Finite Element Formulation for Static Shape Control of a Thin Euler-Bernoulli Beam Using Piezoelectric Actuators

Aerospace ◽  
2004 ◽  
Author(s):  
Eric J. Ruggiero ◽  
John Singler ◽  
John A. Burns ◽  
Daniel J. Inman
Keyword(s):  
Finite Element ◽  
Active Control ◽  
Shape Control ◽  
High Surface ◽  
Weak Form ◽  
Element Formulation ◽  
Bernoulli Beam ◽  
Control Scheme ◽  
Surface Precision ◽  
Euler Bernoulli Beam

The main component of future space satellites will be an ultra-large, ultra-low mass aperture for high bandwidth communication or high quality imaging from on-orbit. Such an aperture will require an extremely high surface precision tolerance in order to be effective, especially for imaging purposes. Such tight surface precision tolerances dictate the use of an active control scheme to enable tight control of the shape of the aperture. Further, by integrating an active control scheme during the fabrication process, the aperture will become multi-functional and enable many scientific endeavors. One possible method for analyzing ultra-flexible space structures is through the use of the finite element method. Although many commercial packages are available, careful design of a tailored finite element solver can reveal important information about the system, such as where sensors should be placed on the structure. As an illustrative example, this work formulates the weak form of the equation of motion governing the dynamics of a cantilevered, Euler-Bernoulli beam. In particular, static shape control will be implemented on such a beam using a mathematically formulated LQR controller.

FINITE ELEMENT ANALYSIS OF PIEZOELECTRIC ELASTICITY WITH SINGULAR INPLANE ELECTROELASTIC FIELDS

2006 ◽  
Vol 03 (01) ◽  
pp. 115-135 ◽  
Author(s):  
MENG-CHENG CHEN ◽  
JIAN-JUN ZHU ◽  
K. Y. SZE
Keyword(s):  
Finite Element ◽  
Ad Hoc ◽  
Wedge Angle ◽  
Electric Displacement ◽  
Weak Form ◽  
Efficient Solutions ◽  
Element Formulation ◽  
Displacement Fields ◽  
Element Analysis ◽  
Electroelastic Fields

An ad hoc one-dimensional finite element formulation is developed for the eigenanalysis of inplane singular electroelastic fields at material and geometric discontinuities in piezoelectric elastic materials by using the eigenfunction expansion procedure and the weak form of the governing equations for prismatic sectorial domains composed of piezoelectrics, composites or air. The order of the electroelastic singularities and the angular variation of the stress and electric displacement fields are obtained with the formulation. The influence of wedge angle, polarization orientation, material types, and boundary and interface conditions on the singular electroelastic fields and the order of their singularity are also examined. The simplicity and accuracy of the formulation are demonstrated by comparison to several analytical solutions for piezoelectric and composite multi-material wedges. The nature and speed of convergence suggests that the present eigensolution could be used in developing hybrid elements for use along with standard elements to yield accurate and computationally efficient solutions to problems having complex global geometries leading to singular electroelastic states.

Developing a beam formulation for semi-crystalline two-way shape memory polymers

2020 ◽  
Vol 31 (12) ◽  
pp. 1465-1476
Author(s):  
Mohammad-Ali Maleki-Bigdeli ◽  
Majid Baniassadi ◽  
Kui Wang ◽  
Mostafa Baghani
Keyword(s):  
Finite Element ◽  
Shape Memory ◽  
Shape Memory Polymer ◽  
Beam Theory ◽  
Shape Memory Polymers ◽  
Bernoulli Beam ◽  
Von Karman ◽  
Von Kármán ◽  
Euler Bernoulli Beam ◽  
Beam Theories

In this research, the bending of a two-way shape memory polymer beam is examined implementing a one-dimensional phenomenological macroscopic constitutive model into Euler–Bernoulli and von-Karman beam theories. Since bending loading is a fundamental problem in engineering applications, a combination of bending problem and two-way shape memory effect capable of switching between two temporary shapes can be used in different applications, for example, thermally activated sensors and actuators. Shape memory polymers as a branch of soft materials can undergo large deformation. Hence, Euler–Bernoulli beam theory does not apply to the bending of a shape memory polymer beam where moderate rotations may occur. To overcome this limitation, von-Karman beam theory accounting for the mid-plane stretching as well as moderate rotations can be employed. To investigate the difference between the two beam theories, the deflection and rotating angles of a shape memory polymer cantilever beam are analyzed under small and moderate deflections and rotations. A semi-analytical approach is used to inspect Euler–Bernoulli beam theory, while finite-element method is employed to study von-Karman beam theory. In the following, a smart structure is analyzed using a prepared user-defined subroutine, VUMAT, in finite-element package, ABAQUS/EXPLICIT. Utilizing generated user-defined subroutine, smart structures composed of shape memory polymer material can be analyzed under complex loading circumstances through the two-way shape memory effect.

