# A 2-DoF Kinematic Chain Analysis of a Magnetic Spring Excited by Vibration Generator Based on a Neural Network Design for Energy Harvesting Applications

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Laboratory Stands for the Magnetic Spring and Vibration Generator

#### 2.2. The Magnetic Spring as a 2-DoF Kinematic Chain

_{h}

_{1}is the symbolic representation of the mass center for the first joint and the mass of the levitated magnet is m

_{h}

_{2}. In this case, the value of mass m

_{h1}is equal to 0. The stiffnesses of the springs k

_{h}

_{1}(z) and k

_{h}

_{2}(z) have the same value equal to ½ of the magnetic spring’s stiffness. The stiffnesses of the springs k

_{h}

_{1}(z) and k

_{h}

_{2}(z) depend on the position of the magnet. The damping coefficients b

_{h}

_{1}and b

_{h}

_{2}of the magnetic spring are equal and were calculated using optimization in Matlab [17]. The force F

_{z}is the external force caused by the vibration generator movement. The magnetic spring is acting as an inertial generator; therefore, the force F

_{z}is considered an inertial force (1):

_{h}is the the proof mass and the levitated magnet mass, and a

_{v}is the acceleration of the vibration generator [5].

_{i}is the distance between axes z

_{i}and z

_{i-}

_{1}, α

_{i}is the angle between axes z

_{i}and z

_{i-}

_{1}, d

_{i}is the distance between axes x

_{i}and x

_{i-}

_{1}, θ

_{i}is the angle between axes x

_{i}and x

_{i-}

_{1,}d

_{1}* is the displacement of the prismatic joint, and θ

_{2}* is the displacement of the rotational joint. The * sign means that the parameter changes with the time.

**A**

_{i}for each joint formulated based on Table 1 and Table 2 can be presented in (2):

_{i}= cosθ

_{i}, sθ

_{i}= sinθ

_{i}, cα

_{i}= cosα

_{i}, and sα

_{i}= sinα

_{i}.

**T**

_{2}is given by (3):

_{1}is the position of the levitated magnet in the direction of the magnetization, θ

_{2}is the rotation of the magnet around the radial axis of the magnetization, and a

_{2}is the radius of the levitated magnet.

**T**

_{c2}is shown in (4):

_{c2}is the distance between the geometry center of the levitated magnet and the gravity center of the levitated magnet.

**D**is the inertia matrix,

**C**is the Christoffel matrix, F

_{pi}is the potential force or torque acting on the i joint, F

_{i}is the external force or torque acting on the i joint, and F

_{bi}is the damping forces or torques acting on the i joint.

**D**is obtained as a part of a kinetic energy E

_{k}:

**J**

_{vi}is the Jacobian matrix of the linear speed of the i joint,

**J**

_{wi}is the Jacobian matrix of the rotational speed of the i joint, m

_{i}is the mass of the i link,

**I**

_{i}is the moment of inertia of the i link, and q is the joint displacement [15].

_{c1}can be obtained by Equation (7) and the rotational joint gravity center J

_{c2}by Equation (8):

**z**

_{0}is referred to the base coordinate system {0}—[0,0,1] (Figure 2).

**o**

_{c2}is the center of the second gravity center,

**o**

_{1}is the center of the first gravity center, and

**z**

_{1}is referred to in Figure 2.

**expressed in the coordinate system of the the i joint is calculated by Equation (9):**

_{i}_{1}is the mass of the first joint and is equal to 0, m

_{2}is the mass of the levitated magnet, and I

_{2}is the moment of inertia of the levitated magnet calculated along the radius.

**D**[15], as shown in Equation (11):

_{p1}is the potential force acting on the first joint and the spring force of the magnetic force presented in Equation (15), F

_{b1}is the damping force acting on the first joint presented in Equation (13), τ

_{p2}is the potential and damping torque acting on the second joint presented in Equation (16), and τ

_{b2}is the damping torque acting on the second joint presented in Equation (14). The spring constant force and gravitational force are in equilibrium.

