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Forced transverse vibrations of an elastically connected complex simply supported double-beam system

SOUND AND

VIBRATION

Journal of Sound and Vibration264(2003)273–286

https://www.sodocs.net/doc/bc15461846.html,/locate/jsvi

Z.Oniszczuk

Faculty of Mechanical Engineering and Aeronautics,Rzesz!ow University of Technology,ul.W.Pola2,35-959Rzesz!ow,

Poland

Received2January2002;accepted27May2002

Abstract

The present paper is devoted to analyzing undamped forced transverse vibrations of an elastically connected complex double-beam system.The problem is formulated and solved in the case of simply supported beams.The classical modal expansion method is applied to ascertain dynamic responses of beams due to arbitrarily distributed continuous loads.Several cases of particularly interesting excitation loadings are investigated.The action of stationary harmonic loads and moving forces is considered.In discussing vibrations caused by exciting harmonic forces,conditions of resonance and dynamic vibration absorption are determined.The beam-type dynamic absorber is a new concept of a continuous dynamic vibration absorber(CDVA),which can be applied to suppress excessive vibrations of corresponding beam systems.A numerical example is presented to illustrate the theoretical analysis.

1.Introduction

Beam-type structures are widely used in many branches of modern civil,mechanical,and aerospace engineering.Therefore,the dynamics of single beam and beam systems is still a subject of great interest to many investigators.The vibration theory for simple one-dimensional continuous systems as beams and strings is developed in a number of monographs by e.g.,Ziemba [1],Solecki and Szymkiewicz[2],Kaliski[3],Snowdon[4],Fryba[5],Nowacki[6],Timoshenko et al.[7],Craig[8],and Rao[9],among others.As is well known,there are four fundamental models for transversely vibrating beam.These are the Bernoulli–Euler,Rayleigh,shear(Fl.ugge), and Timoshenko models[1–3,7,10].In Ref.[11],the author has discussed free transverse vibrations of two simply supported Bernoulli–Euler beams connected by a Winkler elastic layer. The present paper deals with forced vibrations of this complex beam system.Different problems concerning forced responses of an elastically connected double-beam system(or sandwich beam)

doi:10.1016/S0022-460X(02)01166-5

are considered by many scientists:Dublin and Friedrich [12],Seelig and Hoppmann II [13],Kessel

[14],Kessel and Raske [15],Kozlov [16],Saito and Chonan [17,18],Lu and Douglas [19],Oniszczuk [20,24,35,40–42],Chonan [21,22],Jacquot and Foster [23],Douglas and Yang [25],Irie et al.[26],Dmitriev [27],Hamada et al.[28,29],Yamaguchi [30],Vu [31],Kokhmanyuk [32],

Aida et al.[33],Chen and Sheu [34],Chen and Lin [36],Szcze !s

niak [37],Kawazoe et al.[38],and Vu et al.[39].The analysis of special cases of forced harmonic vibrations allows one to formulate conditions which generate the appearance of a dynamic vibration absorption in a double-beam system.This very interesting phenomenon is of great technical importance and therefore a beam is often applied as a continuous dynamic vibration absorber (CDVA).Theories of beam-type absorbers are presented in a few works by Kessel and Raske [15],Jacquot and Foster [23],Hamada et al.[28,29],Yamaguchi [30],Vu [31],Aida et al.[33],Chen and Sheu [34],Oniszczuk

[35,40,41],Chen and Lin [36],Kawazoe et al.[38],and Vu et al.[39].

In the present paper being an extension of work described in reference [11],exact theoretical general solutions of undamped forced vibrations for a simply supported double-beam system are determined.Then,several cases of particular excitation loadings are studied.The forced vibration analysis of a similar system of elastically connected two strings presented by the author in Refs.

[41,43,44]can be helpful in the investigation of the title system because of the same boundary conditions and the same mathematical procedures applied.The general vibration analysis of an elastically connected double-beam system is complicated and laborious in view of a large variety of possible combinations of boundary conditions [24,35].Vibrations of a general system of two beams governed by arbitrary boundary conditions which are four fundamental homogeneous ones,will be discussed in a future publication.

