1st International Conference
Computational Mechanics and Virtual Engineering 
COMEC 2005
20 – 22 October 2005, Brasov, Romania
ON THE PERIODICAL MOIONS AROUND THE EQUILIBRIUM
POSITION FOR AN AUTOMOBILE WITH NONLINEAR SUSPENSION
Nicolae–Doru Stănescu1
1
University of Piteşti, Piteşti, ROMANIA, e-mail s_doru@yahoo.com
Abstract: This paper is an extension of a previous work. Basing on a generalized Lindstedt–Poincaré method in two
dimensions case purposed by the author, we construct an approximate solution of the small perturbation around the
equilibrium position. Comparative to the linear case, in our paper an infinite series of harmonic terms will appear in the
general solution.
Keywords: nonlinear suspension, harmonic terms, Lindstedt–Poincaré
1. INTRODUCTION
In reference [20] the moving equation for the considered automobile were obtained as (notations and description
in [20])
2
m1z1  k 2 z 2  z1    2 z 2  z1 3  3 2 z 2  z1 2 z 20  3 2 z 2  z1 z 20
 k1 z1 ;
2
.
m2 z2  k 2 z 2  z1    2 z 2  z1 3  3 2 z 2  z1 2 z 20  3 2 z2  z1 z 20
For such a system we obtained only one equilibrium position, name it:
z1  0 ; z 2  0 .
(1)
(2)
2. PRESENTATION OF THE METHOD. OBTAINING THE CANONICAL FORM FOR A
TWO SECOND ORDER DIFFERENTIAL EQUATIONS SYSTEM
We shall suppose that we have to solve the two nonlinear second order differential equations system
x  f1 x, y   0 ; y  f 2 x, y   0 .
(3)
In addition, the functions f1 and f 2 are analytical and the following developments can be written
f 1 x, y  


 f1
f
 2 f1
 2 f1
1   2 f1
2
2
x 1
y 
x

2
xy

y
 ,
x 0,0 
y 0,0 
2!  x 2
xy
y 2 0,0  




0
,
0
0
,
0




 f2
f
 2 f2
2 f2
1  2 f2
2
2
x 2
y 
x

2
xy

y
   ,
2
x 0,0 
y 0,0 
2!  x 2
xy

y
0,0 
0,0 
0,0  

where we have assumed that the critical point is given by
x, y   0,0
The system (3) can be written as
f
f
f
f
x  1
x 1
y  f1 x, y   0 ; y  2
x 2
y  f 2 x, y   0 ,
x 0,0 
y 0,0 
x 0,0 
y 0,0 
the notations are being obvious.
Let us now consider the transformation
x  1   2 ; y  1   2 .
Results:
f 2 x, y  
1
(4)
(5)
(6)
(7)
(8)
1  2 
 f1
f
 1   2   1  1   2   f1* 1 ,  2   0 ;
x 0,0 
y 0,0 
f
1   2    f 2 1   2   f 2* 1 ,  2   0 ,
1  2  2
x 0,0 
y 0,0 
with:
f1* 1 , 2   f1 1  2, 1   2  ; f 2* 1 ,  2   f 2 1  2, 1   2  .
Multiplying the first relation (9) by  , the second relation (9) by  and adding the results, we find

    1  1     f1

x 0,0 
 
(9)
(10)

 f1
f
f

  2
  2
y 0,0 
x 0,0 
y 0,0  



f
f
f
f

  2    1
 2 1
2 2
  2

x 0,0 
y 0,0 
x 0,0 
y 0,0  


(11)
  f1* 1 ,  2    f 2* 1 ,  2   0 .
Multiplying now the first relation (9) by  , the second relation (9) by  and adding the results, we deduce
 


f
f
f
f

  2  1    1
 2 1
2 2
  2

x 0,0 
y 0,0 
x 0,0 
y 0,0  




f
f
f
f

  2    1
  1
  2
  2

x 0,0 
y 0,0 
x 0,0 
y 0,0  


(12)
  f1* 1 ,  2    f 2* 1 ,  2   0 .
We wish that in the expression (11) the term in  2 vanishes, and in the expression (12) the term in  1 vanishes,
too. Results:
f
f
f
f
  1
 2 1
2 2
  2
 0;
x 0,0 
y 0,0 
x 0,0 
y 0,0 
(13)
 f1
 f2
2  f1
2  f2
 


