# New Approach for Vibration Suppression through Restrictors on Towering Steel Columns with Supporting Frame

Authors: Wei Tan; Xiantao Fan; Le Xu; Yang Wang
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

Towering steel column with supporting frame is a typical equipment in chemical process engineering. Here, displacement restrictors are proposed to restrict the displacement of the towering and slender equipment while making the column to be structurally nonlinear and statically indeterminate. This study investigated on along-wind and cross-wind vibration characteristics of such equipment with the restrictors experimentally and numerically. A field test is carried out to measure the natural frequency and damping ratio of the 42.5-meter-high equipment vibrating in the wind, which is the prototype of the experimental model. A noncontact excitation system was applied on the experimental model to simulate the wind loads. The displacements and strains of the experimental model are collected under different frame types by changing the heights of displacement restrictors. The numerical simulation and experimental results showed that the height of displacement restrictors has a great influence on the vibration intensity of the equipment. An optimum location, recommended as about 40% of the height, could decrease the vibration intensity and enhance the safety of the equipment. Based on the results, a simplified formula in which the natural frequency and the damping ratio dominate the dynamic behavior of along-wind and cross-wind vibration, respectively, is derived from multi-degrees-of-freedom system. It could be furtherly utilized to predict the amplitude ratio of two structures and select a better design featuring an efficient vibration suppression performance. This work presents an important design guide to the frame-supporting towering process equipment and is of great significance to an economical and safety design.


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## Body

## 1. Introduction

When fluid flows around a circular body at a certain velocity, vortices form rhythmically from the leeward side and produce a periodic lift force on the body in a transversal direction. As the approaching flow velocity increases, the vortex shedding frequency is close enough to the body’s natural frequency so that the circular body is induced to respond in the lift direction, which is named vortex-induced vibration (VIV) [1, 2]. If the particular fluid flow is wind, the vibration of a circular body subject to wind is often referred to as cross-wind vibration (CWV) [3]. In addition, due to the violent fluctuation of the along-wind load, the along-wind vibration (AWV) is also significant especially when its velocity is high [4].The self-supporting towering circular equipment is one of the most common structures in the chemical engineering, such as the chemical tower, chimney, and pipelines. Towering self-supporting chemical columns are usually set up outside and tend to suffer CWV and AWV during the whole period of lives. According to Robertson et al. and Peil et al., the CWV can cause deflection and fatigue problems of structures and even lead to collapses [5, 6]. Siringoringo and Fujino described AWV of a suspension bridge tower under moderate wind velocity, in which the results show that under certain angle of attack, the bluff body of the windward tower leg created vortex shedding as indicated by the presence of single-frequency oscillation of the wind in front of leeward tower [7]. Srinivas et al. studied the along-wind response of a tapered chimney, where the wind is considered as a stationary random process composed of spatially varying mean speed on which turbulent fluctuations are superimposed [4]. Ke et al. studied the average wind load on a superlarge cooling tower, and the results could provide references to the structure selection and wind resistance design of such type of steel cooling towers [8]. In addition, the numerical study is becoming the main research method [9]. Narendran et al. studied the vortex-induced vibration in multicolumn offshore platform by an explicit dynamic subgrid-scale model [10]. And there are several original boundary methods applied in the numerical simulation, such as the direct-forcing immersed boundary method, which eliminates the requirement of mesh regeneration at each time step, owing to the movement of the cylinder [11]. Because of the destructiveness of AWV and CWV, researches on the vibration suppression of towering equipment are remarkable to the economical and safe design.Wind tunnel is widely used to study the dynamic behavior of structures under wind loads, especially for such high Reynolds number wind environment [12, 13]. Carril et al. conducted an experimental investigation on lattice towers built in Brazil using a wind tunnel, and a common approach to consider the effect of wind forces on antennas independent of the lattice tower, without considering the effects of their presence on the computation of wind forces, the effects of wind forces, and structural responses were analyzed [14]. Ke et al. studied the multidimensional extreme aerodynamic load under typical four-tower arrangements by using the wind tunnel [15]. Zhou et al. studied AWV and CVW of Guangzhou New TV Tower, in which wind force was measured and used to compute wind -induced responses by the experiment [16]. In addition, field measurement is also applied to study the wind forces and responses of towering structures. Glanville and Kwok presented the results of a field measurement on a steel frame tower, from which the dynamic characteristics, including frequencies, mode shapes, and damping values of the tower were obtained [17]. Another field measurement of the dynamic performance of the Ganton Tower under typhoons was studied by Guo et al. [18].In this work, the study object is a chemical towering process equipment which is generally used in the fluid catalytic cracking unit suffering intensely AWV and CWV. However, unlike the conventional self-supporting equipment, this object is frame supporting, and its shell body is several meters away from the ground resulting that the equipment is becoming more likely to vibrate under wind loads. Thus, an effective approach for vibration suppression is pressing needed. Displacement restrictors installed beneath and above the support are originally proposed to limit the displacement of the equipment, resulting in a structurally nonlinear and statically indeterminate column. Hence, it is important to study the dynamic behavior of the frame-supporting column equipment with displacement restrictor.This work aims to study the dynamic behavior of frame-supporting equipment with displacement restrictor under wind loads and to verify this vibration suppression approach. Both along-wind load and cross-wind load were applied using a noncontact electromagnetic excitation system. The frequency, damping ratio, and mode shape of the experimental model were measured; in the meantime, the displacements and elastic bending strains of the model were gained experimentally and numerically. Combined with the results, a simplified formula in which the natural frequency and the damping ratio dominate the dynamic behavior of along-wind and cross-wind vibration, respectively, is derived from multi-degrees-of-freedom, which is aimed to assist the design of the restrictor. Based on the study results, some feasible optimization suggestions are given for vibration suppression of such equipment.


## 2. Experimental Setup

### 2.1. Field Tests of the Prototype

The frame-supporting towering column in Qingdao, China, shown in Figure1 is the prototype of this work. The parameters are listed in Table 1. As it is shown in Figure 1, the prototype consists of a frame and a cylindrical shell. The shell is fixed on the fourth floor of the frame that the bottom of the shell is 3.1 m away from the ground, which is called H4th in this paper. There were four displacement restrictors installed on the fifth floor in some cases, and that is named H5th.Figure 1
The (a) prototype and (b) cross section of the frame-supporting equipment.
(a)
(b)Table 1
The parameters of the prototype.
Parameters
H
prototype (m)
D
prototype (m)
H
k (m)
W
k(m)
H
4th (m)
H
5th (m)
Values
42.5
1.24
20
3.7
14.3
6.4Modal parameters identification method under ambient excitation was applied in the field tests. The prototype is standing outside and vibrating under the wind loads. The acceleration-time history curve of the prototype was recorded. After eliminating the offset and filtering the data, random decrement technique is utilized. It is the key process to obtain the modal parameters, such as the natural frequency and the damping ratio. The results of the field test are listed in Table2.Table 2
The modal parameters of the prototype.
Modal parameters
Natural frequency,f (Hz)
Damping ratio,ζ
H
4th
1.0018
0.0114
H
5th
1.0173
0.0125According to Table2, it is shown that f is small so as to the critical wind speed is about 7∼8 m/s, which is 4∼6 class in the wind scale. It means that this frame-supporting equipment is prone to vibrate under the cross-wind loads as well. In addition, the damping ratio of the prototype is small, which is implied that the energy dissipation of the prototype is insufficient so that the vibration amplitude will be large considerably. Therefore, it is remarkable to propose a vibration suppression approach and study the dynamics behavior of the towering frame-supporting equipment in depth. A small-scale experiment and numerical simulation were carried out based on the elastic similarity criterion.


