# Equivalent Electronic Circuit of a System of Oscillators Connected with Periodically Variable Stiffness

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## Abstract

**:**

## 1. Introduction

## 2. System Description

## 3. Mathematical Model

#### 3.1. Dimensional Equations of the System

- 1
- ${F}_{\mathrm{Ri}}\left({\dot{x}}_{i}\right)={C}_{i}{\dot{x}}_{i}+\frac{{T}_{i}{\dot{x}}_{i}}{{\left({\dot{x}}_{i}^{2}+{\u03f5}^{2}\right)}^{\frac{1}{2}}}$; $i=1,2$. ${F}_{\mathrm{Ri}}\left({\dot{x}}_{i}\right)$ is the resistance force of the bearings and the term $\frac{1}{{\left({\dot{x}}_{i}^{2}+{\u03f5}^{2}\right)}^{\frac{1}{2}}}$ is the smooth approximation of the function sign(${\dot{x}}_{i}$).
- 2
- ${F}_{\mathrm{Si}}\left({x}_{i}\right)$ is the force due to the magnetic spring.${F}_{\mathrm{Si}}\left({x}_{i}\right)={F}_{M}(\delta -{x}_{i})-{F}_{M}(\delta +{x}_{i})$ = ${F}_{\mathrm{MO}}\left[\frac{1}{{\{1+{d}_{1}(\delta -{x}_{i})\}}^{4}}-\frac{1}{{\{1+{d}_{1}(\delta +{x}_{i})\}}^{4}}\right]$; $i=1,2$. The idea of the above formula is considered by the two assumptions: (i) the repulsive force between two magnets is defined by the simplest expression of the inverse square law, where the dipole expression has been considered. (ii) when ${x}_{i}$ = $\delta $, the first term of the ${F}_{\mathrm{Si}}\left({\dot{x}}_{i}\right)$ expression becomes the unity, when ${x}_{i}$ = $-\delta $, the second term becomes unity, and the expression ${F}_{\mathrm{Si}}\left({\dot{x}}_{i}\right)$ becomes negative value.
- 3
- The stiffness coupling of the considered system, ${K}_{\mathrm{C}}\left(t\right)$ = $\left(\frac{{K}_{1}+{K}_{2}}{2}\right)+\left(\frac{{K}_{1}-{K}_{2}}{2}\right)cos\left(2{\omega}_{\mathrm{e}}t\right)$. It varies periodically having the linear frequency, $f=2\xb7\frac{{\omega}_{\mathrm{e}}}{2\pi}$, where ${\omega}_{\mathrm{e}}$ is the angular frequency of oscillation.

