Dynamics of a non-ideally driven oscillating system under frictional slip

Abstract

The Sommerfeld Effect pertains to the non-linear jump phenomenon near the resonance frequency of an excited structure powered by a non-ideal drive that is incapable to supply the required power to the oscillating sub-system. Often this situation is dealt with by increasing the viscous damping present in the system at the expense of considerable power loss. However, the use of non-viscous damping for the attenuation of the Sommerfeld Effect is considered in very few articles. In this paper, the dynamics of a non-ideal DC motor-driven vibrating system with frictional slip as the dissipation element is presented. Use of the method of multiple time scales is employed to find the response of the system semi-analytically. The non-linear dynamical characteristics of the response as the motor speed crosses the resonance condition is analyzed using the linear stability analysis and subsequently, the obtained response is verified using numerical results. The transition of the operating condition between stable and unstable zones and the post-resonance dynamic behavior of such systems indicated that under proper parametric condition, it is possible to achieve a smooth variation of operating speed via a combination of stick and slip motion. Furthermore, the rate of voltage increment is also found to play a pivotal role in surpassing the critical speed of the system.

Keywords

Sommerfeld effect non-ideal drive dry friction non-linear vibration

1. Introduction

Almost every practical mechanical system is a non-ideal system as the energy source in the system, commonly known as the drive, is capable of delivering only limited amount of power to the driven sub-system. The process of power transfer is governed by the coupled dynamics of the drive and the driven system (Karthikeyan et al., 2015). When a vibrating system is powered by a DC motor, the amplitude of vibration increases as the operating speed of the motor approaches the speed corresponding to the resonance condition for the system. At this point, any additional power supply contributes to significant increment in the amplitude of vibration with negligible increment in the operating speed of the motor. However, for sufficiently large increment in input power, the motor speed suddenly jumps to a much higher value and at once the amplitude of vibration reduces considerably. While coasting down from a higher operating speed, the system exhibits similar abrupt variation in the motor speed and the vibration amplitude simultaneously near resonance (Nayfeh and Mook, 1979). This undesired sudden jump in angular velocity of the motor and the amplitude of vibration is known as the Sommerfeld Effect (Cveticanin et al., 2018); it prevents the motor from having a smooth control of operating speed and also exposes the bearing support to possible failure due to the sudden change in loading conditions.

The characterization and control of the Sommerfeld Effect in various mechanical systems have posed a challenge for researchers over the years (Cveticanin and Zukovic, 2015). The jump phenomenon in a DC motor-driven rotor-shaft system under both steady state and transient operating conditions was explored using a semi-analytical method by Bisoi et al.( 2017). Anisotropic supports were found to result in multi-Sommerfeld Effect in this system (Bharti et al., 2019). The behavior of a DC motor-driven base-excited mechanism was examined by Sinha et al.( 2020), and the Sommerfeld effect was observed through multiple resonance zones. Three types of Sommerfeld effects in rotor dynamics were explored by González-Carbajal et al. (2023); Samantaray (2022) developed the concept of a quasi-static torque curve to characterize and predict the appearance of the Sommerfeld effect in vibro-compacting machines. Chen et al. (2023) conducted a comprehensive experimental study on the frequency capture attribute of four unbalanced rotors and found that synchronization before and after the natural frequency is pivotal for the Sommerfeld effect. Varanis et al. (2019, 2021) studied the non-linear jump phenomena in the frequency domain using wavelet analysis. Chaotic response was observed in non-ideal vibrating systems (Belato et al., 2001, 2002; Chiroiu and Dumitriu, 2015) considered a pendulum base-excited along the horizontal direction by a slider-crank mechanism and characterized jump phenomena in chaotic vibration response. Warmiński (2001, 2002) presented a detailed analysis of the regular and chaotic vibration response of non-ideally driven Matthieu-type parametrically excited oscillating systems. A comprehensive review of the Sommerfeld Effect in various non-ideal systems was provided in Balthazar et al. (2004; Cvetićanin (2010).

