Dynamic buckling of a single-degree-of-freedom system with variable mass

Eur. J. Mech. A/Solids 20 (2001) 661–672  2001 Éditions scientifiques et médicales Elsevier SAS. All rights reserved S0997-7538(01)01160-3/FLA

Dynamic buckling of a single-degree-of-freedom system with variable mass Livija Cveticanin ∗ Faculty of Engineering, Trg D. Obradovica 6, 21000 Novi Sad, Yugoslavia (Received 3 February 2000; revised and accepted 30 April 2001) Abstract – In this paper dynamic buckling of the single-degree-of-freedom system with variable mass is analyzed. In the system the mass variation is slow and is a function of slow variable time. Due to mass variation the impact force acts. The motion of the system is described with a nonlinear ordinary differential equation with time variable parameters. A new approximate analytic criterion of dynamic buckling for the non-autonomous systems which have the conservation law of energy type is developed. The conservation law is formed applying the Noetherian approach. The suggested method allows the determination of dynamic buckling load without solving the corresponding nonlinear differential equation of motion. For this value of dynamic load the motion of the system becomes unbounded. The obtained analytic value is compared with the numeric one. It shows a good agreement.  2001 Éditions scientifiques et médicales Elsevier SAS variable mass / dynamic buckling / conservation law

1. Introduction The problem of dynamic buckling has been widely analyzed in the last twenty years. The term dynamic buckling refers to some critical load condition for which large amplitude or divergent oscillations escaping from the associated static position are detected. This is a form of instability that is always associated with some motion. The buckling load is the value for which unstable motion appears. Various analytical and numerical methods for denoting buckling load and algorithms that may be derived, which form criteria for a practical search of a dynamic buckling state are developed. For all of the analytical methods it is common that these criteria allow the determination of exact dynamic buckling load without solving the corresponding nonlinear differential equations of motion. The most widely applied procedure for determining buckling load in conservative systems is the total potential energy approach (Raftoyiannis and Kounadis, 1988). An unbounded motion is possible only if the load λ is such that the system is forced to pass through an unstable equilibrium position which causes the kinetic energy to vanish and thus the total potential energy is zero. The critical state is characterized by a load amplitude λD and a displacement xD as in the static case, but there is also the length of time at which buckling occurs and the velocity and acceleration of the system at the critical state. The phasespace approach is applicable for conservative, autonomous and holonomic systems. This method gives the critical load for the case when the velocity x˙ and the acceleration x¨ vanish. An
alternative but equivalent ˙ 2 + (x) ¨ 2 . For the nonformulation may be written in terms of the phase velocity vk defined as vk = (x) conservative discrete dissipative systems the approximate dynamic buckling load is obtained using the energy balance equation (Kounadis, 1997; Raftoyiannis and Kounadis, 2000). The total potential energy criteria for dynamic buckling of conservative systems is no longer valid. Applying the energy balance equation, the lower ∗ Correspondence and reprints.

E-mail address: [email protected] (L. Cveticanin).

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and upper bound buckling are estimated. Using the qualitative and quantitative analysis the neighborhood of a compound branching point being the boundary between static and dynamic instability is also thoroughly discussed (Kounadis, 1994-a). Under suddenly applied load of infinite duration it is found that dynamic buckling occurring always through a saddle, leads to an escaped motion which is finally attracted by remote stable equilibrium positions (Sophianopoulos, 1996). The inflection point criteria (Kounadis, 1994-b) is also used to obtain dynamic buckling for the simple one-degree-of-freedom dissipative model. It is confirmed via the Liapunov direct method for global stability (Kounadis, 1996). Using the aforementioned method the dynamic buckling load of a cutting mechanism is obtained (Cveticanin and Maretic, 2000). Dynamic instability of simple one-degree-of-freedom systems is also studied under two loading parameters: λ1 associated to a static load and λ2 which represents a dynamic load. The equations necessary to evolute critical load under different forms of dynamic load are presented (Brewer and Godoy, 1991, 1993). The approximate dynamic buckling loads of discrete systems are obtained via the geometric considerations of their energy surface (Gantes et al., 1998). In this paper the boundary of dynamic buckling for one degree of freedom system with variable mass is determined. The variation of the mass of the system represents the adding or separating of the mass particles during the time. Due to mass variation in time an impact force acts. The system is non-autonomous and nonconservative. Only those dynamical systems are considered which are referred to as potential or Lagrangian, regardless of the fact that the Lagrangian function depends on the slow time. For these non-conservative systems the first integral of energy type is denoted. To find the first integral is not an easy task. In this paper the conservation law is obtained by applying Noether’s theory. Using the conservation law of energy type a new criterion for the computation of dynamic buckling load is developed. It is based on the energy approach which was developed for the conservative systems. In this paper it is adopted for the non-conservative systems with Lagrangian. The suggested procedure is applied for the real mechanical systems: a sieve and a rotor of a textile machine. Both machines represent one-degree-of-freedom systems with variable mass and impact force. The mass is increasing or decreasing in time. The mass function is linear and a quadratic function of slow time, respectively. The analytic results for dynamic buckling are compared with numerical ones.

