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II. How To Use ATILA 3.1
Bimorph (4) PROBLEM - 3D Rectangular Plate, PZT4 = 40mm x 6mm x 1mm x 2 pieces, Cantilever Support PROBLEM TYPE We consider here various problem types:
In this analysis, the matrix equation becomes:
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:
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 MethodThis 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 MethodThis method consists in using a truncated Taylor series expansion of Xn+1 and its first derivative:
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:
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 MethodThis 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:
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., CALCULATION
EXAMPLE
Transident -> Wilson -> Calculation -> Postprocess -> Observe the Current and Phase results Note that this pulse is the uniform stress on the bimorph. | ||||||||||||||||||||||||||||||||||||||||||||||