Analysis of the impact-induced two-to-one internal resonance in nonlinear doubly curved shallow panels with rectangular platform * YuryA. Rossikhin, Marina V. Shitikova and Muhammed Salih Khalid J. M. Voronezh State University of Architecture and Civil Engineering Research Center on Dynamics of Solids and Structures 20-letija Oktjabrja Street 84, 394006 Voronezh, RUSSIA Abstract – The problem of the low-velocity impact of an elastic sphere upon a nonlinear doubly curved shallow panel with rectangular platform is investigated. The ap- proach utilized in the present paper is based on the fact that during impact only the modes strongly coupled by the two- to-one internal resonance condition are initiated. Such an approach differs from the Galerkin method, wherein reso- nance phenomena are not involved. Since is it assumed that shell’s displacements are finite, then the local bearing of the shell and impactor’s materials is neglected with respect to the shell deflection in the contact region. In other words, the Hertz’s theory, which is traditionally in hand for solv- ing impact problems, is not used in the present study; in- stead, the method of multiple time scales is adopted, which is used with much success for investigating vibrations of nonlinear systems subjected to the conditions of the inter- nal resonance. Keywords – Impact-induced internal resonance, impact response, doubly curved shallow panels with rectangular platform, method of multiple time scales. I. I NTRODUCTION Doubly curved shallow panels under impact load are encountered frequently in practice [1]. Nonlinear behav- ior of some types of thin-walled doubly curved panels un- der large deformation is very sensitive to their parameters involving their curvatures. Thus, it has been revealed in [2, 3] that such an important nonlinear phenomenon as the occurrence of internal resonances, which are governed by the shell’s parameters, is of fundamental importance in the study of large-amplitude vibrations of doubly curved shal- low shells with rectangular base, simply supported at the four edges and subjected to various dynamic loads. * Some of the results had been presented at the 2015 International Con- ference on Mechanics, Materials, Mechanical and Chemical Engineer- ing (MMMCE’15), Barcelona, Spain, 7-9 April 2015. This research was made possible by the Grant No. 2014/19 as a Government task from the Ministry of Education and Science of the Russian Federation to Voronezh State University of Architecture and Civil Engineering In spite of the fact that the impact theory is substan- tially developed, there is a limited number of papers de- voted to the problem of impact over geometrically nonlin- ear shells. Literature review on this subject could be found in Kistler and Wass [4]. An analysis to predict the transient response of a thin, curved laminated plate subjected to low velocity transverse impact by a rigid object was carried out by Ramkumar and Thakar [5], in so doing the contact force history due to the impact phenomenon was assumed to be a known linear- dependent input to the analysis. The coupled governing equations, in terms of the Airy stress function and shell deformation, were solved using Fourier series expansions for the variables. A methodology for the stability analysis of doubly curved orthotropic shells with simply supported bound- ary condition and under impact load from the viewpoint of nonlinear dynamics was suggested in [1]. The nonlin- ear governing differential equations were derived based on a Donnell-type shallow shell theory, and the displacement was expanded in terms of the eigenfunctions of the linear operator of the motion equation using the Galerkin proce- dure. To analyze the influence of each single mode on the response to impact loading, only one term composed of two half-waves was used in developing the governing equation, whereas the contact force was proposed a priori to be a sine function during the contact duration. The review of papers dealing with the impact response of curved panels and shells shows that a finite element method and such commercial finite element software as ABAQUS and its modifications are the main numerical tools adopted by many researchers [6]–[20], in so doing the load due to low velocity impact was treated as an equiva- lent quasi-static load and Hertzian law of contact was used for finding the peak contact force. Recently a new approach has been proposed for the analysis of the impact interactions of nonlinear doubly curved shallow shells with rectangular base under the low- velocity impact by an elastic sphere [21]. It has been as- INTERNATIONAL JOURNAL OF MECHANICS Volume 9, 2015 ISSN: 1998-4448 329
Transcript
Analysis of the impact-induced two-to-one internalresonance in nonlinear doubly curved shallow panels with
rectangular platform∗
Yury A. Rossikhin, Marina V. Shitikova and Muhammed Salih Khalid J. M.Voronezh State University of Architecture and Civil Engineering
Research Center on Dynamics of Solids and Structures20-letija Oktjabrja Street 84, 394006 Voronezh, RUSSIA
Abstract – The problem of the low-velocity impact ofan elastic sphere upon a nonlinear doubly curved shallowpanel with rectangular platform is investigated. The ap-proach utilized in the present paper is based on the fact thatduring impact only the modes strongly coupled by the two-to-one internal resonance condition are initiated. Such anapproach differs from the Galerkin method, wherein reso-nance phenomena are not involved. Since is it assumed thatshell’s displacements are finite, then the local bearing of theshell and impactor’s materials is neglected with respect tothe shell deflection in the contact region. In other words,the Hertz’s theory, which is traditionally in hand for solv-ing impact problems, is not used in the present study; in-stead, the method of multiple time scales is adopted, whichis used with much success for investigating vibrations ofnonlinear systems subjected to the conditions of the inter-nal resonance.
Keywords – Impact-induced internal resonance, impactresponse, doubly curved shallow panels with rectangularplatform, method of multiple time scales.
I. INTRODUCTION
Doubly curved shallow panels under impact load areencountered frequently in practice [1]. Nonlinear behav-ior of some types of thin-walled doubly curved panels un-der large deformation is very sensitive to their parametersinvolving their curvatures. Thus, it has been revealed in[2, 3] that such an important nonlinear phenomenon as theoccurrence of internal resonances, which are governed bythe shell’s parameters, is of fundamental importance in thestudy of large-amplitude vibrations of doubly curved shal-low shells with rectangular base, simply supported at thefour edges and subjected to various dynamic loads.
∗Some of the results had been presented at the 2015 International Con-ference on Mechanics, Materials, Mechanical and Chemical Engineer-ing (MMMCE’15), Barcelona, Spain, 7-9 April 2015. This research wasmade possible by the Grant No. 2014/19 as a Government task from theMinistry of Education and Science of the Russian Federation to VoronezhState University of Architecture and Civil Engineering
In spite of the fact that the impact theory is substan-tially developed, there is a limited number of papers de-voted to the problem of impact over geometrically nonlin-ear shells. Literature review on this subject could be foundin Kistler and Wass [4].