2. Mathematical model The differential equation of a one-degree-of-freedom system with slow time variable mass is (see Cveticanin, 1998): 

d ∂EK dt ∂ x˙



−

∂EK ∂V (x, τ, λ) + = D + , ∂x ∂x

(1)

where x is the generalized displacement, x˙ = dx/dt is the generalized velocity, V is the potential energy of the system, EK is the kinetic energy of the system,  is the impact (reactive) force (Meshcherski, 1952), λ is the linear parameter of the step force and τ = εt is the slow time parameter, ε  1 is a small parameter, t is time, and D is: 

D=



dm ∂ ∂EK . dt ∂m ∂ x˙

(2)

m(τ )x˙ 2 , 2

(3)

Let us assume the kinetic energy to be: EK =

Dynamic buckling of a single-degree-of-freedom system

663

and the impact force as a linear function of velocity and mass variation (see Cveticanin, 1993) is:  = (p − 1)

dm(τ ) x, ˙ dt

(4)

where p is a constant coefficient. The differential equation of motion is: mx¨ +

∂V (x, λ, τ ) = (p − 1)m ˙ x, ˙ ∂x

(5)

where m(τ ) ≡ m and m ˙ = dm/dt. It is a non-liner differential equation which corresponds to a non-conservative system for which the Lagrangian exists. The Lagrangian of the system (5) is: L = m−p





mx˙ 2 − V (x, λ, τ ) . 2

(6)

The equation (5) describes the vibrations of the machines with variable mass like sieves and centrifuges as well as rotors of the machines in the cable, carpet and textile industry on which the band is winding up and down. 3. Conservation law In this paper using the Noetherian approach (Vujanovic and Jones, 1989) the conservation law of the system (5) with Lagrangian (6) is obtained. The conservation law has the form: 



˙ − m1−p m1−p xF

x˙ 2 + m−p V f − P = const. 2

(7)

The functions: f ≡ f (t, x),

F ≡ F (t, x),

P ≡ P (t, x),

(8)

are arbitrary ones, which satisfy the Noether identity: 



x˙ 2 ∂V ∂V + pm−p−1 mV F (1 − p)m m ˙ ˙ − m−p f − m−p 2 ∂t ∂x   x˙ 2 + m1−p x˙ F˙ − f˙ m1−p + m−p V − P˙ = 0, 2 −p

(9)

where (·) = d/dt and V ≡ V (x, λ, τ ). To obtain the unknown functions (8) which satisfy the condition (9) we apply Killing’s method (Cveticanin, 1994). It is based on separation of the partial differential equations for the same order of the function x˙ from the equation (9). The following Killing’s partial differentialequations are obtained: 1 ∂f = 0, (10) x˙ 3 : − m1−p 2 ∂x x˙ 2 :

1 1 − p −p ∂F ∂f m mf − m1−p = 0, ˙ + m1−p 2 ∂x 2 ∂t

(11)

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L. Cveticanin x: ˙

∂P ∂f ∂F 1−p m − = 0, − m−p V ∂t ∂x ∂x



x˙ 0 :

pm−p−1 mV ˙ − m−p

(12)



∂f −p ∂P ∂V ∂V F− m V− = 0. f − m−p ∂t ∂x ∂t ∂t

(13)

x2 P1 (t) + xP2 (t) + P3 (t), 2

(14)

