INVESTIGATION OF STABILITY ON NONLINEAR OSCILATIONS WITH ONE DEGREE OF FREEDOM

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1 91 Kragujeva J. Math. 25 (2003) INVESTIGATION OF STABILITY ON NONLINEAR OSCILATIONS WITH ONE DEGREE OF FREEDOM Julka Knežević-Miljanović 1 and Ljubia Lalović 2 1 Mathematis faulty of Belgrade, Serbia and Montenegro 2 Faulty of Mehanial Engineering of Kraljevo, Serbia and Montenegro (Reeived May 15, 2003) Abstrat. This paper presents a phase portrait of differential equation of vehile vertial osillations when establishing fore in springs is hanged under nonlinear laws: F = (x + αx 2 ) and F = (x + αx 3 ), > 0, α 0. Analysis of nonlinear osillation stability has been done using the phase portrait analyses. The differential equation of vehile vertial osillations with one degree of freedom when on the vehile suspension mass m is ated the F establishing fore spring that is hanged under the law: F = (x + αx 2 ), > 0, α 0, where x is an generated oordinate of the following form: mx + (x + αx 2 ) = 0, (1)

Mehanial Engineering of Kraljevo, Serbia and Montenegro (Reeived May 15, 2003) Abstrat.

2 92 where m > 0, > 0, α 0 are real parameters. The differential equation (1) is equal to the following system of equations dx dt = y dy dt = m (x + αx2 ). (2) The differential equation of the system phase trajetory (2) is: ydy = m (x + αx2 ) dx. The phase trajetories are integral urves of the differential equation. After integration within the limits from t 0 to t with starting onditions x(t 0 ) = x 0, y(t 0 ) = y 0 we reah the following ([3]): Π(x) = 6m (3x2 + 2αx 3 ) and h = 1 2 y 0 + 6m (3x αx 3 0). The phase trajetories are the following urves: y = ± 2(h Π(x)) or y = ± 2 ( h ) 6m (3x2 + 2αx 3 ). (3) Draw the graphi of funtions (3) and test the stability of vehile motion in the ase of rigid springs when α > 0 (if α = 0.1), and in the ase of soft springs when α < 0 (if α = 0.1). enter. Theorem 1. For the equation (1) the following is taken: (I) For α = 0.1, position of the systems equilibrium are points (0, 0) and ( 10, 0). For x = 0 the funtion Π(x) has its minimum, it means that the point (0, 0) is a For x = 10 the funtion Π(x) has its maximum, it means that the point ( 10, 0) is a saddle (fig. 1) enter. (II) For α = 0.1 the equilibrium positions are the points (0, 0) and (10, 0). For x = 0 the funtion Π(x) has its minimum, it means that the point (0, 0) is a For x = 10 the funtion Π(x) has its maximum, so the point (10, 0) is a saddle (fig. 2).

After integration within the limits from t 0 to t with starting onditions x(t 0 ) = x 0, y(t 0 ) = y 0 we reah the following ([3]): Π(x) = 6m (3x2 + 2αx 3 ) and h = 1 2 y 0 + 6m (3x2 0 + 2αx 3 0).

3 93 fig. 1 fig. 2 a) Change of pot. energy of rigid springs a) Change of pot. energy of soft springs b) Phase trajetories b) Phase trajetories For α > 0 and α < 0 there are two harateristi points, enter O and saddle A. The system motion is stable at enter O environment only, i.e. for the starting value of potential energy Π(x 0 ) = h 0 = 0. For h > h 0 the phase point moves along the phase trajetories that are moved away from the equilibrium positions being suitable to unstable osillatory vehile motion. 2) The differential equation of vertial vehile osillations with one degree of freedom, when F spring establishing fore affets the suspension mass m of the vehile that hanges under the law: F = (x + αx 3 ), > 0, α 0 it reahes the following mx + (x + αx 3 ) = 0. (4) Define the phase trajetories if α > 0 (rigid springs) and if α < 0 (soft springs).

e. for the starting value of potential energy Π(x 0 ) = h 0 = 0.

4 94 The potential energy: Π(x) = The phase trajetories are urves y = ± 2 ( h 4m (2x2 + αx 4 ). ) 4m (2x2 + 2αx 4 ). If α > 0 (α = 0.1) the equilibrium position O(0, 0) is a enter. The phase trajetories are ellipses. Starting value of potential energy is Π(x 0 ) h(x 0 ) : Π(x 0 ) = h 0 = 0 (fig. 3). fig. 3 a) Change of potential energy of rigid springs b) Phase trajetories The theorem is aeptable as follows: Theorem 2. For α > 0 the differential equation (4) has a stable periodial solution. The vehile osillatory motion at arbitrary starting onditions is stable. (II) Now look at the vehile osillatory motion for soft springs (α = 0.1). The points (0, 0), ( 10, 0), ( 10, 0) are equilibrium positions of the system. The urve Π(x) has an extreme values for x = 0, x = 10 and x = 10. For x = 0 it has a minimum, for x = 10 and x = 10 it has a maximum h 0.

3 a) Change of potential energy of rigid springs b) Phase trajetories The theorem is aeptable as follows: Theorem 2. For α > 0 the differential equation (4) has a stable periodial solution.

5 95 The point O(0, 0) is enter, and points A 1 ( 10, 0), A 2 ( 10, 0) are saddles (fig. 4). fig. 4 a) Change of potential energy of soft springs b) Phase trajetories The point O is suitable for osillatory vehile motion. Points A 1 and A 2 are suitable for four half-trajetories. For values h being lose to h 0 values, integral urves are similar to hyperbola form. Around the points A 1 and A 2 the phase plane is divided into four parts. Look at the point A 1. If h > h 0 the phase trajetories are at upper I range and lower IV range. In the first ase we have a stable periodial solution that is suitable to free harmoni vehile osillations. If the phase point is lose to the equilibrium position and if it is on II and IV phase trajetories, it will asymptotially beome loser to the equilibrium position. If the phase point is on the I and III phase trajetories branhes and on the phase trajetories of hyperbole type left to the point A 1 it will be moved away from the equilibration position. Unstable equilibration position of the system is suitable for the point A 1. If the starting potential energy has value of h 1, that rosses the urve Π(x) nowhere nor onerns it, then the phase trajetories onsist of two symmetri urves relating to the x axis that are moved away from both sides to the

For values h being lose to h 0 values, integral urves are similar to hyperbola form. Around the points A 1 and A 2 the phase plane is divided into four parts. Look at the point A 1.

6 96 infinity. Due to the symmetry, the same ould be onluded for the point A 2. Referenes [1] V. I. Arnol d, Ordinary differential equation, Mosow, Siene [2] N. Ćućuz, L.Rusov, Dynamis of motor ars, Privredni Pregled, Belgrade, [3] R. Šćepanović, J. Knežević, Miljanović, Lj. Protić, Differential equation, Mathematis faulty, Belgrade, 1997.

Ćućuz, L.Rusov, Dynamis of motor ars, Privredni Pregled, Belgrade, 1973. [3] R.

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