An analysis to predict the transient response of a thin,curved laminated plate subjected to low velocity transverseimpact by a rigid object was carried out by Ramkumar andThakar [5], in so doing the contact force history due to theimpact phenomenon was assumed to be a known linear-dependent input to the analysis. The coupled governingequations, in terms of the Airy stress function and shelldeformation, were solved using Fourier series expansionsfor the variables.
A methodology for the stability analysis of doublycurved orthotropic shells with simply supported bound-ary condition and under impact load from the viewpointof nonlinear dynamics was suggested in [1]. The nonlin-ear governing differential equations were derived based ona Donnell-type shallow shell theory, and the displacementwas expanded in terms of the eigenfunctions of the linearoperator of the motion equation using the Galerkin proce-dure. To analyze the influence of each single mode on theresponse to impact loading, only one term composed of twohalf-waves was used in developing the governing equation,whereas the contact force was proposed a priori to be a sinefunction during the contact duration.
The review of papers dealing with the impact responseof curved panels and shells shows that a finite elementmethod and such commercial finite element software asABAQUS and its modifications are the main numericaltools adopted by many researchers [6]–[20], in so doing theload due to low velocity impact was treated as an equiva-lent quasi-static load and Hertzian law of contact was usedfor finding the peak contact force.
Recently a new approach has been proposed for theanalysis of the impact interactions of nonlinear doublycurved shallow shells with rectangular base under the low-velocity impact by an elastic sphere [21]. It has been as-
INTERNATIONAL JOURNAL OF MECHANICS Volume 9, 2015
ISSN: 1998-4448 329
sumed that the shell is simply supported and partial dif-ferential equations have been obtained in terms of shell’stransverse displacement and Airy’s stress function. The lo-cal bearing of the shell and impactor’s materials has beenneglected with respect to the shell deflection in the contactregion, and therefore, the contact force is found analyticallywithout utilizing the Hertzian contact law. The equationsof motion have been reduced to a set of infinite nonlinearordinary differential equations of the second order in timeand with cubic and quadratic nonlinearities in terms of thegeneralized displacements. Assuming that only two naturalmodes of vibrations dominate and interact during the pro-cess of impact and applying the method of multiple timescales, the set of equations has been obtained, which al-lows one to find the time dependence of the contact forceand to determine the contact duration and the maximal con-tact force.
Further the approach proposed by Rossikhin et al. [21]has been generalized by the authors [22] for studying theinfluence of the impact-induced three-to-one internal reso-nances on the low velocity impact response of a nonlineardoubly curved shallow shell with rectangular platform.
Such an additional nonlinear phenomenon as the inter-nal resonance could be examined only via analytical treat-ment, since any of existing numerical procedures could notcatch this subtle phenomenon. Moreover, impact-inducedinternal resonance phenomena should be studied in detailas their initiation during impact interaction may lead to thefact that the impacted shell could occur under extreme load-ing conditions resulting in its invisible and/or visible dam-age and even failure.
Because of this and to investigate the mechanism ofimpact-induced internal resonance, in the present paper, asemi-analytical method proposed previously [21, 22] is ap-plied to investigate doubly curved shallow panels with arectangular boundary and simply supported boundary con-dition subjected to impact load, resulting in the impact-induced two-to-one internal resonance.
II. PROBLEM FORMULATION AND GOVERNING
EQUATIONS
Assume that an elastic or rigid sphere of mass M movesalong the z-axis towards a thin-walled doubly curved shellwith thickness h, curvilinear lengths a and b, principle cur-vatures kx and ky and rectangular base, as shown in Fig. 1.Impact occurs at the moment t = 0 with the low velocityεV0 at the pointN with Cartesian coordinates x0, y0, whereε is a small dimensionless parameter.
According to the Donnell-Mushtari nonlinear shallowshell theory [23], the equations of motion in terms of lateral
Figure 1: Geometry of the doubly curved shallow shell
deflection w and Airy’s stress function φ have the form
D
h
(∂4w
∂x4+ 2
∂4w
∂x2∂y2+∂4w
∂y4
)=∂2w
∂x2
∂2φ
∂y2
+∂2w
∂y2
∂2φ
∂x2− 2
∂2w
∂x∂y
∂2φ
∂x∂y
+ky∂2φ
∂x2+ kx
∂2φ
∂y2+F
h− ρw, (1)
1E
(∂4φ
∂x4+ 2
∂4φ
∂x2∂y2+∂4φ
∂y4
)= −∂
2w
∂x2
∂2w
∂y2
+(∂2w
∂x∂y
)2
− ky∂2w
∂x2− kx
∂2w
∂y2, (2)
whereD = Eh3
12(1−ν2) is the cylindrical rigidity, ρ is the den-sity, E and ν are the elastic modulus and Poisson’s ratio,respectively, t is time, F = P (t)δ(x− x0)δ(y − y0) is thecontact force, P (t) is yet unknown function, δ is the Diracdelta function, x and y are Cartesian coordinates, overdotsdenote time-derivatives, φ(x, y) is the stress function whichis the potential of the in-plane force resultants
Nx = h∂2φ
∂y2, Ny = h
∂2φ
∂x2, Nxy = −h ∂2φ
∂x∂y. (3)
The equation of motion of the sphere is written as
Mz = −P (t) (4)
subjected to the initial conditions
z(0) = 0, z(0) = εV0, (5)
where z(t) is the displacement of the sphere, in so doing
z(t) = w(x0, y0, t). (6)
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ISSN: 1998-4448 330
Considering a simply supported shell with movableedges, the following conditions should be imposed at eachedge:at x = 0, a
w = 0,∫ b
0
Nxydy = 0, Nx = 0, Mx = 0, (7)
and at y = 0, b
w = 0,∫ a
0
Nxydx = 0, Ny = 0, My = 0, (8)
where Mx and My are the moment resultants.The suitable trial function that satisfies the geometric
boundary conditions is
w(x, y, t) =p∑p=1
q∑q=1
ξpq(t) sin(pπxa
)sin(qπy
b
), (9)
where p and q are the number of half-waves in x and ydirections, respectively, and ξpq(t) are the generalized co-ordinates. Moreover, p and q are integers indicating thenumber of terms in the expansion.
