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Wind vibration control of stay cables using magnetorheological dampers under optimal equivalent control algorithm
Journal of Sound and Vibration 443 (2019) 732e747
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Wind vibration control of stay cables using magnetorheological dampers under optimal equivalent control algorithm Yu-Liang Zhao, Zhao-Dong Xu*, Cheng Wang Key Laboratory of C&PC Structures of the Ministry of Education, Southeast University, Nanjing 210096, China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 12 April 2018 Received in revised form 6 December 2018 Accepted 10 December 2018 Available online 17 December 2018 Handling Editor: J. Macdonald
In a cable-stayed bridge, the employment of magnetorheological (MR) dampers among stay cables are steadily increasing for mitigation of rain-wind induced vibration. A major problem for the practical application of these devices is to develop a suitable algorithm that effectively suppresses the vibration of stay cables with basic calculations and minimal feedback requirement. This paper discusses an optimal equivalent control algorithm for vibration mitigation of stay cables based on the linear quadratic regulator (LQR). The control algorithm is anticipated to effectively reduce the risk of structural response amplification, which is due to the control forces along the direction of MR damper motion. Furthermore, this algorithm approximates the optimal LQR control forces through equivalent stiffness and damping, resulting in a significant reduction in the calculative effort of the optimal control forces. Hence, it is proved as a great significance for the MR semi-active control system application in the field of practical engineering. In the proposed equivalent control technique, control performance is evaluated by the vibration control problem faced by wind induced stay cable of the Second Nanjing Yangtze River Bridge. The results demonstrate that the MR semi-active control algorithm as proposed in this paper performs better than that of optimal passive control with almost achieving the level of LQR control with minimal feedbacks requirements. © 2018 Elsevier Ltd. All rights reserved.
Keywords: Semi-active algorithm MR damper Optimal equivalent model Stay cables Vibration control
1. Introduction Stay cables as a primary load bearing elements of cable-stayed bridges are prone to vibration under wind and rain excitation due to their small mass density and low inherent damping characteristic leading to greater risks towards life and property [1e4]. Therefore, it is very essential to develop technologies that can effectively control and mitigate the rain-wind induced vibration of stay cables. Usually, various damping devices are installed at the stay cables with the focus to improve the dynamic characteristics and reduce the dynamic responses in the contrary. Viscous damper, tuned mass damper, MR dampers and so on are certain few recent examples which are being applied [5e7]. Semi-active MR damper as a damping device has a significant advantage due its fast response, good controllability, low energy consumption and outstanding adaptability [8e10]. Thus it is proved to be relatively suitable for vibration control of stay-cables. According to the Johnson
* Corresponding author. E-mail addresses: [email protected] (Y.-L. Zhao), [email protected] (Z.-D. Xu), [email protected] (C. Wang). https://doi.org/10.1016/j.jsv.2018.12.016 0022-460X/© 2018 Elsevier Ltd. All rights reserved.
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et al. [11] study on stay cables for the control effect of semi-active control system, the result showed that the MR semi-active control system could obtain a higher modal damping ratio as compared to optimal passive viscous damper. Based on the inverse state space theory, Duan et al. [12] proposed a state transition feedback control algorithm which just requires the feedback of acceleration information to mitigate the vibration of cables equipped with MR damper. Through employing the passive control algorithm, Weber et al. [13] studied the effectiveness of the vibration control of MR damper on stay cables. The result depicted the optimal damping force to be inversely proportional to the modes order, indicating the requirement of larger damping forces to manage the low-order vibration. Weber et al. [14] discovered under the control of the linear quadratic Gaussian (LQG) algorithm designed according to the energy dissipation target, the MR damper could be expressed as a viscous damping force and could effectively control the vibration of cables with arbitrary strength and frequency. Zhou et al. [15] developed a semi-active control strategy based on the modulated homogeneous friction algorithm to mitigate the three-dimensional vibration of stay cables with discrete MR dampers. Christenson et al. [16] proposed a control algorithm based on the feedback of the relative displacement and the damping force of the damper to control the vibration of the stay cable, where the significant reduction of vibration response could reach up to 50%. On the basis of Johnson's Control-Oriented model [17], Li et al. [18] used LQG active control and limit Hrovat semi-active control strategy to discern vibration mitigation of the cables with MR dampers. Applying the energy equivalence principle to control the vibration of cables equipped with MR dampers, Weber and Boston [19] proposed a passive linear optimal frictional damping theory. Zhao and Zhu [20] with a stochastic optimal semi-active control algorithm developed a quasi-integrable Hamiltonian system to mitigate the multimode vibration of cable using MR damper. Caterino [21] implemented a shaking table test of 1/20 scaled wind tower model to investigate the effectiveness of MR damper semi-active control system in mitigating wind-induced vibrations. Chen et al. [22] developed a self-sensing MR damper control system to control the vibration of stay cables. In addition, Ni et al. [23], Park et al. [24], Ok et al. [25] and so on used the neural network and fuzzy control technologies to control the vibration of stay cables. Based on the control force characteristics of clipped LQR, Weber and Distl [26] derived an approximate optimal control solution for the mitigation of multi-mode cable vibration by semi-active damping with negative stiffness. This approximate optimal control solution does not require state estimation by an observer model, and thus it is simple and inexpensive to realize on real stay cable bridges. However, because of the complexity of MR semi-active control system it is difficult to substantiate the on-line real-time intelligence control. Therefore, a credible MR semi-active control system has not been widely applied in vibration mitigation of stay cables. In 2001, Dongting Lake Bridge introduced the first MR dampers on stay cables [27] to control the wind-raininduced vibration. Xu et al. [28] applied MR damper to mitigate the rain-wind induced vibration of the stay cables in West Second Ring Bridge of Hanzhong city. Li et al. [18] had employed MR dampers to control the vibration of cable at Binzhou Yellow River Highway bridge in Shandong Province. Weber and Distl [29] applied MR semi-active control system to control the vibration of Russky Bridge which adopted the damping force tracking algorithm with the consideration of the temperature effect. It is obvious that in practical engineering the MR semi-active control systems were not widely employed in vibration control of stay cables. One of the major reason is the lack of an appropriate control algorithm with easy implementation and good performance in vibration control with basic calculations and minimal feedback requirement. Therefore, this paper proposes an optimal equivalent control algorithm that takes into account the direction of control force. The control algorithm effectively reduces the dynamic response of the stay cables, while being able to significantly ease the calculation effort of the control force. The structure of this paper are as follows. At first, a new single-rod MR damper is designed and fabricated with test for its mechanical properties. Followed by the optimal equivalent control algorithm description. Furthermore, the control effect of the proposed control algorithm is demonstrated by the simulation of a wind-induced stay cable with a controlled MR damper. The results are compared with the optimal passive control and the LQR control. Finally, the conclusions are drawn. 2. Property tests of the single-rod MR damper 2.1. Test description In this paper the MR damper is designed and developed by our research group. It is a new type of single rod MR damper with spring-floating compensation device. The damping device has the advantages of large working stroke and small installation space which can effectively avoid the problem of air cavity during assembly. The damper adopts a new type of isolation volume compensation device to solve the problem of volume compensation of a single rod MR damper as shown in Fig. 1. A floating piston separated the compensation chamber from working chamber in a volume compensation device. In a compensation chamber volume compensation is achieved by changing the volume of the compressible porous rubber. MR damper magnetic circuit mainly includes outer cylinder, piston, MR fluid and damping gap. The gap between the piston and the cylinder is the MR fluid flow channel, at the middle of the piston excitation coil is entangled, the outer cylinder is a magnetic yoke resulting in the formation of closed magnetic circuit. The outer cylinder and the piston are a part of the magnetic circuit and also the main support of the whole structure. The damper is filled with the anti-sedimentation MR fluid developed by our research group [30], which adopts graphene oxide (GO)-coated magnetic particles and surfactantsmodified iron particles as magnetically conductive particles. Thus the anti-sedimentation performance of the damper has been greatly improved. The piston and the piston rod are made of electrical iron DT4, which is high permeable with low
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Fig. 1. Schematic diagram of damper structure.
residual magnetism, rest of other components are made of 45# steel. The excitation coil in the middle of the piston is a 1 mm diameter copper wire with maximum 2 A load current. The magnetic circuit of this single rod MR damper is optimized, and the design parameters shown are shown in Table 1. To obtain the mechanical properties of the damper, the mechanical performance test of the single rod MR damper are conducted in Southeast University laboratory and the test setup are shown in Fig. 2. While conducting the test, the sinusoidal wave form as d ¼ A sinð2pftÞ is used to control the servo-hydraulic testing machine, A is the displacement amplitude, f is the excitation frequency and t is the loading time. Different excitation frequencies f and amplitudes A are applied to the damper to test its mechanical properties under different currents. The data acquisition system takes a 1000 Hz sampling frequency to obtain the displacement and corresponding damping force of each test condition. Afterwards, the corresponding velocity can be obtained by the discrete differential calculation. 2.2. Experimental results For the elimination of abnormal data as a result of experimental equipment instability, the damper mechanical properties are evaluated by averaging 6-loop data outside the initial two and last two laps. Fig. 3 shows the force displacement curve and force velocity curve of different current corresponding to amplitude 7.5 mm and excitation frequency 1 Hz. Fig. 3 shows that while there is no current input in the damper, the force-displacement hysteresis curve is basically an ellipse indicating a viscous damping characteristics of the damper. Further seen, with the increase of the current there is significant increase in maximum output force and the maximum yield force of the damper. There is gradual rectangular depiction in the characteristics of force displacement curve due to a significant increase of Coulomb force. There is a progressive fullness in the force displacement curve with the gradual current increase. This indicates the significant increase in the energy dissipation capacity of the damper with the increase of current. In addition, it is seen from Fig. 3 that with the increase of current at low velocity, the curves of force velocity show obvious hysteric characteristics. When the current reaches 1.0 A, the maximum damping forces of the damper has no significant increase in relation to the increase of current
Table 1 Main design parameters of the damper. Content
Parameter
Content
Parameter
outer diameter of cylinder Inner diameter of cylinder Piston diameter Piston rod diameter Effective gap length Damping gap width Excavated depth
62 mm 52 mm 50 mm 14 mm 50 mm 1.0 mm 14 mm
Excavated width number of field turns Total length of piston Working stroke Spring stiffness Diameter of threading hole Range of working current
50 mm 300 207 mm ±50 mm 236 N/mm 4 mm 0e1.5 A
Fig. 2. Damper performance test photograph.
