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Coupled nonlinear behavior of beam with a moving mass
Applied Acoustics 156 (2019) 367–377
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Coupled nonlinear behavior of beam with a moving mass Anwesa Mohanty ⇑, Moses Prasad Varghese, Rabindra Kumar Behera National Institute of Technology, Rourkela, Odisha, India
a r t i c l e
i n f o
Article history: Received 27 December 2018 Received in revised form 10 May 2019 Accepted 17 July 2019
a b s t r a c t In this article, it is proposed to comprehend the coupled nonlinear analysis of beam under three different boundary conditions, such as (1) cantilever, (2) fixed-fixed and (3) simply supported having moving mass. Due to the beam and mass interaction, coupling term arises and it results kinematic nonlinearities in the system. A fundamental geometric model is developed by considering nonlinearities of the system. Euler-Bernoulli beam assumptions are considered. By applying Hamilton’s principle a coupled mathematical formulation of the desired system is derived. Further Galerkin discretization method is used in the mathematical system to analyse dynamic behaviour by converting it from continuous to discrete problem. The resulting equations are solved using perturbation method. Then MATLAB ODE solver is used to plot various graphs for variation of amplitude and deflection with respect to time in case of both beam and mass. Under the internal resonance condition the time response curves are plotted to analyze the beating phenomenon for the beam and mass. Due to small time steps and very high frequency variation, mass position and tip deflection plots appear dark. Finally, a comparative study is also made for various detuning parameter to show the modal behavior of beam-mass system. Ó 2019 Elsevier Ltd. All rights reserved.
1. Introduction A wide range of work on dynamic response of structures has already been carried out in the past decades. Dynamic analysis is the field of analyzing the dynamic properties of system under vibrational excitation. Moving particles/loads have a noticeable influence on dynamic behavior of structures and force them to vibrate at high velocity. Some of the practical examples in our day-to-day life are bridges, machining operation and beams subjected to pressure operation. For this reason dynamic analysis has got the fundamental attention of the researchers. Among many different cases of study some authors focuses on the loads moving with a constant and non-uniform velocity. For broad knowledge of moving load problems many references are available in the book by Fryba [1]. Michaltsos et al. [2], Ye and Chen [3] studied effect of moving load on dynamic characteristic of simply supported beam (SSB) by changing different parameters. Later in 1997 a comparison study of moving mass and moving force is done by using finite difference method [4]. Nonlinear coupled vibration analysis of beam having moving support under the action of moving mass is solved numerically [5–7]. Different numerical techniques like method of integral equation, Galerkin’s method, approximate method including asymptotic solution are introduced for solution ⇑ Corresponding author at: NIT Rourkela, Odisha- 769008, India. Tel.: +918249373048, fax.: 0661-2472926. E-mail address: [email protected] (A. Mohanty). https://doi.org/10.1016/j.apacoust.2019.07.024 0003-682X/Ó 2019 Elsevier Ltd. All rights reserved.
purpose. In 2000, Ichikawa et al. [8], Hilal and Zibdeh [9] looked over the dynamic characteristics of beam having different boundary conditions loaded with moving mass. For further analysis Kessel and Schlack [10] introduced damping phenomena to SSB subjected to cyclic moving load. Wu et al. [11] presented different approach of finite element method (FEM) to do dynamic analysis of structure having single and double moving load. Initially they used finite element packages for simply supported beam having point moving load and further 3-D structure having moving loads is analyzed. Theoretical analysis of dynamic response for Timoshenko beam under moving mass [12] and accelerated moving mass [13] is carried by implementing different numerical techniques. A comparative study is carried out over the dynamic characteristics of cantilever beam under moving mass [14] using numerical and perturbation method. Further numerical iterative techniques such as Ritz and average acceleration methods are implemented in other research for the modal analysis [15,16]. Some papers applied precise time-step integration method [16] to analyze dynamic response of continuous system having moving loads. Convenient use of spectrogram helps to capture the nonstationary behavior of the signal. Thus, Abdomian’s decomposition and homotopy perturbation methods are used for the numerical analysis of dynamic behavior of beam-mass system [17]. Aziz et al. [18] and Johansson et al. [19] investigated various dynamic analysis considering continuous system and bridges subjected to different loading conditions. In some other cases [18,20] researchers are used spectral element method to get the the exact dynamic response. Analytical solution for both Timoshenko beam [21] and Euler-Bernoulli
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[22] beam considering visco-elastic foundation under different loading conditions are further studied. In 2016 Svedholm et al. [23] studied analytically the vibration characteristics of damped uniform beam having different boundary conditions under moving loads. They derived orthogonality conditions and closed form solution to analyze dynamic response. Jorge et al. [24] investigated the effects of foundation stiffness in case of beam having nonlinear elastic foundation and another work is done by Nguyen et al. [25] in 2016 on the nonlinear foundation by considering nonlinear wrinkle and the linear Pasternak foundation parameters. Some researchers focused on linear and nonlinear vibration analysis of functionally graded (FG) plate under moving load [26,27]. Lal et al. Malekzadeh and Monajjemzadeh [28,29] studied the thermal environment effect on dynamic behavior of FG beam and plate. In most of the cases, authors employed FEM with differential quadrature and newmark’s time integration method [25,28,29] to solve the system of problem. In recent year a numerical analysis of three dimensional beam under moving mass is also carried out considering geometrical non-linearity [30]. As micro and nano-beams are challenges for researchers, different studies are carried out by considering different structures and solution methods. Further study is continued in the field of dynamic analysis by analyzing vibrational characteristics of beam containing FG nanocomposite for various beam theories [31]. Nonlinear behavior of the micro beam system was studied by using differential quadrature and iterative method [32,33]. Sharma and Kaur [34] developed a mathematical model to study the anisotropic, thermo-elastic characteristic of a clamped-clamped micro-beam. Recently, axial and transverse dynamic study was represented by Hosseini and Rahmani [35] for nano-beam excited by constant moving load. Wang et al. [36] contemplated the vibration response of composite beam reinforced with graphene nanoplatelet excited by two traversing mass. Hashemi and Khaniki et al. [37] discussed the dynamic behavior of nano beam system using Erigen’s nanolocal theory excited by nano particle. Although many research works has done on this topic over the past decades, but due to the complexity and real physical applications of moving mass problems, still it is the main point of interest for researchers. In case of moving mass it considers the gravitational and inertia effect of moving load where as in case of moving load no such complexity is there. Present study is focused on the dynamic behavior of beam with general boundary conditions having spring mass system as moving mass.
Fig. 1. Schematic diagram of (a) cantilever beam (b) fixed-fixed and (c) Simply supported beam with moving mass.
of beams. U, V, W denotes the displacements in the undeformed X, Y and Z direction respectively. To introduce the geometrical nonlinearity large displacement field of corresponding beam-mass system is obtained by considering horizontal (u) and vertical (v) deformation as:
U ¼ u v sinðhÞ; where sinðhÞ ¼ h ¼
@x
v ðx; tÞ
V ¼ v ðx; tÞ v ð1 cosðhÞÞ; where cosðhÞ ¼ 1 þ @u @x
ð1Þ
W ¼0
In case of small displacement and rotation only vertical deformation v(x, t) is assumed and the corresponding following displacement field is obtained as: @ U ¼ v sinðhÞ; sinðhÞ ¼ h ¼ @x v ðx; tÞ (2) V ¼ v ðx; tÞ; cosðhÞ ¼ 1 W¼0 Equation of motion The velocity field for both beam and mass is retrieved from respective displacement field by differentiating it with respect to time. Hence in case of large displacement it gives
U t ¼ ut v cosðhÞ
2. Analytical formulation
@
@h @h ; V t ¼ v t v sinðhÞ @t @t
ð3Þ
Whereas in case of small displacement it results In the current research work, assumptions which are taken into considerations are as follow: (i) beam obeys the Euler-Bernoulli beam theory, (ii) moving mass remains in contact with beam during the analysis. Fig. 1 shows the uniform beam-mass system with different boundary conditions such as, fixed-free (cantilever), fixed–fixed and simply supported. The following nomenclatures are used for the theoretical analysis; Length of beam is ‘L’, cross-sectional area ‘a’, mass density ‘q’ and flexural rigidity ‘EI’. The moving mass ‘M’ slides along the beam are attached with a spring of stiffness ‘k’ from the left end. The position of the mass is ‘s(t)’. Here all these dimenssionalized parameters are normalized by dividing required parameters for the proper calculation.
