# 1D-CAE model building and simulation based on modal parameter identification of Electrical-Mechanical coupling system

Soichiro Takata

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Soichiro Takata. 1D-CAE model building and simulation based on modal parameter identification of Electrical-Mechanical coupling system. 11th International Operational Modal Analysis Conference, Institut National de Recherche en Informatique et en Automatique [Inria], May 2025, Rennes, France. hal-05398601

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# 1D-CAE model building and simulation based on modal parameter identification of Electrical-Mechanical coupling system

Soichiro Takata<fnref n="1" />

## ABSTRACT

Infrastructure sensing instruments are well-developed in many fields; however, existing measurement devices mainly use passive sensing methods. The vibration response from infrastructure is fundamental small because of its static structure. Thus, the modal response could not be sufficiently measured using microtremors. This study develops a vibration-active sensing device to detect infrastructure deterioration. Previous active sensing methods mainly used scientific measurement instruments, and structural monitoring using these methods is limited. Thus our development target was a simplified active sensing device composed of an Internet of Things sensor and device. Our active vibration-sensing device contained an electrodynamic vibration speaker to excite the target vibration mode. Thus, the desired vibrational mode can be selectively excited in the principal subcomponent, or component structures. Therefore, the device can sufficiently obtain the desired modal response. In development, the most important aspect is the systematic analysis of the electro-dynamic speaker and structure. The electro-dynamic speaker has its resonance characteristics (electrical circuit and mechanical resonances), which are affected by the structural resonance characteristics. For a more detailed analysis, an electrical-mechanical coupling analysis is an effective method. Recently, 1D-computer-aided engineering (CAE) has been used to simulate and design multi-physics, multi-domain, and large-degree-of-freedom systems. This study focuses on the 1D-CAE technology. The application to an electrodynamic vibration speaker system (i.e., a typical electrical-mechanical coupling system) was considered. In particular, electrical power consumption was considered. First, a low-dimensional model of an electrical dynamic vibration speaker was derived using 1D-CAE. We determined the frequency-response function of the electro-dynamic vibration speaker based on the measurements of the input voltage and output acceleration. The frequency-response function contains distinct resonance peaks. Moreover, modal parameter identification equations were derived based on governing equation of the electrical-mechanical coupling system and parameteric fitting was performed. Consequently, we obtained reasonable identified values. Additionally, the 1D-CAE simulations were performed using the identified parameters. 1D-CAE simulations were used to calculate the state variables in all terminals. Thus, electrical power consumption was estimated using the source voltage and current.

<footnote n="1">¹National Institute of Technology, Tokyo College, takata@tokyo-ct.ac.jp</footnote>
# 1. INTRODUCTION

Infrastructure sensing instruments are well developed in many fields [1-5]. This sensing method is based on physically intensive properties; however, existing measurement devices primarily use passive sensing methods. Active sensing methods primarily use scientific measurement specifications; however, structural monitoring using these methods is limited. The vibration response from infrastructure is fundamental small because of its static structure. Thus, the modal response could not be sufficiently measured by using microtremors. Typically, the operational modal analysis method uses only the seismometer's response. However, there is a problem of decreased accuracy in situations with a low signal-to-noise ratio and large damping. This study develops a vibration-active sensing device for the detection of infrastructure deterioration [6-8]. The active vibration-sensing device contained an electrodynamic vibration speaker to excite the target vibration mode. Thus, the desired vibrational mode can be selectively excited in the principal subcomponent, or component structures and can obtain the desired modal response [9-11]. On the other hand, the previous active sensing device was well used by the PZT transducer, the sampling frequency becomes the high sampling rate. Thus, the measurement unit is required as an expensive instrument. Our proposed method has cost merit because the system was composed of the generally used electrodynamic speaker and measurement instrument [12-13].

In device development, the most important task is the systematic analysis of the electro dynamic speaker and structure. The electro-dynamic speaker has its own resonance characteristics (electrical circuit resonance and mechanical resonance), which are affected by the structural resonance characteristics. For a more detailed analysis, an electrical-mechanical coupling analysis is an effective method.