Substituting (9) in (6) and using (4), we obtain
P (t)=−Mp∑p=1
q∑q=1
ξpq(t) sin(pπx0
a
)sin(qπy0
b
).
(10)In order to find the solution of the set of equations (1)
and (2), it is necessary first to obtain the solution of (2).For this purpose, let us substitute (9) in the right-hand sideof (2) and seek the solution of the equation obtained in theform
φ(x, y, t)=m∑m=1
n∑n=1
Amn(t) sin(mπx
a
)sin(nπy
b
),
(11)where Amn(t) are yet unknown functions.
Substituting (9) and (11) in (2) and using the orthogo-nality conditions of sines within the segments 0 ≤ x ≤ aand 0 ≤ y ≤ b, we have
Amn(t) =E
π2Kmnξmn(t) +
4Ea3b3
(m2
a2+n2
b2
)−2
×∑k
∑l
∑p
∑q
Bpqklmnξpq(t)ξkl(t), (12)
where
Bpqklmn = pqklB(2)pqklmn − p
2l2B(1)pqklmn,
B(1)pqklmn=
∫ a
0
∫ b
0
sin(pπxa
)sin(qπy
b
)sin(kπx
a
)× sin
(lπy
b
)sin(mπx
a
)sin(nπy
b
)dxdy,
B(2)pqklmn=
∫ a
0
∫ b
0
cos(pπxa
)cos(qπy
b
)cos(kπx
a
)× cos
(lπy
b
)sin(mπx
a
)sin(nπy
b
)dxdy,
Kmn =(kym2
a2+ kx
n2
b2
)2(m2
a2+n2
b2
)−2
.
Substituting then (9)–(12) in (1) and using the orthog-onality condition of sines within the segments 0 ≤ x ≤ aand 0 ≤ y ≤ b, we obtain an infinite set of coupled non-linear ordinary differential equations of the second order intime for defining the generalized coordinates
ξmn(t) + Ω2mnξmn(t) +
8π2E
a3b3ρ(13)
×∑p
∑q
∑k
∑l
Bpqklmn
(Kkl−
12Kmn
)
× ξpq(t)ξkl(t) +32π4E
a6b6ρ
×∑r
∑s
∑i
∑j
∑k
∑l
∑p
∑q
Brsijmn
× Bpqklijξrs(t)ξpq(t)ξkl(t)
+4Mabρh
sin(mπx0
a
)sin(nπy0
b
)×∑p
∑q
ξpq(t) sin(pπx0
a
)sin(qπy0
b
)=0,
where Ωmn is the natural frequency of the mnth mode ofthe shell vibration defined as
Ω2mn =
E
ρ
[π4h2
12(1− ν2)
(m2
a2+n2
b2
)2
+Kmn
]. (14)
The last term in each equation from (13) describes theinfluence of the coupled impact interaction of the targetwith the impactor of the mass M applied at the point withthe coordinates x0, y0.
It is known [24, 25] that during nonstationary exci-tation of thin bodies not all possible modes of vibrationwould be excited. Moreover, the modes which are stronglycoupled by any of the so-called internal resonance condi-tions are initiated and dominate in the process of vibration,in so doing the types of modes to be excited are dependenton the character of the external excitation.
Thus, in order to study the additional nonlinear phe-nomenon induced by the coupled impact interaction due to(13), we suppose that only two natural modes of vibrationsare excited during the process of impact, namely, Ωαβ andΩγδ .
Then the set of equations (13) is reduced to the follow-ing two nonlinear differential equations:
p11ξαβ + p12ξγδ + Ω2αβξαβ + p13ξ
2αβ + p14ξ
2γδ
+ p15ξαβξγδ+p16ξ3αβ+p17ξαβξ
2γδ=0, (15)
INTERNATIONAL JOURNAL OF MECHANICS Volume 9, 2015
ISSN: 1998-4448 331
p21ξαβ + p22ξγδ + Ω2γδξγδ + p23ξ
2γδ + p24ξ
2αβ
+ p25ξαβξγδ+p26ξ3γδ+p27ξ
2αβξγδ=0, (16)
where coefficients pij (i, j = 1, 2) depending on impactor’smass, the coordinates of the impact point, as well as on theparameters of the target and the numbers of the inducedmodes have the following form:
p11 = 1 +4Mρhab
s21, p22 = 1 +
4Mρhab
s22,
p12 = p21 =4Mρhab
s1s2,
s1 = sin(απx0
a
)sin(βπy0
b
),
s2 = sin(γπx0
a
)sin(δπy0
b
),
while all other coefficients pij (i = 1, 2, j = 3, 4, ..., 7)governed only by the shell parameters and the numbers oftwo interacting impact-induced modes are presented in Ap-pendix.
III. METHOD OF SOLUTION
In order to solve a set of two nonlinear equations (15)and (16), we apply the method of multiple time scales [26]via the following expansions:
ξij(t) = εX1ij(T0, T1) + ε2X2
ij(T0, T1), (17)
where ij = αβ or γδ, Tn = εnt are new independentvariables, among them: T0 = t is a fast scale characterizingmotions with the natural frequencies, and T1 = εt is a slowscale characterizing the modulation of the amplitudes andphases of the modes with nonlinearity.
Considering that
d2
dt2ξij = ε
(D2
0X1ij
)+ ε2
(D2
0X2ij + 2εD0D1X
1ij
),
where ij = αβ or γδ, and Dni = ∂n/∂Tni (n = 1, 2, i =
0, 1), and substituting the proposed solution (17) in (15)and (16), after equating the coefficients at like powers ofε to zero, we are led to a set of recurrence equations tovarious orders:to order ε
p11D20X
11 + p12D
20X
12 + Ω2
1X11 = 0, (18)
p21D20X
11 + p22D
20X
12 + Ω2
2X12 = 0; (19)
to order ε2
p11D20X
21 + p12D
20X
22 + Ω2
1X21 = −2p11D0D1X
11
− 2p12D0D1X12 − p13(X1
1 )2
− p14(X12 )2 − p15X
11X
12 , (20)
p21D20X
21 + p22D
20X
22 + Ω2
2X22 = −2p21D0D1X
11
− 2p22D0D1X12 − p23(X1
1 )2
− p24(X12 )2 − p25X
11X
12 , (21)
where for simplicity is it denoted X11 = X1
αβ , X12 = X1
γδ ,X2
1 = X2αβ , X2
2 = X2γδ , Ω1 = Ωαβ , and Ω2 = Ωγδ.
