Dynamics and Stability of a Two Degree of Freedom Oscillator With an Elastic Stop

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Dynamics and Stability of a Two Degree of Freedom Oscillator With an Elastic Stop

Madeleine Pascal

To cite this version:

Madeleine Pascal. Dynamics and Stability of a Two Degree of Freedom Oscillator With an Elastic

Stop. Journal of Computational and Nonlinear Dynamics, American Society of Mechanical Engineers

(ASME), 2005, 1 (1), pp.94-102. �10.1115/1.1961873�. �hal-00342874v2�

Madeleine Pascal

Laboratoire Systèmes Complexes, Université d’Evry Val d’Essonne et CNRS FRE 2494, 40 rue du Pelvoux, 91020 Evry cedex, France e-mail: mpascal@iup.univ-evry.fr

Dynamics and Stability of a Two Degree of Freedom Oscillator With an Elastic Stop

A two degree of freedom oscillator with a colliding component is considered. The aim of the study is to investigate the dynamic behavior of the system when the stiffness obstacle changes to a finite value to an infinite one. Several cases are considered. First, in the case of rigid impact and without external excitation, a family of periodic solutions are found in analytical form. In the case of soft impact, with a finite time duration of the shock, and no external excitation, the existence of periodic solutions, with an arbitrary value of the period, is proved. Periodic motions are also obtained when the system is submitted to harmonic excitation, in both cases of rigid or soft impact. The stability of these periodic motions is investigated for these four cases.

Keywords: Nonlinear Vibrations, Two Degree of Freedom Oscillator, Rigid and Soft Impact, Periodic Motion, Stability, Forced and Unforced System

1 Introduction

Vibrating systems with clearance between the moving parts are frequently encountered in technical applications. These systems with impacts are strongly nonlinear; they are usually modeled as a spring-mass system with amplitude constraint. Such systems have been the subject of several investigations, mainly in the simplest case of a one degree of freedom system

关

1–4

兴

and more seldom for multidegree of freedom systems

关

5–8

兴

. On the other hand, the system behavior during the contact between the moving parts can be described as rigid impact, usually associated with a restitution coefficient, or modeled as soft impact, with a finite time duration of the shock. Several other parameters like damping, external ex- citation, influence the behavior of the system.

The work is the continuation of a previous paper

关

9

兴

, in which a two degree of freedom oscillator is considered. The nonlinearity in this case comes from the presence of two fixed stops limiting the motion of one mass. Assuming no damping and no external excitation, the behavior of the system is investigated when the obstacles stiffness changes from a finite value to an infinite one. In both cases, a family of symmetrical periodic solutions, with two impacts per period, is obtained in analytical form.

In the present paper, a two degree of freedom system in the presence of one fixed obstacle is considered. Assuming that no damping occurs, we investigate four cases: Unforced system with rigid impact, unforced system with soft impact, forced system

共

with harmonic excitation

兲

with rigid impact and, at last, forced system with soft impact. In all cases, periodic solutions are found and stability results of these particular motions are obtained.

2 Problem Formulation

The system under consideration

共

Fig. 1

兲

is a generalization of the double oscillator investigated in the paper

关

9

兴

. It consists of two masses m

₁

and m

₂

connected by linear springs of stiffness k

₁

and k

₂

. The displacement z

₁

of the mass m

₁

is limited by the presence of a fixed stop. When z

₁

is greater than the clearance, a contact of the first mass with the stop occurs; this contact gives rise to a restoring force associated to a spring stiffness k

₃

. The mathematical model of the system is given by:

Mz¨ + Kz = F + P cos

共␻

t +

␸兲

, z =

共

z

₁

,z

₂兲^t

, F =

冉

^{0 f}

^共

^z

^1兲冊

^,

共

1

兲

f

共

z

₁兲

=

再

^{− 0 k}

^3共

^z

¹

^{− 1}

^兲

^{z z}

^11⬎⬍

^{1 1}

冎

^{, M =}

^冉

^{m 0}

¹

^{m 0}

^2冊

^,

K =

冉

^{− k k}

¹1

k

₁

⁻ + ^k

¹

k

₂冊

P =

共^PP12兲

^and

^␸

are the amplitude and the phase angle of the har- monic excitation.