The mathematical framework for non-ideal systems, outlined in Cveticanin et al. (2018), involves a damping term representing energy dissipation. Most articles use viscous damping for this purpose. Increasing energy dissipation is often recommended to mitigate the Sommerfeld Effect (Samantaray et al., 2010). Recently, Iskakov et al.( 2022) showed that joint linear and nonlinear damping can weaken and eliminate the Sommerfeld effect. However, large viscous damping causes high energy dissipation which is not economical. Several attempts to explore the system’s response to various non-viscous damping have also been made. Castão et al. (2008, 2011); Piccirillo et al. (2014) investigated the attenuation of the Sommerfeld effect in a mechanical oscillator connected with an unbalanced non-ideal motor using the Magneto-Rheological Damper (MRD). Felix et al. (2013) used viscoelastic material as a dynamic vibration absorber, Kossoski et al. (2018) employed Shape Memory Alloys (SMA) as actuators, Brogin et al. (2022) proposed a controller based fuzzy Takagi–Sugeno modeling, Petrochino et al. (2023) used an electromagnetic absorber to mitigate the Sommerfeld effect, and of late, Jha and Dasgupta, (2020) proposed a numerically verified novel approach that uses a parametrically varying fractional order parameter of external damping to gradually reduce the nonlinear jump phenomenon. The use of active magnetic bearings in a non-ideal internally damped rotor system, as explored in Dasgupta (2022), revealed that time-delayed feedback through AMB is crucial in suppressing non-linear jump phenomena. There are a handful of such other articles (Jha and Dasgupta, 2022) where non-viscous damping is considered.

An alternative method of elimination of the Sommerfeld Effect is discussed in Chakraborty et al. (2019) where only dry friction is considered as the mode of dissipation of energy. The minimum value of frictional parameter that allows nonslip operating condition until the motor speed crosses the natural frequency of the system is found. With this frictional parameter, the system is found to achieve a smooth transition of motor speed as the oscillating system goes from nonslip condition to slip condition where the oscillation starts and thus, the system avoids the Sommerfeld effect. However, the dry friction present in the system causes stick-slip motion of the mass (Lopez Arteaga et al., 2004; Shaw, 1986). A recent study with frictional damping was presented in Zhang et al. (2021) where, the authors introduced a unique dynamical model for a vibrating mechanical system that involves a single exciter and one outer ring. They examined the impact of sliding dry friction on the vibration amplitude and phase difference between the exciter and the excited sub-system. Another study of such system with frictional slip is presented in Lima and Sampaio (2020) although, the non-linear coupling between the rotational and oscillating degrees of freedom was not considered. Therefore, in slip phase, the coupled dynamics of the motor and the oscillating sub-system is yet remains unexplored.

In this paper, the dynamics of a non-ideal DC motor driven oscillating system is considered where only frictional slip is present as the dissipating element. In this study, a spring-mass system equivalent to a non-ideal DC motor mounted on top of a flexible support is considered. The novelty of this article lies in the approach of explaining the jump phenomenon and the dynamics of the system using the eigenvalue analysis. By effectively combining the system dynamics obtained during the stick phase (Chakraborty et al., 2019) and the stable response obtained in the slip phase, a seamless transition through resonance is found to be possible. Also, the critical rate of voltage increment for the escape of the motor speed from the capture zone is a major finding reported in this article.

This paper is structured as follows—a basic introduction to the Sommerfeld effect in non-ideal systems and a number of prominent earlier works in this field is discussed in Section 1. In Section 2, the mathematical model of the DC motor driven oscillating system with frictional slip as dissipation is formulated. Section 3 deals with the approximate analytical solution of the mathematical model for both constant voltage input and increasing voltage input. The steady-state operating speed of the motor and the amplitude of vibration are analytically obtained in this section. Section 4 focuses on the comparison of the response which is found analytically in the previous section and the numerical results obtained in by simulating the system in MATLAB. Finally, Section 5 concludes this article.

2. Problem definition

In this section, the problem at hand is defined. Initially, the system’s properties, including the mechanical properties of each component, operational constraints, frictional conditions, and so forth are elaborated upon. Subsequently, the equation of motion for the mechanical system which is semi-analytically solved in the next section is found.

2.1. System description

The examined system that is schematically shown in Figure 1 comprises of a motor of mass M, placed on a frictional surface and connected to the fixed end by a linear spring of stiffness k. During operation, the unbalance present in the rotor generates oscillating forces along x − and y − directions; as a result, the motor block exhibits vibration along the x − axis while constantly maintaining contact with the frictional surface. The analysis of the system is conducted under the assumption of continuous slip phase of the motor block, while neglecting the unbalanced mass torque and motor block tipping.

