In-Plane Responses of Axially Moving Plates Subjected to Arbitrary Edge Excitations

Transcription

1 International Jornal of echanical and echatronics Engineering Vol:7, o:7, 13 In-Plane Responses of Axially oing Plates Sjected to Aritrary Edge Excitations T. H. Yong and Y. S. Cio International Science Index, echanical and echatronics Engineering Vol:7, o:7, 13 waset.org/plication/16414 Astract The free and forced in-plane irations of axially moing plates are inestigated in this work. The plate possesses an internal damping of which the constittie relation oeys the Kelin-Voigt model, and the excitations are aritrarily distrited on two opposite edges. First, the eqations of motion and the ondary conditions of the axially moing plate are deried. Then, the extended Ritz method is sed to otain discretized system eqations. Finally, nmerical reslts for the natral freqencies and the mode shapes of the in-plane iration and the in-plane response of the moing plate sjected to aritrary edge excitations are presented. It is osered that the symmetry class of the mode shapes of the in-plane iration disperses gradally as the moing speed gets higher, and the - and -components of the mode shapes elong to different symmetry class. In addition, large response amplitdes haing shapes similar to the mode shapes of the plate can e excited y the edge excitations at the resonant freqencies and with the same symmetry class of distrition as the -components. Keywords Aritrary edge excitations, axially moing plates, in-plane iration, extended Ritz method. A I. ITRODUCTIO XIALLY moing wes are freqently encontered applications in indstries, sch as magnetic tapes, driing elts, paper sheets, and plastic wes, etc. In order to raise prodction efficiency, the moing speed of the we gets faster and faster, which worsens the iration prolem of the we. Therefore, a ast nmer of researches concerning the dynamic ehaiors of moing wes can e fond in the literatre [1]-[6]. Becase the lowest natral freqency and critical speed of the ot-of-plane iration of the moing we are lower than those of the in-plane iration, most of the references in literatre deal with the ot-of-plane iration of the moing wes. In literatre, the references concerning the in-plane iration of rectanglar plates are mch lesser than those pertaining to the ot-of-plane iration of rectanglar plates. Hyde et al. [7] stdied the free in-plane iration of rectanglar plates ndergoing plane stress deformation throgh the Ritz discretization. The natral freqencies and mode shapes of rectanglar plates with three different kinds of ondary conditions were presented in this work. The free in-plane irations of a rectanglar plate nder comined static pressre/tension edge loads were inestigated y Waer [8]. In this paper, the in-plane deformation de to static edge loads is T. H. Yong and Y. S. Cio are with ational Taiwan Uniersity of Science and Technology, Taipei 16, Taiwan (phone: ; fax: ; ( thyong@mail.ntst.ed.tw, 131@ mail.ntst.ed. tw). determined first y soling the plane stress prolem, and the free in-pane iration is analyzed y the Rayleigh-Ritz method. A mathematical model was deeloped y Farag and Pan [9] for the prediction of the forced response of rectanglar plates with all edges clamped to in-plane point force excitations. All the aoe three references dealt with the in-plane iration of a stationary rectanglar plate, and the references dealing with the in-plane iration of axially moing rectanglar plates are een less. Shin et al. [1] stdied the free in-plane iration of an axially moing plate y the Galerkin method. The elastic plate is sjected to a constant tension along two mass-transport ondaries, and two sets of ondary conditions, which are free and fixed constraints in the lateral direction at two mass-transport ondaries are discssed in this stdy. To athors knowledge, no research work on the in-plane response of axially moing plates sjected to aritrary edge excitations can e fond in the literatre. Therefore, this work inestigates the free and forced in-plane irations of axially moing plates. The plate possesses an internal damping of which the constittie relation oeys the Kelin-Voigt model, and the excitations are aritrarily distrited on two opposite edges. First, the eqations of motion and the ondary conditions of the axially moing plate are deried. Then, the extended Ritz method is sed to otain discretized system eqations. Finally, nmerical reslts for the natral freqencies and the mode shapes of the in-plane iration and the in-plane response of the moing plate sjected to aritrary edge excitations are presented. II. EQUATIOS OF OTIO Consider a rectanglar plate of dimension a moing in the x-direction with a constant speed V. The plate is simply spported on two opposite edges x = and a, and is free on the other two edges y = and. The plate is sjected to an excitation f ( y, t) aritrarily distrited on two simply spported edges, as shown in Fig. 1. The plate possesses an internal damping of which the constittie relation oeys the Kelin-Voigt model, i.e., ε σ = E( ε + c ) (1) where σ and ε are the stress and strain of the plate, respectiely, and t is a temporal ariale. E and c are Yong s modls and internal damping coefficient of the plate, International Scholarly and Scientific Research & Innoation 7(7) scholar.waset.org/ /16414

