well as the stability and behavior of the system motion. The impact of various parameters, like excitation speed, damping and fraction coefficients, linear and nonlinear stiffness of springs, and masses that acted on the motion and stability of the system is analyzed and studied.

The mathematical model and governing equations

The main aim of the current section is to describe the investigated model. Therefore, two masses $m_{1}$ and $m_{2}$ are considered to be in motion on a belt with constant speed $v_{0}$ , see Figure 1(a) and Table 1. These masses are connected with nonlinear damped springs, in which one of these masses is connected to a wall. They are exposed to an excited force resulting from the movement of the belt, which results in relative movement between the masses, linear/nonlinear stiffness of the springs $k_{1} x + α_{1} x^{3}$ and $k_{2} x + α_{2} x^{3}$ , a damper with viscous damping due to the damping coefficients $c_{j} (j = 1,2)$ , friction forces $F_{j} (ω_{j})$ that acting between masses $m_{j}$ , and the belt which they are dependent on the relative velocity $ω_{j}$ among two of them. Therefore, we can write the controlling system of motion as listed below.

Figure 1.

(a) Self-sustaining, friction-generating, linked oscillators. (b) Illustration of the forces acting on the blocks.

Table 1.

Nomenclatures.

Nomenclature	The meaning
SS	Stick-slip
DOF	Degrees-of-freedom
SMD	Spring mass damper
EOM	Equations of motion
ODE	Ordinary differential equations
CIS	Critical instability speed
MSA	Multiple scales approach
HPM	Homotopy perturbation method
$v_{0}$	Excitation speed
$m_{1}, m_{2}$	The masses of the block on the conveyor belt
$k_{1}, k_{2}, α_{1}, α_{2}$	Stiffness of the springs
$c_{j} (j = 1,2)$	Damping coefficients
$ω_{j}$	The relative speed of the block and conveyor belt motion
$μ_{j s}$	The static friction’s coefficient
$μ_{j m}$	The kinetic friction’s coefficient
$v_{j m}^{*}$	The velocity at which the friction forces reach to its maximum value
$F_{j} (ω_{j})$	Friction forces, acts between the mass and the belt
$f_{0 j}$	The maximum static friction force
$v_{b 0}$	The critical instability of system’s speed
$v_{b 1}$	The critical stick-slip system’s speed
$F_{S 1}$	The force of the first linear and nonlinear spring acting on the first block
$F_{S 2}$	The force of the second linear and nonlinear spring acting on the second mass
$F_{D 1}$	The force of the first damper affecting the first blocks
$F_{D 2}$	The force of the second damper acting between the two masses

The EOM can be obtained from Newton’s second law. Since the investigated system consists of two blocks, to obtain the desired equations, one can apply this law to these blocks as follows:

m_{1} a_{x_{1}} = \sum F_{x_{1}}, m_{2} a_{x_{2}} = \sum F_{x_{2}},

(1)

where

a_{x_{1}} = {\ddot{x}}_{1}

and

a_{x_{2}} = {\ddot{x}}_{2}

are, respectively, the accelerations of the considered blocks in the directions

x_{1}

and

x_{2}

, while

\sum F_{x_{1}}

and

\sum F_{x_{2}}

are the applied forces on these blocks (see Figure 1(b)) and they have the forms

\begin{array}{l} \sum F_{x_{1}} = F_{1} (ω_{1}) + F_{S 2} + F_{D 2} - F_{S 1} - F_{D 1}, \\ \sum F_{x_{2}} = F_{2} (ω_{2}) - F_{S 2} - F_{D 2} . \end{array}

(2)

Here, $F_{S 1}$ represents the force of the first linear and nonlinear spring acting on the first block in terms of the fixation point. The force $F_{S 2}$ is the force of the second linear and nonlinear spring acting on the second mass, which is located between both masses. $F_{D 1}$ is the force of the first damper affecting the first blocks in terms of installation and $F_{D 2}$ is the force of the second damper acting between the two masses. These forces take the following forms:

\begin{array}{l} F_{S 1} = k_{1} x_{1} + α_{1} x_{1}^{3}, \\ F_{S 2} = k_{2} (x_{2} - x_{1}) + α_{2} {(x_{2} - x_{1})}^{3}, \\ F_{D 1} = c_{1} {\dot{x}}_{1}, \\ F_{D 2} = c_{2} ({\dot{x}}_{2} - {\dot{x}}_{1}) . \end{array}

(3)