A. Solution of Equations at Order of ε
Following Rossikhin et al. [27], we seek the solutionof (18) and (19) in the form:
where A1(T1) and A2(T1) are unknown complex func-tions, cc is the complex conjugate part to the precedingterms, and A1(T1) and A2(T1) are their complex conju-gates, ω1 and ω2 are unknown frequencies of the coupledprocess of impact interaction of the impactor and the target,and α1 and α2 are yet unknown coefficients.
Substituting (22) and (23) in (18) and (19) and gather-ing the terms with eiω1T0 and eiω2T0 yields(−p11ω
21 − p12α1ω
21 + Ω2
1
)A1e
iω1T0
+(−p11ω
22 − p12α2ω
22 + Ω2
1
)A2e
iω2T0 +cc= 0, (24)
(−p21ω
21 − p22α1ω
21 + α1Ω2
2
)A1e
iω1T0
+(−p21ω
22 − p22α2ω
22 + Ω2
2α2
)A2e
iω2T0 +cc= 0. (25)
In order to satisfy equations (24) and (25), it is a needto vanish to zero each bracket in these equations. As a re-sult, from four different brackets we have
α1 = −p11ω21 − Ω2
1
p12ω21
, (26)
α1 = − p21ω21
p22ω21 − Ω2
2
, (27)
α2 = −p11ω22 − Ω2
1
p12ω22
, (28)
α2 = − p21ω22
p22ω22 − Ω2
2
. (29)
Since the left-hand side parts of relationships (26) and(27), as well as (28) and (29) are equal, then their right-hand side parts should be equal as well. Now equating thecorresponding right-hand side parts of (26), (27) and (28),(29) we are led to one and the same characteristic equationfor determining the frequencies ω1 and ω2:(
Ω21 − p11ω
2) (
Ω22 − p22ω
2)− p2
12ω4 = 0, (30)
whence it follows that
ω21,2 =
(p22Ω2
1 + p11Ω22
)±√
∆2 (p11p22 − p2
12), (31)
INTERNATIONAL JOURNAL OF MECHANICS Volume 9, 2015
ISSN: 1998-4448 332
∆ =(p22Ω2
1 − p11Ω22
)2+ 4 Ω2
1Ω22p
212.
Reference to relationships (31) shows that as the im-pactor mass M → 0, the frequencies ω1 and ω2 tend tothe natural frequencies of the shell vibrations Ω1 and Ω2,respectively. Coefficients s1 and s2 depend on the num-bers of the natural modes involved in the process of impactinteraction, αβ and γδ, and on the coordinates of the con-tact force application x0, y0, resulting in the fact that theirparticular combinations could vanish coefficients s1 and s2
and, thus, coefficients p12 = p21 = 0.
B. Solution of Equations at Order of ε2 for the Case ofImpact-Induced 2:1 Internal Resonance
Now substituting (22) and (23) in (20) and (21), weobtain
p11D20X
21 + p12D
20X
22 + Ω2
1X21
= −2iω1(p11 + α1p12)eiω1T0D1A1
−2iω2(p11 + α2p12)eiω2T0D1A2
−(p13 + α21p14 + α1p15)A1
[A1e
2iω1T0 + A1
]−(p13 + α2
2p14 + α2p15)A2
[A2e
2iω2T0 + A2
]−2 [p13 + α1α2p14 + (α1 + α2)p15]A1
×[A2e
i(ω1+ω2)T0 + A2ei(ω1−ω2)T0
]+ cc,
(32)p21D
20X
21 + p22D
20X
22 + Ω2
2X22
= −2iω1(p21 + α1p22)eiω1T0D1A1
−2iω2(p21 + α2p22)eiω2T0D1A2
−(p23 + α21p24 + α1p25)A1
[A1e
2iω1T0 + A1
]−(p23 + α2
2p24 + α2p25)A2
[A2e
2iω2T0 + A2
]−2 [p23 + α1α2p24 + (α1 + α2)p25]A1
×[A2e
i(ω1+ω2)T0 + A2ei(ω1−ω2)T0
]+ cc.
(33)Reference to equations (32) and (33) shows that the
following impact-induced two-to-one internal resonancecould occur:
ω1 = 2ω2. (34)
Condition (34) could be initiated by the impactor dur-ing the impact interaction with the target, i.e., the doublycurved shallow panel, since the frequencies ω1 and ω2 de-pend not only on the natural frequencies of two inducedmodes of the vibration of the target, but they depend alsoby the mass of the striker and the position of the impact,what it is evident from (31).
Thus, when the frequencies ω1 and ω2 are coupledby the impact-induced two-to-one internal resonance (34),equations (32) and (33) could be rewritten in the followingform:
p11D20X
21 + p12D
20X
22 + Ω2
1X21
= B1 exp(iω1T0) +B2 exp(iω2T0)
+Reg + cc,
(35)
p21D20X
21 + p22D
20X
22 + Ω2
2X22
= B3 exp(iω1T0) +B4 exp(iω2T0)
+Reg + cc,
(36)
where all regular terms are designated by Reg, and
B1 = −2iΩ21ω−11 D1A1 −
(p13 + α2
2p14 + α2p15
)A2
2,
B2 = −2iΩ21ω−12 D1A2 − 2[p13 + α1α2p14
+ (α1 + α2) p15]A1A2,
B3 = −2iΩ22ω−11 α1D1A1 −
(p23 + α2
2p24 + α2p25
)A2
2,
B4 = −2iΩ22ω−12 α2D1A2 − 2[p23 + α1α2p24
+ (α1 + α2) p25]A1A2.
Let us show that the terms with the exponentsexp(±iωiT0) (i = 1, 2) produce circular terms. For thispurpose we choose a particular solution in the form
X21 p = C1 exp(iω1T0) + cc,
X22 p = C2 exp(iω1T0) + cc, (37)
orX2
1 p = C ′1 exp(iω2T0) + cc,X2
2 p = C ′2 exp(iω2T0) + cc, (38)
where C1, C2 and C ′1, C ′2 are arbitrary constants.Substituting the proposed solution in (35) and (36) or
in (37) and (38), we are led to the following sets of equa-tions, respectively:p12ω
21 (α1C1 − C2) = B1,
p21ω21
(−C1 + 1
α1C2
)= B3,
(39)
or p12ω22 (α2C
′1 − C ′2) = B2,
p21ω22
(−C ′1 + 1
α2C ′2
)= B4.