### 2.2. Experimental Model

#### 2.2.1. Model of the Frame-Supporting Equipment

An aero-elastic model of the towering structure was manufactured in this study to make sure that the experimental model and prototype had the same frequency, mode shape, and damping ratio. It was not demanded to ensure that each frequency and mode shape of the prototype and the model is same, because the first mode dominates the dynamic behavior of towering structure, while the higher mode contributed little to it; thus the first frequency and the first mode shape of the prototype and the model should be the same [19].The geometric scaling was set as 1 : 25, so the model was 1.7 meters high. The model was made of aluminum alloy and mainly consisted of a center bar, several cylindrical shells, and a supporting frame. The center bar was solid with a diameter of 6 mm. This solid bar provided the stiffness of the structure and lowered the frequency at the same time.As Figure2(a) shows, the frame has seven levels and are denoted as H1st to H7th from the bottom up. Four bolts and nuts were set at the same height as restrictors to limit the displacement of the shell in four main directions. H4th is the support where the shell was supported, and no restrictor was set at this height. The restrictor, placed from H5th to H7th, was denoted as upper restrictor and the restrictor, placed from H1st to H3rd, was denoted as lower restrictor. Four vices were used as the test stand to fix the bottom of the frame.Figure 2
Scheme of (a) the experimental model, (b) experimental setup configuration, (c) testing arrangements, and (d) cross section of the experimental model.
(a)
(b)
(c)
(d)In this work, an electromagnetic excitation system was originally introduced to simulate the wind loads. The excitation system was programmable and able to simulate along-wind load and cross-wind load. The consistency of the electromagnetic excitation system and the wind load would be furtherly explained in the next section. The exciter was noncontact so that the model can vibrate freely under excitation. As shown in Figure2(b), the test equipment used in this work consisted of a model mounted on the test stand, an electromagnetic excitation system, and a signal processing system. Figure 2(c) shows the arrangement of the measurement point. Strain gauges were set both at the frame legs and the support (H4th). To avoid stress concentration, smooth transition was created at the frame leg and two gauges were set at each side of the leg. At the support (H4th), gauges were set along the two-orthogonal directions. A data acquisition system (DH3820) was used to record the strain signals at a sampling frequency of 50 Hz. The elastic bending strain gained from experiment at the frame leg was denoted as εbottom. The elastic bending strain gained from experiment at the support (H4th) was denoted as εsupport.The accelerometers were installed at the top of the model, and two accelerometers were aligned orthogonally along the major axes of the model. To balance the weight of the accelerometer, additional weight was set at the opposite side. A data acquisition system (DH5908) was used to acquire the acceleration signals at a sampling frequency of 100 Hz. The acceleration gained from experiment was denoted asatop and was processed with frequency integration algorithm to gain the displacement of the model denoted as ytop.


### 2.3. Wind Load Generation in Exciter

#### 2.3.1. Along-Wind Load

The along-wind load could be divided into the mean wind force and the fluctuating wind force, and it can be calculated through the frequency-domain method or time-domain method. The frequency-domain method is easy to apply, but it could not calculate the transient response of the structure. The time-domain method could obtain the time history curve of the vibration, and it is an important research method to simulate the random process. Scholars proposed many time-domain methods to simulate the along-wind load including linear filter method, harmony superposition method, inverse fast Fourier transfer method, and wavelet analysis [20–22]. The harmony superposition method used in this work is based on a solid foundation in mathematics and is effective in simulating the along-wind load [23]. The along-wind force for mass point i could be written as [24, 25](1)Fit=∑j=1i∑k=1NHijωk2Δωcosωkt+θijωk+ϕjk,where N is the division number, Δω=ωu−ωl/N is the frequency increment, ωu is the upper limit frequency, ωl is the lower limit frequency, θijωk=arctanImHijωk/ReHijωk is the phase angle of Hijωk, ϕjk is the random number ranging from 0 to 2π, and Hijωkis gained from Cholesky decomposition of the power spectrum density function (PSD) of the along-wind load:(2)Spω=Hω⋅H¯Tω,where Spω is the PSD of the along-wind load. There are many wind speed spectrums proposed through field measurements and wind tunnel experiments, such as von Kármán spectrum, Davenport spectrum, Kaimal spectrum, and Simiu spectrum. Among them, the Davenport wind speed spectrum applied in this work is widely used to calculate the wind loads:(3)Svω=4Kv¯102x2ω1+x24/3,where K is the terrain rough factor, v¯10 is the mean wind speed at 10 meters above the ground, and ω is the frequency of the wind and x=600ω/πv¯10.The time-domain along-wind load was programmed and applied through the electromagnetic vibration exciter. However, the only disadvantage is that the wind force could only be exerted on the top, and the sensitiveness and tendency analysis are still unaffected. When a load was applied to the top of a structure ofn nodes, the displacement of the top at a certain time could be solved as(4)Y=p0ϕTmϕωn2,where p0 is the applied, ϕ is the mode shape vector,ϕ=φ1⋯φnT, m=diagm11⋯mnnis the lumped mass matrix, and ωn is the natural circular frequency of the structure.The displacement ratio is defined as the displacement divided by the height of the structure. The displacement ratio of the prototype and the experimental model is(5)YprototypeHprototype=p0HprototypeϕTmϕωn2,YmodelHmodel=p0HmodelϕTmϕωn2,where Yprototype is the displacement of prototype, Hprototype is the total height of the prototype, Ymodel is the displacement of the experimental model, and Hmodel is the total height of the model.Since the model and the prototype have the same mode shape and natural frequency, the applied loadp0 must multiply a reduced factor δ before being applied on the model to ensure that the two structures had the same displacement ratio. The lumped mass matrixes of the two structures were assumed to be changing in a linear trend described as Mprototype=βMmodel where β is the amplification coefficient of the mass. The reduced factor δ was then solved as(6)δ=HmodelβHprototype.


#### 2.3.2. Cross-Wind Load

Due to the complexity of CWV, current research usually uses the semiempirical model to simplify the fluid-solid interaction in CWV. Through wind-tunnel experiments, the parameters of semiempirical model are gained. According to a large number of tests, many models of the CWV are proposed by scholars [26–29]. Considering that the motion of structure in CWV is nearly sinusoidal, the harmonic force model was presented. The cross-wind load can be conservatively estimated by the following equation:(7)FLt=q0DCLcosωvt,where q0 represents the basic wind pressure of the local area, D is the column outer diameter, CL is root mean square (RMS) lift coefficient, ωv=2πfv, and fv is the vortex shedding frequency.This model has its defects that the fluid-solid interaction was not considered and the lock-in effect cannot be explained. However, this model has a clear mechanical conception and is easy to calculate. Europe Code and Chinese Code use this model to calculate the CWV [30, 31].In order to study the vibration of the structure under cross-wind load when resonating, this work applied the harmonic force model to simulate the cross-wind load. The cross-wind load when resonating is written as(8)FLt=12ρvc2DCLcosωct,where vc=ωcD/2πSt is the critical wind velocity, St is the Strouhal number, ωc=2πfc, and fc is the natural frequency of the structure.As for the towering frame structures, the Reynolds number is in the supercritical regime when CWV occurs [12]. According to wind tunnel experiments [32], the RMSs of the lift coefficient CL and St are almost constant regardless of the Reynolds number. In this work, CL and St are taken as the constant values. Hence, equation (8) is simplified as(9)FLt=fc2Ccosωct,where C is a constant number.According to equation (9), F(t) is proportional to fc2. The frequency of the experimental model is the only factor that changes when the restrictor height is changing. Here, the excited load is multiplied by an amplification factor γ which is defined as(10)γ=frestrictorfnone−restrictor2,where frestrictor represents the natural frequency when the displacement restrictor is equipped andfnone−restrictor is the natural frequency when there is only the support.There were several CWV of chemical towers occurred in China recently, with the amplitudes of the vibrations ranging fromHprototype/600 to Hprototype/300 due to different types of the structures. In this work, the cross-wind load was set as a sinusoidal force to excite the structure to vibrate under certain amplitude. The vibration amplitude of the experimental model without restrictor was set to be Hmodel/300. After setting restrictors, the frequency changed, and the excited load was multiplied by the amplification factor γ.


## 2.1. Field Tests of the Prototype

The frame-supporting towering column in Qingdao, China, shown in Figure1 is the prototype of this work. The parameters are listed in Table 1. As it is shown in Figure 1, the prototype consists of a frame and a cylindrical shell. The shell is fixed on the fourth floor of the frame that the bottom of the shell is 3.1 m away from the ground, which is called H4th in this paper. There were four displacement restrictors installed on the fifth floor in some cases, and that is named H5th.Figure 1
The (a) prototype and (b) cross section of the frame-supporting equipment.
(a)
(b)Table 1
The parameters of the prototype.
Parameters
H
prototype (m)
D
prototype (m)
H
k (m)
W
k(m)
H
4th (m)
H
5th (m)
Values
42.5
1.24
20
3.7
14.3
6.4Modal parameters identification method under ambient excitation was applied in the field tests. The prototype is standing outside and vibrating under the wind loads. The acceleration-time history curve of the prototype was recorded. After eliminating the offset and filtering the data, random decrement technique is utilized. It is the key process to obtain the modal parameters, such as the natural frequency and the damping ratio. The results of the field test are listed in Table2.Table 2
The modal parameters of the prototype.
Modal parameters
Natural frequency,f (Hz)
Damping ratio,ζ
H
4th
1.0018
0.0114
H
5th
1.0173
0.0125According to Table2, it is shown that f is small so as to the critical wind speed is about 7∼8 m/s, which is 4∼6 class in the wind scale. It means that this frame-supporting equipment is prone to vibrate under the cross-wind loads as well. In addition, the damping ratio of the prototype is small, which is implied that the energy dissipation of the prototype is insufficient so that the vibration amplitude will be large considerably. Therefore, it is remarkable to propose a vibration suppression approach and study the dynamics behavior of the towering frame-supporting equipment in depth. A small-scale experiment and numerical simulation were carried out based on the elastic similarity criterion.


