student assignments solution

1. DEPARTMENTS OF MECHANICAL ENGINEERING, ME
MODELING AND STOCHASTIC ANALYSIS OF NONLINEAR
FRACTIONAL QUARTER CAR SUSPENSION SYSTEMS
Figure 1: Schematic diagram of SDOF quarter car model
The integer order nonlinear quarter car model
Following is established for the integer order of the suspension system dynamical equation
( )
1 0
o c
mx k x x F
+ − + = (a)
where m is mass of the body, x is the vertical acceleration of the mass, 1
1 16,000 .
k N m−
= is the
suspension stiffness coefficient, o
x the road excitation, x the body’s vertical displacement and c
F
is hysteretic nonlinear damping and stiffness force, which is dependent on the relative
displacement and velocity given by:
( ) ( ) ( )
3 3
2 1 2
c o o o
F k x x c x x c x x
= − + − + − (b)
where 240
m kg
= and 3
2 30,000 .
k N m−
= − , 1
1 250 . .
c N s m−
= and 3 3
2 25 . .
c N s m
− −
= − are hysteretic
nonlinear damping force with constants [h].
Considering Eq. (a), Eq. (b) can be rewritten as
( ) ( ) ( ) ( )
3 3
1 2 1 2 0
o o o o
mx k x x k x x c x x c x x
+ − + − + − + − = (3)
Assuming the relative vertical displacement o
y x x
= − , and the road profile ( )
o
x F t
= . We have
the capability to simplify equation of motion (3) to

2. DEPARTMENTS OF MECHANICAL ENGINEERING, ME
( )
3 3
y y y y y F t
   
+ + + + = (c)
Where 1
k
m
 = , 2
k
m
 = , 1
c
m
 = and 2
c
m
 = .
The states are chosen such that 1 2
,
y y y y
= = . Thus, the state space model for the system can be written
as
( )
,
1 2
3 3
2 1 1 2 2
y y
y y y y y F t
   
=
= − − − − + (d)
In this instance, we used a three-dimensional chaotic system as stochastic excitation.
,
5
3 4 5
4 3 5
.
5 3 4 5
,
1 2
3 3
2 1 1 2 2
,
3
,
4
5
ey
y
y
y y
y y
y y y y y
y ay dy
y by y
y cy y k
    +
=
= − − − −
= −
= − +
= − + +
(e)
Where; 4, 1 4, 7, 1,
c d a b k
= = = = = e the tuning parameter and 5
y is the stochastic noise.
Stability of the Equilibrium Points
The equilibrium points of the system (e) can be calculated by equating the state equations to 0.
,
5
3 4 5
4 3 5
.
5 3 4 5
0 ,
2
3 3
0 1 1 2 2
0 ,
0 ,
0
ey
y
y
y y
y
y y y y
ay dy
by y
cy y k
    +
=
= − − − −
= −
= − +
= − + +
(f)
The system has finite equilibrium points. The equilibrium points of the system are found a
 
1 0,0,2.5466,2.5466, 4.2426
E = −
 
2 0,0, 2.5466, 2.5466,4.2426
E = − −
 
3 0,0,0,0,1
E = .
The Jacobian matrix of the proposed quarter car model at equilibrium points is found as

3. DEPARTMENTS OF MECHANICAL ENGINEERING, ME
2 2
1 2
5 4
5 3
5 4 5 3 3 4
0 1 0 0 0
3 3 0 0
0 0
0 0
0 0
y c y e
J a dy dy
y b y
y y y y y y c
  
 
 
− − − −
 
 
= − −
 
−
 
 
−
 
The characteristics equation is computed by  
det 0.
I J
 − = Then solving the characteristics
equation for each equilibrium points will obtain the system Eigen-values. The Eigen-values for the
system are given by Table
Table The Equilibrium points and Eigen values of the System
S.No
.
Equilibrium point Eigen values
1  
1 0,0,2.5466,2.5466, 4.2426
E = − 1,2 3
4 5
0.5209 8.1483 , 4.0377,
3.2038, 5.3488.
i
 
 
= −  =
= − =
2  
2 0,0, 2.5466, 2.5466,4.2426
E = − − 1,2
3,4 5
0.5209 8.1483 ,
3.1231 6.3298 , 10.7611.
i
i