(40)
From the sets of equations (39) and (40) it is evi-dent that the determinants comprised from the coefficientsstanding at C1, C2 and C ′1, C ′2 are equal to zero, therefore,it is impossible to determine the arbitrary constants C1, C2
and C ′1, C ′2 of the particular solutions (37) and (38), whatproves the above proposition concerning the circular terms.
In order to eliminate the circular terms, the terms pro-portional to eiω1T0 and eiω2T0 should be vanished to zeroputtingBi = 0 (i = 1, 2, 3, 4). So we obtain four equationsfor defining two unknown amplitudes A1(t) and A2(t).However, it is possible to show that not all of these fourequations are linear independent from each other.
For this purpose, let us first apply the operators(p22D
20 + Ω2
2) and (−p12D20) to (35) and (36), respec-
tively, and then add the resulting equations. This procedurewill allow us to eliminate X2
2 . If we apply the operators(−p12D
20) and (p11D
20 +Ω2
1) to (35) and (36), respectively,
INTERNATIONAL JOURNAL OF MECHANICS Volume 9, 2015
ISSN: 1998-4448 333
and then add the resulting equations. This procedure willallow us to eliminate X2
1 . Thus, we obtain[(p11p22−p2
12)D40 +(p11Ω2
2+p22Ω21)D2
0 +Ω21Ω2
2
]X2
1
=[(p22D
20 + Ω2
2)B1 − p12D20B3
]exp(iω1T0)
+[(p22D
20 + Ω2
2)B2 − p12D20B4
]exp(iω2T0)
+ Reg + cc,(41)[
(p11p22−p212)D4
0 +(p11Ω22+p22Ω2
1)D20 +Ω2
1Ω22
]X2
2
=[−p12D
20B1 + (p11D
20 + Ω2
1)B3
]exp(iω1T0)
+[−p12D
20B2 + (p11D
20 + Ω2
1)B4
]exp(iω2T0)
+ Reg + cc.(42)
To eliminate the circular terms from equations (41) and(42), it is necessary to vanish to zero the terms in eachsquare bracket. As a result we obtain
(Ω22 − p22ω
21)B1 + p12ω
21B3 = 0
p12ω21B1 + (Ω2
1 − p11ω21)B3 = 0
(43)
and (Ω2
2 − p22ω22)B2 + p12ω
22B4 = 0
p12ω22B2 + (Ω2
1 − p11ω22)B4 = 0
(44)
From equations (43) and (44) it is evident that the de-terminant of each set of equations is reduced to the char-acteristic equation (30), whence it follows that each pairof equations is linear dependent, therefore for further treat-ment we should take only one equation from each pair inorder that these two chosen equations are to be linear inde-pendent. Thus, for example, taking the first equations fromeach pair and considering relationships (27) and (29), wehave
B1 + α1B3 = 0, (45)
B3 + α2B4 = 0. (46)
Substituting values of B1-B4 in (45) and (46), we ob-tain the following solvability equations:
2iω1D1A1 +b1k1A2
2 = 0, (47)
2iω2D1A2 +b2k2A1A2 = 0, (48)
where
ki =Ω2
1 + α2iΩ
22
ω2i
(i = 1, 2),
b1 = p13 + α22p14 + α2p15 + α1(p23 + α2
2p24 + α2p25),
b2 = 2 p13 + α1α2p14 + (α1 + α2)p15
+α2[p23 + α1α2p24 + (α1 + α2)p25] .
Let us multiply equations (47) and (48) by A1 andA2, respectively, and find their complex conjugates. Afteradding every pair of the mutually adjoint equations with
each other and subtracting one from another, as a result weobtain
2iω1
(A1D1A1−A1D1A1
)+b1k1
(A2
2A1+A22A1
)=0,
(49)
2iω1
(A1D1A1+A1D1A1
)+b1k1
(A2
2A1−A22A1
)=0,
(50)
2iω2
(A2D1A2−A2D1A2
)+b2k2
(A1A
22+A1A
22
)=0,
(51)
2iω2
(A2D1A2+A2D1A2
)+b2k2
(A1A
22−A1A
22
)=0.
(52)Representing A1(T1) and A2(T1) in equations (49)–
(52) in the polar form
Ai(T1) = ai(T1)eiϕi(T1) (i = 1, 2), (53)
we are led to the system of four nonlinear differential equa-tions in a1(T1), a2(T1), ϕ1(T1), and ϕ2(T1)
(a21). = − b1
k1ω1a1a
22 sin δ, (54)
ϕ1 −b1
2k1ω1a−1
1 a22 cos δ = 0, (55)
(a22). =
b2k2ω2
a1a22 sin δ, (56)
ϕ2 −b2
2k2ω2a1 cos δ = 0, (57)
where δ = 2ϕ2−ϕ1, and a dot denotes differentiation withrespect to T1.
From equations (54) and (56) we could find that
b2k2ω2
(a21). +
b1k1ω1
(a22). = 0 (58)
Multiplying equation (58) by MV0 and integratingover T1, we obtain the first integral of the set of equations(54)–(57), which is the law of conservation of energy,
MV0
(b2k2ω2
a21 +
b1k1ω1
a22
)= K0, (59)
where K0 is the initial energy.Considering that K0 = 1
2 MV 20 , equation (59) is re-
duced to the following form:
b2k2ω2
a21 +
b1k1ω1
a22 =
V0
2. (60)
Let us introduce into consideration a new functionξ(T1) in the following form:
a21 =
k2ω2
b2E0ξ(T1), a2
2 =k1ω1
b1E0 [1− ξ(T1)] , (61)
where E0 = V0/2.