The substitution of equations (2) and (3) in equation (1) yields the following EOM:

\begin{array}{l} m_{1} {\ddot{x}}_{1} - k_{2} (x_{2} - x_{1}) - α_{2} {(x_{2} - x_{1})}^{3} - c_{2} ({\dot{x}}_{2} - {\dot{x}}_{1}) + k_{1} x_{1} + α_{1} x_{1}^{3} + c_{1} {\dot{x}}_{1} - F_{1} (ω_{1}) = 0, \\ m_{2} {\ddot{x}}_{2} + k_{2} (x_{2} - x_{1}) + α_{2} {(x_{2} - x_{1})}^{3} + c_{2} ({\dot{x}}_{2} - {\dot{x}}_{1}) - F_{2} (ω_{2}) = 0 . \end{array}

(4)

The dry friction equations of the model under study can be written as follows:⁵⁰

F_{j} (ω_{j}) = F_{0 j} S g n (v_{0} - {\dot{x}}_{j}) - \frac{3 F_{0 j}}{4 v_{j m}^{*}} (v_{0} - {\dot{x}}_{j}) + \frac{F_{0 j}}{4 {v_{j m}^{*}}^{3}} {(v_{0} - {\dot{x}}_{j})}^{3}; F_{0 j} = μ_{j s} - μ_{j m} (j = 1,2) .

(5)

Here, $ω_{j} = v_{0} - {\dot{x}}_{j}$ symbolizes the block’s relative velocity and conveyor belt motion, $F_{0 j}$ the maximum static friction force, $μ_{j m}$ and $μ_{j s}$ are, respectively, the coefficients of static and kinetic friction, and $v_{j m}^{*}$ indicates the velocity at which the friction forces reach to their maximum value.^50,51 The curves of the previous equation can be seen in Figure 2, which reveal the variation of the functions $F_{j}$ versus ${\dot{x}}_{j}$ . Thus, friction’s function must accommodate the following:

\begin{array}{l} | μ (0) | \leq μ_{j s}, | μ (v_{j m}) | = μ_{j m}, | μ^{'} (v_{j m}) | = 0, \\ μ (- ω_{j}) = - μ (ω_{j}), μ^{'} (- ω_{j}) = μ^{'} (ω_{j}); μ^{'} = d μ / d ω_{j} . \end{array}

(6)

Figure 2.

Friction model.

It should be observed that while the block is at rest in relation to the conveyor belt, the relative velocity $ω_{j} = v_{0} - {\dot{x}}_{j}$ will equal zero. As the block begins to slide, the likelihood for friction to increase first reduces, and then rises.

Analysis of the SS and critical speeds

If the parameter $ω_{0}$ is defined according to $τ = ω_{0} t$ , then the dimensionless variables and parameters can be written as

\begin{array}{l} ω_{0} = \frac{τ}{t}, x_{j} = y_{j} L, ω_{j}^{2} = \frac{κ_{j}}{m_{1} ω_{0}^{2}}, ω_{3}^{2} = \frac{κ_{2}}{m_{2} ω_{0}^{2}}, γ_{j} = \frac{α_{j} L^{2}}{m_{1} ω_{0}^{2}}, γ_{3} = \frac{α_{3} L^{2}}{m_{2} ω_{0}^{2}}, \\ β_{j} = \frac{c_{j}}{m_{1} ω_{0}}, β_{3} = \frac{c_{2}}{m_{2} ω_{0}}, f_{j} = \frac{F_{j}}{m_{j} L ω_{0}^{2}}, {\dot{X}}_{j} = ω_{0} {\dot{x}}_{j} (j = 1,2) . \end{array}

(7)

Consequently, it is possible to rewrite the EOM (4) in the following dimensionless forms:

\frac{d^{2} y_{1}}{d τ^{2}} = ω_{2}^{2} (y_{2} - y_{1}) + γ_{2} {(y_{2} - y_{1})}^{3} - γ_{1} y_{1}^{3} - ω_{1}^{2} y_{1} + β_{2} (\frac{d y_{2}}{d τ} - \frac{d y_{1}}{d τ}) - β_{1} \frac{d y_{1}}{d τ} + f_{1},

(8)

\frac{d^{2} y_{2}}{d τ^{2}} = f_{2} - ω_{3}^{2} (y_{2} - y_{1}) - γ_{3} {(y_{2} - y_{1})}^{3} - β_{3} (\frac{d y_{2}}{d τ} - \frac{d y_{1}}{d τ}),

(9)

where

\begin{array}{l} f_{j} (ω_{j}) = f_{0 j} s g n (v_{0} - \frac{d X_{j}}{d τ}) - \frac{3 f_{0 j}}{4 v_{j}^{*}} (v_{0} - \frac{d X_{i}}{d τ}) : \frac{f_{0 j}}{4 {v_{j}^{*}}^{3}} {(v_{0} - \frac{d X_{j}}{d τ})}^{3} : \\ f_{0 j} = μ_{j s} - μ_{j m} (j = 1,2), \end{array}