2.Formulation of the problem

The scheme of vibratory system considered is depicted in Fig.1.An elastically connected double-beam system consists of two parallel beams of the same length,which are joined by an elastic layer modelled as a Winkler massless foundation.Both beams are homogeneous,prismatic and slender,what makes it possible to apply the classical Bernoulli–Euler beam theory in deriving the equations of system motion.For simplicity of the analysis,all four ends of beams are assumed to be simply supported.The beams are subjected to arbitrarily distributed transverse continuous loads.Small undamped vibrations of the system are discussed.

Fig.1.The dynamic model of an elastically connected complex simply supported double-beam system.

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274

The transverse vibrations of a generally loaded double-beam system are governed by the following differential equations[13–15,20,24,29,33,35,40,41],based on the Bernoulli–Euler theory:

K1w iv

tm1.w1tkew1àw2T?f1ex;tT;

K2w iv

tm2.w2tkew2àw2T?f2ex;tT;e1Twhere w i?w iex;tTis the transverse beam de?ection;f i?f iex;tTis the exciting distributed continuous load;x;t are the spatial co-ordinate and the time,respectively;E i is Young’s modulus of elasticity;F i is the cross-sectional area of the beam;J i is the moment of inertia of the beam cross-section;K i is the?exural rigidity of the beam;h;k are the thickness and the stiffness modulus of a Winkler elastic layer,respectively;l is the length of the beam;r i is the mass density;

K i?E i J i;m i?r i F i;’w i?@w i=@t;w0

?@w i=@x;i?1;2:

The boundary conditions for simply supported beams are:

w ie0;tT?w00

i e0;tT?w iel;tT?w00

el;tT?0;i?1;2:e2T

3.Solution of the problem

The forced vibrations of beams subjected to arbitrarily distributed continuous loads are determined by applying the classical modal expansion method[24,35,40,41].Particular solutions of non-homogeneous differential equations(1)representing forced vibrations of a double-beam system are assumed in the following form[24,35,40,41]:

w1ex;tT?

X N

n?1X2

i?1

X1inexTS inetT?

X N

n?1

X nexT

i?1

S inetT?

X N

n?1

sinek n xT

i?1

S inetT;

w2ex;tT?

X N

n?1X2

i?1

X2inexTS inetT?

X N

n?1

X nexT

i?1

a in S inetT?

X N

n?1

sinek n xT

i?1

a in S inetT;e3T

where

a in?eK1k4

n tkàm1o2

Tkà1?keK2k4

tkàm2o2

Tà1?Oà2

eO2

11n

ào2

?O2

20eO2

22n

ào2

Tà1;

a1n a2n?àm1mà1

2?àM1Mà1

;a1n>0;a2n o0;e4T

i?1;2;K?kl;k n?là1n p;M i?m i l?r i F i l;n?1;2;3;y O2iin?eK i k4ntkTmà1i??K i là3en pT4tK Mà1i;O2i0?kmà1i?KMà1i;

O4 120?O2

?k2em1m2Tà1?K2eM1M2Tà1;

o21;2n?0:5feO211ntO222nT8?eO211nàO222nT2t4O4120 1=2g;o1n o o2n;e5TZ.Oniszczuk/Journal of Sound and Vibration264(2003)273–286275

o2 1;2n ?0:5f?eK1k4

tkTmà1

teK2k4

tkTmà1

8e?K1k4

tkTmà1

teK2k4

tkTmà1

à4k4

em1m2Tà1?K1K2k4

tkeK1tK2T T1=2g;

X1inexT?X nexT;X2inexT?a in X nexT;X nexT?sinek n xT;e6TX nexTis the known mode shape function for a simply supported single beam and S inetTare the unknown time functions corresponding to the natural frequencies o in[11].All quantities mentioned above are de?ned in Ref.[11],in which the free vibration problem of the title system is considered.

