A piezoelectric ultrasonic transducer dynamics modeling method based on bond graph theory

Abstract

The application discloses a piezoelectric ultrasonic transducer dynamics modeling method based on a bond graph theory, and mainly comprises the following steps: 1, respectively constructing an equivalent circuit model of an electrical end and an equivalent transmission model of a mechanical end of the piezoelectric ultrasonic transducer; 2, constructing a power bond graph model of the transducer according to the bond graph theory; 3, constructing a mechanical-electrical coupling state space equation of the transducer from the power bond graph model; 4, using the state space equation to establish a transmission model among an input electric signal, a system state variable, an amplitude and a vibration speed output variable of the transducer system under simple harmonic excitation, and describing a coupling dynamics behavior. Compared with a traditional dynamics modeling method, the method can comprehensively consider the mechanical-electrical coupling characteristics of the piezoelectric ultrasonic transducer, can provide an effective characterization carrier of the dynamics behavior of the piezoelectric ultrasonic transducer, can reveal the energy conversion law between the electrical end and the mechanical end, and has good practical value and engineering significance in the research, characteristic analysis and control of the mechanical-electrical coupling dynamics behavior.

Description

Technical Field

[0001] This invention relates to the fields of electromechanical coupling dynamics modeling, dynamic characteristic analysis, and transducer control of sandwich piezoelectric ultrasonic transducers, and proposes a new method for electromechanical coupling dynamics modeling of sandwich piezoelectric ultrasonic transducers based on power bond graph theory. Background Technology

[0002] Piezoelectric ultrasonic transducers utilize the inverse piezoelectric effect of piezoelectric ceramics to convert high-frequency electrical signals into high-frequency mechanical vibration outputs, and are widely used in micro-nano manipulation, ultrasound-assisted precision machining, manufacturing, and biomedicine. Since the piezoelectric ceramic is directly excited by an external circuit, while the mechanical amplitude transformer is indirectly driven through coupling, the piezoelectric transducer is a typical electromechanical coupling device, and its dynamic behavior is influenced by the coupling of multiple factors from both the electrical and mechanical ends. To describe its energy conversion and transfer laws, reveal the causes and evolution of its dynamic behavior, and establish an efficient control strategy, the establishment of an analytical model of its electromechanical coupling dynamics is essential.

[0003] Currently, for traditional sandwich piezoelectric ultrasonic transducers, dynamic models are often established using energy principle analysis, finite difference method, electromechanical equivalent method, and transfer matrix method. Energy principle analysis is significantly affected by the transducer's geometry and cross-sectional envelope. The finite difference method only considers the mechanical end of the transducer and does not take into account the electrical characteristics of the piezoelectric material, thus failing to accurately predict the electromechanical coupling characteristics and dynamic behavior of the transducer system. The equivalent circuit method requires experimental testing to determine electrical parameters before using the constructed equivalent circuit model for analysis, and similarly cannot reflect the electromechanical coupling characteristics of the transducer. The transfer matrix method, based on modular thinking, breaks down the transducer into several beam, rod, or plate structural units, first constructing dynamic models for each unit, and then connecting all the units together using interface connection conditions between adjacent units to construct the dynamic model of the entire system. This model is often a complex system of linear equations involving many variables, making it difficult to solve.

[0004] Bond graphs, as a modeling method for multi-energy-domain coupled systems, are particularly suitable for complex problems involving multiple energy sources, such as mechanical, electrical, and hydraulic systems. The bond graph method represents components with similar properties from different domains using the same type of bond graph primitives, connecting them through bonds that transfer power. Based on the interrelationships between these components, a holistic model of the system is established. This method can not only establish a transmission model between outputs and inputs but also reveal internal energy coupling and conversion relationships, facilitating the study of multi-energy-coupled systems. Summary of the Invention

[0005] The purpose of this invention is to provide a dynamic modeling method for sandwich piezoelectric ultrasonic transducers based on bond graph theory, thereby addressing the aforementioned shortcomings in existing technologies. For transducers, complex systems involving electrical, mechanical, and acoustic energy coupling, a bond graph model of the transducer is established using this method. This leads to a mathematical model of the input and output, which not only intuitively describes the dynamic characteristics of the transducer but also reveals the intrinsic relationship between output vibration characteristics and input voltage, internal mechanical damping, stiffness, etc.

