Abstract
1. In the present paper, a numerical method of getting differential coefficients of high order is presented.
A continuously differentable function ƒ is approximated by the linear combination of pyramid-shaped base function {Φi} and coefficients {αi} as
ƒ≈∑iαiΦi
Then, nth derivated function ∂nƒ/∂xk, …∂xlis also approximated with {Φi} and other coefficients {α∂xk, …∂xli} Using theory of distribution, coefficients {α∂xk, …∂xli} is expressed in terms of {αi} After all, nth derivated function is approximated by following form.
∂n/∂xk, …∂xlƒ≈∑iα∂xk, …∂xliΦi=∑ijαjN∂xk, …∂xlijΦi
α∂xk, …∂xli=∑jN∂xk, …∂xlijαj
2. The method is applicated for forced vibration of a beam.
Partial differential equations of beam vibration contain two differentials of 2nd order and 4th order ; ∂2/∂t2 and ∂4/∂x4. Though it is said to be difficult to build a space-time element applicable for such problem, it is quite easy to apply a space-time element for FEM analysis using the present method.