Substituting the assumed solutions(3)into system(1)results in the relationships

X N n?1X n

i?1

?.S inteO211nàO210a inTS in ?mà11f1;

X N n?1X n

i?1

?.S inteO222nàO220aà1inTS in a in?mà12f2:

Taking expressions(4)into consideration gives

X N n?1X n

i?1

e.S into2in S inT?mà11f1;

X N

n?1

X n

i?1

e.S into2in S inTa in?mà12f2:

Multiplying the above relationships by the eigenfunction X m then integrating them with respect to x from0to l and applying the classical orthogonality condition[11]

Z l 0X m X n d x?

Z l

sinek m xTsinek n xTd x?c d mn;

c?c2

n ?

Z l

d x?

Z l

sin2ek n xTd x?0:5l;e7T

where d mn is the Kronecker delta function:d mn?0for m a n;and d mn?1for m?n;one gets a set of equations

X2 i?1e.S into2in S inT?2Mà11

Z l

f1X n d x;

i?1

e.S into2in S inTa in?2Mà12

Z l

f2X n d x

from which after some manipulation the following two in?nite sequences of ordinary differential equations for the unknown time functions are obtained:

.S in to2

S in?H inetT;i?1;2;n?1;2;3;y;e8T

where

H1netT?d1n

Z l

0?a2n Mà1

f1ex;tTàMà1

f2ex;tT sinek n xTd x;

H2netT?d2n

Z l

0?a1n Mà1

f1ex;tTàMà1

f2ex;tT sinek n xTd x;e9T

d1n?àd2n?2ea2nàa1nTà1?2O2

10eo2

ào2

Tà1:

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Particular solutions of Eq.(8)satisfying homogeneous initial conditions are [24,35,40–42]S in et T?o à1in Z l 0

H in es Tsin ?o in et às T d s ;i ?1;2:e10TFinally,the forced vibrations of an elastically connected simply supported double-beam system are found to be in the form

w 1ex ;t T?X N n ?1sin ek n x TX 2i ?1o à1in

Z t 0

H in es Tsin ?o in et às T d s ;w 2ex ;t T?X

N n ?1sin ek n x TX 2i ?1a in o à1in Z t 0H in es Tsin ?o in et às T d s :e11T

Solutions (11)are suf?ciently versatile to allow determination of the dynamic response of this system due to an arbitrary exciting transversal loading as well as stationary loads and moving forces.For simplicity of further analysis and discussion it is assumed that the exciting load acts only on the ?rst beam (see Fig.2),whilst the other one is not loaded;i.e.,f 1ex ;t Ta 0;f 2ex ;t T?0:Thus the time functions (9)take the simpler form H in et T?b in Z l

f 1ex ;t Tsin ek n x Td x ;i ?1;2;e12T

where b 1n ?a 2n d 1n M à11?2a 2n ?ea 2n àa 1n TM 1 à1;b 2n ?a 1n d 2n M à11?2a 1n ?ea 1n àa 2n TM 1 à1:

If the exciting load is a harmonic function of time f 1ex ;t T?f ex Tsin ept T;then one gets H in et T?b in sin ept TZ l

0f ex Tsin ek n x Td x ;i ?1;2;

e13T

where f ex Tis the arbitrary function of spatial co-ordinate x ;and p is the frequency of the exciting harmonic load.

Next relationships (10)can be transformed to the form S in et T?b in o à1in

Z l 0f ex Tsin ek n x Td x Z t 0sin eps Tsin ?o in et às T d s ?b in F n eo 2in àp 2Tà1?sin ept Tàp o à1in sin eo in t T ;e14Twhere F n ?R l 0f ex Tsin ek n x Td x ;i ?1;2:

Fig.2.An elastically connected complex double-beam system subjected to distributed continuous harmonic load.

Z.Oniszczuk /Journal of Sound and Vibration 264(2003)273–286277

A detailed forced vibration analysis is now performed for two interesting cases of the exciting harmonic loadings acting stationarily:uniform distributed continuous load and concentrated force.The classical case of a moving concentrated force is also discussed.