3 Unforced System

Let us consider the system without external excitation

共

P = 0

兲

. 3.1 Rigid Impact. When the stiffness obstacle k

₃

tends to infinity, a rigid impact of the first mass against the stop occurs.

Starting from initial positions z

₀

=

共^1y兲

and initial velocities z˙

₀

=

共^wu兲^共

^u

^⬎

⁰

^兲

, corresponding to a contact of the first mass against the stop, assuming a perfect elastic impact, the new positions z

_c

and the new velocities z˙

_c

after the shock are obtained by:

冉

^{z z˙}

^cc冊

⁼

冉

^{0 I E 0}

冊冉

^{z z˙}

⁰0冊

^{, I =}

冉

^{1 0 0 1}

冊

^{, E =}

冉

^{− 1 0 0 1}

冊 ^共

²

^兲

After the impact, the system performs a free motion defined by:

冉

^{z z˙}

冊

^{= C}

^共

^t

^兲冉

^{z z˙}

^cc冊

^{, C}

^共

^t

^兲

⁼

冉^⌫⌫¹3^共共

^t t

^兲兲 ^⌫⌫²1^共共

^t t

^兲兲冊

^,

^⌫i

⁼

^⌳

^B

^i共

^t

^兲⌳−1

共

i = 1,2,3

兲 共

3

兲

⌳

=

冉␭

¹

1 ␭

¹

2冊

^{, B}

^1共

^t

^兲

⁼

冉

^{C 0}

^1共

^t

^兲

^{0 C}

2共

t

兲冊

^,

B

₂共

t

兲

= 冢 ^S

^␻1

⁰

^共1

^t

^兲

^S

^␻2

⁰

^共2

^t

^兲

冣 ^,

^共

⁴

^兲

B

₃共

t

兲

=B ˙

1共

t

兲

, C

_i共

t

兲

= cos

␻i

t, S

_i共

t

兲

= sin

␻i

t

共

i= 1, 2

兲

.

In these formulas,

共␻1

,

␻2兲

are the roots of the characteristic equation:

⌬共␻²兲⬅

det

共

K− M

␻²兲

= 0 while

⌿i

=

共_␭¹_i兲

are defined by

共

K −M

␻_i²兲⌿i

= 0

共

i= 1 , 2

兲

.

The following properties for the

⌫i

matrices hold:

⌫12共

t

兲

−

⌫2共

t

兲⌫3共

t

兲

= I,

⌫i共

t

兲⌫j共

t

兲

=

⌫j共

t

兲⌫i共

t

兲

for i, j = 1,2,3

共

5

兲

Moreover, the coefficients C

_ij共

t

兲

of the 4 by 4 matrix C

共

t

兲

satisfy the property:

C

ij共

t

兲

= C

_i−2,j−2共

t

兲

,

共

i, j = 3,4

兲 共

6

兲

Let us investigate if for a set of initial conditions Z

₀

=

共^zz˙⁰0兲

^{, z}

0

=

共¹_y兲

, z˙

₀

=

共^u⬎_w⁰兲

, it is possible to obtain a periodic solution of pe- riod T, with one impact per period.

The free motion performed by the system after the rigid impact finishes at time t=T when z

₁共

T

兲

= 1 and z˙

₁共

T

兲⬎

0. Let us denote by Z

_f

=

共^zz˙^ff兲

the positions and the velocities reached by the system at t=T.

The condition to obtain such a periodic motion is given by:

Z

_f⬅

C

共

T

兲

Z

_c

= Z

₀

, Z

_c

= H

₀

Z

₀

, H

₀

=

冉

^{0 I E 0}

冊 ^共

⁷

^兲

Let us introduce the position Z

_s

reached by the system from the initial position Z

₀

after a backward motion of duration T: Z

_s

=C

共

−T

兲

Z

₀

. The condition

共

7

兲

is equivalent to: Z

_s

=Z

_c

.