2 International Jornal of echanical and echatronics Engineering Vol:7, o:7, 13 International Science Index, echanical and echatronics Engineering Vol:7, o:7, 13 waset.org/plication/16414 respectiely. Therefore, the stress-strain relations for the plane stress state are gien y E ε ε x y σ x = [( ε ) c( )] x + νε y + + ν 1 ν (a) E ε y ε x σ y = [( ε ) c( )] y + νε x + + ν 1 ν () E ε xy σ xy = [ ε xy + c ] (c) 1+ ν where ν is Poisson s ratio of the plate. In terms of the in-plane displacement components, the strain energy of the plate can e expressed as Eh U = (1 ν ) a [( ) + ( ) 1 ν + ν( )( ) + ( + ) ] dxdy where h is the thickness of the plate, and and are the in-plane displacement components of the plate in the x- and y-directions, respectiely. The kinetic energy of the axially moing plate is gien y ρh T = a [( V + + V ) + ( + V ) ] dxdy where ρ is the mass density of the plate. The irtal work done y the edge excitation is written as δw K = x= a δ x= ] (3) (4) f ( y, t)[ δ dy (5) The irtal work done y the internal damping force is gien y δw d Ehc = 1 ν a [( + ν ) δ( ) + ( + 1 ν ν ) δ( ) + ( + ) δ( + )] dxdy The irtal work done y the momentm of the plate moing across the ondaries is fond to e δ = (6) a a ρhv[( V + + V ) δ + ( + V ) δ ] dy (7) The extended Hamilton principle for non-conseratie systems reqires t t1 ( δt δu + δwk + δwd δ ) dt = (8) where t 1 and t are two time instants at which the irtal displacements δ and δ anish. Sstitting (3) (7) into (8) and going throgh ariational calcls maniplations yield the eqations of motion and ondary conditions for the axially moing plate, ρh( + V + V Eh ) 1 ν Eh Ehc ( + ) (1 + ν) x y 1 ν ( + ν ) ( + ν ) Ehc ( + ) = f ( y, t)[ δ( x a) δ( x)] (1 + ν) (9a) ρh( + V + V ) Eh ( + ν ) 1 ν where Eh Ehc ( + ) (1 + ν ) x y 1 ν Ehc ( + ) = (1 + ν ) x, ( + ν ) (9) at x = and a, = and = (1a) x at y = and, = and = (1) y, and y xy are the in-plane stress resltants of the plate. Eqations (1a) and (1) can e redced to those otained y Shin et al. [1] if the internal damping force terms are deleted. Eqations (1a) and (1) constitte a system of the second order partial differential eqations in terms of the displacement components and. Since the exact soltion of them is not feasile, to otain approximate soltions in the finite dimensional space, discretization of the system eqations is carried ot. De to the ondary conditions of the prolem, it is ery difficlt to find the comparison fnctions, the ones that satisfy oth the geometric and natral ondary conditions, as the trial fnctions. Therefore, the extended Ritz method, which admits the admissile fnctions satisfying the geometric ondary conditions only as the trial fnctions, is adopted in conjnction with the extended Hamilton principle. Assme the displacement components and to e of the forms, x m y n ( x, y, t) = A m n mn ( t)( ) ( ), = = a (11a) ( m + 1) πx y n ( x, y, t) = B m n mn ( t) sin ( ) = = a (11) where A mn (t) and B mn (t) are ndetermined fnctions. Sstitting (11a) and (11) into (8) and non- dimensionalizing all parameters and ariales yield the following eqation, xy International Scholarly and Scientific Research & Innoation 7(7) scholar.waset.org/ /16414