Case1:Stationary exciting harmonic loads.To begin with the general case of distributed continuous harmonic load is considered.The?rst beam is subjected to arbitrarily distributed load f1ex;tT?fexTsineptTacting on the entire length of the beam as is shown in Fig.2.

The forced vibrations of the beam system are determined from the general solutions(11)by using relations(13)and(14)

w1ex;tT?

X N

n?1sinek n xTA1n sineptTt

i?1

B in sineo in tT

;

w2ex;tT?

X N

n?1sinek n xTA2n sineptTt

i?1

a in B in sineo in tT

;e15T

where

A1n?2F n Mà1

1eO2

22n

àp2T?eo2

àp2Teo2

àp2T à1;

A2n?2F n Kà1O4120?eo21nàp2Teo22nàp2T à1;

B1n?2a2n F n Mà1

1p?ea1nàa2nTo1neo2

àp2T à1;

B2n?2a1n F n Mà1

1p?ea2nàa1nTo2neo2

àp2T à1;

F n?

Z l

fexTsinek n xTd x;k n?là1n p;i?1;2:e16TThe solutions obtained using Eq.(15)are composed of two parts.The?rst part being a function of sineptTdenotes the steady state(pure)forced vibrations of the system,and the other one containing the terms sineo in tTrepresents the free vibration produced by the application of exciting loading.Neglecting the free response,and assuming that only the steady state response has a practical signi?cance,the forced vibrations of an elastically connected double-beam system are found to be in the form

w1ex;tT?sineptT

X N

n?1A1n sinek n xT;w2ex;tT?sineptT

X N

n?1

A2n sinek n xT:e17T

Discussing the steady state vibration amplitudes A1n;A2n(16)a number of interesting and important conclusions may be drawn.This analysis leads to the fundamental conditions

(a)Condition of resonance:p?o in;i?1;2;n?1;2;3;y

(b)Condition of dynamic variation absorption:

p2?p2

n ?O2

22n

?eK2k4

tkTmà1

??K2là3en pT4tK Mà1

;

A1n?0;A2n?à2F n Kà1?à2Kà1

Z l

0fexTsinek n xTd x:e18T

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With the application of harmonic forces,a dynamic vibration absorption phenomenon occurs and the second beam acts as a dynamic vibration absorber in relation to the?rst one.Relationship

(18)is the basic condition of a dynamic vibration absorption which can be used to optimal design

a complex system of two beams.Optimum values of tuning parameters of a dynamic absorber are found by a proper choice of the elastic layer stiffness modulus k;?exural rigidity K2?E2J2and mass of the second beam M2?m2l?r2F2l:The dynamic absorption eliminates any selected harmonic component A1n of the?rst beam vibrations.The dynamical damper reduces forced vibrations of the?rst beam but never eliminates them completely[35,41].

Case1.1:Uniform distributed harmonic load.The uniform distributed harmonic continuous load f1ex;tT?f sineptTacts on the?rst beam(see Fig.3).f and p are the amplitude and exciting frequency of the load,respectively.

After performing calculations on the basis of solutions(11),(13)and(14),the forced vibrations are received in the form

w1ex;tT?

enTsinek n xTA1n sineptTt

i?1

B in sineo in tT

;n?1;3;5;y;

w2ex;tT?

enTsinek n xTA2n sineptTt

i?1

a in B in sineo in tT

;e19T

where

A1n?4FeM1n pTà1eO2

22n àp2T?eo2

àp2Teo2

àp2T à1;

A2n?4FeKn pTà1O4

120?eo2

àp2Teo2

àp2T à1;

B1n?4a2n FeM1n pTà1p?ea1nàa2nTo1neo2

àp2T à1;

B2n?4a1n FeM1n pTà1p?ea2nàa1nTo2neo2

àp2T à1;

F?fl;i?1;2;k n?là1n p;n?1;3;5;y:e20TOmitting the terms of free response,the steady state forced vibrations of beams are found to be w1ex;tT?sineptT

enT

A1n sinek n xT;

w2ex;tT?sineptT

enT

A2n sinek n xT;n?1;3;5;y:e21T

Fig.3.An elastically connected complex double-beam system subjected to uniform distributed harmonic continuous load.

