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Non-wind-induced nonlinear damping and stiffness on slender prisms: a forced vibration-pressure balance
Engineering Structures 207 (2020) 110107
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Engineering Structures journal homepage: www.elsevier.com/locate/engstruct
Non-wind-induced nonlinear damping and stiffness on slender prisms: a forced vibration-pressure balance
T
⁎
Zengshun Chena,b, Yemeng Xua, K.T. Tseb, , Liang Huc, K.C.S. Kwokd, Bubryur Kime a
School of Civil Engineering, Chongqing University, Chongqing 400044, China Department of Civil and Environmental Engineering, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China c NatHaz Modeling Laboratory, University of Notre Dame, Notre Dame, IN 46556, USA d School of Civil Engineering, The University of Sydney, NSW 2006, Australia e Department of Architectural Engineering, Kyungil University, Gyeongbuk, South Korea b
A R T I C LE I N FO
A B S T R A C T
Keywords: Nonlinear damping Nonlinear stiffness Forced vibration test Wind-off force Bluff body aerodynamics
The non-wind-induced additional nonlinear damping and stiffness of a spring-suspension system (under wind-off conditions) would significantly influence the prediction of aeroelastic response. This study proposes a forced vibration technique to identify the non-wind-induced nonlinear damping and stiffness of both vertical and inclined prisms. The forced vibration-pressure test was performed to observe the non-wind-induced pressure at first. It was observed that the pressure was remarkably affected by the amplitude of structural oscillation, and the corresponding non-wind-induced force is thus nonlinear. A procedure is then proposed to identify the nonwind-induced nonlinear damping and stiffness of the prisms from the tested nonlinear forces. The results yielded by the proposed procedure in terms of a vertical prism has been verified by comparing the response predicted by the unsteady aerodynamic force with/without the non-wind-induced nonlinearities with the measured response. Moreover, the non-wind-induced nonlinearities of inclined prisms have been tested by using the proposed forced vibration technique, which also suggests a mathematical model of the identified results for the purpose of convenience.
1. Introduction In wind engineering, spring-suspension systems have been widely used to evaluate wind actions on structures [2,16]. It has been confirmed that this type of system inevitably exhibits nonlinear behaviors practically due to its inherent nonlinear damping and stiffness [8]. Although the nonlinear damping and stiffness may be of small quantity and vary slowly with the amplitude of oscillation, they make difference in the system dynamic behaviors [10]. Under wind-off conditions (at zero-wind speed), an oscillating model would accelerate its surrounding air and generate aerodynamic forces on the model. This type of aerodynamic force induced by the interaction between the oscillating model and its surrounding air under wind-off conditions is termed as nonwind-induced aerodynamic force in the present study. The non-windinduced aerodynamic force is an important source of the physical nonlinearity of a spring-suspension system. Moreover, the non-windinduced aerodynamic force attributes to a considerable portion of the experimentally tested wind force on structures. Cao and Ge [1] have
found that the non-wind-induced aerodynamic force of a bridge deck section can be as much as approximately 67.5% of the structural damping whereas the aerodynamic stiffness also accounts for a considerable percentage (12.8%) of the structural stiffness. Gao and Zhu [7] have also demonstrated that the non-wind-induced aerodynamic damping of a bridge deck is considerable. The non-wind-induced aerodynamic damping, like aerodynamic damping during galloping, is totally caused by model motion. It is self-excited and negative. This suggests that the exclusion of the non-wind-induced aerodynamic damping will result in underestimation in predicting wind-induced vibration of structures. It is thus of great importance to identify the nonwind-induced aerodynamic force on structures. Few studies have investigated the non-wind-induced aerodynamic forces on structures. Gao and Zhu [7] have identified the non-windinduced aerodynamic damping and stiffness of a two-dimensional bridge deck section using an energy equivalent method based on free decay responses. The aerodynamic damping and stiffness were used for predicting nonlinear galloping of the bridge deck. Thereafter, Cao and
⁎ Corresponding author at: Department of Civil and Environmental Engineering, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China. E-mail addresses: [email protected] (Z. Chen), [email protected] (K.T. Tse), [email protected] (K.C.S. Kwok).
https://doi.org/10.1016/j.engstruct.2019.110107 Received 6 March 2019; Received in revised form 23 October 2019; Accepted 16 December 2019 0141-0296/ © 2019 Elsevier Ltd. All rights reserved.
Engineering Structures 207 (2020) 110107
Z. Chen, et al.
(Fig. 1). The model was forced to oscillate harmonically in the crosswind direction by an actuator that was controlled by a signal generator and a power amplifier. The angle of inclination of the test model was α = 0o , 5o , 10o, 15o , 20o and 30o , as shown in Fig. 1 (c). The amplitude of oscillation varied from around 2% to 30% in terms of the width of the test model. The tip response of the test model was observed by a laser displacement sensor (LDS) installed underneath of the test rig. The sampling frequency and duration were 500 Hz and 110 s, respectively. The frequency of oscillation was 7.8 Hz, which was the same with that in an aeroelastic test [5]. It should be noted that for aeroelastic test, the experimental error propagation may closely affect the accuracy in response prediction. Details for considering the experimental error can be found in literature [11].
Ge [1] have obtained the non-wind-induced aerodynamic damping and frequency of two-dimensional tri-box deck sections using a piecewise fitting method. Xu and Zhang [14] and Zhang and Xu [15] have numerically investigated non-wind-induced damping and added mass of vibrating bridge deck sections. The effect of amplitude of vibration on the damping and added mass has been discussed. A mathematical model was developed to model aerodynamic forces acting on the deck sections at zero wind speed. These studies have greatly advanced our understanding on non-wind-induced aerodynamic forces on two-dimensional deck sections. However, the previous identification methods are based on free decay responses observed from free decay tests, which would vary with the initial amplitude of the free decay tests. Thus, several free decay tests are often required to obtain acceptable nonwind-induced damping and frequency ratios. Furthermore, previous studies have focused on non-wind-induced aerodynamic force on twodimensional deck sections, while the non-wind-induced aerodynamic force on three-dimensional prisms has not been investigated yet. This study aims at proposing a forced vibration test technique to identify the non-wind-induced aerodynamic forces on slender prisms. Both vertical prisms and inclined prisms are of concern, because (1) many practical structures, such as the pylons of the Alamillo Bridge in Spain, the Hong Shan Bridge in China, have been built with an angle of inclination, not only for the architecture purpose but also the superior load-bearing capacity, and (2) the wind along mountain slopes acting on vertical structures is equivalent to the horizontal wind flowing on an inclined structure [6]. In this study, Section 2 proposes the forced vibration test to observe the non-wind-induced pressure on both vertical and inclined slender prisms. Section 3 presents the experimental results of the non-wind-induced pressure and the force-displacement relationship of a vertical prism. The non-wind-induced aerodynamic damping and mass of the prism are identified from the test results by a procedure proposed. This identification is then verified. In Section 4, the non-wind-induced damping and stiffness of inclined prisms are determined by using the forced vibration technique. Section 5 summarizes the main findings of the study.
