Modal analysis of isotropic beams in peridynamics

Modal analysis in continuum mechanics is widely used in damage detection, quality control and to validate numerical simulations. Peridynamics is a reformulation of continuum mechanics that allow discontinuities, such as cracks and voids, to arise and propagate without any special techniques, however, modal analysis in peridynamics has not been explored in detail. In this study we simulate the modal response of an aluminium plate with clamped-free boundary conditions in peridynamics, and compare results to a FE simulation and an experimental modal test. The natural frequencies from peridynamics agree better with the experimental results than frequencies from finite-elements for all modes except mode 1. Differences between peridynamic and experimental results ranged (in magnitude) from 0.13% to 10.38%, and between peridynamic and finite-element results from 0.23% to 9.72%. All mode shapes calculated in peridynamics corresponded to those from finite-elements and experimental measurements.


Introduction
Natural frequencies and mode shapes, together called modes, are determined by the material properties i.e. mass, damping, stiffness, and object's boundary conditions. If either of these properties change, then the modes of an object will change. This is used to detect damage in structures, control manufacturing quality, validate simulations etc. [1]- [4].
Peridynamics (PD) is a reformulation of classical continuum mechanics (CM) that allows discontinuities, such as cracks in damaged structures, to be modelled without any special techniques. PD was first introduced in the bond-based form [5] and later extended to state-based form [6], in this article we only use the state-based PD. Modal analysis in PD has been explored in [7], where a good agreement between a two-dimensional PD simulation, classical mechanics solution, and finite-element (FE) solution was found. In this study we compare the results from PD modal analysis to FE and experimental results.
Continuum mechanics theory uses partial differential equations to represent strain and stress between particles. Since partial derivatives are undefined along discontinuities, cracks can't be modeled without the additional assumptions of fracture mechanics. PD is a non-local extension of CM, that uses integral equations and deformation to describe relations between particles, therefore relations remain valid in the presence of discontinuities. Furthermore, elastic PD model reproduces CM model as the non-local interaction distance goes to zero, if the motion, constitutive model and nonhomogeneities are sufficiently smooth [8]. A PD body consists of an infinite number of particles, each associated with some volume and identified by its a coordinate . A particle interacts through bonds 〈 − 〉 with a family of particles , the interaction range is described by the horizon , and all interactions vanish beyond the horizon. In three dimensions, this means that a particle interacts with all particles within a sphere with a radius > 0 around itself. The particle is influenced by the collective deformation of all particles in its family . This interaction results in the force density vector acting at particle . Similarly, a particle is influenced by the collective deformation of its family and the force density vector acting at is , this vector can be viewed as the force exerted by the particle on the particle . Force density vectors are stored in an infinitedimensional array, called the force vector state, . When an object undergoes displacement , the position vector describes a particle's position in the deformed configuration and the relative position vectors in the deformed configuration − are stored in an array called the deformation vector state, . Since force vector state ( , ) depends on the relative deformation of , it can also be written as ( , ) = ( ( , )), where -force scalar state. The peridynamic equation of motion is , where -density, ̈ -acceleration, and -external force density. Cracks in a body are introduced by breaking bonds when some criterion is reached. The simplest damage model is the critical stretch i.e. a bond is broken when its stretched past some critical value. A very short description of peridynamics is given here, so for further information readers are referred to [9]- [11]. Modal analysis is solved by solving the eigenvalue problem that arises from the equation of motion of some object. In matrix form this equation is: where -force vector, -mass matrix, -damping matrix, -stiffness matrix, -displacement vector. We consider undamped free vibrations, so the ̇ term vanishes and force vector is equal to zero, and the equation of motion is reduced to: This equation is solved as generalized eigenvalue problem: where -eigenvectors, -eigenvalues. To solve the generalized eigenvalue problem a shift-inverse transformation is used. The original problem is shifted closer to some value and inverted so that the smallest eigenvalues of the original problem becomes the smallest eigenvalues of the transformed problem: Each eigenvalue corresponds to one natural frequency , and eigenvector corresponds to the mode shape of . Natural frequencies are related to eigenvalues through: For further discussion about structural dynamics see [12], and for eigenvalue problems see [13].

