Comparison of Characteristic Failure Frequency Models for Ice Induced Vibrations by Natalie Baddour

COMPARISON OF CHARACTERISTIC FAILURE FREQUENCY MODELS FOR ICE INDUCED VIBRATIONS SUMIN JEONG and NATALIE BADDOUR∗ Department of Mechanical Engineering University of Ottawa 161 Louis Pasteur Ottawa, ON, K1N 6N5, Canada e-mail: nbaddour@uottawa.ca Abstract Several modelling approaches have been proposed for the modelling of ice-induced vibrations. Part of the differences in these approaches stem from the difference in the failure mode of the ice, whether in compression or bending. Within the spectre of crushing-failure models, different analytical models have been proposed. To investigate the differences, similarities and ease of implementation of models based on the crushing failure, two specific crushing failure frequency models are chosen for solution, simulation and comparison. These models are solved and then simulated with the same set of parameters. Corrections to one of the model are proposed and it is shown that although based on a similar modelling approach, the models do not predict the same behaviour.

1. Introduction Ice-induced vibrations (IIVs) have been investigated through analytical and numerical modelling. Historically, modelling approaches to IIVs were broadly characterized into two types, a characteristic failure frequency approach or a negative damping type approach. These theoretical models were proposed by Keywords and phrases: vibrations, ice-induced vibrations, modelling, ice models. ∗Corresponding author

Received October 21, 2010

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attempting to define the origin of IIVs, however the analysis is complicated by the involvement of several uncertain aspects of environment and circumstance. It was first proposed that crushed ice tends to break into a certain size and that the fracture size and velocity of an ice sheet determine a characteristic failure frequency which in turn decides the forcing frequency [1]. Matlock et al. [2] also supported the idea that there is a characteristic failure frequency that is a property of the ice, basing this on observations of field data. However, no physical process for this phenomenon was proposed. The characteristic failure frequency model is the first main type of IIV model and the essence of the model is that the ice force only depends on properties of the ice and not on properties of the structure. In contrast to the model with a characteristic failure frequency mechanism, Blenkarn [3] explained IIVs as self-excited vibrations due to negative damping, a second approach to IIVs. According to the self-excited model, IIVs originate from the interaction between a flexible structure and ice crushing forces that tend to decrease with increasing stress rate. In this modelling approach, the structure plays a very definitive role in the ice force and the ice force is not purely dependent on properties of the ice. M채채tt채nen, who is one of the big contributors of the selfexcited model, attempted to further the acceptance of the self-excited IIV model through his field and laboratory experiments [4]. Xu and Wang [5] also suggested an ice force oscillator model based on the self-excited model. During dynamic icestructure interaction, it is possible that both the structure and the ice sheet experience deformations. After a failure event, therefore, the structure and the ice edge both rebound against each other. Even if the structure is very rigid, the ice edge can still be active. Typically a model of self-excitation is useful to model this type of iceaction scenario whereas the characteristic frequency models would be unable to capture the role of the structure in developing the total ice force [6]. Contrary to the self-excited model, the structure does not play an active role in the characteristic failure frequency models. This means that the ice force is not constrained by the proposed negative damping effect. In the characteristic failure frequency model, the frequencies of IIVs are determined by the properties and velocities of the ice in contact with the structure. The relevant properties of the ice include salinity, thickness, temperature, density, and grain structure [7]. Along with these properties, other uncertain factors can be involved in the ice failure mechanism, which further complicates the analysis of IIVs. With this reasoning, it follows that every ice sheet can fail at different frequencies. By restricting the

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properties of the ice sheet, researchers have tried to explain the ice failure mechanism which takes place at the contact zone between the ice and structure. The details, however, vary from theory to theory due in part to uncertainties in the modelling of the ice itself. As research in the field progressed, different physical processes to explain the origin of the ice forces were proposed. In particular, attention was turned to whether the ice was considered to fail in crushing (compression) or bending. It is thought that compressive (crushing) failures occur as a result of ice interacting with a narrow vertical (cylindrical) structure [8]. Tsinker defines crushing as the complete failure of granularization of the solid ice sheet into particles of grains or crystal dimension; no cracking, flaking or any other failure mode occurs during pure crushing. One of the earliest published methods for predicting ice loads due to crushing failure was developed by Korzhavin in the 1940s [8]. Korzhavin proposed a formula for calculating the design ice load for slender vertical structures and his formula is not a description of the mechanics of the ice failure process or of the dynamic ice force acting on the structure. It is, however, a useful formula for the ice force in crushing failure and dynamic models of ice forces due to ice crushing failures are based on this formulation. Since the structure plays a strong role in the crushing failure of ice, crushing failure models tend to fall into the category of self-excited models, where the ice force cannot be constructed as a pre-determined function of time. Specific models that correspond to ice in crushing failure are given in Toyama et al. [9] and Huang and Liu [10]. Even though these two articles model the same physical phenomena, there are differences in their modelling approaches. The concept of adding ice-breaking cones to cylindrical structures was proposed in the late 1970â&#x20AC;&#x2122;s, changing the effective shape of the structure from cylindrical to conical. The ice force on a conical structure is smaller than the force on a cylindrical structure of similar size [7, 11-13]. It is thought that the main reason for the reduction in the ice force is that a well-designed cone can change the ice failure mode from crushing to bending. It is now considered that the primary failure mechanism for ice interacting with a conical structure is that of bending failure. In this failure mode, the ice sheets impacting on a cone fails by bending and typically the ice breaks almost simultaneously in each event of ice failure. The broken ice pieces are cleared away before the next failure event. With conical structures and ice failing in bending, analytical models for these are more likely to fall into the characteristic failure frequency models that were initially proposed in the 70sâ&#x20AC;&#x2122; since