(10)

In accordance with the ordinary differential equations’ (ODE) theory, one can transform a set of higher-order ODF to another one of first-order ODE. Then, the formulas listed below can be used to achieve this objective

\begin{array}{l} x = y_{1}, z = y_{2}, \\ y = \frac{d x}{d τ} = \frac{d y_{1}}{d τ}, \frac{d y}{d τ} = \frac{d^{2} y_{1}}{d τ^{2}}, \\ w = \frac{d z}{d τ} = \frac{d y_{2}}{d τ}, \frac{d w}{d τ} = \frac{d^{2} y_{2}}{d τ^{2}} . \end{array}

(11)

The following system of differential equations is generated by inserting equation (11) into equations (8) and (9).

\begin{array}{l} \frac{d x}{d τ} = y, \\ \frac{d^{2} y_{1}}{d τ^{2}} = \frac{d y}{d τ} = ω_{2}^{2} (z - x) + γ_{2} {(z - x)}^{3} - γ_{1} x^{3} - ω_{1}^{2} x + β_{2} (w - y) - β_{1} y + f_{1}, \\ \frac{d z}{d τ} = w, \\ \frac{d^{2} y_{2}}{d τ^{2}} = \frac{d w}{d τ} = f_{2} - ω_{3}^{2} (z - x) - γ_{3} {(z - x)}^{3} - β_{3} (w - y) . \end{array}

(12)

Because of the mutual action of friction and spring forces, the block will remain in equilibrium as long as the excitation speed is sufficiently high.⁵² As a result, the parameters of system and the equilibrium conditions $s g n (ω_{i}) = - 1$ and $d x_{j} / d τ = 0$ must be hold. The system parameters can be selected in the forms $μ_{1 s} = 150.6, μ_{1 m} = 21.2, μ_{2 s} = 80.55, μ_{2 m} = 11.3, v_{1 m} = 3.6 m / s, v_{2 m} = 3.6 m / s$ . We can assess the system’s stability at the equilibrium point using Lyapunov’s stability theory. The Lyapunov first-order approximation theory states that linear analysis can be employed to evaluate the system stability close to its point of equilibrium. Moreover, the linear analysis of the braking system is more complicated due to the friction model’s discontinuity at zero relative velocity. Assuming the relative velocity is always positive,⁵² we can use Lyapunov’s stability theory to analyses the robustness of the considered system’s equilibrium. At $d x_{j} / d τ = 0,$ the possible expressions of equilibrium points can be represented in the forms

χ_{x} = Θ, χ_{y} = 0, χ_{z} = Ξ, χ_{w} = 0 .

(13)

If the point of equilibrium is moved to the origin zero point, the resulting equations are as follows:

\begin{array}{l} \bar{x} = x + Θ, \bar{y} = y, \\ \bar{z} = z + Ξ, \bar{w} = w . \end{array}

(14)

Therefore, the previous equation (12) becomes

\begin{array}{l} \frac{d \bar{x}}{d τ} = \bar{y}, \\ \frac{d \bar{y}}{d τ} = ω_{2}^{2} (\bar{z} - \bar{x}) + γ_{2} {(\bar{z} - \bar{x})}^{3} - γ_{1} {\bar{x}}^{3} - ω_{1}^{2} \bar{x} + β_{2} (\bar{w} - \bar{y}) - β_{1} \bar{y} + f_{1}, \\ \frac{d \bar{z}}{d τ} = \bar{w}, \\ \frac{d \bar{w}}{d τ} = f_{2} - ω_{3}^{2} (\bar{z} - \bar{x}) - γ_{3} {(\bar{z} - \bar{x})}^{3} - β_{3} (\bar{w} - \bar{y}) . \end{array}

(15)

One can determine the Jacobian matrix from equation (15) for the system’s first-order approximation equation at $\bar{x} = \bar{y} = \bar{z} = \bar{w} = 0$ as follows:

A = (\begin{array}{c} 0 & 1 & 0 & 0 \\ a_{21} & a_{22} & a_{23} & a_{24} \\ 0 & 0 & 0 & 1 \\ a_{41} & a_{42} & a_{43} & a_{44} \end{array}),