_{1}is the linear damping of the spring and ${\dot{d}}_{1}$ is the linear velocity of the levitated magnet.

_{2}is the rotational damping of the spring, and ${\dot{\theta}}_{2}$ is the rotational velocity of the levitated magnet.

_{m}-axis (Figure 3). The exterior boundaries were set to Neumann boundaries and the interior boundaries between magnets were set to natural boundaries.

_{m}-axis in the range of 0° to 10°. The force approximation and torque equation by Matlab are presented, respectively, in Equations (15) and (16):

_{1}is the position of the levitated magnet, θ

_{2}is the angle of the rotational variation of the levitated magnet, p

_{f00}, …, p

_{f70}are coefficients of the force approximation equation, and p

_{t00}, …, p

_{t60}are coefficients of the torque approximation equation. These coefficients are shown, respectively, in Table 3 and Table 4.

^{−3}Nm are due to the low values of the magnetic field intensity during the rotational movement of the levitated magnet. The highest value of the torque is reached for the highest rotational angle of the levitated magnet. The torque value is the lowest when the levitated magnet is in the middle position between the fixed magnets. Therefore, the entire spring is affected more by the applied force than the torque.

_{m}for a 2-DoF magnetic spring in the z

_{0}-axis (Figure 2) is calculated as:

_{m}= d

_{c1}+a

_{2}sin(θ

_{2})

_{c1}is the linear displacement of the levitated magnet, a

_{2}is the radius of the levitated magnet, and θ

_{2}is the angular position of the levitated magnet.

_{T}(18), derived from the induced voltage, is included in the external force Equation (1):

_{L}is the the load resistance, R

_{C}is the resistance of the one coil, and e is the induced voltage.

_{m}is the displacement of the magnet, and ϕ is the magnetic flux.

_{z}(1) is given by:

_{m}is the relative displacement of the levitated magnet in coil, and p

_{1}, p

_{2}, and p

_{3}are coefficients of the magnetic flux approximation equation. These coefficients are shown in Table 5 for each coil, respectively.

#### 2.3. The Neural Network Model of the Vibration Generator

**w**

_{i}and parameters (thresholds) within the same network to minimize the sum of the squared error functions. The weight of an input is the number which, when multiplied by the input

**x**

_{i}, gives the weighted input. The function g is the unit’s activation function:

**X**and the output variable

**Y**is achieved by adjusting the parameters and weights to reduce errors. The process of finding a set of weights so that the network produces the desired output for a given input is called training. Neural networks learn the relations between different input and output patterns.

**x**is defined as a vector of frequency and amplitude of the current whose waveform was obtained by signal generator AGILENT 33210a amplified by amplifier IRS2092. The amplitude of the current was calculated using fast Fourier transform (FFT). The output y(

**x**) is expressed by the amplitude of the vibration generator obtained by the FFT applied to the signal of laser distance meters LK-G32 (25). The training dataset for the network has 140 samples and the validation dataset has 70 samples. The feedforward backpropagation neural network is composed of the input layer with two neurons arranged in the first hidden layer and other two neurons arranged in the second hidden layer using a Log-sigmoid transfer function (logsig), and an output layer with hyperbolic tangent sigmoid transfer function (tangsig), as shown in Figure 5.

**W**

_{i}is an array containing weights to layer 1 from input 1,

**W**

_{h}is an array containing weights to the hidden layer,

**W**

_{o}is an array containing weights to the output layer,

**b**

_{i}is an array containing bias values to layer 1,

**b**

_{h}is an array containing bias values to hidden layer 2, and

**b**

_{o}is an array containing bias values to the output layer. The weights and biases are shown in Table 6.

