INFLUENCE OF FLOW ON ICE-INDUCED VIBRATION OF STRUCTURES Sumin Jeong, Natalie Baddour Mechanical Engineering Department, University of Ottawa 770 King Edward Av., Ottawa, ON, K1N 6N5, sjeon045@uottawa.ca, nbaddour@uottawa.ca Abstract: Ice-induced vibrations (IIV) of structures in cold regions are very common phenomena. Although numerous investigations have been conducted in the past, the effect of fluid flow on IIV has not been fully studied. In this paper, a new mathematical model of IIV including the flow effect is proposed using Matlock’s IIV model combined with Morrison’s equation. The model is based on a single degree-of-freedom system with a discrete ice forcing function which is defined by the characteristic failure mechanism. A typical ice forcing function is chosen and approximated using a Fourier series to simulate the new model. By simulation of various numerical data, the range and influence of the flow effect on IIV are defined in this paper. Keywords: Flow Effect, Ice-Induced Vibration, Morrison’s Equation, Offshore Structures 1. INTRODUCTION Offshore structures built in ice-covered waters are subject to ice-induced vibrations (IIV) which originate from the interaction between the structures and the ice sheets [1]. Water flow, wind or thermal expansion drive the ice sheet surrounding the structure, and the moving ice sheet transfers force to the structure. The structure is elastically displaced by the ice until the elastic restoring force of the structure exceeds the ice strength. Part of the ice sheet then breaks and the structure bounces back in the opposite direction. As a result, IIV can cause fatigue damage and sometimes a resonance condition, which is hazardous on the structure [2]. Analysis of the IIV mechanism on structures, therefore, is very crucial from both structural and design points of views, since IIV can severely threaten the safety of structures. In spite of its long research history, there has been no consensus about the IIV mechanism. Peyton (1968) suggested that the velocity of a uniform-thickness ice sheet is related to the generated ice force based on his experimental records [1]. Neil (1976) proposed that crushed ice tends to break into a certain size, determining a characteristic failure frequency which in turn decides the forcing frequency [1]. In opposition to the characteristic failure mechanism, Blenkarn (1970) explained IIV as a self-excited vibration due to negative damping [3]. According to the self-excited model, IIV originates from the interaction between a flexible structure and decreasing ice crushing forces with increasing stress rate [4]. Määttänen (1980, 1981, and 1983) supported the self-excited IIV model through his field and laboratory

experiments [3]. In this paper, the characteristic failure model is adopted because of its simplicity for analytical study. An ice forcing function, therefore, is treated as a series of discrete events for the simulation. Since structures subjected to IIV are located in water, the influence of the flow is another important factor to analyze. The flow effect on IIV, however, has not been fully studied in past research. In this paper, a single degree-of-freedom model with a discrete forcing function, Matlock’s IIV Model, is developed to include the flow effect [1]. The flow effect is taken into account using Morrison’s equation. Together with Matlock’s IIV model and Morrison’s equation, a new mathematical model is proposed. In accordance with the new model, the simplest ice forcing function based on the characteristic failure mechanism is used in the new model in order to reduce analytical complexity. Various numerical data are used in simulations with the new mathematical model to investigate how the IIV characteristics of the structure change with the inclusion of the flow effect. 2. MODELING OF IIV Modeling of IIV can be categorized into two distinct methods: the first one simulates IIV as a continuous interaction due to the self-excited effect, and the other explains IIV as a series of discrete events [2]. Matlock et al. (1971) proposed an intuitive discrete-event IIV model, as shown in Figure 1, which is based on the characteristic failure mechanism. Because of its simplicity, Matlock’s model has been widely used in IIV research.