2.2. Governing equations

The mathematical model of a non-ideal DC motor driven oscillating system is derived in Cveticanin et al. (2018) using Lagrangian equations. However, frictional damping (Romano and Garcia, 2008) and electromagnetic coupling (Lima et al., 2018) were not considered in that article. This article introduces a term for frictional slip and incorporates the equation for electromagnetic coupling, resulting in differential equation (1) those describe the mechanical response and electro-mechanical coupling properties of the system for a given voltage input V(t) = V₀ + δV ⋅ t

(M + m) \ddot{x} + k x + μ_{k} ((M + m) g - m e {\dot{φ}}^{2} \sin φ + m e \ddot{φ} \cos φ) \cdot s g n (\dot{x}) = m e (\ddot{φ} \sin φ + {\dot{φ}}^{2} \cos φ)

(1a)

(J + m e^{2}) \ddot{φ} + c_{φ} \dot{φ} - m e \ddot{x} \sin φ = κ I

(1b)

L \dot{I} + R I + κ \dot{φ} = V_{0} + δ V \cdot t

(1c)

The above equations are normalized by defining a non-dimensional displacement and current terms, y and ι, respectively, using characteristic length X and characteristic current I₀ such that, y = x/X and ι = I/I₀, and introducing a rescaled time variable τ = ω₀t. The normalized forms of the equation are

y^{″} + y + ϵ μ_{k} (f - φ^{' 2} \sin φ + φ^{″} \cos φ) s g n (y^{'}) = ϵ (φ^{″} \sin φ + φ^{' 2} \cos φ)

(2a)

φ^{″} + ϵ ζ_{φ} φ^{'} - ϵ {η y}^{″} \sin φ = ϵ γ ι

(2b)

ϵ χ ι^{'} + ι + Γ φ^{'} = v + β τ

(2c)

where

3. Approximate analytical solution

Due to the non-linear and coupled nature of differential equations as presented in equation (2), the derivation of a closed-form solution is found to be exceedingly challenging. Consequently, an approximate solution is commonly sought in such cases. In this section, the system’s responses are studied using the method of multiple-time scales.

3.1. Constant voltage input

In this operating condition β = 0. The method of multiple time-scales is used for the analysis of this system where, $| ϵ | ≪ 1$ as the unbalance mass is significantly smaller than the mass of the motor. Multiple temporal scales are introduced as τ_j = ϵ^jτ such that, d/dτ = ∑_jϵ^jD_j where, D_j = d/dτ_j. The dependent variables are asymptotically expanded in the powers of ϵ as y = ∑_jϵ^jy_j, φ = ∑_jϵ^jφ_j, and ι = ∑_jϵ^jι_j where j = 0, 1, 2, …. After substituting these expressions in equation (1), expanding them and separately collecting the terms with the same powers of ϵ, the perturbative equations are obtained. The $O (1)$ and $O (ϵ)$ equations are written as

D_{0}^{2} y_{0} + y_{0} = 0 \Rightarrow y_{0} = A e^{i τ_{0}} + \tilde{A} e^{- i τ_{0}} = B \cos (τ_{0} + ψ)

(3a)

D_{0}^{2} φ_{0} = 0 \Rightarrow φ_{0} = c τ_{0}

(3b)

ι_{0} + Γ D_{0} φ_{0} = v

(3c)

\begin{array}{l} D_{0}^{2} y_{1} + y_{1} = - 2 D_{1} D_{0} y_{0} - μ_{k} {(f - (D_{0} φ_{0})}^{2} \sin φ_{0} + D_{0}^{2} φ_{0} \cos φ_{0}) s g n (D_{0} y_{0}) + {(D_{0} φ_{0})}^{2} \cos φ_{0} + D_{0}^{2} φ_{0} \sin φ_{0} \end{array}

(4a)

D_{0}^{2} φ_{1} = - 2 D_{1} D_{0} φ_{0} - ζ_{φ} D_{0} φ_{0} + η D_{0}^{2} y_{0} \cdot \sin φ_{0} + γ ι_{0}

(4b)

ι_{1} = - χ D_{0} ι_{0} - Γ D_{1} φ_{0} - Γ D_{0} φ_{1}

(4c)

Here, the normalized motor speed c in equation (3b) is independent of τ₀. A detuning parameter σ is introduced (Cveticanin et al., 2018) to study the near resonance response of the system such that, c ≈ 1 + ϵσ. In the method of multiple time-scale analysis, higher order terms only add a minor correction on lower order terms, which in this case is achieved by eliminating the secular term in τ₀ in the $O (ϵ)$ equations by setting the coefficient of $e^{i τ_{0}}$ in the equation (4a) to zero. Therefore, the solvability condition is obtained as