#### 2.4. The Simulink Model of Magnetic Spring Based on the Input Signal of the Vibration Generator Obtained by the Neural Network

_{z}shown in Figure 8. In Figure 7, the transposition of the Jacobian of the second joint mass center

**J**

_{c2}

^{T}is obtained by Equation (8). The potential force presented in Equation (15) and damping force for the first joint presented in Equation (13) are contained in block F

_{1}. The potential torque presented in Equation (16) and damping torque for the second joint presented in Equation (14) are contained in block τ

_{2}. The inversion of the inertia matrix

**D**presented in Equation (10) and Christoffel matrix

**C**presented in Equation (11) are contained, respectively, in blocks

**D**

^{−1}and

**C**. The result of the model is vector

**q**of the linear and rotational movement of the levitated magnet. In integration blocks, 1/s acceleration and velocity of the levitated magnet are integrated. The movement of the magnet is limited by the magnetic spring design. The derivation block du/dt is a derivative of the position of the levitated magnet velocity.

**x**) contains the input–output function for the vibration generator model by the ANN presented in Equation (25). The block sin(2πft) contains a sinusoidal function with the input frequency f and simulation time t. The m

_{2}contains the mass of the levitated magnet.

_{2}is 1.77 × 10

^{−3}kg and the inertia moment I

_{2}of the levitated magnet around the axis perpendicular to linear movement is calculated from the magnet’s height and radius and equals 1.24 × 10

^{−8}kgm

^{2}. The distance a

_{2}between the geometry center of the levitated magnet and the point on which the movement was measured is equal to the radius of the levitated magnet 5 × 10

^{−3}m. The distance a

_{c2}between the geometry center of the levitated magnet and the gravity center of the levitated magnet is assumed to be equal to 5 × 10

^{−4}m. The linear damping coefficients b

_{h1}and b

_{h2}were calculated in the optimization process and each equals 0.045 Ns/m. The linear damping coefficient of the whole magnetic spring b

_{1}is the sum of the linear damping coefficients b

_{h1}and b

_{h2}and it is equal to 0.09 Ns/m. The rotational damping coefficient b

_{2}of the whole magnetic spring is equal to 2 × 10

^{−7}Nms/rad.

_{T}is equal to 0. For the simulation with the coil transducer force, F

_{T}is calculated based on Equation (18). Transducer force depends on the velocity of the magnet in the coil. The voltage is calculated by Equation (19) and the magnetic flux by Equation (21). The generated power is calculated by Equation (22). The voltage and electrical power depend on the velocity of the levitated magnet. The magnetic flux depends on the position of the levitated magnet. The position and velocity are obtained from the simulation in Simulink. The load resistance equals the resistance of the coil, which is 24 Ω.

#### 2.5. The FEM Transient Model of the Magnetic Spring

_{L}was calculated. The possible eddy currents that could be induced in the magnets were omitted in this model.

## 3. Results and Discussion

#### 3.1. The Displacement Results

#### 3.2. The Theoretical Electrical Power Outcome

#### 3.3. Future Research

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Measuring station for the vibration generator and movable magnet position. (1) LK-G32, (2) LK-G152, (3) magnetic spring, (4) vibration generator [17], (

**b**) magnetic spring (1) the casing of the magnetic spring, (2) the levitated magnet, (3) fixed magnets.

**Figure 2.**Magnetic spring: (

**a**) section view of coils, fixed and levitated magnets, (

**b**) 2-DoF kinematic chain.

**Figure 4.**The force (

**a**) and torque (

**b**) as a function of the linear and rotational position of the levitated magnet.

**Figure 5.**A multi-layer feedforward ANN of the vibration generator model with an input layer, two hidden layers, and an output layer.

**Figure 6.**Learning curves of neural network for presented neural network model of the vibration generator.

**Figure 7.**Block diagram by Simulink/Matlab of the magnetic spring and vibration generator simulation model.

**Figure 8.**Block diagram by Simulink/Matlab of the vibration generator simulation model based on the neural network model.

