Analytical method for cantilevered Fe-Gallium alloy magnetostrictive vibration energy harvesting model

Abstract

The present invention discloses an analytical analysis method for a cantilevered iron-gallium alloy magnetostrictive vibration energy harvesting model. First, a vibration energy harvesting model is made. The vibration energy harvesting model includes a horizontally arranged composite beam, which includes an additional layer of an iron-gallium alloy material and a structural layer of an aluminum material. One end of the composite beam is fixed on a base, and the other end of the composite beam is connected to a resonance adjustment block; an induction coil is spirally wound on the composite beam, and the induction coil is connected to a current measurement module; then, up and down reciprocating impacts are applied to the base to make the composite beam vibrate; finally, the output voltage and output power of the vibration energy harvesting model are calculated using a fixed formula. The significant effect of the technical solution of the present invention is that the output voltage and output power of the cantilevered iron-gallium alloy vibration energy harvesting model can be explicitly expressed, and the calculation results can be obtained relatively quickly, thereby providing a decision-making basis for related designs.

Description

Technical Field

[0001] The invention relates to a vibration energy collection technology of a magnetostrictive material, and in particular to a vibration energy analysis method. Background Art

[0002] In the field of energy harvesting technology, vibration energy is the most common energy in the environment and can be applied to a wide range of occasions. Therefore, vibration energy harvesting technology has always been a hot topic for domestic and foreign scholars. Among vibration energy harvesting technologies, piezoelectric, electromagnetic, electrostatic, and magnetostrictive are the most in-depth. As an emerging magnetostrictive material, iron-gallium alloy has the advantages of large strain, low brittleness, and high tensile strength, and can withstand large tension during bending, which has attracted widespread attention.

[0003] At present, most of the research on vibration energy harvesting of iron-gallium alloys uses cantilever beams for theoretical modeling, and the method uses the principles of structural dynamics and the Runge-Kutta method to calculate numerical solutions. However, the numerical solution has the disadvantages of inaccurate results, large errors, and relatively long time consumption when using numerical calculation tools such as Mathematica to solve. Summary of the invention

[0004] The present invention provides an analytical analysis method for a cantilevered iron-gallium alloy magnetostrictive vibration energy collection model, which is performed in the following steps:

[0005] Step 1: Make a vibration energy harvesting model

[0006] The vibration energy harvesting model comprises a horizontally arranged composite beam, the composite beam comprises an additional layer and a structural layer bonded to each other, the additional layer is made of an iron-gallium alloy material, the structural layer is made of aluminum, the structural layer is located below the additional layer, the additional layer and the structural layer have a length of L, the additional layer and the structural layer have a width of b, and the additional layer has a thickness of h g ;

[0007] The composite beam is arranged horizontally, one end of the composite beam is fixed on a base, and the other end of the composite beam is connected to a resonance adjustment block;

[0008] An induction coil is spirally wound on the composite beam, and the induction coil is connected to a current measurement module;

[0009] A coordinate system is established with the neutral plane of the composite beam as the x-axis direction, the fixed end of the composite beam as the origin of the x-axis, and the vertical direction as the y-axis direction;

[0010] Step 2: Apply reciprocating impacts up and down to the base to make the composite beam vibrate, and the current measurement module continuously measures the induced current output by the induction coil, and calculates the x-axial stress σ of the iron-gallium alloy according to formula ①g , and magnetic induction intensity B:

[0011]

[0012] E g is the Young's modulus of the Fe-Ga alloy;

[0013] μ is the magnetic permeability of the Fe-Ga alloy

[0014] d is the magnetomechanical coupling piezomagnetic coefficient of the Fe-Ga alloy;

[0015] ε is the strain of the composite beam;

[0016] N is the number of turns of the induction coil wound on the composite beam;

[0017] i(t) is the induced current output by the induction coil;

[0018] t represents time;

[0019] Step 3: According to Hamilton's principle, we have formula ②;

[0020]

[0021] in:

[0022] δ is the Dirac delta function;

[0023] TT, V, W nc denote kinetic energy, potential energy and virtual work done by non-conservative forces respectively;

[0024] TT is calculated according to the following formula ③:

[0025]

[0026] in:

[0027] m is the mass per unit length of the composite beam;

[0028] x is the coordinate value of the composite beam at the x-coordinate;

[0029] w rel (x, t) is the y-direction displacement of the composite beam relative to the base at the x-coordinate at time t; w b (t) is the base displacement at time t;