## 2.2. Experimental Model

### 2.2.1. Model of the Frame-Supporting Equipment

An aero-elastic model of the towering structure was manufactured in this study to make sure that the experimental model and prototype had the same frequency, mode shape, and damping ratio. It was not demanded to ensure that each frequency and mode shape of the prototype and the model is same, because the first mode dominates the dynamic behavior of towering structure, while the higher mode contributed little to it; thus the first frequency and the first mode shape of the prototype and the model should be the same [19].The geometric scaling was set as 1 : 25, so the model was 1.7 meters high. The model was made of aluminum alloy and mainly consisted of a center bar, several cylindrical shells, and a supporting frame. The center bar was solid with a diameter of 6 mm. This solid bar provided the stiffness of the structure and lowered the frequency at the same time.As Figure2(a) shows, the frame has seven levels and are denoted as H1st to H7th from the bottom up. Four bolts and nuts were set at the same height as restrictors to limit the displacement of the shell in four main directions. H4th is the support where the shell was supported, and no restrictor was set at this height. The restrictor, placed from H5th to H7th, was denoted as upper restrictor and the restrictor, placed from H1st to H3rd, was denoted as lower restrictor. Four vices were used as the test stand to fix the bottom of the frame.Figure 2
Scheme of (a) the experimental model, (b) experimental setup configuration, (c) testing arrangements, and (d) cross section of the experimental model.
(a)
(b)
(c)
(d)In this work, an electromagnetic excitation system was originally introduced to simulate the wind loads. The excitation system was programmable and able to simulate along-wind load and cross-wind load. The consistency of the electromagnetic excitation system and the wind load would be furtherly explained in the next section. The exciter was noncontact so that the model can vibrate freely under excitation. As shown in Figure2(b), the test equipment used in this work consisted of a model mounted on the test stand, an electromagnetic excitation system, and a signal processing system. Figure 2(c) shows the arrangement of the measurement point. Strain gauges were set both at the frame legs and the support (H4th). To avoid stress concentration, smooth transition was created at the frame leg and two gauges were set at each side of the leg. At the support (H4th), gauges were set along the two-orthogonal directions. A data acquisition system (DH3820) was used to record the strain signals at a sampling frequency of 50 Hz. The elastic bending strain gained from experiment at the frame leg was denoted as εbottom. The elastic bending strain gained from experiment at the support (H4th) was denoted as εsupport.The accelerometers were installed at the top of the model, and two accelerometers were aligned orthogonally along the major axes of the model. To balance the weight of the accelerometer, additional weight was set at the opposite side. A data acquisition system (DH5908) was used to acquire the acceleration signals at a sampling frequency of 100 Hz. The acceleration gained from experiment was denoted asatop and was processed with frequency integration algorithm to gain the displacement of the model denoted as ytop.


## 2.2.1. Model of the Frame-Supporting Equipment

An aero-elastic model of the towering structure was manufactured in this study to make sure that the experimental model and prototype had the same frequency, mode shape, and damping ratio. It was not demanded to ensure that each frequency and mode shape of the prototype and the model is same, because the first mode dominates the dynamic behavior of towering structure, while the higher mode contributed little to it; thus the first frequency and the first mode shape of the prototype and the model should be the same [19].The geometric scaling was set as 1 : 25, so the model was 1.7 meters high. The model was made of aluminum alloy and mainly consisted of a center bar, several cylindrical shells, and a supporting frame. The center bar was solid with a diameter of 6 mm. This solid bar provided the stiffness of the structure and lowered the frequency at the same time.As Figure2(a) shows, the frame has seven levels and are denoted as H1st to H7th from the bottom up. Four bolts and nuts were set at the same height as restrictors to limit the displacement of the shell in four main directions. H4th is the support where the shell was supported, and no restrictor was set at this height. The restrictor, placed from H5th to H7th, was denoted as upper restrictor and the restrictor, placed from H1st to H3rd, was denoted as lower restrictor. Four vices were used as the test stand to fix the bottom of the frame.Figure 2
Scheme of (a) the experimental model, (b) experimental setup configuration, (c) testing arrangements, and (d) cross section of the experimental model.
(a)
(b)
(c)
(d)In this work, an electromagnetic excitation system was originally introduced to simulate the wind loads. The excitation system was programmable and able to simulate along-wind load and cross-wind load. The consistency of the electromagnetic excitation system and the wind load would be furtherly explained in the next section. The exciter was noncontact so that the model can vibrate freely under excitation. As shown in Figure2(b), the test equipment used in this work consisted of a model mounted on the test stand, an electromagnetic excitation system, and a signal processing system. Figure 2(c) shows the arrangement of the measurement point. Strain gauges were set both at the frame legs and the support (H4th). To avoid stress concentration, smooth transition was created at the frame leg and two gauges were set at each side of the leg. At the support (H4th), gauges were set along the two-orthogonal directions. A data acquisition system (DH3820) was used to record the strain signals at a sampling frequency of 50 Hz. The elastic bending strain gained from experiment at the frame leg was denoted as εbottom. The elastic bending strain gained from experiment at the support (H4th) was denoted as εsupport.The accelerometers were installed at the top of the model, and two accelerometers were aligned orthogonally along the major axes of the model. To balance the weight of the accelerometer, additional weight was set at the opposite side. A data acquisition system (DH5908) was used to acquire the acceleration signals at a sampling frequency of 100 Hz. The acceleration gained from experiment was denoted asatop and was processed with frequency integration algorithm to gain the displacement of the model denoted as ytop.


## 2.3. Wind Load Generation in Exciter

### 2.3.1. Along-Wind Load

The along-wind load could be divided into the mean wind force and the fluctuating wind force, and it can be calculated through the frequency-domain method or time-domain method. The frequency-domain method is easy to apply, but it could not calculate the transient response of the structure. The time-domain method could obtain the time history curve of the vibration, and it is an important research method to simulate the random process. Scholars proposed many time-domain methods to simulate the along-wind load including linear filter method, harmony superposition method, inverse fast Fourier transfer method, and wavelet analysis [20–22]. The harmony superposition method used in this work is based on a solid foundation in mathematics and is effective in simulating the along-wind load [23]. The along-wind force for mass point i could be written as [24, 25](1)Fit=∑j=1i∑k=1NHijωk2Δωcosωkt+θijωk+ϕjk,where N is the division number, Δω=ωu−ωl/N is the frequency increment, ωu is the upper limit frequency, ωl is the lower limit frequency, θijωk=arctanImHijωk/ReHijωk is the phase angle of Hijωk, ϕjk is the random number ranging from 0 to 2π, and Hijωkis gained from Cholesky decomposition of the power spectrum density function (PSD) of the along-wind load:(2)Spω=Hω⋅H¯Tω,where Spω is the PSD of the along-wind load. There are many wind speed spectrums proposed through field measurements and wind tunnel experiments, such as von Kármán spectrum, Davenport spectrum, Kaimal spectrum, and Simiu spectrum. Among them, the Davenport wind speed spectrum applied in this work is widely used to calculate the wind loads:(3)Svω=4Kv¯102x2ω1+x24/3,where K is the terrain rough factor, v¯10 is the mean wind speed at 10 meters above the ground, and ω is the frequency of the wind and x=600ω/πv¯10.The time-domain along-wind load was programmed and applied through the electromagnetic vibration exciter. However, the only disadvantage is that the wind force could only be exerted on the top, and the sensitiveness and tendency analysis are still unaffected. When a load was applied to the top of a structure ofn nodes, the displacement of the top at a certain time could be solved as(4)Y=p0ϕTmϕωn2,where p0 is the applied, ϕ is the mode shape vector,ϕ=φ1⋯φnT, m=diagm11⋯mnnis the lumped mass matrix, and ωn is the natural circular frequency of the structure.The displacement ratio is defined as the displacement divided by the height of the structure. The displacement ratio of the prototype and the experimental model is(5)YprototypeHprototype=p0HprototypeϕTmϕωn2,YmodelHmodel=p0HmodelϕTmϕωn2,where Yprototype is the displacement of prototype, Hprototype is the total height of the prototype, Ymodel is the displacement of the experimental model, and Hmodel is the total height of the model.Since the model and the prototype have the same mode shape and natural frequency, the applied loadp0 must multiply a reduced factor δ before being applied on the model to ensure that the two structures had the same displacement ratio. The lumped mass matrixes of the two structures were assumed to be changing in a linear trend described as Mprototype=βMmodel where β is the amplification coefficient of the mass. The reduced factor δ was then solved as(6)δ=HmodelβHprototype.


