Dynamic response of a double Euler Bernoulli beam due to a moving constant load

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1 Journal of Sound and Vibration 297 (26) JOURNAL OF SOUND AND VIBRATION Dynamic repone of a double Euler Bernoulli beam due to a moving contant load M. Abu-Hilal Department of Mechanical and Indutrial Engineering, Applied Science Private Univerity, Amman 93, Jordan Received 3 March 25; received in revied form 9 January 26; accepted 6 March 26 Available online 27 June 26 Abtract In thi paper, the dynamic repone of a double-beam ytem travered by a contant moving load i tudied. The ytem conit of two elatic homogeneou iotropic Euler Bernoulli beam. The two imply upported primatic beam are identical, parallel one upon the other and connected continuouly by a vicoelatic layer. The dynamic deflection of both beam are given in analytical cloed form. The effect of the moving peed of the load and the damping and the elaticity of the vicoelatic layer on the dynamic repone of the beam are invetigated in detail. Alo everal plot of the deflection of the beam are given and dicued for different value of peed parameter, damping ratio, and tiffne parameter. Furthermore, the force tranmitted between the beam i determined and the effect of the ytem parameter on thi force are tudied. Plot of the maximum repone of the beam are preented. r 26 Elevier Ltd. All right reerved.. Introduction Beam are very important element in civil, mechanical, and aeronautical engineering. The vibration problem of ingle beam i very good developed and explored in detail in hundred of contribution. On the other hand, there are only few contribution dealing with the vibration of double-beam ytem, becaue of the difficulty in olving the governing coupled partial differential equation. Onizczuk [] tudied the free vibration of two parallel imply upported beam continuouly joined by a Winkler elatic layer. The eigenfrequencie and mode hape of vibration of the conidered double-beam ytem have been found uing the claical aumed mode ummation. Alo the preented theoretical analyi i illutrated by a numerical example, in which the effect of phyical parameter characterizing the vibrating ytem on the natural frequencie i invetigated. Vu et al. [2] preented an exact method for olving the vibration of a double-beam ytem ubjected to a harmonic excitation. The tudied ytem conit of a main beam with an applied force, and an auxiliary beam, with a ditributed pring and dahpot in parallel between the two beam. Onizczuk [3] analyzed undamped forced tranvere vibration of an elatically connected imply upported double-beam ytem. The modal expanion method i applied to acertain dynamic repone of beam due to arbitrarily ditributed continuou load. Several cae of excitation loading are invetigated. Shamalta and Matrikine [4] invetigated the teady-tate dynamic repone of an embedded addre: hilal22@yahoo.com X/$ - ee front matter r 26 Elevier Ltd. All right reerved. doi:.6/j.jv