#### 3.2. Non-Dimensional Equations of the System

## 4. Numerical Results

#### 4.1. Behavior of the Dry-Friction and Resistance Force Terms

#### 4.2. Behavior of Magnetic Spring Force Term

#### 4.3. Time-Series and Phase-Space Plots of the Considered System

#### 4.4. Bifurcation Diagrams and the Corresponding Maximal Lyapunov Exponent

#### 4.5. Co-Existence of Two Attractors

## 5. Equivalent Circuit of the Coupled Mechanical Oscillators

- 1.
- ${f}_{\mathrm{Ri}}\left({z}_{i}\right)$ = ${f}_{\mathrm{LDi}}\left({z}_{i}\right)$ + ${f}_{\mathrm{DFi}}\left({z}_{i}\right)$ = $2{\zeta}_{i}{z}_{i}+\frac{{t}_{i}{z}_{i}}{{\left({z}_{i}^{2}+{a}_{1}^{2}\right)}^{\frac{1}{2}}}$; $i=1,2$. ${f}_{\mathrm{Ri}}\left({z}_{i}\right)$ is the non-dimensional resistance force of the bearings. ${f}_{\mathrm{LDi}}\left({z}_{i}\right)$ is the linear damping force, and ${f}_{\mathrm{DFi}}\left({z}_{i}\right)$ is the force due to the dry-friction. The values of ${\zeta}_{1}$, ${\zeta}_{2}$, ${t}_{1}$, ${t}_{2}$ and ${a}_{1}$ are $0.0868$, $0.1036$, $0.2413$, $0.2880$, and $5.8548\times {10}^{-3}$ respectively.
- 2.
- ${f}_{\mathrm{Si}}\left({y}_{i}\right)$ = ${b}_{i}\left[\frac{1}{{\{1+{D}_{1}({\delta}_{1}-{y}_{i})\}}^{4}}-\frac{1}{{\{1+{D}_{1}({\delta}_{1}+{y}_{i})\}}^{4}}\right]$; $i=1,2$., ${f}_{\mathrm{Si}}\left({y}_{i}\right)$ is the non-dimensional force due to the magnetic spring. The values of ${b}_{1}$, ${b}_{2}$, ${D}_{1}$, and ${\delta}_{1}$ are $45.7143$, $54.5737$, $1.5$, and 1 respectively. To reduce the complexity of the equations, we have chosen the most straightforward expressions of the non-dimensional force due to the magnetic spring, which are in the Equations (7) and (8).
- 3.
- The non-dimensional stiffness couplings of the considered system, ${\beta}_{1}\left(\tau \right)$ = $\left[\left(\frac{1+\beta}{2}\right)+\left(\frac{1-\beta}{2}\right)cos\left(2{\omega}_{\mathrm{nd}}\tau \right)\right]({y}_{1}-{y}_{2})$ and ${\beta}_{2}\left(\tau \right)$ = $\mu \left[\left(\frac{1+\beta}{2}\right)+\left(\frac{1-\beta}{2}\right)cos\left(2{\omega}_{\mathrm{nd}}\tau \right)\right]({y}_{2}-{y}_{1})$. The parameter values are $\beta =11.11$, $\mu =1.1938$. The non-dimensional angular frequency, ${\omega}_{\mathrm{nd}}$, will have to be varied in order to obtain the bifurcation diagram.

#### 5.1. Equivalent Circuit Diagram of the Dry-Friction Term ${f}_{\mathrm{DFi}}\left({z}_{i}\right)$

#### 5.2. Equivalent Circuit Diagram of the Magnetic Spring Force Term ${f}_{\mathrm{Si}}\left({y}_{i}\right)$

#### 5.3. Equivalent Circuit Diagram of ${\beta}_{i}\left(\tau \right)$

## 6. Results Obtained from Simulating the Circuit

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**The Schematic representation of a system composed of two oscillators connected with periodically variable stiffness. (Color online).

**Figure 2.**Evolution of the Dry-Friction terms with varying the (

**a**) state-variable ${\dot{y}}_{1}$ of the first oscillator and (

**b**) state-variable ${\dot{y}}_{2}$ of the second oscillator, respectively. In each figure, the x-axis is the non-dimensional velocities (i.e., state-variables ${\dot{y}}_{1}$ and ${\dot{y}}_{2}$) and the y-axis is the non-dimensional dry-friction terms. The parameter values are: ${a}_{1}=5.8548\times {10}^{-3}$, (

**a**) ${\zeta}_{1}=0.0868$, (

**b**) ${\zeta}_{2}=0.1036$. (Color online).

**Figure 3.**Plots of the resistance force terms ${f}_{\mathrm{R}1}\left({\dot{y}}_{1}\right)$ and ${f}_{\mathrm{R}2}\left({\dot{y}}_{2}\right)$ corresponding to the (

**a**) state-variable ${\dot{y}}_{1}$ and (

**b**) state-variable ${\dot{y}}_{2}$, respectively. In each figure, the x-axis is the non-dimensional velocities (i.e., state-variables ${\dot{y}}_{1}$ and ${\dot{y}}_{2}$) and the y-axis is the non-dimensional force due to the resistance term. The parameter values are: ${a}_{1}=5.8548\times {10}^{-3}$, (

**a**) ${\zeta}_{1}=0.0868$, (

**b**) ${\zeta}_{2}=0.1036$. (Color Online).

**Figure 4.**Plots of the magnetic spring force terms ${f}_{\mathrm{S}1}\left({y}_{1}\right)$ and ${f}_{\mathrm{S}2}\left({y}_{2}\right)$ with the variation of the (