2.2. Pressure measurement The oscillation model would accelerate the surrounding air and in turn generate pressure on the oscillation model, as shown in Fig. 2 (a). There were 162 pressure taps installed on 9 levels of the test model, and the distribution of pressure taps of the test model is depicted in Fig. 2 (b). The pressure was sampled by using a synchronous multi-pressure sensing system (SMPSS). The sampling frequency and duration were 500 Hz and 110 s, respectively. The SMPSS was connected with pitot tube, and pressure coefficients were obtained by the pressure difference between pitot tube and static pressure. It should be emphasized that, the SMPSS for pressure measurement at low wind speeds is inaccurate if takes ‘still’ air as static pressure as the pressure observed by pitot tube is close to the reference pressure. To address this problem, a high static pressure was given to the SMPSS during the test and was deducted from the tested results in data analysis. It should also be noted that the pressure measurement was in synchronism with response measurement by introducing a reference signal, which can be received by the pressure and response measurement systems, as introduced in literature [6]. 3. Experimental results
2. Experimental setups
3.1. Non-wind-induced pressure and force
2.1. Forced vibration
The non-wind-induced pressure (at zero wind speeds) on the test model was observed in Section 2.2. Taking the vertical prism for example, the RMS of non-wind-induced pressure on a lateral face (in crosswind direction) under various amplitudes of oscillation is presented in Fig. 3. From σy / D = 2.63% to σy / D = 9.94%, the maximum RMS pressure is substantially increased by 66% (from around 0.009 to 0.015), indicating that the RMS non-wind-induced pressure is significantly influenced by the amplitude of the model oscillation. This is
The forced vibration wind tunnel tests were performed in the high wind speed section of the CLP Power Wind/Wave Tunnel Facility at the Hong Kong University of Science and Technology. The dimension of the high wind speed section was 3 m (in width) × 2 m (in height). A test model with dimensions of 50.8 mm in width (D) × 50.8 mm in depth (D) × 915 mm in height (H) was mounted on a forced vibration test rig (a)
(b)
(c)
(d)
Test model with pressure taps
Wind
Oscillation direction y
Pivot Point Roller
Test model
Roller Steel strand
Steel strand
Rotating disk
LDS Actuator
y
Actuator
Fig. 1. Forced vibration wind tunnel test: (a) plan view of test rig; (b) stereogram of test rig; (c) test model in wind tunnel; (d) wind acting on oscillation model. 2
(b)
(a) air
Tip Level 9 Level 8
air
oscillation prism
wall
wall
50.8
50.8
Level 7
100 80 72.2 32.2
Engineering Structures 207 (2020) 110107
Z. Chen, et al.
120
Level 6
915
130
Level 5
120
Level 4
air
air
90
Level 3 85.6 85
Level 2 Level 1
wall
Base
wall
pivot
Side face Windward face
Fig. 2. Pressure measurements: (a) diagrammatic sketch of air-induced pressure on oscillation prisms; (b) distribution of pressure taps (units in mm).
increase with the increase in the model oscillation amplitude. The force is not always in-phase with the displacement which indicates nonlinear features. To study the nonlinear features, the non-wind-induced force is decomposed into the aerodynamic damping and additional aerodynamic stiffness components, as introduced in the following section.
as expected because the surrounding air is accelerated more at a larger amplitude of oscillation, resulting in the suction or repulsion around the test model more significant. It is observed from Fig. 3 that the ‘bubble’ like configurations emerge at the top and bottom ends of the test model and the magnitude of RMS pressures in the regions is remarkably larger than that at other regions. This is known as the three-dimensional ‘end’ effect [4,9], which is caused by the suction or reattachment generated by flow separation around the two ends. Fig. 3 also shows that the magnitude of RMS pressures tends to increase with the height as the test model is pivoted to oscillate and the amplitude of oscillation increases with height. The non-wind-induced force was evaluated by integrating the observed pressure with respect to the attribute area. The non-wind-induced force-displacement of the vertical model is depicted in Fig. 4. It is noteworthy that the range of displacement and force remarkably
3.2. Non-wind-induced additional damping and stiffness 3.2.1. Identification procedure The governing equation of motion of the test model per unit height at the elevation z above the ground under the zero-wind speed is expressed as
my¨ (t ) + cy ̇ (t ) + ky (t ) = P0 (t ) where m =
H
∫0 ρs
D 2ϕ2 (z ) dz
Fig. 3. Non-wind-induced RMS pressure on a vertical test model. 3
(1)
is the generalized mass of the test model, ρs
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Fig. 4. Non-wind-induced force-displacement of a vertical test model. z
damping ratios κ 0 and ξ0 as well as forces P0s and P0d are determined as
is sectional mass density, D is the model width, and ϕ (z ) = H is mode shape (the test model assumedly oscillates with the linear mode shape, H is the model height); c and k are the damping and stiffness coefficients; P0 (t ) is the generalized non-wind-induced force obtained in the previous section (Section 3.1). P0 (t ) consists of a non-wind-induced force component and a motion-induced force component,
P0 (t ) = 1/2ρU 2DH [C0 (t ) + CM (t )]
K a y = 1/2ρU 2DHCM
(3)
Ka 3K a = 2ω2m 2ω2ρs HD 2
(4)
which can also be represented by a stiffness term that is in-phase with the oscillating displacement and a damping term that is in-phase with the oscillating velocity,
Ga = κ + iη
(5)
where, κ and η are the non-wind-induced stiffness and damping coefficients, respectively. Substituting Eqs. (3), (4) and (5) into Eq. (2) yields,
η y )̇ ω
y = y ̂ cos(ωt )
(7)
κ =
-3 I1 ρa ω2D 2H 2y ̂
(8)
η=
-3 I1 ρa ω2D 2H 2y ̂
(9)
where the parameters I1 and I2 are defined as
1 T
∫0
I2 = lim
1 T
∫0
T →∞
T
T
ρ d , P0 = 2mωξ0 (y ) y ̇ ρs
(13)
κ = 0.6048(A/ D)2 − 0.0663(A/ D) + 0.4767, R2 = 0.8619
(14)
η = −0.4707(A/ D)0.7394 + 0.02181, R2 = 0.8763
(15)