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Fig. 3. Mechanical characteristic curves of damper, (a) force-displacement (b) force-velocity.
due to the magnetic saturation of the MR fluid. The damper force can reach 1.65 kN at 2.0 A and only 0.79 kN at 0 A indicating the MR damper can be used in the MR semi-active control system for its continuous adjustable damper force. Due to the vibration of the actual stay cable present and the small amplitude, the employed principle of MRD design were under normal conditions (vibration is not severe), with no current input into the MR damper. When the amplitude of the actual cable vibration reaches a certain value the current is controlled by the algorithm and the real-time changing current becomes an input to MR Damper. Therefore, the ratio of maximum response forces for maximum current and zero current were not large which is beneficial to the long-term engineering application of MR dampers. In addition, the figure shows that the curves corresponding to different conditions are relatively full. There are no “tightening” phenomenon caused by vacuum at the maximum displacement indicating the designed volume compensation device performs well, with an effective avoidance of the force-lag phenomenon caused by improper compensation.
3. The optimal equivalent control algorithm 3.1. Equation of notion of cable with MR damper This study focuses on the one-dimensional vibration of a stay cable with MR dampers installed near the cable supporting points as shown in Fig. 4. The selected in-plane vibration responses evaluate the control effect of the MR semi-active control system on the stay cables. The model of the control system is shown in Fig. 4. The equations of motion of the cable with MR dampers can be expressed as
m
v2 wðx; tÞ vwðx; tÞ v2 wðx; tÞ T þc ¼ fc þ 2 sin2 g$fd $dðx xd Þ vt vt 2 vx2
(1)
where, m is the mass per unit length of the stay cable; c is the viscous damping per unit length; T is the static cable tension; wðx; tÞ is the configuration of the stay cable; fc is external dynamic loading on the cable; g is the installation angle of the MR
Fig. 4. Vibration control model of stay cable.
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damper; fd is the damping force of the MR damper; xd is the installation location of the MR damper; dð,Þ is the Dirac delta function. According to Hamilton Principle, the vibration responses of cable can be solved by Galerkin method with the y-direction displacement expressed as a modal coordinate n X
yðx; tÞ ¼
h 4i ðxÞqi ðtÞ ¼ 4q ¼ 41 ðxÞ; 42 ðxÞ; /; 4n ðxÞT ½q1 ðtÞ; q2 ðtÞ; /; qn ðtÞo
(2)
1
where 4i ðxÞ denotes the shape function, qi ðtÞ are generalized displacements, n is the number of the shape functions. Substituting the Eq. (2) into the Eq. (1), the motion Eq. (1) can be expressed as
M* q€ þ C* q_ þ K* q ¼ P* þ jd Fd ðtÞ
(3)
where, M* ¼ ½m*ij is the mass matrix,
m*ij ¼
Z
L 0
m4i ðxÞ4j ðxÞdx
(4)
L is the length of the stay cable. C* ¼ ½c*ij is the damping matrix,
c*ij ¼
Z
L
0
c4i ðxÞ4j ðxÞdx
(5)
K* ¼ ½k*ij is the stiffness matrix,
k*ij
Z ¼
L
0
00
00
EI4i ðxÞ4j ðxÞdx þ
Z
L 0
S0 40i ðxÞ40j ðxÞdx þ 2
Z 0
L
1 0 Z L EAmgy00 @ 4i ðxÞdxA40j ðxÞdx LS0 0
(6)
where E is Young's modulus of the cable material; I is moment of inertia of cable cross section; A is the cross-sectional area of the stay cable, S0 is the initial cable force, ðÞ0 and ð$Þ denote partial derivatives with respect to x, and t , respectively. P* ¼ ½p* ij is the load vector,
p*ij ¼
Z
L 0
fc 4i ðxÞdx
(7)
q ¼ ½q1 ; q2 ; /; qn are the generalized displacements, Fd ðtÞ is control force applied to the stay cable at time t, and the damper load vector jd is
jd ¼ 4T ðxd Þ ¼ ½41 ðxd Þ 42 ðxd Þ /
4n ðxd ÞT
(8)
3.2. Optimization of LQR control coefficient The LQR algorithm is an all-state feedback control algorithm with excellent control effect. Eq. (3) can be expressed as a state space equation:
z_ ¼ Az þ BFd ðtÞ þ DP* where,
0 *I1 * ; _ T; A ¼ z ¼ ½q q * * C M K M 0 0 1 1 ; D¼ B¼ 4 M* M* define the objective function of the system,
(9)
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Z J¼
t2 h
t1
737
i zT Q z þ RF 2d ðtÞ dt (10)
2
the optimal control force can be obtained: opt
Fd
¼ Gz
(11)
where, Q is positive semidefinite state matrix, R is positive scalar weighting of the control effort, and G as gain matrix, respectively. The control effect of LQR control algorithm is related to weight coefficient matrix Q and R. Due to passive characteristic of the MR damper, the equation cannot provide the damping force in the same direction as the motion velocity and is unavoidable in the LQR control algorithm. Therefore, when the LQR algorithm is employed in MR semi-active control system, it is necessary to consider the directivity of the damper force when optimizing the weight coefficient matrix Q and R. For convenience, the weight coefficient Q and R can be respectively expressed as:
* Q ¼a K 0
0 ; R ¼ bI M*
(12)
where a; b are undetermined coefficients, and I is the unit matrix respectively. The following is a case analysis of the AII cable of the Dongting Lake Bridge. It illustrates the optimization process of the stay cable with the main parameters as shown in Table 2. From the bottom support, the control force is applied at a position of 3.125% L. For the simulation of the most unfavorable forced vibrations of the cable, the sinusoidal excitations adopt the first three-order natural frequencies which are applied near the distal end of the cable. The weight coefficient Q and R are optimized by minimizing the dynamic responses at one fourth and the midpoints of the cable. For considering the direction of the control force, the optimal control parameters ðaopti ; bopti Þ are obtained to such that the performance index
Jmin ¼ h
dmax amax þk du;max au;max
(13)
is minimized. Where, h and k are weight coefficients; dmax ; du;max are the maximum displacements of the controlled and uncontrolled stay cable respectively; amax ; au;max are the maximum accelerations of the controlled and uncontrolled stay cable respectively. The parameters a; b are optimized by least squares method and the obtained optimal values are shown in Table 3. At the observation point these optimal aopti and bopti perform well in suppressing of the maximum steady-state displacement. The optimal coefficients aopti and bopti are respectively substituted into the state space equation of the cables according to the ordinal number K of the coefficients ðaopti ; bopti Þ as shown in Table 3 to calculate the gain matrices G, and the first 61-order modal vibration are taken into account. Because the MR Dampers are installed only at one position of the stay cable, and the calculation considers the corresponding structural response of the first 61-order modes, the matrix G degenerates into a 1 122 vector, in which the first 61 elements are the coefficients corresponding to the displacement response of the first 61order modes, and the latter 61 elements are the coefficients corresponding to the velocity response of the first 61-order modes. Fig. 5 shows that the element values of the gain matrix G vary with different optimal coefficients ðaopti ; bopti Þ. In the figure, the z-axis represents the element values of the G matrix; the y-axis represents the ordinal number of the element in the G matrix; the x-axis represents the K-value, which is the ordinal number of the optimal coefficients ðaopti ; bopti Þ as shown in Table 3. It is observed from Fig. 5 that the element values of the gain matrix G corresponding to the same ordinal number of the element in the G matrix are almost unchanged with different optimal coefficients, which shows that the optimal control force corresponding to the same order modes considering in the calculation (first 61-order modes in this case) does not vary with change of the optimal coefficients in Table 3.
Table 2 The main parameters of the stay cable AII. Length (m)
Mass/length (kg/m)
Tension (kN)
Diameter (m)
Angle ( )
The first three frequencies (Hz)
220
85
7115
0.118
22.5
0.66, 1.32, 1.97
Table 3 Optimal values of ðaopti ; bopti Þ. Order K
1
2
3
4
5
6
7
ða; bÞ
(102,107)
(101 106)
(100 105)
(101 104)
(102 103)
(103 102)
(104 101)
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Fig. 5. Element values of G varying with optimal coefficients ðaopti ; bopti Þ. (K is the ordinal number of optimal coefficients ðaopti ; bopti Þ as shown in Table 3).