- 2 i \frac{d A}{d τ_{1}} + \frac{1}{2} e^{i σ τ_{1}} - \frac{2 i μ_{k} f}{π} e^{i ψ} = 0

(5)

It is convenient to represent the complex amplitude term A = (1/2)ae^iΦ, substitute in equation (5) and then separate the real and imaginary parts, resulting in a couple of autonomous equations (6a) and (6b) with the substitution of στ₁ − Φ = θ

\frac{d a}{d τ_{1}} - \frac{1}{2} \sin θ + \frac{2 μ_{k} f}{π} = 0

(6a)

a σ - a \frac{d θ}{d τ_{1}} + \frac{1}{2} \cos θ = 0

(6b)

The fixed points or equilibrium points corresponding to the steady-state response of the dynamical system are found from the condition da/dτ₁ = dθ/dτ₁ = 0. Combining these two conditions yields

σ a = \frac{1}{2} \sqrt{1 - {(\frac{4 μ_{k} f}{π})}^{2}}

(7)

In order to have real, practical results, 4μ_kf/π < 1. Therefore, the necessary condition to maintain continuous slip in this system is found as

μ_{k} < \frac{π m e ω_{0}^{2}}{4 (M + m) g}

(8)

Replacing $B = 2 \sqrt{A \tilde{A}} = a$ from equation (3a) in the equation (4b) and integrating both sides with respect to τ₀, the angular velocity component is obtained as

D_{0} φ_{1} = \frac{η a}{2 (c + 1)} \cos ((c + 1) τ_{0} + ψ) + \frac{η a}{2 (c - 1)} \cos ((c - 1) τ_{0} - ψ) + (γ (v - c Γ) - ζ_{φ} c) τ_{0}

(9)

The angular speed does not increase linearly and diverge with time as the voltage input is constant (V₀). To satisfy this condition, the secular term, that is, the coefficient of the term τ₀ from the right hand side of equation (9) is set to zero. So, when the motor does not operate near resonance, c = γv/(γΓ + ζ_φ). Therefore, if the Sommerfeld Effect did not exist, the motor’s ideal operating speed at that input voltage would have been $ω_{ideal} = κ V / ω_{0} (κ^{2} + c_{φ} R)$ . However, an additional term contributes to the secular term as the motor operates near resonance in the presence of Sommerfeld Effect as the equation (9) becomes

D_{0} φ_{1} = (- ζ_{φ} c + γ (v - Γ c) - \frac{2 η a μ_{k} f}{π}) τ_{0} + \frac{η a}{2 (c + 1)} \cos ((c + 1) τ_{0} + ψ)

(10)

Therefore, the correct expression of the angular velocity component is

c = \frac{γ v}{ζ_{φ} + γ Γ} - \frac{2 η μ_{k} f a}{π (ζ_{φ} + γ Γ)}

(11)

Substituting equation (11) in equation (6b)

\frac{d θ}{d τ_{1}} = \frac{1}{2 a} \cos θ - \frac{Q a}{ϵ} + \frac{W}{ϵ}

(12)

From the steady-state operating condition, that is, dθ/dτ₁ = 0, the normalized amplitude is found as

Q a^{2} - W a + ϵ P = 0 \Rightarrow a = \frac{W \pm \sqrt{W^{2} - 4 ϵ Q P}}{2 Q}

(13)

where

W = \frac{γ v}{ζ_{φ} + γ Γ} - 1 = \frac{κ V}{ω_{0} (κ^{2} + c_{φ} R)} - 1, Q = \frac{2 η μ_{k} f}{π (ζ_{φ} + γ Γ)} = \frac{2 μ_{k} (M + m) g X}{π ω_{0} (\frac{κ^{2}}{R} + c_{φ})}, P = \pm \frac{1}{2} \sqrt{1 - {(\frac{4 μ_{k} f}{π})}^{2}}

3.2. Variable voltage input

For the operating condition where β ≠ 0, similar approximation method is employed to find the response of the system. Equations (3a) and (3b) remain same while the equation (3c) becomes

ι_{0} + Γ D_{0} φ_{0} = v \Rightarrow ι_{0} = v - Γ c + β τ_{0}

(14)

Therefore, near resonance, the equation corresponding to the angular velocity component in