**Figure 10.**The amplitude frequency characteristics of the vibration generator and levitated magnet movement for the amplitude of input current (

**a**) 0.35 A and (

**b**) 0.75 A.

**Figure 11.**The amplitude frequency characteristics of the levitated magnet movement for constant input current (

**a**) 0.35 A and (

**b**) 0.75 A in the frequency range of 80–120 Hz.

**Figure 12.**The amplitude as a function of the frequency of measured vibration generator movement for constant input current (

**a**) 0.35 A and (

**b**) 0.75 A.

**Figure 13.**The amplitude as a function of the frequency of the measured and simulated vibration generator movement for constant input current (

**a**) 0.35 A and (

**b**) 0.75 A.

**Figure 14.**The amplitude as a function of the frequency of the measured and simulated levitated magnet movement for constant input current (

**a**) 0.35 A and (

**b**) 0.75 A.

**Figure 15.**The amplitude as a function of the frequency of the 1-DoF and 2-DoF levitated magnet movement simulation for constant input current (

**a**) 0.35 A and (

**b**) 0.75 A.

**Figure 16.**The displacement amplitude as a function of the frequency of the (

**a**) 1-DoF and (

**b**) 2-DoF levitated magnet for external excitation with constant amplitude of sinusoidal force.

**Figure 17.**The displacement amplitude as a function of the frequency of the levitated magnet was calculated by the 1-DoF and 2-DoF kinematic chain models and the FEM model.

**Figure 18.**The amplitude as a function of the frequency of the generated power was calculated for (

**a**) 1-DoF and (

**b**) 2-DoF for external excitation with constant amplitude of sinusoidal force.

**Figure 19.**The amplitude as a function of the frequency of generated power was calculated for the 1-DoF and 2-DoF kinematic chain models and the FEM model.

i | a_{i} | α_{i} | d_{i} | θ_{i} |
---|---|---|---|---|

1 | 0 | $\frac{\mathsf{\pi}}{2}$ | d_{1} * | 0 |

2 | a_{2} | $-\frac{\mathsf{\pi}}{2}$ | 0 | θ_{2} * |

**Table 2.**Kinematic parameters of the gravity centers of the 2-DoF kinematic chain representing the magnetic spring.

i | a_{i} | α_{i} | d_{i} | θ_{i} |
---|---|---|---|---|

1 | 0 | 0 | d_{c1} * | 0 |

2 | a_{c2} | 0 | 0 | θ_{c2} * |

Coefficients | F_{1} | Coefficients | F_{1} | Coefficients | F_{1} |
---|---|---|---|---|---|

p_{f00} | −2.722 × 10^{−4} | p_{f01} | −3.933 × 10^{−5} | p_{f02} | 1.819 × 10^{−5} |

p_{f10} | −0.528 | p_{f11} | 4.439 × 10^{−4} | p_{f12} | −8.381 × 10^{−5} |

p_{f20} | 5.516 × 10^{−4} | p_{f21} | −3.026 × 10^{−4} | p_{f22} | 2.552 × 10^{−5} |

p_{f30} | −8.839 × 10^{−3} | p_{f31} | −2.663 × 10^{−4} | p_{f32} | 4.041 × 10^{−5} |

p_{f40} | −9.927 × 10^{−5} | p_{f41} | 4.18 × 10^{−5} | p_{f42} | −3.334 × 10^{−6} |

p_{f50} | 3.023 × 10^{−4} | p_{f51} | 2.141 × 10^{−5} | p_{f52} | −1.275 × 10^{−6} |

p_{f60} | −4.222 × 10^{−6} | p_{f61} | −5.648 × 10^{−7} | ||

p_{f70} | 8.719 × 10^{−6} |

Coefficients | τ_{2} | Coefficients | τ_{2} | Coefficients | τ_{2} |
---|---|---|---|---|---|

p_{t00} | −1.453 × 10^{−6} | p_{t01} | −1.218 × 10^{−4} | p_{t02} | −8.364 × 10^{−8} |

