Theoretical and Numerical Analysis of 1 : 1 Main Parametric Resonance of Stayed Cable Considering Cable-Beam Coupling

1School of Transportation and Civil Engineering & Architecture, Foshan University, Foshan, Guangdong 528000, China 2Faculty of Civil and Architectural Engineering, East China University of Technology, Nanchang, Jiangxi 330013, China 3College of Engineering, Department of Architectural Engineering, Kangwon National University, Chuncheon 200-701, Republic of Korea 4College of Civil Engineering, Hunan University, Changsha, Hunan 410082, China


Introduction
The excitation of parametric resonance is dependent on time and is used as a parameter in the vibration equation of the system [1][2][3].According to whether the amplitude and frequency of the excitation change with time or not, the parametric resonance of the stay cable is divided into two cases: When the quality of deck or tower is far greater than the cable, without considering the effect of cables on the deck or tower, as an ideal incentive, the incentive is not affected by the response and the system vibration is simplified as spring-mass [4] and cable-mass [5][6][7][8][9] model, in which the input energy of the stay cable is from the support point; when the vibration of cables and bridge deck or tower are coupled with each other, the system will be coupled with resonance in initial condition of cable-deck or cable-tower system.Finally, the large vibration of these two can be aroused, and the system vibration is simplified as the [10][11][12] model of the cable flexible beam.
In this paper, the displacement time history response and amplitude-frequency characteristics of the cable are mainly discussed in the case of the 1 : 1 main parametric resonance in the cable-beam coupling system, the approximate solutions of motion equations are solved by the method of multiple scales, and the results are verified by numerical simulation.However, the 2 : 1 parametric resonance and 1 : 2 and 1 : 3 superharmonic resonance response problems are not the focus of discussion here.

Theoretical Model of Parametric Resonance
Considering Cable-Beam Coupling

Differential Equation of Motion with Parametric Resonance Considering
Cable-Beam Coupling.Several basic assumptions are made before establishing the equation of motion for the cable-beam composite structure [13]: (1) The material nonlinearity of the cable and beam is not considered.
(2) The gravity sag curve of cable is considered as a parabola.
(3) The bending stiffness, torsional stiffness, and shear stiffness of the cable are not considered.
(4) The change of cable force along the length direction is not considered.
(5) Cable is always in elastic state during vibration.
(6) The axial deformation of beam is not considered.
On the premise of the above assumptions, the model is established in Figure 1, considering the refined model of the cable-beam coupling.
The motion trajectory of the cable and the deck beam is described by local coordinate system o-    and o-    , respectively, as shown in Figure 1.According to the Hamilton principle, the differential equation of motion for the cablebeam coupling structure is [14] For the convenience of research, the variables with subscripts c and b are defined as the variables of the cable and the deck beam, respectively.Variables u and v indicate the longitudinal and transverse vibration displacement, respectively, and they are the function of the position coordinate x and time t.X and Y represent the longitudinal and transverse sag curves function, respectively; moreover,  = 4 0 [/ − (/) 2 ].The meaning of other variables in formula ( 1) is as follows: , , , , , , , ,  0 , , and  are the quality of unit length, natural frequency, damping coefficient, damping ratio, elastic modulus, length, moment of inertia, the cross section area, the transverse initial deflection at the midspan, the sag at the midspan, and gravitational acceleration, respectively.Units of each variable are taken from the international unit system.In addition,  is the tangential tension of the stay cable,   is the dynamic tension of cable in vibration process, N is axial force of deck beam, s is thr coordinate by the arc length, and () is Dirac function: By using the static equilibrium relation of stay cable, formula (1) can be simplified as follows: where  0 is the static tangential tension of stay cable in formula (3); moreover,  0 is static axial tension and   and   are the axial strains of stay cable and beam, respectively.The axial strains can be expressed as follows: where   represents the longitudinal displacement of beam in formula (4); the two-order small quantity of longitudinal elastic strain for the cable and beam can be neglected, because the longitudinal deformation of the cable and beam is much less than that of transverse deformation.It can be approximate to take  ≈  for the stay cable with small sag, and formula (3) can be simplified as In the case of low modal vibration for the cable, the coupling effect of transverse and longitudinal vibration for the cable is not considered.Meanwhile, the first equation of formula ( 5) can be simplified as The second equation of formula ( 5) can be simplified as From the simplified assumptions, the geometric boundary condition of the stay cable is as follows: On both sides of formula ( 6) is integral on [0,   ] range for   at the same time, and it is brought into formula (8), which can be obtained as follows: The axial dynamic force increment of the cable in the process of transverse vibration can be expressed as follows: The axial force of bridge deck beam in vibration is as follows: Equations ( 9) and ( 10) are simultaneous.The third equation of formula ( 5) is simplified; differential equations of motion for the cable-beam coupled vibration are as follows:

Dimensionless Method of Parametric Resonance Equations
Considering Cable-Beam Coupling.The differential equation ( 12) of motion for cable-beam coupled vibrations is derived by using dimensionless method, and the following dimensionless quantities are defined [15][16][17]: where   and   are the first natural frequencies of the cable without considering or considering the sag;   is the first natural frequency of bridge deck beam;  is the sagspan ratio of cable;  2 is the stiffness coefficient of cable;   and   are the span of cable and beam, respectively;  2 is Irvine parameter;   indicates the ratio coefficient of first-order natural frequency for bridge deck beam;  denotes the dimensionless time; V  and V  are the transverse relative displacements of the cable and beam, respectively.  and   are the relative frequencies of the cable and beam, respectively.The expressions of each parameter are 1/2 ; Equation ( 12) is derived by using the dimensionless quantity of the above definition.
When considering that the first-order mode is only the main vibration, and constraint on the boundary condition of formula ( 8) is satisfied, the dimensionless displacement function of the cable is as follows: Moreover, the geometric boundary condition of the fixed end for the bridge deck beam is as follows: The physical boundary condition of the connection between cable and bridge deck beam is as follows: while the dimensionless modal function of in-plane vibration for the deck beam can be expressed as follows: where  1 ,  2 ,  3 , and  4 are constant coefficients; they are determined by formulas ( 17) and (18).Moreover,   is determined by the following characteristic equation: And the dimensionless displacement function of the deck beam can be expressed as follows: Formulas ( 16) and ( 21) are brought into formula (12) by the Galerkin method, on both sides multiplied by sin(  ) and   at the same time, and integral on [0, 1]; it can be obtained by using the mode orthogonality principle: where , and  5 are parameters.

Approximate Solution of the Method of Multiple Scales.
The approximate solution of formula ( 22) is solved by the method of multiple scales, setting the form of solution as follows: Advances in Materials Science and Engineering 5 Before the multiscale method is adopted to solve the equation, the parameters are as follows: (24) Formula ( 23) is brought into formula (22), and the same power coefficient of  is sorted out, which can be obtained as follows:  order is 2 order is 3 order is From formula (27), the first solution of the equation can be expressed as where cc represents the front of the conjugate.Formula (28) is brought into the second equations of formula (27) and can be obtained as follows: It can be obtained by eliminating the periodic term of formula (29): By using the method that the equation contains a nonperiodic term of multiple scales, when the internal resonance is not considered, the solution of formula ( 29) is as follows: (31)

Discussion on Resonance Characteristics of Parametric
Excitation.From formula (31), ( 1  1 /( 2  −  2  ))    0 items contained in the expression of the displacement modal component  2 of the cable will make the first equation show the nonsingular solution when the frequency ratio between the bridge and the cable is satisfied with   :   = 1 : 1.It can be seen from the second equations that −( 5  1 /( 2  −  2  ))    0 items contained in the expression of the displacement modal components  2 of the bridge deck beam will make the second equation show the nonsingular solution, and this moment the cable-beam coupling system will present the 1 : 1 principal parametric resonance phenomenon with displacement modal component  2 and  2 as the main form.
Similarly, when the frequency ratio between the bridge deck beam and the cable is satisfied with the   :   = 2 : 1 and   : At this moment, the cable-beam coupling system will present the large vibration phenomenon of the 2 : 1 principal parametrical resonance and 1 : 2 superharmonic resonance with displacement modal component  2 and  2 as the main form.

Analysis on Numerical Example
4.1.Basic Parameters.The cable in practical engineering of a cable-stayed bridge can be taken as the study object, in order to further verify the characteristics of 1 : 1 main parametrical resonance for cable-beam coupled system.The geometric parameters and material properties of the stay cable are shown in Table 1.  and 3. From cable's dimensionless displacement response time history and spectrum curve in Figure 2, stay cable presents large beat vibration near the equilibrium position at its natural frequency   .Due to the small damping of the cable and the slow decay of parametrical resonance displacement response, the sharp vibration has been continued.From the dimensionless displacement response time history and amplitude-frequency characteristic curve of the deck beam end in Figure 3, beam presents the obvious beat vibration near the equilibrium position at initial displacement, the coupling effect is obvious, and the amplitude-frequency characteristic curve of the deck beam end is in single peak value, which says that it does harmonic vibration at its natural frequency.

