Dynamic stability of axially accelerating Timoshenko beam: Averaging

European Journal of Mechanics A/Solids 29 (2010) 81–90

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Dynamic stability of axially accelerating Timoshenko beam: Averaging method Xiao-Dong Yang a, *, You-Qi Tang a, Li-Qun Chen b, c, C.W. Lim d a

Department of Engineering Mechanics, Shenyang Institute of Aeronautical Engineering, Shenyang 110136, China Department of Mechanics, Shanghai University, Shanghai 200436, China c Shanghai Institute of Applied Mathematics and Mechanics, Shanghai 200072, China d Department of Building and Construction, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong b

a r t i c l e i n f o

a b s t r a c t

Article history: Received 21 February 2008 Accepted 17 July 2009 Available online 24 July 2009

This study investigates dynamic stability in transverse parametric vibrations of an axially accelerating tensioned beam of Timoshenko model on simple supports. The axial speed is assumed as a harmonic fluctuation about the constant mean speed. The Galerkin method is applied to discretize the governing equation into a finite set of ordinary differential equations. The method of averaging is applied to analyze the instability phenomena caused by subharmonic and combination resonance. Numerical examples demonstrate the effects of the mean axial speed, bending stiffness, rotary inertia and shear modulus on the instability boundaries. Ó 2009 Elsevier Masson SAS. All rights reserved.

Keywords: Axially moving beam of Timoshenko model Averaging method Subharmonic resonance Combination resonance Dynamic stability

1. Introduction Axially moving beams are involved in many engineering devices, such as band saws, serpentine belts, magnet tapes and paper in processing. The transverse vibrations associated with these devices have limited their applications (Wickert and Mote, 1988; Abrate, 1992). One major problem is the occurrence of large vibrations, termed as parametric vibrations, due to axial speed fluctuations, ¨z which often happen when the drive motors run at high speed (O ¨ zkaya and Pakdemirli, 2000; O ¨ z, 2001; Chen and Pakdemirli, 1999; O et al., 2004a; Chen and Yang, 2005). For example, the vibration of the blade of band saws results in poor cutting quality. Therefore, understanding transverse vibrations of axially moving beams is important for the design of the devices (Lengoc and McCallion,1995). There are comprehensive studies on such systems. Most of them take the string and Euler beam as simplified model of the axially moving continuum. The parametric vibrations of axially moving strings have been investigated extensively. These researches include numerical simulations (Fung et al., 1997, 1998; Chen et al., 2004b), analytical expressions of steady-state responses (Zhang and Zu, 1999a,b), and chaotic behaviors (Chen et al., 2003). Accounting for the Euler model, Marynowski (2002) and Marynowski and Kapitaniak (2002) used three-term Galerkin discretization to investigate the nonlinear vibration response of an

* Corresponding author. Tel.: þ86 24 89723710; fax: þ86 24 89723727. E-mail address: [email protected] (X.-D. Yang). 0997-7538/$ – see front matter Ó 2009 Elsevier Masson SAS. All rights reserved. doi:10.1016/j.euromechsol.2009.07.003

axially moving beam excited by a changing tension. Marynowski (2004) further studied numerically nonlinear dynamical behavior of an axially moving viscoelastic beam with time-dependent tension based on four-term Galerkin discretization. Yang and Chen (2005) studied numerically bifurcation and chaos of an axially accelerating nonlinear beam based on second-term Galerkin discretization. Perturbation methods, such as multiple scale method and the method of averaging are useful techniques in studying the nonlinear and parametric vibrations of axially moving continuum. ¨ z et al. (1998) employed the method of multiple scales to study O dynamic stability of an axially accelerating beam with small ¨ zkaya and Pakdemirli (2000) combined the bending stiffness. O method of multiple scales and the method of matched asymptotic expansions to construct non-resonant boundary layer solutions for multiple scales and the method of matched asymptotic expansions to construct non-resonant boundary layer solutions ¨z for an axially accelerating beam with small bending stiffness. O ¨ z (2001) applied the method of and Pakdemirli (1999) and O multiple scales to calculate analytically the stability boundaries of an axially accelerating beam under pinned–pinned and clamped– clamped conditions respectively. Parker and Lin (2001) adopted a first-term Galerkin discretization and the perturbation method to study dynamic stability of an axially accelerating beam subjected to a tension fluctuation. Suweken and Van Horssen (2003) applied the method of multiple scales to a discretized system obtained by the Galerkin method to study the dynamic stability of an axially accelerating beam with pinned–pinned ends. Chen et al.