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the ice force can effectively be modelled as only depending on properties of the ice. Once it is known the structure is conical, the effect of the structure on the modelling is in the choice of a bending-failure model and after that it tends to play no role in the analytical model of the ice force itself. With conical structures, the local ice forces were experimentally found to drop to almost zero after each event of ice failure [14, 15] and are consequently modelled analytically with a periodic function. For example, the forcing function suggested by Qu et al. [16], is a simple one degree-of-freedom model corresponding to a periodic ice force. This model is based on their own previous work [14, 15] as well as that of Hirayama and Obara [13]. This model attempts to represent an ice sheet failing through the bending mode and simply models the forcing function as a saw-tooth shaped, periodic force. From a modelling point of view, the bending failure of ice is easier to model since the ice forcing function is periodic and mathematically does not depend on the motion of the structure. This type of model thus becomes a standard forced vibration problem and once we have chosen the form of the external ice force, it is a relatively straightforward problem to solve. However, as previously mentioned, the crushing failure of ice does tend to involve the motion of the structure itself and this type of failure falls into those types of models that were referred to as â&#x20AC;&#x2DC;negative dampingâ&#x20AC;&#x2122; models. The difficulty in the modelling is that the motion of the structure itself is involved in how the ice fails and thus an a priori ice force is more difficult to formulate and requires greater care in the modelling and solution of the problem. Generally speaking, simple one degree of freedom models can be applied only for narrow structures that permit simplification to a SDOF model. More advanced models should also cope with ice-induced vibrations of structures that can arise in many modes. Sophisticated models for modern design should be applicable for different kinds of structures: wide and narrow, stiff and compliant. Phenomena that should be included are (i) changes between simultaneous and non-simultaneous crushing while analyzing any structure that is wider than the ice thickness, (ii) the (elastic) far-field behaviour of an ice sheet, which can contribute to the dynamic icestructure interaction (iii) the multi-degree-of-freedom nature of certain structures and (iv) dynamic soil-structure interaction, potentially including soil damping and also aerodynamic damping. However, in spite of these concerns, single degree of freedom models are still remarkably useful in light of the insights they offer. In particular for the modelling of ice induced vibrations, an understanding of how the

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ice is modelled for a single DOF system can be transferred into more sophisticated models that incorporated additional features of structure and soil. As previously mentioned, several different model types in the literature and even within each type, different specific models have been proposed. This paper attempts to consider, numerically implement and compare two of these crushingfailure based models in order to further the understanding of this particular type of modelling approach to IIVs. As previously mentioned, the modelling for the ice forces due to ice failing in bending is relatively more straightforward, which is why we examine models for ice failing in crushing. In particular, the models proposed by Toyama et al. [9] and Huang and Liu [10] are reproduced and closely compared to each other. Toyama et al. and Huang and Liu are typical crushing failure models, and they both share the same idea of the division of the forcing function into several phases. Both of these models attempt to represent the crushing failure mode of the ice sheet, which leads to the idea of segmenting the forcing function into several phases. As is shown through simulations, care must be taken with these phasedivision type models to ensure the continuity of the predicted motion. Toyoma’s model is developed and solved in detail and corrections to the original model are suggested in order to correct the discontinuities in the resulting motion that the original model produces. Through analytical and numerical solutions of these models using the same ice and structural properties, the models are compared to investigate differences within the fundamental IIV modelling approach of crushing ice failure. 2. Model by Toyama et al. [9] The paper written by Toyama et al., Model Tests on Ice-induced Self-excited Vibration of Cylindrical Structure [9], is a remarkable achievement in the field of ice-induced vibration (IIV) research. The paper proposed dividing the ice forcing into two stages: elastic and crushing phases. As mentioned in the previous section, this paper attempts to model an ice field that is impacting a vertical structure and is thus assumed to be failing in the crushing mode. The phase division of the ice forcing function was suggested by past research such as in Matlock et al. [2], however Toyama et al. actually developed equations of motion according to the phase division. Ranta and Räty [17] published the paper titled “On the Analytic Solution of Ice-induced Vibrations in a Marine Pile Structure” containing the same idea in the same year, but their paper was more focused on the continuity and