2 478 M. Abu-Hilal / Journal of Sound and Vibration 297 (26) v (x,t) ct f (x,t) x v 2 (x,t) k r x Fig.. Double-beam ytem. railway track to a moving train. The model for the track conit of a flexible plate performing vertical vibration, two beam that are connected to the plate by continuou vicoelatic element and an elatic foundation that upport the plate. Two harmonic load that move uniformly along the beam decribe the train load. The plate, the beam and the elatic foundation are employed to model a concrete lab of an embedded track, the rail and the ground reaction, repectively. Double-beam ytem interconnected only at dicrete point are invetigated in Ref. [5,6]. Alo double-tring and double-rod ytem are tudied in Ref. [7 9]. In thi paper, the dynamic behavior of a double-beam ytem travered by a contant moving load i tudied. The ytem i compoed of two elatic primatic homogeneou iotropic Euler Bernoulli beam a hown in Fig.. The beam are parallel one upon the other arranged and connected continuouly by a vicoelatic layer. Thi layer i modeled a a ditributed pring damper ytem. The top beam i travered by a load and i deignated a the primary beam. The bottom beam i deignated a the econdary beam. In general, the two beam may be different and differently upported. But in order to decouple the governing partial differential equation in a imple way, two retriction are made: (a) the beam mut be identical and (b) the boundary condition of the ame ide of the ytem mut be the ame [2]. The ytem tudied in thi paper conit of two identical imply upported beam. The dynamic deflection of both beam are given in analytical cloed form. The effect of the moving peed of the load and the damping and the elaticity of the vicoelatic layer on the dynamic repone of the beam are invetigated in detail. Alo everal plot of the maximum repone of the beam veru a peed parameter are preented. Furthermore, the force tranmitted between the beam i determined and the effect of the ytem parameter on thi force are tudied. 2. Mathematical formulation 2.. Mathematical model and governing equation The tranvere vibration of the double-beam ytem hown in Fig. i governed by the two coupled partial differential equation. EIv þ kðv v 2 Þþrð_v _v 2 Þþm v ¼ f ðx; tþ, () EIv 2 þ kðv 2 v Þþrð_v 2 _v Þþm v 2 ¼, (2) where EI i the flexural rigidity of the beam, E i Young modulu of elaticity, I i the moment of inertia of the cro-ectional area of the beam, m i the ma per unit length of the beam k i the pring contant of the vicoelatic layer, r i the damping coefficient of the vicoelatic layer, and v i (x,t) i the tranvere deflection of the ith beam at poition x and time t. A prime denote differentiation with repect to poition x and a dot denote differentiation with repect to time t. The boundary condition of the tudied imply upported beam are v ð; tþ ¼v 2 ð; tþ ¼, (3)

3 In order to decouple Eq. () and (2), let Thu, v ðl; tþ ¼v 2 ðl; tþ ¼, (4) EIv EIv Adding Eq. () and (2) and conidering Eq. (7) lead to Subtituting Eq. (8) into Eq. (2) and conidering Eq. (7) yield where the forcing function f n ðx; tþ i defined a ð; tþ ¼EIv 2ð; tþ ¼, (5) ðl; tþ ¼EIvðL; tþ ¼. (6) 2 vðx; tþ ¼v ðx; tþþv 2 ðx; tþ. (7) v ðx; tþ ¼vðx; tþ v 2 ðx; tþ. (8) EIv þ m v ¼ f ðx; tþ. (9) EIv 2 þ 2kv 2 þ 2r_v 2 þ m v 2 ¼ f n ðx; tþ, () f n ðx; tþ ¼kv þ r_v. () So, two uncoupled partial differential equation (Eq. (9) and ()) are obtained. Thee equation can be eaier olved than the coupled Eq. () and (2). Firt, Eq. (9) i olved, then the obtained olution v(x,t) i ubtituted into Eq. () to obtain the forcing function f n ðx; tþ in order to be able to olve Eq. (). The outcoming olution of thi equation repreent the deflection v 2 (x,t) of the econdary beam. The repone v (x,t) of the primary beam may be then obtained from Eq. (8). Note that Eq. () i identical to the governing partial differential equation of the forced vibration of an Euler Bernoulli beam on a vicoelatic foundation, wherea Eq. (9) i that of an undamped Euler Bernoulli beam Solution of Eq. (9) In modal form, v(x,t) may be given a M. Abu-Hilal / Journal of Sound and Vibration 297 (26) vðx; tþ ¼ X n¼ X n ðxþy n ðtþ, (2) where y n (t) i the generalized deflection of the nth mode and X n (x) i the nth eigenfunction of a imply upported beam defined a X n ðxþ ¼in k n x (3) with k n ¼ np L. (4) Subtituting Eq. (2) into Eq. (9) and then multiplying by X j (x), and integrating with repect to x between and L lead to y n þ o 2 n y n ¼ Q n ðtþ (5) with the circular frequency of the nth mode ffiffiffiffiffiffi ffiffiffiffiffiffi k n o n ¼ ¼ k 2 EI n, (6) m m n