3 International Jornal of echanical and echatronics Engineering Vol:7, o:7, 13 International Science Index, echanical and echatronics Engineering Vol:7, o:7, 13 waset.org/plication/16414 [ ] U + ( V [ G] + c[ K ]) U + ([ K ] + V [ K ]) U F( τ ) (1) e e g = (1 ν ρa where the non-dimensional parameters V = V, E E E c = c, τ = t, and a prime denotes (1 ν ) ρa ( 1 ν ) ρa a differentiation with respect to τ. The mass matrix [ ], the moing-speed dependent matrix [G], the elastic stiffness matrix [ K e ], and the geometric stiffness matrix [ K g ] hae the following forms, [ ] [] [] [ ] =, [ ] = [ [ ] = ] [ G ] [] G, [] [ G ] [ K ] [ K ] [ K g ] [] K e, [ K g ] =, [ K ] [ K ] [] [ K g ] in which the lock matrices [ ], [ ], [ K ], and [ K ] T are symmetric, and [ K ] = [ K ]. Therefore, the mass matrix [ ] and the elastic stiffness matrix K ] are ) [ e symmetric, the moing-speed dependent matrix [G] and the geometric stiffness matrix [ K g ] are neither symmetric nor skew-symmetric. The response ector U and the forcing ector F (τ) in (1) hae the following forms, A f ( τ) U =, F ( τ) = B where A and B are colmn matrices formed y all and B mn (τ) /, and f (τ) is a colmn matrix. A mn (τ) / Eqation (1) goerns the in-plane forced response of an axially moing plate. To find the natral freqencies and the mode shapes of the free in-plane iration, neglect the nonhomogeneos forcing term on the right-hand side of the eqation, and rewritten the set of the second order differential eqations into a set of first order differential eqations, which yield the following eigenale prolem, s [ ] [ ] [ ] [ I ] ( V Z + [ G] + c [ K ]) ([ K ] + V [ K ]) e e g Z = (13) [ I ] [ ] where [I ] is an identity matrix. The eigenales of the aoe eqation are in pairs of complex conjgates, s j 1, j = λ j ± iω j, j=1,,, L, where L = ( + 1)( + 1) is the total degrees of freedom of the discretized system. The eigenectors of the aoe eqation are also in pairs of complex conjgates, Z j 1, j = X j ± iy j, j=1,,, L. The imaginary part of the eigenale ω j is the non- dimensionalized natral freqency of the jth mode of the axially moing plate, while ς = λ / ω is the damping ratio of the jth mode of the axially j j j moing plate. If the real parts of all eigenales are smaller than or eqal to, the axially moing plate is stale; otherwise, the axially moing plate is nstale. Therefore, the critical speeds for the in-plane iration of the plate can e otained when the real part of one eigenale ecomes positie. To calclate the in-plane forced response of the axially moing plate, assme the edge excitation is simply harmonic in time, i.e., f ( y, t) = q( y) sin Ωt, where Ω is the exciting freqency. The forcing ector F (τ) in (1) will hae the form F ( τ) = Q sin Ωτ, where Q is a constant ector, and Ω = Ω (1 ν ) ρa / E. To match the form of the forcing ector, assme the response ector to e of the form, U τ) = U s sin Ωτ + U cosωτ (14) ( c where U s and U c are the sine and cosine components of the response ector, respectiely. Sstitting (14) into (1) yields the following eqation, ( Q = [ K ] + [ ] Ω [ ] Ω [ ] + e V K g ) ( V G c[ Ke ]) Ω( [ ] + V G c[ K ]) ([ K ] + V [ K ] Ω [ ] e e g U ) U s c (15) Once U s and U s are soled, the fnctions A ) and B mn (τ) are determined, from which the in-plane responses and are calclated. Since the response ector contains the sine and cosine components, the fnctions A ) and B ) mn (τ consist of the sine and cosine components also, i.e., Amn ( τ ) s c = A sin Ωτ + cosωτ mn Amn Bmn ( τ ) s c = B sin Ωτ + cosωτ mn Bmn mn (τ mn (τ (16a) (16) s c s c where A mn, A mn, B mn, and B mn are constants. The responses and can e written in the amplitde-phase angle form, where the amplitdes ( ξ, η, τ) = ( ξ, η)sin( Ωτ φ ) (17a) C ( ξ, η, τ) = ( ξ, η)sin( Ωτ φ ) (17) C International Scholarly and Scientific Research & Innoation 7(7) scholar.waset.org/ /16414