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stability of IIVs. The method of Toyama et al., therefore, is more frequently adopted by other IIV researchers. However, inspite of its pioneering work, the Toyama et al. paper has several parts that require improvement. The Toyama et al. paper derives equations for a one degree-of-freedom system, thus the equation of motion is given as m x + cx + kx = f (t ),

(1)

where m, c, k and f (t ) are a mass, damping coefficient, stiffness and forcing function, respectively. Toyama et al. refer to their model as a self-excited model. Toyama et al. define the ice force as two separate phases. An elastic phase is the period for which the structure and ice move at the same velocity due to negligibly small deformations of the ice. Thus x = z and x = 0, and the elastic ice force from

equation (1) can be written as f (t ) = k ( x ) + cz ,

(2)

where z is the velocity of the ice. The second phase, a crushing phase, starts at the maximum value, Fm , of f (t ) . The crushing force, Fc , is defined to be constant and smaller than Fm . From equation (1), the equation of motion in the crushing phase is given by m x + cx + kx = Fc .

(3)

Both equations (2) and (3) present an ice force that is independent of loading rate, therefore the Toyama et al. The equation of motion for the elastic phase can be obtained by substituting equation (2) into (1) as m x + cx + kx = kx + cz . (4) Since x = 0 during the elastic phase, x = z .

Taking integrals on both sides to obtain the displacement of the system,

â&#x2C6;Ť dt = â&#x2C6;Ť dt dx

which is x(t ) = z (t ) + C1 ,

(5)

where C1 is the initial displacement. The displacements of the structure and ice are

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the same during the elastic phase. When f (t ) reaches Fm , the crushing phase starts. Toyama et al. assume that their model has zero or very small damping, so that the system is an under-damped system. For the crushing phase, equation (3) has the following homogeneous solution xh (t ) = e −ζωnt ( A1 cos ωd t + A2 sin ωd t ),

(6)

where ζ, ωn and ωd are the damping ratio, natural frequency and damped natural frequency, respectively, which are defined as ζ=

c , 2mωn

ωn =

k m

and ωd = ωn 1 − ζ 2 ,

respectively. Since Fc is constant, the particular solution is x p (t ) = C2

(7)

x p = x p = 0.

(8)

and

Substituting equations (7) and (8) into equation (3) kx p = Fc

or F x p = C2 = c . k

(9)

Therefore, the total solution is expressed as x(t ) = xh (t ) + x p (t ) = e − ζωnt ( A1 cos ωd t + A2 sin ωd t ) +

With the following initial conditions [9], x0 =

( Fm − cz ) k

and x 0 = z ,

Fc . k

(10)

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the A1 can be calculated from equation (10) as x(0 ) = A1 +

Fc F − cz = m k k

(11)

or A1 =

1 ( F − Fc − cz ). k m

(12)

The velocity can be obtained by taking the derivative of equation (10) x (t ) = − ζω n e −ζωnt ( A1 cos ωd t + A2 sin ωd t ) + ωd e −ζωnt (− A1 sin ωd t + A2 cos ωd t )

(13)

and substituting the initial condition x (0) = − ζω n A1 + ωd A2 = −

1 ζω n ( f m − f c − cz ) + ωd A2 = z k

(14)

or A2 =

ζω n z ( F − Fc − cz ) + . kωd m ωd

(15)

The solutions of the elastic and crushing phases are thus completely derived. Toyama et al. proposed that IIVs can be described as alternatively repeating solutions of the elastic and crushing phases, corresponding to equations (5) and (10). The unique work done by Toyama et al. is of calculating the duration of each phase. One complete period consists of the elastic period, te , followed by the crushing period, tc , ttotal = te + tc .