3.3. Optimal equivalent model After the above optimization, under the LQR algorithm MR semi-active control system can theoretically realize optimal semi-active control. In order to meet the requirements of engineering applications the control algorithm can obtain the ideal control effect with the possible minimal calculation and feedbacks. Therefore, an equivalent calculation method is required to approximate the LQR optimal control force. Setting G ¼ ½G1 ; G2 , and the optimal control force can be expressed as
q opt F d ðtÞ ¼ ½G1 ; G2 _ ¼ G1 q G2 q_ q
(14)
according to Eq. (2), the y-direction displacement can be expressed as
8 n X > > > 4i ðxd Þ,qi ðtÞ ¼ 4d q > yd ðxd ; tÞ ¼ < 1
n X > > > _ ðx ; tÞ ¼ 4i ðxd Þ,q_ i ðtÞ ¼ 4d q_ y > d d :
(15)
1
substituting Eq. (15) into Eq. (14) opt 1 _ F d ¼ G1 41 d ½4d q G2 4d ½4d q 1 1 _ ¼ G1 4d yd ðxd ; tÞ G2 4d yd ðxd ; tÞ
(16)
defining the equivalent stiffness Ke ¼ G1 41 , and equivalent damping Ce ¼ G2 41 . d d The Eq. (16) can be rewritten as opt
F d ðtÞ ¼ Ke yd ðxd ; tÞ Ce y_d ðxd ; tÞ
(17)
assuming that there exist an optimal vector L ¼ ½L1 ; L2 ; /; Ln1 ; Ln T to make L4d ¼ In set up, that is
2
3
6 7 6 7 Li 4d ¼ Ii ¼ 60; /; 0 ; 1; 0; /; 0 7 4|fflfflfflffl{zfflfflfflffl} |fflfflfflffl{zfflfflfflffl}5 i1
ni
(18) 1n
However, the a condition which satisfies optimal vector is difficult to obtain, afterwards the quasi-optimal vector L* ¼ ½L*1 ;
L*2 ; /; L*n1 ; L*n
is defined for quasi-optimal control. defining G ¼ L* 4d I, followed by
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Wi ¼ kGi k ¼ L*i 4d Ii ¼
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h
T
tr L*i 4d Ii L*i 4d Ii
739
(19)
where trðÞ is the sum of the diagonal elements of the matrix. When Wi gets to the minimum value, an optimal value L*i is obtained. Hereafter the equivalent coefficients of the equivalent model of the optimal control force can be expressed as
Ke ¼ G1 L* ;
Ce ¼ G2 L*
(20)
Considering the AII stay cable of Dongting Lake Bridge as an example, the main parameters are shown in Table 2. The parameters of weight matrix are adopted as aopti ¼ 101 , bopti ¼ 104 respectively where the first 61-order modal vibration are taken into account in the calculation which satisfies the mode requirement of structural vibration control. In order to improve the convergence rate of the solution of the cable response, a “0” order mode is adopted according to the suggestion of Johnson et al. [15]. The quasi-optimal vector L* is obtained by using the above calculation method, and then the equivalent stiffness Ke and equivalent damping Ce are obtained as Ke ¼ 293:01 kN=m and Ce ¼ 10:69 kN,s=m, respectively. The relationship curve of L30 and kG30 k is shown in Fig. 6, and it can be seen that L30 has a quasi-optimal value of 0.02 making kG30 k a minimum value of 0. 993. The number of modes at which the analysis is truncated will affect the quasi-optimal vector L* and 4d, which will eventually affect the equivalent stiffness Ke and equivalent damping Ce. Fig. 7 shows the relation between the Ke and Ce obtained by the quasi-optimal vector and the ‘Number of modes’. Here, the ‘Number of modes’ is the number of modes at which the analysis is truncated. It is found that the quasi-optimal Ke and Ce corresponding to different ‘Number of modes’ are different. At first, both quasi-optimal Ke and Ce increase with the increase of ‘Number of modes’ and gradually decreases after reaching the maximum value. For obtaining optimal equivalent stiffness and damping for the quasi-optimal control, the equivalency is analyzed on the vibration responses of the stay cable corresponding to different ‘Number of modes’. The external excitation is
P * ðtÞ ¼ F sinð2pftÞ
(21)
where F is the amplitude of external load, and the value are 300 N, 600 N, 900 N, 1200 N and 1500 N, respectively. f is the excitation frequency, where the first three-order natural frequencies of the cable are taken. The equivalent stiffness and damping of different ‘Number of modes’ shown in Fig. 7 are adopted to respective calculations. The maximum steady-state displacements of observation points and corresponding ‘Number of modes’ are shown in Fig. 8. It gives a conclusion from Fig. 8 that there exist the optimal equivalent stiffness Ke and damping Ce to obtain the optimal control effect at the cable system. Furthermore, optimal equivalent stiffness Ke and damping Ce corresponding to the optimal ‘Number of modes’ has no significant variation with the load amplitude at same excitation frequency. Under the excitation of the first two-order frequencies, the optimal ‘Number of modes’ is the same which is also depicted as 12-order. The corresponding quasi optimal equivalent stiffness and damping are Ke ¼ 293:01 kN=m, Ce ¼ 10:69 kN,s=m, which also effectively control the maximum steady-state displacement under the excitation of third order natural frequencies.
Fig. 6. The relationship between L*30 and kG30 k.
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Fig. 7. The relation between quasi-optimal Ke, Ce and ‘Number of modes’, (a) equivalent damping versus ‘Number of modes’ (b) equivalent stiffness versus ‘Number of modes’.
4. Dynamic responses analysis on the semi-active controlled cables The J16 stay cable of the Second Nanjing Yangtze River Bridge was selected for detailed studies. The stochastic windinduced vibration responses were simulated to evaluate the control effect of the control algorithm in vibration mitigation of the stay cable with MR damper. The main parameters of the J16 stay cable are listed in Table 4. 4.1. Wind field simulation The continuous wind velocity field of stay cables is often viewed as a three-dimensional stochastic process that can be expressed as
8 < U ¼ UðzÞ þ uðy; z; tÞ v ¼ vðy; z; tÞ : m ¼ mðy; z; tÞ
(22)
where, the (x, y, z) is a right-handed coordinate system with the x-axis in the direction of the mean wind field, U is the instantaneous wind speed, UðzÞ is the mean wind speed in the longitudinal direction, u, v, m are the turbulent fluctuations in the longitudinal, lateral, and vertical direction directions, respectively, and t is the time. Here, the discussion focuses on the wind-induced vibration responses of the stay cables under the along-wind stochastic wind field. Harmony superposition method and linear filtering method are the two classical simulation methods for the one-dimensional multivariate stochastic process. Here the linear filtering method is adopted to simulate the stochastic wind field for its high efficiency and rigorous digital signal processing. Assuming that the bottom of the cable is 40 m away from the horizontal plane and 56 simulation nodes are distributed from bottom to top along the cable direction, sequentially numbered and spaced at 5 m. The wind speed at 20 m height is set to be 32.6 m/s where the value of AR model order is 4 with the Karman constant is set to be 0.4 while the roughness is set to be 0.3 and the Kaimal spectrum is set to be the target spectrum. Through the simulation calculation, the wind velocity time history curve and the power spectrum of wind velocity at node No. 30 are obtained and shown in Fig. 9. Fig. 9 shows that the power spectrum curve fits well with the target power spectrum curve. 4.2. Semi-active control strategy based on an optimal equivalent model The stochastic wind field simulated above is employed as the external excitation as explained in the motion Eq. (9). where the instantaneous wind load is
pðz; tÞ ¼
rv2 ðz; tÞCH D sin2 q 2
(23)
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Fig. 8. Maximum steady displacement of the observation point under the excitation of the first three-order natural frequencies. (a) first-order natural frequency excitation (b) second-order natural frequency excitation (c) third-order natural frequency excitation.
Table 4 The main parameters of the stay cable J16. Length (m)
Mass/length (kg/m)
Tension (kN)
Diameter (m)
Angle ( )
The first three frequencies (Hz)
280
80.78
8020
0.213
26.5
0.71, 1.39, 2.16
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Fig. 9. The wind velocity time-history and the power spectrum at node No. 30, (a) the wind velocity time history curve at node No.30 (b) the power spectrum of wind velocity at node No. 30.
where, r is the atmospheric density and denotes 1:23 kg=m3 , vðx; tÞ is the instantaneous wind speed at time t, CH is the windward resistance coefficient of the stay cable expressed as 1.01, D is the equivalent diameter of the stay cable, which is 0.213 m, inclination q is 26.5 . Based on LQR algorithm discussed above the optimal equivalent control algorithm is adopted to control the vibration of stay cables. It applies a simple tracking process on the LQR control force with the minimal points at the cable. Here, a clipped optimal strategy based on optimal equivalent model is applied.