(16)

where

\begin{array}{l} a_{21} = - 1.5 - \frac{189}{2} Θ^{2} - \frac{243}{2} {(Θ - Ξ)}^{2}, \\ a_{22} = 0.123 - 0.008 {V_{0}}^{2}, \\ a_{23} = 0.75 + \frac{243}{2} {(Θ - Ξ)}^{2}, \\ a_{24} = 0.001, \\ a_{41} = 0.536 + \frac{1215}{14} {(Θ - Ξ)}^{2}, \\ a_{42} = 0.176 - 0.012 {V_{0}}^{2}, \\ a_{43} = - 0.536 - \frac{1215}{14} {(Θ - Ξ)}^{2}, \\ a_{44} = - 0.001 . \end{array}

(17)

Considering the following values of variables

\begin{array}{l} μ_{1 s} = 150.6, μ_{1 m} = 21.2, μ_{2 s} = 80.55, μ_{2 m} = 11.3, v_{1 m} = 3.6 m / s, \\ v_{2 m} = 3.6 m / s, κ_{1} = 10, κ_{2} = 5, m_{1} = 5 k g, m_{2} = 3 k g, c_{1} = 30, \\ c_{2} = 10, L = 3 m, α_{1} = 30, α_{2} = 10, ω_{0} = 2 . \end{array}

(18)

The following formula gives us the matrix $A$ characteristic equation

| A - λ I | = λ^{4} + Γ_{1} λ^{3} + Γ_{2} λ^{2} + Γ_{3} λ + Γ_{4} = 0,

(19)

where

\begin{array}{l} Γ_{1} = - 0.122 + 0.008 v_{0}^{2}, \\ Γ_{2} = 2.036 + 0.001 v_{0}^{2} + 302.786 Θ^{2} - 416.572 Θ Ξ + 208.286 Ξ^{2}, \\ Γ_{3} = - 0.197 + 0.019 v_{0}^{2} + Θ^{2} (3.030 v_{0}^{2} - 31.872) + Θ Ξ (63.797 - 6.059 v_{0}^{2}) \\ + Ξ^{2} (3.030 v_{0}^{2} - 31.899), \\ Γ_{4} = 0.402 + Θ^{2} (115.715 + 8201.25 Θ^{2}) + Ξ^{2} (65.090 + 8201.25 Θ^{2}) \\ - Θ Ξ (16402.5 Θ^{2} - 130.179), \end{array}

\begin{array}{l} Θ = \frac{9334503.075}{\sqrt[3]{(ρ + λ)}} - δ, Ξ = ϵ - τ + η - θ, τ = \frac{0.002}{\sqrt[3]{Γ}}, \\ ϵ = \frac{14.441}{\sqrt[3]{Γ}}, Γ = υ + 3.703 \times 10^{- 18} λ, λ = \sqrt{υ}, η = \frac{5250657.98}{χ}, \\ υ = 4.395 \times 10^{46} + ρ^{2}, ρ = - 4.038 \times 10^{24} + 1.367 \times 10^{24} v_{0} - 4.366 \times 10^{22} v_{0}^{3}, \\ θ = 1.059 \times 10^{- 8} χ, χ = \sqrt[3]{γ - 0.0078125 λ + ξ - κ + \frac{Δ Δ}{\sqrt[3]{Γ^{2}}} - \frac{ψ}{\sqrt[3]{Γ^{2}}} + μ - α + \sqrt{δ}}, \end{array}

\begin{array}{l} σ = - 1.4947254 \times 10^{7} + 5060475 v_{0} - 161633 v_{0}^{3}, \\ γ = - 1.105 \times 10^{23} + 4.152 \times 10^{22} v_{0} - 1.484 \times 10^{21} v_{0}^{3}, \\ δ = 1.223 \times 10^{44} + {(γ - 0.0078125 λ + ξ - κ + \frac{Δ Δ}{\sqrt[3]{Γ^{2}}} - \frac{ψ}{\sqrt[3]{Γ^{2}}} + μ - α)}^{2}, \\ Δ Δ = 2.706 \times 10^{26} v_{0}, ψ = 8.642 \times 10^{24} {v_{0}}^{3}, ξ = \frac{1.272 \times 10^{27}}{Γ}, \\ κ = \frac{7.992 \times 10^{26}}{\sqrt[3]{Γ^{2}}}, μ = \frac{197.939 λ}{\sqrt[3]{Γ^{2}}}, α = 5.3464 \times 10^{19} \sqrt[3]{Γ} . \end{array}

According to the Hurwitz’s theory, the basic critical instability speed (CIS) of the system under consideration may be obtained from $Γ_{1}$ ’s solution of, that is, $Γ_{1} = - 0.122 + 0.008 v_{0}^{2} = 0$ , and therefore, the system will reach to a critical point of instability at $v_{b 0} = 3.9855 m / s$ . Furthermore, the system reaches a stable equilibrium point when the excitation speed exceeds the CIS. With the increase of the excitation speed, the system becomes more stable and the SS motion becomes less likely. Therefore, it is possible to determine the critical SS speed $v_{b 1}$ as follows:⁵³

v_{b 1} = \frac{(\sum_{j = 1}^{2} μ_{j s} - \sum_{j = 1}^{2} μ_{j m})}{\sqrt{2 π (\sum_{j = 1}^{2} c_{j}) \sqrt{(k_{1} + k_{2} + α_{1} + α_{2}) (m_{1} + m_{2})}}},

