Glossary

24.1 Sources of nonlinearity in piezoelectric systems

3 min read•august 9, 2024

Piezoelectric systems aren't always straightforward. They can get funky due to various nonlinearities. These include material quirks, shape changes, and weird interactions between electrical and mechanical parts.

Understanding these nonlinearities is crucial for designing better energy harvesters. They can make things tricky, but they also open up new possibilities for harvesting energy across a wider range of conditions.

Material and Geometric Nonlinearities

Fundamental Types of Nonlinearities

Top images from around the web for Fundamental Types of Nonlinearities

  • Frontiers | Field-Induced Nonlinearities in Silicon Waveguides Embedded in Lateral p-n Junctions View original

    Is this image relevant?

  • Piezoelectric energy harvesting from a PMN–PT single nanowire - RSC Advances (RSC Publishing ... View original

    Is this image relevant?

  • Frontiers | Field-Induced Nonlinearities in Silicon Waveguides Embedded in Lateral p-n Junctions View original

    Is this image relevant?

1 of 3

Top images from around the web for Fundamental Types of Nonlinearities

  • Frontiers | Field-Induced Nonlinearities in Silicon Waveguides Embedded in Lateral p-n Junctions View original

    Is this image relevant?

  • Piezoelectric energy harvesting from a PMN–PT single nanowire - RSC Advances (RSC Publishing ... View original

    Is this image relevant?

  • Frontiers | Field-Induced Nonlinearities in Silicon Waveguides Embedded in Lateral p-n Junctions View original

    Is this image relevant?

1 of 3
  • arises from the inherent properties of piezoelectric materials changing under varying stress or strain conditions
  • occurs when the structure's shape significantly changes during deformation, altering its stiffness and dynamic behavior
  • Large deformation manifests when displacements become comparable to the structure's dimensions, leading to nonlinear strain-displacement relationships
  • Duffing oscillator serves as a mathematical model representing systems with cubic nonlinearity in their restoring force (spring-mass systems)

Implications of Material and Geometric Nonlinearities

  • Material nonlinearity affects the piezoelectric coefficients, causing them to vary with applied stress or electric field strength
  • Geometric nonlinearity introduces additional terms in the equations of motion, complicating the system's dynamic response
  • Large deformation necessitates the use of nonlinear strain measures and can lead to phenomena like snap-through buckling
  • Duffing oscillator behavior includes multiple equilibrium positions, jump phenomena, and chaotic responses under certain conditions

Mathematical Representations and Analysis

  • Material nonlinearity often modeled using higher-order terms in constitutive equations (D=εE+αE2+βE3D = εE + αE^2 + βE^3, where D is electric displacement, E is electric field, ε is permittivity, α and β are nonlinear coefficients)
  • Geometric nonlinearity represented through nonlinear strain-displacement relations (von Kármán plate theory for moderately large deflections)
  • Large deformation analysis requires consideration of Green-Lagrange strain tensor and second Piola-Kirchhoff stress tensor
  • Duffing oscillator equation: x¨+δx˙+αx+βx3=Fcos⁡(ωt)\ddot{x} + δ\dot{x} + αx + βx^3 = F\cos(ωt), where x is displacement, δ is damping coefficient, α and β are stiffness coefficients, F is forcing amplitude, and ω is forcing frequency

Damping and Boundary Condition Nonlinearities

Nonlinear Damping Mechanisms

  • Damping nonlinearity occurs when energy dissipation depends nonlinearly on system motion or state
  • Coulomb damping results from dry friction, producing a constant opposing force independent of velocity
  • Quadratic damping arises in fluid systems, with damping force proportional to the square of velocity
  • Structural damping involves complex interactions within materials, often modeled using frequency-dependent damping coefficients

Boundary Condition Nonlinearities

  • Boundary condition nonlinearity emerges from changes in support conditions during system motion
  • Contact nonlinearity occurs when a structure impacts a rigid stop or another flexible structure
  • Sliding supports introduce friction-based nonlinearities at the boundaries
  • Nonlinear springs or dashpots at boundaries create force-displacement or force-velocity nonlinearities