 
 0.
x 0,0 
y 0,0 
x 0,0 
y 0,0 
Let us observe that the two equations (13) are one and the same using the substitutions    and    .
The general problem is: what conditions must fulfill the equation
(14)
Auv  Bu 2  Cv 2  0
to have real solutions for u and v . We can see the equation (14) as a second-degree equation in the unknown u ,
and its discriminate is
(15)
  A2  4BC v 2 .
The equation has real solutions if and only if
(16)
A 2  4BC  0 .
Recalling the equations (13) for which:
f
f
f
f
; B 1
; C 2
,
A  1
 2
(17)
x 0,0 
x 0,0 
y 0,0 
y 0,0 


the condition (16) reads
2
2
 f



 f1
f
 f2
 2
    f1
  2  f2
4 1
 0.
 y

 x 0,0  
y 0,0  x 0,0 
y 0,0  x 0,0 
0,0  



The condition (18) is the same with the condition that the secular equation
 f1

x 0,0 
 f2
x 0,0 
 f1
y 0,0 
0
 f2

y 0,0 
(18)
(19)
2
has real roots.
Assuming now that the conditions (18) is fulfilled, the system (7) can be written
A   B   f  ,    0 ; A   B   f  ,    0
1 1
1 1
1
1
2
2 2
2 2
2
1
2
(20)
or, equivalently,
(21)
1  a111  f1 1 ,  2   0 ; 2  a 22 2  f 2 1 ,  2   0 .
We shall name the system (21) the canonical form for a system of two nonlinear second order differential
equations for applying a Lindstedt–Poincaré type method.
3. PERIODICAL SOLUTION
Let us consider the two nonlinear second order differential equations system, brought in the canonical form
u  a11u  f u, v   0 ; v  a 22v  gu, v   0 ,
(22)
with a11  0 , a22  0 .
We shall assume that the functions f and g are analytical in a neighborhood of the critical point
u, v   0,0
so we can write the following developments
1u 2  b 1uv  b 1v 2  b 1u 3  b 1u 2 v  b 1uv 2  b 1v 3   ;
f u, v   b20
11
02
30
21
12
03
2 u 2  b 2 uv  b 2 v 2  b 2 u 3  b 2 u 2 v  b 2 uv 2  b 2 v 3   .
gu, v   b20
11
02
30
21
12
03
(23)
(24)
We shall note:
1  a11 ;  2  a 22 ;   1 2
and we shall assume that the solution of the system (22) can be written as
ut   u1 t    2u2 t    3u3 t    ; vt   v1 t    2 v 2 t    3 v3 t   
and, in addition:
1   01  11   2 21  ;  2   02  12   2 22  ,
where  is a small dimensionless parameter. We shall introduce the variable
  t
and results:
2
2
d 2u
d 2v
2 d u
2 d v




;
.
dt 2
d 2 dt 2
d 2
Replacing the precedent relations in (22), we find:
2
 01   11   02   12   d 2 u1   2 u 2   
d

(25)
(26)
(27)
(28)
(29)


 

 b   u   u  v   v    b   v   v   
.
 b   u   u    b   u   u   v   v   
 b   u   u  v   v    b   v   v      0 ,
         d v   v   
d
      v   v    b   u   u   
 b   u   u  v   v    b   v   v   
.
 b   u   u    b   u   u   v   v   
 b   u   u  v   v    b   v   v      0 .
1 u   2 u   
  01   11  2 u1   2 u 2    b20
1
2
2
1
11
2
1
1
30
1
12
1
3
2
1
2
1
21
2
2
1
1
02
2
2
2
2
1
2
2
2
1
1
03
2
11
02
2
01
2
2
11
2
30
2
12
2
1
3
2
2
1
2
20
1
2
1
2
2
2
21
2
1
2
2
2
2
1
2
2
2
1
(30b)
2
2
2
1
2
2
1
2
02
2
2
2
2
1
3
2
12
2
11
2
1
2
01
(30a)
2
2
2
1
2
2
1
2
2
1
2
03
2
2
1
2
3
2
Equating with 0 the coefficients for  ,  ,  3 , … from the system (30), one obtains:
2
2
 01
 02
d 2 u1
d
2
2
2
2
  01
u1  0 ;  01
 02
d 2 v1
d
2
2
  02
v1  0 ,
3
(31)
2
2
 01
 02
2
2
 01
 02
d 2u2
d 2
d 2 v2
d 2