(20)

where

(\sum_{j = 1}^{2} μ_{j s} - \sum_{j = 1}^{2} μ_{j m})

symbolizes the dissimilarity between the dynamic force and the static friction force,

(\sum_{j = 1}^{2} c_{j})

represents the system’s damping, and

κ_{1} + κ_{2} + α_{1} + α_{2}

corresponds to the block’s mass and the system’s stiffness.

The critical SS speed for the system under study is $2.4722$ at $\sum_{j = 1}^{2} μ_{j s} - \sum_{j = 1}^{2} μ_{j m} = 198.6500,$ in addition to the above-mentioned values of $m_{j}, c_{j}, k_{j}$ , and $α_{j}$ . If the speed is smaller than $2.4722$ , the SS motion is produced which causes sticking of the block on the belt. We find that SS motion occurs in the system when the feeding speed is less than the CIS and the crucial SS speed. Next, we will analyze the simulation’s output to learn more about how different system parameters influence the SS’s stability and motion.^54,55 The model’s significance arises from the wide range of domains in which it can be used, particularly physics and engineering subjects revolving around vibrating systems like buildings, ships, and pumps.

Studying the effects of system parameters

We will focus, in this section, on the numerical solutions of the EOM (8) and (9) in view of the MATLAB R2020a. Furthermore, we investigate the impact of different system factors on stability and SS performance in particular. Therefore, we can comprehend the influence of each parameter on the system’s reaction; we shall subdivide this part into a few subsections.

Excitation speed effect

To fully comprehend the influence of the excitation speed $v_{0}$ , we recall the mentioned data above in addition to the system’s basic conditions listed below are taken into consideration.

\begin{array}{l} x = 0, y = 0, w = 0, z = 0, \\ \dot{x} = 0, \dot{y} = 0, \dot{w} = 0, \dot{z} = 0. \end{array}

(21)

Portions of Figure 3 are calculated when the excitation speed has the values $v_{0} (= 1.0,2.0,3.2,5)$ and they display the graphs of phase plane which demonstrate the effects of the speed $v_{0}$ on the system’s response, in which the CIS $v_{b 0}$ is $3.9855 m / s$ and the critical SS speed $v_{b 1}$ is $2.4722 m / s$ .

Figure 3.

Describes the diagrams of phase planes for different values of speeds excitation: (a) $v_{0} = 1.0$ , (b) $v_{0} = 2.0$ , (c) $v_{0} = 3.2$ , (d) $v_{0} = 5.0$ .

As observed in Figure 3, the illustrative phase plane diagrams vary remarkably with the use of different excitation speeds, in which they are examined according to various values of the used speeds besides the relations $v_{0} < v_{b 1} < v_{b 0}$ , $v_{b 1} < v_{0} < v_{b 0}$ , and $v_{b 1} < v_{b 0} < v_{0}$ . The extent of the variance and then the charts of phase plane in the corresponding parts (a, b) and (c, d) of Figure 3 demonstrate that the dynamical system under study has been started with asymptotically stable and the SS phase occupies a significant position throughout the entire cycle. In addition, the adhesion phase is still present, where the adhesion phase gradually disappears with the continuous increase of the speed.

The graphics shown in Figure 4 express the equivalent time history of the corresponding ones in Figure 3. These portions are drawn in the planes of displacements and time. According to all of the time history plots, the system displays quasi-periodic motion at different excitation levels and is free of chaotic motion. The drawn curves of portions (a), (b), (c), and (d) start from their initial points and then rise to their certain values to oscillate periodically over the examined time interval. Then, one can conclude that the motion has a stable procedure.

Figure 4.

Describes the diagrams of time histories for different values of speeds excitation: (a) $v_{0} = 1.0$ , (b) $v_{0} = 2.0$ , (c) $v_{0} = 3.2$ , (d) $v_{0} = 5.0$ .

Portions of Figure 5 express the equivalent Poincaré maps of the corresponding ones of Figure 3. These portions are drawn in the planes of displacements and velocities. According to all of the Poincaré plots, the system exhibits periodic motion after a very short time from the start of the motion, at different excitation levels and the system is free of chaotic motion.

Figure 5.

