Dynamic Response Analysis of Composite Tether Structure to Airborne Wind Energy System Under Impulsive Load

applied sciences

Article Dynamic Response Analysis of Composite Tether Structure to Airborne Wind Energy System under Impulsive Load

Kwangtae Ha

Department of Floating Offshore Wind Energy System, University of Ulsan, Ulsan 680-749, Korea; [email protected]; Tel.: +82-52-259-2694

Abstract: The tether structure plays the role of transferring the traction force of an airborne wind energy system (AWES) to the ﬁxed or mobile ground system with less motion and maintains the ﬂying airborne system as a critical component. The implementation of a geometrically tailored tether design in an AWES could avoid unwanted snap-through failure, which can be take place in a conventional tether structure under impulsive loading. This concept relies on the redundant load path of the composite structure composed of tailored length and strength. In this study, the dynamic response of this composite tether structure to airborne wind energy systems, such as a kite wind power system, was analytically investigated. Also, for very long tether applications, an approximate model of the tether response was developed, which resulted in a dramatic reduction of computational efforts while preserving the accuracy quite well compared to the exact solution.

Keywords: tether; failure analysis; airborne wind energy system (AWES); failure analysis; composite

1. Introduction Wind energy systems have gained much attention for their ability to provide clean and environmentally friendly energy resources in contrast to fossil fuels [1,2]. Recently, the geometric dimensions and capacity rate of wind turbines have been getting larger in Citation: Ha, K. Dynamic Response order to meet the required reduction of LCOE (levelized cost of energy). This became Analysis of Composite Tether possible due to the evolution of technologies, such as materials, structural designs, systems, Structure to Airborne Wind Energy aerodynamics, and so on. It is possible to capture more energy from incoming wind System under Impulsive Load. Appl. Sci. 2021, 11, 166. with an increased rotor size since power production is proportional to the square of the http://doi.org/10.3390/app11010166 rotor diameter [3]. Furthermore, the paradigm shift to offshore wind from onshore wind has been accelerating the growth of wind turbine size and capacity due to steadier and Received: 16 November 2020 more stable wind resources, not to mention having the advantages of being far away Accepted: 23 December 2020 from artificial obstacles and potential noise issues [4,5]. However, the CAPEX (capital Published: 26 December 2020 expenditure) of huge wind turbine structures, which includes the material cost, installation cost, transportation cost, and operational and maintenance costs, is becoming increasingly Publisher’s Note: MDPI stays neu- unfavorable. Therefore, the sizes of wind turbines are now approaching an economically tral with regard to jurisdictional clai- feasible limit under current technologies [6]. ms in published maps and institutio- There have been many innovative concepts developed to harvest wind energy effi- nal affiliations. ciently with a fraction of the traditional wind turbine cost. As an emerging technology, airborne wind energy systems (AWES) utilize steadier and stronger wind resource resulting from the high altitude [7–11]. Figure1 shows two different concepts of AWES, Ground-Gen systems, and Fly-Gen systems [12]. Copyright: © 2020 by the author. Li- censee MDPI, Basel, Switzerland. As shown in Figure1, both airborne wind energy systems use a tether structure to This article is an open access article connect ground systems (mobile or fixed) to flying devices without a tower to capture wind distributed under the terms and con- energy at high altitudes above those of traditional tower-based wind turbines. By conditions of the Creative Commons At- trolling tether length, the harvesting height can be adjusted to maximize wind resource tribution (CC BY) license (https:// availability [13]. That is, the tether structure plays the role of transferring the traction force creativecommons.org/licenses/by/ of the AWES to the fixed or mobile ground system with less motion. Therefore, its design 4.0/). life should be able to operate under the cyclic load caused by the repeated reeling motion

