Control of a two-degree-of-freedom system with combined excitations

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Original Research Article

Control of a two-degree-of-freedom system with combined excitations H.S. Bauomy a,c,*, A.T. El-Sayed b a

Department of Mathematics, Faculty of Science, Zagazig University, Zagazig 44519, Egypt Department of Basic Sciences, Modern Academy for Engineering and Technology, Egypt c Department of Mathematics, College of Arts and Science in Wadi Addawasir, Salman Bin Abdulaziz University, P.O. Box 54, Wadi Addawasir 11991, Saudi Arabia b

article info

abstract

Article history:

In this paper, the chaotic dynamics of parametrically, externally and tuned excited sus-

Received 29 May 2013

pended cable is studied with negative cubic velocity feedback. The equations of motion of

Accepted 17 May 2014

this system are exhibited by two-degree-of-freedom system including quadratic and cubic

Available online xxx

nonlinearities. Using the multiple scale perturbation technique, the response of the nonlinear system near the simultaneous primary, sub-harmonic, combined and internal reso-

Keywords:

nance case of this system is extracted up to the second order approximation. The stability of

Suspended cable

the obtained numerical solution is investigated using frequency response equations. The

Active control

effect of different parameters on the vibrating system behavior are investigated and

Tuned excitation

reported. The simulation results are achieved using MATLAB (R2012a) programs. # 2014 Politechnika Wrocławska. Published by Elsevier Urban & Partner Sp. z o.o. All

Stability

rights reserved.

1.

Introduction

The work on nonlinear oscillations of the suspended cables has received considerable attention because their importance in many applications in the fields of communications, electricity, mooring systems, transportation, cable-stayed bridge, and crane-operation systems. A lot of these applications need controllers to avoid disturbances of the system. Perkins [1] examined the effect of one support motion on the three-dimensional nonlinear response. Using the Galerkin method, Perkins constructed a two-degree-of-freedom model to analyze the two-to-one internal resonance, and he analyzed the first-order approximation of the obtained model by the

method of multiple scales. Lee and Perkins [2] found by using a three-degree-of-freedom model that strong coupling between in-plane and out-of-plane components occurred under simultaneous one-to-one and two-to-one internal resonances in suspended cables. Benedettini et al. [3] studied nonlinear oscillations of a four-degree-of-freedom model of the suspended cables under multiple internal resonance conditions. They have established that the response of suspended cables near or away from the first crossover can be large and exhibit very complex behavior due to the simultaneous presence of multiple internal resonances involving several in-plane modes and out-of-plane modes. Luongo and Piccardo [4] studied the nonlinear dynamics of a flexible, elastic, suspended cable driven by mean wind speed, blowing perpendicularly to the cable's

* Corresponding author at: Department of Mathematics, Faculty of Science, Zagazig University, Zagazig 44519, Egypt. Tel.: +20 1227023777. E-mail address: [email protected] (H.S. Bauomy). http://dx.doi.org/10.1016/j.acme.2014.05.007 1644-9665/# 2014 Politechnika Wrocławska. Published by Elsevier Urban & Partner Sp. z o.o. All rights reserved.

Please cite this article in press as: H.S. Bauomy, A.T. El-Sayed, Control of a two-degree-of-freedom system with combined excitations, Archives of Civil and Mechanical Engineering (2014), http://dx.doi.org/10.1016/j.acme.2014.05.007

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Nomenclature x and y the vertical (in-plane) and horizontal (out-ofplane) displacements of the cable respectively D0, D1 and D2 differential operators T0 and (T1, T2) fast and slow time scales respectively (Tn = ent) ; n = 0, 1, 2 the natural frequencies associated with the vx, vy symmetric in-plane and out-of-plane of the cable modes cx ¼ e^cx ; cy ¼ e^cy the linear damping coefficients ^ t ðt ¼ 2; 3Þ the nonlinat ; g t ¼ e^ g t ; ht ¼ e^ ht ; b2 ¼ eb at ¼ e^ ear coefficients f m ¼ e^f m ðm ¼ 1; 2Þ the parametric excitation force amplitude of the cable ^ the external and tuned excitation forces ^ Q ¼ eQ F ¼ eF; amplitude of the cable ^ m positive constants (gains) Gm ¼ e G V, Vi, (i = 1, 2, 3) the excitation frequencies e(0 < e  1) a small perturbation parameter

plane in two-to-one internal resonance and derived a twodegree-of-freedom model. Rao and Iyengar [5] studied the nonlinear response of a suspended cable under periodic excitation. In their study, the two-to-one internal resonance between the symmetric in-plane and out-of-plane modes was investigated by using the two-degree-of-freedom model, and they found that the non-planar motion of a cable with given sag may be activated within a certain region of the external resonance. Zhao et al. [6] examined the coupling dynamics of inclined cables between in-plane and out-of-plane vibrations under one-to-one internal resonances, and studied the dynamic features of cables. Recently, based on the 3-D model formulation, which is not restricted to cables with very small sag, Srinil et al. [7] investigated the nonlinear characteristics of the large amplitude free vibrations of inclined sagged cables, and they observed strong coupling phenomena when the two-to-one internal resonance conditions were activated. Arafat and Nayfeh [8] studied the motion of shallow suspended cables with primary resonance excitation. The method of multiple scales is applied to study nonlinear response of this suspended cables and its stability and the dynamic solutions. Some interesting work on the nonlinear dynamics of cables to harmonic excitations can be found in the review articles by Rega [9,10]. Nielsen and Kierkegaard [11] investigated simplified models of inclined cables under super and combinatorial harmonic excitation and gave analytical and purely numerical results. Zheng et al. [12] considered the super-harmonics and internal resonance of a suspended cable with almost commensurable natural frequencies. Zhang and Tang [13] investigated the chaotic dynamics and global bifurcations of the suspended inclined cable under combined parametric and external excitations. Chen and Xu [14] investigated the global bifurcations of the inclined cable subjected to a harmonic excitation leading to primary resonances with the external damping by using averaging method. EL-Bassiouny [15] made an investigation on the control of the vibration of the crankshaft in internal combustion engines

subjected to both external and parametric excitations via an absorber having both quadratic and cubic stiffness nonlinearities. Lei et al. [16] applied an active control technique to coordinate a kind of two parametrically excited chaotic systems. Kamel and Hamed [17], studied the nonlinear behavior of an inclined cable subjected to harmonic excitation near the simultaneous primary and 1:1 internal resonance using multiple scale method. Abe [18] investigated the accuracy of nonlinear vibration analyses of a suspended cable, which possesses quadratic and cubic nonlinearities, with 1:1 internal resonance. The nonlinear dynamics of suspend cable structures have been studied with 2:1 internal resonances by the authors [19,20]. The out-of-plane dynamic stability of inclined cables subjected to in-plane vertical support excitation is investigated by Gonzalez-Buelga et al. [21]. Chen et al. [22] studied the bifurcations and chaotic dynamics of the parametrically and externally excited suspended elastic cable. Belhaq et al. [23] investigated the control of chaos of one-degree-of-freedom system with both quadratic and cubic nonlinearities subjected to combined parametric and external excitations. Several control methods leading to suppression of chaos have been presented. Pai et al. [24] designed new non-linear vibration absorbers using higherorder internal resonances and saturation phenomena to suppress the steady state vibrations of a linear cantilevered skew aluminum plate subjected to single external force. Higherorder internal resonances are introduced using quadratic, cubic, and/or quartic terms to couple the controller with the plate. The displacement and velocity feedback signals are considered. Berlioz and Lamarque [25] studied the theoretical and experimental investigations of an inclined cable subjected to external sinusoidal forcing leading to primary and sub-harmonic resonances. A theoretical discussion and some numerical results relating to a nonlinear state designed for shallow cable vibration are presented and studied by Faravelli and Ubertini [26]. A sample suspended cable, representing a physical model is considered as the case study, and non collocated feedback, based on active transverse control, is considered as a final application of the state observer. Also, active feedback control for cable vibrations is studied by Ubertini [27] for analytical and numerical models. A suitable dimensional analytical Galerkin model is derived to investigate the effectiveness of the feedback control, which represents the final application of the state observer. Wang and Zhao [28–30] applied different methods to investigate the nonlinear response of the suspend cable with three-to-one internal resonance, and numerical simulations are used to illustrate the chaotic dynamics of the cable. They also extended the previous work to consider the out-of-plane motion of a shallow suspended cable [31]. The three-to-one internal resonance between the third and the first symmetric in-plane modes and the one-to-one internal resonance between the third symmetric in-plane mode and the third symmetric out-of-plane mode are taken in to account. The case of the primary resonance of the first symmetric mode is also considered. Wang and Yau [32,33] applied the differential transformation (DT) method and the Runge–Kutta method, respectively, to investigate the nonlinear dynamic behavior of the probe tip of an atomic force microscope (AFM). Also, they analyzed the bifurcation characteristics of an AFM cantilever system utilizing the DT method.

