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Fig. 2.3
Spring-mass system
One can also consider a model with nonlinear spring dynamics given by
g.x/ D k.1 a
2
x
2
/x j
ax
j <1 model of softening spring
g.x/ D k.1 C a
2
x
2
/x x > x
thres
model of hardening spring
(2.19)
The combination of a hardening spring, a linear viscous damping term and of a
periodic external force F D Acos.!t/ results into the Duffing oscillator
mx C c x C
kx
C ka
2
x
3
D Acos.!t/
(2.20)
The combination of a linear spring, a linear viscous damping term, a dry friction
term and of a zero external force generates the following oscillator model
mx C
kx
C c x C .x; x/ D 0
(2.21)
where
8
<
k
mg
sign.x/ if jx>0j
kx
if jxjD0 and jxj
s
mg
=k
s
mg
sign.x/ if
.x; x/ D
(2.22)
:
x D 0 and jxj >
s
mg
=k
By defining the state variables x
1
D x and x
2
Dx one has
x
1
D x
2
(2.23)
k
c
1
x
2
D
m
x
1
m
x
2
m
.x; x/
For x
2
Dx>0one obtains the state-space description
x
1
D x
2
(2.24)
k
m
x
1
c
m
x
2
k
g
x
2
D