(16)

Based on experimental data, Toyama et al. calculated the elastic and crushing periods [9] as te 1−β = tn πα

(17)

tc 1− β⎞ 1 = 1 to tan −1 ⎛⎜ ⎟, tn π ⎝ α ⎠

(18)

and

where α=

kz , f m ωn

(19)

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β=

Fc Fm

(20)

tn =

2π . ωn

(21)

and

Now that the complete solutions and periods of each phase have been obtained, numerical simulations may be performed. Table 1. Parameters for simulations

Parameter

Model structure

h (mm)

D (mm)

m (kg)

1600

c (kNs/m)

0.2

k (MN/m)

A (m)

44.93

σ c (MPa)

1.4

xmax (mm)

2.1. Numerical simulations

The numerical simulations of the Toyama et al. model were performed, based on the parameters presented in Table 1 which is obtained from Kärnä and Turunen [18]. The IIV data of the channel marker in the Baltic were measured by Nordlund et al. during the winter of 1987 to 1988, and the structural properties were specified. Some structural properties not specified on Kärnä and Turunen [18] are derived from the density of steel, ρ steel = 7859 kg m 3, which is assumed to be the material of the channel marker. In Table 1, h is the thickness of the ice sheet, and σ c is the compressive strength of the ice. The initial or maximum ice force is calculated by Korzhavin’s approach Fm = 0.9

σ c Dh h 1+ 5 = 168.03 ( N m ), D A

(22)

where A is the length of the structure [18]. Additionally, two more parameters, Fc and z , need to be introduced. Equations (19) and (20) can only be used when β is

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larger than 0.5, therefore Fc is arbitrarily chosen as 5700 N so that β becomes 0.755 [9]. Since the velocity of ice is not specified or limited by Toyama et al., the velocity of ice is also randomly chosen from 0.001 to 1 m/s. Applying the parameters to equation (19), α is calculated as 5.229. By substituting α and β into equations (17) and (18), the elastic and crushing periods are obtained as te = 0.042 and tc = 1.002 seconds. Equations (6) and (10) can be evaluated over the periods of te and tc .

Figure 1. Calculated IIV based on Toyama et al. [9].

Figure 1 shows the result calculated rigorously based on the Toyama et al. model. Although β, which determines the applicability of the theory, is larger than 0.5, discontinuities appear between the elastic and crushing phases, as well as between total periods. The discontinuities continue to exist for all ranges of ice velocities even at the very low ice velocity of 0.001 m/s. This indicates that the Toyama et al. method cannot model all cases of IIVs. The Toyama et al. model requires improvement to be applicable to more general cases of IIVs. The discontinuities between the phases are closely related to the initial conditions. The amount of displacement at the end of the elastic phase should be equal to the initial displacement of the crushing phase. Thus equation (2) can be written at t = te as

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f (te ) = kxe + cz = k ( x ⋅ te ) + cz = k ( z ⋅ te ) + cz = Fm

(23)

which follows since x = z for the elastic phase. However, equation (23) does not hold for all cases, especially for the given parameters. Therefore, the initial condition for displacement at the beginning of the crushing phase needs to be changed so that it is the same as the displacement at the end of the elastic phase. According to the definition of the elastic phase by Toyama et al., the velocities of the ice and structure are the same during the elastic phase. In other words, the ice travels the same displacement as the structure does. From the definition, a new initial condition for the displacement is presented as x0 = z ⋅ t e + C3 ,

(24)

where C3 is a constant to compensate the initial displacement of the elastic phase.

Figure 2. IIV based on the modified Toyama et al. method.

Figure 2 is the IIV with the modified initial conditions when C3 is set to be zero. The discontinuities between the phases are eliminated but after one complete period, which consists of one elastic phase and one crushing phase, discontinuities still exist. The end of the crushing phase does not match the beginning of the second elastic phase. If the ice velocity decreases to lower than 0.01 m/s, then those discontinuities disappear. This means that the Toyama et al. method is only applicable at very low ice velocities.

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Figure 3. IIV based on the fully modified Toyama et al. method. C3 of equation (24) can be used to fix the gap between the complete periods.

After each complete period, the end displacement is compensated by C3 = x(ttotal ), and the compensated initial condition becomes x0 = z â&#x2039;&#x2026; te + C3 = z â&#x2039;&#x2026; te + x(ttotal ).

(25)

With the compensated initial conditions, equation (10) is solved every complete period. Figure 3 presents three complete periods plotted by the fully modified Toyama et al. model. When the ice velocity is larger than 0.01 m/s, the vibrations grow as time increases. As long as the ice velocity or the ice force remains the same, the vibration will keep increasing or decreasing because the duration of the elastic and crushing phases are fixed by the ice velocity and the ice force. These modifications were introduced in order to eliminate the non-physical discontinuities between the phases, as shown in Figures 1-3. By comparing these figures, the maximum displacement calculated by the original Toyama et al. model is less than 0.5, while the maximum displacement is about 0.6 by the modified Toyama et al. model. Furthermore, the maximum displacements in Figures 1 and 2 are relatively constant with time, while the maximum displacement in Figure 3 with the fully modified Toyama et al. model is not. Clearly there are still some problems with the Toyama model as the original model and modified model suffer from discontinuities while the fully modified model has a maximum displacement that

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varies in time. The nature of these problems seem to stem from the fact that the duration of the elastic and crushing phases are fixed by the ice velocity and ice force, when intuition suggests that the duration of these separate phases will change as the vibrations evolve. In fact, this is what the next model proposes as a modelling approach. 3. Model by Huang and Liu [10]