**a**) state-variable ${y}_{1}$ and (

**b**) state-variable ${y}_{2}$, respectively. In each of the figures, the x-axis is the non-dimensional displacements (i.e., state-variables ${y}_{1}$ and ${y}_{2}$) and the y-axis is the non-dimensional magnetic spring force terms. The parameter values are: ${D}_{1}=1.5$, ${\delta}_{1}=1$, (

**a**) ${b}_{1}=45.7143$, (

**b**) ${b}_{2}=54.5737$. (Color Online).

**Figure 5.**Comparison the magnetic spring force terms ${f}_{\mathrm{S}1}\left({y}_{1}\right)$ and ${f}_{\mathrm{S}2}\left({y}_{2}\right)$ obtained from the different expressions with the variation of the (

**a**) state-variable ${y}_{1}$ and (

**b**) state-variable ${y}_{2}$. The x-axes are the non-dimensional displacements (i.e., state-variables ${y}_{1}$ and ${y}_{2}$) and the y-axes are the non-dimensional magnetic spring force terms. The parameter values are: ${D}_{1}=1.5$, ${\delta}_{1}=1$, (

**a**) ${b}_{1}=45.7143$, (

**b**) ${b}_{2}=54.5737$. (Color Online).

**Figure 6.**Time-Series waveforms of the considered system for different values of non-dimensional frequency, ${\omega}_{\mathrm{nd}}$. In each of the figure, x-axis is the non-dimensional time and the y-axis is the non-dimensional displacements, ${y}_{1}$ and ${y}_{2}$, and the non-dimensional velocities, ${\dot{y}}_{1}$, ${\dot{y}}_{2}$. The left and the right sides of the upper trace of each figure are the state-variables ${y}_{1}$ and ${\dot{y}}_{1}$ respectively. The left and the right sides of the lower trace of each figure are the state-variables ${y}_{2}$ and ${\dot{y}}_{2}$ respectively. (

**a**) ${\omega}_{\mathrm{nd}}=3.89$, ${y}_{1}$ = ${y}_{2}$ = ${\dot{y}}_{1}$ = ${\dot{y}}_{2}$ = 0, i.e., fixed-point solutions, (

**b**) ${\omega}_{\mathrm{nd}}=4.00$, Period-1 orbit, (

**c**) ${\omega}_{\mathrm{nd}}=4.05$, quasi-periodic orbit, (

**d**) ${\omega}_{\mathrm{nd}}=4.093$, quasi-periodic orbit, (

**e**) ${\omega}_{\mathrm{nd}}=4.098$, chaotic orbit. The parameters are: ${\zeta}_{1}=0.0868$, ${\zeta}_{2}=0.1036$, ${t}_{1}=0.2413$, ${t}_{2}=0.288$, ${b}_{1}=45.7143$, ${b}_{2}=54.5737$, ${a}_{1}=5.8548\times {10}^{-3}$, ${D}_{1}=1.5$, $\beta =11.11$, $\mu =1.1938$, ${\delta}_{1}=1$. The initial condition is chosen to $(0.333,0,0,0)$ to obtain the time-series waveforms. (Color Online).

**Figure 7.**The phase portraits of the considered system for different values of ${\omega}_{\mathrm{nd}}$. x-axis is the non-dimensional displacements ${y}_{1}$, ${y}_{2}$ and the y-axis is the non-dimensional velocities ${\dot{y}}_{1}$, ${\dot{y}}_{2}$, respectively. The left and the right sides of each figure are the phase-portraits of each oscillator. (

**a**) ${\omega}_{\mathrm{nd}}=4.00$, Period-1 orbit, (

**b**) ${\omega}_{\mathrm{nd}}=4.05$, quasi-periodic orbit, (

**c**) ${\omega}_{\mathrm{nd}}=4.093$, quasi-periodic orbit, (

**d**) ${\omega}_{\mathrm{nd}}=4.098$, chaotic orbit. The parameters are: ${\zeta}_{1}=0.0868$, ${\zeta}_{2}=0.1036$, ${t}_{1}=0.2413$, ${t}_{2}=0.288$, ${b}_{1}=45.7143$, ${b}_{2}=54.5737$, ${a}_{1}=5.8548\times {10}^{-3}$, ${D}_{1}=1.5$, $\beta =11.11$, $\mu =1.1938$, ${\delta}_{1}=1$. The initial condition has been chosen at (0.333,0,0,0). (Color Online).