Z.Oniszczuk/Journal of Sound and Vibration264(2003)273–286279

The analysis of the steady state vibration amplitudes A in (20)leads to the fundamental conditions (a)Condition of resonance :p ?o in ;i ?1;2;n ?1;3;5;y :

(b)Condition of dynamic vibration absorption :

p 2?p 2n ?O 222n ?eK 2k 4n tk Tm à12??K 2l à3en p T4tK M à12;

A 1n ?0;A 2n ?à4F eKn p Tà1;n ?1;3;5;y :e22T

Case 1.2:Concentrated harmonic force .The ?rst beam is subjected to the concentrated harmonic force f 1ex ;t T?F et Td ex à0:5l T?F sin ept Td ex à0:5l Tapplied for simplicity at the midspan of the beam (see Fig.4).F and p are the amplitude and frequency of the exciting harmonic force,respectively,and d ex Tis the Dirac delta function.

For this case the forced vibrations of a two-beam system are received in analogous form as (19)w 1ex ;t T?

X en T

sin ek n x TA 1n sin ept TtX 2i ?1B in sin eo in t T"#;n ?1;3;5;y w 2ex ;t T?

X en Tsin ek n x TA 2n sin ept TtX 2i ?1a in B in sin eo in t T"#;e23T

where A 1n ?2b n FM à11eO 222n àp 2T?eo 21n àp 2Teo 22n àp 2T à1;

A 2n ?2b n FK à1O 4120?eo 21n àp 2Teo 22n àp 2T à1;

e24T

B 1n ?2a 2n b n FM à11p ?ea 1n àa 2n To 1n eo 21n àp 2T à1;

B 2n ?2a 1n b n FM à11p ?ea 2n àa 1n To 2n eo 22n àp 2T à1;b n ?sin e0:5n p T?eà1T0:5en à1T;n ?1;3;5;y :

The steady state forced vibrations of the system are (21)w 1ex ;t T?sin ept T

X en TA 1n sin ek n x T;w 2ex ;t T?sin ept TX

en TA 2n sin ek n x T;n ?1;3;5;y :e25T

Fig.4.An elastically connected complex double-beam system subjected to concentrated harmonic force.

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280

The analysis of the steady state vibration amplitudes A in(24)leads to the fundamental conditions

(a)Condition of resonance:p?o in;i?1;2;n?1;3;5;y;

(b)Condition of dynamic vibration absorption:

p2?p2

n ?O2

22n

?eK2k4

tkTmà1

??K2là3en pT4tK Mà1

;

A1n?0;A2n?à2b n FKà1;n?1;3;5;y:e26TCase2:Moving concentrated forces.First,the general case of a moving concentrated force arbitrarily varying in time is presented.The?rst beam is traversed by a point force which moves with a constant velocity v along a beam from the left supportex?0Tto the right supportex?lT(see Fig.5).The exciting loading of a double-beam system is f1ex;tT?FetTdexàvtT;f2ex;tT?0;0o t o T;T?lvà1;where FetTis the concentrated force being an arbitrary function of time;v is the constant velocity of a moving force;dexTis the Dirac delta function;T is the time of load traverse over the beam.

On the basis of relations(10)and(12),the time functions(10)for a moving concentrated force

take the form

S inetT?b in oà1

in Z t

FesTsinep n sTsin?o inetàsT d s;e27T

where

b1n?2a2n?ea2nàa1nTM1 à1;b2n?2a1n?ea1nàa2nTM1 à1;

p n?k n v?là1n p v?n p Tà1;i?1;2;n?1;2;3;y:

Case2.1:Moving constant force.The classical problem of a moving constant concentrated force FetT?F at constant velocity[1,5–7,27,35,37]is discussed.The exciting loading of a double-beam system is f1ex;tT?F dexàvtT;f2ex;tT?0;0o t o T;T?lvà1;where F is the magnitude of a constant force.After calculation of the time functions(27),the forced vibrations of beams(11)are described by the expressions

w1ex;tT?