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(2004a) applied the averaging method to a discretized system via the Galerkin method to present analytically the stability boundaries of axially accelerating viscoelastic beams. Ghayesh and Khadem (2008) discussed the effects of rotary inertia and temperature to the free vibrations of axially moving beam by multiple scale method. In all available studies on axially moving continuum, only string model and Euler beam model are considered. Actually, Euler beam model gives more inaccurate results with increasing slenderness ratio defined by the ratio of length of the beam to the radius of gyration of the cross-section. Timoshenko (1921, 1922) proposed a beam theory which adds the effect of shear as well as the effect of rotation to the Euler–Bernoulli beam. The Timoshenko model is a major improvement for non-slender beams and for highfrequency responses where shear or rotary effects are not negligible (Han et al., 1999). To account for the non-slenderness of the beam, the Timoshenko model is employed in this study to investigate the parametric transverse vibrations of axially accelerating beam on simple supports. This paper is organized as follows. Firstly, the governing equation is derived by the Newton’s second law. The second-order Galerkin method is used to discretize the governing partial differential equation into a set of ordinary differential equations. Then the natural frequencies are obtained for an elastic Timoshenko beam moving in a constant speed. The comparison of results by Timoshenko model and Euler model is presented. When the axial speed is harmonically varying about a mean speed, the instability conditions are analyzed by the averaging method. The effects of the mean axial speed, bending stiffness, rotary inertia and shear modulus to the instability region caused by the resonances are discussed by some numerical examples.

2. Governing equation A uniform axially moving Timoshenko beam, with density r, cross-sectional area A, moment of inertia I, and initial tension P, travels at the time-dependent axial transport speed V(T) between two simple supports separated by distance L. The bending vibration can be described by two variables dependent on axial coordinate X and time T, namely, transverse displacement U(X,T) and a(X,T), the angle of rotation of the cross-section due to the bending moment. According to Newton’s second law, the coupled governing equations are obtained

! d2 U v2 U v2 U va 0 rA 2 ¼ P 2 þ k AG ;  dT vX vX 2 vX

rI

  v2 a v2 a vU 0 a ; ¼ EI þ k AG  vX vT 2 vX 2

(1)

(2)

where I is the area moment of inertia of the cross-section about the neutral axis, k0 the shape factor, E the modulus of elasticity, and G the shearing modulus of the beam. After some derivations, we can decouple the two equations and the governing equation for the transverse displacement is

! v2 EI v2  þ1 k0 AG vT 2 k0 AG vX 2 ! d2 U v2 U rA 2  P 2 ¼ 0: dT vX

v4 U v4 U EI 4  rI 2 2 þ vX vX vT

beam Eq. (3) recovers to that of Euler beam after neglecting the second term and let k0 / N, as

EI

v4 Y d2 U v2 U r þ A  P 2 ¼ 0: 4 2 vX dT vX

(4)

Usually the acceleration of the transverse displacement of Eq. (3) can be cast into the following form:

d2 U v2 U v2 U dV vU v2 U ¼ þ 2V þ þ V2 2: 2 2 vXvT dT vX dT vT vX

(5)

For further study, introduce the dimensionless variables and parameters

sffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffi rA U X P EI u ¼ ; x ¼ ; t ¼ T ; ; v ¼ V ; k ¼ L L P rAL2 PL2 I PI EI c1 ¼ 2 ; c2 ¼ 0 2 2 ; c3 ¼ 0 : k AGL AL kA L G

(6)

Governing equation (3) can be cast into the dimensionless form

v4 u v4 u v2 v2 k 4  c1 2 2 þ c2 2  c3 2 þ 1 vx vx vt vt vx

!

d2 u v 2 u  dt 2 vx2

! ¼ 0;

(7)

where 2 d2 u v2 u v2 u dv vu 2v u ¼ þ 2v : þ þ v vxvt dt vx dt 2 vt 2 vx2

(8)

It is obvious that the dimensionless parameter k denotes the stiffness of the beam, c1 presents the effects of the rotary inertia and c2 associated with c3 accounts for the effects of shear distortion. Equation (7) degenerates into that governs the axially moving Euler beam if c1, c2, c3 are set zero.