Figure 1: Matlock’s IIV Model [2]

The vertical rods on the cart in Figure 1 represent discrete ice forces. The first vertical rod comes in contact with the structure which is represented as the mass M, and then deflects up to the loading phase δ. At the end of the loading phase, the first rod breaks and no force is transmitted to the

structure. The first total period p ends with contact of the second rod, and the second loading phase begins. This process keeps repeating so that the ice forces become a periodic function. The structure is represented as a one degree-of-freedom system consisting of the mass M, damping coefficient c, and spring constant k. All structural properties are assumed to be linear. Defining an ice forcing function is the key factor to analyzing IIV, but the difficulty of IIV analysis is that the actual ice forcing mechanism is a complex non-linear process. In the literature, various linear or non-linear ice forcing functions have been proposed by either field or laboratory experiments. Although mathematical expressions vary from model to model, ice forcing functions can be globally defined as periodic saw-tooth functions. Yue and Bi (1998, 2000) suggested a simple but effective ice forcing function [5] given by

mv + cv + kv = F f (t ) (6)

where m, c, and k represent the mass, damping coefficient, and stiffness of the structure, respectively. The displacement of the system can be evaluated from the magnification factor expressed as Ms =

1 2 2

( k − mω n ) + (cω n ) 2

(7)

where the nth forcing frequency is 2nπ ωn = (8) T so that the forcing frequency is function of n. The magnification factor, therefore, becomes a function of n. The displacement of the structure subjected to IIV without flow effect is derived as 100 ⎡ A ⎧ ⎛ 2nπ ⎞ ⎛ 2nπ ⎞ ⎫⎤ v(t) = 0 + ∑ ⎢ M s ⎨an cos⎜ t + θn ⎟ + bn sin ⎜ t + θn ⎟⎬⎥ (9) k n = 1 ⎢⎣ T T ⎝ ⎠ ⎝ ⎠⎭⎥⎦ ⎩

where θn is the phase shift according to each n. 3. MODELING OF IIV WITH FLOW EFFECT

Figure 2: Ice Forcing Function

⎧ ⎛ t⎞ ⎪ F ⎜1 − ⎟ F (t ) = ⎨ 0 ⎝ τ ⎠ ⎪ 0 ⎩

(0 ≤ t < τ ) (1) (τ ≤ t < T ) Equation (1) describes the saw-tooth function proposed by Yue and Bi. The time τ and T correspond to δ and p of Matlock’s model, respectively. The ice forcing function of Figure 2, therefore, can be combined with Matlock’s model without any contradictions. Using a Fourier series, the ice forcing function F(t) can be written as 100

F f (t ) = A0 +

⎛

∑ ⎜⎜⎝ a

n

n =1

⎛ 2 nπ ⎞ ⎛ 2 nπ ⎞ ⎞ t ⎟ + bn sin ⎜ t ⎟ ⎟⎟ (2) cos⎜ T ⎝ ⎠ ⎝ T ⎠⎠

where A0 =

F0τ 2T

(3)

an =

⎛ ⎛ 2nπ ⋅ τ ⎜⎜1 − cos⎜ ⎝ T 2n π τ ⎝

bn =

F0 F0 T ⎛ 2 nπ ⋅ τ ⎞ − sin ⎜ ⎟ (5) nπ 2n 2 π 2τ ⎝ T ⎠

F0 T 2

2

⎞⎞ ⎟ ⎟⎟ (4) ⎠⎠

and

One hundred summation terms are enough for Ff(t) to approximate 96 percent of the peak values of F(t). The new forcing function Ff(t) replaces F(t) in the equation of motion of Matlock’s model. Since Matlock’s model describes a single degree-of-freedom system, the equation of motion is developed as

Morrison et al. (1950) developed an equation evaluating hydrodynamic forces applied to structures in incompressible flow [6]. IIV of structures is significantly influenced by fluid dynamics because most cases of IIV problems occur in offshore structures. Since vertical monopole structures located in moving or stagnant fluid are efficiently analyzed by using Morrison’s equation, influence of the flow on IIV can be evaluated by combining Morrison’s equation with the IIV model. To apply Morison’s equation to the IIV model, the structure is assumed to have a uniform shape and density for analytical simplicity. The linear Morrison’s equation proposed by Berge and Penzien (1974) is q = CD ρ