 
= − 
=  = −
3  
3 0,0,0,0,1
E = 1,2 3
4 5
0.5209 8.1483 , 1.9083,
8.9083, 4.0.
i
 
 
= −  =
= − =
As shown in the table above, the Eigen values 1,3
 of the equilibrium points 1
E and 3
E are saddle
points and the eigen values 3,4
 of the equilibrium points 2
E are unstable focus which satisfy the
stability condition for chaotic behavior.
The phase portraits and time domain response of system is shown in figure: 2 below under the
parameter values 4, 1 4, 7, 1
c d a b k
= = = = = and the initial conditions[0,0.1,0.1,0.1,0.1] shows
chaotic behavior.

4. DEPARTMENTS OF MECHANICAL ENGINEERING, ME
(a) Phase portrait (b) time series
Figure 2 Phase Portrait and time domain response for the system
The fractional order nonlinear quarter car model
There are three commonly used definition of the fractional order differential operator, viz.
Grunwald-Letnikov, Riemann-Liouville and Caputo .Caputo type fractional calculus is used in this
paper which is defined as
( )
1 ( )
( )
1 ( )
t
o
t g
D g t d
t
t




 
= 
 − −
(g)
Where α is the order of the system: o
t and t are limits of the fractional order equation and ( )
g t is
integer order calculus of the function. Using binomial approximation equation (4) can be modified
as
( ) ( ) ( )
( )
( )
0
0
lim
z t
t p i
t L
i
D g t p d g t ip
 
−
→
−
=
 
 
= −
 
 
 

(h)
Theoretically fractional order differential equations use infinite memory. Hence when we want to
numerically calculate or simulate the fractional order equations, we have to use finite memory
principal, where L is the memory length and p is the time sampling.
( )
1
min ,
1
i i
t L
Z t
p p
a
d d
i

−
 
   
=  
   
   
 
+
 
= −
 
 
(i)
Applying these fractional order approximations in to the integer order model (6) yields the
fractional order nonlinear quarter car model described by (j).
1
2
3
4
5
* ,
5
3 4 5
4 3 5
.
5 3 4 5
,
2
3 3
1 1 2 2
,
,
t
t
t
t
t
e y
y
y
y y
y
y y y y
ay dy
by y
cy y k
D y
D y
D y
D y
D y





    +
=
= − − − −
= −
= − +
= − + +
(j)
Where;  is the fractional order, 4, 1 4, 7, 1,
c d a b k
= = = = = e the tuning parameter and 5
y is the
stochastic noise.

5. DEPARTMENTS OF MECHANICAL ENGINEERING, ME
The frequency-domain method, the Adomian Decomposition Method (ADM) [, and the Adams-
Bashforth-Moulton (ABM) algorithm are the three primary methodologies for the numerical
analysis of the fractional order quarter car model. This section uses the ABM approach whose
accuracy and convergence are more thoroughly discussed in
Let us consider a fractional order dynamical system with order α as
( )
, , 0
D x f t x t T
 =   (k)
Where ( )
0
k k
y yo
= for  
0, 1
k n
 − and T is the finite time.
Equation (11) approaches the expression for the Volterra integral given in
( )
( )
( )
( )
1
1
1
0 0
,
1
!
t
k
n
k
o
k
f x
t
x t x d
k t x



−
=
= +
 −
  (l)
Where  
, : 0, .
n
T
h t nh h N
N
= =  and 1
h

 = .
We can define the discrete form of equation (12) as,
( )
( )
( )
( )
( )
( )
1
'
1 , 1
0
1 , 1 ,
!
k
n
k
o n n j n j n
t h h
x n x f t x n a f t x j
k z z
 
  
−
+ +
 
+ = + + +
 
 
+  +
 
  (m)
Where:
( )
( )
( ) ( ) ( )
1
1
1 1 1
, 1
1 0
2 2 1 1
1 1
j n
n n n j
a n j n j n j j n
j n


  

+
+
+ + +
+
 − − + =


= − + + − − − +  

 = +


( )
( )
( )
( )
1
'
, 1
0 0
1
1 ,
!
k
n n
k
n o j n
k j
t
x n x b f t x j
n
j
k 
−
+
= =
+ = +

 
( ) ( )
( )
, 1 1
j n
h
b n j n j

 

