1st International Conference
Computational Mechanics and Virtual Engineering 
COMEC 2005
20 – 22 October 2005, Brasov, Romania
ON THE STABILITY OF THE EQUILIBRIUM POSITIONS FOR AN
AUTOMOBILE WITH NONLINEAR SUSPENSION
Nicolae–Doru Stănescu1, Ionel Vieru2
1
2
University of Piteşti, Piteşti, ROMANIA, e-mail [email protected]
University of Piteşti, Piteşti, ROMANIA, e-mail [email protected]
Abstract: This paper works with the quarter model of an automobile with nonlinear suspension. The nonlinearity is of third
order at it appears at the stiffness of the spring. We establish the equilibrium positions and we prove that in normal
conditions there exists only equilibrium. Finally we discuss the stability of the equilibrium and we prove that it is simply
stable.
Keywords: nonlinear, suspension, stability
1. INTRODUCTION
We shall consider the automobile modeled by a two degrees of freedom model as it can be seen in figure 1.
The second stiffness is nonlinear, and the elastic force that appears in such an element is
Fe2  k 2 z   2 z 3 ,
(1)
where z is the elongation of the element, and  2 is strictly positive.
At equilibrium, one obtains the equations:
3
3
(2)
; m1 g  k1 z10  k 2 z 20   2 z 20
.
m2 g  k2 z20   2 z20
Representing the forces for the moving status and applying the Newton’s second law, we obtain the equations:
m1z1  k 2 z 2  z1  z 20    2 z 2  z1  z 20 3  k1 z1  z10   m1 g ;
(3)
m2 z2  k 2 z2  z1  z20    2 z2  z1  z20 3  m2 g .
m2 g
z2
k2 ,  2
m2 g
m2 g
k 2  z 2  z1  z 20  
3
k 2 z 20   2 z 20
m1 g
k1
z1
z2
  2  z 2  z1  z 20 3
m1 g z1
m1 g
k1  z1  z10 
k1 z10
Figure 1. The model of a quarter automobile with non–linear suspension
1
Equations (3) in which replace the relations (2) lead us to:
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 z2  z1    2 z2  z1 3  3 2 z2  z1 2 z20  3 2 z2  z1 z20
We denote:
z1  1 ; z 2   2 ; z1   3 ; z 2   4
and we obtain the following system o four first order nonlinear differential equations
d 4
d1
d 2
2
 k 2  2   1    2  2   1 3  3 2  2   1 2 z 20  3 2  2   1 z 20
;
 3 ;
  4 ; m2
dt
dt
dt
d
2
m1 3  k 2  2   1    2  2   1 3  3 2  2   1 2 z 20  3 2  2   1 z 20
 k1 1 .
dt
(4)
(5)
(6)
2. ESTABLISHING THE EQUILIBRIUM POSITIONS
The equilibrium positions are at the intersections of he nullclines. From the equations (6) one obtains:
3  0 ;
4  0 ;
2
k 2  2  1    2  2  1 3  3 2  2  1 2 z 20  3 2  2  1 z 20
 k11  0 ;
2
 k 2  2  1    2  2  1 3  3 2  2  1 2 z 20  3 2  2  1 z 20
 0.
Summing the last two relations (7) results 1  0 and replacing in the third equation (7) we deduce


2
 2  2 22  3 2 2 z 20  3 2 z 20
 k2  0 .
(7)
(8)
We immediately deduce a first solution, name it  2  0 .
The second order equation in (8) has the discriminate
2
  3 22 z 20
 4 2 k 2 .
(9)
The condition that  2 is a positive parameter is an essential one. This condition implies that the discriminate (9)
is a negative one, so the second order equation has no real roots.
 4 k2

If we don’t put the condition that  2 is a positive parameter, it is easy to deduce that for  2  
, 0  the
 3 z 20 
second order equation adds new solutions for (9). It is no the scope of this paper to discuss this situation.
In conclusion, the only equilibrium position is given by
1  0 ;  2  0 ;  3  0 ;  4  0 .
(10)
3. STABILITY OF THE EQUILIBRIUM POSITION
The system (6) can be written
d1
d 2
 3 ;
 4 ;
dt
dt
2
2
d 3
k  k1  3 2 z 20
k  3 2 z 20
 2
1  2
 2  TNL1 ;
dt
m1
m1
2
2
k  3 2 z 20
d 4 k 2  3 2 z 20