4 48 M. Abu-Hilal / Journal of Sound and Vibration 297 (26) where k n ¼ i the generalized tiffne of the nth mode and Z L Z L EIX n X n dx (7) m n ¼ mx 2 n ðxþ dx ¼ ml 2 (8) i the generalized ma of the nth mode. A contant force P travering the beam from the left-hand ide with contant peed o that the load f(x,t) i defined a f ðx; tþ ¼P dðx ctþ, (9) where d(.) denote the Dirac delta function. The generalized force aociated with the nth mode i then given a Q n ðtþ ¼ Z L X n ðxþf ðx; tþ dx ¼ 2P m n ml inðk nctþ. (2) Auming that the beam are originally at ret, i.e. v ðx; Þ ¼v 2 ðx; Þ ¼_v ðx; Þ ¼_v 2 ðx; Þ ¼ (2) the olution of Eq. (5) i then written a y n ðtþ ¼ where h n (t) i the impule repone function defined a ( in o n t o h n ðtþ ¼ n tx; to: Subtituting Eq. (2) and (23) into Eq. (22) yield Z t Z t h n ðt tþq n ðtþ dt, (22) y n ðtþ ¼ 2P in o n ðt tþ in k n ct dt. (24) mlo n Carrying out the integration and ubtituting the obtained reult into Eq. (2) give the olution of Eq. (9) in the form vðx; tþ ¼ 2P X in k n x k n c ml n¼ ðk n cþ 2 in o n t in k n ct. (25) o 2 o n n Note that v(x,t) i independent on the layer damping. (23) 2.3. Deflection of the econdary beam Subtituting Eq. (25) and it time derivative into Eq. () yield f ðx; tþ ¼ 2P X in k n xq ml n ðtþ, (26) n¼ where q n ðtþ ¼ ðk n cþ 2 k k nc in o n t in k n ct þ rk n c½co o n t co k n ctš. (27) o 2 o n n

5 Subtituting Eq. (26) into Eq. () lead to EIv 2 þ 2kv 2 þ 2r_v 2 þ m v 2 ¼ 2P X in k n xq ml n ðtþ. (28) n¼ Solving thi equation yield the deflection v 2 (x,t) of the econdary beam. In modal form, v 2 (x,t) may be given a v 2 ðx; tþ ¼ X n¼ X 2n ðxþy 2n ðtþ, (29) where y 2n (t) i the generalized deflection of the nth mode and X 2n (x) i the nth normal mode defined a X 2n ðxþ ¼in l n x (3) with l n ¼ np L. (3) Subtituting Eq. (29) into Eq. (28) and then multiplying by X j (x), and integrating with repect to x between and L lead to y 2n ðtþþ2o n z n _y 2n ðtþþo 2 n y 2nðtÞ ¼Q nðtþ, (32) where ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi O n ¼ EIl 4 n þ 2k m (33) i the circular frequency of the nth mode and z n ¼ r mo n (34) i the damping ratio of the nth mode. The generalized force aociated with the nth mode Q n i given a Q n ðtþ ¼ Z L X 2j ðxþ 2P X inðk n xþq m n ml n ðtþ dx, n¼ (35) where m n i the generalized ma a defined in Eq. (8). Carrying out the integration and uing the orthogonality relationhip lead to M. Abu-Hilal / Journal of Sound and Vibration 297 (26) Z L X n X k dx ¼ ; nak (36) Q n ðtþ ¼2P m 2 L q nðtþ, (37) where q n (t) i defined in Eq. (27). Auming the beam are originally at ret, the olution of Eq. (32) i then written a y 2n ðtþ ¼ where h n ðtþ i the impule repone function defined for z no a ( h n ðtþ ¼ O dn e z no n t in O dn t; t ; ; to Z t h n ðt tþq nðtþ dt, (38) (39)