Analysis and Modeling Techniques

  • Damping nonlinearity often modeled using velocity-dependent terms (Fd=c1x˙+c2x˙∣x˙∣+c3x˙3F_d = c_1\dot{x} + c_2\dot{x}|\dot{x}| + c_3\dot{x}^3, where F_d is damping force, c_1, c_2, c_3 are damping coefficients)
  • Coulomb damping represented by signum function: Fd=μNsgn(x˙)F_d = μN\text{sgn}(\dot{x}), where μ is friction coefficient and N is normal force
  • Boundary condition nonlinearities incorporated through piecewise functions or inequality constraints in equations of motion
  • Numerical methods like the shooting method or used to analyze systems with damping and boundary nonlinearities

Electromechanical Coupling Nonlinearity

Sources of Electromechanical Coupling Nonlinearity

  • Electromechanical coupling nonlinearity arises from complex interactions between electrical and mechanical domains in piezoelectric systems
  • Material nonlinearity contributes to coupling nonlinearity through stress-dependent piezoelectric coefficients
  • Large electric fields can induce nonlinear polarization effects, altering the electromechanical coupling
  • Geometric nonlinearities affect the distribution of electric field within the piezoelectric material, leading to nonlinear coupling

Effects on System Behavior

  • Coupling nonlinearity results in frequency-dependent
  • Harmonic generation occurs, producing higher-order frequency components in the system response
  • Energy transfer between modes becomes possible due to nonlinear coupling, affecting the system's dynamic behavior
  • Broadband energy harvesting potential increases due to nonlinear coupling effects

Modeling and Analysis Approaches

  • Higher-order terms included in piezoelectric constitutive equations to capture coupling nonlinearity (S=sET+dE+μTE+γE2S = s^ET + dE + μTE + γE^2, where S is strain, T is stress, E is electric field, s^E is elastic compliance, d is piezoelectric strain coefficient, μ and γ are nonlinear coupling coefficients)
  • with nonlinear elements used to model complex geometries and coupling effects
  • Perturbation methods applied to analyze weakly nonlinear coupling in piezoelectric systems
  • Experimental characterization techniques (impedance analysis, laser vibrometry) employed to quantify nonlinear coupling effects in real devices

Key Terms to Review (18)