Please cite this article in press as: H.S. Bauomy, A.T. El-Sayed, Control of a two-degree-of-freedom system with combined excitations, Archives of Civil and Mechanical Engineering (2014), http://dx.doi.org/10.1016/j.acme.2014.05.007

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The results indicate that the probe tip behavior is significantly dependent on the magnitude of the vibrational amplitude. Awrejcewicz et al. [34–36] studied in the first part of the paper chaotic dynamics of continuous mechanical systems such as flexible plates and shallow shells. The considered problems are solved by the Bubnov–Galerkin and higher approximation Ritz methods. New scenarios of transition from regular to chaotic orbits are detected, analyzed and discussed. In second part of the paper both classical and novel scenarios of transition from regular to chaotic dynamics of dissipative continuous mechanical systems are studied. Third part of the paper is devoted to analysis of the hyper, hyperhyper and spatial–temporal chaos of continuous mechanical systems using the Lyapunov exponents. Hsu and Cheng [37] governed a linear dynamical system subjected to combined parametric and forcing excitations of periodic nature by a system of inhomogeneous differential equations with periodic coefficients. They presented an explicit expression for steadystate periodic response of such a system given in terms of the fundamental matrix of the homogeneous system. More recently, a combined excitation problem has been studied. Haquang and Mook [38] reported the non-linear structure vibrations under combined parametric and external excitations. Thampi and Niedzwecki [39] examined the response of a non-linear marine riser to combined excitations by using Markov methods. Patel and Park [40] researched dynamic responses of TLP tethers under combined excitations by a semi-analytical method. Ryu and Isaacson [41] investigated 2D dynamic response of slender maritime structure under regular waves and vessel motions that induces a combined excitation. Kahraman and Blankenship [42] investigated analytically the steady state forced response of a system with clearance subject to commensurate parametric and external forcing excitation by considering a mechanical oscillator with time-varying stiffness and dead zone type clearance nonlinearity. Park and Jung [43] present a numerical analysis of lateral responses of a long slender marine structure under combined parametric and forcing excitations. The results of this study demonstrate that combined excitation needs to be considered for the exact analysis of long slender marine structures subjected to surface platform motions. Wójcicki [44] determine whether one dynamic absorber can reduce the amplitude of the steady-state vibration of a parametric system for natural and parametric resonance frequencies simultaneously. The efficiency of both the conventional dynamic absorber and the parametric absorber is analyzed. Zhang and Meng [45] present a simplified model for the purpose of studying the resonant responses and nonlinear dynamics of idealized electrostatically actuated micro-cantilever based devices in microelectro-mechanical systems (MEMS). The harmonic balance (HB) method is applied to simulate the resonant amplitude frequency responses of the system under the combined parametric and forcing excitations. The aim of this work is to control a two degree of freedom nonlinear differential equations of a suspended cable having quadratic and cubic nonlinearities subjected to mixed excitations via a negative cubic velocity feedback. The method of multiple scales perturbation [46,47] is applied to solve the nonlinear differential equations describing the controlled system up to second order approximations. The behavior of

the system is studied applying Rung–Kutta fourth-order method. The stability of the proposed analytic nonlinear solution near the simultaneous primary, sub-harmonic, combined and internal resonance case is studied numerically. The stability of the system is investigated applying frequency response equations. The effect of different parameters on the steady state responses of the vibrating system is studied and discussed from the frequency response curves.

2.

Mathematical reduction from PDEs to ODEs

Following Benedettini et al. [3], we express the dimensional non-planar equations of motion for a shallow suspended elastic cable hanging at fixed supports and caused by axial excitation and horizontal load is as follows: ! ^ ^1 ^ 1 EA @2 u ^ 1 d2 c ^1 @2 u @u @2 u H 2 ¼ m 2 þ 2^c1 þ L @^t ^2 ^ ^2 dx @x @x @^t ZL "



 2 !#  ^ 1 @u ^ 1 dc ^1 2 ^2 @u @u ^ ^ ^; þ þ F1 cos V1 t þ dx ^ 2 ^ dx ^ ^ @x @x @x

0

^2 ^ 2 EA @2 u ^2 ^2 @2 u @u @2 u H 2 ¼ m 2 þ 2^c2 ^ L @ t ^ ^ ^ @x @x2 @t  2  2 !# ZL " ^ ^ ^1 ^2 @u @u ^1 cos V ^ 1 t þ @u1 dc þ 1  þ F ^ 2 ^ dx ^ ^ @x @x @x 0

^x ^ ^t ^Þ cos V ^ þ Fð dx

(1)

^  1=8, the static equilibrium where, for a sag-to-span ratio b=L ^ profile cðxÞ of the cable may be sufficiently approximated by a parabola. Hence,   ^ ^ mgL2 x ^x ^ cðxÞ ¼ 4b and H ¼ 1 ^ L L 8b

(2)

^ 1 ðx ^; ^tÞ and u ^ 2 ðx ^; ^tÞ denote the in-plane (in-plane; i.e., the Here, u plane defined by the initial static configuration of the cable) and ^ at time ^t, m is the mass out-of-plane displacements at position x ^ per unit length, b is the cable sag, L is the cable span, E is Young's modulus, A is the area of the cross section, the ^ci are viscous damping coefficients, and g is the gravitational acceleration. The boundary conditions are ^1 ¼ u ^ 2 ¼ 0 at x ¼ 0 and x ¼ 1 u

(3)

We introduce non-dimensional quantities defined by ^ x x¼ ; L

ui ¼

^i u ; L

^ b b¼ ; L

c¼

^ c ^ b

and t ¼ g^t

(4)

into (1) and (2), where the characteristic time 1/g will be chosen at the end of the analysis, and obtain the following non-dimensional non-planar equations of motion: Z1 € 1 þ 2c1 u ^ 1  u001 ¼ ðau001 þ bc00 Þ u 0

  1  F1 cos V1 t þ bc0 u01 þ ðu01 2 þ u02 2Þ dx; 2 Z1 ^ 2  u002 ¼ au002 € 2 þ 2c2 u u 0

  1  F1 cos V1 t þ bc0 u01 þ ðu01 2 þ u02 2Þ dx þ FðxÞ cos Vt 2

(5)

Please cite this article in press as: H.S. Bauomy, A.T. El-Sayed, Control of a two-degree-of-freedom system with combined excitations, Archives of Civil and Mechanical Engineering (2014), http://dx.doi.org/10.1016/j.acme.2014.05.007

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where c(x) = 4x(1  x) and the boundary conditions are

yðe; tÞ ¼ y0 ðT0 ; T1 ; T2 Þ þ ey1 ðT0 ; T1 ; T2 Þ þ e2 y2 ðT0 ; T1 ; T2 Þ þ Oðe3 Þ

u1 ¼ u2 ¼ 0 at x ¼ 0 and x ¼ 1

The overdot and prime indicate the derivatives with respect to t and x, respectively, and ^ ^1 H EA 8bEA LF ¼ ; g2 ¼ ; a¼ ; F1 ¼ 2 2 H H mL mgL ^ ^ V V L2 g ^c ; V1 ¼ 1 and V ¼ ¼ g H i g

^ LF F¼ ; H

ci (7)

In order to analyze the nonlinear responses of this system, we first discretize (5) using the Galerkin procedure. To this end, we express the in-plane and out-of-plane displacements u1 and u2 as u1 ðx; tÞ ¼ fn ðxÞxðtÞ;

u2 ðx; tÞ ¼ ’n ðxÞyðtÞ

(8)

where the fn(x) and wn(x) are the modes shapes of the linearized problem, x(t) and y(t) are generalized coordinates. Substituting (8) into (5), the two-degree-of-freedom nonlinear equations can be obtained as follows: € þ 2e2 cx x_ þ ðv2x þ 2e2 V f 1 cos V1 tÞx þ ea2 x2 þ eb2 y2 þ e2 g 2 x3 x þ e2 h2 xy2 ¼ 0

(9)

€ þ 2e2 cy y_ þ ðv2y þ 2e2 V f 2 cos V1 tÞy þ ea3 xy þ e2 g 3 y3 þ e2 h3 x2 y y ¼ 2e2 VF cos Vt;