For F = F1 (t)x + F2 (t),

f = f (t),

P=

and the polynomial potential energy: V = Ax + Bx 2 + Cx 3 + Dx 4 ,

(15)

where A ≡ λA(τ ), B ≡ B(τ ), C ≡ C(τ ), D ≡ D(τ ), the transformed Killing’s equations (11)–(13) are

x2: x:

(16)

F˙1 xm1−p + m1−p F˙2 = xP1 + P2 ,

(17)

fpm−p−1 mD ˙ − m−p f D˙ − 4m−p F1 D − f˙m−p D = 0,

(18)

Cfpm−p−1m ˙ − m−p f C˙ − 3Cm−p F1 − 4m−p F2 D − f˙m−p C = 0,

(19)

1 fpm−p−1 mB ˙ − m−p f B˙ − 2Bm−p F1 − 3Cm−p F2 − f˙m−p B = P˙1 , 2

(20)

˙ − m−p f A˙ − m−p F1 A − 2Bm−p F2 − f˙m−p A = P˙2 , Afpm−p−1 m

(21)

−m−p F2 A = P˙3 .

(22)

x4: x3:

1 1 − p −p m mf ˙ + m1−p F1 − m1−p f˙ = 0, 2 2

x0:

The task is to integrate the equations (16)–(20) and to determine the unknown functions (4). Let us consider some special cases. (a) Let us analyze the system of equations (16)–(22) for the case when: A = A(τ ),

B = const.,

C = 0,

D = const.

(23)

It is F2 = P2 = P3 = 0, 

(24)

 2−p

˙ 2p − 1 m m 3 f, , F1 = f= m0 6 m   ˙2 2p − 1 p+1m f m−p m ¨− , P1 = 6 3 m

(25)

for the following mass variation: 



˙ 2p − 1 m ˙ 3 (p + 1)(1 + 4p) m ˙ 2p − 1 m B= ¨+ 2 m ¨ − 2p m , 3 m 12 m m 9

(26)

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665

where m0 is the initial mass of the system and 2 A˙ 1 + p m ˙ f˙ =− + . f 3A 3 m

(27)

A = λmK ,

(28)

If

where λ and K are constants, from the last equation it is: p=

1 + K. 2

(29)

The conservation law is energy-like: I ≡m

2(1−2p) 3









˙2 p+1m m 2 2p − 1 2p − 1 x 2 x˙ − m ˙ xx ˙ + m ¨− 2 6 6 2 3 m





+ Ax + Bx + Dx 2

4

 

= const.

(30)

(b) For the case when: A = λA1 = const.,

B = const.,

C = 0,

D = const.

(31)

according to the relation (27) it is: 1 p= , 2 and the equation for mass (26) is identically satisfied. The conservation law is of the energy type: I≡

m 2 x˙ + V (x, λ) = const., 2

(32)

(33)

where V = λA1 x + Bx 2 + Dx 4 . Using the so obtained conservation laws, the approximate analytic criteria of buckling is obtained. 4. Approximative analytic criteria of dynamic buckling Let us apply the energy type conservation law of the system (5): I (x, x, ˙ λ, τ ) = const.,

(34)

˙ τ ) + I2 (x, λ, τ ) = const., I1 (x, x,

(35)

i.e.

where I1 has the form of the kinetic energy and I2 of the potential energy. The system is initially at rest and at t = 0 the velocity and the suddenly applied load are zero. It means that: I1 |t =0 = 0,

I2 |t =0 = 0,

(36)

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and I |t =0 = 0.

(37)

I1 + I2 = 0,

(38)

For t > 0:

and I2 = −I1 < 0,

for I1 = 0,

and it means that the term I2 is non-positive during the motion. For the case when x˙ = 0 and I1 = 0, and according to (28) it is: I2 (x, τ, λ) = 0.

(39)

This value corresponds to the maximum values of I2 and the position is unstable. If the motion passes through this position it is unbounded. The condition of extreme value is: ∂I2 (x, τ, λ) = 0. ∂x

(40)

The task is to obtain the values of λD and xD which satisfy the requirements (39) and (40). This value of λD corresponds to the dynamic buckling load. It means that the dynamic buckling load λD and the corresponding coordinate xD are obtained by solving the system of equations (39) and (40). The implicit relationship for λ from (39) is: λ = λ(x, τ ).