[0030] M t is the mass of the resonance adjustment block;

[0031] wrel(L, t) is the y displacement of the composite beam relative to the base at time t and when the x coordinate is L; I t is the moment of inertia;

[0032] V is calculated according to the following formula ④:

[0033]

[0034] in:

[0035] EI is the stiffness of the composite beam at the x coordinate;

[0036] W nc Calculate according to the following formula⑤:

[0037] W nc =W ele +W damp , formula ⑤;

[0038] in:

[0039]

[0040] h c is the distance between the top of the iron-gallium alloy and the neutral plane;

[0041] h b is the distance between the bottom end of the iron-gallium alloy and the neutral plane;

[0042] H(x) is the step function of variable x;

[0043] H(xL) is the step function of the variable xL;

[0044]

[0045] c s is the strain rate damping coefficient;

[0046] I(x) is the moment of inertia of the cross section at the x coordinate;

[0047] c a is the air damping coefficient;

[0048] Substitute formulas ③, ④, and ⑤ into formula ② to obtain formula ⑥:

[0049] in:

[0050]

[0051] Boundary conditions:

[0052] When x=L, we have formula ⑦

[0053]

[0054] When x=0, we have formula⑧

[0055]

[0056] Step 4: Based on Faraday's law of electromagnetic induction, the induced voltage ΔV generated in the Fe-Ga alloy layer with a length of ΔL in the x-axis direction is g (t) Calculated according to formula 9:

[0057]

[0058] in:

[0059] A is the cross-sectional area of ​​the Fe-Ga alloy layer;

[0060] Substitute formula ① into formula ⑨, and let ΔV g (t)→dV g (t), ΔL→dx, and integrate the resulting equation along the x-axis to obtain formula ⑩:

[0061]

[0062] in:

[0063]

[0064] R=R c +R L

[0065] R c is the resistance of the induction coil;

[0066] R L is the resistance of the current measurement module;

[0067] The x-axial strain of the composite beam at thickness y Let ε y =ε, and put it into formula ⑩, we get the following formula

[0068]

[0069] in:

[0070]

[0071] Step 5: Use the Galerkin decomposition method to discretize the motion control equation of the overall system. Calculate w rel (x,t):

[0072]

[0073] in:

[0074] r is the order of the vibration energy harvesting model;

[0075] φ r (x) is the rth order vibration mode of the vibration energy harvesting model;

[0076] q r (t) is the r-th order modal coordinate of the vibration energy harvesting model;

[0077] The formula Substitute formula ⑥ and formula Formula

[0078]

[0079] in:

[0080] θ g =-NdE g h g h pc φ r ′(L);

[0081] C m =2ξω;

[0082]

[0083] The second derivative of the first-order modal coordinates of the vibration energy harvesting model with respect to time t;

[0084] The first derivative of the first-order modal coordinates of the vibration energy harvesting model with respect to time t;

[0085] q(t) is the first-order modal coordinate of the vibration energy harvesting model;

[0086] F(t) is the basic incentive force;

[0087] is the first derivative of the induced voltage with respect to time t;

[0088] h pc is the distance from the center layer to the neutral plane in the thickness direction of the Fe-Ga alloy;

[0089] ω is the free vibration natural frequency of the composite beam;

[0090] φ r ′(L) is the first derivative of the rth vibration mode of the vibration energy harvesting model when x=L;

[0091] ξ is the damping ratio;

[0092] Let the modal force be frequency ω b The simple harmonic force, the steady-state solution of which is given by the formula

[0093]

[0094] in:

[0095] F0 is the modal force amplitude;

[0096] q0 is the modal coordinate amplitude;

[0097] is the phase difference between F(t) and q(t);

[0098] ω b is the excitation frequency;

[0099] V0 is the output voltage;

[0100] Combining formulas Solve the output voltage V0 of the vibration energy harvesting model, as shown in the formula

[0101]

[0102] in:

[0103] BRIEF DESCRIPTION OF THE DRAWINGS

[0104] Figure 1 After applying up and down reciprocating impacts of accelerations of 1g, 2g, 3g, and 4g to the base, power-resistance curves are obtained using the method of the present application and the traditional method respectively. DETAILED DESCRIPTION

[0105] The present invention will be further described below with reference to the embodiments.