(10)

where x and y refer to the vertical (in-plane) and horizontal (out-of-plane) displacements, respectively, cx and cy are viscous damping coefficients, 0 < e  1 is a small parameter. vx and vy, respectively, represent the natural frequencies associated with the anti-symmetric or symmetric in-plane and out-of-plane modes. The other coefficients presented in Eqs. (9) and (10) can be found in paper [3]. Our attention is focused on a suspended cable excited by harmonic, parametric and tuned excitations. The two-degreeof-freedom differential equations describing the nonlinear dynamics of the suspended cable [22] can be written as: ^ 2 y2 þ e^ € þ 2e^cx x_ þ ðv2 þ 2eV^f cos V1 tÞx þ e^ x a2 x2 þ eb g 2 x3 x

The derivatives will be in the form: d ¼ D0 þ eD1 þ e2 D2 dt

(16)

d2 ¼ D20 þ 2eD0 D1 þ e2 ðD21 þ 2D0 D2 Þ dt2

(17)

For the second-order approximation, we introduce three timescales, where Tn = ent and Dn ¼ @T@n ðn ¼ 0; 1; 2Þ Substituting (14)–(17) into (11) and (12) and equating the same power of e leads to: Oðe0 Þ :

ðD20 þ v2x Þx0 ¼ 0

(18a)

ðD20 þ v2y Þy0 ¼ 0

(18b)

OðeÞ : ðD20 þ v2x Þx1 ¼ 2D0 D1 x0  2^cx D0 x0  2x0 V^f 1 cos V1 t ^ sin Vt ^2 y2  g^ 2 x3  h ^ 2 x20  b ^2 x0 y20 þ 2VðF a 0 0 ^ cos V2 t sin V3 tÞ  G ^ 1 ðD0 x0 Þ3 þQ

(19a)

ðD20 þ v2y Þy1 ¼ 2D0 D1 y0  2^cy D0 y0  2y0 V^f 2 cos V1 t ^ cos Vt ^ 3 x0 y0  g^ 3 y30  h ^3 x20 y0 þ 2VðF a ^ cos V2 t cos V3 tÞ  G ^ 2 ðD0 y Þ3 þQ 0

(19b)

Oðe2 Þ :

1

^ cos V2 t sin V3 tÞ þ R1 ^ sin Vt þ Q þ e^ h2 xy2 ¼ 2eVðF

(11)

^ cos V2 t cos V3 tÞ þ R2 ^ cos Vt þ Q ¼ 2eVðF

^2 y y  3^ ^2 ð2x0 y0 y1 þ x1 y20 Þ  2b g 2 x20 x1  h 0 1 (12)

^ 1 x_ 3 and R2 ¼ e G ^ 2 y_ 3 are the control forces where R1 ¼ e G which added to the system and that this system is symmetric with respect to the mirror reflection symmetry x 7! x, y 7!  y. The standard method of perturbation (MSPT) is used to obtain a uniformly valid, asymptotic expansion of the solutions for (11) and (12), taking into account the resonance condition V ffi vx, V1 ffi 2vx, V2  V3 ffi vy and vx ffi 2vy, introducing the s Þ according to detuning parameters s1, s2, s3 and s 4 ðs ¼ e^ V2  V3 ffi vy þ e^ s2 ;

s4 vx ffi 2vy þ e^

ðD20 þ v2x Þx2 ¼ 2D0 D1 x1  ðD21 þ 2D0 D2 Þx0  2^cx ðD0 x1 þ D1 x0 Þ  2x1 V^f 1 cos V1 t  2^ a2 x0 x1

€ þ 2e^cy y_ þ ðv2y þ 2eV^f 2 cos V1 tÞy þ e^ y a3 xy þ e^ g 3 y3 þ e^ h3 x2 y

s1 ; V ffi vx þ e^

(15)

(6)

V1 ffi 2vx þ e^ s3 ;

^ 1 ðD0 x0 Þ2 ðD0 x1 þ D1 x0 Þ  3G

(20a)

ðD20 þ v2y Þy2 ¼ 2D0 D1 y1  ðD21 þ 2D0 D2 Þy0  2^cy ðD0 y1 þ D1 y0 Þ  2y1 V^f 2 cos V1 t  3^ g 3 y20 y1   ^ 3 x0 y1 þ x1 y0  h ^3 ð2y0 x0 x1 þ y1 x20 Þ a ^ 2 ðD0 y Þ2 ðD0 y þ D1 y Þ  3G 0 1 0

(20b)

The general solutions of (18) can be written in the form x0 ¼ A1 ðT1 ; T2 Þexpðivx T0 Þ þ cc

(21a)

y0 ¼ A2 ðT1 ; T2 Þexpðivy T0 Þ þ cc

(21b)

(13)

The asymptotic approximate solution of (11) and (12) is in the form: xðe; tÞ ¼ x0 ðT0 ; T1 ; T2 Þ þ ex1 ðT0 ; T1 ; T2 Þ þ e2 x2 ðT0 ; T1 ; T2 Þ þ Oðe3 Þ (14)

where Am(m = 1, 2) are complex functions in T1 and T2, cc denotes the complex conjugate functions of the previous terms.

Please cite this article in press as: H.S. Bauomy, A.T. El-Sayed, Control of a two-degree-of-freedom system with combined excitations, Archives of Civil and Mechanical Engineering (2014), http://dx.doi.org/10.1016/j.acme.2014.05.007

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Substituting (21) into (19) and using the response condition equation (13) leads to secular terms. Eliminating these secular terms leads to solvability for the first-order approximation:

^ 1 v2 A2 D1 A1  6G ^ 1 v2 A1 A1 D1 A1 2ivx D2 A1 ¼ ½D21 A1  2^cx D1 A1 þ 3G x 1 x þ G^ 1 A1 A21 þ G^ 2 A1 A2 A2 þ G^ 3 A1 A31 þ G^ 4 A1 A21 A2 A2 2

þ G^ 5 A1 A22 A2  þ ½G^ 6 A1 A1 A22 þ G^ 7 A2 A32 expði^ s 4 T1 Þ 2

2 s 4 T1 Þ þ ½G^ 9 A21 A2 expði^ s 4 T1 Þ þ ½G^ 8 A1 A42 expð2i^

g 2 A21 A1  2^ h2 A1 A2 A2 2ivx D1 A1 ¼ ½2^cx ivx A1  3^ 

^ 1 A2 A1  3iv3x G 1

þ

2 s 3 T1 Þ þ ½G^ 10 A1 A1 þ G^ 11 A1 A2 A2 expði^

^2 A2 expði^ ½b s 4 T1 Þ 2

^ s 1 T1 Þ þ ½A1 V^f 1 expði^ s 3 T1 Þ þ ½iVFexpði^

s 3 T1 Þ þ ½G^ 13 A2 expðið^ s 3 þ s^ 4 ÞT1 Þ þ ½G^ 12 A31 expði^ (24a) 2

(22a)

^ 2 A2 A2  2ivy D1 A2 ¼ ½2^cy ivy A2  3^ g 3 A22 A2  2^ h3 A1 A1 A2  3iv3y G 2 " # ^ VQ a3 A1 A2 expðis 4 T1 Þ þ (22b) þ ½^ expðis 2 T1 Þ 2

^ 2 v2 A2 D1 A2  6G ^ 2 v2 A2 A2 D1 A2 2ivy D2 A2 ¼ ½D21 A2  2^cy D1 A2 þ 3G y 2 y þ G^ 13 A1 A1 A2 þ G^ 14 A22 A2 þ G^ 15 A32 A2 þ G^ 16 A1 A1 A22 A2 2

2 þ G^ 17 A21 A1 A2 þ G^ 18 A2  þ ½G^ 19 A21 A1 A2 2 þ G^ 20 A1 A2 A2 expðis 4 T1 Þ þ ½G^ 21 A1 A32 expðis 4 T1 Þ 3 3 þ ½G^ 22 A21 A2 expð2is 4 T1 Þ þ ½G^ 23 A2 expðið2s 4 þ s 3 ÞT1 Þ þ ½G^ 24 A1 A2 expðiðs 4 þ s 3 ÞÞT1 Þ

After eliminating the secular terms, the particular solutions of (19) will be in the form: x1 ¼

y1 ¼

    g^ A3  iv3x G1 A31 a ^ 2 A21 expð2ivx T0 Þ þ 2 1 expð3ivx T0 Þ 2 2 3vx 8vx " # ^ h2 A1 A22 expðiðvx þ 2vy ÞT0 Þ þ ðv2x  ðvx þ 2vy Þ2 Þ # " 2 ^ h2 A1 A2 expðiðvx  2vy ÞT0 Þ þ ðv2x  ðvx  2vy Þ2 Þ " # ^ 2 A2 A2 ^ a2 A1 A1  b þ 2 vx " # A1 V^f 1 þ expðiðV1 þ vx ÞT0 Þ ðv2x  ðV1 þ vx Þ2 Þ # " ^ iVQ þ expðiðV2 þ V3 ÞT0 Þ 2ðv2x  ðV2 þ V3 Þ2 Þ # " ^ iVQ expðiðV2  V3 ÞT0 Þ þ cc þ 2ðv2x  ðV2  V3 Þ2 Þ