Therefore,
T ¼ t 0 =s,s ¼ S=L,A ¼ a=L2 ,m ¼ M=qAL
U mt ¼ U t þ st cosðhÞ; V mt ¼ V t þ st sinðhÞ
The Euler-Bernoulli beam model is assumed for the numerical study. Fig. 2 presents the diagram for small and large deformation
ð4Þ
ð5Þ
Here, U t ; V t ,U mt and V mt are the velocity components in X, Y direction of beam and traversing mass respectively. Subscript ‘m’ stands for moving mass. st is the velocity due to position of moving mass. Neglecting the velocity due to rotation KEb is obtained for large and small displacement respectively as
KEb ¼
3. Theoretical analysis and results 3.1. Displacement field
@h @ v ðx; tÞ ; Vt ¼ @t @t
Similarly in both the cases for large and small displacement velocity of moving mass is obtained as
4
andx ¼ kqAl =mEI: 2
U t ¼ v cosðhÞ
KEb ¼
1 2 1 2
Z
l
0
Z 0
l
2 ut þ v 2t dx
ð6Þ
ðv t Þ2 dx
ð7Þ
KEm can be derived for both the cases as:
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A. Mohanty et al. / Applied Acoustics 156 (2019) 367–377
Fig. 2. Small deflection of beam element and (b) large deflection of beam element.
KEm ¼
1 1 mðU mt Þ2 þ mðV mt Þ2 2 2
ð8Þ
1 2 m st þ u2t þ v 2t þ 2st cosðhÞut þ 2st sinðhÞv t X¼s 2
Z d
ð9Þ
i 1 h 2 m ðst Þ þ ðv t Þ2 þ 2st sinðhÞv t X¼s 2
PEb ¼
1 2
l
0
ð11Þ
Z
Z 0
l
@2v @x2
t2
d
ð13Þ
ðv xx Þ2 dx
ð14Þ
Ldt ¼ 0
t1
Z
t2
t2
Z 0
l
Z
t2
t1
ðutt du þ v tt dv Þdtdx
ð17Þ
v tt þ st v xt þ stt v x þ st v xt þ s2t v xx dv dt m
ðutt þ stt þ 2st uxt þ stt ux Þdudt
ð15Þ
t1
where, L = Kinetic energy (KE) – Potential energy (PE) =ðKEb þ KEm Þ ðPEb þ PEm Þ Cwhere C is the Lagrangian constant. Rl For large deformationC ¼ 12 0 kðx; tÞ v 2x þ ð1 þ ux Þ2 1 dx whereas for small deformation C ¼ 0. For large deformation equation of motion is obtained by substituting Eqs. (6), (9), (11) and (13) into Eq. (15), the resulting equation leads to: Z Z t2 1 l 2 1 d u þ v 2t dx þ m s2t þ u2t þ v 2t þ 2st cosðhÞut þ 2st sinðhÞv t X¼se 2 0 t 2 t1 Z 1 1 l 2 ðv xx cosðhÞ v xx sinðhÞÞ dx mx 2 ð s s e Þ 2 2 2 0 ! Z 1 l ð16Þ kðx;tÞ v 2x þ u2x þ 2ux dx dt ¼ 0 2 0 By integrating each terms of Eq. (16) individually through integration by parts and operating the d operator gives,
1 mx2 ðs se Þ2 dt ¼ 2
d t1
Z
Z
t2
l
ð18Þ
0
Z
Z
t2 t1
mx2 ðs se Þdsdt
Z
t2
0
t1
l
ððv xxxx 2v xxxx ux 3v xxx uxx 2v xx uxxx þ uxxxx v x
2v xx ux uxxx 2v xx u2xx 4v xxx ux uxxx u2x v xxxx þ 4uxx uxxx v x þ2u2xx v xx þ v x ux uxxxx þ v x uxx uxxx þ v xx ux uxxx dv þ 5v xx v xxx þ v xxxx v x þ v xx v x uxxx þ 2v 2xx uxx þ 5v xx v xxx ux þv xxxx v x ux 5uxxx v x v xx uxxxx v 2x du dxdt ! Z t2 Z l @ @2h @2h sinh þ cosh dxdt ¼ 2 @x2 0 @x @x t1 Z
Z
t2
l
d 0
t1
Z
t2
Z
t1
ð20Þ
1 2 k v x þ u2x þ 2ux dxdt 2 l
ððkv xx þ kx v x Þdv ðkx ux þ kuxx þ kx Þdu 1 þ v 2x þ u2x þ 2ux dk dxdt 2
¼
ð19Þ
ðv xx cosðhÞ uxx v x Þ2 dxdt ¼
d
ð12Þ
Here subscripts ‘t’ and ‘x’ denote the derivative with respect to time and distance. The equations of motion for the given system are derived using Hamilton’s principle
Z
t2
m
t1
ðv xx cosðhÞ þ uxx sinðhÞÞ dx 2
0
u2t þ v 2t dxdt ¼
1 2 m st þ u2t þ v 2t þ 2st cosðhÞut þ 2st sinðhÞv t X¼se dt 2 Z t2 ¼ mðstt þ v tt v x þ utt ð1 þ ux Þ þ v t v xt þ uxt ut Þdsdt
Z
Potential energy of beam for large and small deformation respectively is obtained as
Z
t1
Here se represents the equilibrium position of spring-mass subsystem.
1 2
l
t2
ð10Þ
1 PEs ¼ mx2 ðs se Þ2 2
From strain displacement relation e ¼ y
Z
t1
Potential energy due to spring attached to mass is
PEb ¼
1 2
t1
In case of small deformation after substitution of Eq. (5), to simplify the problem, rotary inertia of the beam is dropped from velocity for slender beam. Hence in this case KEm becomes,
KEm ¼
t2
t1
Substituting Eq. (5) for large and small deformation separately in Eq. (8) results
KEm ¼
Z d
0
ð21Þ
Substituting Eqs. (17)–(21) into Eq. (16) and making the coefficient of ds; du; dv ; dk to zero the following four equations are obtained,
stt þ x2 ðs se Þ þ ðutt cosðhÞ þ v tt sinðhÞÞX¼sðtÞ ¼ 0 m stt cosðhÞ þ utt þ 2st uxt þ s2t uxx X¼sðtÞ Z l @ þ utt ðkcosðhÞ þ hxx sinðhÞÞ dx ¼ 0 @x 0 m stt sinðhÞ þ v tt þ 2st v xt þ s2t v xx X¼sðtÞ Z l @ þ v tt ðksinðhÞ hxx cosðhÞÞ dx ¼ 0 @x 0
ð22Þ
ð23Þ
ð24Þ
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A. Mohanty et al. / Applied Acoustics 156 (2019) 367–377
Z l 1 ux þ u2x þ v 2x dx ¼ 0 2 0
ð25Þ
To derive equation of motion for small deformation, Eqs. (7), (10), (11), (14), are substituted into Eq. (15) which results
Z d
1 2
t1
€ i þ ui ðuj Þ stt þ x2 ðs se Þ þ ui jX¼sðtÞ a x
Z
l
0
Z
t2
Z
l
t1
Z
0
ð35Þ For u variation:
ð26Þ
m stt ðui ÞX¼sðtÞ þ ui uj
Z
1 2 v dxdt ¼ 2 t
Z
t2
t1
l
0
v tt dv dxdt
1 2 m st þ v 2t þ 2st v t sinðhÞ X¼sðtÞ dt 2 Z t2 1 2 ¼d m st þ v 2t þ 2st v t v x X¼sðtÞ dt t1 2 Z t2 ¼ mðst dst þ v t dv t þ dðst v t v x ÞÞX¼sðtÞ dt
¼ m
t2
t1
Z
t2
d t1
1 mx2 ðs se Þ2 dt ¼ mx2 2
Z
¼ mx
2
Z
t2
Z
d t1
0
l
1 2 v dxdt ¼ 2 xx ¼
Z
t2
t1
Z
t2
Z
t1
0
X¼sðtÞ
dt
ð28Þ
0
ð29Þ
t1
l
l
v xx dv xx dxdt v xxxx dv dxdt
ð30Þ
ð31Þ
f ¼ stt v x þ ðst Þ2 v xx þ 2st v tx þ v tt
ð33Þ
When his small, the additional vertical movement of any point located at distance y below neutral axis is also small and therefore it is neglected in above equations. To convert partial differential equation to ordinary differential equation, Galerkin’s method is used with a trial function having separate variables. In case of large deformation, variables are assumed to be
uðx; tÞ ¼
v ðx; tÞ ¼
i¼1
and kðx; tÞ ¼
bi ðtÞui ðxÞ; n P i¼1
ki ðtÞui ðxÞ
1 2
xx
ui uj aj þ ui uj ðuk Þx aj ak þ bj bk x
x
dx ¼ 0
ð37Þ
ð38Þ
v ðx; tÞ ¼
n X
ai ðtÞ/i ðxÞ
ð39Þ
h i stt þ x2 ðs se Þ þ ui uj
x X¼s
ðai Þtt aj ¼ 0
ð40Þ
h i nh i aj tt þ stt ui uj aj m ui uj xx X¼s xx X¼s h i h i o 2 þ 2st ui uj aj t þðst Þ ui uj aj x X¼s xx X¼s "Z # "Z # l l ðui Þxx uj dx aj þ ui uj dx aj tt þ 0
xx
ð41Þ