F act d
8 < fMRD;max ¼ Ke xd Ce x_d : fMRD;min
Ke xd Ce x_d fMRD;max & x_d ðKe xd Ce x_d Þ < 0 Ke xd Ce x_d fMRD;max & x_d ðKe xd Ce x_d Þ < 0 else
(24)
where, xd, x_d are the displacement and velocity of the control point respectively; F act d is a real semi-active control force; fMRD;min and fMRD;max are the maximum and minimum control forces applied to the stay cable by the MR damper. The semiactive control strategy is a continuous adjustment of the control force according to the velocity x_d and the direction of the control force F act d at the position of the control point. While the control force and velocity direction are opposite direction, the control force is calculated by the equivalent model according to the displacement and velocity at the control point, whereas at the same direction the control force takes the minimum value. Here the mechanical model of the MR damper is represented by a modified sigmoid model [31],
€ fMRD ¼ C0 u_ þ Fy tanhðkðu_ þ l,sgnðuÞÞÞ
(25)
where, C0 is the viscous coefficient, Fy is the maximum yield force of the MR damper, k, l are the parameters determined by the test, u_, ü are the relative velocity and acceleration of the MR damper at the control point. The typical comparison between the simulated and experimental values of the mechanical properties is shown in Fig. 10. It is indicated that the modified sigmoid model can perfectly represent the mechanical properties of the MR damper.
4.3. Analysis results Newmark b algorithm is applied to calculated the dynamic response of the stay cable where the calculation parameters are set as a ¼ 0:5, b ¼ 0:25, Dt ¼ 0:02s, ensuring that the algorithm is convergent. For the verification of the effectiveness of the proposed semi-active control algorithm, the wind-induced response of the stay cable without control, optimal passive control, semi-active control and LQR control are compared. The optimal passive _ and the damping coefficient of optimal passive control is set as Ct ¼ 43:36 kN,s=m. The calculation control force is fdp ¼ Ct u, damping force in semi-active control are calculated by the semi-active control algorithm as discussed above in Eq. (17), The quasi-optimal equivalent stiffness and equivalent damping are Ke ¼ 253:49 kN=m, Ce ¼ 40:69 kN,s=m, respectively. The values of weight coefficient of the optimized LQR are optimized as Q ¼ ½K; 0; 0; M, R ¼ 105 I.
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Fig. 10. Comparison between experimental results (lines) and simulated results (dots) (a) force-displacement (b) force-velocity.
Under the excitation of stochastic wind load, the time-history curve for displacement, velocity and acceleration responses at 50% relative cable length under the semi-active control and the uncontrolled are compared in Fig. 11. It concludes from the figure that the semi-active control algorithm based on the optimal equivalent model can efficiently mitigate the vibration response of the stay cable with the obvious reduction of displacement, velocity and acceleration response as compared with the uncontrolled state. The time-history curves of displacement, velocity and acceleration responses at 50% relative cable length under the optimal passive control, semi-active control and LQR control are compared in Fig. 12. The control effect of the semi-active control is significantly better as compared to passive control in mitigating vibration of the stay cable with MR damper. The results are slightly inferior than LQR control algorithm. To further analyze the vibration control effect of the semi-active control algorithm, the normalized maximum values of the displacement, velocity and acceleration responses of the stay cable within 100s-200s are compared as listed in Table 5. The results show obvious control efficacy in mitigation vibration of the stay cable at all the optimal passive control, semi-active control and LQR control algorithms as compared to the uncontrolled case. The maximum displacement, velocity and acceleration of semi-active control cable are reduced by 21.4%, 34.2% and 30.0% as compared to the uncontrolled results and by 7.3%, 5.5% and 11.7 while compared to the optimal passive case. The LQR control achieves a 25.9% (maximum displacement), 43.0% (maximum velocity) and 58.3% (maximum acceleration) reduction as compared to the uncontrolled results corresponding to 12.6%, 18.2% and 47.4% reduction over the optimal passive case. It indicates the performance of the semi-active control is superior to those of optimal passive case and worse than those of LQR control. To see a better performance of semi-active control for vibration mitigation, normalized RMS displacement, velocity and acceleration are listed in Table 5. The results indicate that the semi-active control systems perform significantly better in reducing the RMS cable responses than the optimal passive systems. Compared to uncontrolled case, the optimal passive control achieves a decrease in displacement, velocity and acceleration which are 12.87%, 28.89%, and 44.76%, respectively. The semi-active control performs better at about 22.77%, 48.33%, and 52.10% respectively in the RMS displacement, RMS velocity and RMS acceleration corresponding to 11.4%, 27.4% and 13.3% reduction over the optimal passive case. While the LQR control performs bests it is able to further decrease by 26.73%, 56.39% and 56.39%, respectively. Fig. 13 shows the displacement, velocity and acceleration profiles of the controlled and uncontrolled stay cables. These plots clearly demonstrate all the passive control, semi-active control and LQR control have significant control effect in stay cable vibration mitigation. As shown in the Fig. 13, the semi-active control can achieve a much more effective and efficient control than the optimal passive control. Especially in the displacement control, the performance of the semi-active control is nearly equal to the LQR control. Moreover, it is noticed that the semi-active control achieves these performance levels using significantly less feedback points than LQR control. It proves to be of great significance in the application of MR damper in practical engineering. Fig. 14 shows the force displacement trajectories of the MR damper. Here it is observed that the semi-active control shows obvious negative stiffness characteristics [32], and can effectively dissipate vibration energy from the stay cable. Thus, with this optimal equivalent control algorithm, both the displacement and acceleration response of the stay cable can be effectively suppressed using few displacement sensors. 5. Conclusions Based on the newly developed MR damper and the optimal equivalent control algorithm, this paper studies the performance of an semi-active control system. A single rod MR damper with a new type of compensator are designed and fabricated
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Fig. 11. Comparison of the vibration responses of the stay cable under uncontrolled and semi-active control, (a) time history curves of displacement responses (b) time history curves of velocity responses (c) time history curves of acceleration responses.
to test its mechanical properties. Focusing on LQR optimal control algorithm, an optimal equivalent control algorithm is proposed. A major attractive features of the control algorithm is the significant reduction for calculating the optimal control force, thus making it relatively implementable. The effectiveness of the MR damper using this optimal equivalent control algorithm are numerically evaluated with excellent results while comparing the optimal passive control and the LQR control. Based on these results, it is concluded that
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Fig. 12. Comparison of the vibration responses of the stay cable under passive, semi-active and LQR control, (a) time history curves of displacement responses (b) time history curves of velocity responses (c) time history curves of acceleration responses.
(1) The newly developed volume compensation device with a floating piston-spring realizes the requirements for volume change of MR damper through compression of the porous rubber blocks thus effectively avoiding the “tightening” phenomenon in damping force due to improper compensation. (2) The optimal equivalent control algorithm is optimized by contemplating the direction of damping force, which realizes online real-time vibration control of stay cables with limited feedbacks and computation. The max and RMS responses are reduced to about 20%~30% and 25%~50%.
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Table 5 Normalized vibration response of the stay cable under different control algorithms. control algorithm uncontrolled Passive Semi-active LQR
Max RMS Max RMS Max RMS Max RMS
Displacement
Velocity
Acceleration
1.0000 1.0000 0.8482 0.8713 0.7857 0.7723 0.7411 0.7327
1.0000 1.0000 0.6966 0.7111 0.6583 0.5167 0.5700 0.4361
1.0000 1.0000 0.7932 0.5524 0.7004 0.4790 0.4172 0.3636
Fig. 13. Vibration response profiles of stay cable, (a) displacement (b) velocity (c) acceleration.
Fig. 14. Force displacement trajectory of the semi-active MR damper.
(3) The developed optimal equivalent control algorithm performs better than optimal passive case. Under stochastic wind excitation, the maximum and RMS responses of semi-active control structure are reduced to about 10% and 20% as compared to the optimal passive case. Meanwhile, the displacement and velocity responses of the optimal equivalent control are about 5% and 10% more as compared to LQR control. It indicates that the performance in displacement and velocity control almost achieves the level of LQR control.