+ = − + − −
Using the definitions of (k) and (l), the fractional order nonlinear quarter car model can be defined
as,

6. DEPARTMENTS OF MECHANICAL ENGINEERING, ME
( ) ( )
( )
( )
( ) ( )
( )
1
1
1 1 2 1, , 1 2
0
1 0 ' 1 '
2
y n
j n
j
y
h
y n y y n x y j


+
=
 
+ = + + +
 
 +  
 (n)
( ) ( )
( )
( )
( ) ( )
( ) ( )
( )
( )
( ) ( )
( )
( )
( ) ( )
( ) ( )
( )
( )
( ) ( )
( )
2
2
3
' ' '
1 1 2
3
' '
2 2 2 5
3
' '
1 1 2
2, , 1 3
' '
0
2 5
1 1 1
1 0 1 1
2
y
y
n
j n
j
y n y n y n
h
y n y y n e y n
y j y j y j
x
y j e y j

  


  

+
=
 
 
 
− + − + − +
 
 
+ = + − + + +
 
 +  
 
 
− −
 
 
+
 
 
 
− +
 
 

o)
( ) ( )
( )
( )
( ) ( )
( ) ( )
( )
( ) ( )
( ) ( )
( )
3
3
3 4 5
3 3
3, , 1 3 4 5
0
' 1 ' 1 ' 1
1 0
' ' '
2
y
n
j n
y
j
a y n d y n y n
h
y n y
x a y j d y j y j

 +
=
 
+ − + +
 
+ = +  
+ −
 +  
 

(p)
( ) ( )
( )
( )
( ) ( ) ( )
( )
( ) ( ) ( )
( )
4
4
4 3 5
4 4
4, , 1 4 3 5
0
' 1 ' 1 ' 1
1 0
' ' '
2
y
n
j n
y
j
b y n y n y n
h
y n y
x b y j y j y j

 +
=
 
− + + + +
 
+ = +  
+ − +
 +  
 

(q)
( ) ( )
( )
( )
( ) ( ) ( ) ( )
( )
( ) ( )
( )
5
5
5 3 4 5
5 5
5, , 1 5 3 4 5
0
' 1 ' 1 ' 1 ' 1
1 0
' ' ( ) ' ( ) '
2
y
n
j n
y
j
c y n y n y n y n k
h
y n y
x c y j y j y j y j k

 +
=
 
+ + + + + +
 
+ = +  
+ + +
 +  
 

(r)
Where;
( ) ( )
( )
( )
( )
1
'
1 1 1, , 1 2
0
1
1 0
2
n
j n
j
y
y n y y j


+
=
 
+ = +  
 +  
 (s)
( ) ( )
( )
( )
( ) ( )
( ) ( )
( )
( )
( ) ( )
( )
( )
2
3
1 1 2
'
2 2 2, , 1 3
0
2 5
1
1 0
2
n
j n
j
y
y j y j y j
y n y
y j e y j
  

 
+
=
 
 
− − −
 
 
+ = +  
 
 +  
− +
 
 
 
 (t)
( ) ( )
( )
( )
( ) ( )
( ) ( )
( )
( )
3
'
3 3 3, , 1 3 4 5
0
1
1 0
2
n
j n
j
y
y n y a y j d y j y j


+
=
 
+ = + −
 
 +  
 (u)

7. DEPARTMENTS OF MECHANICAL ENGINEERING, ME
( ) ( )
( )
( )
( ) ( ) ( )
( )
4
'
4 4 4, , 1 4 3 5
0
1
1 0
2
n
j n
j
y
y n y b y j y j y j


+
=
 
+ = + − +
 
 +  
 (v)
( ) ( )
( )
( )
( ) ( ) ( ) ( )
( )
5
'
5 5 5, , 1 5 3 4 5
0
1
1 0
2
n
j n
j
y
y n y c y j y j y j y j


+
=
 
+ = + +
 
 +  
 (w)
( )( )
( ) ( ) ( )
1
1
1 1 1
, , 1
1 0
2 2 1 1
1 1
i j n
n n n j
n j n j n j j n
j n


  


+
+
+ + +
+
 − − + =


= − + + − − − +  

 = +


(z)
( ) ( )
( )
, , 1 1 , 0 1,2,3,4,5.
i j n
h
n j n j j n for i

 