**Figure 8.**Poincaré sections of the considered system for different values of ${\omega}_{\mathrm{nd}}$. x-axis is the non-dimensional displacements ${y}_{1}$, ${y}_{2}$ and the y-axis is the non-dimensional velocities ${\dot{y}}_{1}$, ${\dot{y}}_{2}$, respectively. The left (blue color) and the right (red color) sides of each figure are the poincaré sections of each oscillator. (

**a**) ${\omega}_{\mathrm{nd}}=4.00$, Period-1 orbit, (

**b**) ${\omega}_{\mathrm{nd}}=4.05$, quasi-periodic orbit, (

**c**) ${\omega}_{\mathrm{nd}}=4.093$, quasi-periodic orbit, (

**d**) ${\omega}_{\mathrm{nd}}=4.098$, chaotic orbit. The parameters are: ${\zeta}_{1}=0.0868$, ${\zeta}_{2}=0.1036$, ${t}_{1}=0.2413$, ${t}_{2}=0.288$, ${b}_{1}=45.7143$, ${b}_{2}=54.5737$, ${a}_{1}=5.8548\times {10}^{-3}$, ${D}_{1}=1.5$, $\beta =11.11$, $\mu =1.1938$, ${\delta}_{1}=1$. The initial condition is chosen to $(0.333,0,0,0)$. (Color Online).

**Figure 9.**The bifurcation diagrams of the state variables ${y}_{1}$ and ${y}_{2}$, and the corresponding maximal lyapunov exponent of the system. (

**a**) Bifurcation Diagram of ${y}_{1}$: The x-axis is the non-dimensional parameter ${\omega}_{\mathrm{nd}}$ and the y-axis is the sampled value of ${y}_{1}$, (

**b**) Bifurcation Diagram of ${y}_{2}$: The x-axis is the non-dimensional parameter ${\omega}_{\mathrm{nd}}$ and the y-axis is the sampled value of ${y}_{2}$, (

**c**) The maximum lyapunov exponent: The x-axis is the non-dimensional parameter ${\omega}_{\mathrm{nd}}$ and the y-axis is the lyapunov exponent. The parameters are: ${\zeta}_{1}=0.0868$, ${\zeta}_{2}=0.1036$, ${t}_{1}=0.2413$, ${t}_{2}=0.288$, ${b}_{1}=45.7143$, ${b}_{2}=54.5737$, ${a}_{1}=5.8548\times {10}^{-3}$, ${D}_{1}=1.5$, $\beta =11.11$, $\mu =1.1938$, ${\delta}_{1}=1$. The initial condition is considered as $(0.333,0,0,0)$. (Color Online).

**Figure 10.**Co-existing attractors in the bifurcation diagram of ${y}_{1}$: The x-axis is the non-dimensional parameter ${\omega}_{\mathrm{nd}}$ and the y-axis is the sampled value of ${y}_{1}$. The parameters are: ${\zeta}_{1}=0.0868$, ${\zeta}_{2}=0.1036$, ${t}_{1}=0.2413$, ${t}_{2}=0.288$, ${b}_{1}=45.7143$, ${b}_{2}=54.5737$, ${a}_{1}=5.8548\times {10}^{-3}$, ${D}_{1}=1.5$, $\beta =11.11$, $\mu =1.1938$, ${\delta}_{1}=1$. (Color Online).