D D2 u − v (u − v) + C M ρπ u (10) 2 4

where CD and CM are the frictional drag and inertia coefficients, respectively [7], ρ is the fluid density and D is the diameter of the structure. Further, u and u are the velocity and acceleration of the fluid. Equation (10) is linearized based on small motion, implying that the relative velocity of the structure and fluid should be diminutive so that ' CD = C D u − v ≈ constant (11) Morrison’s equation consists of the drag loading from viscous flow and the inertial loading due to inviscid flow. The inertial loading term delivers added mass or virtual mass, related to the velocity of the structure, and can be expressed as ⎛ D 2 ⎞⎟ q I = ⎜ m + C A ρπ v = Mv (12) ⎜ 4 ⎟⎠ ⎝

where CA is the added mass coefficient defined by C A = C M − 1 (13) [7] The terms with bars indicate that the dimensions are given per unit length. Since Morrison’s equation illustrates the drag and inertial loadings in force per unit length, then in order to

apply it to the IIV model, the properties of the structure in equation (6) also need to be written per unit length. The ice forcing function can be changed into force per unit length by dividing the initial force F0 by the length of the structure. Equation (6), per unit length becomes m v + c v + k v = F f (t ) (14) The dynamics of the structure with only the flow effect is developed by replacing the forcing function with equation (10) and m v with equation (12). 2 ⎞ ⎛ ⎜ m + C A ρπ D ⎟v + c v + k v ⎜ 4 ⎟⎠ ⎝

F0 = 0.9

(15)

Equation (15) can be reshaped using equation (11) and (12) into D⎞ D D 2 (16) ⎛ Mv + ⎜ c + CD' ρ ⎟ v + kv = CD' ρ u + CM ρπ u 4 2⎠ 2 ⎝ The left side of equation (16) contains the flow effect on the structure, and the right side describes the flow-induced force. In addition, C'D is assumed to be small, the velocity and acceleration of the flow are assumed to be small so that the flow-induced force is negligible compared to the ice-induced force. The flow thus affects the structure response only via the flow-induced effect on the structure mass and damping parameters and not through an additional flow-induced force. Therefore, the magnification factor of the structure with the flow-induced effect is derived as f

=

1

(k

)

2 − Mω n2

D ⎫ ⎧ + ⎨ω n (c + C D' ρ )⎬ 2 ⎭ ⎩

2

Ms =

( k − m ω n 2 ) 2 + (c ω n ) 2

l

1+ 5

(18)

for comparison. The non-dimensional forms can be developed by multiplying k on both sides.

(19)

0.02

0.01

0

0.01

0

2

4

6

8

Table 1: Parameters for Simulation [8]

The numerical simulations are performed on the data obtained from Kärnä and Turunen (1989) [8]. The IIV data

12

14

The amplitudes of IIV can be calculated by multiplying the coefficient of the Fourier series ‘ An’ which is ⎧⎪ A0 An = ⎨ 2 2 ⎪⎩ a n + bn

n=0 (20) n ≥1

with equation (17) and (18). with flow effect without flow effect

Amplitude (m)

Model structure 41 76 1600 0.2 1 1.4 20

10

Time (sec) Figure 3: Displacement of the Structure

4. NUMERICAL SIMULATION Parameter h (mm) D (mm) m (kg) c (kNs/m) k (MN/m) σc (MPa) vmax (mm)

h D

(17)

Since the magnitude Mf is expressed per unit length, the magnification factor without the flow effect, Ms, also needs to be rewritten as 1

σ c Dh

where ‘l’ is the length of the structure [8]. The maximum displacement of the structure vmax is used to find τ and T of the ice forcing function. By plotting equation (9), τ and T are found as 7 and 10 seconds, respectively.

Displacement (m)

D D2 = CD ρ u − v (u − v) + C M ρπ u 2 4

M

of the channel marker in the Baltic were measured by Nordlund et al during the winter of 1987-1988, and the structural properties are specified [8]. Some structural properties not specified on Kärnä and Turunen (1989) are derived from the density of steel, ρs=7859 kg/m3, which is assumed to be the material of the channel marker. In table 1, h is the thickness of the ice sheet, and σc is the compressive strength of the ice. The initial ice force F0 is calculated by Korzhavin’s approach

0.01

0.005

0

0

10

20

30

40

50

60

70

80

n Figure 4: Flow Effect on IIV

Figure 4 shows that the flow effect lowers the resonance

frequency by 5.8 percent and the amplitude by 22.3 percent. Since the amplitudes of Figure 4 have the same forcing function, the flow effect is only related to the structure. According to equation (17), the flow effect influences the IIV through M and CD' . The frictional drag coefficient CD is 1 for Reynolds number 1,000 to 200,000, therefore CD' takes little part in determining the flow effect [7]. Once the geometry of the structure is fixed, the mass of the structure is the only independent factor influencing the added mass term since CA and D are determined by the geometry of the structure. Based on the mass of the structure, a non-dimensional parameter called the density ratio γρ can be introduced as ⎛ D 2 ⎞⎟ πD 2 M = ⎜⎜ m + C A ρπ ρ (γ ρ + C A ) (21) ⎟= ⎝