4 International Jornal of echanical and echatronics Engineering Vol:7, o:7, 13 International Science Index, echanical and echatronics Engineering Vol:7, o:7, 13 waset.org/plication/16414 C ( ξ, η) = C ( ξ, η) = [ [ and the phase angles φ φ = tan = tan 1 A. Free Viration [ [ m= n= m= n= m= n= m= n= 1 B B s mn c mn A A m= n= m= n= m= n= m= n= s mn c mn x ( ) a x ( ) a y n ( ) ] y ( ) ] m m n + ( m + 1) πx y n sin ( ) ] a ( m + 1) πx y n sin ( ) ] a s x m y n Amn ( ) ( ) a c x m y n Amn ( ) ( ) a s ( m + 1) πx y Bmn sin ( ) a c ( m + 1) πx y Bmn sin ( ) a III. UERICAL RESULTS In order to erify the correctness of the reslts otained in this work, comparison is carried ot with the reslts y Shin et al. [1]. Tale I presents the lowest six natral freqencies of an axially moing rectanglar plate withot internal damping. It is osered that with ==8, the nmerical reslts otained in this work agree excellently with those reported in the reference [1]. The maximm difference etween these two sets of reslts sing the same degrees of freedom is aot.5%. Fig. illstrates the lowest six mode shapes of an axially moing rectanglar plate with internal damping. The -component and the -component of each mode shape are drawn in different plots, respectiely. Since the moing speed of the plate is ery low, the mode shapes almost presere the symmetry class that is owned only y stationary plates. The figre shows that the -components of the first and the forth mode shapes are anti-symmetric in oth the x- and y-directions. The -components of the second and the fifth mode shapes are symmetric in the x-direction and are anti- symmetric in the y-direction. The -component of the third mode shape is anti-symmetric in the x-direction and is symmetric in the y-direction, while that of the sixth mode shape is symmetric in oth the x- and y-directions. The figre shows the opposite trend for the -components. oreoer, the fifth mode shape of the axially moing rectanglar plate is a prely -component mode. The symmetry class of the mode shapes disperses gradally as the moing speed of the plate ecomes higher ecase the moing speed dependent matrix and the geometric stiffness matrix are neither symmetric nor skew-symmetric. Tale II shows the lowest six natral freqencies and modal damping ratios of an axially moing rectanglar plate with internal damping. It is fond that the natral freqencies of the n n + plate with internal damping are slightly lower than those of the plate withot internal damping, and the modal damping ratio is monotonically increasing for higher modes. Fig. 3 depicts the real and the imaginary parts of the eigenales of the lowest seen modes of an axially moing rectanglar plate with internal damping. In Fig. 3 (a), all the imaginary parts of the eigenales, which are the natral freqencies of the axially moing plate, drop as the moing speeds of the plate increase. At V =.486, the imaginary part of the lowest mode redces to zero. oreoer, eering phenomena occr when the imaginary part of the seenth mode meets that of the sixth mode at V =.1587; when the imaginary part of the sixth mode meets that of the fifth mode at V =.1637, and when the imaginary part of the sixth mode meets that of the seenth mode at V = In Fig. 3 (), all the real parts of the eigenales are negatie and hold nearly constant except the followings. The real part of the lowest mode rises and ecomes positie at the speed which the imaginary part of the lowest mode redces to zero. That is the moing plate ecomes nstale at this speed, and this speed is called the first critical speed of the moing plate. The real parts of the fifth, sixth and seenth modes jmp p and down respectiely, and cross each other at the speeds where eering phenomena occr. Becase the internal damping of the plate in this figre is small, Fig. 3 (a) looks ery similar to that otained y Shin et al. [1] in which no internal damping is considered. B. Forced Viration Fig. 4 illstrates the forced response of an axially moing plate sjected to excitations niformly distrited on two simply spported edges at zero and the lowest fie resonant freqencies. All -components of the response plots in the figre are nearly anti-symmetric in the axial direction and are nearly symmetric in the lateral direction, whereas all -components of the response plots show the opposite symmetry class, that is, nearly symmetric in the axial direction and nearly anti-symmetric in the lateral direction. This trend loses gradally for higher resonant freqencies. Becase the response plots possess the same symmetry class of the third mode, the amplitdes of the response plot at the third resonant freqency are one-order larger than those at the other resonant freqencies. Except for that, the response amplitdes generally redce gradally as the resonant freqencies get higher, and the static deformation of the plate sjected to static edge loads are close to the forced response amplitdes at the first resonant freqency. Fig. 5 shows the forced response of an axially moing plate sjected to excitations linearly distrited on two simply spported edges sch that the distrition of the excitations are anti-symmetric in the lateral direction. All -components of the response plots in the figre are almost anti-symmetric in oth axial and lateral directions, whereas all -components of the response plots are almost symmetric in oth directions. Again, this trend loses gradally for higher resonant freqencies. Becase the response plots possess the same symmetry class of International Scholarly and Scientific Research & Innoation 7(7) scholar.waset.org/ /16414