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Acknowledgments This research is financially supported by the National Science Found for Distinguished Young Scholars in China with Grant No. 51625803, Program of Changjiang Scholars of Ministry of Education in China, Ten Thousand Talent Program of Leading Scientists, the Fundamental Research Funds for the Central Universities, Research and Innovation Project for College Graduates of Jiangsu Province with Grant No. KYLX16_0255, A Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions with Grant No.CE02-1-46. These supports are gratefully acknowledged. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27]
[28] [29] [30] [31] [32]
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Zhu, Stochastic optimal semi-active control of stay cables by using magneto-rheological damper, J. Vib. Contr. 17 (2011) 1921e1929. N. Caterino, Semi-active control of a wind turbine via magnetorheological dampers, J. Sound Vib. 345 (2015) 1e17. Z.H. Chen, K.H. Lam, Y.Q. Ni, Enhanced damping for bridge cables using a self-sensing MR damper, Smart Mater. Struct. 25 (2016) 13. Y.Q. Ni, Y. Chen, J.M. Ko, D.Q. Cao, Neuro-control of cable vibration using semi-active magneto-rheological dampers, Eng. Struct. 24 (2002) 295e307. K.S. Park, H.M. Koh, S.Y. Ok, C.W. Seo, Fuzzy supervisory control of earthquake-excited cable-stayed bridges, Eng. Struct. 27 (2005) 1086e1100. S.Y. Ok, D.S. Kim, K.S. Park, H.M. Koh, Semi-active fuzzy control of cable-stayed bridges using magneto-rheological dampers, Eng. Struct. 29 (2007) 776e788. F. Weber, H. Distl, Semi-active damping with negative stiffness for multi-mode cable vibration mitigation: approximate collocated control solution, Smart Mater. Struct. 24 (2015) 115015. Z.Q. Chen, J.M. Ko, Y.Q. Ni, B.F. Spencer, MR damping system on Dongting Lake cable-stayed bridge, smart structures and materials 2003: smart systems and nondestructive evaluation for civil infrastructures, in: Shih-Chi Liu (Ed.), Proceedings of the SPIE, vol. 5057, 2003, pp. 229e235. https:// doi.org/10.1117/12.498072. Z.D. Xu, Y.Q. Guo, J.T. Zhu, F.H. Xu, Intelligent Vibration Control in Civil Engineering Structures, Academic Press, New York, 2016. F. Weber, H. Distl, Amplitude and frequency independent cable damping of Sutong Bridge and Russky Bridge by magnetorheological dampers, Struct. Contr. Hlth. 22 (2015) 237e254. Z.D. Xu, M. Xu, C.L. Sun, Performance tests and microchain model validation of a novel kind of MR fluid with GO-coated iron particles, J. Mater. Civ. Eng. 30 (2018), 040180725. Z.D. Xu, L.F. Sha, X.C. Zhang, H.H. Ye, Design, performance test and analysis on magnetorheological damper for earthquake mitigation, Struct. Contr. 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Contents lists available at ScienceDirect
Journal of Sound and Vibration journal homepage: www.elsevier.com/locate/jsvi
Wind vibration control of stay cables using magnetorheological dampers under optimal equivalent control algorithm Yu-Liang Zhao, Zhao-Dong Xu*, Cheng Wang Key Laboratory of C&PC Structures of the Ministry of Education, Southeast University, Nanjing 210096, China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 12 April 2018 Received in revised form 6 December 2018 Accepted 10 December 2018 Available online 17 December 2018 Handling Editor: J. Macdonald
In a cable-stayed bridge, the employment of magnetorheological (MR) dampers among stay cables are steadily increasing for mitigation of rain-wind induced vibration. A major problem for the practical application of these devices is to develop a suitable algorithm that effectively suppresses the vibration of stay cables with basic calculations and minimal feedback requirement. This paper discusses an optimal equivalent control algorithm for vibration mitigation of stay cables based on the linear quadratic regulator (LQR). The control algorithm is anticipated to effectively reduce the risk of structural response amplification, which is due to the control forces along the direction of MR damper motion. Furthermore, this algorithm approximates the optimal LQR control forces through equivalent stiffness and damping, resulting in a significant reduction in the calculative effort of the optimal control forces. Hence, it is proved as a great significance for the MR semi-active control system application in the field of practical engineering. In the proposed equivalent control technique, control performance is evaluated by the vibration control problem faced by wind induced stay cable of the Second Nanjing Yangtze River Bridge. The results demonstrate that the MR semi-active control algorithm as proposed in this paper performs better than that of optimal passive control with almost achieving the level of LQR control with minimal feedbacks requirements. © 2018 Elsevier Ltd. All rights reserved.
Keywords: Semi-active algorithm MR damper Optimal equivalent model Stay cables Vibration control
1. Introduction Stay cables as a primary load bearing elements of cable-stayed bridges are prone to vibration under wind and rain excitation due to their small mass density and low inherent damping characteristic leading to greater risks towards life and property [1e4]. Therefore, it is very essential to develop technologies that can effectively control and mitigate the rain-wind induced vibration of stay cables. Usually, various damping devices are installed at the stay cables with the focus to improve the dynamic characteristics and reduce the dynamic responses in the contrary. Viscous damper, tuned mass damper, MR dampers and so on are certain few recent examples which are being applied [5e7]. Semi-active MR damper as a damping device has a significant advantage due its fast response, good controllability, low energy consumption and outstanding adaptability [8e10]. Thus it is proved to be relatively suitable for vibration control of stay-cables. According to the Johnson
* Corresponding author. E-mail addresses: [email protected] (Y.-L. Zhao), [email protected] (Z.-D. Xu), [email protected] (C. Wang). https://doi.org/10.1016/j.jsv.2018.12.016 0022-460X/© 2018 Elsevier Ltd. All rights reserved.
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et al. [11] study on stay cables for the control effect of semi-active control system, the result showed that the MR semi-active control system could obtain a higher modal damping ratio as compared to optimal passive viscous damper. Based on the inverse state space theory, Duan et al. [12] proposed a state transition feedback control algorithm which just requires the feedback of acceleration information to mitigate the vibration of cables equipped with MR damper. Through employing the passive control algorithm, Weber et al. [13] studied the effectiveness of the vibration control of MR damper on stay cables. The result depicted the optimal damping force to be inversely proportional to the modes order, indicating the requirement of larger damping forces to manage the low-order vibration. Weber et al. [14] discovered under the control of the linear quadratic Gaussian (LQG) algorithm designed according to the energy dissipation target, the MR damper could be expressed as a viscous damping force and could effectively control the vibration of cables with arbitrary strength and frequency. Zhou et al. [15] developed a semi-active control strategy based on the modulated homogeneous friction algorithm to mitigate the three-dimensional vibration of stay cables with discrete MR dampers. Christenson et al. [16] proposed a control algorithm based on the feedback of the relative displacement and the damping force of the damper to control the vibration of the stay cable, where the significant reduction of vibration response could reach up to 50%. On the basis of Johnson's Control-Oriented model [17], Li et al. [18] used LQG active control and limit Hrovat semi-active control strategy to discern vibration mitigation of the cables with MR dampers. Applying the energy equivalence principle to control the vibration of cables equipped with MR dampers, Weber and Boston [19] proposed a passive linear optimal frictional damping theory. Zhao and Zhu [20] with a stochastic optimal semi-active control algorithm developed a quasi-integrable Hamiltonian system to mitigate the multimode vibration of cable using MR damper. Caterino [21] implemented a shaking table test of 1/20 scaled wind tower model to investigate the effectiveness of MR damper semi-active control system in mitigating wind-induced vibrations. Chen et al. [22] developed a self-sensing MR damper control system to control the vibration of stay cables. In addition, Ni et al. [23], Park et al. [24], Ok et al. [25] and so on used the neural network and fuzzy control technologies to control the vibration of stay cables. Based on the control force characteristics of clipped LQR, Weber and Distl [26] derived an approximate optimal control solution for the mitigation of multi-mode cable vibration by semi-active damping with negative stiffness. This approximate optimal control solution does not require state estimation by an observer model, and thus it is simple and inexpensive to realize on real stay cable bridges. However, because of the complexity of MR semi-active control system it is difficult to substantiate the on-line real-time intelligence control. Therefore, a credible MR semi-active control system has not been widely applied in vibration mitigation of stay cables. In 2001, Dongting Lake Bridge introduced the first MR dampers on stay cables [27] to control the wind-raininduced vibration. Xu et al. [28] applied MR damper to mitigate the rain-wind induced vibration of the stay cables in West Second Ring Bridge of Hanzhong city. Li et al. [18] had employed MR dampers to control the vibration of cable at Binzhou Yellow River Highway bridge in Shandong Province. Weber and Distl [29] applied MR semi-active control system to control the vibration of Russky Bridge which adopted the damping force tracking algorithm with the consideration of the temperature effect. It is obvious that in practical engineering the MR semi-active control systems were not widely employed in vibration control of stay cables. One of the major reason is the lack of an appropriate control algorithm with easy implementation and good performance in vibration control with basic calculations and minimal feedback requirement. Therefore, this paper proposes an optimal equivalent control algorithm that takes into account the direction of control force. The control algorithm effectively reduces the dynamic response of the stay cables, while being able to significantly ease the calculation effort of the control force. The structure of this paper are as follows. At first, a new single-rod MR damper is designed and fabricated with test for its mechanical properties. Followed by the optimal equivalent control algorithm description. Furthermore, the control effect of the proposed control algorithm is demonstrated by the simulation of a wind-induced stay cable with a controlled MR damper. The results are compared with the optimal passive control and the LQR control. Finally, the conclusions are drawn. 2. Property tests of the single-rod MR damper 2.1. Test description In this paper the MR damper is designed and developed by our research group. It is a new type of single rod MR damper with spring-floating compensation device. The damping device has the advantages of large working stroke and small installation space which can effectively avoid the problem of air cavity during assembly. The damper adopts a new type of isolation volume compensation device to solve the problem of volume compensation of a single rod MR damper as shown in Fig. 1. A floating piston separated the compensation chamber from working chamber in a volume compensation device. In a compensation chamber volume compensation is achieved by changing the volume of the compressible porous rubber. MR damper magnetic circuit mainly includes outer cylinder, piston, MR fluid and damping gap. The gap between the piston and the cylinder is the MR fluid flow channel, at the middle of the piston excitation coil is entangled, the outer cylinder is a magnetic yoke resulting in the formation of closed magnetic circuit. The outer cylinder and the piston are a part of the magnetic circuit and also the main support of the whole structure. The damper is filled with the anti-sedimentation MR fluid developed by our research group [30], which adopts graphene oxide (GO)-coated magnetic particles and surfactantsmodified iron particles as magnetically conductive particles. Thus the anti-sedimentation performance of the damper has been greatly improved. The piston and the piston rod are made of electrical iron DT4, which is high permeable with low
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Fig. 1. Schematic diagram of damper structure.