Recently, 1D-computer-aided engineering (CAE) was used to simulate and design multi-physics, multi-domain, and large-degree-of-freedom systems [14-17]. Therefore, the 1D-CAE is suitable for the simulation and state estimation of an electro-dynamic vibration speaker. This study focuses on the 1D-CAE technology. The application to an electrodynamic vibration speaker system (i.e., a typical electrical-mechanical coupling system) was considered with emphasis on electrical power consumption.

<figure><img src="image_2.png" /><figcaption>**Figure 1.** Prototype of vibration sensing actuation device. The device includes the Internet of Things (IoT) sensor, relay circuit, white noise generator, and electro-dynamic speaker.</figcaption></figure>

First, a low-dimensional model of an electrical dynamic vibration speaker was constructed using a 1D-CAE tool. Furthermore, we measured the frequency-response function of the electrodynamic vibration speaker based on input voltage and output acceleration. The frequency-response function exhibited clear resonance peaks. Moreover, modal parameter identification equations were derived based on the
direct current (DC) relationship of the governing equation of the electrical-mechanical coupling system, and parameter fitting was performed. Consequently, we obtained reasonable identified values. Additionally, the 1D-CAE simulations were performed using the identified parameters. 1D-CAE simulations were used to the state variables in all terminals. Thus, electrical power consumption was estimated using the source voltage and current.

## 2. THEORETICAL FORMULATION

### 2.1. 1D-CAE model

In this section, we introduce the 1D-CAE models of an electro-dynamic speaker to activate excitation. The 1D-CAE models are illustrated in Figure 2. The model schematics were generated using the OpenModelica software program. The systems are composed of an electrical source, electrical load (i.e., electrical components), an electrical-mechanical transducer, and a mechanical load (i.e., mechanical component). Model (a) considers the electrical resistance, electrical-mechanical transducer, and mechanical stiffness. The model represents the features of the frequency-response function in the low-frequency band. Model (b) is composed of the electrical resistance, electrical-mechanical transducer, mechanical stiffness, mechanical damper, and mechanical mass. This model considers the contribution of the mechanical resonance characteristics of the electrodynamic speaker. Model (c) considers the inductor in Model (b). The model assumes the contribution of the electrical load to the mechanical resonance characteristics of the electrodynamic speaker. This study verified the frequency-response function of each model.

<figure><img src="image_3.png" /><figcaption>Figure 2. Electrical-mechanical coupled system for excitation function in vibration-sensing-actuation device. The schematics were generated using the OpenModelica software program.</figcaption></figure>

### 2.2. Governing equation

This section describes the governing equations. First, we consider the model shown in Figure 2 (a). The governing equations were derived from the electrical circuit equation and equilibrium of force in the mechanical system. The governing equation becomes a coupled equation: The electrical equation obeys Ohm's law; thus, the input voltage $e_d$ to the system is equal to the product of the current $i_d$ and the resistance $R_d$. Additionally, the force equilibrium obeys action-reaction law, and the stiffness reaction force $k_f x_f$ is equal to the generative force $A i_d$ by the electro-mechanical transducer. Herein, the spring constant is $k_f$, displacement of the electrodynamic speaker is $x_f$, and force factor is $A$. Thus, the governing equation of the circuit in Figure 2 (a) is Eq. (1).

$$ \begin{cases} R_d i_d = e_d \\ k_f x_t = A i_d \end{cases} \qquad (1) $$

The frequency-response function of Eq. (1) can be rewritten as Eq. (2). The frequency response was obtained from the ratio between the input voltage and the output acceleration.
$$ \frac{\ddot{X}_t(\omega)}{E_d(\omega)} = -\frac{A\omega^2/R_d}{k_f} \qquad (2) $$

In this section, we consider the model shown in **Figure 2** (b). Governing equations were derived from the electrical circuit equation and equation of motion in the mechanical system. The circuit equation is the same as Eq. (1). The equation for the mechanical system is expanded using the inertial and damping forces. The governing equation of the circuit shown in **Figure 2** (b) is given by Eq. (3). Herein, the damping constant is $r_f$, velocity of the electrodynamic speaker is $dx_f/dt$, mass is $m_t$, and the acceleration of the electrodynamic speaker is $d^2x_f/dt^2$.