1  2
 2  TNL 2 ,
dt
m2
m2
where TNL stay for nonlinear terms.
Denoting:
2
2
2
2
k  k1  3 2 z 20
k  3 2 z 20
k  3 2 z 20
k  3 2 z 20
; a 32  2
; a 41  2
; a 42   2
,
a31   2
m1
m1
m2
m2
the characteristic equation reads
 0
1
0
0  0
1
0
a 31 a 32   0
a 41 a 42 0  
or, equivalently
4  2 a 42  a31   a31a42  a 41a32  0 ,
(11)
(12)
(13)
(14)
2
where  stays for the complex rate of growth.
Let’s denote:
A  a 42  a31  ; B  a31a 42  a 41a32 .
(15)
The equation (14) is a two-squared one. The second order equation corresponding to it has the discriminate
  A 2  4B  a 42  a31 2  4a 41a32 .
Recalling the expressions (12) we observe that
a 31  0 ; a 32  0 ; a 41  0 ; a 42  0 ,
so the discriminate is strictly nonnegative.
In addition we have
(16)
(17)
(18)
A  A 2  4B ;
therefore the roots of the two-squared equation attached to the characteristic equation are both real and negative.
We deduced that the characteristic equation has four pure imaginary roots; therefore the equilibrium is a simply
stable one.
The character of simply stable of the equilibrium doesn’t depend of the characteristics of he car: masses,
distribution of the masses, stiffness etc.
3. CONCLUSION
In our paper we proved that for an automobile with such a nonlinear suspension there exists only one equilibrium
position. In addition, we proved that the equilibrium is simply stable.
We recall that the essential condition for our deduction is that the parameter  2 is strictly positive. If this
assumption is not fulfilled the system can have more than one equilibrium position and the study becomes more
complicated.
The condition  2  0 is also a sufficient condition for the system has one and only one static equilibrium
position. Indeed, the first relation (2) can be written as
k
m g
3
z 20
 2 z 20  2  0 .
(19)
2
2
From the Hudde method we know that the equation (19) has one and only one real root if and only if its
discriminate is strictly positive. The discriminate reads

k 
m g
1  k3
  4 2   27  2   2  4 2  27 m 2 g  .

 2   2
2 
 2 

It is now obviously that  2  0 implies   0 , and the system has only one equilibrium position.
In our next work we shall study the case of small oscillations around the equilibrium.
3
2
(20)
REFERENCES
[1] Pandrea N., Pârlac S., Popa, D.: Modele pentru studiul vibraţiilor automobilelor, Editura Tiparg, Piteşti,
2001.
[2] Pandrea N., Stănescu, N.–D.: Mecanica, Editura Didactică şi Pedagogică, Bucureşti, 2002.
[3] Teodorescu P. P.: Sisteme mecanice. Modele clasice, 4 vol., Editura Tehnică, Bucureşti, 1984, 1988, 1997,
2002.
[4] Lurie A. I.: Analytical Mechanics, Springer–Verlag, Berlin, Heidelberg, New York, Barcelona, Hong Kong,
London, Milan, Paris, Tokyo, 2002.
[5] Nayfeh A. H., Mook D. T.: Nonlinear Oscillation, John Wiley & Sons, New York, 1979.
[6] Minorsky N.: Nonlinear Oscillations, Van Nostrand Reinhold, New York, 1962
[7] Bellman R.: Stability Theory of Differential Equations, Dover, New York, 1969.
[8] Cunningham W. J.: Introduction to Nonlinear Analysis, McGraw–Hill, New York, 1958.
[9] Guckenheimer J., Holmes P.: Nonlinear Oscillations, Dynamical Systems, and Bifurcation of Vector Fields,
Springer–Verlag, New York, Berlin, Heidelberg, London, Paris, Tokyo, Hong Kong, 1983.
[10] Arrowsmith D. K., Place C. M.: An Introduction to Dynamical Systems, Cambridge University Press, 1994.
[11] Carleson L., Gamelin T. W.: Complex Dynamics, Springer–Verlag, New York, Heidelberg, London, Paris,
1993.
[12] Hale J. K., Koçak H.: Dynamics and Bifurcations, Springer–Verlag, New York, Berlin, Heidelberg,
London, Paris, Tokyo, Hong Kong, Barcelona, Budapest, 1991.
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[13] Kuznetsov Y. A.: Elements of Applied Bifurcation Theory, Springer–Verlag, New York, Berlin,
Heidelberg, London Paris, Tokyo, Hong Kong, Barcelona, Budapest, 1995.
[14] Perko L.: Differential Equations and Dynamical Systems, Springer–Verlag, New York, Berlin, Heidelberg,
1996.
[15] Sibirsky K. S.: Introduction to topological dynamics, Noordhoff International Publishing, Leyden, The
Netherlands, 1975.
[16] Stănescu N.–D., Pandrea M., Vieru I., Pandrea N.: Asupra vibraţiilor de săltare ale automobilelor cu
caracteristică elastică neliniară, a IV-a Conferinţă de Dinamica Maşinilor, Braşov, 2005.
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