#!/usr/bin/env python
# -*- coding: utf-8 -*-
"""Mathmatical models governed by ordinary differential equations.
Describes initial value problems as systems of first order ordinary differential
equations.
:Date: 2019-05-25
.. module:: ode_model
:platform: *nix, Windows
:synopsis: Models of ordinary differential equations.
.. moduleauthor:: Daniel Weschke <daniel.weschke@directbox.de>
"""
from __future__ import division, print_function
from numpy import array, cos, sin, dot, square
from numpy.linalg import inv
[docs]def disk(x, t, *p):
"""Rotation of an eccentric disk.
:param x: values of the function
:type x: list
:param t: time
:type t: list
:param `*p`: parameters of the function
* diameter
* eccentricity
* torque
"""
qp1 = x[3]
qp2 = x[4]
qp3 = x[5]
M = array([[1, 0, cos(x[2])], [0, 1, -sin(x[2])], [0, 0, 1]])
y = array([[-2*p[0]*x[3]+sin(x[2])*x[5]**2-2*p[0]*cos(x[2])*x[5]-x[0]], \
[-2*p[0]*x[4]+cos(x[2])*x[5]**2-2*p[0]*sin(x[2])*x[5]-x[1]], \
[p[2]-p[1]*x[1]*sin(x[2])+p[1]*x[0]*cos(x[2])]])
qp46 = dot(inv(M), y)
qp4, qp5, qp6 = qp46.reshape(-1,).tolist() # 2d array to 1d array to list
return qp1, qp2, qp3, qp4, qp5, qp6
[docs]def disk_nm(xn, xpn, xppn, t, *p):
"""Rotation of an eccentric disk.
:param xn: values of the function
:type xn: list
:param xpn: first derivative values of the function
:type xpn: list
:param xppn: second derivative values of the function
:type xppn: list
:param t: time
:type t: list
:param `*p`: parameters of the function
* diameter
* eccentricity
* torque
"""
N = array([[xppn[0]+cos(xn[2])*xppn[2]+2*p[0]*xpn[0]+2*p[0]*cos(xn[2])*xpn[2]-sin(xn[2])*square(xpn[2])+xn[0]],
[xppn[1]-sin(xn[2])*xppn[2]+2*p[0]*xpn[1]-2*p[0]*sin(xn[2])*xpn[2]-cos(xn[2])*square(xpn[2])+xn[1]],
[xppn[2]+p[1]*(-cos(xn[2])*xn[0]+sin(xn[2])*xn[1])-p[2]]])
dN = array([[1, 0, -sin(xn[2]*xppn[2])-2*p[0]*sin(xn[2])*xpn[2]-cos(xn[2])*square(xpn[2])],
[0, 1, -cos(xn[2]*xppn[2])-2*p[0]*cos(xn[2])*xpn[2]+sin(xn[2])*square(xpn[2])],
[-p[1]*cos(xn[2]), p[1]*cos(xn[2]), p[1]*(sin(xn[2])*xn[0]+cos(xn[2])*xn[1])]])
dNp = array([[2*p[0], 0, 2*p[0]*cos(xn[2])-2*sin(xn[2])*xpn[2]],
[0, 2*p[0], -2*p[0]*sin(xn[2])-2*cos(xn[2])*xpn[2]],
[0, 0, 0]])
dNpp = array([[1, 0, cos(xn[2])],
[0, 1, -sin(xn[2])],
[0, 0, 1]])
return N, dN, dNp, dNpp
[docs]def disk_nmmdk(xn, xpn, xppn, t, *p):
"""Rotation of an eccentric disk.
:param xn: values of the function
:type xn: list
:param xpn: derivative values of the function
:type xpn: list
:param xppn: second derivative values of the function
:type xppn: list
:param t: time
:type t: list
:param `*p`: parameters of the function
* diameter
* eccentricity
* torque
"""
rm = array([[xppn[0]+cos(xn[2])*xppn[2]],
[xppn[1]-sin(xn[2])*xppn[2]],
[xppn[2]]])
rmx = array([[0, 0, -sin(xn[2]*xppn[2])],
[0, 0, -cos(xn[2]*xppn[2])],
[0, 0, 0]])
rmxpp = array([[1, 0, cos(xn[2])],
[0, 1, -sin(xn[2])],
[0, 0, 1]])
rd = array([[2*p[0]*xpn[0]+2*p[0]*cos(xn[2])*xpn[2]-sin(xn[2])*square(xpn[2])],
[2*p[0]*xpn[1]-2*p[0]*sin(xn[2])*xpn[2]-cos(xn[2])*square(xpn[2])],
[0]])
rdx = array([[0, 0, -2*p[0]*sin(xn[2])*xpn[2]-cos(xn[2])*square(xpn[2])],
[0, 0, -2*p[0]*cos(xn[2])*xpn[2]+sin(xn[2])*square(xpn[2])],
[0, 0, 0]])
rdxp = array([[2*p[0], 0, 2*p[0]*cos(xn[2])-2*sin(xn[2])*xpn[2]],
[0, 2*p[0], -2*p[0]*sin(xn[2])-2*cos(xn[2])*xpn[2]],
[0, 0, 0]])
rk = array([[xn[0]],
[xn[1]],
[p[1]*(-cos(xn[2])*xn[0]+sin(xn[2])*xn[1])]])
rkx = array([[ 1, 0, 0],
[ 0, 1, 0],
[-p[1]*cos(xn[2]), p[1]*cos(xn[2]), p[1]*(sin(xn[2])*xn[0]+cos(xn[2])*xn[1])]])
f = array([[0], [0], [p[2]]])
return rm, rmx, rmxpp, rd, rdx, rdxp, rk, rkx, f
if __name__ == '__main__':
True