[0106] An analytical analysis method for a cantilevered iron-gallium alloy magnetostrictive vibration energy harvesting model is carried out in the following steps:

[0107] Step 1: Make a vibration energy harvesting model

[0108] The vibration energy harvesting model comprises a horizontally arranged composite beam, the composite beam comprises an additional layer and a structural layer bonded to each other, the additional layer is made of an iron-gallium alloy material, the structural layer is made of aluminum, the structural layer is located below the additional layer, the additional layer and the structural layer have a length of L, the additional layer and the structural layer have a width of b, and the additional layer has a thickness of h g ;

[0109] The composite beam is arranged horizontally, one end of the composite beam is fixed on a base, and the other end of the composite beam is connected to a resonance adjustment block;

[0110] An induction coil is spirally wound on the composite beam, and the induction coil is connected to a current measurement module;

[0111] A coordinate system is established with the neutral plane of the composite beam as the x-axis direction, the fixed end of the composite beam as the origin of the x-axis, and the vertical direction as the y-axis direction;

[0112] Step 2: Apply reciprocating impacts up and down to the base to make the composite beam vibrate, and the current measurement module continuously measures the induced current output by the induction coil, and calculates the x-axial stress σ of the iron-gallium alloy according to formula ① g , and magnetic induction intensity B:

[0113]

[0114] E g is the Young's modulus of the Fe-Ga alloy;

[0115] μ is the magnetic permeability of the Fe-Ga alloy

[0116] d is the magnetomechanical coupling piezomagnetic coefficient of the Fe-Ga alloy;

[0117] ε is the strain of the composite beam;

[0118] N is the number of turns of the induction coil wound on the composite beam;

[0119] i(t) is the induced current output by the induction coil;

[0120] t represents time;

[0121] Step 3: According to Hamilton's principle, we have formula ②;

[0122]

[0123] in:

[0124] δ is the Dirac delta function;

[0125] TT, V, W nc denote kinetic energy, potential energy and virtual work done by non-conservative forces respectively;

[0126] TT is calculated according to the following formula ③:

[0127]

[0128] in:

[0129] m is the mass per unit length of the composite beam;

[0130] x is the coordinate value of the composite beam at the x-coordinate;

[0131] wrel (x, t) is the y-direction displacement of the composite beam relative to the base at the x-coordinate at time t;

[0132] w b (t) is the base displacement at time t;

[0133] M t is the mass of the resonance adjustment block;

[0134] w rel (L, t) is the y-direction displacement of the composite beam relative to the base at time t and when the x-coordinate is L;

[0135] I t is the moment of inertia;

[0136] V is calculated according to the following formula ④:

[0137]

[0138] in:

[0139] EI is the stiffness of the composite beam at the x coordinate;

[0140] W nc Calculate according to the following formula⑤:

[0141] W nc =W ele +W damp , formula ⑤;

[0142] in:

[0143]

[0144] h c is the distance between the top of the iron-gallium alloy and the neutral plane;

[0145] h b is the distance between the bottom end of the iron-gallium alloy and the neutral plane;

[0146] H(x) is the step function of variable x;

[0147] H(xL) is the step function of the variable xL;

[0148]

[0149] c s is the strain rate damping coefficient;

[0150] I(x) is the moment of inertia of the cross section at the x coordinate;

[0151] c a is the air damping coefficient;

[0152] Substitute formulas ③, ④, and ⑤ into formula ② to obtain formula ⑥:

[0153]

[0154] in:

[0155]

[0156] Boundary conditions:

[0157] When x=L, we have formula ⑦

[0158]

[0159] When x=0, we have formula⑧

[0160]

[0161] Step 4: Based on Faraday's law of electromagnetic induction, the induced voltage ΔV generated in the Fe-Ga alloy layer with a length of ΔL in the x-axis direction is g (t) Calculated according to formula 9:

[0162]

[0163] in:

[0164] A is the cross-sectional area of ​​the Fe-Ga alloy layer;

[0165] Substitute formula ① into formula ⑨, and let ΔV g (t)→dV g (t), ΔL→dx, and integrate the resulting equation along the x-axis to obtain formula ⑩:

[0166]

[0167] in:

[0168]

[0169] R=R c +R L

[0170] R c is the resistance of the induction coil;

[0171] R L is the resistance of the current measurement module;

[0172] The x-axial strain of the composite beam 1 at thickness y Let ε y =ε, and put it into formula ⑩, we get the following formula

[0173]

[0174] in:

[0175]

[0176] Step 5: Use the Galerkin decomposition method to discretize the motion control equation of the overall system. Calculate w rel (x,t):

[0177]

[0178] in:

[0179] r is the order of the vibration energy harvesting model;