" # g^ 3 A32  iv3y G2 A32 expð3ivy T0 Þ 8v2y " # ^ a3 A1 A2 þ expðiðvx þ vy ÞT0 Þ ðv2y  ðvx þ vy Þ2 Þ # " ^ h3 A21 A2 þ expðið2vx þ vy ÞT0 Þ ðv2y  ð2vx þ vy Þ2 Þ # " ^ h3 A21 A2 þ expðið2vx  vy ÞT0 Þ ðv2y  ð2vx  vy Þ2 Þ " # ^ VF þ expðiVT0 Þ ðv2y  V2 Þ " # A2 V^f 2 expðiðV1 þ vy ÞT0 Þ þ ðv2y  ðV1 þ vy Þ2 Þ " # A2 V^f 2 þ expðiðV1  vy ÞT0 Þ ðv2y  ðV1  vy Þ2 Þ # " ^ VQ þ expðiðV2 þ V3 ÞT0 Þ þ cc 2ðv2y  ðV2 þ V3 Þ2 Þ

þ ½G^ 25 A21 A2 expðis 3 T1 Þ þ ½G^ 26 A1 A2 expðis 3 T1 Þ 2

(24b) where G d ðd ¼ 1; 2; :::; 26; G ¼ eG^ Þ are constants (see Appendix)

3.

Stability analysis

The stability of the considered system is investigated at the worst resonance case (confirmed numerically), which is the simultaneous primary, sub-harmonic, combined and internal resonance case, has been chosen to study the stability from the second-order approximation solution. From (16) and multiplying both sides by 2ivn(n = x, y) we get (23a)

2ivx

dA1 ¼ e2ivx D1 A1 þ e2 2ivx D2 A1 þ Oðe3 Þ dt

(25a)

2ivy

dA2 ¼ e2ivy D1 A2 þ e2 2ivy D2 A2 þ Oðe3 Þ dt

(25b)

To analyze the solution of equations (22) and (24), it is convenient to express Am in the form a m expði’m Þ Am ¼ (26) 2 where am and wm are the steady state amplitudes and phases of the motions respectively. Substituting (26), (22) and (24) into (25) and equating imaginary and real parts, we obtain: a_ 1 ¼ ½cx a1 þ H1 a31 þ H2 a1 a22 þ H3 a51 þ H4 a31 a22 þ H5 a1 a42    s1 V VF þ 3H6 a21 þ H7 a22 þ F  cos ðu1 Þ 2v2x vx þ ½H8 þ 5H9 a21  sin ðu1 Þ þ ½H10 a1 a2  sin ðu2 Þ þ ½H11 a1 þ H12 a31  cos ðu3 Þ    s3 V f 1 V f 1 G 11 3 2  H a þ a a a þ 1 13 1 1 2 sin ðu3 Þ 4v2x 2vx 8vx

(23b)

Substituting (21), (23) into (20) and using the response condition equation (13) leads to secular terms. Eliminating these secular terms leads to solvability for the second-order approximation:

þ ½5H14 a21 a22 þ H15 a22 þ H16 a42  cos ðu4 Þ     s 4 b2 b2 2 4 þ H17 a21 a22 þ þ þ H a a sin ðu4 Þ 18 2 2 8v2x 4vx þ ½H19 a1 a42 cos ð2u4 Þ þ ½H20 a1 a42  sin ð2u4 Þ   b VQ þ  2 2 a2 sin ðu2  u4 Þ þ ½H21 a22  sin ðu3 þ u4 Þ 8vx vy  2  V f1 þ F cos ðu3  u1 Þ 4v3x

(27a)

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’_ 1 a1 ¼ ½H22 a1 þ H23 a31 þ H24 a1 a22 þ H25 a51 þ H26 a31 a22 þ H27 a1 a42 

a_ 2 ¼ ½cy a2 þ H31 a32 þ H32 a21 a2 þ H33 a52 þ H34 a21 a32 þ H35 a41 a2  þ ½H36 a1 a2 þ H37 a1 a32 þ H38 a31 a2  cos ðu4 Þ " ! # ð2s y  s x Þa3 a3 3 3  a þ H a a þ H a a þ a 1 2 39 1 2 40 1 2 sin ðu4 Þ 8v2y 4vy

þ ½H8 þ H9 a21  cos ðu1 Þ   s1 V VF F  sin ðu1 Þ þ H6 a21 þ H7 a22 þ 2v2x vx

þ ½H41 þ 3H42 a22  cos ðu2 Þ " ! # Vs y Q VQ 2 2 þ H a þ H a þ þ sin ðu2 Þ 43 44 2 1 2vy 4v2y

   s3 V f 1 V f 1 G 11 þ  þ a1 a22 cos ðu3 Þ a1 þ H28 a31  2 4vx 2vx 8vx þ ½H11 a1 þ H29 a31  sin ðu3 Þ

"

    s 4 b2 b2 2 4 þ þ H a a cos ðu4 Þ þ H30 a21 a22 þ 18 2 2 8v2x 4vx

þ

þ ½H14 a21 a22  H15 a22  H16 a42  sin ðu4 Þ

þ

"

þ ½H20 a1 a42  cos ð2u4 Þ þ ½H19 a1 a42  sin ð2u4 Þ  þ

# a3 VF a cos ðu1 þ u4 Þ þ ½H46 a1 a2 sinðu3 þ u4 Þ 2 8vx v2y

"



þ 

b2 VQ a2 cos ðu2  u4 Þ þ ½H21 a22  cos ðu3 þ u4 Þ 8v2x vy

 2  V f1 þ F sin ðu3  u1 Þ 4v3x

# h3 VF a a cos ðu1 Þ þ ½H45 a21 a2  sin ðu3 Þ 1 2 4vx v2y

# a3 VQ a sin ðu2  u4 Þ þ ½H47 a21 a32  cos ð2u4 Þ 1 16v3y

þ ½H48 a21 a32  sin ð2u4 Þ þ ½H49 a32  cos ðu3 þ 2u4 Þ (27b)

þ ½H50 a32  sin ðu3 þ 2u4 Þ

(28a)

Fig. 1 – Time trace of the response and phase-plane diagrams of two modes for non-resonance case (basic case) at selected values: cx = 0.2, a2 = 0.3, b2 = 0.2, g2 = 0.3, h2 = 0.4, cy = 0.1, a3 = 0.1, g3 = 0.1, h3 = 0.1, vx = 7.5, vy = 7.8, V = 1.4, V1 = 5.3, V2 = 4.75, V3 = 3.35, f1 = 4.0, f2 = 3.0, F = 2.0, Q = 4.0. Please cite this article in press as: H.S. Bauomy, A.T. El-Sayed, Control of a two-degree-of-freedom system with combined excitations, Archives of Civil and Mechanical Engineering (2014), http://dx.doi.org/10.1016/j.acme.2014.05.007

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Fig. 2 – Numerical solution of the time response and phase-plane diagrams when V = vx, V2 S V3 ffi vy, V1 = 2vx, vx = 2vy. (a) System without controller. (b) With negative cubic velocity feedback control (G1 = 1.0, G2 = 2.0).

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’_ 2 a2 ¼ ½H51 a2 þ H52 a32 þ H53 a21 a2 þ H54 a52 þ H55 a21 a32 þ H56 a41 a2  " ! # ð2s y  s x Þa3 a3 3 3  þ þ a1 a2 þ H57 a1 a2  H40 a1 a2 8v2y 4vy

u_ 11 ¼ E9 a11 þ E10 u11 þ E11 a21 þ E12 u21

(32b)

a_ 21 ¼ E13 a11 þ E14 u11 þ E15 a21 þ E16 u21

(33a)

H58 a1 a32

u_ 21 ¼ E17 a11 þ E18 u11 þ E19 a21 þ E20 u21

(33b)

H38 a31 a2  sin ðu4 Þ

þ cos ðu4 Þ þ ½H36 a1 a2 þ " ! # Vs y Q VQ 2 2 þ    3H43 a2  H44 a1 cos ðu2 Þ 2vy 4v2y þ ½H41 þ H42 a22  sin ðu2 Þ þ ½H59 a21 a2  cos ðu3 Þ " # a3 VF a2 sin ðu1 þ u4 Þ þ ½H46 a1 a2  cos ðu3 þ u4 Þ þ 8vx v2y " # a3 VQ þ  a1 cos ðu2  u4 Þ þ ½H48 a21 a32  cos ð2u4 Þ 16v3y þ ½H47 a21 a32  sin ð2u4 Þ þ ½H50 a32  cos ðu3 þ 2u4 Þ þ ½H49 a32  sin ðu3 þ 2u4 Þ

(28b)

where Ej(j = 5, 6, . . ., 20) are nonlinear parameter in a10, a20, u10, u20, u30, u40 with the parameters cx, vx, V, f1, a2, b2, g2, h2, F, Q, G1, cy, vy, f2, V1, a3, g3, h3, G2 (see Appendix). The stability of a particular equilibrium solution was determined by examining the real part of the Jacobian matrix of the right hand sides of Eqs. (32) and (33).