[0006] The specific implementation steps of this method are as follows:

[0007] Step 1: Establish the equivalent circuit model of the transducer's electrical terminals, and based on the principle of energy conservation, establish the bond graph model of the transducer system's electrical terminals;

[0008] Step 2: Establish an equivalent transfer model of the transducer's mechanical end, and based on the principle of energy conservation, establish a bond graph model of the transducer system's mechanical end;

[0009] Step 3: Based on the principle of energy conservation and the electromechanical energy conversion relationship, establish an electromechanical coupling bond diagram model of the transducer system;

[0010] Step 4: Establish the state-space equations of the piezoelectric ultrasonic transducer system using its electromechanical coupling bond graph model;

[0011] Step 5: Construct a transmission model between state variables, vibration, vibration velocity output and the input electrical signal of the transducer system using state-space equations, describe the electromechanical coupling dynamic behavior characteristics of the transducer system, and reveal the energy conversion law between its electrical and mechanical ends.

[0012] Furthermore, step 1 includes the following steps:

[0013] Step 1.1: Based on the working principle of the sandwich piezoelectric ultrasonic transducer, the piezoelectric ceramic vibrator is equivalent to an equivalent circuit consisting of the static branch containing the static capacitor C0 and the dynamic branch containing the dynamic resistor R1, dynamic capacitor C1 and dynamic inductor L1 connected in parallel; establish its equivalent circuit model.

[0014] Step 1.2: Based on the piezoelectric ceramic size parameters and material property parameters, and based on the equivalent circuit of the transducer and its dynamic admittance model under resonant operation, solve for the equivalent resistance, capacitance and inductance parameters of the transducer's electrical terminals;

[0015] Step 1.3: Based on the principle of energy conservation, the equivalent voltage source, inductance, capacitance and resistance parameters of the piezoelectric ultrasonic transducer electrical terminals in Step 1.1 are converted into potential source Se, inertial element 1, capacitive element C and resistive element R, respectively; according to Kirchhoff's laws, the equipotential junction 0 junction or the equicurrent junction 1 junction is selected for connection, thereby establishing the bonding diagram model of the electrical terminals of the piezoelectric transducer system;

[0016] Furthermore, step 2 includes the following steps:

[0017] Step 2.1: Equivalently model the front cover plate, clamping flange and mechanical amplitude transformer of the transducer's mechanical end components as a mass-stiffness-damping model;

[0018] Step 2.2: Calculate the equivalent stiffness and equivalent damping of each component of the transducer by combining the tensile and compressive stiffness and viscous damping theories in mechanics of materials, and calculate the equivalent mass using the concept of lumped mass.

[0019] Step 2.3: Convert the equivalent mass, stiffness, and damping of the transducer in Step 2.2 into inertial element I, resistive element R, and capacitive element C, and connect them by equipotential junction 0 or equicurrent junction 1 according to the constant velocity connection relationship between each element, thereby establishing the mechanical end bond diagram model of the transducer.

[0020] Furthermore, step 3 includes the following steps: for the electrical end power bond graph model and mechanical end power bond graph model of the piezoelectric ultrasonic transducer established in steps 1 and 2, the electromechanical coupling relationship between the two is represented by introducing a gyroscope GY, and the two models are connected together to form an electromechanical coupling bond graph model.

[0021] Furthermore, step 4 includes the following steps:

[0022] Step 4.1: Take the generalized momentum p of the inertial element and the generalized displacement q of the capacitive element in Step 3 as the state variables of the system, and take the system input voltage and the transducer output vibration velocity as the input and output variables respectively. Based on the interrelationship between the various port elements, the state space equation expression corresponding to the electromechanical coupling dynamic model of the transducer system can be constructed.