There are several reasons why the Huang and Liu model [10] is chosen for an in-depth review after reviewing Toyama et al. First, the Huang and Liu paper is more recent and builds on the efforts of previous research. Second, both models by Huang and Liu, as well as and Toyama et al. attempt to model an ice sheet that is failing by crushing. Both these models are applicable to narrow structures that can be simplified as a one degree-of-freedom model. In addition, the model proposed by Huang and Liu is a more advanced phase-division model along the lines of the Toyama et al. model. Huang and Liu divide IIVs into three phases according to the velocity and deflection of a contacting ice sheet while Toyama et al. separate the phases of IIVs into two phases, based on the ice forcing function. Contrary to the Toyama et al. model, which uses fixed phases, each phase of the Huang and Liu model is determined on-the-fly as the evolution of the IIVs is calculated. Finally, Huang and Liu provide detailed parameters of their simulations so that other researchers can easily reproduce their results.

Figure 4. Matlock et al. IIV model (from [1]).

Huang and Liu define ice forces as loading, extrusion, and separation phases. The definitions of the loading and extrusion phases are similar to those of the elastic and crushing phases of Toyama et al. with the exception of the conditions of each phase. Huang and Liu disagree with the idea of Toyama et al. that the structural and

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ice velocities are the same during the elastic phase. Instead, they suggest that both loading and extrusion phases occur when the structural velocity is less than that of the ice. The phase division between the loading and extrusion phases depends on whether the deflection of the ice exceeds the failure deflection, δ f , or not. If the amount of ice deflection is less than the failure deflection, the structure is under the loading phase. The extrusion phase starts once the ice deflection is greater than the failure deflection. The ice deflection is derived directly from Figure 4 [10] as δ = x0 + z t − x(t ) − p(n − 1),

(26)

where x0 is the initial displacement of the ice sheet and n is the number of the tooth in contact. The ice force at the loading phase is expressed as f (t ) = k c δ(t ) + Fe , where Fe is the residual force from the previous crush and kc is the effective contact stiffness. During the extrusion phase, the ice force remains at Fe , which corresponds to the crushing phase of Toyama et al. [9]. The separation phase begins when the structural velocity is greater than the ice velocity, therefore no force is transmitted during the separation phase. Combining the ice forces of the each phase with a one degree-of-freedom system, the equation of motion is given by [10] ⎧k c [ x0 + z t − x − p(n − 1)] + Fe , ⎪⎪ m x + cx + kx = ⎨ Fe , ⎪ ⎪⎩0,

0 ≤ δ < δf

and x ≤ z ,

δ f ≤ δ < p and x ≤ z , (27) x > z .

Huang and Liu normalize equation (27) by using Δ=

F fmax , k

(28)

where F fmax is the maximum failure ice force [10]. x, x0 , p and δ can be normalized by dividing by Δ. Other terms are normalized as ~ kc = kc k ,

~ Fe = Fe F fmax and

~z = z ω Δ n

in which the tildes over the letters indicates that the terms are normalized. Finally,

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normalized time is introduced as

~ t = ωn t .

Figure 5. Ice crushing strength vs. strain rate (from [10]).

Using the normalized factors, equation (27) becomes the non-dimensional equation given by ~ ~ ~ ~ ~ ⎧k c [ ~ x0 + ~z t − ~ x−~ p (n − 1)] + Fe , 0 ≤ δ < δ f and ~ x ≤ ~z , ⎪ ~ ~ ⎪~ x + 2ζ~ δf ≤ δ < ~ p and ~ x ≤ ~z , (29) x + ~ x = ⎨ Fe , ⎪ ~ x > ~z , ⎪⎩0, x and ~ x are second and first derivatives of the normalized displacement, where ~ ~ ~ ~ ~ x = d 2 ~ x d t 2 and ~ x = d ~ x d t [10]. The failure deflection, δ f , is defined as ~ ( z z ) − F~ σ ~ f r t c δ f ( z r z t ) = , (30) ~ kc where z r is the relative velocity between the structure and ice sheet, and z t is the transitional relative ice velocity corresponding to ε t of Figure 5 which is the ~ , is transitional strain rate. In addition, the normalized crushing strength, σ f expressed as

~ in which σ fd ~ by σ . fmax

μ ~ ~ ⎧ ~ ( z z ) = ⎪(1 − σ fd ) ( z r z t ) + σ fd , z r z t ≤ 1 (31) σ ⎨ f r t ~ ) ( z z )ν + σ ~ , 1 < z z ⎪⎩(1 − σ fd r t fd r t ~ again correspond to Figure 5 and are normalized by dividing and σ fb

