Bond Graph Modeling and Simulation of Hybrid Piezo-Flexural-Hydraulic Actuator ^†

Abstract

In this study, a hybrid piezo-flexural-hydraulic actuator is modeled and simulated using bond graph methodology. The hybrid actuator comprises piezoelectric stack actuator, mechanical flexural amplifier, and hydraulic piston actuator. The piezoelectric stack actuator produces electrically controllable displacement. This displacement is amplified by a cascading combination of flexural amplifier and hydraulic actuator. A domain-independent bond graph model for the proposed hybrid actuator is developed. Using this bond graph, a mathematical model and a state space representation for the hybrid actuator are derived. The bond graph model is simulated using a 20-sim bond graph simulation software. The results of the simulation provide displacement characteristics and sensitivity analysis for each component and the hybrid actuator as a whole. The study plays a significant role in understanding the dynamic behavior of a multi-domain system using the bond graph methodology.

1. Introduction

Piezoelectric materials have the ability to generate electrical charge from applied mechanical stress. This phenomenon is called the piezoelectric effect. These materials are used in synthesizing sensors and transducers that are used for measuring force, acceleration, etc. Materials exhibiting piezoelectric effect also demonstrate the inverse piezoelectric effect where an applied electric field produces an internal mechanical stress, and in turn, displacement. This makes piezoelectric materials critical in the production of microactuators and micromanipulators. These actuators provide several advantages in terms of faster response, high positioning accuracy, no backlash or static friction thus low wear and tear, and compact design. They require less actuation power and do not create any stray magnetic fields [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15].

Piezoelectric actuators are electro-mechanical actuators that produce electrically controllable displacement in the order of nanometres to micrometers. Despite very small displacements, these actuators are capable of generating significant force and precision-controlled displacement, and thus, they are used in microscopic positioning applications such as fine adjusting lenses, mirrors, etc. To overcome the limitation of small displacement, these actuators are cascaded with flexural displacement amplification system, hydraulic displacement amplification system, etc. [1,5].

This study presents a hybrid piezo-flexural-hydraulic actuator that produces electrically controllable displacement in the range of a few centimeters by cascading both a mechanical flexural amplifier and a hydraulic piston actuator. This study aims at developing a domain-independent model for the hybrid actuator using bond graph methodology. Using this bond graph, a mathematical model and a state space representation for the hybrid actuator are obtained to understand its dynamic behavior. The bond graph is simulated using a 20-sim simulation to attain displacement and sensitivity analysis.

2. Hybrid Piezo-Hydraulic Actuator

The following section delineates the operating principle and the mathematical modeling for each individual component and the hybrid actuator as a whole.

2.1. Operating Principle of Hybrid Piezo-Hydraulic Actuator

The proposed hybrid actuator comprises a piezoelectric stack actuator, a mechanical flexural amplifier, and a hydraulic piston actuator. These components are cascaded in series in order to achieve a significant amplification of displacement. The displacement amplification factor is in the order of 100s, thus making the hybrid actuator suitable for a wide variety of applications. The design of each component is scrutinized for optimum throughput in terms of displacement.

Since a single layer of piezoceramic produces an insufficient displacement for reasonable voltage input, the multiple layers are stacked together and divided by electrodes, and each layer is polarized along its thickness and its axis of motion to construct a piezo-stack actuator.

The piezoelectric stack actuator is cascaded with a flexure-guided displacement amplifier, designed with a protective preload. This provides a larger displacement compared to the piezoelectric stack actuator in a more compact package. The amplified actuator provides advantages like high stroke, high force, compactness, and design flexibility.

The amplified piezoelectric actuator is then cascaded with a hydraulic piston actuator. This further amplifies the output displacement as the diameter of the output piston is smaller compared to that of the input piston. The hydraulic actuator produces force significantly higher than that of the pneumatic or electric actuator and has a high stability due to the incompressibility of fluids.

2.2. Mathematical Modeling of Hybrid Piezo-Hydraulic Actuator

A mathematical model for each component and the hybrid actuator as a whole is presented in the following section. This helps in deriving the displacement amplification factor at each stage and also understanding the dynamic behavior of the hybrid actuator.

2.2.1. Piezoelectric Stack Actuator

An electro-mechanical model for the piezoelectric stack actuator is based on the conversion of electric potential to mechanical force.