Appl. Sci. 2021, 11, 166. https://doi.org/10.3390/app11010166 https://www.mdpi.com/journal/applsci Appl. Sci. 2021, 11, x FOR PEER REVIEW 2 of 12

controlling tether length, the harvesting height can be adjusted to maximize wind resource availability [13]. That is, the tether structure plays the role of transferring the traction force of the AWES to the fixed or mobile ground system with less motion. Therefore, Appl. Sci. 2021, 11, 166 its design life should be able to operate under the cyclic load caused by the repeated2 ofreel- 12 ing motion and to endure the specific breaking strength limit entailed by a certain form of snap loading caused by unexpected impulsive forces [14]. Since the tether is an extremely andcritical to endure component, the speciﬁc it should breaking be designed strength limitaccording entailed to bya fail-safe a certain rule form [15]. of snap To avoid loading the causedaccidental by unexpected failure of the impulsive tether under forces [unexpected14]. Since the snap tether loading, is an extremely the tether critical should compo- be de- nent,signed it should to increase be designed the capacity according of energy to a fail-safedissipation rule [15]. To avoid the accidental failure of the tether under unexpected snap loading, the tether should be designed to increase the capacity of energy dissipation.

(a) Ground-Gen system (b) Fly-Gen system

FigureFigure 1. 1.Examples Examples of of airborne airborne wind wind energy energy systems systems (AWES) (AWES) [12 [12].].

InIn this this paper, paper, the the AWES AWES is simply is simply modeled mode asled a two-pointas a two-point mass structuremass structure with a tetherwith a connection,tether connection, where thewhere small the masssmall representsmass represents a flying a flying airborne airborne system system such such as a as kite a kite or flyingor flying aircraft, aircraft, and and the the other other big big mass mass corresponds corresponds to to a grounda ground station station with with little little move- move- ment.ment. The The benefit benefit of of this this tailoring tailoring concept, concept, applicable applicable to to the the airborne airborne wind wind turbine turbine system system underunder impulsive impulsive tensile tensile loading, loading, was was analytically analytically investigatedinvestigated inin aa previousprevious studystudy [16[16].]. Often,Often, there there are are cases cases in in which which the the tether tether length length can can increase increase to to up up to to 1000 1000 m m to to access access windwind resources resources at at extremely extremely high high altitudes altitudes above above 500 500 m. m. In In this this work, work, an an approximate approximate techniquetechnique for for such such a a long long tether tether was was developed developed in in order order to to reduce reduce the the computational computational costs costs dramaticallydramatically whilewhile providingproviding goodgood accuracy,accuracy, even even when when compared compared to to the the original original full full modeling.modeling. TheThe approximated tether tether modeling modeling de developedveloped in in this this paper paper could could be applicable be appli- cableto the to mooring the mooring systems systems of floating of floating wind windturbine turbine systems. systems. The efficient The efficient and light and design light designwill bepossible will bepossible by applying by applying optimization optimization techniques techniques such as suchgenetic as geneticalgorithms algorithms or gradi- orent-based gradient-based optimization optimization to maximize to maximize the energy the energy capacity capacity or to find or to the find advanced the advanced tether tethergeometry geometry [17–21] [17 –21].

2.2. Concept Concept of of a a Tailored Tailored Tether Tether TheThe original original concept concept of of the the tether tether to to be be used used for for AWES AWES application application was was addressed addressed in in detail in [16,22], where a one-dimensional flexible composite was proposed. The tether detail in [16,22], where a one-dimensional flexible composite was proposed. The tether structure consisted of a matrix and high-strength fibers and was designedwith supplemen- structure consisted of a matrix and high-strength fibers and was designedwith supple- tary and additional load paths in case of sequential failures. Figure2a shows the schematic Appl. Sci. 2021, 11, x FOR PEER REVIEW mentary and additional load paths in case of sequential failures. Figure3 of 12 2a shows the sche- concept modeled in drawing software and Figure2b shows the physically manufactured matic concept modeled in drawing software and Figure 2b shows the physically manu- demonstration of the tether made of the elastomer matrix and high-strength glass fiber. factured demonstration of the tether made of the elastomer matrix and high-strength glass fiber.

x2 O

(a) (b)

FigureFigure 2. 2. AA schematic schematic concept concept of of a a tether tether structure structure ( (aa)) and and its its physically physically manufactured manufactured example example (b). (b).