[0023] Step 4.2: Substitute the specific parameter values ​​calculated in Steps 1.2 and 2.2 into the state-space equations in Step 5. This will establish a transmission model between the transducer system's state variables, vibration, and vibration velocity output and the transducer system's input, thereby describing the electromechanical coupling dynamic behavior characteristics of the transducer system and revealing the energy conversion law between its electrical and mechanical ends.

[0024] Compared with the prior art, the present invention has the following positive effects:

[0025] (1) A dynamic model of piezoelectric transducer based on power bond graph theory is proposed, which fully considers the electromechanical coupling characteristics of the transducer and avoids the problems of the traditional finite difference method starting only from the mechanical end of the transducer and not considering the electrical characteristics of the piezoelectric material; the equivalent circuit method converts the transducer as an electrical model, which fails to reflect its mechanical characteristics; the transfer matrix method generates a relatively complex set of linear equations, which is not easy to solve.

[0026] (2) The electromechanical coupling dynamic model of the piezoelectric ultrasonic transducer constructed by the proposed method can describe the electromechanical coupling dynamic behavior characteristics of the transducer system and reveal the energy conversion law between its electrical end and mechanical end.

[0027] (3) The state-space equation constructed by the bond graph electromechanical coupling model can establish a transmission model between state variables, vibration, vibration velocity output and transducer system input. It can not only be used to study the influence of input excitation signal on transducer output response, but also to understand the influence of internal factors such as mechanical loss, inertial force, stress distribution on transducer output response.

[0028] (4) This electromechanical coupling dynamic model has good practical value and engineering significance in the study of transducer dynamic behavior, characteristic analysis and control method development. Attached Figure Description

[0029] Figure 1 This is a structural diagram of the transducer in this invention;

[0030] Figure 2 This is an equivalent model diagram of the transducer in this invention;

[0031] Figure 3 This is the bond diagram model of the equivalent circuit of the transducer in this invention;

[0032] Figure 4 This is a partial bond graph model of the transducer mass-spring-damping model in this invention;

[0033] Figure 5 This is the overall bond graph model of the transducer system in this invention. Detailed Implementation

[0034] This invention provides a dynamic modeling method for piezoelectric ultrasonic transducers based on bond graph theory. To facilitate the description of its specific implementation process and make the invention easier to understand, a detailed introduction will be given below with reference to the accompanying drawings and specific examples.

[0035] like Figure 1 As shown, the transducer in this example mainly consists of a rear cover plate, two piezoelectric ceramics, a front cover plate, a flexible clamping mechanism, and an amplitude transformer. The specific dimensions (ignoring the electrode plates whose thickness is negligible) and material parameters of each part are shown in Tables 1 and 2 below.

[0036] Table 1 Main geometric parameters of the transducer

[0037]

[0038] Table 2. Material parameters of various structures of the transducer

[0039]

[0040] Based on the working principle of the sandwich piezoelectric ultrasonic transducer, the piezoelectric ceramic vibrator is equivalently represented as an equivalent circuit consisting of a static branch containing a static capacitor and a dynamic branch composed of a series-connected dynamic resistor, capacitor, and inductor. The mechanical amplitude transformer is equivalently represented as a mass-stiffness-damping model. Since the piezoelectric ceramic vibrates bidirectionally along the longitudinal direction during operation, and the output end of the amplitude transformer is the working end of the transducer, only the section from the piezoelectric ceramic to the output end of the amplitude transformer is modeled here, excluding the rear cover plate. The overall equivalent model of the transducer is as follows: Figure 2 As shown, C0 is the static capacitor, which is a clamping capacitor formed by the electrical parallel connection of piezoelectric ceramic sheets; R l C1 represents the dynamic resistance, characterizing the dielectric and mechanical losses of the piezoelectric ceramic; C1 and L1 represent the dynamic capacitance and dynamic inductance, respectively. M is the equivalent mass of the amplitude transformer section, characterizing the magnitude of the inertial force; k is the equivalent stiffness of the amplitude transformer, characterizing the magnitude of the force required for deformation; c is the equivalent damping of the amplitude transformer, characterizing the stress distribution at each contact surface. By combining the parameter values ​​in Tables 1 and 2 and substituting them into the relevant theoretical formulas, the specific values ​​of each parameter can be obtained.