[0180] φ r (x) is the rth order vibration mode of the vibration energy harvesting model;

[0181] q r (t) is the r-th order modal coordinate of the vibration energy harvesting model;

[0182] The formula Substitute formula ⑥ and formula Formula

[0183]

[0184] in:

[0185] θ g =-NdE g h g h pc φ r ′(L);

[0186] C m =2ξω;

[0187]

[0188] The second derivative of the first-order modal coordinates of the vibration energy harvesting model with respect to time t;

[0189] The first derivative of the first-order modal coordinates of the vibration energy harvesting model with respect to time t;

[0190] q(t) is the first-order modal coordinate of the vibration energy harvesting model;

[0191] F(t) is the basic incentive force;

[0192] is the first derivative of the induced voltage with respect to time t;

[0193] h pc is the distance from the center layer to the neutral plane in the thickness direction of the Fe-Ga alloy;

[0194] ω is the free vibration natural frequency of the composite beam;

[0195] φ r ′(L) is the first derivative of the rth vibration mode of the vibration energy harvesting model when x=L;

[0196] ξ is the damping ratio;

[0197] Let the modal force be frequency ω b The simple harmonic force, the steady-state solution of which is given by the formula

[0198]

[0199] in:

[0200] F0 is the modal force amplitude;

[0201] q0 is the modal coordinate amplitude;

[0202] is the phase difference between F(t) and q(t);

[0203] ω b is the excitation frequency;

[0204] V0 is the output voltage;

[0205] Combining formulas Solve the output voltage V0 of the vibration energy harvesting model, as shown in the formula

[0206]

[0207] in:

[0208]

[0209] The output power P of the vibration energy harvesting model is calculated according to the following formula:

[0210] The first-order vibration displacement is A tip , A tip =q0φ(L):

[0211] φ(L) is the first-order vibration mode of the vibration energy harvesting model when x=L;

[0212]

[0213] Output voltage V0: Substitute the expression of q0 into formula 15

[0214]

[0215] Output Power Substituting V0 expression into:

[0216]

[0217] Test example:

[0218] The base is subjected to reciprocating impacts of 1g, 2g, 3g, and 4g accelerations, respectively, and the output power under different loads is solved by the method of the above embodiment (analytical solution power), and the power-resistance curve is obtained, as shown in FIG. Figure 1 As shown;

[0219] According to the same impact conditions (applying 1g, 2g, 3g, and 4g accelerations to the base, respectively), the corresponding numerical power solution is calculated by the traditional method to obtain the voltage-power curve, as shown in Figure 1 As shown;

[0220] from Figure 1 It can be seen that the result (analytic solution power) obtained by the method of this scheme is highly consistent with the corresponding numerical solution power calculated by the traditional method under the same conditions, indicating that the method of the present invention is suitable for analyzing the cantilevered iron-gallium alloy magnetostrictive vibration energy harvesting model. The analytical solution obtained by this method is an explicit expression with clearer physical meaning; while the numerical solution obtained by the traditional method is generally solved by matrix method, and the required parameters are generally solved by iteration, so it takes a long time.

[0221] Beneficial effect: The analytical analysis method of the cantilevered iron-gallium alloy magnetostrictive vibration energy harvesting model of the present invention can explicitly express the output voltage and output power of the cantilevered iron-gallium alloy vibration energy harvesting model, and can obtain the calculation results relatively quickly, thereby providing a decision-making basis for related designs.

[0222] Finally, it should be noted that the above description is only a preferred embodiment of the present invention. Under the guidance of the present invention, ordinary technicians in this field can make various similar expressions without violating the purpose and claims of the present invention, and such changes all fall within the scope of protection of the present invention.