A tailored structural member was conceptually designed to generate a yield-type response with applicability to the arrest of a moving body. The structural member comprised a primary load path, called the primary element, connected in parallel to a longer secondary load path, called the secondary element, at a pair of common nodes. Both primary and secondary elements were combined to form the connector element, which had the sum of the cross-sectional areas of both elements. When external loads were applied to the system in the longitudinal direction, the main elements and connectors were subjected to the tension loads. If all the main elements had the same geometric and material distributions, the failure probabilities of the main elements were the same. In reality, however, there exists an unpredictable progressive failure sequence starting from the main elements, meaning that the secondary elements break after all the main elements break. Figure 3 shows the progressive failure mechanism of the tether structure with ten main elements. As shown in Figure 3, complete failure occurs after the failure of all the main elements and the ensuing failure of the one of secondary elements. Figure 3 also shows the traditional tether structure made of a single element with constant cross-sectional areas along the longitudinal direction with the same total length. The failure responses of both traditional and tailored tether structures are shown in Figure 3 as straight line and zigzag-type (yield-type) responses, respectively. In Figure 3, the areas under the traditional and tailored tether structures represent the mechanical work (force times displacements) required to reach complete failure for each structure. They show that more energy is required to completely break the tailored tether structure compared to the traditional structure because of the larger area (Ats) compared to the area (Aus) of the untailored traditional tether structure.

Figure 3. Comparison of response prediction.

Appl. Sci. 2021, 11, x FOR PEER REVIEW 3 of 12

x2 O

(a) (b) Figure 2. A schematic concept of a tether structure (a) and its physically manufactured example (b).

A tailored structural member was conceptually designed to generate a yield-type response with applicability to the arrest of a moving body. The structural member com-

Appl. Sci. 2021, 11, 166 prised a primary load path, called the primary element, connected in3 ofparallel 12 to a longer secondary load path, called the secondary element, at a pair of common nodes. Both primary and secondary elements were combined to form the connector element, which had A tailored structural member was conceptually designed to generate a yield-type re- the sum of the cross-sectional areas of both elements. When external loads were applied sponse with applicability to the arrest of a moving body. The structural member comprised ato primary the system load path, in calledthe longitudinal the primary element, direction, connected the inmain parallel elements to a longer and secondary connectors were sub- loadjected path, to called the tension the secondary loads. element, If all the at amain pair ofelements common had nodes. the Both same primary geometric and and material secondarydistributions, elements the were failure combined probabilities to form the of connector the main element, elements which were had the the sumsame. of In reality, how- theever, cross-sectional there exists areas an of unpredictable both elements. When progressiv external loadse failure were appliedsequence to the starting system from the main in the longitudinal direction, the main elements and connectors were subjected to the tensionelements, loads. meaning If all the main that elements the secondary had the same elements geometric break and materialafter all distributions, the main elements break. theFigure failure 3 probabilities shows the of progressive the main elements failure were mechan the same.ism In reality, of the however, tether there structure exists with ten main anelements. unpredictable As shown progressive in Figure failure sequence 3, complete starting fail fromure the occurs main elements,after the meaning failure of all the main thatelements the secondary and the elements ensuing break failure after all of the the main one elements of secondary break. Figure elements.3 shows Figure the 3 also shows progressive failure mechanism of the tether structure with ten main elements. As shown in Figurethe traditional3, complete failuretether occurs structure after the made failure of ofa allsing thele main element elements with and constant the ensuing cross-sectional ar- failureeas along of the onethe oflongitudinal secondary elements. direction Figure with3 also the shows same the traditionaltotal length. tether The structure failure responses of madeboth of traditional a single element and withtailored constant tether cross-sectional structures areas are along shown the longitudinalin Figure 3 direc- as straight line and tionzigzag-type with the same (yield-type) total length. Theresponses, failure responses respective of bothly. traditionalIn Figure and 3, tailored the areas tether under the tradi- structurestional and are tailored shown in tether Figure3 structures as straight linerepresen and zigzag-typet the mechanical (yield-type) work responses, (force times displace- respectively. In Figure3, the areas under the traditional and tailored tether structures representments) required the mechanical to reach work complete (force times failure displacements) for each requiredstructure. to reachThey completeshow that more energy failureis required for each to structure. completely They show break that the more tailored energy istether required structure to completely compared break the to the traditional tailoredstructure tether because structure of compared the larger to the area traditional (Ats) co structurempared because to the of area the larger (Aus) area of the untailored (Ats)traditional compared tether to the structure. area (Aus) of the untailored traditional tether structure.