6 482 in which M. Abu-Hilal / Journal of Sound and Vibration 297 (26) qffiffiffiffiffiffiffiffiffiffiffiffi O dn ¼ O n z 2 n (4) i the damped circular frequency of the nth mode. Subtituting Eq. (37) and (39) into Eq. (38) yield y 2n ðtþ ¼ 2P e z no n t Z t m 2 e z no n t in O dn ðt tþq LO n ðtþ dt. (4) dn Carrying out the integration give k n c y 2n ðtþ ¼a n ½ða 3n a 6n a 2n a 4n Þ co o n t þða 2n a 5n a 3n a 7n Þ in o n tš o n þ a n ½ða 8n a 2n a n a 3n Þ co k n ct þða n a 3n a 9n a 2n Þ in k n ctš k n c þ a n ða 2n a 4n a 3n a 6n Þþða n a 3n a 8n a 2n Þ e z no n t co O dn t o n k n c a n ða 2n a 5n þ a 3n a 7n Þ ða n a 3n þ a 9n a 2n Þ e z no n t in O dn t, o n ð42þ where a n ¼ P m 2 LO dn ðk n cþ 2, o 2 n a 3n ¼ a 3n ¼ ðz n O n Þ 2 þðo dn þ o n Þ 2, ðz n O n Þ 2 þðo dn þ o n Þ 2, a 4n ¼ z n O n k ðo dn o n Þro n, a 5n ¼ z n O n ro n þðo dn o n Þk, a 6n ¼ z n O n k þðo dn þ o n Þro n, a 7n ¼ z n O n ro n ðo dn þ o n Þk, a 8n ¼ z n O n k ðo dn k n cþrk n c, a 9n ¼ z n O n rk n c þðo dn k n cþk, a n ¼ z n O n k þðo dn þ k n cþrk n c, a n ¼ z n O n rk n c ðo dn þ k n cþk, a 2n ¼ ðz n O n Þ 2 þðo dn k n cþ 2, a 3n ¼ ðz n O n Þ 2 þðo dn þ k n cþ 2. (43a2m)

7 After ubtituting y 2n (t) a given in Eq. (42) into Eq. (29), the deflection of the econdary beam i known. The deflection of the primary beam i obtained from Eq. (8), where v(x,t) and v 2 (x,t) are given in Eq. (25) and (29), repectively The Force tranmitted between the beam The force tranmitted between the two beam i defined a F T ¼ kðv v 2 Þþrð_v _v 2 Þ. (44) Uing Eq. (8), (25), (29), and (42) yield for the force tranmitted: F T ¼ P X ml n¼ in k n x O dn M. Abu-Hilal / Journal of Sound and Vibration 297 (26) ½½a n a 3n a 8n a 2n Š co k n ct þ½a 9n a 2n a n a 3n Š in k n ct ½a 9n a 2n þ a n a 3n Še z no n t in O dn t þ½a 3n a 4n þ a 2n a 5n Še z no n t co O dn t, where a 8n to a 3n are defined in Eq. (43) and a 4n ¼ kz n O n þ rðo 2 n þ O dnk n cþ, a 5n ¼ kz n O n rðo 2 n O dnk n cþ. ð45þ (46a,b) 3. Reult and dicuion In order to illutrate the obtained analytical olution, the dimenionle deflection vðl=2; Þ v i ¼ ; i ¼ ; 2 (47) v veru the dimenionle time are given for both beam where only the firt term of ummation i conidered. The ymbol v deignate the tatic deflection at mid-pan of a imply upported beam loaded with a tatic force P at point x ¼ L/2 and i defined a v ¼ P L 3 48EI. (48) The dimenionle time i defined a ¼ ct L. (49) Thu when ¼ the force i at the left-hand ide of the beam and when ¼ the force i at the right-hand ide of the beam. The contant force P enter the upper beam at ret from the left-hand ide at poition x ¼ and move to the right with contant peed c. The effect of peed i repreented by the dimenionle peed parameter a, where a ¼ c, (5) c cr with the critical peed c cr defined a [] c cr ¼ o L p, (5) where ffiffiffiffiffiffi o ¼ p 2 EI. (52) L m

8 484 M. Abu-Hilal / Journal of Sound and Vibration 297 (26) (a).6.4 (a2) (b).6.4 (b2) (c).6.4 (c2) (d).6.4 (d2) (e).6.4 (e2) Fig. 2. Dimenionle dynamic deflection veru the normalized time for primary beam ( v ) and econdary beam ( v 2 ) for a peed factor a ¼ :, (a i ) b ¼ :, (b i ) b ¼, (c i ) b ¼, (d i ) b ¼, (e i ) b ¼, i ¼ ; 2, ( ) z ¼, ( ) z ¼ :, ( ) z ¼ :5.