5 International Jornal of echanical and echatronics Engineering Vol:7, o:7, 13 International Science Index, echanical and echatronics Engineering Vol:7, o:7, 13 waset.org/plication/16414 the first and the forth modes, the amplitdes of the response plot at the first resonant freqency are at least two-order larger than those at the other resonant freqencies, and the amplitdes of the response plot at the forth resonant freqency are larger than those at the second and the third resonant freqencies. The response plot at the first resonant freqency is similar to the static deformation plot sjected to static edge loads, and the response amplitdes at the first resonant freqency are aot one hndred times of the static deformation sjected to static edge loads, which is the inerse of the internal damping factor c. ote that the amplitdes of the response plot at the first resonant freqency in this figre are een larger than those of the plate sjected to niformly distrited edge excitations at the third resonant freqency, as shown in Fig. 3 (c). The forced response of an axially moing plate sjected to excitations linearly, anti-symmetrically distrited on the middle hales of two simply spported edges is depicted in Fig. 6. The distrition of the excitations is again anti- symmetric in the lateral direction, and the intensity of the excitation is twice as that in Fig. 5. All corresponding plots in oth figres look similar, t the amplitdes in this figre are a little it smaller than those in the preios figre. The effect of the internal damping on the forced response of an axially moing plate is shown in Fig. 7. In this figre, the forced response of the plate considered in Fig. 5 t with a larger internal damping factor is illstrated. Comparing etween oth figres, it is osered that the response plots in oth figres look similar, and the response plots at the first, the second, and the forth resonant freqencies in this figre deiate more from the mode shapes of the plate than the corresponding plots in Fig. 5. Becase the response plots possess the same symmetry class of the first and the forth modes, the amplitde ratios etween Figs. 5 and 7 at the first and the forth resonant freqencies are aot 1:1, which is approximately inersely proportional to the internal damping ratio etween these two figres. Bt the amplitdes etween these two figres at the other resonant freqencies are of the same orders. The effect of the aspect ratio on the forced response of an axially moing plate is presented in Fig. 8. In this figre, the forced response of a sqare plate is considered. Becase the aspect ratio of the plate is smaller, the natral freqencies of the plate rise and the mode shapes of the plate change also. The -component of the first mode shape is anti-symmetric in oth the x- and y-directions. The -components of the second and the forth mode shapes are symmetric in the x-direction and are anti-symmetric in the y-direction. The -components of the third and the fifth mode shapes are anti-symmetric in the x-direction and are symmetric in the y-direction. The -components of the mode shapes hae the opposite class of symmetry. Becase the distrition of the edge excitation is linear, anti-symmetric in the lateral direction, the amplitdes of the response at the first resonant freqency is the highest among those at the lowest fie resonant freqencies. In addition, the amplitdes of the response at the first resonant freqency in this figre are mch smaller than those at the first resonant freqency in Fig. 5 (a) ecase the aspect ratio of the plate in this figre is smaller than that of the plate considered in Fig. 5, and hence the plate considered in this figre is more stiffer. IV. COCLUSIOS The free and forced in-plane irations of axially moing plates sjected to aritrarily distrited edge excitations on two opposite, simply-spported edges are stdied y the extended Ritz method in this work. merical reslts for the natral freqencies, mode shapes and the critical speed of the free in-plane iration and the response amplitdes of the forced in-plane iration are presented for different system parameters and the distrition of the edge excitations. The nmerical reslts for the natral freqencies, mode shapes and the critical speed of the free in-plane iration are compared with [1], and excellent agreement of the nmerical reslts etween these two works is osered. From these nmerical reslts, the following conclsions can e drawn: 1) The mode shapes of the corresponding stationary plate presere symmetry class, and the -components and -components of the mode shapes elong to the opposite symmetry class. The symmetry class of the mode shapes of the free in-plane iration disperses gradally as the moing speed of the plate ecomes higher. ) The natral freqencies of the plate with internal damping are slightly lower than those of the plate withot internal damping, and the modal damping ratio is monotonically increasing for higher modes. 3) At the first critical speed, the imaginary part of the lowest mode redces to zero, and the real part of the lowest mode ecomes positie. Veering phenomena occr among the fifth, sixth and seenth modes when two imaginary parts of these modes meet each other, and the real parts of these modes jmp p and down respectiely and cross each other at the speeds where eering phenomena occr. 4) For an axially moing plate sjected to excitations symmetrically distrited in the lateral direction on two simply spported edges, all -components of the response plots are nearly anti-symmetric in the axial direction and are nearly symmetric in the lateral direction. For an axially moing plate sjected to excitations anti-symmetrically distrited in the lateral direction on two simply spported edges, all -components of the response plots are nearly anti-symmetric in oth the axial and the lateral directions. The -components of the response plots show the opposite symmetry class. This trend loses gradally for higher resonant freqencies and faster moing speeds. 5) If the response plot possesses the same symmetry class as the mode shape of a mode, the amplitdes of the response plot at that resonant freqency will e mch larger than those at the other resonant freqencies. Except for that, the response amplitdes generally redce gradally as the resonant freqencies get higher. 6) The effect of the internal damping is ery effectie only when the symmetry class of the edge excitation meets the International Scholarly and Scientific Research & Innoation 7(7) scholar.waset.org/ /16414