**Figure 11.**The equivalent circuit diagram of the force due to the dry-friction ${f}_{\mathrm{DF}1}\left({z}_{1}\right)$. The resistance values are: ${R}_{104}$ = ${R}_{105}$ = ${R}_{106}$ = ${R}_{1}$ = ${R}_{2}$ = ${R}_{3}$ = ${R}_{4}$ = 100 k$\mathsf{\Omega}$, k = $0.2413$. $OA1$ to $OA4$ are the Op-Amps, ${M}_{1}$ to ${M}_{3}$ are the multipliers, ${q}_{1}$ is the dc voltage source with the value of ${a}_{1}^{2}$. K is the gain of a proportional block. (Color Online).

**Figure 12.**The equivalent circuit diagram of the force due to the dry-friction ${f}_{\mathrm{DF}2}\left({z}_{2}\right)$. The resistance values are: ${R}_{119}$ = ${R}_{120}$ = ${R}_{121}$ = ${R}_{9}$ = ${R}_{1}1$ = ${R}_{3}$ = ${R}_{1}$ = 100 k$\mathsf{\Omega}$, k = $0.2413$. $OA4$ to $OA8$ are the four Op-Amps, ${M}_{4}$, ${M}_{5}$, and ${M}_{6}$ are the three multipliers, ${q}_{1}$ is the dc voltage source with the value of ${a}_{1}^{2}$. K is the gain of a proportional block. (Color Online).

**Figure 13.**The equivalent circuit diagram of the force due to the magnetic spring ${f}_{\mathrm{S}1}\left({y}_{1}\right)$. The resistance values are: ${R}_{3}$ = $2.50$ k$\mathsf{\Omega}$, ${R}_{4}$ = $18.26$ k$\mathsf{\Omega}$, $R=100$ k$\mathsf{\Omega}$. $OA9$ is the Op-Amp, ${M}_{7}$ and ${M}_{8}$ are the two multipliers. (Color Online).

**Figure 14.**The equivalent circuit diagram of the force due to the magnetic spring ${f}_{\mathrm{S}2}\left({y}_{2}\right)$. The resistance values are: ${R}_{11}$ = $14.91$ k$\mathsf{\Omega}$, ${R}_{12}$ = $2.1$ k$\mathsf{\Omega}$, $R=100$ k$\mathsf{\Omega}$. $OA10$ is the Op-Amp, ${M}_{9}$ and ${M}_{1}0$ are the two multipliers. (Color Online).

**Figure 15.**The equivalent circuit diagram of the non-dimensional spring force term ${\beta}_{1}\left(\tau \right)$. The resistance values are: R = ${R}_{9}$ = ${R}_{10}={R}_{11}={R}_{13}={R}_{12}={R}_{19}={R}_{20}{R}_{21}=100$ k$\mathsf{\Omega}$. The capacitor $C=1$ nF. OA11, OA12, OA13, and OA14 are the four Op-Amps, ${M}_{1}1$ is the multiplier. ${V}_{\mathrm{in}}$ is the sine wave originating from a function generator. ${V}_{\mathrm{DC}}$ is a dc voltage source of value $6.055$ V. (Color Online).

**Figure 16.**The equivalent circuit diagram of the non-dimensional spring force term ${\beta}_{2}\left(\tau \right)$. The resistance values are: R = ${R}_{84}$ = ${R}_{85}={R}_{83}={R}_{86}={R}_{87}={R}_{88}={R}_{89}={R}_{90}=100$ k$\mathsf{\Omega}$. The capacitor $C=1$ nF. OA15, OA16, OA17, and OA18 are the four Op-Amps, ${M}_{12}$ is the multiplier. ${V}_{\mathrm{in}1}$ is the sine wave. ${V}_{\mathrm{DC}1}$ is a dc voltage source of value $7.2284$ V. (Color Online).