INTERNATIONAL JOURNAL OF MECHANICS Volume 9, 2015
ISSN: 1998-4448 334
sumed that the shell is simply supported and partial dif-ferential equations have been obtained in terms of shell’stransverse displacement and Airy’s stress function. The lo-cal bearing of the shell and impactor’s materials has beenneglected with respect to the shell deflection in the contactregion, and therefore, the contact force is found analyticallywithout utilizing the Hertzian contact law. The equationsof motion have been reduced to a set of infinite nonlinearordinary differential equations of the second order in timeand with cubic and quadratic nonlinearities in terms of thegeneralized displacements. Assuming that only two naturalmodes of vibrations dominate and interact during the pro-cess of impact and applying the method of multiple timescales, the set of equations has been obtained, which al-lows one to find the time dependence of the contact forceand to determine the contact duration and the maximal con-tact force.
Further the approach proposed by Rossikhin et al. [21]has been generalized by the authors [22] for studying theinfluence of the impact-induced three-to-one internal reso-nances on the low velocity impact response of a nonlineardoubly curved shallow shell with rectangular platform.
Such an additional nonlinear phenomenon as the inter-nal resonance could be examined only via analytical treat-ment, since any of existing numerical procedures could notcatch this subtle phenomenon. Moreover, impact-inducedinternal resonance phenomena should be studied in detailas their initiation during impact interaction may lead to thefact that the impacted shell could occur under extreme load-ing conditions resulting in its invisible and/or visible dam-age and even failure.
Because of this and to investigate the mechanism ofimpact-induced internal resonance, in the present paper, asemi-analytical method proposed previously [21, 22] is ap-plied to investigate doubly curved shallow panels with arectangular boundary and simply supported boundary con-dition subjected to impact load, resulting in the impact-induced two-to-one internal resonance.
II. PROBLEM FORMULATION AND GOVERNING
EQUATIONS
Assume that an elastic or rigid sphere of mass M movesalong the z-axis towards a thin-walled doubly curved shellwith thickness h, curvilinear lengths a and b, principle cur-vatures kx and ky and rectangular base, as shown in Fig. 1.Impact occurs at the moment t = 0 with the low velocityεV0 at the pointN with Cartesian coordinates x0, y0, whereε is a small dimensionless parameter.
According to the Donnell-Mushtari nonlinear shallowshell theory [23], the equations of motion in terms of lateral
Figure 1: Geometry of the doubly curved shallow shell
deflection w and Airy’s stress function φ have the form
D
h
(∂4w
∂x4+ 2
∂4w
∂x2∂y2+∂4w
∂y4
)=∂2w
∂x2
∂2φ
∂y2
+∂2w
∂y2
∂2φ
∂x2− 2
∂2w
∂x∂y
∂2φ
∂x∂y
+ky∂2φ
∂x2+ kx
∂2φ
∂y2+F
h− ρw, (1)
1E
(∂4φ
∂x4+ 2
∂4φ
∂x2∂y2+∂4φ
∂y4
)= −∂
2w
∂x2
∂2w
∂y2
+(∂2w
∂x∂y
)2
− ky∂2w
∂x2− kx
∂2w
∂y2, (2)
whereD = Eh3
12(1−ν2) is the cylindrical rigidity, ρ is the den-sity, E and ν are the elastic modulus and Poisson’s ratio,respectively, t is time, F = P (t)δ(x− x0)δ(y − y0) is thecontact force, P (t) is yet unknown function, δ is the Diracdelta function, x and y are Cartesian coordinates, overdotsdenote time-derivatives, φ(x, y) is the stress function whichis the potential of the in-plane force resultants
Nx = h∂2φ
∂y2, Ny = h
∂2φ
∂x2, Nxy = −h ∂2φ
∂x∂y. (3)
The equation of motion of the sphere is written as
Mz = −P (t) (4)
subjected to the initial conditions
z(0) = 0, z(0) = εV0, (5)
where z(t) is the displacement of the sphere, in so doing
z(t) = w(x0, y0, t). (6)
INTERNATIONAL JOURNAL OF MECHANICS Volume 9, 2015
ISSN: 1998-4448 335
Figure 2: Phase portrait: Ω1 = 2Ω2
velocity along the vertical lines δ = ±πn (n = 0, 1, 2, ...)has the aperiodic character, while in the vicinity of the lineξ = 1/3 it possesses the periodic character.
IV. INITIAL CONDITIONS
In order to construct the final solution of the problemunder consideration, i.e. to solve the set of equations (54)–(57) involving the functions a1(T1), a2(T1), or ξ(T1), aswell as ϕ1(T1), and ϕ2(T1), or δ(T1), it is necessary to usethe initial conditions
w(x, y, 0) = 0, (72)
w(x0, y0, 0) = εV0, (73)
b2k2ω2
a21(0) +
b1k1ω1
a22(0) = E0. (74)
The two-term relationship for the displacement w (9)within an accuracy of ε according to (17) has the form
w(x, y, t) = ε[X1αβ(T0, T1) sin
(απxa
)sin(βπyb
)+X1
γδ(T0, T1) sin(γπxa
)sin(δπyb
)]+O(ε2).
(75)Substituting (22) and (23) in (75) with due account for
(53) yields
w(x, y, t) = 2εa1(εt) cos [ω1t+ ϕ1(εt)]
+a2(εt) cos [ω2t+ ϕ2(εt)]
sin(απxa
)sin(βπyb
)+2ε
α1a1(εt) cos [ω1t+ ϕ1(εt)]
+α2a2(εt)cos[ω2t+ϕ2(εt)]
sin(γπxa
)sin(δπyb
)]+O(ε2).
(76)Differentiating (76) with respect to time t and limiting
ourselves by the terms of the order of ε, we could find thevelocity of the shell at the point of impact as follows
w(x0, y0, t) = −2εω1a1(εt) sin [ω1t+ ϕ1(εt)]
+ω2a2(εt) sin [ω2t+ ϕ2(εt)]s1
−2εα1ω1a1(εt) sin [ω1t+ ϕ1(εt)]
+α2ω2a2(εt) sin [ω2t+ ϕ2(εt)]s2 +O(ε2).