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Table 2. Normalized parameters for simulations

Parameter

Value

0.1

ζ ~ σ fd ~ σ

0.04

0.5

ν ~ p

–2

~ Fe ~z

0.7 0.5

10 0.2

The closed form solutions of equation (29) are given by Huang and Liu. For the loading phase, ~ ~⎧ F − 2ζ ~ z ⎤ ⎪⎡ ~ ~ ~ ⎥ cos ωl tl x ( tl ) = e − ζ tl ⎨⎢ ~ x0l (1 − k r ) − e 2 ⎥⎦ kq ⎪⎩⎢⎣

1 ωl

~ ⎡ ⎛ ζF 2ζ 2 ⎞⎤ ~ ⎫⎪ ⎢~ x 0l + ζ ~ x0l (1 − k r ) − 2e − k r ~z ⎜1 − 2 ⎟⎥ sin ωl tl ⎬ ⎜ kq k q ⎟⎠⎦⎥ ⎪⎭ ⎣⎢ ⎝

~ ⎛ F ~ 2ζ ~z ⎞ + 2e + k r ⎜ ~ x0l + ~z tl − 2 ⎟ , ⎜ k q ⎟⎠ kq ⎝

(32)

~ ~ ~ where the subscripts 0l mean the initial values of the loading phase and tl = t − t0l ; ~ ~ ~ ωl2 = k q2 − ζ 2 ; k q2 = 1 + k ; and k r = k 1 + k . The solution of the extrusion phase

is given by ~ ~ ~ ~ 1 ~ ~ ~ ~ x ( te ) = e − ζ te ⎧⎨( ~ x0e − Fe ) cos ωe te + [ x0e + ζ( ~x0e − Fe )] sin ωe ~te ⎫⎬ + Fe (33) ω ⎩ ⎭ e

in which the terms with the subscripts 0e are again the initial values of the extrusion ~ ~ ~ phase ωe2 = 1 − ζ 2 , and te = t − t0e . In Huang and Liu [10], the sin term of the ~ ~ extrusion phase is misprinted as [~ x 0e + ζ( ~ x0e − Fe ) sin ωe te ] , and is corrected in

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equation (33). Finally, the separation phase is expressed as ~ 1 ~ ~ ~ ~ x ( ts ) = e − ζ ts ⎡⎢ ~ x0 s cos ωe ts + ( x0 s + ζ ~x0 s ) sin ωe ~ts ⎤⎥ , ω ⎣ ⎦ e

(34)

where the subscripts 0s indicate the initial values of the separation phase, and ~ ~ ~ ts = t − t0s . 3.1. Numerical simulations

Since equation (29) is a nonlinear equation, the numerical simulations are performed with a small time step. Beginning with the initial displacement and velocity, the deflection of the ice, δ, and the relative velocity, ~z , are calculated at every time step to determine which phase the system is in. To know the deflection of the ice and the relative velocity, however, the solutions, equations (32), (33) and (34), must be solved previously. Therefore, the phase conditions and solutions should technically be evaluated simultaneously, but the actual algorithm calculates the phase conditions based on the values from the previous time step because of the nonlinearity and mutual dependency. Once the phase is selected, the initial values of the phase are determined from the end values of the previous phase. The initial values of the phase remain the same until a phase change occurs. With this algorithm, the parameters given in Table 2 are applied to obtain the non-dimensional response of the system and ice force.

Figure 6. Response and ice force of the Huang and Liu model.

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Figure 7. Response and ice force (from [10]). x 0 = 0, and the ice Figure 6 is plotted with the initial conditions ~ x0 = 0 and ~ z t , where ~ z t is again the normalized transitional relative ice velocity ~z = 2.2 ~

velocity. The response of the Huang and Liu model does not display the discontinuity which was observed in the Toyama et al. model. It is evident that all initial values of each phase are properly determined so that the ends of the each phase correspond to the beginning of the next phase. The calculated response, however, is notably different from the response presented in Huang and Liu [10], Figure 7. Although the ice forces of Figures 6 and 7 illustrate similar behaviour, the magnitudes and frequencies of the responses are not identical. There are several possible reasons for this disagreement. The algorithm used to reproduce the Huang and Liu model is very sensitive to changes in the parameters. The precision of simulation programs, therefore, can significantly influence the calculation results. In addition, the algorithm itself may be different from the one used by Huang and Liu, of which no details are given. Since the Huang and Liu model includes nonlinearity, there is the possibility that a small difference in the beginning can lead to a considerable difference in the final stage. 4. Comparison of the Models

Two different modelling approaches of IIVs due to ice failing in crushing have been introduced. These modelling approaches have one common feature which is that they attempt to model the crushing of the ice by a phase-division type of approach to the forcing function. Although expressions vary from model to model, the foundation of the models is a one degree-of-freedom system based on the