9 M. Abu-Hilal / Journal of Sound and Vibration 297 (26) The effect of the layer tiffne k i repreented by the dimenionle parameter b ¼ kl4 EI. (53) Fig. 2 how the dimenionle deflection for the primary ( v ) and econdary ( v 2 ) beam for a peed factor a ¼ : and different value of damping ratio z and tiffne parameter b. From Fig. 2a e it i obervable that dependent on the dimenionle time, the deflection v may be decreaed or increaed by increaing the value of the damping ratio z. Alo the abolute maximum value of v i reached hort after the load croe the mid-pan of the beam. Furthermore, the figure how that increaing the value of the tiffne parameter b lead to a decreae in the dimenionle deflection v, where value of b maller than. or higher than 4 have irrelevant influence on the dimenionle deflection v. In the abence of damping, very mall value of b mean that the two beam are decoupled (weak elatic coupling). However, increaing the value of b lead to increaing the coupling between the two beam where very high value of b lead to rigid-coupling of the beam. From Fig. 2a e it i noticeable that the value of v by rigid-coupling (2e ) are reduced down to 5% of the value of v for the uncoupled ytem (2a ). Furthermore, the damping effect become maller with increaing the value of the tiffne parameter b. Fig. 2a 2 e 2 how the dimenionle dynamic deflection v 2 for the econdary beam. From thee figure it i oberved that increaing the value of tiffne parameter b increae the deflection v 2. Very high value of b (Fig. 2e 2 ) caue rigid-coupling between the two beam o that they vibrate together a a unit. Conequently, the normalized deflection v 2 equal v in thi cae. For very mall value of tiffne parameter b (Fig. 2-a 2 ), which mean weak elatic coupling, the dimenionle deflection v 2 i increaed by increaing the value of the damping ratio z. That i becaue increaing the value of z increae the coupling between the two beam o that higher mechanical energy will be tranmitted from the primary to the econdary beam. Conequently, the excitation force acting on the econdary beam are increaed. From Fig. 2a 2 e 2 it i oberved that the effect of damping on the normalized deflection v 2 become maller by increaing the value of the tiffne parameter b. Fig. 3 how dimenionle deflection for the primary ( v ) and econdary ( v 2 ) beam for a peed factor a ¼ :25 and different value of damping ratio z and tiffne parameter b. The figure how that, in general, the ytem behave a in the cae for a peed factor a ¼ :. But from Fig. 3 it i obervable that the abolute maximum value of v and v 2 occur at the time at which the load travered ca. 4% of the beam length. However the abolute maximum value of v 2 occur a little earlier for relative mall value of b. Fig. 3a c how that the abolute maximum value of v occur at a later time by increaing the damping ratio z. From the figure it i furthermore obervable that increaing the damping ratio may, depending on the dimenionle time, increae or decreae the value of v. To be pecific, from Fig. 3a it i evident to ee that for value of between and.5, increaing the value of z decreae the value of v. However, for value of between ca..5 and ca..8, increaing the value of z lead to an increae in the value of v. Fig. 3a 2 c 2 how that the abolute maximum value of v 2 occur at an earlier time by increaing the damping ratio z. Fig. 4 how the dimenionle deflection for the primary ( v ) and econdary ( v 2 ) beam for a peed factor a ¼ :5 and different value of damping ratio z and tiffne parameter b. The figure how that increaing the value of the tiffne parameter b decreae the deflection v. In the cae of rigid-coupling (Fig. 4e ), the value of v i reduced down to 5% of v for the uncoupled beam. Alo a previouly tated, the value of v may be decreaed or increaed by increaing the value of damping ratio z. The abolute maximum value of v i reached in the time interval in which the load travel between.5 and.7 of the beam pan. Thi value i delayed by increaing the value of the damping ratio z. Fig. 4a 2 e 2 how that increaing the value of the tiffne parameter b increae the dimenionle deflection v 2. In the cae of rigid-coupling, v 2 equal v. Furthermore, it i oberved that the abolute maximum value of v 2 i reached after the load travered 6% of the beam length, but before the load left the beam. Thi value occur earlier by increaing the value of the damping ratio.