FigureFigure 3. Comparison3. Comparison of response of response prediction. prediction.

3. Application of a Tailored Tether to AWES In this work, a tailored tether structure with characteristics of progressive failure (or yield-type failure response) was applied for the purpose of arresting an airborne wind energy system under snap loading; the analytical modeling of the failure response is described in this section.

3.1. Simpliﬁcation of an AWES with a Tailored Structure The AWES model was simpliﬁed as a two-degree-of-freedom system with two longi- tudinally moving masses, M and m, and two masses connected by a tether, as shown in Figure4. Appl. Sci. 2021, 11, x FOR PEER REVIEW 4 of 12

3.1. Simplification of an AWES with a Tailored Structure The AWES model was simplified as a two-degree-of-freedom system with two lon- gitudinally moving masses, M and m, and two masses connected by a tether, as shown in Figure 4. The displacement and velocity components of an airborne system were represented as x (lower case) and v (lower case) and the station components as X (upper case) and V (upper case). In this study, only the longitudinal motion was considered. When the distance between the two masses is less than the total tether length, the effect of the external load on the structure is negligible. Otherwise—that is, when the distance between the two mass systems is greater than the tether length—the tether structure is subjected to impul- Appl. Sci. 2021, 11, 166 4 of 12 sive (or snap) loading. The response of the tailored tether structure under impulsive loading was analytically modeled as described below.

Figure 4. SimpliﬁedSimplified AWES modelmodel connectedconnected withwith aa compositecomposite tether.tether.

3.2. DynamicThe displacement Response Mode andl velocity of Simplified components AWES Model of an airborne system were represented as x (lowerBased on case) the andsimplified v (lower model case) shown and the in stationFigure 4, components the motion asof Xthe (upper airborne case) system and V(m) (upper can be case). expressed In this by study, Equation only (1) the [23]. longitudinal motion was considered. When the distance between the two masses is less than the total tether length, the effect of the external 𝑚𝑥 =𝐹 −𝑃(𝛿)=𝐹load on −𝐾 the𝛿 structure − (𝑖 is − 1)(𝑙 negligible. −𝑙)] = 𝐹 Otherwise—that−𝐾𝑢 −𝑢 −(𝑖−1)(𝑙 is, when −𝑙 the)] distance between the (1) = 𝐹 −𝐾𝑥−𝑋−𝐿−(𝑖−1)(𝑙two mass systems is greater−𝑙)] = than 𝐹 −𝐾 the(𝑥 tether − 𝑋 length—the −𝐿) tether structure is subjected to impulsive (or snap) loading. The response of the tailored tether structure under impulsive loadingwhere P was is analytically the internal modeled load, δ asis describedthe end displacement, below. p is the primary element, s is