X N

n?1sinek n xTA1n sinep n tTt

i?1

B in sineo in tT

;

w2ex;tT?

X N

n?1sinek n xTA2n sinep n tTt

i?1

a in B in sineo in tT

;e28T

Fig.5.An elastically connected complex double-beam system subjected to a moving concentrated force.

Z.Oniszczuk/Journal of Sound and Vibration264(2003)273–286281

where

A 1n ?2FM à11eO 222n àp 2n T?eo 21n àp 2n Teo 22n àp 2n T à1;

A 2n ?2FK à1O 4120?eo 21n àp 2n Teo 22n àp 2n T à1;

e29T

B 1n ?2a 2n FM à11p n ?ea 1n àa 2n To 1n eo 21n àp 2n T à1;

B 2n ?2a 1n FM à11p n ?ea 2n àa 1n To 2n eo 22n àp 2n T à1;p n ?k n v ?l à1n p v ?n p T à1;T ?lv à1;0o t o T :

Case 2.2:Moving harmonic force.The second interesting problem of a moving concentrated force is the uniform motion of a harmonic force F et T?F sin ept T[1,5,7,36,43].The exciting loading of a double-beam system is now the following:f 1ex ;t T?F sin ept Td ex àvt T;f 2ex ;t T?0;0o t o T ;T ?lv à1;where F and p are the amplitude and frequency of the exciting harmonic force,respectively.

The forced vibrations of a two-beam system are expressed by the relations w 1ex ;t T?

X N n ?1

sin ek n x TA 1n sin ep n t Tsin ept TtB 1n cos ep n t Tcos ept TtX 2i ?1C in sin eo in t T"#;w 2ex ;t T?

X N n ?1sin ek n x TA 2n sin ep n t Tsin ept TtB 2n cos ep n t Tcos ept TtX 2i ?1a in C in sin eo in t T"#;e30T

where A 1n ?2FM à11

ea 1n m 1n n 1n u 2n àa 2n m 2n n 2n u 1n T?ea 1n àa 2n Tm 1n m 2n n 1n n 2n à1;A 2n ?2FM à12

em 1n n 1n u 2n àm 2n n 2n u 1n T?ea 2n àa 1n Tm 1n m 2n n 1n n 2n à1;B 1n ?4FM à11pp n ea 1n m 1n n 1n àa 2n m 2n n 2n T?ea 2n àa 1n Tm 1n m 2n n 1n n 2n à1;

B 2n ?4FM à12

pp n em 1n n 1n àm 2n n 2n T?ea 1n àa 2n Tm 1n m 2n n 1n n 2n à1;C 1n ?4a 2n FM à11

m 2n n 2n pp n ?ea 2n àa 1n Tm 1n m 2n n 1n n 2n à1;C 2n ?4a 1n FM à12

m 1n n 1n pp n ?ea 1n àa 2n Tm 1n m 2n n 1n n 2n à1;m in ?o 2in àep n àp T2;n in ?o 2in àep n tp T2;u in ?o 2in àp 2n àp 2;

i ?1;2;n ?1;2;3;y ;p n ?k n v ?l à1n p v ?n p T à1;T ?lv à1;0o t o T :

4.Numerical example

To illustrate theoretical considerations presented,the forced vibrations of a double-beam system depicted in Fig.3due to an harmonic uniformly distributed load (see Case 1.1)are discussed in detail.The exciting loading of the system is f 1ex ;t T?f sin ept T;f 2ex ;t T?0;where f and p are the amplitude and frequency of exciting harmonic load.