Weak Form Finite Element Formulation for the Helmholtz Equation

2004 ◽  
Vol 41 (1) ◽  
pp. 1-9 ◽  
Author(s):  
Lian Yuh Tio ◽  
Andrew A. P. Gibson ◽  
Bernice M. Dillon ◽  
Lionel E. Davis
Keyword(s):  
Finite Element ◽  
Helmholtz Equation ◽  
Weak Form ◽  
Finite Element Formulation ◽  
Element Formulation

Analysis of bimorph piezoelectric beam energy harvesters using Timoshenko and Euler–Bernoulli beam theory

2012 ◽  
Vol 24 (2) ◽  
pp. 226-239 ◽  
Author(s):  
Gang Wang
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Slenderness Ratio ◽  
Beam Theory ◽  
Bernoulli Beam ◽  
Energy Harvesters ◽  
Bernoulli Beam Theory ◽  
Spectral Finite Element ◽  
Euler Bernoulli Beam ◽  
Element Method

Single-degree-of-freedom lumped parameter model, conventional finite element method, and distributed parameter model have been developed to design, analyze, and predict the performance of piezoelectric energy harvesters with reasonable accuracy. In this article, a spectral finite element method for bimorph piezoelectric beam energy harvesters is developed based on the Timoshenko beam theory and the Euler–Bernoulli beam theory. Linear piezoelectric constitutive and linear elastic stress/strain models are assumed. Both beam theories are considered in order to examine the validation and applicability of each beam theory for a range of harvester sizes. Using spectral finite element method, a minimum number of elements is required because accurate shape functions are derived using the coupled electromechanical governing equations. Numerical simulations are conducted and validated using existing experimental data from the literature. In addition, parametric studies are carried out to predict the performance of a range of harvester sizes using each beam theory. It is concluded that the Euler–Bernoulli beam theory is sufficient enough to predict the performance of slender piezoelectric beams (slenderness ratio > 20, that is, length over thickness ratio > 20). In contrast, the Timoshenko beam theory, including the effects of shear deformation and rotary inertia, must be used for short piezoelectric beams (slenderness ratio < 5).

Spectral finite element for vibration analysis of cracked viscoelastic Euler–Bernoulli beam subjected to moving load

2015 ◽  
Vol 226 (12) ◽  
pp. 4259-4280 ◽  
Author(s):  
Vahid Sarvestan ◽  
Hamid Reza Mirdamadi ◽  
Mostafa Ghayour ◽  
Ali Mokhtari
Keyword(s):  
Finite Element ◽  
Vibration Analysis ◽  
Moving Load ◽  
Bernoulli Beam ◽  
Spectral Finite Element ◽  
Euler Bernoulli Beam

scholarly journals A Finite Element Formulation for Kirchhoff Plates in Strain-gradient Elasticity

2019 ◽  
Author(s):  
Alireza Beheshti
Keyword(s):  
Finite Element ◽  
Strain Gradient ◽  
Virtual Work ◽  
Hermite Polynomials ◽  
Gradient Elasticity ◽  
Weak Form ◽  
Element Formulation ◽  
Strain Gradient Elasticity ◽  
Rectangular Plates ◽  
Kirchhoff Plates

The current contribution is centered on bending of rectangular plates using the finite element method in the strain-gradient elasticity. To this aim, following introducing stresses and strains for a plate based on the Kirchhoff hypothesis, the principle of the virtual work is adopted to derive the weak form. Building upon Hermite polynomials and by deeming convergence requirements, four rectangular elements for the static analysis of strain-gradient plates are presented. To explore the performance of the proposed elements, particularly in small scales, some problems are solved and the results are compared with analytical solutions.

Weak Form Quadrature Element Analysis of Diaphragm Walls

2014 ◽  
Vol 580-583 ◽  
pp. 380-385
Author(s):  
Ye Li ◽  
Hong Zhi Zhong
Keyword(s):  
Finite Element ◽  
Earth Pressure ◽  
Weak Form ◽  
Element Formulation ◽  
Element Analysis ◽  
Complex Load ◽  
Beam Elements ◽  
Finite Element Software ◽  
Load Distributions ◽  
Diaphragm Walls

In combination with Rankine's earth pressure theory, a weak form quadrature element formulation is established for analysis of diaphragm walls. Results are compared with those of Paroi2, a finite element software package for diaphragm walls, to demonstrate the effectiveness and the advantages of the present formulation. Accurate results are obtained with only a few weak form quadrature beam elements, contrasting with dense finite element division that is needed for complex load distributions over the diaphragm wall.

Sign in / Sign up

Export Citation Format

Share Document