3. Natural frequencies for the beam with constant moving speed Substituting Eq. (8) into (7) yields

v4 u v4 u k 4  c1 2 2 þ vx vx vt

v2 v2 c2 2  c3 2 þ 1 vt vx ! 2 dv vu v u ¼ 0: þ  dt vx vx2

!

v2 u v2 u v2 u þ 2v þ v2 2 2 vxvt vt vx ð9Þ

The Galerkin method is employed to simplify Eq. (9). Under given boundary conditions, the solution of Eq. (10) may be expanded into the series separated by space and time variables

uðx; tÞ ¼ qðtÞT 4ðxÞ;

(10)

where the infinite column matrixes q(t) and 4(x) are assembled respectively by the generalized displacements and the assumed eigenfunctions, namely

rI

ð3Þ

If the rotary inertia effect and shear distortion effect are not considered, the governing equation of axially moving Timoshenko

qðtÞ ¼ ðq1 ðtÞ; q2 ðtÞ; .qN ðtÞÞT ; 4ðxÞ ¼ ð41 ðtÞ; 42 ðtÞ; .4N ðtÞÞT : (11) Substituting the appropriate derivatives of Eq. (10) into (9), one obtains the residual

X.-D. Yang et al. / European Journal of Mechanics A/Solids 29 (2010) 81–90



 € T 4 þ 2vq_ T 40 þ v2  1 qT 400 þ vq € T 400 _ T 40  c1 q Rðx; tÞ ¼ q h   € T 400 þ 4vv_ q_ T 400 qT 4 þ v2  1 q þ kqT 40000 þ c2 z T

T

€ 40 þ 4€ þ 2v_ 2 qT 400 þ 2v€ vqT 400 þ 5v_ q vq_ 40 þ 2v 0 qT 40 i h   _ T 4% þ 2vq_ T 4% þ v2  1 qT 40000 þ0 vqT 40 yx  c3 vq i € T 400 ; þq (12)

where the elements of infinite matrices L, B1, B2 are determined by

B1ij

Z1

B2ij (13)



i2 p2 ; i ¼ j 0; isj # ( hð1Þiþj 1 ð1Þij 1 ; isj j þ ði; j ¼ 1; 2; .; NÞ ¼ iþj ij 0; i ¼ j # ( hð1Þiþj 1 ð1Þij 1 ; isj j2 d þ ði; j ¼ 1; 2; .; NÞ ¼ iþj ij 0; i ¼ j

Lij ¼

where dot denotes differentiation with respect to t and prime stands for differentiation with respect to x. If the weighting functions are also chosen 4(x), then the Galerkin method requires that the residual (12) should satisfy

Rðx; tÞ4ðxÞT dx ¼ 0TN ;

83

  €  2vB1 q_  v2  1 Lq  vB € _ 1 q þ ðc1 þ c2 þ c3 ÞLq q  2 €  4vv_ Lq_  2v_ 2 Lq  2v€ vLq þðk þ c3 ÞL q þ c2 z q  v2 Lq   €  4€ _ 2q _ 1q q 0 vB1 q  c3 p2 vB vB1 q_  2vB1 0 5vB  ð15Þ þ 2p2 vB2 q_ þ v2 L2 q ¼ 0;

0

(16)

where 0N is an infinite zero column matrix. Substituting Eq. (12) into Eq. (13) and transposing the resulting equation, one obtains the Galerkin discretization of the governing equation (9) for the axially moving Timoshenko beam on simple supports