Claims

1. An analytical method for cantilevered iron-gallium alloy magnetostrictive vibration energy harvesting model, characterized in that Follow these steps: Step 1: Make a vibration energy harvesting model The vibration energy harvesting model comprises a horizontally arranged composite beam, the composite beam comprises an additional layer and a structural layer bonded to each other, the additional layer is made of an iron-gallium alloy material, the structural layer is made of aluminum, the structural layer is located below the additional layer, the additional layer and the structural layer have a length of L, the additional layer and the structural layer have a width of b, and the additional layer has a thickness of h g ; The composite beam is arranged horizontally, one end of the composite beam is fixed on a base, and the other end of the composite beam is connected to a resonance adjustment block; An induction coil is spirally wound on the composite beam, and the induction coil is connected to a current measurement module; A coordinate system is established with the neutral plane of the composite beam as the x-axis direction, the fixed end of the composite beam as the origin of the x-axis, and the vertical direction as the y-axis direction; Step 2: Apply reciprocating impacts up and down to the base to make the composite beam vibrate, and the current measurement module continuously measures the induced current output by the induction coil, and calculates the x-axial stress σ of the iron-gallium alloy according to formula ① g , and magnetic induction intensity B: E g is the Young's modulus of the Fe-Ga alloy; μ is the magnetic permeability of the Fe-Ga alloy d is the magnetomechanical coupling piezomagnetic coefficient of the Fe-Ga alloy; ε is the strain of the composite beam; N is the number of turns of the induction coil wound on the composite beam; i(t) is the induced current output by the induction coil; t represents time; Step 3: According to Hamilton's principle, we have formula ②; in: δ is the Dirac delta function; TT, V, W nc denote kinetic energy, potential energy and virtual work done by non-conservative forces respectively; TT is calculated according to the following formula ③: in: m is the mass per unit length of the composite beam; x is the coordinate value of the composite beam at the x-coordinate; w rel (x, t) is the y-direction displacement of the composite beam relative to the base at the x-coordinate at time t; w b (t) is the base displacement at time t; M t is the mass of the resonance adjustment block; w rel (L, t) is the y-direction displacement of the composite beam relative to the base at time t and when the x-coordinate is L; I t is the moment of inertia; V is calculated according to the following formula ④: in: EI is the stiffness of the composite beam at the x coordinate; W nc Calculate according to the following formula⑤: W nc = W ele +W damp , Formula ⑤; in: h c is the distance between the top of the iron-gallium alloy and the neutral plane; h b is the distance between the bottom end of the iron-gallium alloy and the neutral plane; H(x) is the step function of variable x; H(xL) is the step function of the variable xL; c s is the strain rate damping coefficient; I(x) is the moment of inertia of the cross section at the x coordinate; c a is the air damping coefficient; Substitute formulas ③, ④, and ⑤ into formula ② to obtain formula ⑥: in: Boundary conditions: When x=L, we have formula ⑦ , Formula ⑦; When x=0, we have formula⑧ Step 4: Based on Faraday's law of electromagnetic induction, the induced voltage ΔV generated in the Fe-Ga alloy layer with a length of ΔL in the x-axis direction is g (t) Calculated according to formula 9: in: A is the cross-sectional area of ​​the Fe-Ga alloy layer; Substitute formula ① into formula ⑨, and let ΔV g (t)→dV g (t), ΔL→dx, and integrate the resulting equation along the x-axis to obtain formula ⑩: in: R=R c +R L R C is the resistance of the induction coil; R L is the resistance of the current measurement module; The x-axial strain of the composite beam at thickness y Let ε y =ε, and put it into formula ⑩, we get the following formula in: Step 5: Use the Galerkin decomposition method to discretize the motion control equation of the overall system. Calculate w rel (x,t): in: r is the order of the vibration energy harvesting model; φ r (x) is the rth order vibration mode of the vibration energy harvesting model; q r (t) is the r-th order modal coordinate of the vibration energy harvesting model; The formula Substitute formula ⑥ and formula Formula in: i g =-NdE g h g h pc f r ′(L); C m =2out; The second derivative of the first-order modal coordinates of the vibration energy harvesting model with respect to time t; The first derivative of the first-order modal coordinates of the vibration energy harvesting model with respect to time t; q(t) is the first-order modal coordinate of the vibration energy harvesting model; F(t) is the basic incentive force; is the first derivative of the induced voltage with respect to time t; h pc is the distance from the center layer to the neutral plane in the thickness direction of the Fe-Ga alloy; ω is the free vibration natural frequency of the composite beam; φ r ′(L) is the first derivative of the rth vibration mode of the vibration energy harvesting model when x=L; ξ is the damping ratio; Let the modal force be frequency ω b The simple harmonic force, the steady-state solution of which is given by the formula in: F0 is the modal force amplitude; q0 is the modal coordinate amplitude; is the phase difference between F(t) and q(t); ω b is the excitation frequency; V0 is the output voltage; Combining formulas Solve the output voltage V0 of the vibration energy harvesting model, as shown in the formula in:

2. The analytical analysis method of the cantilevered iron-gallium alloy magnetostrictive vibration energy harvesting model according to claim 1 is characterized in that The output power P of the vibration energy harvesting model is calculated according to the following formula:

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