(41)

It describes the so-called dynamic curve. Substituting (41) into (39) we find the identity: 



I2 x, τ, λ(x, τ ) = 0.

(42)

Differentiating the relation (42) it is: ∂I2 ∂I2 ∂λ + = 0. (43) ∂x ∂λ ∂x From the equation (43) the dependence of λ and x is defined. The maximal value of this function is the critical load. Due to the relation (40) it is: ∂I2 ∂λ = 0, for = 0. (44) ∂x ∂λ It is the necessary condition of extremum of the λ−x curve. It is maximum for: ∂ 2λ < 0. ∂x 2 Differentiating equation (43) we obtain: ∂ 2 I2 ∂λ ∂I2 ∂ 2 λ ∂ 2 I2 + + 2 = 0, ∂x 2 ∂x∂λ ∂x ∂λ ∂x 2

(45)

Dynamic buckling of a single-degree-of-freedom system

667

i.e. according to (44): ∂I2 ∂ 2 λ ∂ 2 I2 = − . ∂x 2 ∂λ ∂x 2 Since ∂I2 /∂λ represents the external forces linearly dependent on λ the value is negative. Using also the relation (45) we obtain: ∂ 2 I2 (xD , λD ) < 0. ∂x 2 It means that the solution (xD , λD ) is of the above system of equations correspond to the point of intersection of the curve described with equation (40) and the curve described with equation (41). At the same time this point is due to (44) and (45), the maximum in the dynamic curve (41). 5. Dynamic buckling in the system with p = 1/2 Let us consider the dynamic properties of the system where the mass is varying in time. The mass variation is slow. Due to mass variation the reactive force appears. The case when p = 1/2 is analyzed. The differential equation of motion is: 1 dm d (mx) ˙ + k1 (x − x0 ) − k3 (x − x0 )3 = x˙ + P , (46) dt 2 dt where P is a step force which acts on the system and x0 is the imperfection. The potential energy of the system is: k3 k1 (47) V = (x − x0 )2 − (x − x0 )4 − P (x − x0 ). 2 4 Substituting equation (47) into quation (33) the conservation law is: I=

k3 m 2 k1 x˙ + (x − x0 )2 − (x − x0 )4 − P (x − x0 ) = const. 2 2 4

(48)

For x(0) ˙ = 0 and x(0) = x0 it is I = 0. Let us introduce the dimensionless parameters: m , m0 P , λ= Lk1

M=

ξ=

x , L

ξ0 =

x0 , L

X = ξ − ξ0 ,

ω2 =

T = ωt.

k1 , m0

γ3 =

k3 L2 , k1 (49)

The dimensionless differential equation is: d ˙ + X − γ3 X 3 = 1 M˙ X˙ + λ. (M X) dT 2

(50)

The corresponding conservation law is according to (48): M ˙ 2 1 2 γ3 4 X + X − X − λX = 0, 2 2 4

(51)

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where I1 =

M ˙2 X , 2

(52)

1 γ3 I2 = X 2 − X 4 − λX, 2 4

(53)

∂I2 = X − γ3 X 3 − λ. ∂X

(54)

and the corresponding first derivative is:

5.1. Analytic method for denoting buckling From the relation (53) and the condition (39) the dynamic curve has the form: γ3 1 λ = X − X3 . 2 4

(55)

From equation (54) and the relation (40) it is: 



and

1 X2 = √ , γ3

λ = X 1 − γ3 X 2 .

(56)

The curve (56) has zeros for: X1 = 0 and a maximum: 2 , λs = √ 3 3γ3

(57)

for 1 . 3γ3 (Xs , λs ) are the static buckling coordinates. The dynamic curve (55) has two zeros: Xs = √



X3 = 0,

X4 =

and a maximum: 1 λD = 3 for





XD =

(58)

2 , γ3

2 , 3γ3

(59)

2 . 3γ3

(60)

(XD , λD ) represents the dynamic buckling value. It depends on the value of γ3 . The same pair of values is obtained as the intersection point of dynamic curve (55) and the curve (56). In figure 1 both curves for γ3 = 1

Dynamic buckling of a single-degree-of-freedom system

669

Figure 1. λ−X curves according the equation (54) and equation (55) and their intersection point (λD , XD ).

are plotted. The position of dynamic buckling is obtained as the intersection between the two curves. The intersection point is: XD = 0.8165,

λD = 0.2732.