where, is the coefficient of determination, which can present the quality of curve fitting. To verify the identified nonlinearities in Fig. 5, a hybrid aeroelasticpressure balance (HAPB) wind tunnel test carried out in a previous study [5] is utilized herein. In this test, both the unsteady force by pressure integration and the wind-induced response were synchronously observed. The two components of the aerodynamic force (windinduced and non-wind-induced forces) were identified by the procedure presented above. The structural parameters of the HAPB system include: the sectional density was 277.65 kg/m3; the damping ratio was 0.78%; the nonlinear physical damping ratio and frequency were ξe = −0.006045A−0.0399 + 0.01574 and fe = 0.7821A−0.005517 + 7.085, respectively, which were identified by a wavelet method [3]. Substituting the force and structural parameters (mass, physical damping and stiffness) into Eq. (1), the response of the test model can be calculated by using a step-by-step iteration method and compared with the experimentally observed response. Fig. 6 (a) compares the response predicted by the observed force (without considering the effect of non-wind-induced nonlinearities), the observed force but excluding all the non-wind-induced nonlinearities (with considering the effect of non-wind-induced nonlinearities), and the observed response. It is noted that the response to the observed
(6)
Jointly using Eqs. (6) and (7) and taking into account the orthogonality of trigonometric functions, one can obtain
T →∞
ξ0 (y ) = η (y )
R2
In a forced vibration test, the test model oscillates harmonically with the tip response as,
I1 = lim
(12)
3.2.2. Identification results and verification The non-wind-induced damping and stiffness are determined according to the forced vibration-pressure measurement technique and the identification procedure introduced above. The non-wind-induced damping and stiffness of the vertical test model are shown in Fig. 5. It is observed in Fig. 5 (a) that the non-wind-induced damping of the vertical prism is always negative and decreases with the increase in the amplitude of oscillation ( A/ D ). Since the negative damping increases the structural response, neglecting the non-wind-induced damping would overestimate the response. On the other hand, Fig. 5 (b) shows that the non-wind-induced stiffness of the vertical prism is positive and increases with the amplitude of oscillation. It is also observed that both the non-wind-induced damping and stiffness are nonlinear and vary slowly with the amplitude of oscillation. The curves in Fig. 5 are weighted by polynomials in Eqs. (14) and (15) using the least mean square method.
whose dimensionless form Ga is
P0 (t ) = 1/2ρU 2DHC0 (t ) + 2mω2 (κy +
ρ ρ s , P0 = κ 0 (y ) s y ρ ρs
(2)
where ρ is the density of air; U is the wind velocity at a reference point and it is equivalent to pressure difference as mentioned in Section 2; C0 (t ) and CM (t ) are the total non-wind force and motion-induced force coefficients at the height of z . A complex aerodynamic impedance K a has been introduced in literature [12,13],
Ga =
κ 0 (y ) = κ (y )
P0 (t ) cos(ωt ) dt
(10)
P0 (t ) sin(ωt ) dt
(11)
Based on Eqs. (10) and (11), the non-wind-induced stiffness and 4
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Fig. 5. Non-wind-induced damping and stiffness of the vertical test model: (a) nonlinear damping; (b) nonlinear stiffness.
4. Nonlinearities of inclined prisms
wind force is substantially larger than the observed response, and the difference is remarkable at high wind speeds. This is because the oscillation at high wind speeds as well as the associated amplitude- dependent non-wind-induced nonlinearities are much larger than those at low wind speeds. In contrast, the response predicted by the observed wind force but excluding non-wind-induced nonlinearities agrees well with the experimental results. Moreover, the comparison between the response to the observed force excluding the non-wind-induced damping, to the same observed force but excluding the additional stiffness, and the observed response is presented in Fig. 6 (b). It shows that the response predicted only excluding the non-wind-induced damping is slightly larger at low wind speeds but is much larger at high wind speeds than the experimental response whereas the response with only the non-wind-induced stiffness excluded is substantially smaller than the observed response. The observation in Fig. 6 suggests that (1) the non-wind-induced nonlinearities may significantly influence the wind-induced response of a prism especially at high wind speeds. The non-wind-induced nonlinear damping can be as large as 20% of the physical damping of the oscillation model. (2) Ignoring the non-wind-induced damping will overestimate the structural response predicted, whereas neglecting the non-wind-induced stiffness will underestimate the response. (3) The effect of the non-wind-induced damping on the response predicted is more significant than the effect of non-wind-induced stiffness. (4) It is verified that the forced vibration-pressure measurement technique is capable of identifying non-wind-induced additional damping and stiffness.
The non-wind-induced nonlinearities of inclined test models are determined by using the forced vibration-pressure measurement technique verified through the vertical test model in the previous section. The non-wind-induced damping and stiffness are presented in Figs. 7 and 8, respectively. Fig. 7 shows that all the damping coefficients of inclined prisms are negative. Despite the fluctuations of the coefficients, the general trends in all the subfigures are almost the same. The coefficients tend to decrease with the increase in the amplitude of oscillation, whereas the magnitude of damping tends to decrease at large inclinations (e.g. α = 30o ). Fig. 8 indicates that the stiffness of inclined test models is positive, and their values tends to increase at large inclinations (e.g. α = 30o ). The results in Figs. 7-8 can be used for predicting the aeroelastic responses of inclined prisms, which are anticipated to improve the accuracy of the prediction. To facilitate the convenient use, the results are fitted by polynomials as 0.9165 + 0.00144, α = 5o , R2 = 0.6334 ⎧− 0.4965(A/ D) ⎪ 0.6338 o + 0.0392, α = 10 , R2 = 0.6623 ⎪− 0.3761(A/ D) ⎪ η = − 0.479(A/ D)0.9625 − 0.00285, α = 15o , R2 = 0.8349 ⎨ ⎪− 0.5414(A/ D)0.9168 − 0.00792, α = 20o, R2 = 0.8395 ⎪ ⎪− 0.4568(A/ D)1.291 − 0.03837, α = 30o, R2 = 0.4251 ⎩
(16)
Fig. 6. Verification for the identified non-wind-induced nonlinearities: (a) including and excluding the nonlinearities; (b) excluding the non-wind-induced damping or stiffness. 5
Engineering Structures 207 (2020) 110107
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Fig. 7. Non-wind-induced damping coefficients of inclined prisms.