It results for the determination of the four scalar parameters

共

y , u, w , T

兲

the four scalar equations:

共

− H

₀

+ C

共

− T

兲兲

Z

₀

= 0, Z

₀

=

共

1, y, u,w

兲^t

or equivalently:

共⌫1

− I

兲

z

₀

−

⌫2

z˙

₀

= 0

−

⌫3

z

₀

+

共⌫1

− E

兲

z˙

₀

= 0

⌫i

=

⌫i共

T

兲 共

i = 1,2,3

兲 共

8

兲

Taking into account the properties

共

5

兲

of the

⌫i

matrices, the sys- tem

共

8

兲

leads to:

z

₀

=

共⌫1

− I

兲⁻¹⌫2

z˙

₀

共

9

兲 共

E + I

兲

z˙

₀

= 0

The last equation of

共

9

兲

reduces to w= 0. From the first one, y and u are obtained in terms of the period:

y =

共␻2

t

₂

−

␻1

t

₁兲␭1␭2

␭2␻2

t

₂

−

␭1␻1

t

₁

, u =

共␭1

−

␭2兲␻1␻2

t

₁

t

₂

␭2␻2

t

₂

−

␭1␻1

t

₁

,

t

_i

= tan

冉^␻

²

ⁱ

^T

冊 ^共

^{i = 1,2}

^{兲 共}

¹⁰

^兲

In case of rigid impact, a family of periodic solutions is obtained for which the initial conditions are defined in terms of the period and the initial velocity w of the nonimpacting mass is zero. For these particular motions, the conditions giving the positions and the velocities after the shock can be formulated by:

z

_c

= z

₀

, z˙

_c

= − z˙

₀ 共

11

兲

These results are similar to the results obtained in Ref.

关

9

兴

. The system considered in this previous paper is a symmetrical system with respect to the position z

₁

of the first mass and it can be

expected that the obtained results are due to this property. But it is not the real explanation because the system investigated now is not symmetrical.

3.2 Soft Impact. Let us assume that the stiffness obstacle is bounded. The mathematical model of the system is given by:

For z

₁⬍

1

Mz¨ + Kz = 0

共

12

兲

For z

₁⬎

1

Mz¨ + K

₁

z = K

₃

K

₁

=

冉

^k

¹

^{− + k}

1

^k

³

k

₁

⁻ + ^k

¹

k

₂冊

^K

³

⁼

冉

^{k 0}

³冊 ^共

¹³

^兲

Let us assume that the initial conditions are given by: Z

₀

=

共

1 ,y, u, w

兲^t

u

⬎

0.

A periodic solution is defined in two steps:

– For 0

艋

t

艋␶

, z

₁⬎

1, the system is defined by the motion equations

共

13

兲

. The time duration

␶

of this constraint motion is defined by the condition:

z

₁共␶兲

= 1

共

14

兲

Let us denote by Z

_c

=Z

共␶兲

=

共

1 ,y

_c

, u

_c

, w

_c兲^t

the value of the parameters at the end of shock, with the condition u

_C⬍

0.

– For

␶艋

t

艋␶

+T, a free motion obtained from Eqs.

共

12

兲

and initial conditions Z

_c

occurs. This motion finishes when z

₁共␶

+T

兲

= 1. Let us denote by Z

_f

=Z

共␶

+T

兲

=

共

1 ,y

_f

,u

_f

, w

_f兲^t

the value of the parameters at the end of free motion

共

u

_f⬎

0

兲

. The condition to obtain a periodic orbit of period

␶

+T is given by

Z

f

= Z

₀ 共

15

兲

The piecewise linear systems

共

12

兲

and

共

13

兲

give the two parts of the motion in analytical form.