After using the boundary conditions for cantilever, fixed–fixed and simply supported, the eigen functions are derived as follows; For cantilever beam:
ui ¼ coshðki xÞ cosðki xÞ
cosðki lÞ þ coshðki lÞ ðsinhðki xÞ sinðki lÞ þ sinhðki lÞ ð42Þ
sinðki xÞÞ For fixed–fixed beam:
ui ¼ ðsinhðki xÞ sinðki xÞÞ þ
sinhðki lÞ sinðki lÞ ðcoshðki xÞ cosðki lÞ coshðki lÞ
cosðki xÞÞ
ð43Þ
For simply supported beam:
ai ðtÞui ðxÞ;
i¼1 n P
stt bj þ 2st bj t þ s2t ui uj bj
¼0
where
n P
x X¼sðtÞ
0
ð32Þ
0
X¼sðtÞ
bj tt þ ui uj
By using the above formulation, Eqs. (31) and (32) become
ðv xxxx þ v tt Þdx ¼ 0
mf jx¼sðtÞ þ
tt
For the small deformation, vertical displacement is assumed as
ðs se Þdsdt
v variation: l
ui uj aj
For kvariation:
Z l
ðs se Þdðs se Þdt
stt ðtÞ þ x2 ðsðtÞ se Þ þ v tt v x jX¼sðtÞ ¼ 0 Z
aj þ
x
Further substituting Eqs. (27)–(30) into Eq. (26) and equating the coefficient ofds and dv to zero following equations are obtained, For s variation:
For
Z l
stt aj þ 2st aj t
i¼1
0
Z
X¼sðtÞ
ui uj bj tt þ ðui Þxx uj bj þ ðui Þx uj ðuk Þx kj bk þ xx 0 ðui Þx uj ðuk Þxx ðul Þxx bj ak al þ bj bk bl dx ¼ 0
t1 t2
þ ui u0j
x
Z l
ðstt þ v tt v x Þds þ stt v x þ s2t v xx þ 2st v tx þ v tt dv t2
tt
v variation:
m ui uj
Z
aj
þ ðui Þxx uj aj xx X¼sðtÞ xx 0 þðui Þx uj kj þ ðui Þx uj ðuk Þx kj ak ðui Þt uj ðuk Þtt aj ak þ bj bk tt ðui Þx uj ðuk Þxx ðul Þxx aj ak al þ aj bk bl dx ¼ 0 ð36Þ For
t1
þs2t ui uj
ð27Þ
t2
Z
X¼sðtÞ
d
t1
ðai Þtt aj þ ðbi Þtt bj ¼ 0
v 2t dx þ
Each terms of Eq. (26) is integrated separately using integration by parts followed by the operation of variational operatord, which gives
d
X¼sðtÞ
1 2 m st þ v 2t þ 2st sinðhÞv t X¼se 2 ! Z 1 1 l 2 2 2 v dx dt ¼ 0 mx ðs se Þ 2 2 0 xx t2
In the formulations Eqs. (22)–(25), by implementing variable separation method given in Eq. (34) it results, For s variation:
ð34Þ
ui ¼ sinðki xÞ
ð44Þ
Considering 1st mode ðu1 Þof desired beam as basis function, Eqs. (40) and (41) reduce to:
stt þ x2 s þ c1 ða1 Þtt a1 ¼ 0
ð45Þ
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A. Mohanty et al. / Applied Acoustics 156 (2019) 367–377
ða1 Þtt þ x21 a1 þ mc2 stt a1 þ 2mc2 st ða1 Þt þ 2mc2 sða1 Þtt ¼ 0
where c1 ¼ u1 ðu1 Þx X¼se , c2 ¼ R l 0
Rl
u1 ðu1 Þx jX¼s
e
ðu1 Þ2 dxþmðu1 Þ2 j
@ 2 u2 @T 20
,
þ x21 u2 ¼ 2
ðu 0
2 ð u Þ dx þ mðu1 Þ2
1 0
ð47Þ
X¼se
The final set of equation of motion for both large and small deformation is obtained in Eqs. (35)–(38) and Eqs. (45) and (46) respectively. As method of multiple scales (MMS) is cumbersome to find the solution for large deformation case, average acceleration method can be suitably used to study modal analysis. However for small deformation case the nonlinearity arises due to the interaction between the beam and mass can be solved using MMS [38]. The MMS is implemented in Eqs. (45) and (46) to get the desired solution. Time scale for formulation is defined by using
T n ¼ en t
dT 2
¼
@2 @T 20
þ 2e
@2
ð49Þ
@T 21
Neglecting the higher order terms uniform expansion for s and
@T 20
ð55Þ
s1 ¼ P1 ðT 1 ÞeixT 0 þ P 1 ðT 1 ÞeixT 0 u1 ¼ P 2 ðT 1 Þeix1 T 0 þ P2 ðT 1 Þeix1 T 0
ð56Þ
Here P 1 , P 2 are complex variable and P1 ,P2 are complex conjugate. Substituting Eq. (56) into the right hand side (RHS) of Eqs. (54) and (55), results
RHSð25Þ ¼ 2i
@P 1 xeixT 0 þ c1 P22 x21 e2ix1 T 0 þ c1 P 2 P2 x21 þ 2i @T 1
@P1 2 xeixT 0 þ c1 P2 x21 e2ix1 T 0 þ c1 P2 P2 x21 @T 1
ð57Þ
@P2 x1 eix1 T 0 þ mc2 2x21 þ x2 þ 2xx1 P1 P 2 eiðx1 þxÞT 0 @T 1 @P 2 þ mc2 2x21 þ x2 2xx1 P1 P 2 eiðx1 xÞT 0 þ 2i x1 eix1 T 0 @T 1 þ mc2 2x21 þ x2 þ 2xx1 P1 P 2 eiðx1 þxÞT 0 þ mc2 2x21 þ x2 2xx1 P1 P 2 eiðx1 xÞT 0
RHSð26Þ ¼ 2i
d @ @ ¼ þe dt @T 0 @T 1 2
@ 2 u1
The solution of Eqs. (52) and (53) are
ð48Þ
where e is scaling parameter, n = 0,1,2. . . Here two time scale is considered i.e, T 0 and T 1 as the nonlinearities have a very small effect. By using chain rule, derivative with respect to ‘t’ transformed into ‘T 0 ’and ‘T 1 ’ which is written as follows:
d
@ 2 u1 @ 2 s1 @s1 @u1 mc2 u1 2mc2 @T 0 @T 1 @T 0 @T 0 @T 20
2mc2 s1
X¼se
2 1 Þxx dx
ðx1 Þ2 ¼ R l
ð46Þ
To eliminate the terms tends to secular terms, coefficient of eixT 0 andeix1 T 0 are setting to zero. Internal resonance is assumed to investigate the coupled characteristics of the system as,
a1 is assumed to be,
x ¼ 2x1 þ er
sðtÞ ¼ s1 ðT 0 ; T 1 Þ þ es2 ðT 0 ; T 1 Þ
where ris a small detuning parameter. While ris zero we get perfect internal resonance 1:2. Inserting Eq. (59) into Eqs. (57) and (58) and making the coefficient of secular term to zero, it gives
a1 ¼ a1 ðT 0 ; T 1 Þ þ ea2 ðT 0 ; T 1 Þ
ð50Þ
Substituting above asymptotic expansion into time derivatives provides
st ¼
@s1 @s2 @s1 @s2 þe þe þ e2 @T 0 @T 0 @T 1 @T 1 @ 2 s1
stt ¼
@T 20
þe
@ 2 s2 @T 20
þ 2e
att ¼
@ 2 a1 @T 20
þe
@ 2 a2 @T 20
@ 2 s1 @ 2 s2 þ 2e2 @T 0 @T 1 @T 0 @T 1
þ 2e
@ 2 a1 @ 2 a2 þ 2e2 @T 0 @T 1 @T 0 @T 1
@T 20 @ 2 u1 @T 20
@T 20
þ x2 s1 ¼ 0
ð52Þ
þ x2 u 1 ¼ 0
ð53Þ
þ x2 s2 ¼ 2
P 1 ðT 1 Þ ¼
1 p ðT 1 Þeiu1 ðT 1 Þ 2 1
P 2 ðT 2 Þ ¼
1 p ðT 1 Þeiu2 ðT 1 Þ 2 2
ð51Þ
@ 2 s1 @ 2 u1 c1 u1 @T 0 @T 1 @T 20
ð60Þ
Setting real and imaginary part of Eq. (60) to zero, results
@p1 1 c1 p22 x21 ¼ sinð2/2 /1 rT 1 Þ @T 1 4 x p1
ð54Þ
@ u1 1 c1 p22 x21 ¼ cosð2/2 /1 rT 1 Þ 4 x @T 1
@p2 1 mc2 x2 2xx1 þ 2x21 ¼ p1 p2 sinð2/2 /1 rT 1 Þ 4 @T 1 x1 p2
e1 order: @ 2 s2
@P2 þ mc2 2x21 þ x2 2xx1 P1 P2 eirT 1 ¼ 0 @T 1
Complex variables are taken in polar form as
Putting the above equation into Eqs. (45) and (46), and setting the coefficient of e0 ande1 to zero, gives e0 order:
@ 2 s1
@P1 þ c1 P22 x21 eirT 1 ¼ 0 @T 1
2ix1
@ a1 @ a2 @ a1 @ a2 þe þe þ e2 @T 0 @T 0 @T 1 @T 1
at ¼
2ix
ð59Þ
@ u2 1 mc2 x2 2xx1 þ 2x21 ¼ p1 p2 cosð2/2 /1 rT 1 Þ 4 @T 1 x1 ð61Þ
where, p1 ; u1 are the modal amplitudes and phase angle for mass p2 ; u2 are the amplitudes and phases for beam deflection
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Table 1 Comparison of frequency.