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crushing failure model. Therefore, if the same parameters are applied, the predictions of the models can be compared. Two sets of the parameters have been used to run the numerical simulations, Tables 1 and 2. Since the Huang and Liu model uses normalized parameters, Table 2, it should be reversible to the original equation to be applicable the parameters given in Table 1. Huang and Liu, however, do not provide enough information for the original equation, therefore the other model, the Toyama et al. model, is normalized to utilize Table 2. The Toyama et al. model can be normalized by using Δ as for the other model. From the equation of motion of the Toyama et al. model [9], ⎧kx + cz , m x + cx + kx = ⎨ ⎩ Fc ,

0 ≤ t < te , te ≤ t < tc .

(35)

Equation (35) is divided by Δ and m. Fe is added to the elastic force and t e , tc and Fc are replaced with τ, T and Fe , respectively in order to synchronize the ice force

with the ice forces of the other two models. Normalization is performed through the same procedure as the other model and expressed as ~ ~ ~ ⎧⎪k ~ x + 2ζ ~z + Fe , 0 ≤ t < ~τ , ~ ~ ~ (36) x + 2ζ x + x = ⎨ ~ ~ ~τ ≤ ~ ⎪⎩ Fe , t < T. Responses of the non-dimensional Toyama et al. model are obtained from the modified approach which is proposed in the first section to remove discontinuities. For the elastic phase, the response is calculated as ~ ~ ~ x (t ) = ~ z t + C3 . (37)

Figure 8. Response of the Toyama et al. model.

134

SUMIN JEONG and NATALIE BADDOUR The response for the crushing phase is expressed as ~ ~ ~ ~ ~ ~ x ( t ) = e − ζ t [ A3 cos 1 − ζ 2 t + A4 sin 1 − ζ 2 t ] + Fe ,

(38)

C3 , A3 and A4 can be calculated from the initial values of each phase.

The response of the normalize Toyama et al. model corresponding to the same ice force is plotted in Figure 8. By using the same τ, T , Fe and Fm , the ice forcing functions of the models have identical frequency and magnitude. The responses of the two models, however, shown in Figures 6 and 8 completely differ from each other even though the same structural properties are applied, displaying the different models of the ice force built into each model. The Toyama et al. model calculates the magnitude of the response to be much larger than the other two models. The difference in response for the Toyama model compared to the Huang and Liu model is unexpected as both models attempt to represent the crushing failure of the ice field. The Toyama model predicts a response of higher amplitude and lower frequency than the Huang and Liu model. In other words, there are large discrepancies between the predictions of the individual models, in particular since the two models are based on the crushing failure of ice - the same modelling approach for the same physical phenomena. Furthermore, as discussed in the section on the Toyama model, even with the modifications to ensure continuity, there was some difficulty in the variation of the maximum amplitude of the Toyama model perhaps suggesting that the assumption of fixed length phases may be questionable. What this really suggests is that concrete measurement data needs to be made available in the literature so that past and future proposed models can be verified, in particular for models of the crushing-failure type. 5. Conclusions

In this paper, two IIV models of the crushing failure type are solved, simulated and compared in detail. These models have in common that they both consider the ice forcing function to occur in separate phases and both are models of ice failing in crushing by impacting a cylindrical structure. These phases depend on the structure properties and the structure response. The first model, by Toyoma and colleagues, divides the ice forcing function into two phases, elastic and crushing. These phases are taken to be of fixed duration and those authors proposed a way to calculate the duration of each phase. However, upon implementation of this model, it became obvious that this particular model introduces discontinuity into the predicted motion

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of the structure, a clearly non-physical result. To correct this, a modification to this model is proposed in order to ensure that the predicted structural motion remains continuous. However, this modification results in the maximum amplitude of the response varying in time, suggesting that there may still be some problems with the model and perhaps the assumption of phases of constant duration may be questionable. The second model considered in this paper is by Huang and Liu. This model is also a characteristic failure frequency model that considers the ice forcing function to occur in separate phases and is also a crushing-failure model for cylindrical structures. However, in contrast to the Toyama model, the forcing function is separated into three separate phases, loading, extrusion and separation (where the ice loses contact with the structure). Furthermore, these phases are not considered to be of fixed duration but their changing duration is calculated as part of the solution to the problem. Since the phases are no longer of fixed duration, the model becomes essentially nonlinear and a time-stepping approach needs to be taken in order to numerically solve the equation of motion. This model does not suffer from the same discontinuity problem found in the Toyama model and no modifications were necessary from that point of view. Since the Toyama and Huang and Liu model both suggest the idea of phase-division for the ice forcing function and both model crushing failures, it might be reasonable to expect that their models would have similar predictions for the structural response, but this is not the case. With the improved Toyama model, the numerical simulations were conducted with the same parameters and ice force in order to enable a fair comparison with the Huang and Liu. The numerical simulations of both models presented completely different responses although they were simulated using the same parameters and ice force. The Toyama model was the easiest to implement. The Toyama model considers phase division of the ice force but needed modification to ensure a physical result with no discontinuities in the structural response. The Huang and Liu model considers a more complicated phase division and becomes nonlinear, forcing a more complicated numerical approach to the solution of the problem. These observations are made on the implementation and theoretical aspects of these models, yet no experimental data were found in the open literature that could be used to compare the accuracy of these models in predicting the real behaviour of the system. What has been demonstrated here is that the various theoretically proposed models do not predict the same behaviour, even when the same parameters have