For simplicity of the analysis,it is assumed that both beams are geometrically and physically identical,then the values of the parameters from reference [11]can be used in the numerical

Z.Oniszczuk /Journal of Sound and Vibration 264(2003)273–286

282

calculations:

E ?E i ?1?1010N m à2;

F ?F i ?5?10à2m 2;J ?J i ?4?10à4m 4;i ?1;2;K 0?K i ?E i J i ?4?106N m 2;K ?kl ;k ?2?105N m à2;l ?10m ;M ?M i ?m i l ?1?103kg ;m ?m i ?r i F i ?1?102kg m à1;r ?r i ?2?103kg m à3:The steady state harmonic responses of the system are described by relations (21)w 1ex ;t T?sin ept T

X en TA 1n sin ek n x T;w 2ex ;t T?sin ept TX en T

A 2n sin ek n x T;n ?1;3;5;y ;where

A 1n ?4F eMn p Tà1eO 222n àp 2T?eo 21n àp 2Teo 22n àp 2T à1;

A 2n ?4F eKn p Tà1O 4120?eo 21n àp 2Teo 22n àp 2T à1;

O 222n ?eK 0k 4n tk Tm

à1??K 0l à3en p T4tK M à1?0:5eo 21n to 22n T;O 4120?k 2m à2?K 2M à2;o 21n ?K 0k 4n m à1;o 22n ?eK 0k 4n t2k Tm à1;

F ?fl ;i ?1;2;K ?kl ;k n ?l à1n p ;n ?1;3;5;y :

These solutions are expressed only by the symmetric mode shapes of vibration [11]because of the symmetry of the applied load.Performing calculations of component amplitudes A 1n and A 2n versus the exciting frequency p ;the resonant diagram can be built.The resonant diagram shown in Fig.6characterizes the progress of steady state forced harmonic vibrations of the system,and

Fig.6.The resonant diagram of the steady state forced harmonic vibrations of an elastically connected complex double-beam system subjected to uniform distributed harmonic load.

Z.Oniszczuk /Journal of Sound and Vibration 264(2003)273–286283

comprises the ?rst two resonance curves.The full lines 11,13represent the amplitudes of the ?rst beam vibration components A 11;A 13;and the broken lines 21,23describe the amplitudes of the second beam vibration components A 21;A 23:The resonances take place when the excitation frequency of harmonic load is equal to the one of the natural frequencies of the system p ?o 11;o 21;o 13;o 23;then the corresponding amplitudes A 11;A 21;A 13;A 23tend to in?nity.The frequencies p 1;p 3denote the tuned exciting frequencies for which the dynamic vibration absorption is realized,and the amplitudes A 11;A 13are suppressed.These frequencies are evaluated from the condition of dynamic vibration absorption (22)

p 2n ?O 222n ??K 0l à3en p T4tK M à1?0:5eo 21n to 22n T;

which leads to the beam amplitudes

A 1n ?0;A 2n ?à4F eKn p Tà1;n ?1;3;5;y :

The dynamic absorption phenomenon is of great practical importance and can be applied to reduce forced harmonic vibrations of elastically connected double-beam systems.

5.Conclusions

The undamped forced transverse vibrations of an elastically connected complex simply supported double-beam system have been considered.The modal expansion method is applied to receive the dynamic response of beams caused by arbitrarily distributed continuous loads.General solutions obtained are used to determine forced vibrations for several interesting cases of stationary exciting harmonic loadings and moving concentrated forces.Analyzing responses due to exciting harmonic forces the conditions of resonance and dynamic vibration absorption are formulated.Tuning parameters found can be employed to optimum design of a dynamic absorber of the beam type.The beam-type dynamic absorber is a new concept of a continuous dynamic vibration absorber (CDVA),which can be used to suppress excessive vibrations of corresponding beam systems.It is well known that DVAs are of great practical importance in engineering applications [3,4,7,9,45–47],and among them CDVAs play considerable role.Many recent studies have been devoted to beam-type [15,23,28–31,33–36,38–41],string-type [35,41,43],membrane-type [35,48],plate-type [35,49],and shell-type [50]continuous dynamic vibration absorbers.

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