3.3. Validation
Natural Frequency(rad/sec)
Present code
Exact Method [39]
x1 x2 x3
20.476 128.330 359.301
20.460 128.235 359.004
Table 2 Comparison of normalized deflection. Velocity(m/sec)
Present code
Ref. [40]
132
1.7133
1.7324
The effect of speed is represented by speed parameter,
a ¼ c=ccr
ð62Þ
where c is the speed and ccr critical speed of the mass which is represented as
ccr ¼
x1 l p
ð63Þ
3.2. Numerical analysis A beam having geometrical dimensions ð0:5 0:01 0:001Þm and the material properties E = 69 GPa, q ¼ 2712 kg=m3 is considered for the numerical analysis.
To validate the present formulation and MATLAB code at first fundamental frequencies generated from the code is compared with the exact solution obtained by Rao for cantilever beam given in Table 1. As for the second validation comparison is made between the maximum normalized deflections obtained from present study and from the Ref. for simply supported beam excited by single traversing load (100kN). The maximum deflection and corresponding velocity is mentioned in Table 2. It can be noticeable from the table that present formulation has good agreement with previous published results. 4. Results Analysis of beam having different boundary conditions is done by using perturbation method. Only 1st mode for the mentioned boundary conditions is taken for the analysis purpose. Natural frequencies are calculated by using Eq. (47). For all the three cases (cantilever, fixed–fixed and simply supported beams) the initial value for mass displacement (s0) and tip deflection (vt0) are taken as 0.00001 and 0.1 respectively. p20 is taken as 0.5vt0. It was found that Runge-Kutta iteration method do not produce efficient result. So to maximize accuracy and efficiency, the final ordinary differen-
Fig. 3. For cantilever m = 1.0,se = 0.5, s10 =0.00001, v t0 =0.1,r = 0.0002 (a) perturbation solution for mass and (b) perturbation solution for beam (c) mass position and (d) tip deflection.
A. Mohanty et al. / Applied Acoustics 156 (2019) 367–377
Fig. 4. m = 0.1, se = 0.5, s10 =0.00001,
373
v t0 =0.1, r=0.095 (a) perturbation solution for mass and (b) perturbation solution for beam (c) mass position and (d) tip deflection.
tial Eq. (61) are solved using Jacobi iteration method. Stiff ODE solver is used to plot the modal amplitude and the variation of mass and beam deflection. The time response in Figures (3–8) shows the beating phenomenon for the mass and the beam by changing different parameter under internal resonance case. As mentioned in Nonlinear Oscillation by Nayfeh [41] detuning parameter is calculated. Various graphs are plotted by changing detuning parameter r and taking e ¼ 1for each boundary condition. Fig. 3 shows the response curve for moving mass at equilibrium position and beam response due to the effect of moving mass for the case of cantilever beam. Here equilibrium position (se) of moving mass is 0.5 and non-dimenssionalized mass ratio, m is 1. Fig. 3 (a) and (b) presents the fluctuation of modal amplitudeðp1 Þ in case of mass and in case of beamðp2 Þ respectively. The value of r is used to compare various solutions. In this case it is considered as 0.0002. Fig. 3(c) and (d) are the numerical solution obtained from Eqs. (45) and (46) by using stiff ODE23s solver. In Fig. 3 maximum deflection of mass is 0.028 and minimum is 0 whereas in case of beam maximum deflection is 0.05 and minimum deflection is 0.015. Fig. 4 shows the similar response curve by taking the mass ratio as 0.1 andr ¼ 0:095. It is found that for same initial value by changing the parameters like mass ratio andr, value of amplitude changes. Maximum deflection obtained is 0.031 and minimum deflection is 0 in case of mass and in case of beam maximum
deflection is 0.05 and minimum is 0.048. By comparing Figs. 3 and 4 it is found that time period decreases by reducing nondimenssionalized mass ratio and detuning parameter. Fig. 5 is obtained the variation curve of moving mass and beam for the case of fixed–fixed condition. Here position of moving mass se = 0.9 and non-dimensionalized mass ratio m = 0.1 is taken and r is considered as 0.0085. It produces maximum amplitude 0.16 and minimum amplitude 0 in case of mass and maximum amplitude is 0.1 and minimum amplitude is 0.04 in case of beam. Fig. 5(c) and (d) are plotted for the results obtained from Eqs. (45) and (46). Similar curve is obtained in Fig. 6 for r ¼ 0and m = 0.1. In Fig. 6 it is found that after time period 250, amplitude decreases gradually. In this condition for mass maximum deflection is 0.125 and minimum one is zero while maximum is 0.1 and minimum is 0.06 in case for beam. Fig. 7 shows the variation of modal amplitude, mass position and beam deflection for moving mass and beam in case of simply supported end condition. Here equilibrium position of moving mass and non-dimenssionalized mass ratio is selected as se = 0.9 and m = 1 respectively.r is chosen as 0.0085. Fig. 7(c) and (d) are the results for the numerical solution obtained for simply supported beam. Maximum and minimum amplitude of mass is 0.0017 and 0 respectively and for beam maximum amplitude is 0.1 and minimum is 0.09997. Fig. 8 shows the plot by taking m as unity and mass position 0.5. It is found that maximum ampli-
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Fig. 5. For fixed–fixed beam, m = 0.1, Se = 0.9, s10 =0.00001, beam deflection.
Fig. 6. m = 0.1, Se = 0.9,s10 = 0.00001,
v t0 =0.1, (a)and (b) perturbation solutionr = 0.0085 for mass and beam respectively, (c) mass position and (d)
v t0 =0.1,r = 0 (a) perturbation solution for mass and (b) perturbation solution for beam (c) mass position and (d) Beam deflection.
A. Mohanty et al. / Applied Acoustics 156 (2019) 367–377
Fig. 7. For simply supported beam m = 1, Se = 0.9, s10 =0.00001, position and (d) Beam deflection.
v t0 =0.1,r ¼ 0:0085
375
(a) perturbation solution for mass and (b) perturbation solution for beam (c) mass
Fig. 8. For simply supported beam m = 1, Se = 0.5, s10 =0.00001, v t0 =0.1,r ¼ 0:095 (a) perturbation solution for mass and (b) perturbation solution for beam (c) mass position and (d) Beam deflection.
376
A. Mohanty et al. / Applied Acoustics 156 (2019) 367–377
Fig. 9. Variation of non-dimensionalized deflection with respect to non-dimensionalized time for cantilever beam, m = 1.
various estimations ofa in case of fixed–fixed beam. For variation of afrom 0.25 to 1 the maximum change in amplitude is 55.33%. When time reaches 0.5 the corresponding amplitude is 1.05 fora = 0.25 and whena = 1 the maximum amplitude is 1.63 at time 0.7. When time reaches around 0.9 maximum deflection observed fora = 1.6. Fig. 11 shows the variation of non-dimensionalized mid-point displacement with respect to time for various estimations ofa in case of simply supported beam. With small value ofa i:e 0:25, maximum dynamic deflection of 1.123 is observed at 40% of time completion. Whereas fora ¼ 1:6, maximum deflection is obtained at time ratio 0.9. When achanges from 0.25 to 1 with variation of time the maximum change in amplitude obtained is 52.37%. It is felt that for all the mentioned boundary condition of beam, the mid-point displacement shifts towards right when speed of moving load increases.
5. Conclusion In this current research, the dynamic characteristics of beammass systems are explored for various boundary conditions. The system consists of a moving mass connected with spring that controls the movement of mass. Perturbation solution is obtained for the desired system considering kinematic nonlinearities. Detuning parameter is used for the comparison purpose. It is found that for small value ofr, system works near the resonance and by taking r = 0 it comes x ¼ 2x1 (from Eq. (59)). Some generalized observation are obtained from this study are given below.
Fig. 10. Variation of non-dimensionalized deflection with respect to non-dimensionalized time taking m = 1 for fixed–fixed beam.
1. The period of beat is different for various boundary conditions. 2. In all these cases for the response of beam deflection, the minimum amplitude variation of the beat never goes to zero where as in case of moving mass minimum variation of amplitude tends to zero. 3. In all the graphs both mass position and beam deflection appear dark due to small time steps and very high frequency variation. 4. For higher nondimensional time, amplitude of system increases with increase in velocity parametera.