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been used. This strongly suggests that the development of theoretical models of IIVs should be made along with experimental studies. Theoretical models of IIVs without comparing the actual IIV data cannot predict the real physics of IIVs. However, it would appear that no major experimental, widely disseminated IIV research has been conducted since the early stages of IIV research. This would also point to the need for another perspective on IIVs in order to solve the inconsistency of the existing IIV models which are based on the crushing failure. Furthermore, it would be most helpful to the modelling community if any experimental data on ice inducted vibrations could be made widely and freely available in order to help with the development of accurate theoretical models. References [1]

D. S. Sodhi, Ice-induced vibrations of structures, Special Report 89-5, International Association for Hydraulic Research Working Group on Ice Forces, 1989.

[2]

G. Matlock, W. Dawkins and J. Panak, Analytical model for ice-structure interaction, J. Eng. Mech. Div (1971), 1083-1092.

[3]

K. A. Blenkarn, Measurement and analysis of ice forces on cook inlet structures, Proceedings of Offshore Technology Conference, OTC 1261, 1970, pp. 365-378.

[4]

D. S. Sodhi and C. E. Morris, Characteristic frequency of force variations in continuous crushing of sheet ice against rigid cylindrical structures, Proceedings of Cold Regions Science and Technology 12 (1986), 1-12.

[5]

J. Xu and L. Wang, Ice force oscillator model and its numerical solutions, Proceedings of 7th International Conference on Offshore Mechanics and Arctic Engineering, 1988, pp. 171-176.

[6]

M. Määttänen, Ice-induced vibrations of structures self-excitation, Special Report 895, International Association for Hydraulic Research Working Group on Ice Forces, 1989.

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A. Barker, G. Timco, H. Gravesen and P. Vølund, Ice loading on Danish wind turbines, Part 1: Dynamic model tests, Proceedings of Cold Regions Science and Technology 41 (2005), 1-23.

[8]

Gregory Tsinker, Marine Structures Engineering, 1st ed., Chapman-Hall, New York, 1995.

[9]

Y. Toyama, T. Sensu, M. Minami and N. Yashima, Model tests on ice-iceduced selfexcited vibration of cylindrical structures, Proceedings of 7th International Conference on Port and Ocean Engineering under Arctic Conditions 2 (1983), 834-844.

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[10]

G. Huang and P. Liu, A dynamic model for ice-induced vibration of structures, J. Offshore Mech. Arctic Eng. 131(1) (2009), 011501-1-011501-6.

[11]

R. Frederking and J. Schwartz, Model test of ice forces on fixed and oscillating cones, Cold Regions Science and Technology 6 (1982), 61-72.

[12]

D. S. Sodhi, C. E. Morris and G. F. N. Cox, Dynamic analysis of failure modes on ice sheets encountering sloping structures, Proceedings of the 6th International Conference on Offshore Mechanics and Arctic Engineering 4 (1987), 281-284.

[13]

I. Hirayama and K. Obara, Ice forces on inclined structures, Proceedings of the 5th International Offshore Mechanics and Arctic Engineering (1986), 515-520.

[14]

Q. Yue and X. Bi, Full-scale test and analysis of dynamic interaction between ice sheet and conical structure, Proceedings of 14th International Association for Hydraulic Research (IAHR) Symposium on Ice, 2, 1998.

[15]

Q. Yue and X. Bi, Ice induced jacket structure vibration, J. Cold Regions Eng. 14 (2000), 81-92.

[16]

Y. Qu, Q. Yue, X. Bi and T. Kärnä, A random ice force model for narrow conical structures, Cold Regions Sci. Tech. 45 (2006), 148-157.

[17]

M. Ranta and R. Räty, On the analytic solution of ice-induced vibrations in a marine pile structure, Proceedings of 7th International Conference on Port and Ocean Engineering under Arctic Conditions 2 (1983), 901-908.

[18]

T. Kärnä and R. Turunen, Dynamic response of narrow structures to ice crushing, Proceedings of Cold Regions Science and Technology 17 (1989), 173-187.