stiffness of vertical and inclined prisms have been identified by using the proposed forced vibration-pressure measurement technique. The test results on the non-wind-induced pressure and force-displacement of a vertical prism have been discussed. On the basis of the non-windinduced force given by the forced vibration-pressure measurement technique, the non-wind-induced damping and stiffness have been identified by a procedure proposed. The proposed procedure in terms of the vertical prism have been verified by the comparison between the predicted response with and without the non-wind-induced nonlinearities and the measured response. After the verification, the nonwind-induced nonlinearities of inclined prisms were determined by the proposed technique. The main findings in the study are summarized as follows.
2 o ⎧ 0.331(A/ D) + 0.080(A/ D) + 0.4789, α = 5 , R2 = 0.88 ⎪ 2 o 2 ⎪ 0.136(A/ D) + 0.132(A/ D) + 0.4485, α = 10 , R = 0.9151 ⎪ κ = − 0.142(A/ D)2 + 0.115(A/ D) + 0.449, α = 15o , R2 = 0.644 ⎨ ⎪ 0.354(A/ D)2 + 0.047(A/ D) + 0.4372, α = 20o, R2 = 0.8463 ⎪ ⎪3.404(A/ D)2 − 0.356(A/ D) + 0.4445, α = 30o, R2 = 0.887 ⎩
(17)
5. Concluding remarks In this study, the non-wind-induced additional damping and
Fig. 8. Non-wind-induced stiffness coefficients of inclined prisms. 6
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• The • • •
cstc2019jcyj-msxm0639), the Key project of Technological Innovation and Application Development in Chongqing (Grant No.: cstc2019jscxgksb0188), the Fundamental Research Funds for the Central Universities (Project No.: 2019CDXYTM0032), a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. 16242016) and the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. 2019R1G1A1095215 and 2019H1D3A1A01101442).
non-wind-induced pressure on prisms increases remarkably with the increase in the amplitude of the structural oscillation. The non-wind-induced aerodynamic force appears nonlinear. The non-wind-induced damping of prisms is negative, whereas the non-wind-induced stiffness is positive. The exclusion of non-windinduced damping would overestimate the response of prisms predicted; In contrast, the exclusion the non-wind-induced stiffness would underestimate the response. The non-wind-induced nonlinear damping can be as much as 20% of the structural damping and is not negligible in predicting wind-induced response of aeroelastic structures. A forced vibration technique is capable of identifying the non-windinduced damping and added mass of prisms and is recommended for determining non-wind-induced nonlinearities of bluff bodies.
References [1] Cao F, Ge Y. Air-induced nonlinear damping and added mass of vertically vibrating bridge deck section models under zero wind speed. J Wind Eng Ind Aerodyn 2017;169:217–31. [2] Cermak JE. Wind-tunnel development and trends in applications to civil engineering. J Wind Eng Ind Aerodyn 2003;91:355–70. [3] Chen ZS, Tse KT. Identification of physical nonlinearities of a hybrid aeroelastic–pressure balance. Nonlinear Dyn 2019;98:95–111. [4] Chen ZS, Tse KT, Hu G, Kwok KCS. Experimental and theoretical investigation of galloping of transversely inclined slender prisms. Nonlinear Dyn 2018;91:1023–40. [5] Chen ZS, Tse KT, Kwok KCS, Kim B, Kareem A. Modelling unsteady self-excited wind force on slender prisms in a turbulent flow. Eng Struct 2020;202:109855. [6] Chen ZS, Tse KT, Kwok KCS, Kareem A. Aerodynamic damping of inclined slender prisms. J Wind Eng Ind Aerodyn 2018;177:79–91. [7] Gao G-Z, Zhu L-D. Nonlinear mathematical model of unsteady galloping force on a rectangular 2: 1 cylinder. J Fluids Struct 2017;70:47–71. [8] Gao G, Zhu L. Nonlinearity of mechanical damping and stiffness of a spring-suspended sectional model system for wind tunnel tests. J Sound Vib 2015;355:369–91. [9] Lin N, Letchford C, Tamura Y, Liang B, Nakamura O. Characteristics of wind forces acting on tall buildings. J Wind Eng Ind Aerodyn 2005;93:217–42. [10] Noël J-P, Kerschen G. Nonlinear system identification in structural dynamics: 10 more years of progress. Mech Syst Sig Process 2017;83:2–35. [11] Rizzo F, Caracoglia L, Montelpare S. Predicting the flutter speed of a pedestrian suspension bridge through examination of laboratory experimental errors. Eng Struct 2018;172:589–613. [12] Steckley A. Motion-induced wind forces on chimneys and tall buildings. Western University; 1989. [13] Vickery B, Steckley A. Aerodynamic damping and vortex excitation on an oscillating prism in turbulent shear flow. J Wind Eng Ind Aerodyn 1993;49:121–40. [14] Xu F, Zhang Z. Numerical simulation of windless-air-induced added mass and damping of vibrating bridge decks. J Wind Eng Ind Aerodyn 2018;180:98–107. [15] Zhang M, Xu F. Nonlinear vibration characteristics of bridge deck section models in still air. J Bridge Eng 2018;23:04018059. [16] Zhou Y, Kareem A. Aeroelastic balance. J Eng Mech 2003;129:283–92.
