PIBRAC-WP320: Initial Results of FEA Study

PIBRAC-WP320 Initial results of FEA study Pedro Portela, Pedro Bandeira, J. F. Silva Gomes October 2005 1 Introduction and Scope The purpose of this document is to report on the results of the Finite Elements analysis performed on the high power- PIBRAC piezoelectric motor. 1 2 Geometry, material properties and assumptions This chapter will address the data used to develop the FE model. This includes the geometric data, provided by SAGEM, the mechanical and piezoelectric material properties and some assumptions regarding the behavior and operating conditions of the piezoelectric motor. 2.1 Geometry The fundamental geometric characteristics and dimensions of the piezoelectric motor were taken from version 7 (seven) of the design spreadsheet. This development of this spreadsheet is of the responsibility of SAGEM. A summary of the geometric design parameters is liste in table 2.1. The significance of these parameters can be verified in figure 2.1 where a 1 8sector of the motor stator is represented. For convenience, the motor exterior and interior diameter are also represented. Parameter name Parameter symbol Value Leave mean radius rm36.36 Leave delta radius △rm9.09 Leave exterior radius re40.91 Leave interior radius ri31.82 Tangential ceramic thickness ec2.54 Exterior leave thickness em5.07 Tangential ceramic width △rc8.82 Leave length lm7.4 Tangential ceramic length lc18.16 Rigid layer thickness ei1.03 Elastic layer thickness ee0.87 Table 2.1: Geometric data compiled from the design model (Mod`ele 7 spreadsheet.) 2 40.91 R (Re) 9.08 (Lc/2) 1.01 9x 100 8.82 (DRc) 9.09 (DRm) Figure 2.1: Technical drawing of the sector used in the analysis. 3 2.2 Material properties In figure 2.2 the five different material assignments are shown. The wedges, the metallic block and the rigid layer are made of an isotropic steel. The elastic layer is made of a yet undefined material but must have a tangential stiffness of Kt= 8.787N/m. Metallicblock Piezoelectricstacks Wedges Elasticlayer Rigidlayer Figure 2.2: Illustration of the five different material assignments. The mechanical properties of these materials are listed in table 2.2. The properties of the piezoceramic materials are listed separately in the following pages. Material name Young Modulus Poisson ratio Density Metallic block 2.1E011 0.3 7700 Wedges 2.1E11 0.3 7700 Rigid layer 2E11 0.3 7850 Elastic layer 2.48E9 0.2 785 Table 2.2: Mechanical properties of the isotropic materials used in the model.) The was determined by acknowledging that the tangential stiffness Ktcan be given by 4 Kt=GA ee (2.1) where G : is the shear modulus of the material A : is the surface area of the elastic layer ee: is the thickness of the elastic layer Additionally, the elastic layer’s density is estimated to be roughly 2% of the density of steel. The piezoceramic mechanical properties can be considered orthotropic. E=  7.69E10 7.69E10 5.09E3  Pa The coordinate system associated to the piezoelectic elements is shown in figure 2.3. x z y Figure 2.3: Material coordinate system for the piezoelectric elements. The shear modulus is G= 1.7E10 and the density is ρ= 7700kg/m3V. The dielectric matrix is given by D=  7.04E−9 0 0 0 7.04E−9 0 0 0 5.95e−9  F/m The electric field will be applied in the polling direction (z). Thus, the piezoelectric coefficient matrix as required by ABAQUS is mostly zero, except for three terms that correspond to the 5 effect produced by an electric field applied in the z or 3 direction. The piezoelectric stress coefficients are given by e3=  9.86 0 0 0−2.8 0 0 0 14.7  C/m2 2.3 Loads and boundary conditions For the eigenvalue extraction step, no loads were applied to the model, but two different sets of boundary conditions were applied. The first set, considers that the cyclic symmetry planes are clamped. This means that all the displacement components of every node on this surface are matched by an equal and opposite displacement on the surface adjacent to it. Alternatively, the displacement DOF’s on these surfaces can be matched by a constraint equation that represents cyclic symmetry. The displacements of one surface (the slave surface) are equal to the displacements of the master surface. This is the most realistic boundary condition and so it was also used in parallel to the clamped boundary condition. The results are discussed in the next chapter. Boundarysurfaces Figure 2.4: FE model mesh and boundary surfaces. 6 3 Analysis results and discussion 3.1 Eigenvalue analysis Eigenvalue analysis was performed to verify the mode shapes and eigenfrequency of the piezoelectric motor both in the tangential and longitudinal or normal direction. The PIBRAC piezoelectric motor must have a tangential and normal mode of vibration at the same frequency of 35kHz. 3.1.1 Tangential mode Figure 3.1 represents the tangential eigenmode of the sector subjected to clamped boundary conditions. Figure 3.1: Tangential eigenmode with Clamped boundary surfaces. Frequency 34.810 kHz Figure 3.2 represents the tangential eigenmode of the sector subjected to cyclic symmetry conditions. The cyclic symmetry mode for this eigenmode is four. This suggests that this is a local mode in four or the eight sector of the stator. This is coherent with the motor’s operation. 7 Figure 3.2: Tangential eigenmode with cyclic symmetric boundary conditions. Frequency 36.079 kHz. The displacement fields for both cases are very similar. However there is a slight difference of 1.269kHz (3.6%) between both models that is explained by the difference in stiffness caused by the boundary conditions. 3.1.2 Normal mode To determine the normal mode of vibration, the rotor and an additional stator was added to the assembly. The additional stator has an interface to the normal piezoelectric stack actuator that is responsible for the excitation of the longitudinal or normal mode. The material used for the rotor model was steel and cyclic symmetry boundary conditions were used on the side surfaces. Because the stator is free to rotate around the motor axis, the corresponding degree of freedom was released on this interface surface. All the others were restrained. Although the geometry used for the model is currently out-of- date the results of this first analysis is shown in figure 3.3. 8