Mathematically, a piezoelectric stack actuator can be represented as a mass spring damper system. The system can be represented using a second-order differential equation and is as follows:

$$m_p\ddot{x_p}+c_p\dot{x_p}+k_p{x}_p=F_p$$

(1)

$$m_p\ddot{x_p}+c_p\dot{x_p}+k_p{x}_p=k_p·d_{33}·V_0$$

(2)

where $m_p$ is mass, $k_p$ is stiffness, and $c_p$ is the damping of the piezoelectric stack actuator, $V_0$ is the applied voltage, and $d_{33}$ is the piezoelectric constant along the z-axis.

The expression provided in the above equation can be modified as shown in the equation below to represent the displacement produced by the piezoelectric stack actuator along the x-axis.

$$x_p\left(t\right)=\frac{k_p{d}_{33}{V}_0}{k_p}-\frac{m_p\ddot{x_p}}{k_p}-\frac{c_p\dot{x_p}}{k_p}$$

(3)

2.2.2. Flexural Displacement Amplifier

Since the piezoelectric stack actuator is embedded inside a flexural amplifier whose stiffness resists its motion, the equation can be modified to determine the displacement of the flexural amplifier along the x-axis and can be represented as follows:

$$x_a\left(t\right)=\frac{k_p{d}_{33}{V}_0}{k_p+k_s}-\frac{m_p\ddot{x_a}}{k_p+k_s}-\frac{c_p\dot{x_a}}{k_p+k_s}$$

(4)

where $k_s$ is the stiffness of the flexural amplifier along the x-axis, and $x_a$ is the displacement of the flexural amplifier along the x-axis [1].

The displacement along the x-axis is amplified through the flexural amplifier and the output as displacement along the z-axis. The mathematical expression for the total displacement along the z-axis after the flexural amplifier can be represented as follows:

$$z_a=A.F_f·x_a$$

(5)

where $z_a$ is the displacement of the amplified piezoelectric actuator along the x-axis, and $A.F_f$ is the amplification factor of the flexural displacement amplifier.

A three-dimensional model of the flexural amplifier is built in solid edge and imported into ANSYS workbench to perform finite element analysis. The stiffness of the flexural amplifier along the x-axis and its displacement amplification factor are determined using FEA. For the amplification of displacement, it must be noted that the output piston diameter is smaller compared to the input piston diameter.

A mathematical expression for the hydraulic amplification factor can be derived based on Pascal’s law in fluid dynamics and is as follows:

$$P_1=P_2; Volume=constant$$

(6)

$$\frac{F_1}{A_1}=\frac{F_2}{A_2}; A_1·l_1=A_2·l_2$$

(7)

$$\frac{l_2}{l_1}=\frac{A_1}{A_2}=\frac{d_1^2}{d_2^2}=A.F_h$$

(8)

where $P_1$ and $P_2$ are pressures at input and output pistons, respectively, $F_1$ and $F_2$ are forces exerted at input and output pistons, respectively, $A_1$ and $A_2$ are the areas of input and output pistons, respectively, $d_1$ and $d_2$ are the diameters of input and output pistons, respectively, $l_1$ and $l_2$ are the displacements of input and output pistons, respectively, and $A.F_h$ represents the displacement amplification factor for the hydraulic piston actuator.

Since the hydraulic piston actuator is cascaded with flexural amplifier, the total displacement of the hybrid actuator is computed as a product of the hydraulic piston actuator’s displacement amplification factor and the displacement of the flexural amplifier along the z-axis. Thus, the mathematical expression for the total hybrid actuator displacement can be written as follows:

$$z_h=A.F_h\cdot z_a$$

(9)

where $z_h$ is the total displacement of the hybrid actuator, $z_a$ is the displacement at the amplified piezoelectric actuator, and $A.F_h$ represents the displacement amplification factor for hydraulic piston actuator.

Table 1. Hydraulic actuator parameters.

Parameter	Value
Chamber diameter	41.00 mm
Chamber depth	4.00 mm
Input piston diameter	40.00 mm
Output piston diameter	4.50 mm

represents the dimensional specifications for the hydraulic piston actuator.

Table 2. Piezo-hydraulic actuator parameters.

Parameter	Value
Mass of piezo-stack actuator, mp	3.8 g
Stiffness of piezo-stack actuator, kp	50 N/μm
Damping of piezo-stack actuator, cp	150 Ns/m
Capacitance of piezo-stack actuator, C	1.55 μF
Electromechanical coupling, d33	0.1334 μm/V
Stiffness of flexural amplifier along x-axis, ks	28.903 N/μm
Amplification factor for flexural amplifier, A.Ff	6.39
Amplification factor for hydraulic actuator, A.Fh	79.02

presents parameters taken into consideration for modeling the piezoelectric stack actuator, the mechanical flexural actuator, and the hydraulic piston actuator.