References
Fig. 11. Variation of nondimensionalized deflection with respect to nondimensionalized time taking m = 1 for simply supported beam.
tude is 0.12 and minimum is 0 for mass deflection and for beam deflection, maximum amplitude is 0.1 and minimum value is 0.04. By comparing Figs. 7 and 8 it is ensured that amplitude is more when mass is at mid position of beam and decreases when mass moves towards the end. Fig. 9 shows time histories representation of displacement in dimensionless form for cantilever beam. It is observed that there is decrease in trend of deflection when load traverses from fixed to free end for different values of a ¼ 0:55; 1 and 1:6. Fig. 10 demonstrates the graphical variation of nondimensionalized mid-point displacement with respect to time for
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Contents lists available at ScienceDirect
Applied Acoustics journal homepage: www.elsevier.com/locate/apacoust
Coupled nonlinear behavior of beam with a moving mass Anwesa Mohanty ⇑, Moses Prasad Varghese, Rabindra Kumar Behera National Institute of Technology, Rourkela, Odisha, India
a r t i c l e
i n f o
Article history: Received 27 December 2018 Received in revised form 10 May 2019 Accepted 17 July 2019
a b s t r a c t In this article, it is proposed to comprehend the coupled nonlinear analysis of beam under three different boundary conditions, such as (1) cantilever, (2) fixed-fixed and (3) simply supported having moving mass. Due to the beam and mass interaction, coupling term arises and it results kinematic nonlinearities in the system. A fundamental geometric model is developed by considering nonlinearities of the system. Euler-Bernoulli beam assumptions are considered. By applying Hamilton’s principle a coupled mathematical formulation of the desired system is derived. Further Galerkin discretization method is used in the mathematical system to analyse dynamic behaviour by converting it from continuous to discrete problem. The resulting equations are solved using perturbation method. Then MATLAB ODE solver is used to plot various graphs for variation of amplitude and deflection with respect to time in case of both beam and mass. Under the internal resonance condition the time response curves are plotted to analyze the beating phenomenon for the beam and mass. Due to small time steps and very high frequency variation, mass position and tip deflection plots appear dark. Finally, a comparative study is also made for various detuning parameter to show the modal behavior of beam-mass system. Ó 2019 Elsevier Ltd. All rights reserved.
1. Introduction A wide range of work on dynamic response of structures has already been carried out in the past decades. Dynamic analysis is the field of analyzing the dynamic properties of system under vibrational excitation. Moving particles/loads have a noticeable influence on dynamic behavior of structures and force them to vibrate at high velocity. Some of the practical examples in our day-to-day life are bridges, machining operation and beams subjected to pressure operation. For this reason dynamic analysis has got the fundamental attention of the researchers. Among many different cases of study some authors focuses on the loads moving with a constant and non-uniform velocity. For broad knowledge of moving load problems many references are available in the book by Fryba [1]. Michaltsos et al. [2], Ye and Chen [3] studied effect of moving load on dynamic characteristic of simply supported beam (SSB) by changing different parameters. Later in 1997 a comparison study of moving mass and moving force is done by using finite difference method [4]. Nonlinear coupled vibration analysis of beam having moving support under the action of moving mass is solved numerically [5–7]. Different numerical techniques like method of integral equation, Galerkin’s method, approximate method including asymptotic solution are introduced for solution ⇑ Corresponding author at: NIT Rourkela, Odisha- 769008, India. Tel.: +918249373048, fax.: 0661-2472926. E-mail address: [email protected] (A. Mohanty). https://doi.org/10.1016/j.apacoust.2019.07.024 0003-682X/Ó 2019 Elsevier Ltd. All rights reserved.
purpose. In 2000, Ichikawa et al. [8], Hilal and Zibdeh [9] looked over the dynamic characteristics of beam having different boundary conditions loaded with moving mass. For further analysis Kessel and Schlack [10] introduced damping phenomena to SSB subjected to cyclic moving load. Wu et al. [11] presented different approach of finite element method (FEM) to do dynamic analysis of structure having single and double moving load. Initially they used finite element packages for simply supported beam having point moving load and further 3-D structure having moving loads is analyzed. Theoretical analysis of dynamic response for Timoshenko beam under moving mass [12] and accelerated moving mass [13] is carried by implementing different numerical techniques. A comparative study is carried out over the dynamic characteristics of cantilever beam under moving mass [14] using numerical and perturbation method. Further numerical iterative techniques such as Ritz and average acceleration methods are implemented in other research for the modal analysis [15,16]. Some papers applied precise time-step integration method [16] to analyze dynamic response of continuous system having moving loads. Convenient use of spectrogram helps to capture the nonstationary behavior of the signal. Thus, Abdomian’s decomposition and homotopy perturbation methods are used for the numerical analysis of dynamic behavior of beam-mass system [17]. Aziz et al. [18] and Johansson et al. [19] investigated various dynamic analysis considering continuous system and bridges subjected to different loading conditions. In some other cases [18,20] researchers are used spectral element method to get the the exact dynamic response. Analytical solution for both Timoshenko beam [21] and Euler-Bernoulli
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[22] beam considering visco-elastic foundation under different loading conditions are further studied. In 2016 Svedholm et al. [23] studied analytically the vibration characteristics of damped uniform beam having different boundary conditions under moving loads. They derived orthogonality conditions and closed form solution to analyze dynamic response. Jorge et al. [24] investigated the effects of foundation stiffness in case of beam having nonlinear elastic foundation and another work is done by Nguyen et al. [25] in 2016 on the nonlinear foundation by considering nonlinear wrinkle and the linear Pasternak foundation parameters. Some researchers focused on linear and nonlinear vibration analysis of functionally graded (FG) plate under moving load [26,27]. Lal et al. Malekzadeh and Monajjemzadeh [28,29] studied the thermal environment effect on dynamic behavior of FG beam and plate. In most of the cases, authors employed FEM with differential quadrature and newmark’s time integration method [25,28,29] to solve the system of problem. In recent year a numerical analysis of three dimensional beam under moving mass is also carried out considering geometrical non-linearity [30]. As micro and nano-beams are challenges for researchers, different studies are carried out by considering different structures and solution methods. Further study is continued in the field of dynamic analysis by analyzing vibrational characteristics of beam containing FG nanocomposite for various beam theories [31]. Nonlinear behavior of the micro beam system was studied by using differential quadrature and iterative method [32,33]. Sharma and Kaur [34] developed a mathematical model to study the anisotropic, thermo-elastic characteristic of a clamped-clamped micro-beam. Recently, axial and transverse dynamic study was represented by Hosseini and Rahmani [35] for nano-beam excited by constant moving load. Wang et al. [36] contemplated the vibration response of composite beam reinforced with graphene nanoplatelet excited by two traversing mass. Hashemi and Khaniki et al. [37] discussed the dynamic behavior of nano beam system using Erigen’s nanolocal theory excited by nano particle. Although many research works has done on this topic over the past decades, but due to the complexity and real physical applications of moving mass problems, still it is the main point of interest for researchers. In case of moving mass it considers the gravitational and inertia effect of moving load where as in case of moving load no such complexity is there. Present study is focused on the dynamic behavior of beam with general boundary conditions having spring mass system as moving mass.
Fig. 1. Schematic diagram of (a) cantilever beam (b) fixed-fixed and (c) Simply supported beam with moving mass.
of beams. U, V, W denotes the displacements in the undeformed X, Y and Z direction respectively. To introduce the geometrical nonlinearity large displacement field of corresponding beam-mass system is obtained by considering horizontal (u) and vertical (v) deformation as:
U ¼ u v sinðhÞ; where sinðhÞ ¼ h ¼
@x
v ðx; tÞ
V ¼ v ðx; tÞ v ð1 cosðhÞÞ; where cosðhÞ ¼ 1 þ @u @x
ð1Þ
W ¼0
In case of small displacement and rotation only vertical deformation v(x, t) is assumed and the corresponding following displacement field is obtained as: @ U ¼ v sinðhÞ; sinðhÞ ¼ h ¼ @x v ðx; tÞ (2) V ¼ v ðx; tÞ; cosðhÞ ¼ 1 W¼0 Equation of motion The velocity field for both beam and mass is retrieved from respective displacement field by differentiating it with respect to time. Hence in case of large displacement it gives
U t ¼ ut v cosðhÞ
2. Analytical formulation
@
@h @h ; V t ¼ v t v sinðhÞ @t @t
ð3Þ
Whereas in case of small displacement it results In the current research work, assumptions which are taken into considerations are as follow: (i) beam obeys the Euler-Bernoulli beam theory, (ii) moving mass remains in contact with beam during the analysis. Fig. 1 shows the uniform beam-mass system with different boundary conditions such as, fixed-free (cantilever), fixed–fixed and simply supported. The following nomenclatures are used for the theoretical analysis; Length of beam is ‘L’, cross-sectional area ‘a’, mass density ‘q’ and flexural rigidity ‘EI’. The moving mass ‘M’ slides along the beam are attached with a spring of stiffness ‘k’ from the left end. The position of the mass is ‘s(t)’. Here all these dimenssionalized parameters are normalized by dividing required parameters for the proper calculation.