residual magnetism, rest of other components are made of 45# steel. The excitation coil in the middle of the piston is a 1 mm diameter copper wire with maximum 2 A load current. The magnetic circuit of this single rod MR damper is optimized, and the design parameters shown are shown in Table 1. To obtain the mechanical properties of the damper, the mechanical performance test of the single rod MR damper are conducted in Southeast University laboratory and the test setup are shown in Fig. 2. While conducting the test, the sinusoidal wave form as d ¼ A sinð2pftÞ is used to control the servo-hydraulic testing machine, A is the displacement amplitude, f is the excitation frequency and t is the loading time. Different excitation frequencies f and amplitudes A are applied to the damper to test its mechanical properties under different currents. The data acquisition system takes a 1000 Hz sampling frequency to obtain the displacement and corresponding damping force of each test condition. Afterwards, the corresponding velocity can be obtained by the discrete differential calculation. 2.2. Experimental results For the elimination of abnormal data as a result of experimental equipment instability, the damper mechanical properties are evaluated by averaging 6-loop data outside the initial two and last two laps. Fig. 3 shows the force displacement curve and force velocity curve of different current corresponding to amplitude 7.5 mm and excitation frequency 1 Hz. Fig. 3 shows that while there is no current input in the damper, the force-displacement hysteresis curve is basically an ellipse indicating a viscous damping characteristics of the damper. Further seen, with the increase of the current there is significant increase in maximum output force and the maximum yield force of the damper. There is gradual rectangular depiction in the characteristics of force displacement curve due to a significant increase of Coulomb force. There is a progressive fullness in the force displacement curve with the gradual current increase. This indicates the significant increase in the energy dissipation capacity of the damper with the increase of current. In addition, it is seen from Fig. 3 that with the increase of current at low velocity, the curves of force velocity show obvious hysteric characteristics. When the current reaches 1.0 A, the maximum damping forces of the damper has no significant increase in relation to the increase of current
Table 1 Main design parameters of the damper. Content
Parameter
Content
Parameter
outer diameter of cylinder Inner diameter of cylinder Piston diameter Piston rod diameter Effective gap length Damping gap width Excavated depth
62 mm 52 mm 50 mm 14 mm 50 mm 1.0 mm 14 mm
Excavated width number of field turns Total length of piston Working stroke Spring stiffness Diameter of threading hole Range of working current
50 mm 300 207 mm ±50 mm 236 N/mm 4 mm 0e1.5 A
Fig. 2. Damper performance test photograph.
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Fig. 3. Mechanical characteristic curves of damper, (a) force-displacement (b) force-velocity.
due to the magnetic saturation of the MR fluid. The damper force can reach 1.65 kN at 2.0 A and only 0.79 kN at 0 A indicating the MR damper can be used in the MR semi-active control system for its continuous adjustable damper force. Due to the vibration of the actual stay cable present and the small amplitude, the employed principle of MRD design were under normal conditions (vibration is not severe), with no current input into the MR damper. When the amplitude of the actual cable vibration reaches a certain value the current is controlled by the algorithm and the real-time changing current becomes an input to MR Damper. Therefore, the ratio of maximum response forces for maximum current and zero current were not large which is beneficial to the long-term engineering application of MR dampers. In addition, the figure shows that the curves corresponding to different conditions are relatively full. There are no “tightening” phenomenon caused by vacuum at the maximum displacement indicating the designed volume compensation device performs well, with an effective avoidance of the force-lag phenomenon caused by improper compensation.
3. The optimal equivalent control algorithm 3.1. Equation of notion of cable with MR damper This study focuses on the one-dimensional vibration of a stay cable with MR dampers installed near the cable supporting points as shown in Fig. 4. The selected in-plane vibration responses evaluate the control effect of the MR semi-active control system on the stay cables. The model of the control system is shown in Fig. 4. The equations of motion of the cable with MR dampers can be expressed as
m
v2 wðx; tÞ vwðx; tÞ v2 wðx; tÞ T þc ¼ fc þ 2 sin2 g$fd $dðx xd Þ vt vt 2 vx2
(1)
where, m is the mass per unit length of the stay cable; c is the viscous damping per unit length; T is the static cable tension; wðx; tÞ is the configuration of the stay cable; fc is external dynamic loading on the cable; g is the installation angle of the MR
Fig. 4. Vibration control model of stay cable.
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damper; fd is the damping force of the MR damper; xd is the installation location of the MR damper; dð,Þ is the Dirac delta function. According to Hamilton Principle, the vibration responses of cable can be solved by Galerkin method with the y-direction displacement expressed as a modal coordinate n X
yðx; tÞ ¼
h 4i ðxÞqi ðtÞ ¼ 4q ¼ 41 ðxÞ; 42 ðxÞ; /; 4n ðxÞT ½q1 ðtÞ; q2 ðtÞ; /; qn ðtÞo
(2)
1
where 4i ðxÞ denotes the shape function, qi ðtÞ are generalized displacements, n is the number of the shape functions. Substituting the Eq. (2) into the Eq. (1), the motion Eq. (1) can be expressed as
M* q€ þ C* q_ þ K* q ¼ P* þ jd Fd ðtÞ
(3)
where, M* ¼ ½m*ij is the mass matrix,
m*ij ¼
Z
L 0
m4i ðxÞ4j ðxÞdx
(4)
L is the length of the stay cable. C* ¼ ½c*ij is the damping matrix,
c*ij ¼
Z
L
0
c4i ðxÞ4j ðxÞdx
(5)
K* ¼ ½k*ij is the stiffness matrix,
k*ij
Z ¼
L
0
00
00
EI4i ðxÞ4j ðxÞdx þ
Z
L 0
S0 40i ðxÞ40j ðxÞdx þ 2
Z 0
L
1 0 Z L EAmgy00 @ 4i ðxÞdxA40j ðxÞdx LS0 0
(6)
where E is Young's modulus of the cable material; I is moment of inertia of cable cross section; A is the cross-sectional area of the stay cable, S0 is the initial cable force, ðÞ0 and ð$Þ denote partial derivatives with respect to x, and t , respectively. P* ¼ ½p* ij is the load vector,
p*ij ¼
Z
L 0
fc 4i ðxÞdx
(7)
q ¼ ½q1 ; q2 ; /; qn are the generalized displacements, Fd ðtÞ is control force applied to the stay cable at time t, and the damper load vector jd is
jd ¼ 4T ðxd Þ ¼ ½41 ðxd Þ 42 ðxd Þ /
4n ðxd ÞT
(8)
3.2. Optimization of LQR control coefficient The LQR algorithm is an all-state feedback control algorithm with excellent control effect. Eq. (3) can be expressed as a state space equation:
z_ ¼ Az þ BFd ðtÞ þ DP* where,
0 *I1 * ; _ T; A ¼ z ¼ ½q q * * C M K M 0 0 1 1 ; D¼ B¼ 4 M* M* define the objective function of the system,
(9)
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Z J¼
t2 h
t1
737
i zT Q z þ RF 2d ðtÞ dt (10)
2
the optimal control force can be obtained: opt
Fd
¼ Gz
(11)
where, Q is positive semidefinite state matrix, R is positive scalar weighting of the control effort, and G as gain matrix, respectively. The control effect of LQR control algorithm is related to weight coefficient matrix Q and R. Due to passive characteristic of the MR damper, the equation cannot provide the damping force in the same direction as the motion velocity and is unavoidable in the LQR control algorithm. Therefore, when the LQR algorithm is employed in MR semi-active control system, it is necessary to consider the directivity of the damper force when optimizing the weight coefficient matrix Q and R. For convenience, the weight coefficient Q and R can be respectively expressed as:
* Q ¼a K 0
0 ; R ¼ bI M*
(12)
where a; b are undetermined coefficients, and I is the unit matrix respectively. The following is a case analysis of the AII cable of the Dongting Lake Bridge. It illustrates the optimization process of the stay cable with the main parameters as shown in Table 2. From the bottom support, the control force is applied at a position of 3.125% L. For the simulation of the most unfavorable forced vibrations of the cable, the sinusoidal excitations adopt the first three-order natural frequencies which are applied near the distal end of the cable. The weight coefficient Q and R are optimized by minimizing the dynamic responses at one fourth and the midpoints of the cable. For considering the direction of the control force, the optimal control parameters ðaopti ; bopti Þ are obtained to such that the performance index
Jmin ¼ h
dmax amax þk du;max au;max
(13)
is minimized. Where, h and k are weight coefficients; dmax ; du;max are the maximum displacements of the controlled and uncontrolled stay cable respectively; amax ; au;max are the maximum accelerations of the controlled and uncontrolled stay cable respectively. The parameters a; b are optimized by least squares method and the obtained optimal values are shown in Table 3. At the observation point these optimal aopti and bopti perform well in suppressing of the maximum steady-state displacement. The optimal coefficients aopti and bopti are respectively substituted into the state space equation of the cables according to the ordinal number K of the coefficients ðaopti ; bopti Þ as shown in Table 3 to calculate the gain matrices G, and the first 61-order modal vibration are taken into account. Because the MR Dampers are installed only at one position of the stay cable, and the calculation considers the corresponding structural response of the first 61-order modes, the matrix G degenerates into a 1 122 vector, in which the first 61 elements are the coefficients corresponding to the displacement response of the first 61order modes, and the latter 61 elements are the coefficients corresponding to the velocity response of the first 61-order modes. Fig. 5 shows that the element values of the gain matrix G vary with different optimal coefficients ðaopti ; bopti Þ. In the figure, the z-axis represents the element values of the G matrix; the y-axis represents the ordinal number of the element in the G matrix; the x-axis represents the K-value, which is the ordinal number of the optimal coefficients ðaopti ; bopti Þ as shown in Table 3. It is observed from Fig. 5 that the element values of the gain matrix G corresponding to the same ordinal number of the element in the G matrix are almost unchanged with different optimal coefficients, which shows that the optimal control force corresponding to the same order modes considering in the calculation (first 61-order modes in this case) does not vary with change of the optimal coefficients in Table 3.
