Simulation Of The Electromechanical Device Response To Vibration Impact

Mathematical modelling

The mathematical model is presented in the form of a synthesis of three subsystems, schematically shown in Figure 2.

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Let's consider each of the subsystems separately and their interaction with each other.

Magnetic subsystem

The magnetic subsystem consists of 3 main elements:

- source of magnetic flux– coil $N_W$with electric current $i_i\left(t\right)$;

- magnetic resistance of the air gap with constant and variable components ${\mathfrak{R}}_G=\left(d\right(t)/({\mu }_{\mathrm{0}}{S}_G\left)\right)$;

- magnetic resistance of the magnetic circuit ${\mathfrak{R}}_C=l_C/\left({\mu }_r{\mu }_{\mathrm{0}}{S}_C\right)$.

According to Ampere’s law, the magnetomotive force is determined by the equation:

$F_{\Phi }\left(t\right)=N_W[I_{\text{DC}}+I_i(t\left)\right]={\oint }_C\stackrel{\to }{H}\cdot d\stackrel{\to }{l}$,(1)

here $\stackrel{\to }{H}$– is the vector of the magnetic field strength, $\stackrel{\to }{dl}$– is the vector of elementary displacement along the integration contour C, marked in Figure 1 with a dashed line. From Figure 1 follow that contour integral can be represented as:

${\oint }_C\stackrel{\to }{H}\cdot d\stackrel{\to }{l}=H_C{l}_C+\mathrm{2}{H}_{C}d\left(t\right)$,(2)

here $l_C$– is the average length of the magnetic flux path

For simplicity, we will assume that the cross-sectional area of the magnetic circuit and the air gap are the same. Then, for magnetic flux density (magnetic induction $B$) and magnetic field strength ( $H$) we obtain, $H_C=B/\left({\mu }_r{\mu }_{\mathrm{0}}\right)$ for magnetic circuit and $H=B/{\mu }_{\mathrm{0}}$ for air gap, substituting (2) into (1) we arrive at the result (Zakaria, et al., 2010):

$N_W{i}_i\left(t\right)=\left(\frac{l_C}{{\mu }_r}+\mathrm{2}d\left(t\right)\right)\frac{B\left(t\right)}{{\mu }_{\mathrm{0}}}$,(3)

therefore, the following relation is valid for the magnetic field induction:

$B\left(t\right)=\frac{{\mu }_{\mathrm{0}}{N}_W{i}_i\left(t\right)}{\mathrm{2}\left[d\right(t)+d_C]}$,(4)

where $d_C=l_C/\left(\mathrm{2}{\mu }_r\right)$ - is the length of the path of the magnetic flux in the core of the magnetic circuit and along yoke. We consider the magnetic permeability of the material of the yoke and the magnetic circuit to be the same. (Tatevosyan, et al., 2018).

Electrical subsystem

For the sake othe f simplicity, we neglect the losses in the voltage source, i.e., direct voltage source is ideal. Active resistance of the winding $R_W$ is taken equal to the resistance at direct current. Thus, almost all energy is released in the shut resistor $R_S$. Applying the second Kirchgoff’s law for mesh, shown in Figure 2(b), we obtain the following relationship:

$U_{\text{DC}}=(R_S+R_W)(I_{\text{DC}}+i_l(t\left)\right)+\frac{d\lambda \left(d\right(t),i_t(t\left)\right)}{dt}$,(5)

where $d\lambda \left(d\right(t),i_t(t\left)\right)=N_W\Phi \left(t\right)$– is the magnetic coupling. From equation (4) for magnetic field induction we obtain the following for the magnetic flux:

$\Phi \left(t\right)=S_{C}B\left(t\right)=\frac{{\mu }_{\mathrm{0}}{S}_C{N}_W{i}_t\left(t\right)}{\mathrm{2}\left[d\right(t)+d_C]}$.(6)

Thus, the basic relationship for the electrical part of the subsystem can be written as:

$U_{\text{DC}}=(R_S+R_W)(I_{\text{DC}}+i_i(t\left)\right)+\frac{{\mu }_{\mathrm{0}}{S}_C{N}_W^{\mathrm{2}}}{\mathrm{2}\left[d\right(t)+d_C]}\frac{d{i}_i\left(t\right)}{dt}+\frac{{\mu }_{\mathrm{0}}{S}_C{N}_W^{\mathrm{2}}(I_{\text{DC}}+i_i(i\left)\right)}{\mathrm{2}\left[d\right(t)+d_C{]}^{\mathrm{2}}}\frac{d\left[d\right(t\left)\right]}{dt}$.(7)

From this ratio, it can be concluded about the mutual influence of a time-varying air gap $d\left(t\right)$ on the operation of the magnetization circuit and the mechanical subsystem (Azin, et al., 2017).