ðE5  lÞ E6

E7 E8



E9

ðE  lÞ E E 10 11 12

¼0 (34)

E13

E ðE  lÞ E 14 15 16



E17

E18 E19 ðE20  lÞ

where u1 = s1T1  w1, u2 = s2T1  w2, u3 = s3T1  2w1, u4 = s4T1 + g1  2w2 and H‘(‘ =1, 2, . . ., 59) are constants (see Appendix).

3.1.

l4 þ r1 l3 þ r2 l2 þ r3 l þ r4 ¼ 0

Stability at the fixed point

The steady-state solution of our dynamical system corresponding to the fixed point of (27) and (28) is obtained when a_ m ¼ 0 and u_ m ¼ 0, which in turn correspond to 1 1 1 (29) ’_ 1 ¼ s^ 1 ¼ s^ 3 ¼ s x ; ’_ 2 ¼ s^ 2 ¼ s^ 4 þ s x ¼ s y 2 2 2 Hence, the frequency response equations (FRE) for practical case (a1 6¼ 0, a2 6¼ 0) as follows: s 2x þ E1 s x þ E2 ¼ 0;

s 2y þ E3 s y þ E4 ¼ 0

(30)

where E1, E2, E3 and E4 are constants in the parameters cx, vx, V, f1, a2, b2, g2, h2, F, Q, G1, cy, vy, f2, V1, a3, g3, h3, G2, a1, a2.

3.2.

Nonlinear solution

To determine the stability of the non-linear solution, one lets am ¼ am0 þ am1 ;

um ¼ um0 þ um1

(31)

where am0 and um0 are the solutions of (27) and (28) and am1, um1 are perturbations which are assumed to be small compared with am0 and um0. Substituting (31) into (27) and (28) and keeping only the linear terms in am1 and um1, we obtain a_ 11 ¼ E5 a11 þ E6 u11 þ E7 a21 þ E8 u21

Then:

(32a)

(35)

where r1, r2, r3 and r4 are constants in the parameters Ej(j = 5, 6, . . ., 20). If the real part of each eigenvalues is negative, the corresponding equilibrium solution is asymptotically stable otherwise become unstable. According to the Routh-Hurwitz criterion, the necessary and sufficient conditions for all the roots of (35) to possess negative real parts are: r1 > 0, r1r2  r3 > 0, r3 ðr1 r2  r3 Þ  r21 r4 > 0 and r4 > 0.

4.

Numerical results

4.1.

Active control effect

The numerical solution of the given system of Eqs. (11) and (12) is determined by using a fourth order Rung–Kutta algorithm to study the behavior of the system. Fig. 1 illustrates the response and the phase-plane for the non-resonant system behavior, where the maximum steady state amplitudes x and y are about 30% and 70% of the external excitation amplitude F respectively, this case can be regarded as a basic case. Fig. 2 shows that the time history and the phase phaseplane for, one of the worst resonance case, the simultaneous primary, sub-harmonic, combined and internal resonance case where V ffi vx, V1 ffi 2vx, V2  V3 ffi vy and vx ffi 2vy (Table 1 shows the results of the worst resonance conditions). It is

Fig. 3 – Effect of the gains G1 and G2.

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Fig. 4 – Effect of system parameters on FRE of the first mode x at (a1 6¼ 0 and a2 6¼ 0). (a) Effects of the detuning parameter sx. (b) Effects of the excitation force f1. (c) Effects of the natural frequency vx. (d) Effects of the damping coefficient cx. (e) Effects of the nonlinear coefficient a2. (f) Effects of the nonlinear coefficient g2. (g) Effects of the nonlinear coefficient b2. (h) Effects of the parameters h2 and Q. (i) Effects of the external force F. (j) Effects of the gain G1.

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Fig. 4. (Continued ).

Table 1 – Summary of the worst resonance cases. Resonance cases

%x

%y

Fig. no

Remarks

Without controller Non-resonant vx 6¼ vy 6¼ V 6¼ V1 6¼ V2 6¼ V3 Primary resonance V ffi vy Sub harmonic resonance V1 ffi 2vy Internal resonance vx ffi 4vy Combined resonance V1  V2  V3 ffi vy Simultaneous resonance V = vx, V2  V3 ffi vy, V1 = 2vx, vx = 2vy

observed from this figure that the steady-state amplitudes are increased to about 3333% and 1000% respectively of that value shown in Fig. 1. In this resonance case we apply active control (a negative cubic velocity feedback) and we found that the amplitudes of the system are decreased. The effectiveness of the controller is represented by (Ea = steady state amplitude of the system without controller/steady state amplitude of the system with controller), where (Ea = 5000% for the amplitude x and Ea = 3000% for the amplitude y). Fig. 3 shows the comparison between the three different types of negative velocity feedback (linear or its quadratic or

100 1666 1000 333 1350 3333

100 666 500 666 670 1000

Fig. 1

Fig. 2a

Multi-limit Chaotic Multi-limit Multi-limit Multi-limit Chaotic

cycle cycle cycle cycle

cubic value). Also it can be seen from the figure that best of controller to reduced the amplitudes of the system when using negative cubic velocity feedback.

4.2.

Response curves and effects of different parameters

The frequency response equation (FRE) given by Eq. (30) are nonlinear algebraic equations of a1 against sx and a2 against sy. These equations are solved numerically as shown in Figs. 4 and 5. Some figures possess hard and soft effects. This bending leads to multi-valued solutions and to jump phenomenon.

Table 2 – Numerical solutions of frequency response equations. Parameters

Effect

Fig. 4(a) (a1, sx)

Parameters

Effect

Fig. 5(a) (a2, sy)

Excitation force f1 Natural frequency vy Damping coefficient cy and excitation force F Non-linear parameter a3 Non-linear parameter g3 Non-linear parameter h3 Excitation force Q

M.I. M.D. S.A.

Fig. 5(b) Fig. 5(c) Fig. 5(d)

M.D. M.D. M.D. M.I.

Fig. Fig. Fig. Fig.

Gain G2

M.D.

Fig. 5(i)

Excitation force f1 Natural frequency vx Damping coefficient cx

M.I. M.D. M.D.

Fig. 4(b) Fig. 4(c) Fig. 4(d)

Non-linear parameter a2 Non-linear parameter g2 Non-linear parameter b2 Non-linear parameter h2 and Excitation force Q Excitation force F Gain G1

M.I. with H&S M.D. M.I. S.A.

Fig. Fig. Fig. Fig.

M.I. M.D.

Fig. 4(i) Fig. 4(j)

4(e) 4(f) 4(g) 4(h)

5(e) 5(f) 5(g) 5(h)

M.I. denotes that the amplitude is monotonic increasing function in the parameter. M.D. denotes that the amplitude is monotonic decreasing function in the parameter. S.A. denotes that the amplitude is saturation. H&S means that the parameter has hardening and softening nonlinearity effects.

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Fig. 5 – Effect of system parameters on FRE of the second mode y at (a1 6¼ 0 and a2 6¼ 0). (a) Effects of the detuning parameter sy. (b) Effects of the excitation force f2. (c) Effects of the natural frequency vx. (d) Effects of the parameters cy and F. (e) Effects of the nonlinear coefficient a3. (f) Effects of the nonlinear coefficient g3. (g) Effects of the nonlinear coefficient h3. (h) Effects of the tuned force Q. (i) Effects of the gain G2.

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found that all predictions from analytical solutions are in good agreement with the numerical simulation.

6.

Conclusions

The vibrations of second order nonlinear differential equations of a suspended cable system subjected to external, parametric and tuned excitation forces can be controlled via a negative cubic velocity feedback to the system. Multiple time scale perturbation technique is useful to determine approximate solutions for the differential equations describing the system up to second order approximation. To study the stability of the system, the frequency response equations are applied. The effects of the different parameters of the system are studied numerically. From the above study the following may be concluded:

Fig. 5. (Continued ).