### 2.3.2. Cross-Wind Load

Due to the complexity of CWV, current research usually uses the semiempirical model to simplify the fluid-solid interaction in CWV. Through wind-tunnel experiments, the parameters of semiempirical model are gained. According to a large number of tests, many models of the CWV are proposed by scholars [26–29]. Considering that the motion of structure in CWV is nearly sinusoidal, the harmonic force model was presented. The cross-wind load can be conservatively estimated by the following equation:(7)FLt=q0DCLcosωvt,where q0 represents the basic wind pressure of the local area, D is the column outer diameter, CL is root mean square (RMS) lift coefficient, ωv=2πfv, and fv is the vortex shedding frequency.This model has its defects that the fluid-solid interaction was not considered and the lock-in effect cannot be explained. However, this model has a clear mechanical conception and is easy to calculate. Europe Code and Chinese Code use this model to calculate the CWV [30, 31].In order to study the vibration of the structure under cross-wind load when resonating, this work applied the harmonic force model to simulate the cross-wind load. The cross-wind load when resonating is written as(8)FLt=12ρvc2DCLcosωct,where vc=ωcD/2πSt is the critical wind velocity, St is the Strouhal number, ωc=2πfc, and fc is the natural frequency of the structure.As for the towering frame structures, the Reynolds number is in the supercritical regime when CWV occurs [12]. According to wind tunnel experiments [32], the RMSs of the lift coefficient CL and St are almost constant regardless of the Reynolds number. In this work, CL and St are taken as the constant values. Hence, equation (8) is simplified as(9)FLt=fc2Ccosωct,where C is a constant number.According to equation (9), F(t) is proportional to fc2. The frequency of the experimental model is the only factor that changes when the restrictor height is changing. Here, the excited load is multiplied by an amplification factor γ which is defined as(10)γ=frestrictorfnone−restrictor2,where frestrictor represents the natural frequency when the displacement restrictor is equipped andfnone−restrictor is the natural frequency when there is only the support.There were several CWV of chemical towers occurred in China recently, with the amplitudes of the vibrations ranging fromHprototype/600 to Hprototype/300 due to different types of the structures. In this work, the cross-wind load was set as a sinusoidal force to excite the structure to vibrate under certain amplitude. The vibration amplitude of the experimental model without restrictor was set to be Hmodel/300. After setting restrictors, the frequency changed, and the excited load was multiplied by the amplification factor γ.


## 2.3.1. Along-Wind Load

The along-wind load could be divided into the mean wind force and the fluctuating wind force, and it can be calculated through the frequency-domain method or time-domain method. The frequency-domain method is easy to apply, but it could not calculate the transient response of the structure. The time-domain method could obtain the time history curve of the vibration, and it is an important research method to simulate the random process. Scholars proposed many time-domain methods to simulate the along-wind load including linear filter method, harmony superposition method, inverse fast Fourier transfer method, and wavelet analysis [20–22]. The harmony superposition method used in this work is based on a solid foundation in mathematics and is effective in simulating the along-wind load [23]. The along-wind force for mass point i could be written as [24, 25](1)Fit=∑j=1i∑k=1NHijωk2Δωcosωkt+θijωk+ϕjk,where N is the division number, Δω=ωu−ωl/N is the frequency increment, ωu is the upper limit frequency, ωl is the lower limit frequency, θijωk=arctanImHijωk/ReHijωk is the phase angle of Hijωk, ϕjk is the random number ranging from 0 to 2π, and Hijωkis gained from Cholesky decomposition of the power spectrum density function (PSD) of the along-wind load:(2)Spω=Hω⋅H¯Tω,where Spω is the PSD of the along-wind load. There are many wind speed spectrums proposed through field measurements and wind tunnel experiments, such as von Kármán spectrum, Davenport spectrum, Kaimal spectrum, and Simiu spectrum. Among them, the Davenport wind speed spectrum applied in this work is widely used to calculate the wind loads:(3)Svω=4Kv¯102x2ω1+x24/3,where K is the terrain rough factor, v¯10 is the mean wind speed at 10 meters above the ground, and ω is the frequency of the wind and x=600ω/πv¯10.The time-domain along-wind load was programmed and applied through the electromagnetic vibration exciter. However, the only disadvantage is that the wind force could only be exerted on the top, and the sensitiveness and tendency analysis are still unaffected. When a load was applied to the top of a structure ofn nodes, the displacement of the top at a certain time could be solved as(4)Y=p0ϕTmϕωn2,where p0 is the applied, ϕ is the mode shape vector,ϕ=φ1⋯φnT, m=diagm11⋯mnnis the lumped mass matrix, and ωn is the natural circular frequency of the structure.The displacement ratio is defined as the displacement divided by the height of the structure. The displacement ratio of the prototype and the experimental model is(5)YprototypeHprototype=p0HprototypeϕTmϕωn2,YmodelHmodel=p0HmodelϕTmϕωn2,where Yprototype is the displacement of prototype, Hprototype is the total height of the prototype, Ymodel is the displacement of the experimental model, and Hmodel is the total height of the model.Since the model and the prototype have the same mode shape and natural frequency, the applied loadp0 must multiply a reduced factor δ before being applied on the model to ensure that the two structures had the same displacement ratio. The lumped mass matrixes of the two structures were assumed to be changing in a linear trend described as Mprototype=βMmodel where β is the amplification coefficient of the mass. The reduced factor δ was then solved as(6)δ=HmodelβHprototype.


## 2.3.2. Cross-Wind Load

Due to the complexity of CWV, current research usually uses the semiempirical model to simplify the fluid-solid interaction in CWV. Through wind-tunnel experiments, the parameters of semiempirical model are gained. According to a large number of tests, many models of the CWV are proposed by scholars [26–29]. Considering that the motion of structure in CWV is nearly sinusoidal, the harmonic force model was presented. The cross-wind load can be conservatively estimated by the following equation:(7)FLt=q0DCLcosωvt,where q0 represents the basic wind pressure of the local area, D is the column outer diameter, CL is root mean square (RMS) lift coefficient, ωv=2πfv, and fv is the vortex shedding frequency.This model has its defects that the fluid-solid interaction was not considered and the lock-in effect cannot be explained. However, this model has a clear mechanical conception and is easy to calculate. Europe Code and Chinese Code use this model to calculate the CWV [30, 31].In order to study the vibration of the structure under cross-wind load when resonating, this work applied the harmonic force model to simulate the cross-wind load. The cross-wind load when resonating is written as(8)FLt=12ρvc2DCLcosωct,where vc=ωcD/2πSt is the critical wind velocity, St is the Strouhal number, ωc=2πfc, and fc is the natural frequency of the structure.As for the towering frame structures, the Reynolds number is in the supercritical regime when CWV occurs [12]. According to wind tunnel experiments [32], the RMSs of the lift coefficient CL and St are almost constant regardless of the Reynolds number. In this work, CL and St are taken as the constant values. Hence, equation (8) is simplified as(9)FLt=fc2Ccosωct,where C is a constant number.According to equation (9), F(t) is proportional to fc2. The frequency of the experimental model is the only factor that changes when the restrictor height is changing. Here, the excited load is multiplied by an amplification factor γ which is defined as(10)γ=frestrictorfnone−restrictor2,where frestrictor represents the natural frequency when the displacement restrictor is equipped andfnone−restrictor is the natural frequency when there is only the support.There were several CWV of chemical towers occurred in China recently, with the amplitudes of the vibrations ranging fromHprototype/600 to Hprototype/300 due to different types of the structures. In this work, the cross-wind load was set as a sinusoidal force to excite the structure to vibrate under certain amplitude. The vibration amplitude of the experimental model without restrictor was set to be Hmodel/300. After setting restrictors, the frequency changed, and the excited load was multiplied by the amplification factor γ.