6 International Jornal of echanical and echatronics Engineering Vol:7, o:7, 13 same symmetry class of the mode shape at that resonant freqency. Otherwise, its effect is insignificant. TABLE I THE ATURAL FREQUECIES OF A UDAPED AXIALLY OVIG PLATE ode no. Present work a / =3, V =.1, c = [1] Error %.131 %.34 %.54 % % % International Science Index, echanical and echatronics Engineering Vol:7, o:7, 13 waset.org/plication/16414 TABLE II THE ATURAL FREQUECIES AD ODAL DAPIG RATIOS OF A DAPED AXIALLY OVIG PLATE a / =3, V =.1, c =.1 ode no ω j ς j Fig. 1 An axially moing plate sjected to aritrarily distrited excitations () nd mode (c) 3 rd mode (d) 4 th mode (a) 1 st mode International Scholarly and Scientific Research & Innoation 7(7) scholar.waset.org/ /16414

7 International Jornal of echanical and echatronics Engineering Vol:7, o:7, 13 International Science Index, echanical and echatronics Engineering Vol:7, o:7, 13 waset.org/plication/16414 (e) 5 th mode (f) 6 th mode Fig. The mode shapes of a damped axially moing plate a / =3,V =.1, c =.1 (a) Imaginary part () Real part Fig. 3 The eigenales cres of the lowest seen modes of an axially moing plate a / =3, c =.1 (a) Ω / ω1 = () Ω / ω1 =.9999 International Scholarly and Scientific Research & Innoation 7(7) scholar.waset.org/ /16414

8 International Jornal of echanical and echatronics Engineering Vol:7, o:7, 13 International Science Index, echanical and echatronics Engineering Vol:7, o:7, 13 waset.org/plication/16414 (c) Ω / ω1 =3.749 (d) Ω / ω1 =3.837 (f) Ω / ω1 = Fig. 4 The forced response plots of an axially moing plate sjected to niformly distrited excitations (a) Ω / ω1 = a / =3, V =.1, c =.1 (e) Ω / ω1 =5.418 () Ω / ω1 =.9999 International Scholarly and Scientific Research & Innoation 7(7) scholar.waset.org/ /16414