10 486 M. Abu-Hilal / Journal of Sound and Vibration 297 (26) (a) (a2) (b) (b2) (c) (c2) (d) (d2) (e) (e2) Fig. 3. Dimenionle dynamic deflection veru the normalized time for primary beam ( v ) and econdary beam ( v 2 ) for a peed factor a ¼ :25, (a i ) b ¼ :, (b i ) b ¼, (c i ) b ¼, (d i ) b ¼, (e i ) b ¼, i ¼ ; 2, ( ) z ¼, ( ) z ¼ :, ( ) z ¼ :5.

11 M. Abu-Hilal / Journal of Sound and Vibration 297 (26) (a).8 (a2) (b).8 (b2) (c).8 (c2) (d).8 (d2) (e).8 (e2) Fig. 4. Dimenionle dynamic deflection veru the normalized time for primary beam ( v ) and econdary beam ( v 2 ) for a peed factor a ¼ :5, (a i ) b ¼ :, (b i ) b ¼, (c i ) b ¼, (d i ) b ¼, (e i ) b ¼, i ¼ ; 2, ( ) z ¼, ( ) z ¼ :, ( ) z ¼ :5.

12 488 M. Abu-Hilal / Journal of Sound and Vibration 297 (26) (a) (a2) (b) (b2) (c) (c2) (d) (d2) (e) (e2) Fig. 5. Dimenionle dynamic deflection veru the normalized time for primary beam ( v ) and econdary beam ( v 2 ) for a peed factor a ¼, (a i ) b ¼ :, (b i ) b ¼, (c i ) b ¼, (d i ) b ¼, (e i ) b ¼, i ¼ ; 2, ( ) z ¼, ( ) z ¼ :, ( ) z ¼ :5.

13 M. Abu-Hilal / Journal of Sound and Vibration 297 (26) (a) (b) (c) α α α Fig. 6. Maximum deflection v ;max and v 2;max veru the peed parameter a for different value of tiffne parameter b and damping ratio z calculated at x ¼ L/2: (yy) v ;max, z ¼, ( ) v ;max,z ¼ :, ( ) v 2;max, z ¼, ( ) v 2;max, z ¼ :, (a) b ¼, (b) b ¼, (c) b ¼. Fig. 5 how dimenionle deflection for the primary ( v ) and econdary ( v 2 ) beam for a peed factor a ¼ and different value of damping ratio z and tiffne parameter b. The figure how that, in general, the ytem behave a in the cae for a peed factor a ¼ :5. Alo for very mall (decoupled) and very high (rigidcoupled) value of the tiffne parameter b, the abolute maximum value of v i reached after the load depart from the beam. For the cae of elatic coupling, thi value i reached hort before the load left the beam. That i becaue a variation of the tiffne parameter lead to a variation of the circular natural frequency O n, given in Eq. (33). Fig. 5a 2 e 2 how that the abolute maximum value of v 2 i reached after the load depart from the beam. Fig. 6 how the normalized maximum deflection of the primary beam v ;max ¼ v ;max =v and the econdary beam v 2;max ¼ v 2;max =v veru the peed parameter a for different value of tiffne parameter b and damping ratio z calculated at x ¼ L/2. From the Figure the following characteritic can be noted:. The maximum deflection v ;max and v 2;max of a double-beam are maller than v max of a ingle beam a preented by Fryba [] and Peterev []. 2. For mall value of tiffne parameter b, an increae in the damping lead to a decreae in v ;max of the primary beam and an increae in v 2;max of the econdary beam. However, for higher value of b, the ytem behave reverely. 3. Increaing the value of the tiffne parameter b lead to a decreae in v ;max and an increae in v 2;max at the ame time. 4. For very high value of b, the maximum deflection v ;max and v 2;max coincide with each other and take value equal to :5 v max of a ingle beam. Fig. 7 how the normalized force tranmitted F T ¼ F T =P between the primary and econdary beam calculated at x ¼ L/2 veru the dimenionle time for different value of peed factor a, damping ratio z, and tiffne parameter b. From the figure the following can be noted:. Increaing the damping lead to an increae in F T,max by relative mall value of tiffne parameter b (Fig 7a c) and to a decreae in F T,max by higher value of b (Fig. 7d f). 2. By increaing the damping, the maximum value of the force tranmitted F T,max occur at an earlier time. 3. The effect of damping i high by mall value of tiffne parameter and low by higher value of thi parameter. 4. In general, increaing the peed parameter lead to an increae in the maximum value of the force tranmitted.