Sketches the Poincaré maps of the relevant aspects in Figure 3, where (a) $v_{0} = 1.0$ , (b) $v_{0} = 2.0$ , (c) $v_{0} = 3.2$ , (d) $v_{0} = 5.0$ .

If the excitation velocity follows the relation $6 \leq v_{0} \leq 8$ , we can conclude that the system is stable (Figure 6). The drawn curves have been plotted in parts (a, c, e) and (b, d, f), which reveal that the curves of the phase plane and the corresponding sections of Poincaré, respectively, at $v_{0} (= 6,7,8)$ . They are showing how the SS phase becomes entrained. It is evident from these parts that the adhesion phase melts as the excitation velocity rises.

Figure 6.

(a, c, e) shows the curves of the phase plane, while (b, d, f) are the corresponding relevant sections of Poincaré: (a, b) $v_{0} = 6.0$ , (c, d) $v_{0} = 7.0$ , and (e, f) $v_{0} = 8.0$ .

Influence of the damping coefficients

This subsection is provided to examine the good influence of the damping coefficients $c_{1}$ and $c_{2}$ on the dynamical system response. The obtained results are drawn in the portions of Figures 7–11 according to the data (18), the initial conditions (21), and the value $v_{0} = 3.2$ of the excitation speed.

Figure 7.

Sketches the phase plane curves when $c_{1}$ has distinct values: (a) $c_{1} = 10.0$ , (b) $c_{1} = 20.0$ , (c) $c_{1} = 30.0$ , (d) $c_{1} = 35.0$ , (e) $c_{1} = 40.0$ , and (f) $c_{1} = 45.0$ .

Figure 8.

Sketches the time history curves when $c_{1}$ has distinct values: (a) $c_{1} = 10.0$ , (b) $c_{1} = 20.0$ , (c) $c_{1} = 30.0$ , (d) $c_{1} = 35.0$ , (e) $c_{1} = 40.0$ , and (f) $c_{1} = 45.0$ .

Figure 9.

Represents the Poincaré sections of the same considered values of Figure 6: (a) $c_{1} = 10.0$ , (b) $c_{1} = 20.0$ , (c) $c_{1} = 30.0$ , (d) $c_{1} = 35.0$ , (e) $c_{1} = 40.0$ , and (f) $c_{1} = 45.0$ .

Figure 10.

Describes the influence of $c_{2}$ on the phase plane curves: (a) $c_{2} = 2.0$ , (b) $c_{2} = 5.0$ , (c) $c_{2} = 10$ , (d) $c_{2} = 15.0$ , (e) $c_{2} = 20.0$ , and (f) $c_{2} = 25$ .

Figure 11.

Describes the influence of $c_{2}$ on the phase plane curves: (a) $c_{2} = 2.0$ , (b) $c_{2} = 5.0$ , (c) $c_{2} = 10$ , (d) $c_{2} = 15.0$ , (e) $c_{2} = 20.0$ , and (f) $c_{2} = 25$ .

Therefore, parts of Figure 7 are graphed when the damping coefficient $c_{1}$ has the values $10,20,30,35,40$ , and $45$ . The included curves in this diagram demonstrate that as $c_{1}$ grows, the sticking phase becomes shorter, and it is hidden to some extent at $c_{1} = 45$ .

One can then conclude that the damping coefficients have a significant impact on the system behavior, whether or not the sticking phase rises. In other words, increasing the damping of the system reduces the adhesion phase to a certain extent. An exploration of the parts of Figure 7 reveals that when the value of $c_{1}$ increases, the outer loops get smaller and closer to the limit cycles, which are often approaching the equilibrium points, as seen in portions (c, d, e, f). Each of them demonstrates how the system becomes stable quickly and eventually achieves an asymptotically stable state.

The corresponding time histories of the drawn curves in Figure 7 are included in the parts of Figure 8 at the same considered values of the damping parameter $c_{1}$ . Based on the drawn waves in parts (a) and (b) of this figure, we can say that these waves have periodic behaviors. On the other hand, the curves of the reminder portions reveal a very fast oscillation at the beginning of the interval of time and then they approach to equilibrium points, as seen in Figure 8(c)–(f). However, Poincaré sections of the drawn curves of Figure 7 are included in the parts of Figure 9 at the same considered values of the damping parameter $c_{1}$ . The sketched curves assert that the motion is asymptotically stable and without chaos.

We will now investigate the behavior of the system when $c_{2}$ has different values. To achieve this aim, curves of Figure 10 are graphed when $c_{2} (= 2,5,10,15,20,25)$ . The motion of the system exhibits a variety of dynamical behaviors, as explored in this figure.

