VM129

VM129
Numerical Differentiation and Integration

Overview

Reference:	Any Basic Calculus Book
Analysis Type(s):	First and Second Derivatives and Integrals Using APDL
Element Type(s):	None
Input Listing:	vm129.dat

Test Case

Given a sine wave f(t) = A sin ωt, find the maximum value of the first and second derivatives. For the same sine wave, find the values of the two integrals:

Figure 181: Numerical Differentiation and Integration

Definitions:

A = 1.2732

ω = frequency = π/72 rad/sec

Analysis Assumptions and Modeling Notes

Arrays for one cycle of 145 data points (one data point per second) are generated, starting from a value of 0 up to 144 (representing time, t, in sec). The *VOPER command is used to obtain the first and second derivatives, and the single and double integrals.

Results Comparison

	Target	Mechanical APDL	Ratio
df/dt (Max)	5.555 x 10^-2	5.554 x 10^-2	1.000
d²f/dt² (Max)	2.424 x 10^-3	2.422 x 10^-3	0.999
l₁	29.18	29.17	1.000
l₂	381.7	381.6562	1.000

max df/dt is obtained as DERIV1 array parameter E.

max d²f/dt² is obtained as DERIV2 array parameter G

l₁ and l₂ are obtained from array parameters F and H (at row 37 corresponding to t = 36 sec)