Therefore,
T ¼ t 0 =s,s ¼ S=L,A ¼ a=L2 ,m ¼ M=qAL
U mt ¼ U t þ st cosðhÞ; V mt ¼ V t þ st sinðhÞ
The Euler-Bernoulli beam model is assumed for the numerical study. Fig. 2 presents the diagram for small and large deformation
ð4Þ
ð5Þ
Here, U t ; V t ,U mt and V mt are the velocity components in X, Y direction of beam and traversing mass respectively. Subscript ‘m’ stands for moving mass. st is the velocity due to position of moving mass. Neglecting the velocity due to rotation KEb is obtained for large and small displacement respectively as
KEb ¼
3. Theoretical analysis and results 3.1. Displacement field
@h @ v ðx; tÞ ; Vt ¼ @t @t
Similarly in both the cases for large and small displacement velocity of moving mass is obtained as
4
andx ¼ kqAl =mEI: 2
U t ¼ v cosðhÞ
KEb ¼
1 2 1 2
Z
l
0
Z 0
l
2 ut þ v 2t dx
ð6Þ
ðv t Þ2 dx
ð7Þ
KEm can be derived for both the cases as:
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A. Mohanty et al. / Applied Acoustics 156 (2019) 367–377
Fig. 2. Small deflection of beam element and (b) large deflection of beam element.
KEm ¼
1 1 mðU mt Þ2 þ mðV mt Þ2 2 2
ð8Þ
1 2 m st þ u2t þ v 2t þ 2st cosðhÞut þ 2st sinðhÞv t X¼s 2
Z d
ð9Þ
i 1 h 2 m ðst Þ þ ðv t Þ2 þ 2st sinðhÞv t X¼s 2
PEb ¼
1 2
l
0
ð11Þ
Z
Z 0
l
@2v @x2
t2
d
ð13Þ
ðv xx Þ2 dx
ð14Þ
Ldt ¼ 0
t1
Z
t2
t2
Z 0
l
Z
t2
t1
ðutt du þ v tt dv Þdtdx
ð17Þ
v tt þ st v xt þ stt v x þ st v xt þ s2t v xx dv dt m
ðutt þ stt þ 2st uxt þ stt ux Þdudt
ð15Þ
t1
where, L = Kinetic energy (KE) – Potential energy (PE) =ðKEb þ KEm Þ ðPEb þ PEm Þ Cwhere C is the Lagrangian constant. Rl For large deformationC ¼ 12 0 kðx; tÞ v 2x þ ð1 þ ux Þ2 1 dx whereas for small deformation C ¼ 0. For large deformation equation of motion is obtained by substituting Eqs. (6), (9), (11) and (13) into Eq. (15), the resulting equation leads to: Z Z t2 1 l 2 1 d u þ v 2t dx þ m s2t þ u2t þ v 2t þ 2st cosðhÞut þ 2st sinðhÞv t X¼se 2 0 t 2 t1 Z 1 1 l 2 ðv xx cosðhÞ v xx sinðhÞÞ dx mx 2 ð s s e Þ 2 2 2 0 ! Z 1 l ð16Þ kðx;tÞ v 2x þ u2x þ 2ux dx dt ¼ 0 2 0 By integrating each terms of Eq. (16) individually through integration by parts and operating the d operator gives,
1 mx2 ðs se Þ2 dt ¼ 2
d t1
Z
Z
t2
l
ð18Þ
0
Z
Z
t2 t1
mx2 ðs se Þdsdt
Z
t2
0
t1
l
ððv xxxx 2v xxxx ux 3v xxx uxx 2v xx uxxx þ uxxxx v x
2v xx ux uxxx 2v xx u2xx 4v xxx ux uxxx u2x v xxxx þ 4uxx uxxx v x þ2u2xx v xx þ v x ux uxxxx þ v x uxx uxxx þ v xx ux uxxx dv þ 5v xx v xxx þ v xxxx v x þ v xx v x uxxx þ 2v 2xx uxx þ 5v xx v xxx ux þv xxxx v x ux 5uxxx v x v xx uxxxx v 2x du dxdt ! Z t2 Z l @ @2h @2h sinh þ cosh dxdt ¼ 2 @x2 0 @x @x t1 Z
Z
t2
l
d 0
t1
Z
t2
Z
t1
ð20Þ
1 2 k v x þ u2x þ 2ux dxdt 2 l
ððkv xx þ kx v x Þdv ðkx ux þ kuxx þ kx Þdu 1 þ v 2x þ u2x þ 2ux dk dxdt 2
¼
ð19Þ
ðv xx cosðhÞ uxx v x Þ2 dxdt ¼
d
ð12Þ
Here subscripts ‘t’ and ‘x’ denote the derivative with respect to time and distance. The equations of motion for the given system are derived using Hamilton’s principle
Z
t2
m
t1
ðv xx cosðhÞ þ uxx sinðhÞÞ dx 2
0
u2t þ v 2t dxdt ¼
1 2 m st þ u2t þ v 2t þ 2st cosðhÞut þ 2st sinðhÞv t X¼se dt 2 Z t2 ¼ mðstt þ v tt v x þ utt ð1 þ ux Þ þ v t v xt þ uxt ut Þdsdt
Z
Potential energy of beam for large and small deformation respectively is obtained as
Z
t1
Here se represents the equilibrium position of spring-mass subsystem.
1 2
l
t2
ð10Þ
1 PEs ¼ mx2 ðs se Þ2 2
From strain displacement relation e ¼ y
Z
t1
Potential energy due to spring attached to mass is
PEb ¼
1 2
t1
In case of small deformation after substitution of Eq. (5), to simplify the problem, rotary inertia of the beam is dropped from velocity for slender beam. Hence in this case KEm becomes,
KEm ¼
t2
t1
Substituting Eq. (5) for large and small deformation separately in Eq. (8) results
KEm ¼
Z d
0
ð21Þ
Substituting Eqs. (17)–(21) into Eq. (16) and making the coefficient of ds; du; dv ; dk to zero the following four equations are obtained,
stt þ x2 ðs se Þ þ ðutt cosðhÞ þ v tt sinðhÞÞX¼sðtÞ ¼ 0 m stt cosðhÞ þ utt þ 2st uxt þ s2t uxx X¼sðtÞ Z l @ þ utt ðkcosðhÞ þ hxx sinðhÞÞ dx ¼ 0 @x 0 m stt sinðhÞ þ v tt þ 2st v xt þ s2t v xx X¼sðtÞ Z l @ þ v tt ðksinðhÞ hxx cosðhÞÞ dx ¼ 0 @x 0
ð22Þ
ð23Þ
ð24Þ
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Z l 1 ux þ u2x þ v 2x dx ¼ 0 2 0
ð25Þ
To derive equation of motion for small deformation, Eqs. (7), (10), (11), (14), are substituted into Eq. (15) which results
Z d
1 2
t1
€ i þ ui ðuj Þ stt þ x2 ðs se Þ þ ui jX¼sðtÞ a x
Z
l
0
Z
t2
Z
l
t1
Z
0
ð35Þ For u variation:
ð26Þ
m stt ðui ÞX¼sðtÞ þ ui uj
Z
1 2 v dxdt ¼ 2 t
Z
t2
t1
l
0
v tt dv dxdt
1 2 m st þ v 2t þ 2st v t sinðhÞ X¼sðtÞ dt 2 Z t2 1 2 ¼d m st þ v 2t þ 2st v t v x X¼sðtÞ dt t1 2 Z t2 ¼ mðst dst þ v t dv t þ dðst v t v x ÞÞX¼sðtÞ dt
¼ m
t2
t1
Z
t2
d t1
1 mx2 ðs se Þ2 dt ¼ mx2 2
Z
¼ mx
2
Z
t2
Z
d t1
0
l
1 2 v dxdt ¼ 2 xx ¼
Z
t2
t1
Z
t2
Z
t1
0
X¼sðtÞ
dt
ð28Þ
0
ð29Þ
t1
l
l
v xx dv xx dxdt v xxxx dv dxdt
ð30Þ
ð31Þ
f ¼ stt v x þ ðst Þ2 v xx þ 2st v tx þ v tt
ð33Þ
When his small, the additional vertical movement of any point located at distance y below neutral axis is also small and therefore it is neglected in above equations. To convert partial differential equation to ordinary differential equation, Galerkin’s method is used with a trial function having separate variables. In case of large deformation, variables are assumed to be
uðx; tÞ ¼
v ðx; tÞ ¼
i¼1
and kðx; tÞ ¼
bi ðtÞui ðxÞ; n P i¼1
ki ðtÞui ðxÞ
1 2
xx
ui uj aj þ ui uj ðuk Þx aj ak þ bj bk x
x
dx ¼ 0
ð37Þ
ð38Þ
v ðx; tÞ ¼
n X
ai ðtÞ/i ðxÞ
ð39Þ
h i stt þ x2 ðs se Þ þ ui uj
x X¼s
ðai Þtt aj ¼ 0
ð40Þ
h i nh i aj tt þ stt ui uj aj m ui uj xx X¼s xx X¼s h i h i o 2 þ 2st ui uj aj t þðst Þ ui uj aj x X¼s xx X¼s "Z # "Z # l l ðui Þxx uj dx aj þ ui uj dx aj tt þ 0
xx
ð41Þ
After using the boundary conditions for cantilever, fixed–fixed and simply supported, the eigen functions are derived as follows; For cantilever beam:
ui ¼ coshðki xÞ cosðki xÞ
cosðki lÞ þ coshðki lÞ ðsinhðki xÞ sinðki lÞ þ sinhðki lÞ ð42Þ
sinðki xÞÞ For fixed–fixed beam:
ui ¼ ðsinhðki xÞ sinðki xÞÞ þ
sinhðki lÞ sinðki lÞ ðcoshðki xÞ cosðki lÞ coshðki lÞ
cosðki xÞÞ
ð43Þ
For simply supported beam:
ai ðtÞui ðxÞ;
i¼1 n P
stt bj þ 2st bj t þ s2t ui uj bj
¼0
where
n P
x X¼sðtÞ
0
ð32Þ
0
X¼sðtÞ
bj tt þ ui uj
By using the above formulation, Eqs. (31) and (32) become
ðv xxxx þ v tt Þdx ¼ 0
mf jx¼sðtÞ þ
tt
For the small deformation, vertical displacement is assumed as
ðs se Þdsdt
v variation: l
ui uj aj
For kvariation:
Z l
ðs se Þdðs se Þdt
stt ðtÞ þ x2 ðsðtÞ se Þ þ v tt v x jX¼sðtÞ ¼ 0 Z
aj þ
x
Further substituting Eqs. (27)–(30) into Eq. (26) and equating the coefficient ofds and dv to zero following equations are obtained, For s variation:
For
Z l
stt aj þ 2st aj t
i¼1
0
Z
X¼sðtÞ
ui uj bj tt þ ðui Þxx uj bj þ ðui Þx uj ðuk Þx kj bk þ xx 0 ðui Þx uj ðuk Þxx ðul Þxx bj ak al þ bj bk bl dx ¼ 0
t1 t2
þ ui u0j
x
Z l
ðstt þ v tt v x Þds þ stt v x þ s2t v xx þ 2st v tx þ v tt dv t2
tt
v variation:
m ui uj
Z
aj
þ ðui Þxx uj aj xx X¼sðtÞ xx 0 þðui Þx uj kj þ ðui Þx uj ðuk Þx kj ak ðui Þt uj ðuk Þtt aj ak þ bj bk tt ðui Þx uj ðuk Þxx ðul Þxx aj ak al þ aj bk bl dx ¼ 0 ð36Þ For
t1
þs2t ui uj
ð27Þ
t2
Z
X¼sðtÞ
d
t1
ðai Þtt aj þ ðbi Þtt bj ¼ 0
v 2t dx þ
Each terms of Eq. (26) is integrated separately using integration by parts followed by the operation of variational operatord, which gives
d
X¼sðtÞ
1 2 m st þ v 2t þ 2st sinðhÞv t X¼se 2 ! Z 1 1 l 2 2 2 v dx dt ¼ 0 mx ðs se Þ 2 2 0 xx t2
In the formulations Eqs. (22)–(25), by implementing variable separation method given in Eq. (34) it results, For s variation:
ð34Þ
ui ¼ sinðki xÞ
ð44Þ
Considering 1st mode ðu1 Þof desired beam as basis function, Eqs. (40) and (41) reduce to:
stt þ x2 s þ c1 ða1 Þtt a1 ¼ 0
ð45Þ
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A. Mohanty et al. / Applied Acoustics 156 (2019) 367–377
ða1 Þtt þ x21 a1 þ mc2 stt a1 þ 2mc2 st ða1 Þt þ 2mc2 sða1 Þtt ¼ 0
where c1 ¼ u1 ðu1 Þx X¼se , c2 ¼ R l 0
Rl
u1 ðu1 Þx jX¼s
e
ðu1 Þ2 dxþmðu1 Þ2 j
@ 2 u2 @T 20
,
þ x21 u2 ¼ 2
ðu 0
2 ð u Þ dx þ mðu1 Þ2
1 0
ð47Þ
X¼se
The final set of equation of motion for both large and small deformation is obtained in Eqs. (35)–(38) and Eqs. (45) and (46) respectively. As method of multiple scales (MMS) is cumbersome to find the solution for large deformation case, average acceleration method can be suitably used to study modal analysis. However for small deformation case the nonlinearity arises due to the interaction between the beam and mass can be solved using MMS [38]. The MMS is implemented in Eqs. (45) and (46) to get the desired solution. Time scale for formulation is defined by using
T n ¼ en t
dT 2
¼
@2 @T 20
þ 2e
@2
ð49Þ
@T 21
Neglecting the higher order terms uniform expansion for s and
@T 20
ð55Þ
s1 ¼ P1 ðT 1 ÞeixT 0 þ P 1 ðT 1 ÞeixT 0 u1 ¼ P 2 ðT 1 Þeix1 T 0 þ P2 ðT 1 Þeix1 T 0
ð56Þ
Here P 1 , P 2 are complex variable and P1 ,P2 are complex conjugate. Substituting Eq. (56) into the right hand side (RHS) of Eqs. (54) and (55), results
RHSð25Þ ¼ 2i
@P 1 xeixT 0 þ c1 P22 x21 e2ix1 T 0 þ c1 P 2 P2 x21 þ 2i @T 1
@P1 2 xeixT 0 þ c1 P2 x21 e2ix1 T 0 þ c1 P2 P2 x21 @T 1
ð57Þ
@P2 x1 eix1 T 0 þ mc2 2x21 þ x2 þ 2xx1 P1 P 2 eiðx1 þxÞT 0 @T 1 @P 2 þ mc2 2x21 þ x2 2xx1 P1 P 2 eiðx1 xÞT 0 þ 2i x1 eix1 T 0 @T 1 þ mc2 2x21 þ x2 þ 2xx1 P1 P 2 eiðx1 þxÞT 0 þ mc2 2x21 þ x2 2xx1 P1 P 2 eiðx1 xÞT 0
RHSð26Þ ¼ 2i
d @ @ ¼ þe dt @T 0 @T 1 2
@ 2 u1
The solution of Eqs. (52) and (53) are
ð48Þ
where e is scaling parameter, n = 0,1,2. . . Here two time scale is considered i.e, T 0 and T 1 as the nonlinearities have a very small effect. By using chain rule, derivative with respect to ‘t’ transformed into ‘T 0 ’and ‘T 1 ’ which is written as follows:
d
@ 2 u1 @ 2 s1 @s1 @u1 mc2 u1 2mc2 @T 0 @T 1 @T 0 @T 0 @T 20
2mc2 s1
X¼se
2 1 Þxx dx
ðx1 Þ2 ¼ R l
ð46Þ
To eliminate the terms tends to secular terms, coefficient of eixT 0 andeix1 T 0 are setting to zero. Internal resonance is assumed to investigate the coupled characteristics of the system as,
a1 is assumed to be,
x ¼ 2x1 þ er
sðtÞ ¼ s1 ðT 0 ; T 1 Þ þ es2 ðT 0 ; T 1 Þ
where ris a small detuning parameter. While ris zero we get perfect internal resonance 1:2. Inserting Eq. (59) into Eqs. (57) and (58) and making the coefficient of secular term to zero, it gives
a1 ¼ a1 ðT 0 ; T 1 Þ þ ea2 ðT 0 ; T 1 Þ
ð50Þ
Substituting above asymptotic expansion into time derivatives provides
st ¼
@s1 @s2 @s1 @s2 þe þe þ e2 @T 0 @T 0 @T 1 @T 1 @ 2 s1
stt ¼
@T 20
þe
@ 2 s2 @T 20
þ 2e
att ¼
@ 2 a1 @T 20
þe
@ 2 a2 @T 20
@ 2 s1 @ 2 s2 þ 2e2 @T 0 @T 1 @T 0 @T 1
þ 2e
@ 2 a1 @ 2 a2 þ 2e2 @T 0 @T 1 @T 0 @T 1
@T 20 @ 2 u1 @T 20
@T 20
þ x2 s1 ¼ 0
ð52Þ
þ x2 u 1 ¼ 0
ð53Þ
þ x2 s2 ¼ 2
P 1 ðT 1 Þ ¼
1 p ðT 1 Þeiu1 ðT 1 Þ 2 1
P 2 ðT 2 Þ ¼
1 p ðT 1 Þeiu2 ðT 1 Þ 2 2
ð51Þ
@ 2 s1 @ 2 u1 c1 u1 @T 0 @T 1 @T 20
ð60Þ
Setting real and imaginary part of Eq. (60) to zero, results
@p1 1 c1 p22 x21 ¼ sinð2/2 /1 rT 1 Þ @T 1 4 x p1
ð54Þ
@ u1 1 c1 p22 x21 ¼ cosð2/2 /1 rT 1 Þ 4 x @T 1
@p2 1 mc2 x2 2xx1 þ 2x21 ¼ p1 p2 sinð2/2 /1 rT 1 Þ 4 @T 1 x1 p2
e1 order: @ 2 s2
@P2 þ mc2 2x21 þ x2 2xx1 P1 P2 eirT 1 ¼ 0 @T 1
Complex variables are taken in polar form as
Putting the above equation into Eqs. (45) and (46), and setting the coefficient of e0 ande1 to zero, gives e0 order:
@ 2 s1
@P1 þ c1 P22 x21 eirT 1 ¼ 0 @T 1
2ix1
@ a1 @ a2 @ a1 @ a2 þe þe þ e2 @T 0 @T 0 @T 1 @T 1
at ¼
2ix
ð59Þ
@ u2 1 mc2 x2 2xx1 þ 2x21 ¼ p1 p2 cosð2/2 /1 rT 1 Þ 4 @T 1 x1 ð61Þ
where, p1 ; u1 are the modal amplitudes and phase angle for mass p2 ; u2 are the amplitudes and phases for beam deflection
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Table 1 Comparison of frequency.
