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Locomotion in liquid
The movement in liquid is totally inspired from animal locomotion and is divided into locomotion inside liquid and locomotion at the liquid surface. The design of miniature robots in aquatic medium depends on the liquid properties; the forces were acting on and some factors influencing the locomotion. The forces acting on miniature robots inside liquid are thrust, drag, weight buoyancy and hydrodynamic lift (Figure 1.15).
The thrust here is defined as the force generated by moving portions of the miniature robot body on the fluid surrounding it, to move miniature robot forward. The inertial force depends on the body mass while the buoyancy force is generated by fins and it depends on the mass of fluid displaced. The hydrodynamic lift is generated by fins to supplement buoyancy and balance the vertical forces when horizontal motion is demanded. The drag is the resistive force exerted by the fluid on its body and it consists of viscous drag and pressure drag. Viscous drag is skin friction between the miniature robot and boundary layer of water. The pressure drag exerted by distortions of flow around miniature robot body and energy lost in the vortices formed by the fins as they generate lift or thrust. At the liquid surface, the forces acting on miniature robots are the same but in this case the miniature robots use the surface tension of liquid which represents the work required to deform a liquid over a unit area [(BIEWENER, 2003)] to generate lift, buoyancy and thrust, or they use the mass density of liquid to generate these forces. An important factor influencing the locomotion for miniature robots in liquid is the Reynolds number (Re) that describes the viscous versus inertial forces. At low Re viscous forces reign, but at high Re inertial forces dominate. This has important consequences for the propulsive mechanisms and design for locomotion at low and high Re [(BIEWENER, 2003)]. The Froude number (Fr) is also an important factor that describes the propulsive efficiency of the miniature robots. It is the ratio of the useful propulsive power over the total power expended by the mobile miniature robots. As already mentioned the design of miniature robots in aquatic medium depends on surface tension of liquid, density of liquid, Reynolds number and the forces were acting on, locomotion principles itself are not influenced by the type of liquid. So, the study of locomotion principles in any type of liquid is the same, we then consider the case of water in the following. As the movement in liquid for mobile miniature robots is totally inspired from animal locomotion, a study of animal locomotion in water is given in this section, with some applications in the field of piezoelectric miniature robots.
Locomotion inside water
Locomotion inside water includes swimming or non-swimming locomotion. The latter includes specialized actions such as flying and gliding, as well as jet propulsion. In this section, we will describe only the swimming locomotion because it is the most used in the field of miniature robot and it is divided into fish swimming mechanisms (at high and moderate Re) and micro-organisms swimming mechanisms (at low Re).
Fish swimming mechanisms
To aid in the description of fish swimming mechanisms; Figure 1.16 identifies the elements of fish. The fish swims either by body and/or caudal fin (BCF) movements or using median and/or paired fin (MPF) propulsion. The BCF propulsion has been classified into five swimming modes: Anguilliform, subcarangiform, carangiform, thurnniform and ostraciform. The latter is the only oscillatory BCF mode; it is characterized by the pendulum oscillation of the caudal fin, while the body remains essentially rigid. The others are undulatory BCF modes. In anguilliform mode, ondulations of large amplitude are obtained by whole body.
Micro-organisms swimming mechanisms
All the micro swimming mechanics such as flagella, spermatozoa, cilia, and amoeba crate in one way or another traveling wave, advanced in the opposite direction of the micro organisms locomotion. The swimming mechanics of micro-organisms are divided into flagellar and ciliary swimming.
Flagellar swimming is the simplest swimming method for micro system and is produced by a sinusoidal or helical wave in an elastic tail. In contrast of flagella, cilia beat in asymmetrical fashion i.e. by orienting the cilia parallel to the flow during the recovery stroke much lower drag is produced than when they beat in a more perpendicular orientation during the propulsive stroke [(BIEWENER, 2003)].
Piezoelectric actuators are used in [(Kosa, et al., 2007), (Kosa, et al., 2008)] for creating the travelling wave needed for the movements of swimming micro organisms (Figure 1. 19).
Locomotion at the water surface
Locomotion at the water surface is divided into two different locomotion principles: striding on the water surface like water striders and running on the water surface like a basilisk lizard. Water striders take advantage of their size by using the surface tension of water to generate forces in order to step over the surface of water. These forces increase with the depth of the unwetted limb of the water striders [(BIEWENER, 2003)]. Basilisk lizard have a weight which is larger than the surface tension can support, it takes advantage of the mass density of the water, which exerts a reactive force when running rapidly with its webbed feet [(BIEWENER, 2003)]. As example we take the water strider miniature robot walking on water describes in [(Suhr, et al., 2005)] (Figure 1.20).
Case of piezoelectric patches bonded on a beam
Our system studied is composed of non-collocated piezoelectric patches bonded on a thin beam (having a very large dimension (x) with respect to the two others) has the same width as the piezoelectric patches, as shown in Figure 3. 4. Electric field and electric displacement are uniform across the piezoelectric thicknesses and aligned on the normal to the mid-plane (z-direction), also the piezoelectric patches are polarized in z-direction.