[0041] Table 3. Equivalent parameters of the transducer

[0042]

[0043]

[0044] Based on the fundamental principle of the law of conservation of energy in a system, the voltage source, resistor, inductor, and capacitor in the equivalent circuit are represented by the potential source Se, resistive element R, inertial element I, and capacitive element C, respectively. Then, according to Kirchhoff's laws, equipotential junctions ("0") and equicurrent junctions ("1") are selected to connect them, thus forming a partial bond graph model of the equivalent circuit, as shown below. Figure 3 As shown.

[0045] Similarly, the equivalent mass, stiffness, and damping in the mass-stiffness-damping model are represented by inertial element I, capacitive element C, and resistive element R, respectively. Then, based on the constant velocity connection condition, they are connected using constant flow junctions ("1"), thus constructing a bond diagram model for the mechanical part, as shown below. Figure 4 As shown.

[0046] Furthermore, since the transducer internally uses electromechanical coupling energy conversion to drive the mechanical amplitude transformer, a gyroscope GY is chosen here to represent the electromechanical conversion relationship, ultimately... Figure 3 and Figure 4 The bond graph models in the diagrams are linked together to form the overall bond graph model of the transducer, see... Figure 5 .

[0047] To perform system simulation of the transducer, a mathematical model of the system needs to be established. State-space equations can not only describe the relationship between the system's input and output, but also help reveal the relationship between internal system variables and the output. Therefore, this invention chooses the method of constructing state-space equations based on the system bond graph. The specific steps for establishing the state equations are as follows:

[0048] Let the state variables of the system be

[0049] x = [p5 q7 q] 10 p 12 ] T =[L1f5 C1e7 e 10 / k Mf 12 ] T (1)

[0050] Based on the integral causal relationships of the basic components of the system bond graph model, the following relationship can be obtained:

[0051]

[0052] The system's input variable u and output variable y are respectively

[0053] u=[e1] (3)

[0054] y = [f 12 (4)

[0055] p5 represents the magnetic flux of the dynamic inductor, q7 represents the charge stored in the dynamic capacitor, and q 10 p represents the amount of deformation of the simulated spring. 12 e1 represents the momentum of the amplitude transformer, e1 represents the input voltage of the transducer system, and f represents the input voltage of the transducer system. 12 This indicates the vibration velocity at the output end.

[0056] like Figure 3 As shown, the potential variables and current variables corresponding to each connection in the bond diagram have the following relationship:

[0057]

[0058] Let the state-space equation be

[0059]

[0060] Where parameters A, B, C, and D are

[0061]

[0062] B = [1 0 0 0] T (8)

[0063] C = [0 0 0 1 / M] (9)

[0064] D = 0 (10)

[0065] The transducer system can be dynamically simulated based on the state-space equation, and its output vibration velocity and vibration displacement can be observed when the external input voltage changes.

[0066] Furthermore, the dynamic modeling method based on bond graph theory proposed in this invention differs from other modeling methods. Other methods treat the transducer system as a "black box," focusing only on the influence of input voltage on output velocity / displacement. In contrast, the dynamic model established in this invention considers the electromechanical coupling phenomenon within the transducer system and the influence of stiffness and damping between contact surfaces. By changing the values ​​of its own state variable parameters in the state-space equations, it is possible to explore how factors such as equivalent electrical parameters, damping, and stiffness within the system affect the system output.

[0067] Finally, it should be noted that although the specific embodiments described above are for selected examples, the underlying principles can be extended to transducers with the same or similar structures. After reading the above, any modifications or implementations that do not depart from the spirit and claims of this invention will fall within the scope of protection of this invention.