The curves of all these figures consist of two branches represent stable (solid line) and unstable (dashed lines) solutions. The following table shows the all effects of the different parameters of the system. Table 2.

5. Comparison between time response solutions of the perturbation and the numerical methods Fig. 6 shows the comparison between analytical solution from the second order approximation given by the set of equations (27) and (28) at the same parameters magnitudes used for the numerical solution of equations (11) and (12). The dashed lines show the modulation of the amplitudes for the generalized coordinates x and y. However, the continuous lines represent the time history of vibrations which were obtained numerically as solutions of the original equations of the system. We

(1) The worst resonance case of the system is the simultaneous primary, sub-harmonic, combined and internal resonance case V ffi vx, V1 ffi 2vx, V2  V3 ffi vy and vx ffi 2vy. (2) Negative cubic velocity feedback active controller is the best one for the reported worst resonance case as it reduces the vibration dramatically. (3) The effectiveness of the best controller at the reported worst resonance case are about Ea = 5000 for x and Ea = 3000 for y. (4) The steady-state amplitudes of both modes are monotonic increasing functions in the parameters f1, f2, F, Q, a2 and b2 while decreasing functions in the coefficients cx, vx, vy, G1, G2, a3, g2, g3 and h3. (5) Also, for the nonlinear parameter a2 the steady-state amplitude is making hardening and softening effects. (6) The parameters h2, cy have no significant effects on the steady state amplitudes denoting the occurrence of saturation phenomenon.

X-Amplitude

Perturbation Analysis Numerical Solution

0.5 0 -0.5 0

50

100

150

200 Time

250

300

350

400

0

50

100

150

200 Time

250

300

350

400

Y-Amplitude

0.5 0 -0.5

Fig. 6 – Time responses for the main system with controller. Please cite this article in press as: H.S. Bauomy, A.T. El-Sayed, Control of a two-degree-of-freedom system with combined excitations, Archives of Civil and Mechanical Engineering (2014), http://dx.doi.org/10.1016/j.acme.2014.05.007

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Appendix

! ! ! ! ^ 1 þ 9v6 G ^2 ^2 a ^2 ^3 ^2 ^22 4^ a2 b 2b g 22 þ 12^ g 2 iv3x G ^ h3 h h x 1 ^ ^ 3 ¼ 3^ ^5 ¼  ¼ ; G G ; G ; 4 v2x ðv2x þ 2vx vy Þ 8v2x ðv2x  v2y Þ 2ðv2x  v2y Þ ! ! ! ^ ^2 ^ 2 g^ 3 þ 2b ^2 iv3 G ^ h ^2 iv3y G 2b ^ h2 g^3 þ h ^2 h ^2 ^2 ^2 ^ ^ ^22 a h 2^ a3 h 2^ a2 h 2b y 2  2 þ þ 22 2 ; G^ 8 ¼  G7 ¼ ; G^ 6 ¼ 2 2 2 2 2 2 ðvx  vy Þ ðvx þ 2vx vy Þ vx 8vy vx 4vy ð4vy  4vx vy Þ ! ! ! ! ^ 1 V^f ^ 1 iðV1 þ vx Þv2 V^f ^2 h 2^ a2 h ^2 2b ^3 h ^2 a ^2 ^ 3^ g 2 V^f 1 þ 3G 4^ h2 V^f 2 ^ g 2 V^f 1 þ iv3x G x 1 1 ^ ^   ¼ ¼ ¼ ; G ; G ; G ; G^ 9 ¼ 10 11 12 ð4v2y  4vx vy Þ ð4v2x  4vx vy Þ 3v2x 8v2x ðV21 þ 2V1 vx Þ ðV21  4v2y Þ 0 1 " ! ! # ^ 2 þ 9G ^ 2 v6 ^2 V^f ^2 3^ g 23 þ 12i^ g 3 v3y G 2b ^ 23 ^3 ^ ^2 h ^3 a h 2^ a2 a 2^ a3 b 2 yA ^ 2 ^ 16 ¼ @ ^ 14 ¼ ; G ¼ ; G ¼ þ ; G ; G ; G^ 13 ¼ 15 17 v2x v2x 8v2y ðv2x  v2y Þ ðV21  2V1 vy Þ ðv2y  ðvx þ vy Þ2 Þ " " # # 2   ^3 ð5vx  2vy Þ^ a3 h ^ h23 2V2 ^f 2 10^ h3 a ^2 ^ 20 ¼ þ ; ; G^ 19 ¼ ; G G^ 18 ¼ 2 2 2 2 2ðvx  vy Þ 4vx ðvx  vy Þðvx þ 2vy Þ 3vx ðV1  4v2y Þ ! ! ^2 ^2 ^2 3iðvx þ vy Þv2y a ^3 G ^ 3 iv3y G ^ a3 g^3 þ a ^3 h ^2 ^3 ^3 h ^2 a a 3^ g3a 4^ h3 b ^ 22 ¼ þ þ  þ ; G ; G^ 21 ¼ ð4v2y þ 4vx vy Þ ðv2x  2vx vy Þ v2x ðv2x  2vx vy Þ 8v2y ð4v2y þ 4vx vy Þ ! ^2 ^3 ð6v2y þ v2x  vx vy Þ^ g3h ivy ðv2x  13vx vy þ 6v2y Þ^ h3 G ^2 2^ h3 h  þ ; G^ 23 ¼ 8vx ðvx  vy Þv2y ð4v2y þ 4vx vy Þ 8vx ðvx  vy Þ " " # # ! ! 3iðV1  vy Þv2y ivy ^ 3 V^f 2 ^ 3 V^f 2 a a 3 1 ^ ^ ^ ^ ^   þ g^ 3 Vf 2 þ  G2 Vf 2 ; G 25 ¼ ; G 24 ¼ 8 ðV21 þ 2V1 vy Þ ðV21 þ 2V1 vy Þ ðv2x  2vx vy Þ ðV21 þ 2V1 vy Þ 8v2y ! ! h ^3 V^f 2 h ^3 V^f 2 h ^3 V^f 2 h ^3 V^f 2 þ þ ; G^ 27 ¼ G^ 26 ¼ 2 2 2 ðV1 þ 2V1 vy Þ ð4vx  4vx vy Þ ðV1  2V1 vy Þ ð4v2x þ 4vx vy Þ G^ 1 ¼