– For 0

艋

t

艋␶

, the constraint motion is deduced from a modal analysis of system

共

13

兲

:

Z

共

t

兲

= H

共

t

兲共

Z

₀

− d

兲

+ d, d =

共

d

₁

, d

₂

,0,0

兲^t

d

₁

= k

₃共

k

₁

+ k

₂兲

k

₁

k

₂

+ k

₃共

k

₁

+ k

₃兲

, d

₂

= k

₃

k

₁

k

₁

k

₂

+ k

₃共

k

₁

+ k

₂兲 共

16

兲

H

共

t

兲

=

冉

^{H H}

¹3^共共

^t t

^兲兲

^H H

²₁^共共

^t t

^兲兲冊

^{, H}

^i共

^t

^兲

⁼

^⌺

^G

^i共

^t

^兲⌺−1

共

i = 1,2,3

兲

,

⌺

=

冉␮

¹

1 ^␮

1

²冊 ^共

¹⁷

^兲

G

₁共

t

兲

=

冉

^c

¹

⁰

^共

^t

^兲

^c

2

⁰

共

t

兲冊

^{, G}

²

⁼ 冢 ^s

^␴1

⁰

^共1

^t

^兲

^s

^2␴

⁰

^共2

^t

^兲

冣 ^{, G}

³

^{= G ˙}

¹

^,

共

18

兲

c

i共

t

兲

= cos

共␴i

t

兲

, s

i共

t

兲

= sin

共␴i

t

兲

,

共

i = 1,2

兲

In these formulas,

␴1

,

␴2

,

⌽1

=

共_␮¹₁兲

^,

⌽2

=

共^␮₁²兲

define the char- acteristic frequencies and the eigenvectors of the constraint

Fig. 1 Double oscillator

system

共

13

兲

. For the H

_i

matrices, the properties

共

5

兲

obtained for the

⌫i

matrices hold, together with the property H

_ij共

t

兲

=H

_i−2,j−2共

t

兲

,

共

i, j= 3,4

兲

for the coefficients H

_ij共

t

兲

of the ma- trix H

共

t

兲

.

– For

␶艋

t

艋␶

+T, the free motion is obtained from Z

共

t

兲

=C

共

t

−

␶兲

Z

_c

where the matrix C is defined by formulas

共

3

兲

and

共

4

兲

and Z

_c

=H

共␶兲共

Z

₀

−d

兲

+d.

Let us introduce the positions and the velocities Z

_s

=

共

z

_1s

, z

_2s

, z˙

_1s

, z˙

_2s兲^t

of the system after a backward motion of dura- tion T from the initial position Z

₀

. The condition

共

15

兲

of period- icity is reformulated as

Z

_s⬅

C

共

− T

兲

Z

₀

= Z

_c 共

19

兲

Z

_c

=

冉

^{z z˙}

^cc冊

⁼

冉

^H

¹

^H

^共

^z

3⁰共

⁻ z

₀

^d −

^0兲

d ⁺

₀兲

^H +

²

H ^z˙

⁰₁

z˙ ⁺

₀

^d

⁰冊

^{, d}

⁰

⁼

^共

^d

¹

^,d

^2兲t

Z

_s

=

冉

^{z z˙}

^ss冊

⁼

冉

⁻

^⌫⌫¹

^z

3⁰

z

₀

⁻ +

^⌫⌫²

^z˙

1⁰

z˙

₀冊

^{, H}

ⁱ

^{= H}

^i共␶兲

^,

^⌫i

⁼

^⌫i共

^T

^兲

共

i = 1,2,3

兲 共

20

兲

The condition

共

19

兲

is equivalent to

X

₁

= X

₂

, Y

₁

= Y

₂ 共

21

兲

X

₁⬅

z

c

− z

₀

=

共

H

₁

− I

兲共

z

₀

− d

₀兲

+ H

₂

z˙

₀

, X

₂⬅

z

s

− z

₀

=

共⌫1

− I

兲

z

₀

−

⌫2

z˙

₀

Y

₁⬅

z˙

_c

+ z˙

₀

= H

₃共

z

₀

− d

₀兲

+

共

H

₁

+ I

兲

z˙

₀

, Y

₂⬅

z˙

_s

+ z˙

₀

= −

⌫3

z

₀

+

共⌫1

+ I

兲

z˙

₀ 共

22

兲

From the properties

共

5

兲

of

⌫i

and H

_i

, we deduce:

Y

_i

= P

_i

X

_i 共

i = 1,2

兲

P

₁

= H

₂⁻¹共

H

₁

+ I

兲

, P

₂

= −

⌫2

−1共⌫1

+ I

兲 共

23

兲

The condition

共

21

兲

leads to

X

₁

= X

₂

, P

₁

X

₁

= P

₂

X

₂ 共

24

兲

Two possible cases of periodic solutions can be deduced from

共

24

兲

, namely:

X

₁

= X

₂

= 0 or X

₁

= X

₂

, det

共

P

₁

− P

₂兲

= 0

共

25

兲

3.3 Existence of Periodic Motions (Soft Impact). Let us dis- cuss the first conditions

共

25

兲

. In this case, from

共

22

兲

, we deduce:

z

_c

= z

₀

, z˙

_c

= − z˙

₀ 共

26

兲

The condition

共

14

兲

is fulfilled and the initial conditions are ob- tained from the equations:

共

H

₁

− I

兲共

z

₀

− d

₀兲

+ H

₂

z˙

₀

= 0

共⌫1

− I

兲

z

₀

−

⌫2

z˙

₀

= 0

共

27

兲

This system provides four scalar equations for the determination of the five parameters

共

y, u, w,

␶

, T

兲

. It results that, as in the case of rigid impact, T and hence the period can be chosen arbitrarily.

Moreover, the conditions

共

11

兲

and

共

26

兲

obtained at the end of the shock are the same for both rigid and soft impacts. From

共

27

兲

, we deduce:

z˙

₀

= − H

₂⁻¹共

H

₁

− I

兲共

z

₀

− d

₀兲

关⌫1

− I +

⌫2

H

₂⁻¹共

H

₁

− I

兲兴

z

₀

=

⌫2

H

₂⁻¹共

H

₁

− I

兲

d

₀ 共

28

兲

The last equation

共

28

兲

, after the elimination of y, provides a rela- tion F

共␶

, T

兲

= 0 between the time duration

␶

of the shock and the

time duration T of the free motion.

-The other case X

₁

= X

₂

, det

共

P

₁

− P

₂兲

= 0 leads to no solution

共

see Appendix A

兲

.

In both cases

共

soft or rigid impact

兲

, a family of periodic mo- tions is obtained, with an arbitrary value of the period. Moreover, the conditions

共

26

兲

obtained at the end of the shock for soft impact are consistent with Newton rules of rigid impact

共

11

兲

with a res- titution coefficient equal to one, i.e., with assumption of ideal elastic impact. This rather remarkable result has already been ob- tained for the symmetrical system of Ref.

关

9

兴

.

4 Forced System

Let us assume that the two masses are subjected to harmonic external excitations of period 2

␲

/

␻

, constant amplitudes P

₁

, P

₂

and constant phase angle

␸

. From the results obtained in the pre- vious paragraph, where a family of periodic orbits is found with an arbitrary value of the period, it can be expected that for the forced system, periodic solutions of period 2

␲

/

␻

exist.

4.1 Rigid Impact. Let us investigate the case of rigid impact.

Starting from the initial conditions Z

₀

=

共

1 ,y , u, w

兲^t 共

u

⬎

0

兲

, the conditions Z

_c

=

共

1 ,y

_c

, u

_c

, w

_c兲^t

after the shock are obtained from

共

2

兲

and the free motion performed by the system is given by:

z =

⌫1共

t

兲共

z

₀

− R cos

␸兲

+

⌫2共

t

兲共

z˙

c

+ R

␻

sin

␸兲

+ R cos

共␻

t +

␸兲

z˙ =

⌫3共

t

兲共

z

₀

− R cos

␸兲

+

⌫1共

t

兲共

z˙

c

+ R

␻

sin

␸兲

− R

␻

sin

共␻

t +

␸兲

共

29

兲

where R =

共

R

₁

, R

₂兲^t

is the amplitude of the response defined by:

R

₁

= A

₁

+ A

₂

, R

₂

=

␭1

A

₁

+

␭2

A

₂

A

_i

= P

₁

+

␭i

P

₂

共␻i2

−

␻²兲共

m

₁

+

␭i2

m

₂兲

,

共

i = 1,2

兲 共

30

兲

The free motion finishes at time t =T when z

₁共

T

兲

= 1, z˙

₁共

T

兲⬎

0. Let us denote by Z

_f

=

共

z

_f

, z˙

_f兲^t

the positions and the velocities reached by the system at this time. The condition to obtain a periodic motion of period T is:

Z

_f

= Z

₀

or :

z

₀

=

⌫1共

T

兲共

z

₀

− R cos

␸兲

+

⌫2共

T

兲共

z˙

c

+ R

␻

sin

␸兲

+ R cos

共␻

T +

␸兲

z˙

₀

=

⌫3共

T

兲共

z

₀

− R cos

␸兲

+

⌫1共

T

兲共

z˙

c

+ R

␻

sin

␸兲

− R

␻

sin

共␻

T +

␸兲

z˙

c

= Ez˙

₀ 共

31

兲

Let us assume that T = 2

␲

/

␻

,

␸

= 0, z˙

_c

= −z˙

₀

. A periodic motion of period 2

␲

/

␻

is obtained if the initial conditions Z

₀

=

共

1 ,y, u, 0

兲^t

are defined by the system:

共⌫1

− I

兲共

z

₀

− R

兲

−

⌫2

z˙

₀

= 0

共

32

兲

⌫3共

z

₀

− R

兲

−

共⌫1

+ I

兲

z˙

₀

= 0

⌫i

=

⌫i共

2

␲

/

␻兲

,

共

i = 1,2,3

兲

Taking into account the properties

共

5

兲

of the

⌫i

matrices, this system reduces to

z˙

₀

=

⌫2

−1共⌫1

− I

兲共

z

₀

− R

兲 共

33

兲

and the corresponding values of y and u are obtained:

y = R

₂

+

共␻2

t

₂

−

␻1

t

₁兲␭1␭2

␭2␻2

t

₂

−

␭1␻1

t

₁ 共

1 − R

₁兲

,

共

34

兲

u =

共

1 − R

₁兲共␭1

−

␭2兲␻1␻2

t

₁

t

₂

␭2␻2

t

₂

−

␭1␻1

t

₁

, t

_i

= tan

冉^␻␻^i␲冊 ^共

^{i = 1,2}

^兲

Remark: In more general cases, the impact is described by a

restitution coefficient r

共

0

⬍

r

⬍

1

兲

. The initial conditions and the phase angle related to a periodic solution of period 2

␲

/

␻

can also be obtained in analytical form

关

10

兴

. A similar solution has been studied in paper

关

6

兴

.

4.2 Soft Impact. When the stiffness obstacle is bounded, the motion equations of the system are given by:

Mz¨ + K

₁

z = K

₃

+ P cos

共␻

t +

␸兲

z

₁⬎

1

Mz¨ + Kz = P cos

共␻

t +

␸兲

z

₁⬍

1

共

35

兲

From the initial condition Z

₀

=

共

1 , y, u, w

兲^t共

u

⬎

0

兲

, the solution is defined in two parts:

– For 0

⬍

t

⬍␶

, z

₁⬎

1, the solution is given by:

z = H

₁共

t

兲共

z

₀

− d

₀

− Q cos

␸兲

+ H

₂共

t

兲共

z˙

₀

+ Q

␻

sin

␸兲

+ d

₀

+ Q cos

共␻

t +

␸兲

z˙ = H

₃共

t

兲共

z

₀

− d

₀

− Q cos

␸兲

+ H

₁共

t

兲共

z˙

₀

+ Q

␻

sin

␸兲

− Q

␻

sin

共␻

t +

␸兲 共

36

兲

Q=

共

Q

₁

, Q

₂兲^t

is the response amplitude defined by:

Q

₁

= P

₁共

1 +

␮22兲

+ P

₂共␮1

+

␮2兲 共␴12

−

␻²兲共

m

₁

+ m

₂␮12兲

,

共

37

兲

Q

₂

= P

₁共␮1

+

␮2兲

+ P

₂共

1 +

␮12兲

共␴22

−

␻²兲共

m

₂

+ m

₁␮22兲

The time duration

␶

of this motion is obtained from the con- dition z

₁共␶兲

= 1. Let us denote Z

_c

=

共

1 ,y

_c

, u

_c

, w

_c兲^t

the value of the parameters at t=

␶共

u

_c⬍

0

兲

.

– For

␶⬍

t

⬍␶

+T, the motion of the system is defined by:

z =

⌫1共

t −

␶兲共

z

_c

− R cos

␺兲

+

⌫2共

t −

␶兲共

z˙

_c

+ R

␻

sin

␺兲

+ R cos

共␻

t +

␸兲

z˙ =

⌫3共

t −

␶兲共

z

_c

− R cos

␺兲

+

⌫1共

t −

␶兲共

z˙

_c

+ R

␻

sin

␺兲

− R

␻

sin

共␻

t +

␸兲 共

38

兲

where R=

共

R

₁

, R

₂兲^t

is defined by

共

30

兲

and

␺

=

␻␶

+

␸

. This motion finishes at time t=

␶

+T when z

₁共␶

+T

兲

= 1, z˙

₁共␶

+T

兲

⬎

0. If Z

_f

=

共

z

_f

, z˙

_f兲^t

denote the positions and the velocities reached by the system at this time, the condition to obtain a periodic motion of period

␶

+T is Z

_f

=Z

₀

. Let us assume that

␶

+T = 2

␲

/

␻

,

␸

= −

␻␶

/ 2. At the end of the first part of the motion

共

t=

␶兲

, the positions and the velocities are given by:

z

c

= H

₁共

z

₀

− d

₀

− Q cos

␸

˜

₀兲

+ H

₂共

z˙

₀

− Q

␻

sin

␸

˜

₀兲

+ d

₀

+ Q cos

␸

˜

₀

z˙

c

= H

₃共

z

₀

− d

₀

− Q cos

␸

˜

₀兲

+ H

₁共

z˙

₀

− Q

␻

sin

␸

˜

₀兲

− Q

␻

sin

␸

˜

₀

␸

˜

₀

=

␻␶

/2, H

i

= H

i共␶兲 共

i = 1,2,3

兲 共

39

兲

At the end of the second part of the motion

共

t= 2

␲

/

␻兲

, we obtain:

z

_f

=

⌫1共

z

_c

− R cos

␸

˜

₀兲

+

⌫2共

z˙

_c

+ R

␻

sin

␸

˜

₀兲

+ R cos

␸

˜

₀

z˙

_f

=

⌫3共

z

_c

− R cos

␸

˜

₀兲

+

⌫1共

z˙

_c

+ R

␻

sin

␸

˜

₀兲

+ R

␻

sin

␸

˜

₀

⌫i

=

⌫i共

2

␲

/

␻

−

␶兲 共

i = 1,2,3

兲 共

40

兲

The conditions of periodicity are reformulated as:

X ˜

1

= X ˜

2

, Y ˜

1

= Y ˜

2 共

41

兲

X ˜

1⬅

z

_c

− z

₀

=

共

H

₁

− I

兲共

z

₀

− d

₀

− Q cos

␸

˜

₀兲

+ H

₂共

z˙

₀

− Q

␻

sin

␸

˜

₀兲

Y ˜

1⬅

z˙

_c

+ z˙

₀

= H

₃共

z

₀

− d

₀

− Q cos

␸

˜

₀兲

+

共

H

₁

+ I

兲共

z˙

₀

− Q

␻

sin

␸

˜

₀兲 共

42

兲

X ˜

2⬅

z

_s

− z

₀

=

共⌫1

− I

兲共

z

₀

− R cos

␸

˜

₀兲

−

⌫2共

z˙

₀

− R

␻

sin

␸

˜

₀兲

Y ˜

2⬅

z˙

_s

+ z˙

₀

= −

⌫3共

z

₀

− R cos

␸

˜

₀兲

+

共⌫1

+ I

兲共

z˙

₀

− R

␻

sin

␸

˜

₀兲

Z

_s

=

共

z

_s

, z˙

_s兲^t

= C

共

− T

兲

Z

₀ 共

43

兲

As in the case of an unforced system taking into account the properties of the H

_i

and

⌫i

matrices, the solution of system

共

41

兲

is given by:

X ˜

1

= X ˜

2

= 0, ˜ Y

1

= Y ˜

2

= 0

共

44

兲

We deduce for the forced system the existence of a periodic mo- tion of period 2

␲

/

␻

for which the conditions at the end of the shock are z

_c

=z

₀

, z˙

_c

= −z˙

₀

. The time duration

␶

of the shock and the initial conditions

共

y , u, w

兲

are obtained from the first part of sys- tem

共

44

兲

:

z˙

₀

= Q

␻

sin

␸

˜

₀

− H

₂⁻¹共

H

₁

− I

兲共

z

₀

− d

₀

− Q cos

␸

˜

₀兲 关⌫2−1共⌫1

− I

兲

+ H

₂⁻¹共

H

₁

− I

兲兴

z

₀

=

共

Q − R

兲␻

sin

␸

˜

₀

+

⌫2−1共⌫1

− I

兲

R cos

␸

˜

₀

+ H

₂⁻¹共

H

₁

− I

兲共

d

₀

+ Q cos

␸

˜

₀兲 共

45

兲

These formulas give the relations

共

28

兲

when Q=R = 0

共

unforced system

兲

.

5 Stability of Periodic Motions (Rigid Impact) 5.1 Unforced System. Let us consider a periodic motion of period T related to initial conditions z

₀₀

=

共y¹0兲

^{, z˙}

00

=

共^u₀⁰兲

, where z

₀₀

, z˙

₀₀

are defined by:

共⌫1

− I

兲

z

₀₀

−

⌫2

z˙

₀₀

= 0

⌫i

=

⌫i共

T

兲

−

⌫3

z

₀₀

+

共⌫1

− E

兲

z˙

₀₀

= 0

共

46

兲

Let us consider the perturbed motion defined by a set of new initial conditions

z

₀

= z

₀₀

+ dz

₀

z˙

₀

= z˙

₀₀

+ dz˙

₀

where dz

₀

=

冉

^{0 y}

冊

^{, dz˙}

⁰

⁼

冉

^{w u}

冊 ^共

⁴⁷

^兲

This motion is defined for t

⬎

0, by:

z =

⌫1共

t

兲

z

₀

+

⌫2共

t

兲

Ez˙

₀

z˙ =

⌫3共

t

兲

z

₀

+

⌫1共

t

兲

Ez˙

₀

This motion ends at t=T +dT, when z

₁共

T +dT

兲

= 1 and z˙

₁共

T +dT

兲

⬎

0. Let us denote by z

_f

, z˙

_f

the positions and the velocities of the system at this time.

z

_f

=

⌫1共

T + dT

兲

z

₀

+

⌫2共

T + dT

兲

Ez˙

₀

z˙

_f

=

⌫3共

T + dT

兲

z

₀

+

⌫1共

T + dT

兲

Ez˙

₀

Assuming small perturbations dz

₀

, dz˙

₀

of the initial conditions, dz

_f

= z

_f

− z

₀₀

=

⌫1

dz

₀

+

⌫2

Edz˙

₀

+ L

₁

dT

dz˙

_f

= z˙

_f

− z˙

₀₀

=

⌫3

dz

₀

+

⌫1

Edz˙

₀

+ L

₂

dT

共

48

兲

L

₁

=

⌫

˙

1

z

₀₀

+

⌫

˙

2

Ez˙

₀₀