It is observed that when the value of $c_{2}$ rises, the sticking phase shortens until it vanishes at a value $c_{2} = 25$ . The aforementioned phenomenon demonstrates how the system damping significantly affects whether the sticking phases increase or decrease. Therefore, lowering the sticking period to a specific extent can be achieved by increasing the system damping. It is observed that as the value of $c_{2}$ increases, the rings diverge from each other and their adhesion decreases until it gradually fades when $c_{2} = 25$ . Moreover, a stable point of equilibrium is noted in Figure 10(d)–(f) when $c_{2}$ equals $15,20,$ and $25$ , respectively. In other words, the system has gradually acquired the state of being asymptotically stable after passing through a short period of time. The corresponding time histories of the drawn curves in Figure 10 are included in the parts of Figure 11 at the same considered values of the damping parameter $c_{2}$ that assert the above discussion. It is noted that the examined system reaches its point of equilibrium in Figure 11(d)–(f), where the waves have decayed into their original forms faster than the graphed curves in Figure 11(a)–(c).

Effect of the masses friction coefficient

Additionally, we examine the impact of different static friction coefficient values $μ_{1 m} (= 5,21.2,60)$ and $μ_{2 m} (= 3,11.3,50)$ , due to the masses $m_{1}$ and $m_{2}$ on the diagrams of phase planes. The considered values of the model’s parameters in equation (16) are taken into account, besides the value of $3.2$ the excitation speed $v_{0}$ . Figures 12 and 13 are drawn when $μ_{1 m}$ and $μ_{2 m}$ have the values above, respectively. According to Figure 12, the sticking phase lengthens when the gap between the coefficients of the static and dynamic frictions decreases. Instead of assisting in reducing the sticking phase, this result emphasizes reducing the gap between the static and dynamic coefficients. The limit cycles and the outer loops move toward each other as the difference approaches zero. As a consequence, one of the best ways to improve system stability is to minimize the variation in friction coefficients between static and dynamic situations. Inspection of the plotted curves of Figure 13 reveals that the system reaches to the point of equilibrium when $μ_{2 m} (= 50)$ , as demonstrated in parts (e) and (f).

Figure 12.

Examines the action of $μ_{1 m}$ on the behavior of the motion: (a, b) at $μ_{1 m} = 5$ , (c, d) at $μ_{1 m} = 21.2$ , (e, f) at $μ_{1 m} = 60$ .

Figure 13.

Examines the action of $μ_{2 m}$ on the behavior of the motion: (a, b) at $μ_{2 m} = 3$ , (c, d) at $μ_{2 m} = 11.3$ , (e, f) at $μ_{2 m} = 50$ .

Equivalent stiffness effect on the masses

This subsection is devoted to demonstrate the dynamical behavior of the examined model on the basis of the phase plane plots when the stiffness $κ_{1}, α_{1}$ and $k_{2}, α_{2}$ have various values. Therefore, Figures 14–17 are plotted, respectively, when $κ_{1} (= 5,10,30)$ , $α_{1} (= 10,30,60)$ , $κ_{2} (= 1,5,20)$ , and $α_{2} (= 5,10,60)$ , in addition to the value of excitation speed $v_{0} = 3.2$ and the other data in equation(16).

Figure 14.

Portrays the change of the stiffness values for the linear spring: (a, b) $κ_{1} = 5$ , (c, d) $κ_{1} = 10$ , (e, f) $κ_{1} = 30$ .

Figure 15.

Represents the effects of different nonlinear spring stiffness settings: (a, b) $α_{1} = 10$ , (c, d) $α_{1} = 30$ , (e, f) $α_{1} = 60$ .

Figure 16.

Depicts the phase plane charts when: (a, b) $κ_{2} = 1$ , (c, d) $κ_{2} = 5$ , (e, f) $κ_{2} = 20$ .

Figure 17.

Explores the variation of $α_{2}$ on the phase plane plots when: (a, b) $α_{2} = 5$ , (c, d) $α_{2} = 10$ , (e, f) $α_{2} = 30$ .

The phase plane and time history curves have been impacted by the change of the stiffness nonlinear spring compared to the linear one, as explored in parts of Figures 14 and 15. According to the included curves in Figure 14, one can say that there is no variation to some extent when $κ_{1}$ varies. It is observed that the phases of adhesion decrease as the stiffness coefficient of the nonlinear spring increases, as shown in Figure 15. The point of equilibrium has been found in parts (c, d) and (e, f) at $α_{1} = 30$ and $60$ of the same figure.