 10^ a22 ^ ;G2 ¼ 3v2x

!           3h2 v2y h2 cy 3g 2 cx 3v2x G1 6g 2 G1 6G1 h2 3g 2 V h V cx V ; ; ; ;  H ¼ H ¼  H ¼  H ¼ G F ; H7 ¼ 2 3 F; H8 ¼ 2 F; 2 3 4 5 2 ; H6 ¼ 2 2 2 3 16vx 8vx 2vx 8vx 8 4vx 64 32 32vx ! !     3VG1 h VQ cx V f 1 23V f 1 G1 3G1 ðV1 þ vx Þvx V f 1 13g 2 V f 1 3g 2 V f 1 F ; H10 ¼  2 2 ; H11 ¼  þ H ¼  H ¼  ; ; ; H9 ¼ 12 13 16 2v2x 64 64v3x 8vx vy 8ðV21 þ 2V1 vx Þ 8vx ðV21 þ 2V1 vx Þ !       3b2 v2y b vy G2 3b G1 b cx b cy 9g b h a3 G9 b h G6 ; H15 ¼  2 2 þ 2 2 ; H16 ¼ ; G2 þ 2  2 23  ; H17 ¼  2 32 þ 2 2 þ H14 ¼  2 2 64 8vx 4vx 32vx 64vx 64vx 16vx vy 16vx 16vx vy 16vx ! !     h2 vy G2 5b h 3b2 g 3 b g h2 g 3 h22 V f 1 b2 G 13 þ 2 3 2 ; H19 ¼ þ þ ; H20 ¼ ; ; H21  H18 ¼  2 32  2 2 2 3 32vx 32vx vy 64vx vy 128vx 128vx vy 32vx ð4vy  4vx vy Þ 16vx 4vx           15g 22 c2 V2 f12 G1 3g 2 b2 a3 G2 h2 9v3x 2 3g 2 h2 G4 G ; ; ; ; ; H ¼  þ H ¼   þ H ¼  þ H ¼   H22 ¼  x þ 23 24 25 26 2vx 8v3x 8vx 8vx 16v2x vy 8vx 4vx 256v3x 256 1 32v3x 32vx ! !   g2V f 1 3g 2 V f 1 11V f 1 G1 3G1 ðV1 þ vx Þvx V f 1 h2 G5 þ þ H ¼ ; H28 ¼ H27 ¼  2 3  ; ; 29 32vx 32vx 64v3x 64 8vx ðV21 þ 2V1 vx Þ 8ðV21 þ 2V1 vx Þ ! !       3cy g 3 3v2y G2 3g b G9 b h G6 cx h ^3 3g G2 3G2 h3 ; H31 ¼ ; H34 ¼  ;;  2 23   ; H32 ¼ ; H33 ¼  3 H30 ¼  2 32  2 2 64vx 16vx 16vx vy 16vx 8vy 8 4vy 32 16 ! ! !   3h3 v2x G1 cx a3 23G2 a3 3ðvx þ vy Þvy a3 G2 3v2x a3 G1  ; H38 ¼ ; H36 ¼ ; H37 ¼  ;; H35 ¼ 2 2 2 32vy 8vy 128 16ðvx  2vx vy Þ 64v2y ! ! !   3cy VQ 2h3 b2 þ a3 h2 a3 g 3 3g 3 a3 4h3 b2 3a3 g 2 G 20 9G2 VQ þ þ þ H ¼ þ H ¼ H ¼ ;; ; ; ; H39 ¼ 40 41 42 4v2y 32vx v2y 128v3y 16vy ðv2x  2vx vy Þ 16vy v2x 64vx v2y 16vy 32 0  1 ! ! ! v2x  13vx vy þ 6v2y h3 G2 h3 V f 1 G 27 G 26 a3 V f 1 G 25 3g 3 VQ h VQ A; ; þ  þ ; H44 ¼ 3 3 ; H45 ¼ ; H46 ¼ ; H47 ¼ @ H43 ¼ 3 2 2 32vy 16vy 8vx vy 8vy 8vy 16vx vy 4vy 256vx ðvx  vy Þ  0 1 !  6v2y þ v2x  vx vy g 3 h3 2h3 h2 A; H49 ¼  1  3ðV1  vy Þvy þ G2 V f 2 ; ; H48 ¼ @ 64 8ðV21 þ 2V1 vy Þ 256vx ðvx  vy Þv3y 32vy ð4v2y þ 4vx vy Þ ! ! ! ! c2y G 19 a23 3 1 a3 b2 G 15 3g 3 G 14 h3    V f ; H ¼  H ¼   þ H ¼  þ g ; ; ;; H50 ¼ 51 52 53 3 2 2vy 2vy 32vx v2y 8vy 8vy 32v3y 8vy 4vy 64v3y 8vy V21 þ 2V1 vy ! ! ! 9v3y G22 15g 23 h2 3g h G 17 G 18 þ ; H55 ¼  3 33  ; H56 ¼  3 3  ;; H54 ¼  3 256vy 256 32vy 32vy 32vy 32vy !     a3 g 3 a3 h2 a3 h2 3g 3 a3 h b 11a3 G2 3ðvx þ vy Þvy a3 G2 G 27 G 26   3 22 ; H58 ¼  ; H59 ¼     H57 ¼ 3 2 2 2 2 128vy 32vx vy 8vy ð4vy þ 4vx vy Þ 16vy ðvx  2vx vy Þ 4vy vx 128 16ðvx  2vx vy Þ 8vy 8vy H1 ¼

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 E5 ¼

 cx þ 3H1 a210 þ H2 a220 þ 5H3 a410 þ 3H4 a210 a220 þ H5 a420 þ ð6H6 a10 Þcosðu10 Þ þ ð10H9 a10 Þsinðu10 Þ þ ðH10 a20 Þsinðu20 Þ    sxV f 1 V f 1 G 11 2 2 þ ðH11 þ 3H12 a210 Þcosð2u10 Þ þ  3H a þ a þ sinð2u10 Þ þ ð10H14 a10 a220 Þcosð2u20  u10 Þ 13 10 2v2x 2vx 8vx 20  þ ð2H17 a10 a220 Þsinð2u20  u10 Þ þ ðH19 a420 Þcosð4u20  2u10 Þ þ ðH20 a420 Þsinð4u20  2u10 Þ 

E6 ¼

  sx V VF V2 f 1  3H6 a210 þ H7 a220 þ F  þ F sinðu10 Þ þ ðH8 þ 5H9 a210 Þcos ðu10 Þ  ð2H11 a10 þ 2H12 a310 Þ sin ð2u10 Þ 2v2x vx 4v3x    sx V f 1 V f 1 G 11 a10 þ 2H13 a310 þ  a10 a220 cosð2u10 Þ þ ð5H14 a210 a220 þ H15 a220 þ H16 a420 Þsinð2u20  u10 Þ þ 2 vx vx 4vx     ð2s y  s x Þb2 b2 2 2 2 4 4 4  H17 a10 a20 þ þ þ H a a 18 20 20 cosð2u20  u10 Þ þ ð2H19 a10 a20 Þsinð4u20  2u10 Þ  ð2H20 a10 a20 Þcosð4u 20  2u10 Þ 8v2x 4vx    b VQ þ  2 2 a20 cosðu10  u20 Þ þ ðH21 a220 Þcosð2u20 þ u10 Þ 8vx vy

   G 11 E7 ¼ 2H2 a10 a20 þ 2H4 a310 a20 þ 4H5 a10 a320 þ ð2H7 a20 Þcosðu10 Þ þ sinðu20 Þ þ a10 a20 sinð2u10 Þ þ ð10H14 a210 a20 þ 2H15 a20 4vx     ð2s y  s x Þb2 b2 3 2 3 3 þ 4H16 a20 Þcosð2u20  u10 Þ þ 2H17 a10 a20 þ 2 þ þ 4H a a 20 18 20 sinð2u 20  u10 Þ þ ð4H19 a10 a20 Þcosð4u20  2u10 Þ 8v2x 4vx    b VQ sinðu10  u20 Þ þ ð2H21 a20 Þsinð2u20 þ u10 Þ þ ð4H20 a10 a320 Þsinð4u20  2u10 Þ þ  2 2 8vx vy  E8 ¼ ðH10 a10 a20 Þcosðu20 Þ  2ð5H14 a210 a220 þ H15 a220 þ H16 a420 Þsinð2u20  u10 Þ     ð2s y  s x Þb2 b2 2 4 4 4 þ þ H a a þ 2 H17 a210 a220 þ 18 20 cosð2u20  u10 Þ  ð4H19 a10 a20 Þsinð4u20  2u 10 Þ þ ð4H20 a10 a20 Þcosð4u20  2u10 Þ 20 8v2x 4vx    b2 VQ 2 a  u Þ þ ð2H a Þcosð2u þ u Þ cosðu þ 20 10 20 21 20 10 20 8v2x vy  H24 a220 H27 a420 s x H22   3H23 a10   5H25 a310  3H26 a10 a220   ð2H9 Þcosðu10 Þ  ð2H6 Þsinðu10 Þ E9 ¼ a10 a10 a10 a10      sx V f V f1 G 11 2 H11   2 1þ a20 cosð2u10 Þ  þ 3H29 a10 sinð2u10 Þ  ð2H30 a220 Þcosð2u20  u10 Þ þ 3H28 a10  2vx a10 2vx a10 8vx a10 a10      4 4 H a H a 20 19 20 20 cosð4u20  2u10 Þ þ sinð4u20  2u10 Þ  ð2H14 a220 Þsinð2u20  u10 Þ  a10 a10 

E10 ¼

     a2 V2 f 1 sx V f 1 V f 1 H8 sx V VF þ H9 a10 sinðu10 Þ  H6 a10 þ H7 20 þ F  þ F cosðu Þ þ  þ 10 vx a10 4v3x a10 a10 a10 2a10 v2x v2x vx      ð2s  s Þb H18 a420 G b y x 2 11 2 2 2 a20 sinð2u10 Þ  ð2H11 þ 2H29 a210 Þcosð2u10 Þ  H30 a10 a220 þ þ þ a sinð2u20  u10 Þ þ 2H28 a210  20 4vx 8v2x a10 4vx a10 a10   H15 a220 H16 a420  cosð2u20  u10 Þ  ð2H20 a420 Þsinð4u20  2u10 Þ  ð2H19 a420 Þcosð4u20  2u10 Þ þ H14 a10 a220  a10 a10      H21 a220 b2 VQ þ a20 sinðu10  u20 Þ  sinðu10 þ 2u20 Þ 2 8vx vy a10 a10 