(61)

5.2. Numerical simulation To prove the correctness of the analytic value of the dynamic buckling load it is compared with the numerical one. The numerical procedure for obtaining dynamic buckling load is based on finding the value of λD such that for a small change in λ the response in X presents a large change. This approach requires numerical integration of equation (46), i.e. (50) in order to obtain the response in time. This procedure is general and has a direct physical meaning. There are no approximations involved other than the numerical accuracy with which the response in time is compared. The method of Runge–Kutta is applied for the solving of the differential equation (50). 5.3. Dynamic buckling of a sieve for separation of particles In figure 2 the model of a sieve for separation of particles is shown. The mass of the system is continually varying due to pouring of the material. On the upper side the fresh material drops on the sieve, and on the lower side it leaves the system. It causes vibrations of the system. Due to addition and separation of the mass the impact force acts. The mass variation is (Cveticanin, 1998): m = m0 + m1 ± ρSvt,

Figure 2. The model of a sieve.

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L. Cveticanin

where m0 is the initial mass of the sieve, m1 is the initial mass of the material, ρ is the density of material, S is the surface on which the mass is adding or pouring out, v is the velocity of pouring of the material. Using the dimensionless parameters (49) the equation is: M = (1 + µ1 ± τ ), where µ1 = m1 /m0 , ε = ρS/ρ0 S0 , ρ0 S0 is the unit mass of the sieve. The mass variation is a linear function of slow dimensionless time and let us assume the following numerical values: M = 10 ± τ,

(62)

τ = 0.1T .

(63)

where

The case when the mass increases (figure 3a) and also the case when mass decreases (figure 3b) are analyzed. The displacement-time diagrams for various values of parameter λ are analyzed. The initial conditions for ˙ T = 0 are X(0) = 0.2, X(0) = 0 and the parameter of the system is γ3 = 1. The mass variation is bounded in time. It is in the time interval (0, 100). The parameter λ has the values: 0; 0.075; 0.192; 2.2; 2.5; 3. In figure 3a and figure 3b it can be seen that for λ = 0 the vibrations are around the zero position and that the maximal value of amplitude of vibrations is 0.2. For 0 < λ < 0.192 the vibrations are around the new values of attraction points and the maximal displacement is 0.2. The period of vibrations are the same for all values of λ in the aforementioned interval. For λ = 0.192, the value which corresponds to X = 0.2 in figure 1, the motion is X(T ) = 2. For 0.192 < λ < 0.3 the system vibrates and the minimal value of displacement is 0.2. In this interval increasing the λ increases the period of vibrations and also the amplitude of vibrations. For λ > 0.3 dynamic buckling occurs and the motion is unstable. It is worth to say that the regions of stable and unstable motion are the same independently of the mass variation properties.

(a)

(b)

Figure 3. Variation of the dynamic displacement X against dimensionless time T for various values of the loading λ for the: (a) linear increasing mass variation; (b) linear decreasing mass variation.

Dynamic buckling of a single-degree-of-freedom system

671

5.4. Dynamic buckling of the textile machine rotor Let us analyze the motion of a symmetrical Jeffcott rotor with variable mass (figure 4). The model of the rotor is a shaft-disc system on which the band is winding up or down. The mass of the shaft is negligible in comparison to the mass of the disc. The mass variation of the disc is according to (Cveticanin, 1991): h m = m0 ± ρLvt, 2 where m0 is the initial mass of the disc of the rotor, h is the thickness of the band, ρ is the density of band material, v is the velocity of winding up and down of the band. Introducing the dimensionless parameters (49) the mass variation is a square function of the slow time: M = (1 ± τ )2 ,

(64)

where ε = h/2π R0 and ω = v/R0 . Let us assume that: √ τ = 3T /2 2.

(65)

The elastic properties of the shaft are non-linear. Due to mass variation vibrations in x direction occur. The motion of the system is described with the equation (50). In figure 5a and figure 5b the displacement-time diagrams for increasing and decreasing mass, respectively, and for various values of parameter λ are plotted.