## 3. Numerical Simulation Setup

The experimental model has a complex structure which makes it difficult to get experiment data in some place. Therefore, the numerical simulation was carried out using ANSYS 15.0 to get relevant values which are unable to gain from experiment. The numerical simulation was used to gainytop, εsupport, and εbottom at the testing point to verify the correctness of the simulation. The maximum elastic bending strain of the support (H4th) as shown in Figure 3(b) and the shell above the frame top as shown in Figure 3(c) were gained from the simulation results to evaluate the vibration suppression effectiveness after setting restrictors and were denoted as εsupport max and εshell, respectively. Elastic bending strains gained from experiment and simulation were together referred as ε. The numerical simulation was based on the damping values gained from experiment. The Rayleigh damping matrix was used in ANSYS to simulate the dynamic behavior of the structure under different loads.Figure 3
FEA model of the structure: (a) FEA model of frame, (b) location ofεsupport max, (c) location of εshell, and (d) Location of εbottom.
(a)
(b)
(c)
(d)


## 4. Results and Discussion

In this part, the symbols used are emphasized as follows:ytop is the top displacement of the frame-supporting structure, mm. εsupport is the strain of the support in H4th, με. εsupport max is the max strain of the support in H4th, με. εbottom is the strain in the bottom of the frame, με. εshell is the strain in the shell which contacted with the restrictor, με.


### 4.1. Frequency

The frequency of the experimental model was obtained from the acceleration response under free vibration test using fast Fourier transformation (FFT), of which the curves are shown in Figure4. Spectral peaks indicated that the first mode dominated the response of the structure at 0.854 Hz. The first frequency of structure is shown in Table 3. Compared with Table 2, the natural frequency of the experiment model is consistent with the prototype, which verified the validity of the experimental model. In addition, the experimental data were compared with the numerical result and showed good agreement. As Table 3 shows, the frequency increased with a maximum value, 30.3%, after setting restrictors. Comparing the values of H1st∼H3rd, the frequency remained the same when changing lower restrictor height and the frequency only increased 4.8% compared with structure without restrictor (H4th). The frequency of the structure enlarged with increasing of upper restrictor height when comparing H5th to H7th. The H1stH5th, H1stH6th, and H1stH7th showed the frequency of the experimental model when setting both upper and lower restrictors with the lower one fixed at H1st. Comparing the last three columns with 5th to 7th columns, the frequency increased a little when setting two restrictors compared with one upper restrictor condition. Based on the results, it can be concluded that setting restrictors increased the frequency of the structure, and the frequency was determined by the height of upper restrictor.Figure 4
Time history curves in the model test.
(a)
(b)Table 3
The first frequencies of the structure.
Frequency
H
1st
H
2nd
H
3rd
H
4th
H
5th
H
6th
H
7th
H
1st
H
5th
H
1st
H
6th
H
1st
H
7th
Test (Hz)
0.895
0.895
0.895
0.854
0.971
1.022
1.113
0.977
1.025
1.115
Simulation (Hz)
0.921
0.919
0.913
0.834
1.016
1.082
1.156
1.019
1.082
1.158
Error (%)
2.823
2.612
1.972
2.398
4.429
5.545
3.720
4.093
5.286
3.738


### 4.2. Mode Shape

Eight additional accelerometers were set at different height to measure the mode shape of the structure. Mode shape was determined from the ratio of acceleration signals which were recorded simultaneously. The mode shapes measured by the test were compared with those calculated by simulation. As shown in Figures5(a) and 5(b), two kinds of structure were chosen to verify the numerical simulation which agreed well with the experiment results. Therefore, as shown in Figures 5(c)–5(f), the simulation results were used to study the mode shapes of the frame-supporting structural under different restrictor heights. The mode shape was divided into two regions: beneath the restrictor was denoted as the supported region and above the restrictor was denoted as the free vibration region. Figures 5(c) and 5(d) show the mode shape of structure after setting one-layer restrictors. Figure 5(c) shows that increasing the height of the lower restrictor smoothed the mode shape only in supported region with a same critical point of 0.385. Figure 5(d) shows that changing the height of upper restrictor changes the mode shape in both supported region and free vibration region with uncertain critical points of 0.268∼0.290, indicating that setting one-layer upper restrictors increased the stiffness of the whole structure compared with that of structure without restrictor (H4th). Figure 5(e) exhibits the mode shapes of structure with two restrictors, of which the upper one is fixed on H7th. It can be observed that the mode shape is smoothing through the upper restrictor but with the same critical point. Figure 5(f) also presents the mode shapes of structure with two restrictors, of which the lower one is fixed on H1st, in which the critical points are various. Comparing all the results in Figure 5, the mode shape became smoother after setting restrictors and was mainly determined by the height of upper restrictor.Figure 5
Mode shapes of the structure under different restrictor heights: (a) verified through the values inH4th, (b) verified through the values in H1stH5th, (c) one lower restrictor, (d) one upper restrictor, (e) lower restrictor height = H1st, and (f) upper restrictor height = H7th.
(a)
(b)
(c)
(d)
(e)
(f)


### 4.3. Damping Ratio

The frame-supporting structure was firstly excited using a sinusoidal force. After achieving stable vibration, the excited force was removed and the structure began to vibrate freely in a decay mode. Using this free vibration decay curve, the damping ratio was measured and these values were used in numerical simulation. The damping ratio of structure with different numbers of restrictors and heights were tested under various excited amplitudes. Results showed that the damping ratio of this structure did not change with different displacements. The damping ratio measured by experiments is shown in Figure6. Compared with Table 2, the damping ratio of the model was almost the same with the prototype. Moreover, the damping ratio of the structure increased after setting restrictors compared with no restrictor (H4th). The damping ratio in the same color indicated that changing the lower restrictor did not influence the damping ratio of the structure. As discussed in Figure 5, the height of the lower restrictor had little influence on mode shape, and due to the same vibration mode, the damping ratio of different lower restrictors remained the same. Comparing the damping ratios of different upper restrictor heights, the damping ratios enlarged with the increasing of the restrictor height and the increasing rates were 14%, 34%, and 60%, respectively. It is also illustrated that the upper restrictors play a primary role in vibration suppression.Figure 6
Damping ratio of structure under different restrictor height.