# " N X ð1Þnþk 1 ð1Þnk 1 €n  2v q_ k k q þ nþk nk k ¼ 1;ksn # " N   X ð1Þnþk 1 ð1Þnk 1 2 2 2 qk  v  1 n p qn  v_ k þ nþk nk k ¼ 1;ksn  €n þ ðk þ c3 Þn4 p4 qn þ c2 z þ ðc1 þ c2 þ c3 Þn2 p2 q qn  €n  4vvn _ 2 p2 q_ n  2v_ 2 n2 p2 qn  2v€  v2 n2 p2 q vn2 p2 qn 8 # " < N X ð1Þnþk 1 ð1Þnk 1 €k q k  c2 5v_ þ : nþk nk k ¼ 1;ksn # " N X ð1Þnþk 1 ð1Þnk 1 q_ k þ 4€ v k þ nþk nk k ¼ 1;ksn # " N X ð1Þnþk 1 ð1Þnk 1 0 qk þ 2v k þ nþk nk k ¼ 1;ksn # 9 " = N X ð1Þnþk 1 ð1Þnk 1 qk þ0 v k þ ; nþk nk k ¼ 1;ksn 8 " # < N X ð1Þnþk 1 ð1Þnk 1 2 2 p qk  c3 v_ k d þ : nþk nk k ¼ 1;ksn # " N X ð1Þnþk 1 ð1Þnk 1 2 2 p q_ k þ 2v k d þ nþk nk k ¼ 1;ksn 9 = 2 4 4 þv n p qn ¼ 0; (14) ; where n ¼ 1,2,3,.N. Eqs. (14) can be cast into the matrix and vector form,

0 B A ¼ B @

0 0 0 ðkc3 þc3 v20 ÞL2 þðv20 1ÞL c2

I 0 0

2v0 ðB1 þp2 c3 B2 Þ

c2

0 I 0

Iþðc1 þc2 þc3 c2 v20 ÞL



c2

and the parameter d in Eq. (14) and so forth (16) is defined by

(

d¼

k=2; k is even; k þ 1=2; k is odd:

ðk ¼ 1; 2; /; NÞ:

(17)

Introduce new variable

0

1 q B q_ C C y ¼ B @q € A: 0 q

(18)

Consider the axially moving beam with constant speed v0. The terms involving derivatives of velocity in Eq. (15) can be neglected and after substitutive of Eq. (18), it may be transformed to the following concise form,

y_ ¼ Ay;

(19)

where In Eq. (20) 0 and I are infinite zero and unit matrix, respectively. The infinite matrix A describes the dynamics of transverse vibration of a Timoshenko beam moving at a constant speed. When the speed is lower than the critical speed, matrix A has only pure imaginary eigenvalues, iu1, iu2, and so on. In present investigation, the authors will consider subharmonic resonance and combination resonance of the first several eigenfrequencies. Therefore only first N terms are retained in the Galerkin expansion (10). This procedure is usually called Nth-term Galerkin truncation. In this case, the infinite matrices 0, I, L, B1, B2 become finite of N  N, and A reduces to the finite matrix of 4N  4N. By seeking eigenvalues of finite matrix A, we can find the first N natural frequencies of the system. Fig. 1 illustrates the changes of the first two natural frequencies with axially moving velocity for different stiffness k. With the increase of the axial velocity the natural frequencies decrease until they vanish, where the system loses its stability of the trivial solution. The dash lines denote the result by 2term Galerkin truncation, solid lines denote that by 4-term Galerkin truncation, and dot lines are that obtained by numerical method without truncation. For the first natural frequency, the 2-term and 4-term Galerkin truncations give almost the same results which are

1 0 0 C C I A: 2v0 B1

(20)

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Fig. 1. The natural frequencies of the axially moving Timoshenko beam with constant speed. (a) The first order natural frequency and (b) the second-order natural frequency.

accurate enough compared with that obtained by numerical method without truncation. However, the 2-term Galerkin method cannot predict the second natural frequency well, especially when axially moving speed is high. It can be proved that the higher order Galerkin truncation method must be used to obtain higher order natural frequencies. Also, higher values of stiffness k drive us to employ higher order truncation to satisfy the accuracy.

4. Averaging method We consider subharmonic resonance and combination resonance about the first two natural frequencies of the system. Therefore the first two terms are considered in the Galerkin expansion (10). It is assumed that the axially moving speed is characterized as a simple harmonic fluctuation about the constant mean speed, i.e.,

Fig. 2. Effect of mean axial speed v0 on the stability boundaries. (a) First subharmonic resonance, (b) second subharmonic resonance and (c) combination resonance.

