ME 522 Homework 6
General Nonlinear Analyses
Distributed: April 15, 2014
Due: April 29, 2014
Problem 1: In class we examined the case of a softening spring. Now consider the similar
case of a hardening spring:
x + 2x3 = f
where f=1. For the parts below, you can do them by hand or with Matlab/Maple/Excel.
(a) Use full Newton-Raphson with 1 load step to solve for x within 1%. Sketch the “graphical
interpretation” as we did in class.
(b) Attempt to use modified Newton with 1 load step to solve for x within 1%. Sketch the
“graphical interpretation”.
(c) Use modified Newton with 2 equal load steps to solve for x within 1%.
Problem 2: Consider a system which consists of two hardening springs in series. Each
spring has a stiffness k = 2 + 0.4(Δx)2 .
x2
x1
r
(a) Show that the governing nonlinear equations for the system are:
2x1 + .4x31 − 2(x2 − x1 ) − .4(x2 − x1 )3 = 0 = G1
2(x2 − x1 ) + .4(x2 − x1 )3 − r = 0 = G2
(b) Find
∂G
1
∂x1
∂G2
∂x1
∂G1
∂x2
∂G2
∂x2
(Solution:
KL =
KN L = 1.2
= KL + KN L
4 −2
−2
2
x21 + (x2 − x1 )2 −(x2 − x1 )2
−(x2 − x1 )2
(x2 − x1 )2
.)
(c) Write a program to solve for x1 and x2 using full Newton with 1 load step. Assume the
solution has converged when
| ΔXn |
< 10−4
| Xn |
1
and
ΔXn · (R − F)n−1
< 10−8
ΔX1 · (R − F)0
(This takes only four steps. Do not let your program try more than 10 steps.)
2