p_{t10} | 3.055 × 10^{−6} | p_{t11} | −1.97 × 10^{−6} | p_{t12} | 2.304 × 10^{−7} |

p_{t20} | 9.485 × 10^{−6} | p_{t21} | −8.295 × 10^{−6} | p_{t22} | −1.351 × 10^{−7} |

p_{t30} | −4.63 × 10^{−7} | p_{t31} | 9.043 × 10^{−8} | p_{t32} | −2.009 × 10^{−8} |

p_{t40} | −2.757 × 10^{−6} | p_{t41} | 4.552 × 10^{−7} | p_{t42} | 1.484 × 10^{−8} |

p_{t50} | 3.719 × 10^{−8} | p_{t51} | 4.386 × 10^{−9} | ||

p_{t60} | 1.853 × 10^{−7} |

**Table 5.**Coefficients of approximating polynomials of the magnetic flux for first coil ϕ

_{1}and second coil ϕ

_{2}.

Coefficients | ϕ_{1} | Coefficients | ϕ_{2} |
---|---|---|---|

p_{1} | −2.267 × 10^{−3} | p_{1} | 2.265 × 10^{−3} |

p_{2} | −1.418 × 10^{−1} | p_{2} | −1.161 × 10^{−1} |

p_{3} | −8.599 | p_{3} | −7.507 |

Name | Values Obtained by the ANN Model | Details |
---|---|---|

W_{i} | [2.7365 0.028886; 17.1944 −0.055445] | Weights to layer 1 from input 1 |

W_{h} | [−0.53955 2.742; −9.1492 9.123] | Weights to the hidden layer |

W_{o} | [−7.2542 9.7889] | Weights to the output layer |

b_{i} | [2.6625; 13.5824] | Bias to layer 1 |

b_{h} | [−0.11812; −3.5193] | Bias to layer 2 |

b_{o} | [4.0911] | Bias to the output layer |

External Amplitude [mm] | 1-DoF Power [nW] | 2-DoF Power [nW] First Peak | 2-DoF Power [nW] Second Peak |
---|---|---|---|

0.0001 | 1.958 × 10^{−4} | 1.401 × 10^{−4} | 1.208 × 10^{−4} |

0.0005 | 5.775 × 10^{−3} | 3.516 × 10^{−3} | 3.000 × 10^{−3} |

0.001 | 2.356 × 10^{−2} | 1.407 × 10^{−2} | 1.199 × 10^{−2} |

0.005 | 5.982 × 10^{−1} | 3.519 × 10^{−1} | 2.998 × 10^{−1} |

0.01 | 2.396 | 1.408 | 1.202 |

0.05 | 59.931 | 35.340 | 32.007 |

0.1 | 277.931 | 137.643 | 88.231 |

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## Share and Cite

**MDPI and ACS Style**

Bijak, J.; Lo Sciuto, G.; Kowalik, Z.; Trawiński, T.; Szczygieł, M.
A 2-DoF Kinematic Chain Analysis of a Magnetic Spring Excited by Vibration Generator Based on a Neural Network Design for Energy Harvesting Applications. *Inventions* **2023**, *8*, 34.
https://doi.org/10.3390/inventions8010034

**AMA Style**

Bijak J, Lo Sciuto G, Kowalik Z, Trawiński T, Szczygieł M.
A 2-DoF Kinematic Chain Analysis of a Magnetic Spring Excited by Vibration Generator Based on a Neural Network Design for Energy Harvesting Applications. *Inventions*. 2023; 8(1):34.
https://doi.org/10.3390/inventions8010034

**Chicago/Turabian Style**

Bijak, Joanna, Grazia Lo Sciuto, Zygmunt Kowalik, Tomasz Trawiński, and Marcin Szczygieł.
2023. "A 2-DoF Kinematic Chain Analysis of a Magnetic Spring Excited by Vibration Generator Based on a Neural Network Design for Energy Harvesting Applications" *Inventions* 8, no. 1: 34.
https://doi.org/10.3390/inventions8010034