(77)
Substituting (76) in the first initial condition (72) yields
a1(0) cosϕ1(0) + a2(0) cosϕ2(0) = 0, (78)
α1a1(0) cosϕ1(0) + α2a2(0) cosϕ2(0) = 0. (79)
From equations (78) and (79) we find that
cosϕ1(0) = 0, cosϕ2(0) = 0, (80)
INTERNATIONAL JOURNAL OF MECHANICS Volume 9, 2015
ISSN: 1998-4448 336
whence it follows that
ϕ1(0) = ±π2, ϕ2(0) = ±π
2, (81)
andcos δ0 = cos [2ϕ2(0)− ϕ1(0)] = 0, (82)
i.e.,δ0 = ±π
2± 2πn. (83)
The signs in (81) should be chosen considering the factthat the initial amplitudes are positive values, i.e. a1(0) >0 and a2(0) > 0. Assume for definiteness that
ϕ1(0) = −π2, ϕ2(0) =
π
2. (84)
Substituting now (77) in the second initial condition(73) with due account for (84), we obtain
ω1(s1 + α1s2)a1(0)− ω2(s1 + α2s2)a2(0) = E0. (85)
From equations (74) and (85) we could determine theinitial amplitudes
a2(0) =ω1(s1 + α1s2)ω2(s1 + α2s2)
a1(0)− E0
ω2(s1 + α2s2), (86)
c1a21(0) + c2a1(0) + c3 = 0, (87)
where
c1 = 1 +b1k2ω1(s1 + α1s2)2
b2k1ω2(s1 + α2s2)2,
c2 = −b1k2(s1 + α1s2)2E0
b2k1ω2(s1 + α2s2)2,
c3 =b1k2E
20
b2k1ω1ω2(s1 + α2s2)2− k2ω2E0
b2.
From equations (86) and (87) it is evident that the ini-tial magnitudes depend on the mass and the initial velocityof the impactor, on the coordinates of the point of impact,as well as on the numbers of the two modes induced by theimpact.
Considering (82), from (66) we find the value of con-stant G0
G0 = 0 (88)
Reference to (67) shows that G0 could be zero in threecases: at ξ0 = 0, ξ0 = 1, or when cos δ0 = 0. Theabove analysis of the phase portrait has revealed that thecase ξ0 = 0 is not realized. As for the case ξ0 = 1, thenthe solution for the phase modulated motion takes the formof (71). However, for the found magnitudes of the initialphase difference δ0 (83), the value of tan
(δ02 + π
4
)in (71)
is either equal to zero or to infinity, what means that thiscase could not be realized as well.
That is why in further treatment we will analyze onlythe third case, resulting in the amplitude modulated motion(70) with
δ(T1) = δ0 = const. (89)
Thus, we have determined all necessary constants fromthe initial conditions, therefore we could proceed to theconstruction of the solution for the contact force.
V. CONTACT FORCE
Substituting relationship (77) differentiated one timewith respect to time t in (4), we could obtain the contactforce P (t)
P (t) = 2εMω2
1a1(εt) cos [ω1t+ ϕ1(εt)]
+ω22a2(εt) cos [ω2t+ ϕ2(εt)]
s1
+2εMα1ω
21a1(εt) cos [ω1t+ ϕ1(εt)]
+α2ω22a2(εt) cos [ω2t+ ϕ2(εt)]
s2 +O(ε2).
(90)
From equations (55) and (57) with due account for (89)it follows that
ϕ1(T1) = const = ϕ1(0),ϕ2(T1) = const = ϕ2(0).
(91)
Considering (91) and (84), equation (90) is reduced to
P (t) = 2εMω22
8(s1 + α1s2)a1(εt) cosω2t
−(s1 + α2s2)a2(εt)
sinω2t.(92)
Substituting (61) in (92), we finally obtained
P (t) = 2εMω22
√E0
8(s1 + α1s2)
×√
k2ω2b2
√ξ(εt) cosω2t
− (s1 + α2s2)√
k1ω1b1
√1− ξ(εt)
sinω2t,
(93)
where the function ξ(εt) is defined by (70).Since the duration of contact is a small value, then P (t)
could be calculated via an approximate formula, which isobtained from (92) at εt ≈ 0
P (t) ≈ 16εMω22
(cosω2t− 1
8 æ)
×(s1 + α1s2)a1(0) sinω2t+O(ε2),(94)
where the dimensionless parameter æ
æ =(s1 + α2s2)(s1 + α1s2)
a2(0)a1(0)
(95)
is defined by the parameters of two impact-induced modescoupled by the two-to-one internal resonance (34), as wellas by the coordinates of the point of impact and the initialvelocity of impact.
The dimensionless time τ = ω2t dependence of thedimensionless contact force P ∗
P ∗(τ) ≈(
cos τ − 18
æ)
sin τ, (96)
INTERNATIONAL JOURNAL OF MECHANICS Volume 9, 2015
ISSN: 1998-4448 337
Figure 3: Dimensionless time dependence of the dimen-sionless contact force
where
P ∗(τ) =P (t)
16εMω22(s1 + α1s2)a1(0)
,
is shown in Figure 3 for the different magnitudes of theparameter æ: 0.008, 1, 2, and 4.
Reference to Figure 3 shows that the decrease in theparameter æ results in the increase of both the maximalcontact force and the duration of contact. In other words,from Figure 3 it is evident that the peak contact force andthe duration of contact depend essentially upon the param-eters of two impact-induced modes coupled by the two-to-one internal resonance (34).
VI. PARTICULAR CASE
It has been emphasized in the above reasoning that theinitial amplitudes and phases, as well as the contact forcedepend essentially on the location of the point of impact,i.e., on the coordinates of this point.
Let us consider below a particular case when one ofthe coefficients s1 or s2 vanishes to zero due to a partic-ular combination the position of the impact point and thenumbers of impact-induced modes of vibration. We willexamine below the case s1 6= 0 and s2 = 0, since anotherone, when s1 = 0 and s2 6= 0, could be treated in a similarway.
Thus for this particular case the coefficients p22 = 1,p12 = p21 = 0, and nonlinear differential equations (15)and (16) are reduced to the following set of equations:
p11ξαβ + Ω2αβξαβ + p13ξ
2αβ + p14ξ
2γδ
+ p15ξαβξγδ+p16ξ3αβ+p17ξαβξ
2γδ=0, (97)
ξγδ + Ω2γδξγδ + p23ξ
2γδ + p24ξ
2αβ
+ p25ξαβξγδ+p26ξ3γδ+p27ξ
2αβξγδ=0, (98)
where only one coefficient p11 depend on impactor’s mass,the coordinates of the impact point, as well as on the param-eters of the target and the numbers of the induced modes.