$$ \left\{ \begin{array}{l} R_d i_d = e_d \\ m_t \frac{d^2 x_t}{dt^2} + r_f \frac{dx_t}{dt} + k_f x_t = A i_d \end{array} \right. \qquad (3) $$

The frequency-response function of Eq. (3) can be rewritten as Eq. (4). The frequency response was obtained from the ratio between the input voltage and the output acceleration.

$$ \frac{\ddot{X}_t(\omega)}{E_d(\omega)} = -\frac{A\omega^2/R_d}{-m_t\omega^2 + k_f + ir_f\omega} \qquad (4) $$

We consider the model shown in **Figure 2** (c). Governing equations were derived from the electrical circuit equation and equation of motion in the mechanical system. Herein, the circuit equation is expanded using the voltage by the force factor and the voltage by the inductance. The governing equation of the circuit depicted in **Figure 2** (c) is Eq. (5), where $L_d$ is the inductance.

$$ \left\{ \begin{array}{l} A \frac{dx_t}{dt} + L_d \frac{di_d}{dt} + R_d i_d = e_d \\ m_t \frac{d^2 x_t}{dt^2} + r_f \frac{dx_t}{dt} + k_f x_t = A i_d \end{array} \right. \qquad (5) $$

The frequency-response function of Eq. (5) can be rewritten as Eq. (6). The frequency response was obtained from the ratio between the input voltage and the output acceleration.

$$ \frac{\ddot{X}_t(\omega)}{E_d(\omega)} = \frac{A\omega^2}{-j\omega A^2 - (j\omega L_d + R_d)(-m_t\omega^2 + k_f + ir_f\omega)} \qquad (6) $$

### 2.3. Identification method

In this section, the explanation of the identification method is provided using the paradigm of the case presented in **Figure 2** (c). First, the electrical resistance $\hat{R}_d$ was obtained based on direct measurements using an analog multimeter. Furthermore, the spring constant $\hat{k}_f$ and force factor $\hat{A}$ were obtained from the DC relationship (see, Eq. (1)). A fitting analysis was performed using Eq. (2) using the experimental data in the low-frequency band and eigen-frequency $f_1$. Moreover, the mass was obtained from the mechanical resonant frequency $f_0$, and the damping constant $\hat{r}_f$ was obtained using the half-power method. Finally, inductance $\hat{L}_d$ is obtained from the following eigen-frequency $f_2$:

$$ f_2 = \frac{1}{2\pi} \frac{\hat{A}}{\sqrt{\hat{m}_t L_d}} \qquad (7) $$

## 3. EXPERIMENTS

### 3.1. Frequency-response function measurements

The frequency response was measured using a fast Fourier transform (FFT) analyzer (Ono Sokki, DS-2000). The input-swept sine voltage was generated by the signal output function of the FFT analyzer, and the response was measured using an acceleration pick-up (Ono Sokki, NP-3211). The measurement
conditions were as follows: the sampling frequency was 25.6 kHz, and the voltage ranges were 1.41 V (input voltage) and 0.141 V (response acceleration). The frequency-response functions are shown in **Figure 3**. **Figure 3** (a) shows the frequency responses plotted on logarithmic scales, and **Figure 3** (b) shows the corresponding responses plotted on linear scales. The obtained eigen-frequencies were the $f_0 = 300$ Hz, $f_1 = 100$ Hz, and $f_2 = 480$ Hz.

## 3.2. Identification results

The identified parameters are listed in **Table 1**. These parameters agree with physical sensations. The fitted results of the frequency-response function are shown in **Figure 4** using the fitted parameters and each supposed model are shown in **Figure 4**. Herein, the blue circle shows the experimental values and the black-dotted line shows the case of Eq. (2) (that is, the model in **Figure 2** (a)), the red-dotted line represents Eq. (4) (that is, the model in **Figure 2** (b)), and the blue-dotted line represents Eq. (6) (the model shown in **Figure 2** (c)). The results of the model shown in **Figures 2** (a) and 2 (b) agree well with the experimental results in the low-frequency band at 100 Hz. In addition, the results of the model shown in **Figure 2** (b) are in good agreement with the mechanical resonance curve at approximately 300 Hz. The results of the model shown in **Figure 2** (c) agree with the inductance resonance curve at approximately 480 Hz.