O (ϵ)

becomes

D_{0} φ_{1} = (γ v - (ζ_{φ} + γ Γ) c + β γ τ_{0} - \frac{1}{2} η a \sin (σ τ_{1} - ψ)) τ_{0} + \frac{1}{2 (c + 1)} η a \cos ((c + 1) τ_{0} + ψ)

(15)

So, from the secular term, the normalized motor speed is obtained as

c = 1 + ϵ σ \approx \frac{1}{ζ_{φ} + γ Γ} (γ v + β γ τ_{0} + \frac{1}{2} η a \sin ψ - \frac{1}{2} η a ϵ σ τ_{0} \cos ψ) + O (ϵ^{2})

(16)

For this study, it is assumed that the operating speed is near resonance and it is characterized by the Detuning parameter. As long as $ϵ σ \sim O (ϵ)$ the operating speed remain stuck near resonance. So, for the motor speed to escape from the captured sub-resonance operating zone, the order of the term ϵσ has to be higher than $O (ϵ)$ . The limiting condition is obtained as

\lim_{τ_{1} \to \infty} σ = 1

(17)

For τ₁ → ∞, the terms with linear function of time τ₁ dominate the remaining $O (ϵ)$ terms. As a result, the limiting condition is found as $β γ / \frac{1}{2} η a \cos ψ = ϵ$ and hence, the condition for the rate of voltage increment rate such that the motor speed eventually crosses the resonance condition is

δ V \geq \frac{m^{2} e^{2} ω_{0}^{3} R}{2 κ (M + m)} \times a \cos ψ

(18)

4. Results and discussion

In this section, the amplitude of steady-state vibration and the stability criteria, that is, the property of the eigenvalues, obtained semi-analytically in the previous section are compared with the numerical results. Equation (1) is first converted into state-space form, which consists of five first order differential equations. These equations are then solved in MATLAB using the ode45 routine with the initial conditions

x (0) = \dot{x} (0) = φ (0) = \dot{φ} (0) = I (0) = 0

. The system parameters are listed in Table 1.

Table 1.

System parameters and their values.

System parameters	Description	Value
M	Mass of the motor	5 kg
m	Unbalancing mass	0.05 kg
k	Stiffness of the structure	8080 N/m
e	Eccentricity	0.1 m
J	Moment of inertia of the rotor	0.02 N.m²
κ	Motor constant	0.5
R	Armature resistance	10 Ω
L	Armature inductance	0.001 H
g	Gravitational acceleration	9.81 m/s²
c _φ	Coefficient of rotational damping	0.001 N.m.s/rad
μ _k	Coefficient of kinetic friction	-

For this system, the condition that ensures continuous slip is μ_k < μ_crit = 0.126, that is, if the coefficient of kinetic friction is set less than the critical value, the motor block exhibits pure slip motion during steady state condition; although, it may stick or undergo stick-slip motion for a negligible period during the transient operating condition at the beginning. The natural frequency of this vibrating system is ω_n = 40 rad/s and in ideal condition, the voltage for which the motor reaches this speed is found to be V_ideal = 20.8 V. However, as the DC motor under consideration is non-ideal, the operating speed does not reach the steady state resonance speed at this voltage. For sub-resonance operating condition, that is, $W < 0$ , the amplitude of vibration is

a \approx \frac{W - W (1 - \frac{2 ϵ Q P}{W^{2}})}{2 Q} = \frac{ϵ P}{W} + O (ϵ^{2})

(19)

In this condition,

P < 0

to have positive values of a. Also, the amplitude is very low

(O (ϵ))

. The eigenvalues of the equivalent linearized system are

λ = - \frac{μ_{k} f W}{π ϵ P} \pm \frac{W}{ϵ} \sqrt{{(\frac{μ_{k} f}{π P})}^{2} - 1 - \frac{ϵ P Q}{W^{2}}} \approx \frac{1}{ϵ} (- \frac{μ_{k} f W}{π P} \pm \frac{W}{2} \sqrt{- 5 + \frac{1}{1 - {(\frac{μ_{k} f}{π})}^{2}}})

(20)

The eigenvalues are real if $μ_{k} f / π > 2 / \sqrt{5}$ which is not possible in order to maintain the continuous slip condition as found in expression (8). Thus, the roots are complex and the equilibrium point is stable Focus in this situation. For μ_k = 0.1 and V = 19.5 V which is sub-resonance operating voltage, the numerical response of the system shows that the amplitude of displacement as seen in Figure 2(a) is 2.9 mm which is significantly low as the motor attains a steady state operating speed 37.1 rad/s as shown in Figure 2(b). Using the analytical expressions from equation (19) and equation (11), the amplitude of vibration and the steady-state motor speed are approximately found to be 2.5 mm and 36.6 rad/s, respectively.