The present study has not only advanced our understanding but also presented applicable models of the non-wind-induced additional damping and stiffness of both vertical and inclined prisms, which can be used for predicting the aeroelastic response. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgement The authors appreciate the use of the testing facility, as well as the technical assistance provided by the CLP Power Wind/Wave Tunnel Facility at the Hong Kong University of Science and Technology. The authors would also like to express sincere thanks to the Design and Manufacturing Services Facility (Electrical and Mechanical Fabrication Unit) of the Hong Kong University of Science and Technology for their help in manufacturing the test device. The work described in this paper was supported by the 111 Project of China (Grant No. B18062), the National Natural Science Foundation of China (Grant No.: 51908090), the Natural Science Foundation of Chongqing, China (Grant No.:
7
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Engineering Structures journal homepage: www.elsevier.com/locate/engstruct
Non-wind-induced nonlinear damping and stiffness on slender prisms: a forced vibration-pressure balance
T
⁎
Zengshun Chena,b, Yemeng Xua, K.T. Tseb, , Liang Huc, K.C.S. Kwokd, Bubryur Kime a
School of Civil Engineering, Chongqing University, Chongqing 400044, China Department of Civil and Environmental Engineering, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China c NatHaz Modeling Laboratory, University of Notre Dame, Notre Dame, IN 46556, USA d School of Civil Engineering, The University of Sydney, NSW 2006, Australia e Department of Architectural Engineering, Kyungil University, Gyeongbuk, South Korea b
A R T I C LE I N FO
A B S T R A C T
Keywords: Nonlinear damping Nonlinear stiffness Forced vibration test Wind-off force Bluff body aerodynamics
The non-wind-induced additional nonlinear damping and stiffness of a spring-suspension system (under wind-off conditions) would significantly influence the prediction of aeroelastic response. This study proposes a forced vibration technique to identify the non-wind-induced nonlinear damping and stiffness of both vertical and inclined prisms. The forced vibration-pressure test was performed to observe the non-wind-induced pressure at first. It was observed that the pressure was remarkably affected by the amplitude of structural oscillation, and the corresponding non-wind-induced force is thus nonlinear. A procedure is then proposed to identify the nonwind-induced nonlinear damping and stiffness of the prisms from the tested nonlinear forces. The results yielded by the proposed procedure in terms of a vertical prism has been verified by comparing the response predicted by the unsteady aerodynamic force with/without the non-wind-induced nonlinearities with the measured response. Moreover, the non-wind-induced nonlinearities of inclined prisms have been tested by using the proposed forced vibration technique, which also suggests a mathematical model of the identified results for the purpose of convenience.
1. Introduction In wind engineering, spring-suspension systems have been widely used to evaluate wind actions on structures [2,16]. It has been confirmed that this type of system inevitably exhibits nonlinear behaviors practically due to its inherent nonlinear damping and stiffness [8]. Although the nonlinear damping and stiffness may be of small quantity and vary slowly with the amplitude of oscillation, they make difference in the system dynamic behaviors [10]. Under wind-off conditions (at zero-wind speed), an oscillating model would accelerate its surrounding air and generate aerodynamic forces on the model. This type of aerodynamic force induced by the interaction between the oscillating model and its surrounding air under wind-off conditions is termed as nonwind-induced aerodynamic force in the present study. The non-windinduced aerodynamic force is an important source of the physical nonlinearity of a spring-suspension system. Moreover, the non-windinduced aerodynamic force attributes to a considerable portion of the experimentally tested wind force on structures. Cao and Ge [1] have
found that the non-wind-induced aerodynamic force of a bridge deck section can be as much as approximately 67.5% of the structural damping whereas the aerodynamic stiffness also accounts for a considerable percentage (12.8%) of the structural stiffness. Gao and Zhu [7] have also demonstrated that the non-wind-induced aerodynamic damping of a bridge deck is considerable. The non-wind-induced aerodynamic damping, like aerodynamic damping during galloping, is totally caused by model motion. It is self-excited and negative. This suggests that the exclusion of the non-wind-induced aerodynamic damping will result in underestimation in predicting wind-induced vibration of structures. It is thus of great importance to identify the nonwind-induced aerodynamic force on structures. Few studies have investigated the non-wind-induced aerodynamic forces on structures. Gao and Zhu [7] have identified the non-windinduced aerodynamic damping and stiffness of a two-dimensional bridge deck section using an energy equivalent method based on free decay responses. The aerodynamic damping and stiffness were used for predicting nonlinear galloping of the bridge deck. Thereafter, Cao and
⁎ Corresponding author at: Department of Civil and Environmental Engineering, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China. E-mail addresses: [email protected] (Z. Chen), [email protected] (K.T. Tse), [email protected] (K.C.S. Kwok).
https://doi.org/10.1016/j.engstruct.2019.110107 Received 6 March 2019; Received in revised form 23 October 2019; Accepted 16 December 2019 0141-0296/ © 2019 Elsevier Ltd. All rights reserved.
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(Fig. 1). The model was forced to oscillate harmonically in the crosswind direction by an actuator that was controlled by a signal generator and a power amplifier. The angle of inclination of the test model was α = 0o , 5o , 10o, 15o , 20o and 30o , as shown in Fig. 1 (c). The amplitude of oscillation varied from around 2% to 30% in terms of the width of the test model. The tip response of the test model was observed by a laser displacement sensor (LDS) installed underneath of the test rig. The sampling frequency and duration were 500 Hz and 110 s, respectively. The frequency of oscillation was 7.8 Hz, which was the same with that in an aeroelastic test [5]. It should be noted that for aeroelastic test, the experimental error propagation may closely affect the accuracy in response prediction. Details for considering the experimental error can be found in literature [11].
Ge [1] have obtained the non-wind-induced aerodynamic damping and frequency of two-dimensional tri-box deck sections using a piecewise fitting method. Xu and Zhang [14] and Zhang and Xu [15] have numerically investigated non-wind-induced damping and added mass of vibrating bridge deck sections. The effect of amplitude of vibration on the damping and added mass has been discussed. A mathematical model was developed to model aerodynamic forces acting on the deck sections at zero wind speed. These studies have greatly advanced our understanding on non-wind-induced aerodynamic forces on two-dimensional deck sections. However, the previous identification methods are based on free decay responses observed from free decay tests, which would vary with the initial amplitude of the free decay tests. Thus, several free decay tests are often required to obtain acceptable nonwind-induced damping and frequency ratios. Furthermore, previous studies have focused on non-wind-induced aerodynamic force on twodimensional deck sections, while the non-wind-induced aerodynamic force on three-dimensional prisms has not been investigated yet. This study aims at proposing a forced vibration test technique to identify the non-wind-induced aerodynamic forces on slender prisms. Both vertical prisms and inclined prisms are of concern, because (1) many practical structures, such as the pylons of the Alamillo Bridge in Spain, the Hong Shan Bridge in China, have been built with an angle of inclination, not only for the architecture purpose but also the superior load-bearing capacity, and (2) the wind along mountain slopes acting on vertical structures is equivalent to the horizontal wind flowing on an inclined structure [6]. In this study, Section 2 proposes the forced vibration test to observe the non-wind-induced pressure on both vertical and inclined slender prisms. Section 3 presents the experimental results of the non-wind-induced pressure and the force-displacement relationship of a vertical prism. The non-wind-induced aerodynamic damping and mass of the prism are identified from the test results by a procedure proposed. This identification is then verified. In Section 4, the non-wind-induced damping and stiffness of inclined prisms are determined by using the forced vibration technique. Section 5 summarizes the main findings of the study.