<figure><img src="image_4.png" /><figcaption>**Figure 3.** Measured frequency responses (Bode diagrams). The subfigures on the top represent the amplitude (output acceleration divided by the input voltage), and the subfigure on the bottom represent phase differences.</figcaption></figure>

<table><caption>**Table 1.** Summary of the identified parameters</caption><thead><tr><th>Variable</th><th>R<sub>d</sub>[Ω]</th><th>L<sub>d</sub>[H]</th><th>A[N/A]</th><th>m<sub>t</sub>[kg]</th><th>k<sub>f</sub>[N/m]</th><th>r<sub>f</sub>[Ns/m]</th></tr></thead><tbody><tr><td>Value</td><td>4</td><td>0.0028</td><td>117.8089</td><td>0.6136</td><td>2.1801 × 10<sup>6</sup></td><td>242.8812</td></tr></tbody></table>
<figure><img src="image_5.png" /><figcaption>**Figure 4.** Identification result comparisons between Fig. 2 (a) (without mass and inductor), Fig. 2 (b) (without inductor), and Fig. 2 (c) (full model).</figcaption></figure>

## 4. SIMULATIONS

This section describes the estimation results of electrical power consumption using the 1D-CAE model which are supposed to be the identified parameters in the previous section. The calculation conditions were as follows: the simulation algorithm was Runge-Kutta, the input voltage signal type was white-noise, the standard deviation of the input white noise is 1, and the discrete-time step was 0.0001 s (i.e., the sampling frequency was 10 kHz).

The calculation results are listed in **Table 2**. The current consumptions were also calculated. The maximum current consumption in shown in **Figure 2** (b). The minimum current consumption in shown in **Figure 2** (c). This implies that these results represent the area under the frequency-response function (i.e., the mean-square value of the response spectrum under whitenoise excitation conditions).

In case the model shown in **Figure 2** (b), the frequency-response function has a constant gain amplitude over the mechanical resonance frequency in the frequency band. Consequently, the current consumption was estimated to be large. In contrast, in the model shown in **Figure 2** (c), the amplitude of the frequency-response function decreased in the frequency region above the mechanical resonance frequency. Thus, the current consumption was estimated to be small.

<table><caption>**Table 2.** Summary of the estimated results of consumption current</caption><thead><tr><th rowspan="2">Model</th><th colspan="2">Estimation result of consumption current</th></tr><tr><th>Standard deviation [A]</th><th>Variance [A<sup>2</sup>]</th></tr></thead><tbody><tr><td>Fig. 2 (a)</td><td>0.0430</td><td>0.0019</td></tr><tr><td>Fig. 2 (b)</td><td>0.0595</td><td>0.2737</td></tr><tr><td>Fig. 2 (c)</td><td>0.0035</td><td>0.00001</td></tr></tbody></table>

## 5. CONCLUSIONS

In this paper, we discussed the 1D-CAE model construction and simulations based on the modal parameter identification of an Electrical-mechanical coupling system. The following results were obtained:
(1) The simplified 1D model of an electrical, dynamic vibration speaker was built using a 1D-CAE tool. The simplified model comprised an electrical resistance, inductor, electro-mechanical transducer, a mechanical mass, a spring, and a damper. We considered three types of models (without mass and inductor, without inductor, and with full model).

(2) An identification algorithm was constructed using the governing equations of a simplified 1D-model. The obtained parameters were in agreement with the physical sensation.

(3) Frequency-response function simulations were conducted using the identified parameters. Consequently, the identified frequency-response functions are in good agreement with the experimental values in the case which did not include the inductor model.

(4) A comparison of the estimation of the consumption current was conducted for the three types of models. Consequently, the maximum consumption current was obtained without the inductor model.

## ACKNOWLEDGEMENTS

This work was supported by JST Grant Number JPMJPF2115.

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