Figure 4. The model of a textile machine rotor.

(a)

(b)

Figure 5. Variation of the dynamic displacement X against dimensionless time T for various values of the loading λ for the square function of: (a) increasing mass; (b) decreasing mass.

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˙ The initial conditions are X(0) = 0.81, X(0) = 0 and the parameter of the system is γ3 = 1. The mass variation is in the time interval (0, 1). It can be seen that the mass variation has no influence on the position and the value of dynamic buckling. For λ = 0.26 the motion is stable and for λ = 0.28 it is unbounded. 6. Conclusion It can be concluded: 1. For the non-conservative system with variable mass and impact force for which Lagrangian exists it is possible to obtain the conservation law using the Noether approach. 2. Based on the energy criteria a new approach for dynamic buckling of the non-conservative system with slow variable mass is developed. Using the first integral, which is the conservation law of energy type the boundary of dynamic buckling is obtained. 3. Comparisons of numerical results with those of other analyses obtained via numerical simulation show the reliability of the proposed approach. The motion is stable for λ < λD and unstable for λ > λD . 4. For the case when p = 1/2 and the arbitrary mass variation function, the conservation law of energy type exists. Due to the fact that the mass variation is slow it has no influence on the buckling properties of the system. The buckling properties correspond to those of the system with constant mass, i.e. τ = 0. References Brewer, A.T., Godoy, L.A., 1991. On interaction between static and dynamic loads in instability of symmetric of asymmetric structural systems 147, 105–114. Brewer, A.T., Godoy, L.A., 1993. Interaction between static and dynamic loads in instability of structural systems with quadratic and cubic nonlinearities. Journal of Sound and Vibration 166, 31–43. Cveticanin, L., 1991. Oscillations of a textile machine rotor on which the textile is wound up. Mechanism and Machine Theory 26, 253–260. Cveticanin, L., 1993. Conservation laws in systems with variable mass. Journal of Applied Mechanics 60, 954–959. Cveticanin, L., 1994. Some conservation laws for orbits involving variable mass and linear damping. AIAA Journal of Guidance, Control and Dynamics 17, 209–211. Cveticanin, L., 1998. Dynamics of Machines with Variable Mass. Gordon and Breach, London. Cveticanin, L., Maretic, R., 2000. Dynamic analysis of a cutting mechanism. Mechanism and Machine Theory 35, 1391–1411. Gantes, C., Kounadis, A.N., Mallis, J., 1998. Approximate dynamic buckling loads of discrete systems via geometric considerations of their energy surface. J. Comput. Mech. 21, 398–402. Kounadis, A.N., 1994-a. A qualitative analysis for the local and global dynamic buckling and stability of autonomous discrete systems. Quarterly Journal of Mechanics and Applied Mathematics 47, 269–295. Kounadis, A.N., 1994-b. On the failure of static stability analyses of nonconservtive systems in regions of divergence instability. Int. Journal of Solid Structures 31, 2099–2120. Kounadis, A.N., 1996. Nonlinear dynamic buckling of a simple model via the Liapunov direct method. Journal of Sound and Vibration 193, 1091– 1097. Kounadis, A.N., 1997. Dynamic of imperfection sensitive nonconservative dissipative systems under follower loading. Facta Universitatis, Universitatis Series Mechanics, Automatic Control and Robotics 2, 189–209. Meshcherski, I.V., 1952. Rabotji po mehanike tel peremennoj massji. Gos. Izd. Tehn. Teor. Lit., Moscow. Raftoyiannis, I.G., Kounadis, A.N., 1988. Dynamic buckling of limit-point systems under step loading. Dynamics and Stability of Systems 3, 219– 234. Raftoyiannis, I.G., Kounadis, A.N., 2000. Dynamic buckling of 2-DOF systems with mode interaction under step loading. Int. J. Non-linear Mech. 35, 531–542. Sophianopoulos, D.S., 1996. Static and dynamic stability of a single-degree-of-freedom autonomous system with distinct critical points. Structural Engineering and Mechanics 4, 529–540. Vujanovic, B.D., Jones, S.E., 1989. Variational Methods in Nonconservative Phenomena. Academic Press, New York.