The 1 : 1 Principal Parametrical Resonance Response
We could find that, from the dimensionless displacement response amplitude of cable-beam coupled system, due to deck beam's large quality and stiffness, its energy during the vibration is larger than the energy during cable's parametrical resonance process when its excitation amplitude of vertical initial displacement is small; at this moment, the constraints of the cable on the deck beam movement are very small, while the effect of the deck beam on the cable movement is large, so cable presents large beat vibration, and deck beam presents small beat vibration at given initial conditions.displacement response time history and amplitude-frequency characteristic curve of coupling system are shown in Figures 4 and 5.

When the
From Figures 5 and 6, compared with working condition   (0) = 3.3 × 10 −4 , parametrical resonance's coupling effect of cable-beam composite structure is much more obvious under this working condition.Both cable and deck beam present obvious beat vibration phenomenon, but cable's beat vibration has greater displacement response; the maximal displacement amplitude is increased from 1.18% of the cable length to 2.42%, which says that cable midspan's parametrical resonance displacement amplitude is also increased as the excitation amplitude of vertical initial displacement of deck beam end increased; this is fully consistent with the principle of conservation of energy.

When the Working Conditions
(0) = 5.618 × 10 −6 and   (0) = 3.3 × 10 −8 .Under this working condition, the actual initial displacements of cable midspan and deck beam end are about 0.001 m and 10 −6 m.The dimensionless displacement response time history and amplitude-frequency characteristic curve of coupling system are shown in Figures 6 and 7.
From Figures 6 and 7, under the condition of beam's small initial displacement, both cable and beam present obvious beat vibration phenomenon; the coupling effect is obvious, but at this point the cable's displacement amplitude is small and there is no significant vibration, which is only a slight stable "beat" vibration in the vicinity of the equilibrium position.Analysis of the reason is as follows: from the multiscale differential equations (29) of stay cable   and beam's displacement components  2 and  2 , resonance terms of both parametrical resonance and primary resonance − 11  1  1  (  −  ) 0 and − 1  1     0 are related to the coefficient  11 affected by deck beam's excitation amplitude.By derivation, we conclude that the coefficient 32/ 3 which is the difference between coefficients  1 and  11 is much affected by sag.So the main parametrical resonance will be not significant vibration under beam's small initial condition.

Conclusions
(1) Based on the establishment of the model of the cablebeam coupling dimensionless parametrical resonance, the main parametrical resonance characteristics of cable-beam coupled system are discussed emphatically in terms of the frequency ratio relation of the deck beam and cable meets   :   = 1 : 1.Then, the possibility of the occurrence of large main parametrical resonance of cable is proved theoretically.(2) The nonlinear main parametrical resonance response of cable is irrelevant to its initial displacement, which is increased with the bigger initial displacement of the deck beam end; also, the cable midspan displacement is increased obviously and presents obvious beat vibration phenomenon.
(3) When the vertical initial displacement of the deck beam end is 10 −6 m or even smaller, both the cable and beam present obvious beat vibration phenomenon, but, at this time, the cable displacement amplitude is small and there is no significant vibration, only making a slight stable "beat" vibration in the vicinity of the equilibrium position, which is different from the situation of 2 : 1 parametrical resonance.
of the second equation of formula (31) led to the equations showing nonsingular solution.
Characteristics of Cable-Beam Coupled System 4.2.1.When the Working Conditions   (0) = 5.618 × 10 −6 and   (0) = 3.33 × 10 −4 .Under this working condition, the actual initial displacements of cable midspan and deck beam end are about 0.001 m and 0.01 m.The dimensionless displacement response time history and amplitude-frequency characteristic curve of coupled system are shown in Figures 2

Figure 2 :
Figure 2: Dimensionless displacement response time history and amplitude-frequency characteristic curve of the cable midspan.

Figure 3 :
Figure 3: Dimensionless displacement response time history and amplitude-frequency characteristic curve of the deck beam end.

Frequency 4 Figure 4 :
Figure 4: Dimensionless displacement response time history and amplitude-frequency characteristic curve of the cable midspan.

Figure 5 :
Figure 5: Dimensionless displacement response time history and amplitude-frequency characteristic curve of the deck beam end.

Figure 6 :
Figure 6: Dimensionless displacement response time history and amplitude-frequency characteristic curve of the cable midspan.

Figure 7 :
Figure 7: Dimensionless displacement response time history and amplitude-frequency characteristic curve of the deck beam end.

Table 1 :
Parameter of cables and bridge deck beam.