X.-D. Yang et al. / European Journal of Mechanics A/Solids 29 (2010) 81–90

85

Fig. 3. Effect of k on the stability boundaries. (a) First subharmonic resonance, (b) second subharmonic resonance and (c) combination resonance.

v ¼ v0 þ 3v1 sinðUtÞ;

(21)

where bookkeeping device 3 is a small dimensionless parameter accounting for the fact that the harmonic fluctuation of the speed is very small. The fluctuation frequency is U and amplitude v1. Substituting Eq. (18) into (15) and substituting Eq. (21) into the result yield

y_ ¼ ðA þ v1 A1 sin ut þ uv1 A2 cos utÞy;

B A1 ¼ B @

2v0

0 0 0



ð1c2 u2 ÞLþc3 L2



0 0 0 2½ð12c2 u2 ÞB1 þc3 p2 B2 

c2

c2

0 0 0 2v0 L

0 1 0 C C 0 A; 2B1 (23)

0

0 0 B A2 ¼ B @ 0 2 ð1c2 u ÞB1 þc3 p2 B2

0 0 0 4v0 L

c2

0 0 0 5B1

1 0 0C C 0 A: 0

T AT ¼ U ¼



u1 J

(24)

0 0 0 1

1 0 0 0

;

(25)

1 0 0C C: 1A 0

(26)

Introduce new dependent variable vector and new time variable

x ¼ Ty;

s ¼ Ut:

(27)

The governing equation (22) is transformed into the form with respect with new variables

x_ ¼

1

Sx þ

U

v1

U

sin sCx þ v1 cos sDx;

(28)

where

C ¼ T 1 A1 T;

D ¼ T 1 A2 T:

(29)

We’ll apply the averaging method to Eq. (28) to study the stability caused by parametric resonance. Hence, the speed fluctuation frequency is assumed to be

U ¼ 2un þ 3D;

n ¼ 1; 2;

(30)

or

U ¼ u1 þ u2 þ 3D;



u2 J

0 B1 J ¼ B @0 0

S ¼ T 1 AT;

In the above equations, 0, L, B1 and B2 are 2  2 matrices determined by Eqs. (16). Thus there exists canonical transformation 8  8 matrix T such that

T

0

(22)

where the 8  8 matrix A is defined by Eq. (20) and A1 and A2 are defined

0

where

(31)

where 3 is a bookkeeping parameter to denote the small feature of the fluctuation frequency away from the resonance point.

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X.-D. Yang et al. / European Journal of Mechanics A/Solids 29 (2010) 81–90

Fig. 4. Effect of c1 on the stability boundaries. (a) First subharmonic resonance, (b) second subharmonic resonance and (c) combination resonance.

The elements of dependent variable x can cast into the amplitude-phase forms

x4ðn1Þþ1 ¼ an cos fn ;

gn2 ¼ an

x4ðn1Þþ2 ¼ an sin fn ;

x4ðn1Þþ3 ¼ an cos fn ;

x4ðn1Þþ4 ¼ an sin fn ;

(33)

  a_ n ¼ 3 gn1 cos fn þ gn2 sin fn ; (34) ðn ¼ 1; 2Þ;

where

gn1 ¼ an

v1

U

þan v1 cos s

sin s

4  X

11 12 Cnm cos fm þ Cnm sin fm



m¼1 4  X m¼1

 D 12 D11 an sin fn ; nm cos fm þ Dnm sin fm þ

un

21 22 Cnm cos fm þ Cnm sin fm



m¼1 4  X

22 D21 nm cos fm þ Dnm sin fm



m¼1

depending on the type of resonances, i.e., the first or the second subharmonic resonances or the combination resonance. Substituting Eq. (32) into Eq. (28) and neglecting the other elements of x, yield

  an q_ n ¼ 3 gn2 cos fn  gn1 sin fn ;

4  X

sin s

D  an cos fn : un

where

un u un s þ qn ; fn ¼ n s þ qn ; or fn ¼ s þ qn u1 þ u2 2 u1 2 u2

U

þ an v1 cos s ð32Þ

fn ¼

v1

(35)