Mechanical subsystem

The mechanical subsystem is considered as a vibrational link with one degree of freedom.

External mechanical action is considered in the form of a time-depended force $F\left(t\right)$. The force of magnetic attraction of the yoke to the main magnetic circuit is denoted by $F_M\left(t\right)$.

According to Newton's second law, the equation of motion for the mechanical part can be represented as (Afanasyev & Borisova, 2016):

$m_m\frac{d^{\mathrm{2}}w\left(t\right)}{d{t}^{\mathrm{2}}}+b_S\frac{dw}{dt}+k_{S}w\left(t\right)+F_M\left(t\right)=F\left(t\right)$,(8)

here $w\left(t\right)=d\left(t\right)-d_{\mathrm{0}}$ is the displacement of the yoke from the equilibrium position.

The expression for the force of the magnetic attraction can be obtained from Maxwell’s concept for the forces acting on bodies in a uniform magnetic field with induction:

$F_M\left(t\right)=\frac{\mathrm{1}}{{\mu }_{\mathrm{0}}}{B}^{\mathrm{2}}\left(t\right)S_C$(9)

Substitution expression for $B\left(t\right)$ from (4) into (9) we get:

$F_M\left(d\right(t),I_{\text{DC}},i_i(t\left)\right)=\frac{{\mu }_{\mathrm{0}}{S}_C}{\mathrm{4}}\frac{N_W^{\mathrm{2}}{i}_t^{\mathrm{2}}\left(t\right)}{[d_C+d(t){]}^{\mathrm{2}}}=\frac{{\mu }_{\mathrm{0}}{S}_C}{\mathrm{4}}\frac{N_W^{\mathrm{2}}{i}_t^{\mathrm{2}}\left(t\right)}{[d_C+d_{\mathrm{0}}+w(t){]}^{\mathrm{2}}}$.(10)

From (10) it is $F_M\left(t\right)$ proportional to the square of the total current $i_t\left(t\right)$, flowing through the winding of the electromagnet and, time-varying square of the air gap $d\left(t\right)$.

Substituting (10) to (8) we obtain the equation of motion for the mechanical subsystem:

$m_m\frac{d^{\mathrm{2}}w\left(t\right)}{d{t}^{\mathrm{2}}}+b_S\frac{dw}{dt}+k_{S}w\left(t\right)+\frac{{\mu }_{\mathrm{0}}{S}_C}{\mathrm{4}}\frac{N_W^{\mathrm{2}}{i}_i^{\mathrm{2}}\left(t\right)}{[d_C+d_{\mathrm{0}}+w(t){]}^{\mathrm{2}}}=F\left(t\right)$(11)

Relation (11) shows that the electric magnetic subsystems mutually influence each other, i.e., relations (7) and (11) are interrelated.

Simulation model

The numerical implementation of the electromechanical device model is obtained in the form of a Simscape model, Figure 3 (Mathworks, 2021).

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The “Variable Reluctance” and “Electromagnetic Converter” blocks represent converters, which are gateways that convert energy from one subsystem to another. Each physical subsystem is represented as a set of interacting blocks with lumped parameters. The model is calculated using a built-in constant step solver - ode14x, which combines Newton's method and extrapolation methods to calculate the solution in the next step.

The simulation experiment was conducted with three types of mechanical impacts:

- Random noise – to determine resonance conditions.

- Harmonic, sinusoidal action - to determine the parameters of the system states during transient processes.

- Sinusoidal with a linearly increasing frequency - to determine the conditions for eliminating the negative effect of transient processes on the system.

The numerical values of the parameters are presented in table 1.