Reference to (97) and (98) shows that, despite from(15) and (16), the linear parts of (97) and (98) are uncou-pled. Therefore using expansions (17) and the method ofmultiple time scales, we arrive at the following set of re-currence equations to various orders:to order ε
p11D20X
11 + Ω2
1X11 = 0, (99)
D20X
12 + Ω2
2X12 = 0; (100)
to order ε2
p11D20X
21 + Ω2
1X21 = −2p11D0D1X
11 − p13(X1
1 )2
− p14(X12 )2 − p15X
11X
12 , (101)
D20X
22 + Ω2
2X22 = −2D0D1X
12 − p23(X1
1 )2
− p24(X12 )2 − p25X
11X
12 , (102)
where for simplicity once again is it denoted X11 = X1
αβ ,X1
2 = X1γδ , X
21 = X2
αβ , X22 = X2
γδ, Ω1 = Ωαβ , andΩ2 = Ωγδ .
The solution of (99) and (100) has the form
X11 = A1(T1) eiω1T0 + A1(T1)e−iω1T0 , (103)
X12 = A2(T1) eiΩ2T0 + A2(T1)e−iΩ2T0 , (104)
where ω21 = Ω2
1/p11.Substituting (103) and (104) in (101) and (102) yields
p11D20X
21 + Ω2
1X21 = −2iω1p11D1A1e
iω1T0
−p13A21e
2iω1T0 − p13A1A1 − p14A22e
2iΩ2T0
−p14A2A2 − p15A1A2ei(ω1+Ω2)T0
+A1A2ei(ω1−Ω2)T0 + cc,
(105)
D20X
22 + Ω2
2X22 = −2iΩ2D1A2e
iω2T0
−p23A22e
2iω2T0 − p23A2A2 − p24A21e
2iω1T0
−p24A1A1 − p25A1A2ei(ω1+ω2)T0
−p25A1A2ei(ω1−ω2)T0 + cc.
(106)
Reference to equations (105) and (106) shows that thefollowing impact-induced two-to-one internal resonancescould occur:
ω1 = 2Ω2, (107)
orΩ2 = 2ω1. (108)
A. Internal Resonance ω1 = 2Ω2
In this case we obtain the following solvability equa-tions:
2iω1D1A1 +p14
p11A2
2 = 0, (109)
INTERNATIONAL JOURNAL OF MECHANICS Volume 9, 2015
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2iω2D1A2 + p25A1A2 = 0. (110)
Comparison of solvability equations (109) and (110)with those obtained above for the general case, i.e., with(47) and (48), shows that they are similar in structure withinan accuracy of the coefficients. That is why the solution of(47) and (48) presented in Sec. III B is valid for equations(109) and (110) as well under the proper substitution ofcoefficients b1/k1 and b2/k2 with p14/p11 and p25, respec-tively.
In the case under consideration, the two-term relation-ship for the displacement w (9) within an accuracy of εaccording to (17) has the form
w(x, y, t) = 2εa1(εt) cos [ω1t+ ϕ1(εt)]
× sin(απxa
)sin(βπyb
)+a2(εt)cos[Ω2t+ϕ2(εt)] sin
(γπxa
)sin(δπyb
)+O(ε2).
(111)Differentiating (111) with respect to time t and limiting
ourselves by the terms of the order of ε, we could find thevelocity of the shell at the point of impact as follows
w(x0, y0, t) = −2εω1s1a1(εt) sin [ω1t+ ϕ1(εt)]+O(ε2).(112)
Now substituting (111) and (112) in the initial condi-tions (72)-(74), we could find the initial amplitudes andphases:
ϕ1(0) = −π2, ϕ2(0) =
π
2. (113)
a1(0) =V0
2ω1s1, a2(0) =
√V0p25
2Ω2
(1− V0p11
2ω1s21p14
).
(114)As a result the contact force could be written as
P (t) ≈ 2εMω21a1(0) sinω1t = εMV0ω1s
−11 sinω1t,
= 2εMV0Ω2s−11 sin 2Ω2t. (115)
Reference to (115) shows that the contact force de-pends on the mass of the striker, its initial velocity, the co-ordinates of the point wherein the impact occur ed, as wellas on the frequencies and umbers of two modes coupled bythe impact-induced two-to-one internal resonance (107).
Relationship for the time-dependence of the contactforce (115) obtained above analytically verifies the as-sumption for the contact force proposed in [1] a knownsine-dependent input to the analysis.
Note that another particular case of the impact-inducedtwo-to-one internal resonance (108) could be treated in asimilar way.
As for the particular case when s1 = s2 = 0, then itcould not be realized in this problem, since this conditionresults in the violation of the second initial condition (73).
VII. CONCLUSION
In the present paper, a new approach proposed recentlyby the authors for the analysis of the impact interactionsof nonlinear doubly curved shallow shells with rectangu-lar base under the low-velocity impact by an elastic sphere[21, 22] has been generalized for investigating the impact-induced two-to-one internal resonance.
Such an approach differs from the Galerkin method,wherein resonance phenomena are not involved [1]. Sinceis it assumed that shell’s displacements are finite, then thelocal bearing of the shell and impactor’s materials is ne-glected with respect to the shell deflection in the contactregion. In other words, the Hertz’s theory, which is tradi-tionally in hand for solving impact problems, was not usedin the present study; instead, the method of multiple timescales has been adopted, which is used with much successfor investigating vibrations of nonlinear systems subjectedto the conditions of the internal resonance, as well as to findthe time dependence of the contact force.
It has been shown that the time dependence of the con-tact force depends essentially on the position of the point ofimpact and the parameters of two impact-induced modescoupled by the internal resonance. Besides, the contactforce depends essentially on the magnitude of the initialenergy of the impactor. This value governs the place onthe phase plane, where a mechanical system locates at themoment of impact, and the phase trajectory, along which itmoves during the process of impact.
It is shown that the intricate P (t) dependence atimpact-induced internal resonance (90) give way to rathersimple sine time-dependence, what is an accordance witha priori assumption of some researchers about a sine-likecharacter of the contact force with time [1], [28]–[31].
The procedure suggested in the present paper could begeneralized for the analysis of impact response of platesand shells when their motions are described by three or fivenonlinear differential equations.
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INTERNATIONAL JOURNAL OF MECHANICS Volume 9, 2015
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