II. How To Use ATILA

3.1 Bimorph (4)
VARIOUS PROBLEM TYPES VIEW RESULTS


PROBLEM - 3D Rectangular Plate, PZT4 = 40mm x 6mm x 1mm x 2 pieces, Cantilever Support


PROBLEM TYPE

We consider here various problem types:

Data -> Problem data -> 3D

  • Static - Click Compute Stress -> Pseudo static mode is obtained
  • Modal - Put the number of modes (10) -> Only the resonance modes are obtained
Modal Analysis
Resonance Modes
  • Modal Resantires - Put the number of modes (10) -> Both the resonance and antiresonance modes are obtained
    Modal Resantires Analysis
    Resonance and Antiresonance Modes with Coupling coefficient
    Assign the first resonance mode (Mode 3)
    The animation for the second resonance mode (Mode 6)
  • Harmonic - Click both Compute Stress and Include Losses -> Admittance curve can be obtained
  • Transient

In this analysis, the matrix equation becomes:

  (1)

where cap . and .. denote the first and second time derivative, respectively.  K¢ and K² denote the real and imaginary parts of K respectively.  w0 is the pulsation at which the materials losses are provided. The preceding system of equations may be rewritten as follows:

(2)

This differential equation is solved by an iterative method, taking a constant time step Dt; the successive values of X and Y are noted Xn and Yn and correspond to values calculated at the time nDt.  Three methods are implemented in ATILA: the Central Difference Method, the Newmark Method and the Wilson-q Method.

All methods are based on the same algorithm to calculate Z:

Z and the values of a1, a2, b1, b2, k, l and h depend on the chosen method and are explained below.

a) The Central Difference Method

This method consists in using a second order (parabolic) interpolation along the time axis and deriving the first and second order derivative from it:

Replacing these values in the equation (2) above written attime step n leads to the following parameter values:

Z = Xn+1/Dt2, a1,= ½, a2, = b1,= b2, = k = l = 0 and h = Dt.

Assuming that the initial first derivative of X is zero, the algorithm is supplied with an initial value of Xl:

This method is conditionally stable, i.e., knowing the highest eigen frequency fmax of the lossless system of equations, the time step must satisfy:

p fmax Dt < 1

b) The Newmark Method

This method consists in using a truncated Taylor series expansion of Xn+1 and its first derivative:

(3)

The parameters b and g may be arbitrarily chosen. Introducing these equations in equation (2) above written at time step n+1 leads to the following parameter values:

, a1,= g, a2, =b, b1,= (1-2b)Dt2/2, b2, = (1-g)Dt, k = l = 1 and h = Dt.

Assuming that the initial first derivative of X is zero, the algorithm is supplied with an initialvalue of :

This method will be unconditionally stable, provided that the following inequalities are satisfied:

where fmax is the highest eigen frequency of the lossless system of equations. Note that a value of g > ½ introduces a numerical damping, which is generally avoided. Names are given to usual couples of (g, b): (1/2, 0) stands for the Central Explicit DifferenceMethod, (½, ¼) for the Average Acceleration Method, (1/2,1/6) for the Linear Acceleration Method.

c) The Wilson-q Method

This method is very similar to the Newmark Method, except that the equations (3) are first written for the time step n+q, q³1, instead of n+1 (Dt is replaced with qDt) and the parameters g and b are respectively set to ½ and 1/6. Once Xn+q and its first and second time derivatives are calculated as above, a linear interpolation between Xn and Xn+q gives the solution Xn+1:

This leads to the following parameter values:

, a1,= q/2, a2,= q2/6, b1,= q2Dt2/3, b2 = qDt/2,k = 1, l = q and h = qDt.

The stability of this algorithm is more difficult to determine, but is proven for values of q greater than 1.366.

Important note:

Extending these algorithms to piezoelectricity or magnetostriction may lead to inaccurate results, because of the quasi-static nature of electrical degrees of freedom: the matrix [A] of equation (2) becomes singular, fmax tends to infinity. In the case of the Central Difference Method, the implemented algorithm can deal with this, provided that the loss matrix [B] contains elastic losses only, i.e., and  are both null. In the case of the Newmark Method, a value of g slightly greater than 0.5 will help in damping the spurious oscillations related to electric transients.

CALCULATION EXAMPLE

Transident -> Wilson -> Calculation -> Postprocess -> Observe the Current and Phase results

   

Note that this pulse is the uniform stress on the bimorph.

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