4 ⎠

4

where γρ =

ρs (22) ρ

Figure 5: Density Ratio vs. Magnitude

and ρs is the density of the structure and ρ is the density of the fluid in which the structure is located. Using equation (21) and (17), the relation between the density ratio and the magnification factor is obtained. In Figure 5 and 6, the black contour peaks on the left represent the response with the flow effect and the grey contour peaks on the right represent the response without the flow effect. The flow effect lowers the magnitude factor for all the ranges of the density ratio. The resonance frequency shifting is significant for low density ratio because the added mass term in M becomes relatively larger when the density of the structure is small compared to the density of water. When the density ratio is larger than 8, however, the flow effect has little influence to the frequency change. 5. CONCLUSION IIV is a complex non-linear process which takes place at offshore structures in ice-covered waters. To explain its non-linear mechanism, a variety of IIV models have been proposed, but Matlock’s model has been widely adopted because of its simplicity and clarity. In accordance with Matlock’s model, the ice forcing function proposed by Yue and Bi is used in this paper. The developed model is represented as a Fourier series to calculate the displacement. Since most structures subject to IIV are offshore structures, the structures are influenced by not only the ice forces but also the flow of surrounding water. The flow influences the vibration characteristics of the structure through the added mass and frictional drag terms of Morrison’s equation. The flow-induced force is relatively insignificant compared to the ice-induced force, only the ice-induced force is considered as the forcing function. Together with Matlock’s model, the ice forcing function, and Morrison’s equation, the new IIV model including the flow effect is developed. The numerical simulations are cond ucted on the

Figure 6: Density Ratio vs. Frequency Shift

experimental data cited from Kärnä and Turunen (1989). The amplitudes of IIV with and without the flow effect are compared by assuming the density of the structure as steel. The simulation result indicates that the flow effect changes the amplitudes and frequencies of the IIV. The density of the structure is the most dominant factor influencing the flow effect of IIV. The density ratio of the structure to water is introduced to define the relation between the structural parameter and fluid parameter. According to the results of the simulation, as the density ratio increases, the flow effect is reduced. For a density ratio higher than 8 the flow effect on response is negligible. Since the density of concrete, the most popular construction material, is around 2300 kg/m3, the flow effect can be considerably large in most IIV. REFERENCES [1] S. Sodhi, “Ice-induced Vibrations of Structures,” International

Association for Hydraulic Research Working Group on Ice Forces, Special Report 89-5, pp. 189-221, Feb. 1989. [2] G. Huang and P. Liu, “A Dynamic Model for Ice-induced Vibration of Structures”, 25th International Conference on Offshore Mechanics and Arctic Engineering, 2006-92077, June 2006. [3] S. Sodhi and C. Morris, “Characteristic Frequency of force Vibrations in Continuous Crushing of Sheet Ice against Rigid Cylindrical Structures,” Cold Regions Science and Technology, vol. 12, pp. 1-12, 1986. [4] M. Määttänen, “Ice-induced Vibration of Structure-self-excitation,” International Association for Hydraulic Research Working Group on Ice Forces, Special Report 89-5, pp. 189-221, Feb. 1989. [5] Q. Yan, Y. Qianjin, B. Xiangjun, and K. Tuomo, “ A Random Ice Force Model for Narrow Conical Structures,” Cold regions Science and Technology, vol. 45, pp. 148-157, 2006. [6] J. Wolfram, “On Alternative Approaches to Linearization and Morison’s Equation for Wave Force,” The Royal society, vol. 445, pp. 2957-2974, 1999. [7] J. F. Wilson, “Dynamics of Offshore Structures,” A Wiley-Interscience Publication, 2003. [8] T. Kärnä and R. Turunen, “Dynamic Response of Narrow Structures to Ice Crushing,” Cold Regions Science and Technology, vol 17, pp. 173-187, 1989.