Figures 16 and 17 explore the variation of the phase plane curves and the corresponding time histories when the linear and nonlinear stiffness of the second spring which is connected to the second mass $m_{2}$ , $κ_{2} (= 1,5,20)$ and $α_{2} (= 5,10,30)$ are considered. The stable points of equilibrium are observed in Figure 16(c) and (d) and consequently Figure 16(e) and (f) when $κ_{2} = 5$ and $κ_{2} = 20$ , respectively. On the other hand, one stable point of equilibrium is given when $α_{2} = 5$ , as drawn in Figure 17(a) and (b), while the equilibrium points are missed when $α_{2}$ has any one of the values $α_{2} (= 10,30)$ , as seen in Figure 17(c)–(f).

The effect of the masses’ variation

This part is considered to examine the effect of changing the values of the masses $m_{1}$ and $m_{2}$ of the studied model through some plane graphical plots that represent the relationship between distance and velocity and also time with distance. Therefore, Figures 18 and 19 are drawn when $m_{1} (= 3,5,30)$ and $m_{2} (= 1,3,10)$ , respectively. For this situation, the above numerical values of the parameter of the system in equation (16) are considered along with the value of excitation speed $v_{0} = 3.2$ . It was found that the system has equilibrium points at $m_{1} (= 3,5,30)$ with an unnoticeable increase in the adhesion processes, as seen in Figure 18(a), (c) and (e) for the phase plane and Figure 18(b), (d) and (f) for the corresponding time histories.

Figure 18.

Illustrates of the variation the phase plane curves: (a, b) $m_{1} = 3$ , (c, d) $m_{1} = 5$ , (e, f) $m_{1} = 30$ .

Figure 19.

Illustrates of the variation the phase plane curves: (a, b) $m_{2} = 1$ , (c, d) $m_{2} = 3$ , (e, f) $m_{2} = 10$ .

A quick glance at the depicted curves in Figure 19 demonstrates that there is one point of equilibrium at $m_{2} = 1$ , as drawn in potions (a) and (b). On the other hand, the adhesion processes increase with the increase of the $m_{2}$ values, as explored in Figure 19(c)–(f), in which these processes become maximum at $m_{2} (= 3,10)$ . The reason is going back to when the value of the second block increases, then the friction increases. In other words, when the second block is loaded, the friction force between it and the belt grows. Therefore, the movement becomes more stable. The reason goes to the mutual influence between both blocks, which is mediated by a damper and a spring.

Conclusion

The motion of two nonlinear spring masses subjected to a smooth friction-velocity curve has been investigated. In light of the system friction forces, the governing second order differential EOM have been derived. These equations have been rewritten in the form of first-order differential equations to examine the equilibrium points applying Hurwitz’s theory. The system characteristic equation and Jacobian matrix are employed to estimate the excitation and critical SS speeds. The stability and behavior of the system motion, along with the behavior of the SS movement, have been examined. The motion and stability of the system have been studied and analyzed to assess the effects of numerous factors, including excitation speed, damping and fraction coefficients, linear and nonlinear spring stiffness, and masses. For varied excitation speeds, the dynamical system under study is asymptotically stable, and the SS phase is important throughout the cycle. As speed increases, the adhesion phase gradually vanishes. Regardless of whether the sticking phase rises or not, damping coefficients have a substantial impact on how the system behaves. Mobility and stability of the system are influenced by the connection between two masses. The outcomes are improved with consideration of the stiffness of nonlinear springs.

The paper’s primary arguments are as follows:

1. With the use of the SMD mechanical model, the dynamic properties of system mechanisms are studied. System critical instability and SS velocity are calculated theoretically as $v_{b 0} = 3.9855 m / s$ and $2.4722$ , respectively. However, the excitation velocity is less than $v_{b 0} = 3.9855 m / s$ system instability above a specific point.

2. Critical instability velocity is calculated using the Hurwitz model to be $v_{b 0} = 3.9855 m / s$ , and the system critical SS speed is determined to be $2.4722$ . The system becomes stable if the excitation speed exceeds $v_{b 0} = 3.9855 m / s$ .

3. The behavior of the system is significantly influenced by damping coefficients.

4. The effects of different stiffness values, whether linear or nonlinear, have been examined, where the nonlinear one having an excellent influence over the linear.

5. The investigated model has been affected by the adjustment of damping and friction coefficients, in addition to the various masses of the blocks.

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

Tarek S Amer

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Analysis of the stick-slip behavior of coupled oscillators with dry friction

Abstract

Keywords

Introduction

The mathematical model and governing equations

Analysis of the SS and critical speeds

Studying the effects of system parameters

Excitation speed effect

Influence of the damping coefficients

Effect of the masses friction coefficient

Equivalent stiffness effect on the masses

The effect of the masses’ variation

Conclusion

Footnotes

Declaration of conflicting interests

Funding

ORCID iD

References