Bandwidth: Bandwidth refers to the range of frequencies over which a system, such as a piezoelectric harvester, can effectively operate to convert mechanical energy into electrical energy. In the context of energy harvesting, it relates to how responsive the device is across different excitation frequencies, influencing overall performance and energy conversion efficiency.
Ceramic piezoelectrics: Ceramic piezoelectrics are materials that exhibit piezoelectric properties, allowing them to generate an electric charge in response to mechanical stress, and are commonly used in various applications for energy harvesting. These ceramics often possess high dielectric constants and are characterized by their ability to be easily shaped and molded, making them suitable for a wide range of energy harvesting devices. Their unique properties enable efficient conversion of mechanical energy into electrical energy, which is crucial for various technologies.
Duffing Equation: The Duffing Equation is a nonlinear second-order differential equation that describes the motion of a damped oscillator with a nonlinear restoring force. It is often used to model systems where the stiffness of the system changes with the displacement, making it particularly relevant in the analysis of piezoelectric energy harvesting systems where nonlinearity can arise from various sources.
Dynamic Mechanical Analysis: Dynamic Mechanical Analysis (DMA) is a technique used to measure the mechanical properties of materials as a function of time, temperature, and frequency. This method helps in understanding how materials respond to mechanical stress and strain under various conditions, which is crucial for optimizing performance in energy harvesting applications.
Dynamic Testing: Dynamic testing is a method of evaluating the performance and reliability of piezoelectric systems under varying conditions and stimuli. This type of testing is crucial for understanding how these systems respond to real-world applications, as it involves applying dynamic loads and analyzing the system's output in response. The insights gained from dynamic testing are essential for optimizing energy harvesting efficiency, validating lumped parameter models, and identifying nonlinear behaviors in piezoelectric materials.
Electromechanical Coupling Coefficient: The electromechanical coupling coefficient (k) quantifies the efficiency of converting electrical energy into mechanical energy and vice versa in piezoelectric materials. It indicates how well a material can convert an applied electric field into mechanical deformation, which is crucial for applications in energy harvesting, sensors, and actuators.
Energy Harvesting Devices: Energy harvesting devices are technologies that capture and convert ambient energy from the environment into usable electrical energy. These devices play a vital role in powering small electronics, sensors, and IoT devices by utilizing sources such as vibrations, heat, light, or kinetic energy, ensuring sustainability and reducing dependence on batteries.
Finite Element Analysis: Finite Element Analysis (FEA) is a computational technique used to predict how structures and materials will respond to external forces, vibrations, heat, and other physical effects by breaking down complex objects into smaller, simpler parts called finite elements. This method is essential for understanding the performance and behavior of piezoelectric devices, as it helps in optimizing designs and improving efficiency across various applications.
Geometric nonlinearity: Geometric nonlinearity refers to the phenomenon where the response of a system changes nonlinearly due to large deformations, which affect the geometry of the system itself. In piezoelectric systems, this type of nonlinearity is important because it influences how materials respond to applied forces, altering their performance characteristics and energy harvesting capabilities as they undergo significant shape changes.
Harmonic balance method: The harmonic balance method is a powerful mathematical technique used to analyze and solve nonlinear dynamic systems by approximating their behavior with a series of harmonic functions. This approach simplifies the analysis of systems that exhibit periodic responses, allowing researchers to capture the essential features of nonlinearity in piezoelectric energy harvesting systems and their interactions with external excitations.
Hysteresis: Hysteresis refers to the phenomenon where the state of a system depends on its history, particularly in relation to external stimuli. In piezoelectric systems, this means that the mechanical and electrical responses are not instantaneous and can vary based on prior stress or electric field levels, leading to a looped response curve. This nonlinearity is crucial for understanding energy dissipation in harvesting applications and impacts how these systems are modeled and analyzed.
Material Nonlinearity: Material nonlinearity refers to the behavior of materials that do not exhibit a linear relationship between stress and strain when subjected to external forces. In piezoelectric systems, this nonlinearity can arise due to factors like geometric changes, material properties, and operational conditions, which affect the efficiency and effectiveness of energy harvesting mechanisms.
Nonlinear constitutive equations: Nonlinear constitutive equations are mathematical relationships that describe the response of materials under external forces, highlighting that this response is not directly proportional to the applied forces. These equations take into account the complexity of material behavior, particularly in piezoelectric systems where factors like electric field strength and mechanical stress create a nonlinear relationship in how the material generates electrical energy. Understanding these equations is crucial in analyzing and predicting the performance of piezoelectric devices under varying conditions.
Polymer Piezoelectrics: Polymer piezoelectrics are materials made from polymer-based compounds that exhibit piezoelectric properties, meaning they can generate an electric charge in response to mechanical stress. These materials are often lightweight, flexible, and can be engineered for various applications, making them ideal candidates for energy harvesting systems and a wide range of sensors.
Resonance Frequency: Resonance frequency is the specific frequency at which a system naturally oscillates with greater amplitude due to the alignment of external forces and internal properties. This frequency plays a crucial role in maximizing energy transfer in energy harvesting systems, particularly for piezoelectric devices, allowing them to efficiently convert mechanical energy into electrical energy.
Strain coefficients: Strain coefficients are parameters that describe the relationship between mechanical strain and the electric polarization generated in piezoelectric materials. These coefficients play a crucial role in understanding how these materials respond to mechanical stress and how efficiently they can convert mechanical energy into electrical energy, particularly when considering the nonlinearity observed in piezoelectric systems.
Vibration damping systems: Vibration damping systems are mechanisms or materials designed to reduce or dissipate vibrations in mechanical structures, enhancing their stability and performance. These systems play a crucial role in mitigating the effects of vibrations caused by external forces or internal movements, thereby protecting sensitive components and prolonging the lifespan of structures. In piezoelectric energy harvesting, understanding how these systems interact with nonlinearity is essential for optimizing energy conversion efficiency.
Viscoelasticity: Viscoelasticity is the property of materials that exhibit both viscous and elastic characteristics when deformed. This means that they have the ability to both store energy like a spring (elastic behavior) and dissipate energy as heat (viscous behavior). Understanding viscoelasticity is crucial for analyzing how materials respond under various loading conditions, especially in the context of nonlinearity in piezoelectric systems.
© 2024 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.
Back
Practice QuizGlossary
Practice QuizGlossary
Next guide