1D finite element formulation
In a finite element formulation, the unknowns are the solution values at the nodes of the mesh and the displacement field {u} is related to the corresponding node values {ui} by an interpolation functions. Lagrangian functions are not used in this problem because the solution w (x, t) must be C1-continuous while Lagrange ensures C0 continuity; the choice of Hermite elements satisfies this condition. Thus, with Hermite elements, the solution {u} that depends only on w (x, t) in this case, and it is written as follows on a segment Si: w(x, t) = []{ui}.
Table of contents :
Chapter 1: locomotion principle for piezoelectric miniature robots
1.1. Introduction
1.2. Piezoelectric miniature robot
1.3. Piezoelectric actuators
1.4. Locomotion on a solid substrate
1.4.1. Wheeled locomotion
1.4.2. Walking locomotion
1.4.3. Inchworm locomotion
1.4.4. Inertial drive
1.4.5. Resonant drive
1.4.6. Friction drive
1.4.7. Summary of miniature robots on a solid substrate
1.5. Locomotion in liquid
1.5.1. Locomotion inside water
1.5.1.1. Fish swimming mechanisms
1.4.1.2. Micro-organisms swimming mechanisms
1.5.2. Locomotion at the water surface
1.5.3. Summary of piezoelectric miniature robots inside and on liquid
1.6. Locomotion in air
1.6.1. Flapping wing MAV
1.7. Conclusion and discussion
1.8. References
Chapter 2: Introduction to the numerical modeling of thin structures with piezoelectric patches
2.1. Introduction
2.2. Mechanical equations
2.2.1 Strains
2.2.2 Stresses
2.2.3 Linear elasticity
2.3. Piezoelectricity
2.4. Unknowns to be determined
2.4.1. Bending vibrations of beams
2.4.2. Bending vibrations of plates
2.5. Static equation
2.6. Dynamic equation
2.7. Numerical modeling
2.7.1. Bending vibrations of beams
2.7.2. Bending vibrations of plates
2.7.3. Variational principle
2.7.4. Time discretization: Newmark method
2.8. Conclusion and discussion
2.9. References
Chapter 3: Modeling of non-collocated piezoelectric patches bonded on thin structures
3.1. Introduction
3.2. Constitutive equations
3.2.1 Case of piezoelectric patches bonded on a beam
3.2.1. 1 Mechanical constitutive equations
3.2.1. 2 Piezoelectric constitutive equations
3.2.2 Case of piezoelectric patches bonded on a plate
3.2.2.1 Mechanical constitutive equation
3.2.2.2 Piezoelectric constitutive equation
3.3. Displacement field
3.3.1 Neutral axis
3.3.2 Neutral plane
3.4. Variational formulation
3.4.1 Case of 1D formulation
3.4.2 Case of 2D formulation
3.5. 1D finite element formulation
3.6. 2D finite element formulation
3.7. Numerical equation
3.7.1 Case of beam
3.7.3 Beam-plate numerical equation
3.7.4 Actuator – sensor
3.7.5 Actuator – Actuator
3.8 Conclusion of the chapter
3.9 Appendix: Particular cases
3.10 References
Chapter 4: Experimental validation of models
4.1 Introduction
4.2 Experimented device
4.3 Rayleigh damping
4.4 Validation process
4.4.1 Resonance frequencies validation
4.4.2 Transverse displacement validation
4.4.2.1 Case of beam
4.4.2.2 Case of plate
4.4.2.3 Piezoelectric sensors Validation
4.4.2.4 Piezoelectric capacitance Validation
4.5 Conclusion and discussion
4.6 Appendix
4.7 References
Chapter 5: traveling wave piezoelectric beam robot
5.1 Introduction
5.2 Operation principle
5.2.1 Standing wave and traveling wave
5.2.2 Operation principle case of one mode excitation
5.2.3 Operation principle case of two modes excitation
5.3 Modeling of the piezoelectric beam robot
5.4 Optimal design
5.4.1 Thickness of piezoelectric patches and material used for the beam
5.4.2 Resonance frequency
5.4.3 Optimal operating frequency
5.4.3.1 Case of one mode excitation
5.4.3.1.1 Position 1
5.4.3.1.2 Position 2
5.4.3.1.3 Influence of positions to the performance of the traveling wave
5.4.3.1.4 Actuator-absorber & Absorber-actuator
5.4.3.2 Case of two modes excitation
5.4.3.2.1 Position 1
5.4.3.2.2 Position 2
5.4.3.2.3 Influence of positions to the performance of the traveling wave
5.4.3.2.4 Two modes excitation functionality
5.5 Conclusion and discussion
5.6 Appendix
5.7 References
Chapter 6: Robot manufacturing and experimental measurements
6.1 Introduction
6.2 Fabrication
6.3 Electronic and electric circuits design and realization
6.4 Experimental validation
6.5 Robot characterization
6.6 Significance and benefits
6.7 Conclusion of this chapter
6.8 Appendix
6.9 References
Chapter 7: Overview
7. 1 Introduction
7. 2 Piezoelectric transformers
7. 3 Damping vibration of thin beams and plates
7. 4 Active control of flexible structures
7. 6 Optimization topology
7. 7 Analytical model versus finite element model
7. 8 References
Conclusion and perspectives