2.2. Pressure measurement The oscillation model would accelerate the surrounding air and in turn generate pressure on the oscillation model, as shown in Fig. 2 (a). There were 162 pressure taps installed on 9 levels of the test model, and the distribution of pressure taps of the test model is depicted in Fig. 2 (b). The pressure was sampled by using a synchronous multi-pressure sensing system (SMPSS). The sampling frequency and duration were 500 Hz and 110 s, respectively. The SMPSS was connected with pitot tube, and pressure coefficients were obtained by the pressure difference between pitot tube and static pressure. It should be emphasized that, the SMPSS for pressure measurement at low wind speeds is inaccurate if takes ‘still’ air as static pressure as the pressure observed by pitot tube is close to the reference pressure. To address this problem, a high static pressure was given to the SMPSS during the test and was deducted from the tested results in data analysis. It should also be noted that the pressure measurement was in synchronism with response measurement by introducing a reference signal, which can be received by the pressure and response measurement systems, as introduced in literature [6]. 3. Experimental results
2. Experimental setups
3.1. Non-wind-induced pressure and force
2.1. Forced vibration
The non-wind-induced pressure (at zero wind speeds) on the test model was observed in Section 2.2. Taking the vertical prism for example, the RMS of non-wind-induced pressure on a lateral face (in crosswind direction) under various amplitudes of oscillation is presented in Fig. 3. From σy / D = 2.63% to σy / D = 9.94%, the maximum RMS pressure is substantially increased by 66% (from around 0.009 to 0.015), indicating that the RMS non-wind-induced pressure is significantly influenced by the amplitude of the model oscillation. This is
The forced vibration wind tunnel tests were performed in the high wind speed section of the CLP Power Wind/Wave Tunnel Facility at the Hong Kong University of Science and Technology. The dimension of the high wind speed section was 3 m (in width) × 2 m (in height). A test model with dimensions of 50.8 mm in width (D) × 50.8 mm in depth (D) × 915 mm in height (H) was mounted on a forced vibration test rig (a)
(b)
(c)
(d)
Test model with pressure taps
Wind
Oscillation direction y
Pivot Point Roller
Test model
Roller Steel strand
Steel strand
Rotating disk
LDS Actuator
y
Actuator
Fig. 1. Forced vibration wind tunnel test: (a) plan view of test rig; (b) stereogram of test rig; (c) test model in wind tunnel; (d) wind acting on oscillation model. 2
(b)
(a) air
Tip Level 9 Level 8
air
oscillation prism
wall
wall
50.8
50.8
Level 7
100 80 72.2 32.2
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120
Level 6
915
130
Level 5
120
Level 4
air
air
90
Level 3 85.6 85
Level 2 Level 1
wall
Base
wall
pivot
Side face Windward face
Fig. 2. Pressure measurements: (a) diagrammatic sketch of air-induced pressure on oscillation prisms; (b) distribution of pressure taps (units in mm).
increase with the increase in the model oscillation amplitude. The force is not always in-phase with the displacement which indicates nonlinear features. To study the nonlinear features, the non-wind-induced force is decomposed into the aerodynamic damping and additional aerodynamic stiffness components, as introduced in the following section.
as expected because the surrounding air is accelerated more at a larger amplitude of oscillation, resulting in the suction or repulsion around the test model more significant. It is observed from Fig. 3 that the ‘bubble’ like configurations emerge at the top and bottom ends of the test model and the magnitude of RMS pressures in the regions is remarkably larger than that at other regions. This is known as the three-dimensional ‘end’ effect [4,9], which is caused by the suction or reattachment generated by flow separation around the two ends. Fig. 3 also shows that the magnitude of RMS pressures tends to increase with the height as the test model is pivoted to oscillate and the amplitude of oscillation increases with height. The non-wind-induced force was evaluated by integrating the observed pressure with respect to the attribute area. The non-wind-induced force-displacement of the vertical model is depicted in Fig. 4. It is noteworthy that the range of displacement and force remarkably
3.2. Non-wind-induced additional damping and stiffness 3.2.1. Identification procedure The governing equation of motion of the test model per unit height at the elevation z above the ground under the zero-wind speed is expressed as
my¨ (t ) + cy ̇ (t ) + ky (t ) = P0 (t ) where m =
H
∫0 ρs
D 2ϕ2 (z ) dz
Fig. 3. Non-wind-induced RMS pressure on a vertical test model. 3
(1)
is the generalized mass of the test model, ρs
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Fig. 4. Non-wind-induced force-displacement of a vertical test model. z
damping ratios κ 0 and ξ0 as well as forces P0s and P0d are determined as
is sectional mass density, D is the model width, and ϕ (z ) = H is mode shape (the test model assumedly oscillates with the linear mode shape, H is the model height); c and k are the damping and stiffness coefficients; P0 (t ) is the generalized non-wind-induced force obtained in the previous section (Section 3.1). P0 (t ) consists of a non-wind-induced force component and a motion-induced force component,
P0 (t ) = 1/2ρU 2DH [C0 (t ) + CM (t )]
K a y = 1/2ρU 2DHCM
(3)
Ka 3K a = 2ω2m 2ω2ρs HD 2
(4)
which can also be represented by a stiffness term that is in-phase with the oscillating displacement and a damping term that is in-phase with the oscillating velocity,
Ga = κ + iη
(5)
where, κ and η are the non-wind-induced stiffness and damping coefficients, respectively. Substituting Eqs. (3), (4) and (5) into Eq. (2) yields,
η y )̇ ω
y = y ̂ cos(ωt )
(7)
κ =
-3 I1 ρa ω2D 2H 2y ̂
(8)
η=
-3 I1 ρa ω2D 2H 2y ̂
(9)
where the parameters I1 and I2 are defined as
1 T
∫0
I2 = lim
1 T
∫0
T →∞
T
T
ρ d , P0 = 2mωξ0 (y ) y ̇ ρs
(13)
κ = 0.6048(A/ D)2 − 0.0663(A/ D) + 0.4767, R2 = 0.8619
(14)
η = −0.4707(A/ D)0.7394 + 0.02181, R2 = 0.8763
(15)