As the applied voltage surpasses the ideal critical voltage, the stability condition changes. For operating voltage V above 20.8 V, $W > 0$ . In this operating condition, the parameter $P$ can have both negative and positive sign. If $P < 0$ , then the amplitude of vibration followed by the eigenvalues of the equivalent normalized system are calculated as

a = \frac{W + W (1 - \frac{2 ϵ Q P}{W^{2}})}{2 Q} = \frac{W}{Q} - \frac{ϵ P}{W} + O (ϵ^{2})

(21)

λ \approx - \frac{μ_{k} f Q W}{π (W^{2} - ϵ P Q)} \pm \sqrt{{(\frac{μ_{k} f Q W}{π (W^{2} - ϵ P Q)})}^{2} - \frac{4 P Q}{ϵ}}

(22)

The eigenvalues are real and have different signs. So, the equilibrium point is a Saddle point which is unstable. In Figure 3(b), at a voltage of 22 V, it is observed that the resonance frequency cannot be exceeded by the motor speed, while in Figure 3(a), a very high amplitude of vibration displacement of the motor block is depicted. This is indicative of a typical response characteristic of the unstable operating condition previously characterized by the Saddle point. However, in case of $P > 0$ , real and practical operating conditions are achieved if $W > W_{crit}$ , that is, the applied voltage is greater than a critical finite voltage such that

W_{crit} = \sqrt{4 ϵ P Q} = \sqrt{\frac{4 μ_{k} m e g}{π ω_{0} (\frac{κ^{2}}{R} + c_{φ})} \cdot \sqrt{1 - {(\frac{4 μ_{k} (M + m) g}{π m e ω_{0}^{2}})}^{2}}}

(23)

For this system, $W_{crit} \approx 0.06$ . From equation (23), the critical voltage is found to be V_crit ≈ 22.1 V. For V = 22 V, that is, $W < W_{crit}$ , numerical results indicate that the motor’s operational velocity does not surpass ω₀ (as shown in Figure 3(b)) and remains in an unstable state. Although there is a brief moment when the transient operational speed exceeds ω₀, this condition is unstable, as indicated by the presence of the Saddle equilibrium point identified through the eigenvalues in equation (22). Furthermore, this unstable state is marked by a significantly high amplitude of displacement of approximately 22.8 mm, as shown in Figure 3(a).

It is V = 22.125 V when the motor speed is about to cross the resonance speed (Figures 4(a) and (b)). As V = 22.25 V is applied, that is, $W > W_{crit}$ , the motor speed surpasses ω₀ and settles to a steady state speed ( $\approx 41.8$ rad/s) above ω₀ (Figure 5(b)) and simultaneously, the amplitude of steady state vibration (Figure 5(a)) becomes significantly low ( $\approx 5.9$ mm). As the operating speed crosses the resonance condition, the phase shifts by π (Nayfeh and Mook, 1979) and this is supported by the observation that the parameter $P$ switches from negative to positive. In this operating condition, the amplitude of vibration and the eigenvalues of the equivalent normalized system are same as found in equations (19) and (20), respectively. Therefore, the equilibrium points are stable Focus and the system operates in steady state for V ≥ 22.25 V.

Figure 6 illustrates how the steady-state operating speed changes with the applied voltage. Within the voltage range of 20.8 − 22.1 V, there is minimal fluctuation in angular speed. However, since the operating speed does not attain steady state within this voltage range, that specific section of the plot is not accurate. Nonetheless, the graph shows the Sommerfeld Effect occurring at V = 22.125 V, followed by a gradual increase in steady-state speed with an increase in applied voltage. Additionally, it is worth noting that, as the coefficient of kinetic friction is raised while μ_k < μ_crit, that is, without violating the continuous slip condition, the critical voltage required to cross the resonance point decreases and motor speed jumps to lower steady-state values. However, qualitatively the post-jump motor speed variation with applied voltage is similar in the stable operating region.