Modal analysis
In this study, we compare natural frequencies and mode shapes from PD simulation with the results from FE and experimental modal analysis. A rectangular aluminium plate clamped at one end and free at the other with side lengths of 0.25m×0.05m×0.002m was used as a benchmark, modal test setup is showed in Figure 1. A piezo-electric element located at the end of the beam induced vibrations, and two-dimensional (2D) Polytec PSV-400 laser vibrometer measured the speed of 115 nodes on the plate. Since 2D vibrometer can't measure out-of-plane and longitudinal nodal speed, we could only extract in-plane bending and torsional modes.
Modal analysis requires stiffness and mass matrices. Peridigm [14] with added custom code was used to create them. Then a Matlab script with Arnoldi algorithm [15] calculated natural frequencies and mode shapes. In PD plate was modelled using meshfree approach [16], particle spacing was 0.0005m and the peridynamic horizon was set to 0.00075m. This particle spacing value was selected, because it created the finest mesh for which we could use Matlab "eigs" function without running out of computer's memory. Displacement in all degrees of freedom was fixed for one plane of particles at one end of the plate, creating clamped -free boundary conditions. Material response was modelled using position aware linear solid (PALS) material model [17]. Modulus of elasticity was set to 62.00GPa, shear modulus to 21.83GPa, and density to 2700kg/m 3 .
FE analysis was done in Ansys FE software. The plate was simulated using linear elastic material model with the same material properties as in PD analysis. It was meshed with cubic SOLID185 elements with an edge length of 0.0005m. Displacements were fixed in all degrees of freedom for all nodes on one end plane of the plate, creating clamped-free boundary conditions.

Results and discussion
The first eight modes were determined in PD simulation, FE simulation and experimentally using a laser vibrometer. Their natural frequencies are showed in Table 1. Natural frequency for Mode 5 couldn't be determined experimentally, because it was an out-of-plane bending mode and 2D vibrometer can't measure in the out-of-plane direction. Percent differences between PD, FE solutions, and experimental natural frequencies are showed in Figure 2. Difference between the experimental results (ER) and PD results in absolute values ranged from 0.13% (mode 7) to 10.38% (mode 1), and between ER and FE results from 1.66% (mode 3) to 13.02% (mode 2). PD results agreed better with ER than FE results in all modes except for mode 1. Natural frequencies in torsional modes (modes 3, 6, 8) computed using both PD and FE analysis were closer to ER than frequencies in bending modes. Possibly the value of shear modulus used in simulations were closer to its real value, than the value of modulus of elasticity in simulations to the real modulus of elasticity.
We expected better agreement between FE and PD results, because peridynamics converge to the classical theory as the horizon shrinks, and if the deformation field is sufficiently smooth [8]. The differences in absolute values ranged from 0.23% to 9.72%. However, if split by mode type, it can be seen that natural frequencies of in-plane bending modes agreed the worst with differences ranging from 9.05% (mode 7) to 9.72% (mode 2), next were torsional modes for which differences ranged from 3.18% (mode 3) to 3.70% (mode 8), and the best agreement, difference of 0.23%, was between the results of the out-of-plane bending mode (mode 5). A mesh with four particles in plate's height was used, maybe a better agreement can be achieved with a finer mesh, but mesh convergence wasn't studied here.    measure in the out-of-plane direction. Modes 6, 7, and 8 are the second torsional mode, the fourth inplane bending mode, and the third torsional mode respectively. All modes from PD and FE simulations correspond to each other and to those measured experimentally.

Conclusions
In this study we did a modal analysis of an plate, and compared the results to results from finiteelement simulation and modal test. Peridynamics showed better agreement with experimental results, than finite-element analysis in all modes except mode 1. We expected peridynamic results to have a better agreement with finite-element results, because peridynamics converge to classical mechanics theory when the horizon shrinks. However, better agreement might be achieved with finer mesh. Mode shapes from peridynamic simulation corresponded to those from finite-element simulation and modal test.