### 4.4. Along-Wind Vibration

According to design conditions, the basic wind pressure of the city where the prototype installed was 650 N/m2 so that the wind velocities applied to simulate the along-wind load were set as 18 m/s, 22 m/s, 26 m/s, 30 m/s, and 34 m/s, respectively. Figure 7 shows the time history and power spectral density (PSD) curves of different wind velocities.Figure 7
Time history and PSD of different wind velocities: (a) time history of along-wind load, (b) PSD ofV = 18 m/s, (c) PSD of V = 22 m/s, (d) PSD of V = 26 m/s, (e) PSD of V = 30 m/s, and (f) PSD of V = 34 m/s.
(a)
(b)
(c)
(d)
(e)
(f)Figure7(a) shows the time history of along-wind load for prototype, and the wind velocity was set as 18 m/s. The corresponded PSD curve is shown in Figure 7(b). Analogously, the PSD curves of different wind velocities used to generate along-wind load are shown in Figures 7(c)–7(f). Each PSD curve was compared with theoretical calculation values using equation (3). Results showed that the generated along-wind load matched well with the theory and it can reflect the fluctuation of the wind load. Then after multiplying the reduce factor δ calculated by equation (6), the along-wind load was applied to the structural top using the programmable excitation system. The structure exhibited linear elastic behavior under different along-wind loads, so the results obtained under the maximum along-wind load (Figure 7(f)) were studied. The RMS values of displacement and strains were used to analyze the results.The tested time history curves of acceleration and strain are presented in Figure8. It is apparently that the time history curves in along-wind test are in random coinciding with the properties of wind velocity. Figure 9 shows the time history curves of ytop and εbottom obtained from the test and the simulation. The maximum errors for ytop, εsupport-max, and εbottom were 5.5%, 9.7%, and 8.9% showing numerical results matches well with experimental data. Figure 10(a) shows the comparison of the ytop and strain with one restrictor. After setting one restrictor, it is obvious that ytop and εsupport-max decreased. With the increasing of the upper restrictor height, ytop and εsupport-max decreased 55.7% and 92.1%, respectively, compared with none restrictor (H4th). But ytop and strain remained almost the same when changing the height of lower restrictor, and the values decreased 16.4% and 50.5% correspondingly compared with none restrictor (H4th). εshell and εbottom changed very little with or without restrictors. Since changing lower restrictor height did not influence the results, frame-supporting structure with lower restrictor setting to H1st was selected as a representative to study. As shown in Figure 10(b), compared with none restrictor (H4th), after setting two restrictors, ytop and εsupport-max decreased with increasing of the upper restrictor height, and the values dropped by 56.3% and 90.1%, respectively. However, εshell and εbottom remained almost the same.Figure 8
Time history-acceleration curves in the top of the structure (a) and strain curve in the bottom of the frame structure (b) in along-wind test.
(a)
(b)Figure 9
y
top and εbottom of structure without restrictor: (a) time history curve of ytop and (b) time history curve of εbottom.
(a)
(b)Figure 10
RMSytop and RMS strain under along-wind: (a) one restrictor and (b) two restrictor.
(a)
(b)Comparing Figure10(a) with 10(b), ytop and εsupport-max changed greatly, while εshell and εbottom remained almost the same. More importantly, the changing trends of ytop and εsupport-max always keep the same pace. That is to say that ytop can represent the effectiveness of vibration suppression. Therefore, a simplified formula based on ytop is derived subsequently to assist the design of restrictors when the structure suffering AWV.Due to the fluctuation of the along-wind load, the structure vibrated randomly.ytop was solved by random vibration theory as(11)ytop=∫−∞∞Szω¯dω¯,(12)Sω¯=∑m∑nϕmϕnHm−iω¯Hniω¯Spmpnω¯,where Hm−iω¯ and Hniω¯ are the frequency response functions of mode shape m and n and Spmpnω¯ is the cross spectral density function of random force Pmt,Pnt:(13)Hm−iω¯=1ωm2Mm11−i2ξmω¯/ωm−ω¯/ωm2,Hniω¯=1ωn2Mn11+i2ξnω¯/ωn−ω¯/ωn2.According to Table3 and Figure 6, the towering frame structure is small damping system with sparsity of vibration-mode frequency, and equations (11) and (12) can be simplified as(14)ytop=∫−∞∞∑mϕm12Hm−iω¯Hmiω¯Spmpmω¯dω¯.As forytop, the first-order mode of vibration is the main factor, and the first mode shape has been normalization with the top of the structure which means ϕm12=1. And substituting equation (13) into equation (14), it is simplified as follows:(15)ytop=1ω12M1∫−∞∞11+4ξ12−2ω¯/ω12+ω¯/ω14Sp1p1ω¯dω¯,where M1=ϕ1Tm1ϕ1 and ϕ1=φ1⋯φn1T is the first mode shape, m1=diagm11⋯mnn1 is the lumped mass matrix, and thus M1=∑φi12m1; and the integral term in equation (15) is replaced by fω1,ξ1.After setting the restrictor or changing the height of the restrictor, the mass matrix remained the same, while the first frequency and first-mode shape changed. The ratio ofytop for structure m and structure n was solved as(16)ytopmytopn=ω1n2ω1m2∑φi1,n2∑φi1,m2fω1m,ξ1mfω1n,ξ1n.For decoupling the functionf, numerical simulation was carried out to verify that the ratio of ytop was independent of the damping ratio. The studying objects were structure without restrictor (H4th) and structure with two restrictors (H1stH7th). The damping ratio of H4th changed from 0.01 to 0.016, and the damping of H1stH7th changed from 0.016 to 0.01. The applied along-wind load is shown in Figure 7(f), and it was multiplied by the reduce factor δ. The variation of displacements for two structures after changing the damping ratio under along-wind load was investigated numerically. Comparing with the changing of the damping ratio (60%), the displacement changed a little (10%). The independence of damping was verified. So, equation (16) can be written as(17)ytopmytopn=ω1n2ω1m2∑φi1,n2∑φi1,m2ω1nω1mp.Substituting the values in Figures10(a) and 10(b) to equation (17), the exponent p = 1.In summary, the function model to predict the displacement of two frame-supporting structures in AWV can be written as follows:(18)ytopmytopn=∑φi1,n2∑φi1,m2ω1nω1m3.Since the AWV of frame-supporting structure was determined by the first frequency, increasing frequency could decrease the displacement and strains of the structure. According to Table3 and Figures 10(a) and 10(b), setting upper restrictor increased the frequency greatly and decreased the displacement and strains of the structure. Therefore, setting upper restrictor is the one of the most effective and economic ways to suppress AWV. What is more, the simplified AWV model is derived, which can be conductive to such structural designs.


### 4.5. Cross-Wind Vibration

The cross-wind vibration is caused by the vortex shedding from itself; therefore, it is regular. The time history curves in Figure11 is regular, which can be divided into three parts including starting, stable, and decay vibration regions. The amplitude after stabilization was used to analyze the results. Numerical results were verified using experiment data and the maximum errors for ytop, εsupport, and εbottom were 5.2%, 7.9%, and 5.7%, respectively. Figure 12(a) compared the displacement and elastic strain of structure with one restrictor. As shown in Figure 12(a), compared with none restrictor (H4th), ytop and εsupport max decreased 43.4% and 88.6%, respectively, with the increasing of the upper restrictor height. On the contrary, ytop and strain remained almost the same when changing the height of lower restrictor. When setting upper restrictor, though εbottom and εshell decreased slowly with increasing upper restrictor height, these values were about 27% larger than values of none restrictor (H4th). Changing lower restrictor height did not influence the results, so structure with lower restrictor setting to H1st was selected as a representative to study. As shown in Figure 12(b), compared with none restrictor (H4th), after setting two restrictors, ytop and εsupport max decreased with increasing of the upper restrictor height and reduced 46.9% and 85.3%, respectively. Just as the structure with one restrictor, when setting upper restrictor, though εbottom and εshell decreased slowly with increasing upper restrictor height, these values were about 25% larger than values of none restrictor (H4th). Comparing Figures 12(a) with 12(b), ytop and εsupport max decreased greatly, while εshell and εbottom increased a little. Comparing the reducing values of ytop and εsupport max with the increasing values of εshell and εbottom, these increments were acceptable. Just as the AWV of the structure, the changing trends of ytop and εsupport-max were the same. Hence, ytop also represented the effectiveness of vibration suppression. A simplified formula is also derived to assist the design of restrictors when the structure suffering CWV.Figure 11
Time history-acceleration curves in the top of the structure (a) and strain curve in the bottom of the frame structure (b) in the cross-wind test.
(a)
(b)Figure 12
Amplitude ofytop and strain under cross wind: (a) one restrictor and (b) two restrictor.
(a)
(b)Since the structure was under resonance under cross-wind load, after stabilization, similar to AWV, only one-order mode of vibration is considered, and the harmonic CWV force is written as a force vector in the following equation:(19)Ft=−FL2iexpiω¯t−exp−iω¯t1...0.As for such resonance reaction, the frequency-domain solution is easily available; thus,ytop in CWV is solved as(20)Ynt=12π∫−∞∞Pniω¯Hniω¯expiω¯tdt.Taking equation (13) into equation (20) and ω¯=ωn in resonance, equation (20) is solved as(21)ytop==FL21ω12ϕ1Tm1ϕ1ξ1.Like the AWV, after setting the restrictor or changing the height of the restrictor, the mass matrix remained the same. The ratio ofytop for structure m and structure n was solved as(22)ytopmytopn=FL,mFL,n∑φi1,n2m1∑φi1,m2m1ω1nω1m2ξ1nξ1m.As equation (9) shows, the force FL is linked with the natural frequency, and thus the equation (22) can be simplified as(23)ytopmytopn=∑φi1,n2∑φi1,m2ξ1nξ1m.Equation (23) is the function model to predict the displacement of two frame-supporting structures in CWV. Since the CWV of frame-supporting structure was determined by damping, increasing damping could decrease the displacement and strains of the structure. According to Figures 6, 12(a), and 12(b), setting upper restrictors increased the damping greatly resulting in decreasing the displacement as well as strains of the structure. Therefore, setting upper restrictor is the most effective and economic way to suppress CWV.In summary, the simplified formula to assist the design of restrictors under along wind and cross wind is generalized as follows:(24)ytopmytopn=∑φi1,n2∑φi1,m2ξ1nξ1maω1nω1m3b,where a and b are the coefficients. For AWV, a = 0, and b = 1; for CWV, a = 1, and b = 0.