3. Bond Graph Modeling

Bond graphs provide a graphical description of a physical system using labeled and directed graphs, where vertices represent subsystems/components, and edges represent the ideal energy connection between power ports, i.e., the places where subsystems are interconnected for power interaction.

Since the focus is on power, i.e., the rate of energy transport between components, bond graphs provide methods to model dynamic systems in different domains including electrical, electromagnetic, mechanical, hydraulic, etc., without discrimination. This helps in the analysis, identification, synthesis, and control of multi-domain linear and non-linear dynamic systems [16,17,18,19,20].

Bond graphs are characterized by power variables, namely, effort and flow, and energy variables, namely, momentum and displacement. Half arrows at the end of each power bond show a direction of positive power flow. The power at each bond is represented by the product of effort (generalized force) and flow (generalized velocity). Small strokes perpendicular to a bond denote causality, that is, which amongst the two quantities is the input, i.e., cause, and which is the output, i.e., effect.

3.1. Bond Graph Modeling of Hybrid Piezo-Hydraulic Actuator

Certain assumptions made to facilitate the modeling of the hybrid actuator using bond graph are listed as follows. The electrical and mechanical domains for the piezoelectric ceramic actuator are connected through ‘polarization’. To model the piezoelectric stack actuator, we can assume that all its layers have the same properties. Non-linear components including hysteresis, creep, and temperature dependence are overlooked for this study. The bond graph model for the whole hybrid actuator is represented in Figure 1a and Figure 1b respectively.

Following the direction of the positive power flow in Figure 1a, the input voltage waveform acts on the electromechanical coupling and the stiffness of the PSA to produce a force acting on the lumped PSA represented by a mass–spring–damper system. The displacement produced by the piezoelectric stack is amplified using a mechanical flexural amplifier represented as flexural gain $A.F_f$. The displacement is further amplified by a hydraulic piston actuator represented as hydraulic gain $A.F_h$.

In Figure 1a, $M{S}_e$ represents the modulated effort source, which refers to the voltage input for electric systems and the force input for mechanical systems. $L_2$ represents the mass of PSA, $C_3$ represents the stiffness of PSA, and $R_4$ represents the damping of PSA.

The power variables for each bond include force and velocity represented by ‘$e$’ and ‘$f$’, respectively. The energy variables for each bond include momentum and displacement represented by ‘$p$’ and ‘$q$’, respectively. For example, $e_2$ represents force exerted, $f_2$ represents velocity, $p_2$ represents momentum, and $q_2$ represents the displacement for bond number 2 represented by the mass of the piezoelectric stack actuator.

3.2. Bond Graph Modeling of Hybrid Piezo-Hydraulic Actuator

State space representation helps in the comprehensive and accurate modeling of the complex system, thus aiding in the analysis of its stability and performance. A state space representation can be derived based on the bond graph model of the hybrid actuator.

Using the properties of one-junction, the following equations for effort variables can be derived.

$$e_1=e_2+e_3+e_4$$

(10)

$$e_2=\dot{p_2}; e_3=\frac{q_3}{C_3}$$

(11)

$$e_4=R_4·f_4\Rightarrow e_4=R_4·f_2$$

(12)

$$e_1=\dot{p_3}+\frac{q_3}{C_3}+\frac{R_4}{L_2}$$

(13)

Similarly, using the properties of flow variables, the following equations are derived.

$$f_1=f_2=f_3=f_4$$

(14)

$$f_3=\dot{q_3}\Rightarrow f_1=f_2=\dot{q_3}=f_4$$

(15)

Thus, state space equation can be expressed as follows:

$$\dot{p_2}=-\frac{R_4}{L_2}·p_2-\frac{1}{C_3}·q_3+e_1$$

(16)

$$\dot{q_3}=\frac{1}{L_2}·p_2$$

(17)

In matrix form,

$$\left[\begin{array}{c}\dot{p_2}\\ \dot{q_3}\end{array}\right]=\left[\begin{array}{cc}-\frac{R_4}{L_2}& -\frac{1}{C_3}\\ \frac{1}{L_2}& 0\end{array}\right]·\left[\begin{array}{c}{p}_2\\ q_3\end{array}\right]+\left[\begin{array}{c}1\\ 0\end{array}\right]·\left[e_1\right]$$

(18)

$$\left[q_3\right]=\left[\begin{array}{cc}0& 1\end{array}\right]·\left[\begin{array}{c}{p}_2\\ q_3\end{array}\right]+\left[0\right]·\left[e_1\right]$$

(19)

4. Results

The characteristic plots for input voltage, piezoelectric stack actuator displacement, amplified piezoelectric actuator displacement, and hydraulic piston actuator displacement using 20-sim simulations are shown in Figure 2.