E11 ¼

       ð2s y  s x Þb2 H7 a20 G 11 b2  2H24 a20  2H26 a210 a20  4H27 a320  2 a20 cosð2u10 Þ  2H30 a10 a20 þ 2 þ sinðu10 Þ þ a20 2 a10 4vx 8vx a10 4vx a10  3  3  4H18 a20 2H15 a20 4H16 a20 þ  cosð2u20  u10 Þ  2H14 a10 a20  sinð2u20  u10 Þ  ð4H20 a320 Þcosð4u20  2u10 Þ a10 a10 a10      b2 VQ 2H21 a20 þ ð4H19 a320 Þsinð4u20  2u10 Þ   u Þ þ þ 2u Þ cosðu cosðu 10 20 10 20 8v2x vy a10 a10

       ð2s y  s x Þb2 H18 a420 H15 a220 H16 a420 b2 2 2 þ þ  u Þ  2 H a a   a sinð2u cosð2u20  u10 Þ E12 ¼ 2 H30 a10 a220 þ 20 10 14 10 20 20 8v2x a10 4vx a10 a10 a10 a10      2H21 a220 b2 VQ sinðu10 þ 2u20 Þ a20 sinðu10  u20 Þ  þ ð4H20 a420 Þsinð4u20  2u10 Þ þ ð4H19 a420 Þcosð4u20  2u10 Þ  2 8vx vy a10 a10

Please cite this article in press as: H.S. Bauomy, A.T. El-Sayed, Control of a two-degree-of-freedom system with combined excitations, Archives of Civil and Mechanical Engineering (2014), http://dx.doi.org/10.1016/j.acme.2014.05.007

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   h3 VF E13 ¼ 2H32 a10 a20 þ 2H34 a10 a320 þ 4H35 a310 a20 þ a20 cosðu10 Þ þ ð2H45 a10 a20 Þsinð2u10 Þ þ ð2H44 a10 Þsinðu20 Þ þ ðH36 a20 2 4vx vy    ð2s a3 y  s x Þa3 a20 þ H39 a320 þ 3H40 a210 a20 sinð2u20  u10 Þ þ H37 a320 þ 3H38 a21 a20 Þcosð2u20  u10 Þ þ  2 8vy 4vy    a3 VQ þ ð2H47 a10 a320 Þcosð4u20  2u10 Þ þ ð2H48 a10 a320 Þsinð4u20  2u10 Þ þ ðH46 a20 Þsinðu10 þ 2u20 Þ þ  sinðu10  u20 Þ 3 16vy 

E14 ¼

     h3 VF a10 a20 sinðu10 Þ þ 2H45 a210 a20 cosð2u10 Þ þ H36 a10 a20 þ H37 a10 a320 þ H38 a310 a20 sinð2u20  u10 Þ 2 4vx vy    ð2s y  s x Þa3 a3  a10 a20 þ H39 a10 a320 þ H40 a310 a20 cosð2u20  u10 Þ þ ð2H47 a210 a320 Þsinð4u20  2u10 Þ  2 8vy 4vy    a3 VQ  ð2H48 a210 a320 Þcosð4u20  2u10 Þ þ ðH46 a10 a20 Þcosðu10 þ 2u20 Þ þ  a10 cosðu10  u20 Þ 3 16vy 



E15 ¼



 h3 VF a10 cosðu10 Þ þ ðH45 a210 Þsinð2u10 Þ þ ð2H43 a20 Þsinðu20 Þ 2 4vx vy    ð2s y  s x Þa3 a3  a10 þ 3H39 a10 a220 þ H40 a310 sinð2u20 þ ð6H42 a20 Þcosðu20 Þ þ ðH36 a10 þ 3H37 a10 a220 þ H38 a310 Þcosð2u20  u10 Þ þ 2 8vy 4vy   a VF 3  u10 Þ þ ð3H47 a210 a220 Þcosð4u20  2u10 Þ þ ð3H48 a210 a220 Þsinð4u20  2u10 Þ þ cosð2u20 Þ þ ðH46 a10 Þsinðu10 þ 2u20 Þ 8vx v2y  þ ð3H49 a220 Þcosð4u20 Þ þ ð3H50 a220 Þsinð4u20 Þ  cy þ 3H31 a220 þ H32 a210 þ 5H33 a420 þ 3H34 a210 a220 þ H35 a410 þ



  ðH41 þ 3H42 a220 Þsinðu20 Þ þ

E16 ¼

Vs y Q VQ þ 2vy 4v2y



 þ H43 a220 þ H44 a210 cosðu20 Þ  ð4H47 a210 a320 Þsinð4u20  2u10 Þ

  ð2s y  s x Þa3 a3  a10 a20 þ ð4H48 a210 a320 Þcosð4u20  2u10 Þ  2ðH36 a10 a20 þ H37 a10 a320 þ H38 a310 a20 Þsinð2u20  u10 Þ þ 2 2 8vy 4vy    a3 VF þ H39 a10 a320 þ H40 a310 a20 cosð2u20  u10 Þ  a20 sinð2u20 Þ þ ð2H46 a10 a20 Þcosðu10 þ 2u20 Þ 4vx v2y    a3 VQ 3 3 þ a  u Þ  ð4H a Þsinð4u Þ þ ð4H a Þcosð4u Þ cosðu 10 10 20 49 20 20 50 20 20 16v3y 

E17 ¼

   sy ð2s y  s x Þa3 a3 2 2  2H53 a10  2H55 a10 a220  4H56 a310   þ H a  3H a þ 57 40 20 10 cosð2u20  u10 Þ a20 8v2y 4vy   2H44 a10 þ cosðu20 Þ  ð2H59 a10 Þcosð2u10 Þ  ðH36 þ H58 a220 þ 3H38 a210 Þsinð2u20  u10 Þ þ ð2H48 a10 a220 Þcosð4u20  2u10 Þ a20    a3 VQ cosðu10  u20 Þ  ð2H47 a10 a220 Þsinð4u20  2u10 Þ þ ðH46 Þcosðu10 þ 2u20 Þ þ 3 16vy a20

    ð2s y  s x Þa3 a3 2 3 2  þ þ H a a  H a a E18 ¼ ð2H59 a210 Þsinð2u10 Þ  10 57 10 40 20 10 sinð2u20  u10 Þ þ ðH36 a10 þ H58 a10 a20 8v2y 4vy þ H38 a310 Þcosð2u20  u10 Þ þ ð2H48 a210 a220 Þsinð4u20  2u10 Þ þ ð2H47 a210 a220 Þcosð4u20  2u10 Þ  ðH46 a10 Þsinðu10 þ 2u20 Þ    a3 VQ a10 sinðu10  u20 Þ  3 16vy a20 

E19 ¼

  H53 a210 H59 a210 H51   3H52 a20   5H54 a320  3H55 a210 a20  H56 a310 þ ð6H43 Þcosðu20 Þ  ð2H42 Þsinðu20 Þ  cosð2u10 Þ a20 a20 a20      3 3 ð2s y  s x Þa3 H40 a10 H38 a10 a3 H36 a10   a cosð2u sinð2u20  u10 Þ þ þ 3H a a   u Þ  þ 3H a a þ 10 57 10 20 20 10 58 10 20 8v2y a20 4vy a20 a20 a20 a20     a3 VF H46 a10 þ ð3H48 a210 a20 Þcosð4u20  2u10 Þ  ð3H47 a210 a20 Þsinð4u20  2u10 Þ  sinð2u20 Þ þ cosðu10 þ 2u20 Þ 8vx v2y a20 a20  þ ð3H50 a20 Þcosð4u20 Þ  ð3H49 a20 Þsinð4u20 Þ

Please cite this article in press as: H.S. Bauomy, A.T. El-Sayed, Control of a two-degree-of-freedom system with combined excitations, Archives of Civil and Mechanical Engineering (2014), http://dx.doi.org/10.1016/j.acme.2014.05.007

ACME-230; No. of Pages 17

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archives of civil and mechanical engineering xxx (2014) xxx–xxx



E20 ¼

   H44 a210 H41 þ H42 a20 cosðu20 Þ  ð4H50 a220 Þsinð4u20 Þ sinðu20 Þ  a20 a20    ð2s  s Þa a y x 3 3  þ a10 þ H57 a10 a220  H40 a310 sinð2u20  u10 Þ  2ðH36 a10 þ H58 a10 a220  ð4H49 a220 Þcosð4u20 Þ þ 2 2 8vy 4vy   a3 VF þ H38 a310 Þcosð2u20  u10 Þ  ð4H48 a210 a220 Þsinð4u20  2u10 Þ  ð4H47 a210 a220 Þcosð4u20  2u10 Þ  cosð2u20 Þ 2 4vx vy    a3 VQ  ð2H46 a10 Þsinðu10 þ 2u20 Þ þ a10 sinðu10  u20 Þ 16v3y a20 

Vs y Q VQ  4v2y a20 2vy a20



 3H43 a20 

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