## 4.1. Frequency

The frequency of the experimental model was obtained from the acceleration response under free vibration test using fast Fourier transformation (FFT), of which the curves are shown in Figure4. Spectral peaks indicated that the first mode dominated the response of the structure at 0.854 Hz. The first frequency of structure is shown in Table 3. Compared with Table 2, the natural frequency of the experiment model is consistent with the prototype, which verified the validity of the experimental model. In addition, the experimental data were compared with the numerical result and showed good agreement. As Table 3 shows, the frequency increased with a maximum value, 30.3%, after setting restrictors. Comparing the values of H1st∼H3rd, the frequency remained the same when changing lower restrictor height and the frequency only increased 4.8% compared with structure without restrictor (H4th). The frequency of the structure enlarged with increasing of upper restrictor height when comparing H5th to H7th. The H1stH5th, H1stH6th, and H1stH7th showed the frequency of the experimental model when setting both upper and lower restrictors with the lower one fixed at H1st. Comparing the last three columns with 5th to 7th columns, the frequency increased a little when setting two restrictors compared with one upper restrictor condition. Based on the results, it can be concluded that setting restrictors increased the frequency of the structure, and the frequency was determined by the height of upper restrictor.Figure 4
Time history curves in the model test.
(a)
(b)Table 3
The first frequencies of the structure.
Frequency
H
1st
H
2nd
H
3rd
H
4th
H
5th
H
6th
H
7th
H
1st
H
5th
H
1st
H
6th
H
1st
H
7th
Test (Hz)
0.895
0.895
0.895
0.854
0.971
1.022
1.113
0.977
1.025
1.115
Simulation (Hz)
0.921
0.919
0.913
0.834
1.016
1.082
1.156
1.019
1.082
1.158
Error (%)
2.823
2.612
1.972
2.398
4.429
5.545
3.720
4.093
5.286
3.738


## 4.2. Mode Shape

Eight additional accelerometers were set at different height to measure the mode shape of the structure. Mode shape was determined from the ratio of acceleration signals which were recorded simultaneously. The mode shapes measured by the test were compared with those calculated by simulation. As shown in Figures5(a) and 5(b), two kinds of structure were chosen to verify the numerical simulation which agreed well with the experiment results. Therefore, as shown in Figures 5(c)–5(f), the simulation results were used to study the mode shapes of the frame-supporting structural under different restrictor heights. The mode shape was divided into two regions: beneath the restrictor was denoted as the supported region and above the restrictor was denoted as the free vibration region. Figures 5(c) and 5(d) show the mode shape of structure after setting one-layer restrictors. Figure 5(c) shows that increasing the height of the lower restrictor smoothed the mode shape only in supported region with a same critical point of 0.385. Figure 5(d) shows that changing the height of upper restrictor changes the mode shape in both supported region and free vibration region with uncertain critical points of 0.268∼0.290, indicating that setting one-layer upper restrictors increased the stiffness of the whole structure compared with that of structure without restrictor (H4th). Figure 5(e) exhibits the mode shapes of structure with two restrictors, of which the upper one is fixed on H7th. It can be observed that the mode shape is smoothing through the upper restrictor but with the same critical point. Figure 5(f) also presents the mode shapes of structure with two restrictors, of which the lower one is fixed on H1st, in which the critical points are various. Comparing all the results in Figure 5, the mode shape became smoother after setting restrictors and was mainly determined by the height of upper restrictor.Figure 5
Mode shapes of the structure under different restrictor heights: (a) verified through the values inH4th, (b) verified through the values in H1stH5th, (c) one lower restrictor, (d) one upper restrictor, (e) lower restrictor height = H1st, and (f) upper restrictor height = H7th.
(a)
(b)
(c)
(d)
(e)
(f)


## 4.3. Damping Ratio

The frame-supporting structure was firstly excited using a sinusoidal force. After achieving stable vibration, the excited force was removed and the structure began to vibrate freely in a decay mode. Using this free vibration decay curve, the damping ratio was measured and these values were used in numerical simulation. The damping ratio of structure with different numbers of restrictors and heights were tested under various excited amplitudes. Results showed that the damping ratio of this structure did not change with different displacements. The damping ratio measured by experiments is shown in Figure6. Compared with Table 2, the damping ratio of the model was almost the same with the prototype. Moreover, the damping ratio of the structure increased after setting restrictors compared with no restrictor (H4th). The damping ratio in the same color indicated that changing the lower restrictor did not influence the damping ratio of the structure. As discussed in Figure 5, the height of the lower restrictor had little influence on mode shape, and due to the same vibration mode, the damping ratio of different lower restrictors remained the same. Comparing the damping ratios of different upper restrictor heights, the damping ratios enlarged with the increasing of the restrictor height and the increasing rates were 14%, 34%, and 60%, respectively. It is also illustrated that the upper restrictors play a primary role in vibration suppression.Figure 6
Damping ratio of structure under different restrictor height.