where, is the coefficient of determination, which can present the quality of curve fitting. To verify the identified nonlinearities in Fig. 5, a hybrid aeroelasticpressure balance (HAPB) wind tunnel test carried out in a previous study [5] is utilized herein. In this test, both the unsteady force by pressure integration and the wind-induced response were synchronously observed. The two components of the aerodynamic force (windinduced and non-wind-induced forces) were identified by the procedure presented above. The structural parameters of the HAPB system include: the sectional density was 277.65 kg/m3; the damping ratio was 0.78%; the nonlinear physical damping ratio and frequency were ξe = −0.006045A−0.0399 + 0.01574 and fe = 0.7821A−0.005517 + 7.085, respectively, which were identified by a wavelet method [3]. Substituting the force and structural parameters (mass, physical damping and stiffness) into Eq. (1), the response of the test model can be calculated by using a step-by-step iteration method and compared with the experimentally observed response. Fig. 6 (a) compares the response predicted by the observed force (without considering the effect of non-wind-induced nonlinearities), the observed force but excluding all the non-wind-induced nonlinearities (with considering the effect of non-wind-induced nonlinearities), and the observed response. It is noted that the response to the observed
(6)
Jointly using Eqs. (6) and (7) and taking into account the orthogonality of trigonometric functions, one can obtain
T →∞
ξ0 (y ) = η (y )
R2
In a forced vibration test, the test model oscillates harmonically with the tip response as,
I1 = lim
(12)
3.2.2. Identification results and verification The non-wind-induced damping and stiffness are determined according to the forced vibration-pressure measurement technique and the identification procedure introduced above. The non-wind-induced damping and stiffness of the vertical test model are shown in Fig. 5. It is observed in Fig. 5 (a) that the non-wind-induced damping of the vertical prism is always negative and decreases with the increase in the amplitude of oscillation ( A/ D ). Since the negative damping increases the structural response, neglecting the non-wind-induced damping would overestimate the response. On the other hand, Fig. 5 (b) shows that the non-wind-induced stiffness of the vertical prism is positive and increases with the amplitude of oscillation. It is also observed that both the non-wind-induced damping and stiffness are nonlinear and vary slowly with the amplitude of oscillation. The curves in Fig. 5 are weighted by polynomials in Eqs. (14) and (15) using the least mean square method.
whose dimensionless form Ga is
P0 (t ) = 1/2ρU 2DHC0 (t ) + 2mω2 (κy +
ρ ρ s , P0 = κ 0 (y ) s y ρ ρs
(2)
where ρ is the density of air; U is the wind velocity at a reference point and it is equivalent to pressure difference as mentioned in Section 2; C0 (t ) and CM (t ) are the total non-wind force and motion-induced force coefficients at the height of z . A complex aerodynamic impedance K a has been introduced in literature [12,13],
Ga =
κ 0 (y ) = κ (y )
P0 (t ) cos(ωt ) dt
(10)
P0 (t ) sin(ωt ) dt
(11)
Based on Eqs. (10) and (11), the non-wind-induced stiffness and 4
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Fig. 5. Non-wind-induced damping and stiffness of the vertical test model: (a) nonlinear damping; (b) nonlinear stiffness.
4. Nonlinearities of inclined prisms
wind force is substantially larger than the observed response, and the difference is remarkable at high wind speeds. This is because the oscillation at high wind speeds as well as the associated amplitude- dependent non-wind-induced nonlinearities are much larger than those at low wind speeds. In contrast, the response predicted by the observed wind force but excluding non-wind-induced nonlinearities agrees well with the experimental results. Moreover, the comparison between the response to the observed force excluding the non-wind-induced damping, to the same observed force but excluding the additional stiffness, and the observed response is presented in Fig. 6 (b). It shows that the response predicted only excluding the non-wind-induced damping is slightly larger at low wind speeds but is much larger at high wind speeds than the experimental response whereas the response with only the non-wind-induced stiffness excluded is substantially smaller than the observed response. The observation in Fig. 6 suggests that (1) the non-wind-induced nonlinearities may significantly influence the wind-induced response of a prism especially at high wind speeds. The non-wind-induced nonlinear damping can be as large as 20% of the physical damping of the oscillation model. (2) Ignoring the non-wind-induced damping will overestimate the structural response predicted, whereas neglecting the non-wind-induced stiffness will underestimate the response. (3) The effect of the non-wind-induced damping on the response predicted is more significant than the effect of non-wind-induced stiffness. (4) It is verified that the forced vibration-pressure measurement technique is capable of identifying non-wind-induced additional damping and stiffness.
The non-wind-induced nonlinearities of inclined test models are determined by using the forced vibration-pressure measurement technique verified through the vertical test model in the previous section. The non-wind-induced damping and stiffness are presented in Figs. 7 and 8, respectively. Fig. 7 shows that all the damping coefficients of inclined prisms are negative. Despite the fluctuations of the coefficients, the general trends in all the subfigures are almost the same. The coefficients tend to decrease with the increase in the amplitude of oscillation, whereas the magnitude of damping tends to decrease at large inclinations (e.g. α = 30o ). Fig. 8 indicates that the stiffness of inclined test models is positive, and their values tends to increase at large inclinations (e.g. α = 30o ). The results in Figs. 7-8 can be used for predicting the aeroelastic responses of inclined prisms, which are anticipated to improve the accuracy of the prediction. To facilitate the convenient use, the results are fitted by polynomials as 0.9165 + 0.00144, α = 5o , R2 = 0.6334 ⎧− 0.4965(A/ D) ⎪ 0.6338 o + 0.0392, α = 10 , R2 = 0.6623 ⎪− 0.3761(A/ D) ⎪ η = − 0.479(A/ D)0.9625 − 0.00285, α = 15o , R2 = 0.8349 ⎨ ⎪− 0.5414(A/ D)0.9168 − 0.00792, α = 20o, R2 = 0.8395 ⎪ ⎪− 0.4568(A/ D)1.291 − 0.03837, α = 30o, R2 = 0.4251 ⎩
(16)
Fig. 6. Verification for the identified non-wind-induced nonlinearities: (a) including and excluding the nonlinearities; (b) excluding the non-wind-induced damping or stiffness. 5
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Fig. 7. Non-wind-induced damping coefficients of inclined prisms.