3.3. Validation
Natural Frequency(rad/sec)
Present code
Exact Method [39]
x1 x2 x3
20.476 128.330 359.301
20.460 128.235 359.004
Table 2 Comparison of normalized deflection. Velocity(m/sec)
Present code
Ref. [40]
132
1.7133
1.7324
The effect of speed is represented by speed parameter,
a ¼ c=ccr
ð62Þ
where c is the speed and ccr critical speed of the mass which is represented as
ccr ¼
x1 l p
ð63Þ
3.2. Numerical analysis A beam having geometrical dimensions ð0:5 0:01 0:001Þm and the material properties E = 69 GPa, q ¼ 2712 kg=m3 is considered for the numerical analysis.
To validate the present formulation and MATLAB code at first fundamental frequencies generated from the code is compared with the exact solution obtained by Rao for cantilever beam given in Table 1. As for the second validation comparison is made between the maximum normalized deflections obtained from present study and from the Ref. for simply supported beam excited by single traversing load (100kN). The maximum deflection and corresponding velocity is mentioned in Table 2. It can be noticeable from the table that present formulation has good agreement with previous published results. 4. Results Analysis of beam having different boundary conditions is done by using perturbation method. Only 1st mode for the mentioned boundary conditions is taken for the analysis purpose. Natural frequencies are calculated by using Eq. (47). For all the three cases (cantilever, fixed–fixed and simply supported beams) the initial value for mass displacement (s0) and tip deflection (vt0) are taken as 0.00001 and 0.1 respectively. p20 is taken as 0.5vt0. It was found that Runge-Kutta iteration method do not produce efficient result. So to maximize accuracy and efficiency, the final ordinary differen-
Fig. 3. For cantilever m = 1.0,se = 0.5, s10 =0.00001, v t0 =0.1,r = 0.0002 (a) perturbation solution for mass and (b) perturbation solution for beam (c) mass position and (d) tip deflection.
A. Mohanty et al. / Applied Acoustics 156 (2019) 367–377
Fig. 4. m = 0.1, se = 0.5, s10 =0.00001,
373
v t0 =0.1, r=0.095 (a) perturbation solution for mass and (b) perturbation solution for beam (c) mass position and (d) tip deflection.
tial Eq. (61) are solved using Jacobi iteration method. Stiff ODE solver is used to plot the modal amplitude and the variation of mass and beam deflection. The time response in Figures (3–8) shows the beating phenomenon for the mass and the beam by changing different parameter under internal resonance case. As mentioned in Nonlinear Oscillation by Nayfeh [41] detuning parameter is calculated. Various graphs are plotted by changing detuning parameter r and taking e ¼ 1for each boundary condition. Fig. 3 shows the response curve for moving mass at equilibrium position and beam response due to the effect of moving mass for the case of cantilever beam. Here equilibrium position (se) of moving mass is 0.5 and non-dimenssionalized mass ratio, m is 1. Fig. 3 (a) and (b) presents the fluctuation of modal amplitudeðp1 Þ in case of mass and in case of beamðp2 Þ respectively. The value of r is used to compare various solutions. In this case it is considered as 0.0002. Fig. 3(c) and (d) are the numerical solution obtained from Eqs. (45) and (46) by using stiff ODE23s solver. In Fig. 3 maximum deflection of mass is 0.028 and minimum is 0 whereas in case of beam maximum deflection is 0.05 and minimum deflection is 0.015. Fig. 4 shows the similar response curve by taking the mass ratio as 0.1 andr ¼ 0:095. It is found that for same initial value by changing the parameters like mass ratio andr, value of amplitude changes. Maximum deflection obtained is 0.031 and minimum deflection is 0 in case of mass and in case of beam maximum
deflection is 0.05 and minimum is 0.048. By comparing Figs. 3 and 4 it is found that time period decreases by reducing nondimenssionalized mass ratio and detuning parameter. Fig. 5 is obtained the variation curve of moving mass and beam for the case of fixed–fixed condition. Here position of moving mass se = 0.9 and non-dimensionalized mass ratio m = 0.1 is taken and r is considered as 0.0085. It produces maximum amplitude 0.16 and minimum amplitude 0 in case of mass and maximum amplitude is 0.1 and minimum amplitude is 0.04 in case of beam. Fig. 5(c) and (d) are plotted for the results obtained from Eqs. (45) and (46). Similar curve is obtained in Fig. 6 for r ¼ 0and m = 0.1. In Fig. 6 it is found that after time period 250, amplitude decreases gradually. In this condition for mass maximum deflection is 0.125 and minimum one is zero while maximum is 0.1 and minimum is 0.06 in case for beam. Fig. 7 shows the variation of modal amplitude, mass position and beam deflection for moving mass and beam in case of simply supported end condition. Here equilibrium position of moving mass and non-dimenssionalized mass ratio is selected as se = 0.9 and m = 1 respectively.r is chosen as 0.0085. Fig. 7(c) and (d) are the results for the numerical solution obtained for simply supported beam. Maximum and minimum amplitude of mass is 0.0017 and 0 respectively and for beam maximum amplitude is 0.1 and minimum is 0.09997. Fig. 8 shows the plot by taking m as unity and mass position 0.5. It is found that maximum ampli-
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Fig. 5. For fixed–fixed beam, m = 0.1, Se = 0.9, s10 =0.00001, beam deflection.
Fig. 6. m = 0.1, Se = 0.9,s10 = 0.00001,
v t0 =0.1, (a)and (b) perturbation solutionr = 0.0085 for mass and beam respectively, (c) mass position and (d)
v t0 =0.1,r = 0 (a) perturbation solution for mass and (b) perturbation solution for beam (c) mass position and (d) Beam deflection.
A. Mohanty et al. / Applied Acoustics 156 (2019) 367–377
Fig. 7. For simply supported beam m = 1, Se = 0.9, s10 =0.00001, position and (d) Beam deflection.
v t0 =0.1,r ¼ 0:0085
375
(a) perturbation solution for mass and (b) perturbation solution for beam (c) mass
Fig. 8. For simply supported beam m = 1, Se = 0.5, s10 =0.00001, v t0 =0.1,r ¼ 0:095 (a) perturbation solution for mass and (b) perturbation solution for beam (c) mass position and (d) Beam deflection.
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Fig. 9. Variation of non-dimensionalized deflection with respect to non-dimensionalized time for cantilever beam, m = 1.
various estimations ofa in case of fixed–fixed beam. For variation of afrom 0.25 to 1 the maximum change in amplitude is 55.33%. When time reaches 0.5 the corresponding amplitude is 1.05 fora = 0.25 and whena = 1 the maximum amplitude is 1.63 at time 0.7. When time reaches around 0.9 maximum deflection observed fora = 1.6. Fig. 11 shows the variation of non-dimensionalized mid-point displacement with respect to time for various estimations ofa in case of simply supported beam. With small value ofa i:e 0:25, maximum dynamic deflection of 1.123 is observed at 40% of time completion. Whereas fora ¼ 1:6, maximum deflection is obtained at time ratio 0.9. When achanges from 0.25 to 1 with variation of time the maximum change in amplitude obtained is 52.37%. It is felt that for all the mentioned boundary condition of beam, the mid-point displacement shifts towards right when speed of moving load increases.
5. Conclusion In this current research, the dynamic characteristics of beammass systems are explored for various boundary conditions. The system consists of a moving mass connected with spring that controls the movement of mass. Perturbation solution is obtained for the desired system considering kinematic nonlinearities. Detuning parameter is used for the comparison purpose. It is found that for small value ofr, system works near the resonance and by taking r = 0 it comes x ¼ 2x1 (from Eq. (59)). Some generalized observation are obtained from this study are given below.
Fig. 10. Variation of non-dimensionalized deflection with respect to non-dimensionalized time taking m = 1 for fixed–fixed beam.
1. The period of beat is different for various boundary conditions. 2. In all these cases for the response of beam deflection, the minimum amplitude variation of the beat never goes to zero where as in case of moving mass minimum variation of amplitude tends to zero. 3. In all the graphs both mass position and beam deflection appear dark due to small time steps and very high frequency variation. 4. For higher nondimensional time, amplitude of system increases with increase in velocity parametera.
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Fig. 11. Variation of nondimensionalized deflection with respect to nondimensionalized time taking m = 1 for simply supported beam.
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