+ = − + − −   = (z1)
In order to solve the fractional model described in (10), an efficient numerical simulation is
presented in the next section.
Numerical simulation
In this section we analyse the chaotic behaviour of the suspension system and the numerical
simulations for fractional order system (e) are conducted under stochastic excitation. For the initial
conditions [0, 0.1, 0. 1, 0.1, 0.1] and fractional order  = 0.98, 0.99, 0.995, 0.997, 0.998,1. The 2D state
portraits and the corresponding time domain response of the system are given in Figure 3-8.
q=0.98

8. DEPARTMENTS OF MECHANICAL ENGINEERING, ME
q=0.99
q=0.995
q=0.997

9. DEPARTMENTS OF MECHANICAL ENGINEERING, ME
q=0.998
q=1
The finite time Lyapunov exponents (LEs) of the system (e) are calculated using the Wolf’s
algorithm with the initial conditions as [0, 0.1, 0.1, 0.1,0. 1] and finite time for calculating the LEs
taken as 20000s. The LEs of the system for fractional order α = 0.998 is calculated as
1 0.5418,
L = − 2 3 4
0.5420, 0.3071, 0.1737
L L L
= − = = − and 5 8.9168
L = − . Since the system has
one positive Lyapunov exponent, it is categorized as chaotic system and
1 2 3 4 5 9.8672 0
L L L L L
+ + + + = −  , this shows that the system is dissipative.
In order to understand the dynamical behaviour of the nonlinear quarter car model, we derive and
investigate the bifurcation plots parameters α and b. Dynamic behaviours can be more clearly
observed over a range of parameter values in bifurcation diagrams, which are widely used to
describe transition from periodic motion to chaotic motion in dynamic systems.

10. DEPARTMENTS OF MECHANICAL ENGINEERING, ME
Then, ( )
0.88,1
  and a time step of 0.001
h = is selected. The chaotic bifurcation diagram is
obtained by continuously changing the fractional order α as bifurcating parameter. In figure t can
be seen that when the fractional order α is in the range of 0.919-1.000 dynamic behavior of the
suspension system will be chaotic vibration. (paraphrase)
The fractional order of the system is kept as 0.998
 = . By fixing all the other parameters, b is
varied and the behaviour of the fractional system (10) is investigated.
To see the impact of parameter b1 on the FOSRC system, we derive the bifurcation plots as shown
in Figure 6. The fractional order of the system is fixed at q = 0.99. The FOSRC system shows
chaotic region for 0.026 ≤ b1 ≤ 0.044 and takes routing period doubling limit cycle route to chaos
and period halving exit from chaos. For both the bifurcations shown in Figures 5 , the initial
conditions for the first iteration are kept at [1, 1, 1, 1] and, for every iteration, the initial conditions
are reinitialized to the end values of the state variables. By comparing Figures 3 and 6, we could
say that the bifurcation range of the parameter b1 is more in fractional order than the integer order
SRC system. This also proves that the fractional or- der dynamics show more complex behaviors
than the integer order counter parts.

11. DEPARTMENTS OF MECHANICAL ENGINEERING, ME
(a) Bifurcation of fractional order for α = 0.992
Figure 4 the bifurcation diagram of the system for changes of parameter 2
 .

12. DEPARTMENTS OF MECHANICAL ENGINEERING, ME
Figure 5 the spectrum of the Lyapunov exponents for the system for α = 0.992
When “ 2
 ” between 6.64 and 6.72, for “ 2
 ” between 22.48 and 22.63, for “ 2
 ” between 22.71
and 23.05, for “ 2
 ” between 23.20 and 23.30, for “ 2
 ” between 23.51 and 23.69, and for “ 2
 ”
23.93 and 24.19 it creates chaos. When “ 2
 ” between 6.72 and 8.16 it confirms there is a
discontinuity which is not clearly seen in case of integer order nonlinear quarter car model. Finally,
for “ 2
 ”>30.07 it goes to unbounded. The bifurcation diagram is shown in Figure 4. According
to the spectrum of the Lyapunov exponents of Figure 5, it confirms that the bifurcation diagram
described in Figure 3 of system is unbounded for “ 2
 ”∈ (30.07, ∞].