stiffness of vertical and inclined prisms have been identified by using the proposed forced vibration-pressure measurement technique. The test results on the non-wind-induced pressure and force-displacement of a vertical prism have been discussed. On the basis of the non-windinduced force given by the forced vibration-pressure measurement technique, the non-wind-induced damping and stiffness have been identified by a procedure proposed. The proposed procedure in terms of the vertical prism have been verified by the comparison between the predicted response with and without the non-wind-induced nonlinearities and the measured response. After the verification, the nonwind-induced nonlinearities of inclined prisms were determined by the proposed technique. The main findings in the study are summarized as follows.
2 o ⎧ 0.331(A/ D) + 0.080(A/ D) + 0.4789, α = 5 , R2 = 0.88 ⎪ 2 o 2 ⎪ 0.136(A/ D) + 0.132(A/ D) + 0.4485, α = 10 , R = 0.9151 ⎪ κ = − 0.142(A/ D)2 + 0.115(A/ D) + 0.449, α = 15o , R2 = 0.644 ⎨ ⎪ 0.354(A/ D)2 + 0.047(A/ D) + 0.4372, α = 20o, R2 = 0.8463 ⎪ ⎪3.404(A/ D)2 − 0.356(A/ D) + 0.4445, α = 30o, R2 = 0.887 ⎩
(17)
5. Concluding remarks In this study, the non-wind-induced additional damping and
Fig. 8. Non-wind-induced stiffness coefficients of inclined prisms. 6
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• The • • •
cstc2019jcyj-msxm0639), the Key project of Technological Innovation and Application Development in Chongqing (Grant No.: cstc2019jscxgksb0188), the Fundamental Research Funds for the Central Universities (Project No.: 2019CDXYTM0032), a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. 16242016) and the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. 2019R1G1A1095215 and 2019H1D3A1A01101442).
non-wind-induced pressure on prisms increases remarkably with the increase in the amplitude of the structural oscillation. The non-wind-induced aerodynamic force appears nonlinear. The non-wind-induced damping of prisms is negative, whereas the non-wind-induced stiffness is positive. The exclusion of non-windinduced damping would overestimate the response of prisms predicted; In contrast, the exclusion the non-wind-induced stiffness would underestimate the response. The non-wind-induced nonlinear damping can be as much as 20% of the structural damping and is not negligible in predicting wind-induced response of aeroelastic structures. A forced vibration technique is capable of identifying the non-windinduced damping and added mass of prisms and is recommended for determining non-wind-induced nonlinearities of bluff bodies.
References [1] Cao F, Ge Y. Air-induced nonlinear damping and added mass of vertically vibrating bridge deck section models under zero wind speed. J Wind Eng Ind Aerodyn 2017;169:217–31. [2] Cermak JE. Wind-tunnel development and trends in applications to civil engineering. J Wind Eng Ind Aerodyn 2003;91:355–70. [3] Chen ZS, Tse KT. Identification of physical nonlinearities of a hybrid aeroelastic–pressure balance. Nonlinear Dyn 2019;98:95–111. [4] Chen ZS, Tse KT, Hu G, Kwok KCS. Experimental and theoretical investigation of galloping of transversely inclined slender prisms. Nonlinear Dyn 2018;91:1023–40. [5] Chen ZS, Tse KT, Kwok KCS, Kim B, Kareem A. Modelling unsteady self-excited wind force on slender prisms in a turbulent flow. Eng Struct 2020;202:109855. [6] Chen ZS, Tse KT, Kwok KCS, Kareem A. Aerodynamic damping of inclined slender prisms. J Wind Eng Ind Aerodyn 2018;177:79–91. [7] Gao G-Z, Zhu L-D. Nonlinear mathematical model of unsteady galloping force on a rectangular 2: 1 cylinder. J Fluids Struct 2017;70:47–71. [8] Gao G, Zhu L. Nonlinearity of mechanical damping and stiffness of a spring-suspended sectional model system for wind tunnel tests. J Sound Vib 2015;355:369–91. [9] Lin N, Letchford C, Tamura Y, Liang B, Nakamura O. Characteristics of wind forces acting on tall buildings. J Wind Eng Ind Aerodyn 2005;93:217–42. [10] Noël J-P, Kerschen G. Nonlinear system identification in structural dynamics: 10 more years of progress. Mech Syst Sig Process 2017;83:2–35. [11] Rizzo F, Caracoglia L, Montelpare S. Predicting the flutter speed of a pedestrian suspension bridge through examination of laboratory experimental errors. Eng Struct 2018;172:589–613. [12] Steckley A. Motion-induced wind forces on chimneys and tall buildings. Western University; 1989. [13] Vickery B, Steckley A. Aerodynamic damping and vortex excitation on an oscillating prism in turbulent shear flow. J Wind Eng Ind Aerodyn 1993;49:121–40. [14] Xu F, Zhang Z. Numerical simulation of windless-air-induced added mass and damping of vibrating bridge decks. J Wind Eng Ind Aerodyn 2018;180:98–107. [15] Zhang M, Xu F. Nonlinear vibration characteristics of bridge deck section models in still air. J Bridge Eng 2018;23:04018059. [16] Zhou Y, Kareem A. Aeroelastic balance. J Eng Mech 2003;129:283–92.
The present study has not only advanced our understanding but also presented applicable models of the non-wind-induced additional damping and stiffness of both vertical and inclined prisms, which can be used for predicting the aeroelastic response. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgement The authors appreciate the use of the testing facility, as well as the technical assistance provided by the CLP Power Wind/Wave Tunnel Facility at the Hong Kong University of Science and Technology. The authors would also like to express sincere thanks to the Design and Manufacturing Services Facility (Electrical and Mechanical Fabrication Unit) of the Hong Kong University of Science and Technology for their help in manufacturing the test device. The work described in this paper was supported by the 111 Project of China (Grant No. B18062), the National Natural Science Foundation of China (Grant No.: 51908090), the Natural Science Foundation of Chongqing, China (Grant No.:
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