Claims

1. A piezoelectric ultrasonic transducer dynamics modeling method based on bond graph theory, characterized by: For a sandwich structure piezoelectric ultrasonic transducer consisting of a rear cover plate (1), preload bolts (2), piezoelectric ceramic sheet (3), electrode sheet (4), front cover plate (5), clamping flange (6), and mechanical amplitude transformer (7), an electromechanical coupling dynamic model of this typical structure piezoelectric ultrasonic transducer is established using power bond graph theory. The steps are as follows: Step 1: Establish the equivalent circuit model of the transducer's electrical terminals, and based on the principle of energy conservation, establish the bond graph model of the transducer system's electrical terminals; including the following steps: Step 1.1: Based on the working principle of the sandwich piezoelectric ultrasonic transducer, the piezoelectric ceramic oscillator is equivalent to an equivalent circuit consisting of the static branch where the static capacitor C0 is located and the dynamic branch where the dynamic resistor R1, dynamic capacitor C1 and dynamic inductor L1 are located in parallel, and its equivalent circuit model is established. Step 1.2: Based on the piezoelectric ceramic size parameters and material property parameters, and based on the equivalent circuit of the transducer and its dynamic admittance model under resonant operation, solve for the equivalent resistance, capacitance and inductance parameters of the transducer's electrical terminals; Step 1.3: Based on the principle of energy conservation, the equivalent voltage source, inductance, capacitance and resistance parameters of the piezoelectric ultrasonic transducer electrical terminals in Step 1.1 are converted into potential source Se, inertial element I, capacitive element C and resistive element R, respectively; according to Kirchhoff's laws, the equipotential junction "0" or the equicurrent junction "1" is selected for connection, thereby establishing the bonding diagram model of the electrical terminals of the piezoelectric transducer system; Step 2: Establish an equivalent transfer model for the mechanical end of the transducer, and based on the principle of energy conservation, establish a bond graph model for the mechanical end of the transducer system; including the following steps: Step 2.1: Equivalently model the front cover plate, clamping flange and mechanical amplitude transformer of the transducer's mechanical end components as a mass-stiffness-damping model; Step 2.2: Calculate the equivalent stiffness and equivalent damping of each component of the transducer by combining the tensile and compressive stiffness and viscous damping theories in mechanics of materials, and calculate the equivalent mass using the concept of lumped mass. Step 2.3: Convert the equivalent mass, stiffness and damping of the transducer in Step 2.2 into inertial element I, resistive element R and capacitive element C, and connect them by equipotential junction 0 or equicurrent junction 1 according to the constant velocity connection relationship between each element, thereby establishing the mechanical end bond diagram model of the transducer. Step 3: Based on the principle of energy conservation and the electromechanical energy conversion relationship, establish an electromechanical coupling bond graph model of the transducer system; for the electrical end power bond graph model and mechanical end power bond graph model of the piezoelectric ultrasonic transducer established in Step 1 and 2, the electromechanical coupling relationship between the two is represented by introducing a gyroscope GY, and the two models are connected together to form an electromechanical coupling bond graph model. Step 4: Establish the state-space equations of the piezoelectric ultrasonic transducer system using its electromechanical coupling bond graph model; Step 5: Construct a transmission model between state variables, vibration, vibration velocity output and the input electrical signal of the transducer system using state-space equations, describe the electromechanical coupling dynamic behavior characteristics of the transducer system, and reveal the energy conversion law between its electrical and mechanical ends.

2. The method of claim 1, wherein, Step 4 includes the following steps: Taking the generalized momentum p of the inertial element and the generalized displacement q of the capacitive element in step 3 as the state variables of the system, and taking the system input voltage and the transducer output vibration velocity as the input and output variables respectively, and based on the interrelationship between the various port elements, the state space equation expression corresponding to the electromechanical coupling dynamic model of the transducer system can be constructed.

3. The method of claim 1, wherein, Step 5 includes the following steps: Substituting the specific parameter values ​​calculated in steps 1.2 and 2.2 into the state-space equation in step 5, a transmission model can be established between the state variables, vibration, and vibration velocity output of the transducer system and the input of the transducer system. This allows for the description of the electromechanical coupling dynamic behavior characteristics of the transducer system and the revelation of the energy conversion law between its electrical and mechanical ends.

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