In Eq. (35) Cij and Dij (i,j ¼ 1,2) denote the sub-matrix which are defined by

C ¼



C 11 C 22

 C 12 ; C 22

D ¼



D11 D22

 D12 ; D22

(36)

and their subscripts presents the rows and columns in each 4  4 sub-matrix. From Eq. (34), we can conclude that the fluctuation of amplitudes and phases are very small. Hence the averaging method can be used (Bogoliubov and Mitropolsky, 1961) in this kind of system. Ariaratnam and Namchchivaya (1986), Asokanthan and Ariaratnam (1994) and Chen et al. (2004a) have applied the method of averaging to study axially moving materials. When the fluctuation frequency lies in the neighborhoods of 2un, the principle parametric resonance may arise. In that case the averaged equations take the form

X.-D. Yang et al. / European Journal of Mechanics A/Solids 29 (2010) 81–90

a_ n ¼ 3ðU n v1 cos 2qn þ Vn v1 sin 2qn Þan ;  an q_ n ¼ 3 Vn v1 cos 2qn  Un v1 sin 2qn  uDn an ;

(37)

l2 þ



D un

87

2    Un2 þ Vn2 v21 ¼ 0;

(41)

with the roots

where

!

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2   D l ¼  Un2 þ Vn2 v21  :

12 þ C 21 1 Cnn 22 nn ; Un ¼ D11 nn  Dnn þ 2un 4 ! 11 þ C 22 1 Cnn 21 nn D12 : Vn ¼ nn þ Dnn þ 2un 4

2

(38)

To study the stability of Eqs. (37), two new variables are introduced:

 Un ; Vn   1 U zn ¼ an sin qn þ tan1 n ; Vn 2 

(42)

un

1 2

xn ¼ an cos qn þ tan1

ðn ¼ 1; 2Þ:

(39)

If the roots has at least one root with positive real part, the solution is unstable. Otherwise, the parametric resonance does not happen. The stability boundaries in the neighborhoods of 2un, can be located in the (u, v1) plane. If the speed fluctuation frequency is near enough to the double values of first or second natural frequency, the instability caused by subharmonic resonance will occur. In the case of combination resonance of summation type, one can get the averaged equations

Therefore, Eqs. (37) are expressed in the new variables as

a_ n ¼ 3ðUnm v1 cos 2qn þ Vnm v1 sin 2qn Þan ;

 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi D x_ n ¼ zn ; Un2 þ Vn2 v1 þ

  D an ; an q_ n ¼ 3 Vnm v1 cos 2qn  Unm v1 sin 2qn 

un

 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi D z_ n ¼ xn : Un2 þ Vn2 v1 

un

(40) The stability of the system can be determined by the roots of the characteristic equation corresponding to the linear ordinary differential equation (40). Now we study the characteristic equation

un

where

Unm ¼

! 12 þ C 21 1 Cnm 22 nm D11 ;  D þ nm nm un þ um 4

Fig. 5. Effect of c2 on the stability boundaries. (a) First subharmonic resonance, (b) second subharmonic resonance and (c) combination resonance.

(43)

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Fig. 6. Effect of c3 on the stability boundaries. (a) First subharmonic resonance, (b) second subharmonic resonance and (c) combination resonance.

Vnm

! 11 þ C 22 1 Cnm 12 21 nm Dnm þ Dnm þ ; ¼ un þ um 4

Introduce new variables

(44) m; n ¼ 1; 2:

2n ¼ an cos qn þ ian sin qn ; 2m ¼ am cos qm  iam sin qm : Eq. (43) can be expressed in the new variables as

Fig. 7. The contribution of shear modulus to the natural frequencies. (a) The first order natural frequency and (b) the second-order natural frequency.