Figure 2 depicts the displacement characteristics for the sinusoidal primary voltage input of 15 Vp−p at 1 Hz with a DC offset of 7.5 V. The voltage input from a waveform generator is amplified to 150 Vp−p using a voltage amplifier represented by a voltage gain as observed in Figure 1b. This results in $1 \mathrm{k}{\mathrm{N}}_{\mathrm{p}-\mathrm{p}}$ force being applied to the piezoelectric stack actuator due to the electromechanical coupling represented as $d_{33}$.

As a result, for the applied voltage input of 150 Vp−p, the piezoelectric stack actuator produces a displacement of $20 \mathsf{\mu }{\mathrm{m}}_{\mathrm{p}-\mathrm{p}}$ along the x-axis, which is amplified to $127 \mathsf{\mu }{\mathrm{m}}_{\mathrm{p}-\mathrm{p}}$ by the flexural amplifier along the z-axis. The displacement is further amplified to $1 \mathrm{c}{\mathrm{m}}_{\mathrm{p}-\mathrm{p}}$ by the hydraulic piston actuator along the z-axis.

The displacement amplification factor at each stage can be estimated. The displacement amplification factor for the flexural amplifier is approximately 6.35. The displacement amplification factor for the hydraulic piston actuator is approximately 78.74. The total amplification factor using the hybrid system of mechanical flexure and the hydraulic piston actuator is approximately 500. This significant amplification in displacement by cascading the hybrid mechanical–hydraulic system to the piezoelectric actuator makes it suitable for a wide range of applications.

From Figure 3, the relationship between the piezoelectric stack actuator displacement, the amplified piezoelectric actuator displacement, and the piezo-flexural-hydraulic actuator displacement can be obtained as a linear function of the amplitude of the input voltage.

The displacements and displacement sensitivities, respectively, for hybrid actuator components including the piezoelectric stack actuator, the amplified piezoelectric actuator, and the piezo-hydraulic actuator for the input voltage of 150 Vp−p are provided in

Table 3. Displacements and displacement sensitivities for the input voltage of 150 Vp−p and the DC offset of 75 V.

Parameter	Displacement	Offset	Sensitivity
PSA displacement	20 μmp−p	10 μm	0.1333 μm/V
APA displacement	127 μmp−p	63.5 μm	0.8467 μm/V
PHA displacement	1 cmp−p	0.5 cm	66.66 μm/V

.

5. Conclusions

A piezoelectric actuator has the ability to provide an electrically controlled, precise displacement (stroke). It provides high reliability and maintains a simple design that comprises minimal moving parts, thus requiring no lubrication for its operation.

The piezoelectric stack actuator has the capability to operate billions of times without wear or deterioration. It possesses an exceptional response speed and is only limited by the inertia of the object being moved and the output capability of the electronic driver. When energized, the piezoelectric stack actuator consumes virtually no power and produces very little heat.

However, due to displacements ranging in micrometers, the application of PSA is only limited to micro-positioning and alignment applications. In this study, a multi-domain hybrid piezo-flexural-hydraulic actuator is proposed. The hybrid actuator is modeled using the bond graph methodology to understand its dynamic behavior.

Using cascaded mechanical and hydraulic actuators, the displacement is observed to be amplified by a factor of 100s. Also, due to the small specific weight, i.e., the ratio of the transmitted capacity to the weight of the piezo-hydraulic actuator, the hybrid actuator can be suitable for applications in a variety of industries including automotive, medical, aviation, and consumer electronics.

One of the disadvantages of hybrid piezo-hydraulic actuators is that their hydraulic actuator component may leak fluids, which may lead to a loss of their efficiency. Further study can be conducted to build a more accurate non-linear model of the hybrid actuator that may include hysteresis, backlash, or dead zone due to any of its components.