**Figure 17.**The analog circuit diagram of the considered mechanical system. The resistance values are: R = ${R}_{1}$ = ${R}_{2}$ = ${R}_{3}={R}_{4}={R}_{5}={R}_{6}={R}_{16}={R}_{22}={R}_{23}$ = ${R}_{24}={R}_{25}={R}_{68}={R}_{69}={R}_{71}={R}_{72}={R}_{74}={R}_{81}={R}_{82}={R}_{91}={R}_{92}={R}_{93}$ = 100 k$\mathsf{\Omega}$, ${R}_{30}=576.04$ k$\mathsf{\Omega}$, ${R}_{73}=482.62$ k$\mathsf{\Omega}$, ${R}_{14}=2.50$ k$\mathsf{\Omega}$, ${R}_{15}=18.26$ k$\mathsf{\Omega}$, ${R}_{75}=2.1$ k$\mathsf{\Omega}$, ${R}_{76}=14.91$ k$\mathsf{\Omega}$. The capacitor values: $C={C}_{1}={C}_{3}={C}_{8}=1$ nF. ‘OA’—series are the op-amps with different operations, ${M}_{13}$–${M}_{16}$ are the multipliers. The initial capacitor voltage of ${C}_{1}$ is chosen at $0.333$ V, and the remaining capacitor initial voltages are kept at zero voltage. (Color Online).

**Figure 18.**Time-series waveforms of the four state-variables obtained from the circuit simulation for ${\omega}_{\mathrm{nd}}=3.89$. For each figure, x-axis is the time in sec and the y-axis is the values of ${y}_{1}$, ${y}_{2}$, ${z}_{1}$, and ${z}_{2}$, respectively in V. (

**a**) Time-series waveform of ${z}_{1}$, (

**b**) Time-series waveform of ${y}_{2}$, (

**c**) Time-series waveform of ${y}_{1}$, and (

**d**) Time-series waveform of ${z}_{2}$. The frequency of the sine wave generator is $12.382$ kHz. The initial condition has been chosen at (0.333,0,0,0) to obtain the time-series waveforms. (Color Online).

**Figure 19.**Phase-space diagrams of the circuit when $f=12.732$ kHz. The corresponding value of ${\omega}_{\mathrm{nd}}$ is $4.00$. (

**a**) x-axis is the voltage of ${y}_{1}$ in V and y-axis is the voltage of ${z}_{1}$ in V, (

**b**) x-axis is the voltage of ${y}_{2}$ in V and y-axis is the voltage of ${z}_{2}$ in V. The initial condition has been chosen at $(0.333,0,0,0)$ to obtain the phase portraits. (Color Online).

**Figure 20.**Phase-space diagrams of the circuit when $f=12.892$ kHz. The corresponding ${\omega}_{\mathrm{nd}}=4.05$. (

**a**) x-axis is the voltage of ${y}_{1}$ in V and y-axis is the voltage of ${z}_{1}$, (

**b**) x-axis is the voltage of ${y}_{2}$ in V and y-axis is the voltage of ${z}_{2}$ in V. The initial condition has been chosen at (0.333,0,0,0) to obtain the phase portraits. (Color Online).

**Figure 21.**Phase-space diagrams of the circuit when $f=13.044$ kHz. The corresponding ${\omega}_{\mathrm{nd}}=4.098$. (

**a**) x-axis is the voltage of ${y}_{1}$ in V and y-axis is the voltage of ${z}_{1}$ in V, (

**b**) x-axis is the voltage of ${y}_{2}$ in V and y-axis is the voltage of ${z}_{2}$ in V. The initial condition has been chosen at (0.333,0,0,0) to obtain the phase portraits. (Color Online).

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**MDPI and ACS Style**

Seth, S.; Kudra, G.; Witkowski, K.; Awrejcewicz, J. Equivalent Electronic Circuit of a System of Oscillators Connected with Periodically Variable Stiffness. *Appl. Sci.* **2022**, *12*, 2024.
https://doi.org/10.3390/app12042024

**AMA Style**

Seth S, Kudra G, Witkowski K, Awrejcewicz J. Equivalent Electronic Circuit of a System of Oscillators Connected with Periodically Variable Stiffness. *Applied Sciences*. 2022; 12(4):2024.
https://doi.org/10.3390/app12042024

**Chicago/Turabian Style**

Seth, Soumyajit, Grzegorz Kudra, Krzysztof Witkowski, and Jan Awrejcewicz. 2022. "Equivalent Electronic Circuit of a System of Oscillators Connected with Periodically Variable Stiffness" *Applied Sciences* 12, no. 4: 2024.
https://doi.org/10.3390/app12042024