(45)

X.-D. Yang et al. / European Journal of Mechanics A/Solids 29 (2010) 81–90

2_ n ¼ iuDn 2n þ v1 ðUnm þ iVnm Þ2m ; 2_ m ¼ v1 ðUmn  iVmn Þ2n þ iuDm 2m :

6. Conclusions

(46)

The roots of the characteristic equation corresponding to linear ordinary differential equation (46) are determined by



l2  iD



1



1

un um



lþ

89

D2  ðUnm þ iVnm Þ ðUmn  iVmn Þv21 ¼ 0: un um (47)

The stability boundaries of combination resonance of summation type near the neighborhood of u1 þ u2 can be located in the (u, v1) plane by Eq. (47). Similarly, the analysis for the combination resonance of difference type can be done by the averaging method.

This paper treats dynamic stability of an axially accelerating Timoshenko beam. The natural frequencies of the system with constant axial speed are presented by the Galerkin method. The comparison of results by Timoshenko model and Euler model is presented. Then, the method of averaging is applied to the 2-term Galerkin truncation of the governing equation. The triangle instability regions in the subharmonic resonance and the summation resonance are detected in the (u, v1) plane. The instability range of the axial speed fluctuation frequency increases with the growth the axial speed fluctuation amplitude. The increase of the mean axial speed and the decrease of the stiffness or rotary inertia lead to the increase of instability threshold of the axial speed fluctuation frequency. The contributions of parameters corresponding to the shear distortion are also discussed in the (u, v1) plane.

5. Numerical results The procedure in the previous section can be used to present the stability boundaries in the (u, v1) plane for a set of given parameters. We can determine the instability regions in the subharmonic resonance and combination resonance of summation type, while the combination resonance of difference type cannot be detected. The numerical results show the contributions of some parameters to the instability regions. Fig. 2 illustrate the contribution of dimensionless mean velocity v0 to the instability boundaries in the (u, v1) map. The triangle regions enclosed by the boundary lines denote the area where the system loses its stability under such conditions. Outside regions are stable where no resonance happens. The instability range of speed fluctuation frequency u increases with the growth of speed fluctuation amplitude v1. In Fig. 2, solid lines represent v0 ¼ 1.990, dash–dot lines represent v0 ¼ 2.000, and dash lines are the results of v0 ¼ 2.010 where other parameters are fixed. The comparison of different mean velocity indicates that, with the increase of mean axial speed, the instability regions drift towards the direction of the increasing u in both subharmonic and combination resonances. This conclusion can be checked by the effect of axial speed on the natural frequencies of the system discussed in the former section, which is illustrated by Fig. 1. The effect of dimensionless stiffness is demonstrated in Fig. 3, in which k ¼ 0.799, 0.800, 0.801 are respectively depicted by solid, dash–dot and dash lines. It can be found that the increasing stiffness decreases the threshold of fluctuation frequency to lose stability. This is because the stiffness has changed the natural frequencies of the system as illustrated by Fig. 1. The effect of rotary inertia on the stability boundaries is shown in Fig. 4, in which c1 ¼ 4.200  103, 4.300  103, 4.400  103 are denoted respectively by solid, dash–dot and dash lines. Comparing Fig. 4 with Fig. 3, one finds that the effect of the rotary inertia is similar to that of the mean axial speed. The effects of and c2 are c3 examined in Figs. 5 and 6 respectively. In Fig. 5, c2 ¼ 5.9  105, 6.9  105, 7.9  105 are denoted respectively by solid, dash–dot and dash lines and in Fig. 6, c3 ¼ 0.009, 0.0095, 0.010 are represented respectively by solid, dash–dot and dash lines. Although both c2 and c3 are related to shear distortion according to Eq. (6), they play different roles in the contribution to the stability boundaries in the (u, v1) plane. The increase of c2 leads to expansion of the instability regions and little drift towards the direction of the increasing u, while the increase of c3 does not change the instability regions perceptible but causes the instability regions drift towards the direction of the decreasing u. Actually, as illustrated in Fig. 7, with the increase of the shear modulus, the natural frequencies increase dramatically.

Acknowledgement This investigation is supported by the National Outstanding Young Scientists Fund of China (Project No. 10725209), the National Natural Science Foundation of China (Project No. 90816001, No. 10702045), Shanghai Leading Talents Program, Shanghai Subject Chief Scientist Program (Project No. 09XD1401700), Shanghai Leading Academic Discipline Project (No. Y0103) and the Science Foundation of Education Department of Liaoning Province (No. 2009A572).

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