Table 2 The main parameters of the stay cable AII. Length (m)
Mass/length (kg/m)
Tension (kN)
Diameter (m)
Angle ( )
The first three frequencies (Hz)
220
85
7115
0.118
22.5
0.66, 1.32, 1.97
Table 3 Optimal values of ðaopti ; bopti Þ. Order K
1
2
3
4
5
6
7
ða; bÞ
(102,107)
(101 106)
(100 105)
(101 104)
(102 103)
(103 102)
(104 101)
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Fig. 5. Element values of G varying with optimal coefficients ðaopti ; bopti Þ. (K is the ordinal number of optimal coefficients ðaopti ; bopti Þ as shown in Table 3).
3.3. Optimal equivalent model After the above optimization, under the LQR algorithm MR semi-active control system can theoretically realize optimal semi-active control. In order to meet the requirements of engineering applications the control algorithm can obtain the ideal control effect with the possible minimal calculation and feedbacks. Therefore, an equivalent calculation method is required to approximate the LQR optimal control force. Setting G ¼ ½G1 ; G2 , and the optimal control force can be expressed as
q opt F d ðtÞ ¼ ½G1 ; G2 _ ¼ G1 q G2 q_ q
(14)
according to Eq. (2), the y-direction displacement can be expressed as
8 n X > > > 4i ðxd Þ,qi ðtÞ ¼ 4d q > yd ðxd ; tÞ ¼ < 1
n X > > > _ ðx ; tÞ ¼ 4i ðxd Þ,q_ i ðtÞ ¼ 4d q_ y > d d :
(15)
1
substituting Eq. (15) into Eq. (14) opt 1 _ F d ¼ G1 41 d ½4d q G2 4d ½4d q 1 1 _ ¼ G1 4d yd ðxd ; tÞ G2 4d yd ðxd ; tÞ
(16)
defining the equivalent stiffness Ke ¼ G1 41 , and equivalent damping Ce ¼ G2 41 . d d The Eq. (16) can be rewritten as opt
F d ðtÞ ¼ Ke yd ðxd ; tÞ Ce y_d ðxd ; tÞ
(17)
assuming that there exist an optimal vector L ¼ ½L1 ; L2 ; /; Ln1 ; Ln T to make L4d ¼ In set up, that is
2
3
6 7 6 7 Li 4d ¼ Ii ¼ 60; /; 0 ; 1; 0; /; 0 7 4|fflfflfflffl{zfflfflfflffl} |fflfflfflffl{zfflfflfflffl}5 i1
ni
(18) 1n
However, the a condition which satisfies optimal vector is difficult to obtain, afterwards the quasi-optimal vector L* ¼ ½L*1 ;
L*2 ; /; L*n1 ; L*n
is defined for quasi-optimal control. defining G ¼ L* 4d I, followed by
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Wi ¼ kGi k ¼ L*i 4d Ii ¼
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h
T
tr L*i 4d Ii L*i 4d Ii
739
(19)
where trðÞ is the sum of the diagonal elements of the matrix. When Wi gets to the minimum value, an optimal value L*i is obtained. Hereafter the equivalent coefficients of the equivalent model of the optimal control force can be expressed as
Ke ¼ G1 L* ;
Ce ¼ G2 L*
(20)
Considering the AII stay cable of Dongting Lake Bridge as an example, the main parameters are shown in Table 2. The parameters of weight matrix are adopted as aopti ¼ 101 , bopti ¼ 104 respectively where the first 61-order modal vibration are taken into account in the calculation which satisfies the mode requirement of structural vibration control. In order to improve the convergence rate of the solution of the cable response, a “0” order mode is adopted according to the suggestion of Johnson et al. [15]. The quasi-optimal vector L* is obtained by using the above calculation method, and then the equivalent stiffness Ke and equivalent damping Ce are obtained as Ke ¼ 293:01 kN=m and Ce ¼ 10:69 kN,s=m, respectively. The relationship curve of L30 and kG30 k is shown in Fig. 6, and it can be seen that L30 has a quasi-optimal value of 0.02 making kG30 k a minimum value of 0. 993. The number of modes at which the analysis is truncated will affect the quasi-optimal vector L* and 4d, which will eventually affect the equivalent stiffness Ke and equivalent damping Ce. Fig. 7 shows the relation between the Ke and Ce obtained by the quasi-optimal vector and the ‘Number of modes’. Here, the ‘Number of modes’ is the number of modes at which the analysis is truncated. It is found that the quasi-optimal Ke and Ce corresponding to different ‘Number of modes’ are different. At first, both quasi-optimal Ke and Ce increase with the increase of ‘Number of modes’ and gradually decreases after reaching the maximum value. For obtaining optimal equivalent stiffness and damping for the quasi-optimal control, the equivalency is analyzed on the vibration responses of the stay cable corresponding to different ‘Number of modes’. The external excitation is
P * ðtÞ ¼ F sinð2pftÞ
(21)
where F is the amplitude of external load, and the value are 300 N, 600 N, 900 N, 1200 N and 1500 N, respectively. f is the excitation frequency, where the first three-order natural frequencies of the cable are taken. The equivalent stiffness and damping of different ‘Number of modes’ shown in Fig. 7 are adopted to respective calculations. The maximum steady-state displacements of observation points and corresponding ‘Number of modes’ are shown in Fig. 8. It gives a conclusion from Fig. 8 that there exist the optimal equivalent stiffness Ke and damping Ce to obtain the optimal control effect at the cable system. Furthermore, optimal equivalent stiffness Ke and damping Ce corresponding to the optimal ‘Number of modes’ has no significant variation with the load amplitude at same excitation frequency. Under the excitation of the first two-order frequencies, the optimal ‘Number of modes’ is the same which is also depicted as 12-order. The corresponding quasi optimal equivalent stiffness and damping are Ke ¼ 293:01 kN=m, Ce ¼ 10:69 kN,s=m, which also effectively control the maximum steady-state displacement under the excitation of third order natural frequencies.
Fig. 6. The relationship between L*30 and kG30 k.
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Fig. 7. The relation between quasi-optimal Ke, Ce and ‘Number of modes’, (a) equivalent damping versus ‘Number of modes’ (b) equivalent stiffness versus ‘Number of modes’.
4. Dynamic responses analysis on the semi-active controlled cables The J16 stay cable of the Second Nanjing Yangtze River Bridge was selected for detailed studies. The stochastic windinduced vibration responses were simulated to evaluate the control effect of the control algorithm in vibration mitigation of the stay cable with MR damper. The main parameters of the J16 stay cable are listed in Table 4. 4.1. Wind field simulation The continuous wind velocity field of stay cables is often viewed as a three-dimensional stochastic process that can be expressed as
8 < U ¼ UðzÞ þ uðy; z; tÞ v ¼ vðy; z; tÞ : m ¼ mðy; z; tÞ
(22)
where, the (x, y, z) is a right-handed coordinate system with the x-axis in the direction of the mean wind field, U is the instantaneous wind speed, UðzÞ is the mean wind speed in the longitudinal direction, u, v, m are the turbulent fluctuations in the longitudinal, lateral, and vertical direction directions, respectively, and t is the time. Here, the discussion focuses on the wind-induced vibration responses of the stay cables under the along-wind stochastic wind field. Harmony superposition method and linear filtering method are the two classical simulation methods for the one-dimensional multivariate stochastic process. Here the linear filtering method is adopted to simulate the stochastic wind field for its high efficiency and rigorous digital signal processing. Assuming that the bottom of the cable is 40 m away from the horizontal plane and 56 simulation nodes are distributed from bottom to top along the cable direction, sequentially numbered and spaced at 5 m. The wind speed at 20 m height is set to be 32.6 m/s where the value of AR model order is 4 with the Karman constant is set to be 0.4 while the roughness is set to be 0.3 and the Kaimal spectrum is set to be the target spectrum. Through the simulation calculation, the wind velocity time history curve and the power spectrum of wind velocity at node No. 30 are obtained and shown in Fig. 9. Fig. 9 shows that the power spectrum curve fits well with the target power spectrum curve. 4.2. Semi-active control strategy based on an optimal equivalent model The stochastic wind field simulated above is employed as the external excitation as explained in the motion Eq. (9). where the instantaneous wind load is
pðz; tÞ ¼
rv2 ðz; tÞCH D sin2 q 2
(23)
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Fig. 8. Maximum steady displacement of the observation point under the excitation of the first three-order natural frequencies. (a) first-order natural frequency excitation (b) second-order natural frequency excitation (c) third-order natural frequency excitation.