## 4.4. Along-Wind Vibration

According to design conditions, the basic wind pressure of the city where the prototype installed was 650 N/m2 so that the wind velocities applied to simulate the along-wind load were set as 18 m/s, 22 m/s, 26 m/s, 30 m/s, and 34 m/s, respectively. Figure 7 shows the time history and power spectral density (PSD) curves of different wind velocities.Figure 7
Time history and PSD of different wind velocities: (a) time history of along-wind load, (b) PSD ofV = 18 m/s, (c) PSD of V = 22 m/s, (d) PSD of V = 26 m/s, (e) PSD of V = 30 m/s, and (f) PSD of V = 34 m/s.
(a)
(b)
(c)
(d)
(e)
(f)Figure7(a) shows the time history of along-wind load for prototype, and the wind velocity was set as 18 m/s. The corresponded PSD curve is shown in Figure 7(b). Analogously, the PSD curves of different wind velocities used to generate along-wind load are shown in Figures 7(c)–7(f). Each PSD curve was compared with theoretical calculation values using equation (3). Results showed that the generated along-wind load matched well with the theory and it can reflect the fluctuation of the wind load. Then after multiplying the reduce factor δ calculated by equation (6), the along-wind load was applied to the structural top using the programmable excitation system. The structure exhibited linear elastic behavior under different along-wind loads, so the results obtained under the maximum along-wind load (Figure 7(f)) were studied. The RMS values of displacement and strains were used to analyze the results.The tested time history curves of acceleration and strain are presented in Figure8. It is apparently that the time history curves in along-wind test are in random coinciding with the properties of wind velocity. Figure 9 shows the time history curves of ytop and εbottom obtained from the test and the simulation. The maximum errors for ytop, εsupport-max, and εbottom were 5.5%, 9.7%, and 8.9% showing numerical results matches well with experimental data. Figure 10(a) shows the comparison of the ytop and strain with one restrictor. After setting one restrictor, it is obvious that ytop and εsupport-max decreased. With the increasing of the upper restrictor height, ytop and εsupport-max decreased 55.7% and 92.1%, respectively, compared with none restrictor (H4th). But ytop and strain remained almost the same when changing the height of lower restrictor, and the values decreased 16.4% and 50.5% correspondingly compared with none restrictor (H4th). εshell and εbottom changed very little with or without restrictors. Since changing lower restrictor height did not influence the results, frame-supporting structure with lower restrictor setting to H1st was selected as a representative to study. As shown in Figure 10(b), compared with none restrictor (H4th), after setting two restrictors, ytop and εsupport-max decreased with increasing of the upper restrictor height, and the values dropped by 56.3% and 90.1%, respectively. However, εshell and εbottom remained almost the same.Figure 8
Time history-acceleration curves in the top of the structure (a) and strain curve in the bottom of the frame structure (b) in along-wind test.
(a)
(b)Figure 9
y
top and εbottom of structure without restrictor: (a) time history curve of ytop and (b) time history curve of εbottom.
(a)
(b)Figure 10
RMSytop and RMS strain under along-wind: (a) one restrictor and (b) two restrictor.
(a)
(b)Comparing Figure10(a) with 10(b), ytop and εsupport-max changed greatly, while εshell and εbottom remained almost the same. More importantly, the changing trends of ytop and εsupport-max always keep the same pace. That is to say that ytop can represent the effectiveness of vibration suppression. Therefore, a simplified formula based on ytop is derived subsequently to assist the design of restrictors when the structure suffering AWV.Due to the fluctuation of the along-wind load, the structure vibrated randomly.ytop was solved by random vibration theory as(11)ytop=∫−∞∞Szω¯dω¯,(12)Sω¯=∑m∑nϕmϕnHm−iω¯Hniω¯Spmpnω¯,where Hm−iω¯ and Hniω¯ are the frequency response functions of mode shape m and n and Spmpnω¯ is the cross spectral density function of random force Pmt,Pnt:(13)Hm−iω¯=1ωm2Mm11−i2ξmω¯/ωm−ω¯/ωm2,Hniω¯=1ωn2Mn11+i2ξnω¯/ωn−ω¯/ωn2.According to Table3 and Figure 6, the towering frame structure is small damping system with sparsity of vibration-mode frequency, and equations (11) and (12) can be simplified as(14)ytop=∫−∞∞∑mϕm12Hm−iω¯Hmiω¯Spmpmω¯dω¯.As forytop, the first-order mode of vibration is the main factor, and the first mode shape has been normalization with the top of the structure which means ϕm12=1. And substituting equation (13) into equation (14), it is simplified as follows:(15)ytop=1ω12M1∫−∞∞11+4ξ12−2ω¯/ω12+ω¯/ω14Sp1p1ω¯dω¯,where M1=ϕ1Tm1ϕ1 and ϕ1=φ1⋯φn1T is the first mode shape, m1=diagm11⋯mnn1 is the lumped mass matrix, and thus M1=∑φi12m1; and the integral term in equation (15) is replaced by fω1,ξ1.After setting the restrictor or changing the height of the restrictor, the mass matrix remained the same, while the first frequency and first-mode shape changed. The ratio ofytop for structure m and structure n was solved as(16)ytopmytopn=ω1n2ω1m2∑φi1,n2∑φi1,m2fω1m,ξ1mfω1n,ξ1n.For decoupling the functionf, numerical simulation was carried out to verify that the ratio of ytop was independent of the damping ratio. The studying objects were structure without restrictor (H4th) and structure with two restrictors (H1stH7th). The damping ratio of H4th changed from 0.01 to 0.016, and the damping of H1stH7th changed from 0.016 to 0.01. The applied along-wind load is shown in Figure 7(f), and it was multiplied by the reduce factor δ. The variation of displacements for two structures after changing the damping ratio under along-wind load was investigated numerically. Comparing with the changing of the damping ratio (60%), the displacement changed a little (10%). The independence of damping was verified. So, equation (16) can be written as(17)ytopmytopn=ω1n2ω1m2∑φi1,n2∑φi1,m2ω1nω1mp.Substituting the values in Figures10(a) and 10(b) to equation (17), the exponent p = 1.In summary, the function model to predict the displacement of two frame-supporting structures in AWV can be written as follows:(18)ytopmytopn=∑φi1,n2∑φi1,m2ω1nω1m3.Since the AWV of frame-supporting structure was determined by the first frequency, increasing frequency could decrease the displacement and strains of the structure. According to Table3 and Figures 10(a) and 10(b), setting upper restrictor increased the frequency greatly and decreased the displacement and strains of the structure. Therefore, setting upper restrictor is the one of the most effective and economic ways to suppress AWV. What is more, the simplified AWV model is derived, which can be conductive to such structural designs.


## 4.5. Cross-Wind Vibration

The cross-wind vibration is caused by the vortex shedding from itself; therefore, it is regular. The time history curves in Figure11 is regular, which can be divided into three parts including starting, stable, and decay vibration regions. The amplitude after stabilization was used to analyze the results. Numerical results were verified using experiment data and the maximum errors for ytop, εsupport, and εbottom were 5.2%, 7.9%, and 5.7%, respectively. Figure 12(a) compared the displacement and elastic strain of structure with one restrictor. As shown in Figure 12(a), compared with none restrictor (H4th), ytop and εsupport max decreased 43.4% and 88.6%, respectively, with the increasing of the upper restrictor height. On the contrary, ytop and strain remained almost the same when changing the height of lower restrictor. When setting upper restrictor, though εbottom and εshell decreased slowly with increasing upper restrictor height, these values were about 27% larger than values of none restrictor (H4th). Changing lower restrictor height did not influence the results, so structure with lower restrictor setting to H1st was selected as a representative to study. As shown in Figure 12(b), compared with none restrictor (H4th), after setting two restrictors, ytop and εsupport max decreased with increasing of the upper restrictor height and reduced 46.9% and 85.3%, respectively. Just as the structure with one restrictor, when setting upper restrictor, though εbottom and εshell decreased slowly with increasing upper restrictor height, these values were about 25% larger than values of none restrictor (H4th). Comparing Figures 12(a) with 12(b), ytop and εsupport max decreased greatly, while εshell and εbottom increased a little. Comparing the reducing values of ytop and εsupport max with the increasing values of εshell and εbottom, these increments were acceptable. Just as the AWV of the structure, the changing trends of ytop and εsupport-max were the same. Hence, ytop also represented the effectiveness of vibration suppression. A simplified formula is also derived to assist the design of restrictors when the structure suffering CWV.Figure 11
Time history-acceleration curves in the top of the structure (a) and strain curve in the bottom of the frame structure (b) in the cross-wind test.
(a)
(b)Figure 12
Amplitude ofytop and strain under cross wind: (a) one restrictor and (b) two restrictor.
(a)
(b)Since the structure was under resonance under cross-wind load, after stabilization, similar to AWV, only one-order mode of vibration is considered, and the harmonic CWV force is written as a force vector in the following equation:(19)Ft=−FL2iexpiω¯t−exp−iω¯t1...0.As for such resonance reaction, the frequency-domain solution is easily available; thus,ytop in CWV is solved as(20)Ynt=12π∫−∞∞Pniω¯Hniω¯expiω¯tdt.Taking equation (13) into equation (20) and ω¯=ωn in resonance, equation (20) is solved as(21)ytop==FL21ω12ϕ1Tm1ϕ1ξ1.Like the AWV, after setting the restrictor or changing the height of the restrictor, the mass matrix remained the same. The ratio ofytop for structure m and structure n was solved as(22)ytopmytopn=FL,mFL,n∑φi1,n2m1∑φi1,m2m1ω1nω1m2ξ1nξ1m.As equation (9) shows, the force FL is linked with the natural frequency, and thus the equation (22) can be simplified as(23)ytopmytopn=∑φi1,n2∑φi1,m2ξ1nξ1m.Equation (23) is the function model to predict the displacement of two frame-supporting structures in CWV. Since the CWV of frame-supporting structure was determined by damping, increasing damping could decrease the displacement and strains of the structure. According to Figures 6, 12(a), and 12(b), setting upper restrictors increased the damping greatly resulting in decreasing the displacement as well as strains of the structure. Therefore, setting upper restrictor is the most effective and economic way to suppress CWV.In summary, the simplified formula to assist the design of restrictors under along wind and cross wind is generalized as follows:(24)ytopmytopn=∑φi1,n2∑φi1,m2ξ1nξ1maω1nω1m3b,where a and b are the coefficients. For AWV, a = 0, and b = 1; for CWV, a = 1, and b = 0.


## 5. Conclusion

In this paper, a displacement restrictor is proposed as a method for vibration suppression of structure under wind load and the effectiveness is discussed through the dynamic behavior research of frame-supporting structure under wind load experimentally and numerically. Using electromagnetic excitation system to simulate the along-wind load and cross-wind load is introduced as a new method in this work. The effectiveness and load equivalent method of this excitation system have been shown in this paper. The random along wind and harmonic cross wind can be regenerated perfectly, and the dynamic behavior of the structure could be excited. It is approved that displacement restrictor is an effective approach for vibration suppression through evaluating the displacement and strain. The upper displacement restrictor is indispensable, and two-layer restrictor is recommended. The displacements under AWV and CWV decrease about 50% when the upper restrictor is installed at the 40% height.A simplified formula is derived to assist the design of the displacement restrictor. The accessible modal parameters including modal shape, natural frequency, and damping ratio are applied to calculate the amplitude ratio of two structures and select a better design featuring an efficient vibration suppression performance. This work presented an important design guide to the frame-supporting structure and is of great significance to the economical and safe design.



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*Source: 8761750-2020-02-11.xml*