14 49 M. Abu-Hilal / Journal of Sound and Vibration 297 (26) (a) (b) (c) α =.25 α =.5 α = (d) (e) Fig. 7. Normalized force tranmitted F T;max between the beam veru the dimenionle time for different value of peed factor a, tiffne parameter b, and damping ratio z calculated at x ¼ L/2: ( ) z ¼, (yy) z ¼ :, ( ) z ¼ :5. (a) b ¼ :, (b) b ¼, (c) b ¼, (d) b ¼, (e) b ¼.

15 M. Abu-Hilal / Journal of Sound and Vibration 297 (26) Concluion The dynamic repone for a imply upported homogeneou iotropic double-beam ytem ubject to a moving contant load wa invetigated. The normalized deflection of both beam are obtained in cloed form. Alo everal plot of the deflection of the beam are given and dicued. The effect of the moving peed of the load and the damping and tiffne of the vicoelatic layer on the deflection of the beam are explored. It i found that increaing the damping of the connecting vicoelatic layer may increae or decreae the repone of the beam. However, thi damping ha negligible effect on the deflection of the beam for very high value of the tiffne of the layer. Furthermore, it i found that increaing the tiffne of the layer decreae the deflection of the beam on which the load travel (primary beam) and increae the deflection of the econdary beam. In cae of very high layer tiffne, both beam have the ame deflection ince the beam are then rigid-coupled and behave a a unit. Thi deflection i half the dynamic deflection of a ingle beam travered by a contant load moving with a contant peed. Reference [] Z. Onizczuk, Free tranvere vibration of elatically connected imply upported double-beam complex ytem, Journal of Sound and Vibration (232) (2) [2] H.V. Vu, A.M. Ordonez, B.H. Karnopp, Vibration of a double-beam ytem, Journal of Sound and Vibration (229) (2) [3] Z. Onizczuk, Forced tranvere vibration of an elatically connected complex imply upported double-beam ytem, Journal of Sound and Vibration (264) (23) [4] M. Shamalta, A.V. Metrikine, Analytical tudy of the dynamic repone of an embedded railway track to a moving load, Archive of Applied Mechanic (73) (23) [5] T.R. Hamada, H. Nakayama, K. Hayahi, Free and Forced vibration of elatically connected double-beam ytem, Bulletin of the Japan Society of Mechanical Engineer (983) [6] M. Gu rgo ze, H. Erol, On laterally vibrating beam carrying tip mae, coupled by everal double pring ma ytem, Journal of Sound and Vibration (269) (24) [7] Z. Onizczuk, Tranvere vibration of elatically connected double-tring complex ytem part I: free vibration, Journal of Sound and Vibration (232) (2) [8] Z. Onizczuk, Tranvere vibration of elatically connected double-tring complex ytem part II: forced vibration, Journal of Sound and Vibration (232) (2) [9] H. Erol, M. Gu rgöze, Longitudenal vibration of a double-rod ytem coupled by pring and damper, Journal of Sound and Vibration (276) (24) [] L. Fry ba, Vibration of Solid and Structure Under Moving Load, Noordhoff International, Groningen, 972. [] A.V. Peterev, B. Yang, L.A. Bergman, C.A. Tan, Reviiting the moving force problem, Journal of Sound and Vibration (26) (23) 75 9.