Next, the scenario involving an increase of voltage input during operation is investigated. When the motor is started with an initial applied voltage V₀ = 19.5 V and the voltage is increased at a rate of 0.3 V/s, interesting behavior occurs as the motor speed is found to be captured near resonance speed as shown in Figure 7(b) while the amplitude of oscillation increases with time as shown in Figure 7(a). Remarkably, even when t = 200 s, meaning the applied voltage has reached V = 19.5 + 0.3× 200 = 79.5 V, the motor speed remains stuck near the resonance speed. However, for constant voltage input, the motor is able to pass through resonance at V₀ = 22.25 V. Considering σ ≈ 1 as seen from equation (17), the normalized steady-state amplitude just before the jump is approximately found from equation (7) as a ≈ 0.3. Also, the numerical results after solving equations (6a) and (6b) suggests that, the steady-state value of cos Φ, which is also equal to cos ψ ≈ 0.6. Replacing these values along with the values of the system parameters in equation (18), the critical value of the voltage increment rate is found to be δV_crit = 0.57 V/s. However, as the rate of voltage increment is set as 0.4 V/s, the motor speed, as shown in Figure 8(b) manages to surpass the resonance speed while the oscillation amplitude decreases simultaneously (Figure 8(a)). At t ≈ 9 s when the applied voltage reaches V = 19.5 + 0.4× 9 = 23.1 V, the motor speed crosses the resonance condition. In this scenario, the voltage at which this motor speed exhibits a jump closely aligns with the voltage required for a jump under constant voltage input at V = 22.25 V.

In Chakraborty et al. (2019), researchers determined the critical static friction coefficient $(μ_{s}^{crit})$ , which permits slipping only after the motor speed exceeds resonance. The findings in this paper demonstrate that a steady-state motor speed is attained after passing the resonance condition in the presence of frictional slip. Therefore, the utilization of both non-slip and slip conditions effectively can prevent the occurrence of the Sommerfeld Effect. When the static friction coefficient falls below $μ_{s}^{crit}$ , oscillation is initiated by the motor before reaching the resonance speed. However, one may navigate through resonance by increasing the input voltage at a rate exceeding the minimum advised voltage increment rate, as found in this article; although, a minute surge in motor velocity (39.9 rad/s to 41.8 rad/s as indicated in Figure 6) ensues under these circumstances, as opposed to a significant jump in motor velocity observed in systems with negligible viscous damping (Bharti et al., 2019; Bisoi et al., 2017; Drozdetskaya and Fidlin, 2021; Karthikeyan et al., 2015; Sinha et al., 2018).

5. Conclusion

This article presents a comprehensive study of a DC motor-driven oscillating system with frictional slip as damping. The main outcomes of this article are listed below.

1. The frictional condition that ensures continuous slip is established. When the coefficient of kinetic friction is kept below a critical value, the motor block exhibits pure slip motion during steady-state operation. This critical value is obtained from the solvability condition, as shown in equation (8).

2. Starting from sub-resonance, the motor speed enters the capture zone near resonance and remains there for a specific voltage range. This causes equilibrium points to shift from stable Focus to unstable Saddle points, resulting in Bifurcation. When the motor speed exceeds resonance, equilibrium points switch back to stable Focus from unstable Saddle points, causing another Bifurcation and explaining the non-linear jump phenomenon. The role of the phase switch in steady-state oscillation contributing to the jump phenomenon is also addressed.

3. Numerical findings indicate that raising the coefficient of kinetic friction while maintaining continuous slip leads to a reduced critical jump voltage. This, in turn, results in a decrease in motor speed to lower steady-state levels, as shown in Figure 6. This trend is qualitatively similar to the impact of increasing viscous damping.

4. Raising the voltage during operation is a means to exit the resonance capture region. However, the motor’s speed remains near resonance until the voltage increment rate surpasses a critical threshold. In this condition, any surplus power injected into the system is channeled into the oscillating sub-system, leading to a rise in oscillation amplitude. The transition beyond resonance occurs once the critical voltage increment rate is exceeded, allowing for a smooth adjustment in motor speed. It’s worth noting that the analytical estimate for the critical rate (0.55 V/s) exceeds the numerical result (0.4 V/s).

It’s important to note that during the slip phase, there’s a brief occurrence of stick-slip motion when the coefficient of friction (μ) falls between μ_k (as indicated in equation 8) and $μ_{s}^{crit}$ . The article doesn’t explore the characteristics of the Sommerfeld Effect in the system under stick-slip conditions, providing an opportunity for future research.

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

Pratyaya Chakraborty

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