Table 4 The main parameters of the stay cable J16. Length (m)
Mass/length (kg/m)
Tension (kN)
Diameter (m)
Angle ( )
The first three frequencies (Hz)
280
80.78
8020
0.213
26.5
0.71, 1.39, 2.16
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Fig. 9. The wind velocity time-history and the power spectrum at node No. 30, (a) the wind velocity time history curve at node No.30 (b) the power spectrum of wind velocity at node No. 30.
where, r is the atmospheric density and denotes 1:23 kg=m3 , vðx; tÞ is the instantaneous wind speed at time t, CH is the windward resistance coefficient of the stay cable expressed as 1.01, D is the equivalent diameter of the stay cable, which is 0.213 m, inclination q is 26.5 . Based on LQR algorithm discussed above the optimal equivalent control algorithm is adopted to control the vibration of stay cables. It applies a simple tracking process on the LQR control force with the minimal points at the cable. Here, a clipped optimal strategy based on optimal equivalent model is applied.
F act d
8 < fMRD;max ¼ Ke xd Ce x_d : fMRD;min
Ke xd Ce x_d fMRD;max & x_d ðKe xd Ce x_d Þ < 0 Ke xd Ce x_d fMRD;max & x_d ðKe xd Ce x_d Þ < 0 else
(24)
where, xd, x_d are the displacement and velocity of the control point respectively; F act d is a real semi-active control force; fMRD;min and fMRD;max are the maximum and minimum control forces applied to the stay cable by the MR damper. The semiactive control strategy is a continuous adjustment of the control force according to the velocity x_d and the direction of the control force F act d at the position of the control point. While the control force and velocity direction are opposite direction, the control force is calculated by the equivalent model according to the displacement and velocity at the control point, whereas at the same direction the control force takes the minimum value. Here the mechanical model of the MR damper is represented by a modified sigmoid model [31],
€ fMRD ¼ C0 u_ þ Fy tanhðkðu_ þ l,sgnðuÞÞÞ
(25)
where, C0 is the viscous coefficient, Fy is the maximum yield force of the MR damper, k, l are the parameters determined by the test, u_, ü are the relative velocity and acceleration of the MR damper at the control point. The typical comparison between the simulated and experimental values of the mechanical properties is shown in Fig. 10. It is indicated that the modified sigmoid model can perfectly represent the mechanical properties of the MR damper.
4.3. Analysis results Newmark b algorithm is applied to calculated the dynamic response of the stay cable where the calculation parameters are set as a ¼ 0:5, b ¼ 0:25, Dt ¼ 0:02s, ensuring that the algorithm is convergent. For the verification of the effectiveness of the proposed semi-active control algorithm, the wind-induced response of the stay cable without control, optimal passive control, semi-active control and LQR control are compared. The optimal passive _ and the damping coefficient of optimal passive control is set as Ct ¼ 43:36 kN,s=m. The calculation control force is fdp ¼ Ct u, damping force in semi-active control are calculated by the semi-active control algorithm as discussed above in Eq. (17), The quasi-optimal equivalent stiffness and equivalent damping are Ke ¼ 253:49 kN=m, Ce ¼ 40:69 kN,s=m, respectively. The values of weight coefficient of the optimized LQR are optimized as Q ¼ ½K; 0; 0; M, R ¼ 105 I.
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Fig. 10. Comparison between experimental results (lines) and simulated results (dots) (a) force-displacement (b) force-velocity.
Under the excitation of stochastic wind load, the time-history curve for displacement, velocity and acceleration responses at 50% relative cable length under the semi-active control and the uncontrolled are compared in Fig. 11. It concludes from the figure that the semi-active control algorithm based on the optimal equivalent model can efficiently mitigate the vibration response of the stay cable with the obvious reduction of displacement, velocity and acceleration response as compared with the uncontrolled state. The time-history curves of displacement, velocity and acceleration responses at 50% relative cable length under the optimal passive control, semi-active control and LQR control are compared in Fig. 12. The control effect of the semi-active control is significantly better as compared to passive control in mitigating vibration of the stay cable with MR damper. The results are slightly inferior than LQR control algorithm. To further analyze the vibration control effect of the semi-active control algorithm, the normalized maximum values of the displacement, velocity and acceleration responses of the stay cable within 100s-200s are compared as listed in Table 5. The results show obvious control efficacy in mitigation vibration of the stay cable at all the optimal passive control, semi-active control and LQR control algorithms as compared to the uncontrolled case. The maximum displacement, velocity and acceleration of semi-active control cable are reduced by 21.4%, 34.2% and 30.0% as compared to the uncontrolled results and by 7.3%, 5.5% and 11.7 while compared to the optimal passive case. The LQR control achieves a 25.9% (maximum displacement), 43.0% (maximum velocity) and 58.3% (maximum acceleration) reduction as compared to the uncontrolled results corresponding to 12.6%, 18.2% and 47.4% reduction over the optimal passive case. It indicates the performance of the semi-active control is superior to those of optimal passive case and worse than those of LQR control. To see a better performance of semi-active control for vibration mitigation, normalized RMS displacement, velocity and acceleration are listed in Table 5. The results indicate that the semi-active control systems perform significantly better in reducing the RMS cable responses than the optimal passive systems. Compared to uncontrolled case, the optimal passive control achieves a decrease in displacement, velocity and acceleration which are 12.87%, 28.89%, and 44.76%, respectively. The semi-active control performs better at about 22.77%, 48.33%, and 52.10% respectively in the RMS displacement, RMS velocity and RMS acceleration corresponding to 11.4%, 27.4% and 13.3% reduction over the optimal passive case. While the LQR control performs bests it is able to further decrease by 26.73%, 56.39% and 56.39%, respectively. Fig. 13 shows the displacement, velocity and acceleration profiles of the controlled and uncontrolled stay cables. These plots clearly demonstrate all the passive control, semi-active control and LQR control have significant control effect in stay cable vibration mitigation. As shown in the Fig. 13, the semi-active control can achieve a much more effective and efficient control than the optimal passive control. Especially in the displacement control, the performance of the semi-active control is nearly equal to the LQR control. Moreover, it is noticed that the semi-active control achieves these performance levels using significantly less feedback points than LQR control. It proves to be of great significance in the application of MR damper in practical engineering. Fig. 14 shows the force displacement trajectories of the MR damper. Here it is observed that the semi-active control shows obvious negative stiffness characteristics [32], and can effectively dissipate vibration energy from the stay cable. Thus, with this optimal equivalent control algorithm, both the displacement and acceleration response of the stay cable can be effectively suppressed using few displacement sensors. 5. Conclusions Based on the newly developed MR damper and the optimal equivalent control algorithm, this paper studies the performance of an semi-active control system. A single rod MR damper with a new type of compensator are designed and fabricated
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Fig. 11. Comparison of the vibration responses of the stay cable under uncontrolled and semi-active control, (a) time history curves of displacement responses (b) time history curves of velocity responses (c) time history curves of acceleration responses.
to test its mechanical properties. Focusing on LQR optimal control algorithm, an optimal equivalent control algorithm is proposed. A major attractive features of the control algorithm is the significant reduction for calculating the optimal control force, thus making it relatively implementable. The effectiveness of the MR damper using this optimal equivalent control algorithm are numerically evaluated with excellent results while comparing the optimal passive control and the LQR control. Based on these results, it is concluded that
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Fig. 12. Comparison of the vibration responses of the stay cable under passive, semi-active and LQR control, (a) time history curves of displacement responses (b) time history curves of velocity responses (c) time history curves of acceleration responses.
(1) The newly developed volume compensation device with a floating piston-spring realizes the requirements for volume change of MR damper through compression of the porous rubber blocks thus effectively avoiding the “tightening” phenomenon in damping force due to improper compensation. (2) The optimal equivalent control algorithm is optimized by contemplating the direction of damping force, which realizes online real-time vibration control of stay cables with limited feedbacks and computation. The max and RMS responses are reduced to about 20%~30% and 25%~50%.
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Table 5 Normalized vibration response of the stay cable under different control algorithms. control algorithm uncontrolled Passive Semi-active LQR
Max RMS Max RMS Max RMS Max RMS
Displacement
Velocity
Acceleration
1.0000 1.0000 0.8482 0.8713 0.7857 0.7723 0.7411 0.7327
1.0000 1.0000 0.6966 0.7111 0.6583 0.5167 0.5700 0.4361
1.0000 1.0000 0.7932 0.5524 0.7004 0.4790 0.4172 0.3636
Fig. 13. Vibration response profiles of stay cable, (a) displacement (b) velocity (c) acceleration.
Fig. 14. Force displacement trajectory of the semi-active MR damper.
(3) The developed optimal equivalent control algorithm performs better than optimal passive case. Under stochastic wind excitation, the maximum and RMS responses of semi-active control structure are reduced to about 10% and 20% as compared to the optimal passive case. Meanwhile, the displacement and velocity responses of the optimal equivalent control are about 5% and 10% more as compared to LQR control. It indicates that the performance in displacement and velocity control almost achieves the level of LQR control.
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Acknowledgments This research is financially supported by the National Science Found for Distinguished Young Scholars in China with Grant No. 51625803, Program of Changjiang Scholars of Ministry of Education in China, Ten Thousand Talent Program of Leading Scientists, the Fundamental Research Funds for the Central Universities, Research and Innovation Project for College Graduates of Jiangsu Province with Grant No. KYLX16_0255, A Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions with Grant No.CE02-1-46. These supports are gratefully acknowledged. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27]
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