9 International Jornal of echanical and echatronics Engineering Vol:7, o:7, 13 International Science Index, echanical and echatronics Engineering Vol:7, o:7, 13 waset.org/plication/16414 (c) Ω / ω1 =3.749 (d) Ω / ω1 =3.837 (e) Ω / ω1 =5.418 (f) Ω / ω1 = Fig. 5 The forced response plots of an axially moing plate sjected to linearly, anti-symmetrically distrited excitations a / =3,V =.1, c =.1 (a) Ω / ω1 = () Ω / ω1 =.9999 International Scholarly and Scientific Research & Innoation 7(7) scholar.waset.org/ /16414

10 International Jornal of echanical and echatronics Engineering Vol:7, o:7, 13 International Science Index, echanical and echatronics Engineering Vol:7, o:7, 13 waset.org/plication/16414 (c) Ω / ω1 =3.749 (d) Ω / ω1 =3.837 (e) Ω / ω1 =5.418 (f) Ω / ω1 = Fig. 6 The forced response plots of an axially moing plate sjected to excitations linearly, anti-symmetrically distrited on the middle hales of the edges a / =3,V =.1, c =.1 (a) Ω / ω1 = () Ω / ω1 =.999 International Scholarly and Scientific Research & Innoation 7(7) scholar.waset.org/ /16414

11 International Jornal of echanical and echatronics Engineering Vol:7, o:7, 13 International Science Index, echanical and echatronics Engineering Vol:7, o:7, 13 waset.org/plication/16414 (c) Ω / ω1 =3.59 (d) Ω / ω1 = (f) Ω / ω1 =6.865 Fig 7 The forced response plots of an axially moing plate sjected to linearly, anti-symmetrically distrited excitations a / =3,V =.1, c =.1 (a) Ω / ω1 = () Ω / ω1 =1. (e) Ω / ω1 =5.899 (c) Ω / ω1 =1.31 International Scholarly and Scientific Research & Innoation 7(7) scholar.waset.org/ /16414

12 International Jornal of echanical and echatronics Engineering Vol:7, o:7, 13 [9]. H. Farag, and J. Pan, Free and forced in-plane iration of rectanglar plates, Jornal of Acostical Society of America, ol. 13, pp , [1] C. Shin, W. Kim, and J. Chng, Free in-plane iration of an axially moing memrane, Jornal of Sond and Viration, ol. 7, pp , 4. (d) Ω / ω1 = International Science Index, echanical and echatronics Engineering Vol:7, o:7, 13 waset.org/plication/16414 (e) Ω / ω1 =.379 (f) Ω / ω1 =.391 Fig. 8 The forced response plots of an axially moing plate sjected to linearly, anti-symmetrically distrited excitations a / =1,V =.1, c =.1 REFERECES [1] F. Fng, J. S. Hang, and Y. C. Chen, The transient amplitde of the iscoelastic traelling string: An integral constittie law, Jornal of Sond and Viration, ol. 1, pp , [] L. Q. Chen, and X. D. Yang, Steady-state response of axially moing iscoelastic eams with plsating speed: comparison of two nonlinear models, International Jornal of Solids and Strctres, ol. 4, pp. 35-5, 5. [3] C. Shin, J. Chng, and W. Kim, Dynamic characteristics of the ot-of-plane irations for an axially moing memrane, Jornal of Sond and Viration, ol. 97, pp , 6. [4] K. arynowski, Two-dimensional rheological element in modeling of axially moing iscoelastic we, Eropean Jornal of echanics A/Solids, ol. 5, pp , 6. [5] C. C. Lin, Staility and iration characteristics of axially moing plates, International Jornal of Solids and Strctres, ol. 34, pp , [6] S. Hatami, H. R. Ronagh, and. Azhari, Exact free iration analysis of axially moing iscoelastic plates, Compters & Strctres, ol. 86, pp , 8. [7] K. Hyde, J. Y. Chang, C. Bacca, and J. A. Wickert, Parameter stdies for plane stress in-plane iration of rectanglar plates, Jornal of Sond and Viration, ol. 47, pp , 1. [8] J. Waer, In-plane irations of rectanglar plates nder inhomogeneos static edge load, Eropean Jornal of echanics A/Solids, ol. 11, pp , 199. International Scholarly and Scientific Research & Innoation 7(7) scholar.waset.org/ /16414