2
2
 2  01 11 02
  02 12 01
d u
2
1
d 2
2
  01
u 2  2 01 11u1 
1u 2  b 1u v  b 1 v 2  0 ;
 b20
1
11 1 1
02 1


2
2  0111 02
2
  0212 01
 d
(32)
d 2 v1
2
  02
v2
2
 2 0212 v1 
2 u 2  b 2 u v  b 2 v 2  0 .
 b20
1
11 1 1
02 1
The first relation (31) writes
d u1
1
 2 u1  0
2
d
 02
and it has the solution
 

u1  A1 cos
 1  ,

 02

where A1 and 1 are two constants of integration and they are deduced from the initial conditions.
Analogously, the second equation (31) has the solution
 

v1  A2 cos
  2  ,

 01

with the same observations for A2 and  2 .
The system (32) becomes
2
 

2
2 d u2
2
2  A1
 01
 02
 2  0111 02
  0212 01
cos
 1  
2
2

d
 02
 02

2

(33)
(34)
(35)

  

1  cos2
 1 
 

   02

1 A 2
2
  01
u 2  2 0111 A1 cos
 1   b20

1
2
  02

  

1  cos2
  2 

   01
1 A A cos     cos      b 1 A 2
 b11
 0;
1 2
1
2
2
02

 

2
 02
  01

2
 

2
2 d v2
2
2  A2
 01
 02
 2  0111 02
  0212 01
cos
  2  
2
2
d
 01
  01



(36)
  

1  cos2
 1 
 


   02
2  A 2
2
  02
v 2  2 0212 A2 cos
  2   b20

1

2
 01

  

1  cos2
  2 
   01

2  A A cos     cos      b 2  A 2
 b11
 0.
1 2
1
2
02 2


2
 02
  01

To avoid the secular terms in the solution of he system (36), the following relations must subsist:
2
2  A1
2
2  A2
2  0111 02
  0212 01
 2 0111 A1  0 ; 2  0111 02
  0212 01
 2 0212 A2  0 .
2
2
 02
 01
Results
11  12  0 .
The system (36) has now the solution
  
  


u 2  a1 cos2
 1   a 2 cos2
 1  


   02
   02



 

 



 a3 cos

 1   2   a 4 cos

 1   2   C1 ,
  02  01

  02  01

respectively
4

(37)
(38)
(39a)
  
  


v 2  b1 cos2
 1   b2 cos2
 1  


   02
   02
 

 



 b 3 cos

 1   2   b4 cos

 1   2   C 2 ,
  02  01

  02  01

(39b)
where a i , bi , C j , i  1, 4 , j  1, 2 are constant values.
Analogously, we can discuss the next systems obtained for the coefficients of  3 ,  4 , …
4. THE AUTOMOBILE CASE
Denoting:
2
f 1  k 2 z 2  z1    2 z 2  z1 3  3 2 z 2  z1 2 z 20  3 2 z 2  z1 z 20
 k1 z1 ;
(40)
2
,
f 2  k 2 z 2  z1    2 z 2  z1 3  3 2 z 2  z1 2 z 20  3 2 z 2  z1 z 20
we can easy observe that the condition (19) is fulfilled. Therefore the system (1) can be brought to the canonical
form and the modified Lindstedt–Poincaré method in two dimensions case can be applied.
The system (1) has the general solution given by the similar expressions to formulas (26), (27), (34), (35), (39).
5. CONCLUSION
In our paper we presented a method to determine the periodical solutions around the equilibrium positions for a
two-second order differential equations.
The condition (19) is an essential one. In the case of the quarter of an automobile modeled by system (1), this
condition is always fulfilled.
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