the secondary element, i is the failure stage, L i is the total tether length at the ith stage, 3.2. Dynamic Response Model of Simpliﬁed AWES Model 𝐿 is the tether length of the primary segment, 𝐿 is the tether length of the primary segment,Based 𝐾 is on the the equivalent simpliﬁed tether model stiffness shown inat Figurethe ith 4stage,, the motion 𝑢 is the of thedisplacement airborne system of the (m) can be expressed by Equation (1) [23]. .. mass m, and 𝑢 is the displacement of the mass M. mx = F − P(δ) = F − K δ − (i − 1)l − l = F − K u − u − (i − 1)l − l 1 The1 equationi of motions for pthe station1 systi em1 can2 also be similarlys p expressed as (1) = − − − − ( − ) − = − ( − − ) F1 Ki x X L i 1 𝑀𝑋ls =−𝐹lp +𝑃(𝛿)=−𝐹F1 Ki x +𝐾X (𝑥−𝑋−𝐿Li ) (2) where P is the internal load, δ is the end displacement, p is the primary element, s is the secondaryAt the ith element, stage, thei is global the failure motion stage, of theLi is two- thedegree-of-freedom total tether length at system the ith is stage, givenLp byis thefollowing tether lengthmatrix ofform. the primary segment, Ls is the tether length of the primary segment, Ki is the equivalent tether stiffness at the ith stage, u1 is the displacement of the mass m, and u2 is the displacement of the mass M. The equation of motion for the station system can also be similarly expressed as

.. MX = −F2 + P(δ) = −F2 + Ki(x − X − Li) (2)

At the ith stage, the global motion of the two-degree-of-freedom system is given by following matrix form.

( .. ) m0 x K − K x F + K L .. + i i = 1 i i 0M X −KiKi X −F2 − Ki Li (3) s u 1 ≤ i ≤ (n + 1), (x − X) ∈ L + δi , L + δi Therefore, the general solutions are

xi(t) = qmi(t)+ qMi(t) 1 F1−F2 02 0 0 0 = 2 m+M t − τi + Ai t − τi + Bi + Cicosωi t − τi + Disinωi t − τi 1 1 m + 2 Ki Li + m F1 + F2 mωi 1+ M M (4) m 1 F1−F2 0 2 0 m 0 m 0 Xi(t) = qmi(t) − M qMi(t) = 2 m+M (t − τi ) + Ai t − τi + Bi − M Ci cos ωi t − τi − M Di sin ωi t − τi 1 1 m − 2 Ki Li + m F1 + F2 Mωi 1+ M M Appl. Sci. 2021, 11, 166 5 of 12

where, qmi(t) is the generalized coordinate of the mass m at the ith stage of the speciﬁc time t, qMi(t) is the generalized coordinate of the mass M at the ith stage of the speciﬁc q Ki(m+M) time t, and ωi = ± mM . At the ith stage, the initial conditions are

. 0 0 . 0 0 0 0 0 0 xi τi = xi , xi τi = vi , Xi τi = Xi , Xi τi = Vi (5)

Given the initial conditions from Equation (5), the coefﬁcients of Equation (4) are the following.

0 0 0 0 2 0 0 0 0 mvi + MV mx + MX Ki Li(m + M) − mMωi x − X + MF1 + mF2 M v − V A = i , B = i i , C = − i i , D = i i (6) i + i + i 2 i ( + ) m M m M m(m + M)ωi m m M ωi The maximum elongation at the ith stage is

max δi (t) = xi(t) − Xi(t) − Li q m m 2 m 2 0 F1+ M F2 (7) = Ci 1 + M + Di 1 + M sin ωi t − τi + αi + m Ki(1+ M )

Ci max Pp 1 ≤ i ≤ n where, αi = arctan D and Fi = . i Ps i = (n + 1) 0 f The above equation is valid in the interval τi ≤ t ≤ τi , where the ﬁnal time is given by

 Fmax i m  F + F2  K 1 1 M 1  i K 1+ m C f = i M − i τi arcsin q 2 2 arctan (8) ω C + m + D + m D i  ( i(1 M )) ( i(1 M )) i 

u s Subsequently, if δi > δi+1, the following equations of motion are obtained:

 2 .. )  F1 f f ( u s ) mxi = F1  xei = 2m t − τi + ai t − τi + bi (x − X) ∈ L + δ , L + δ .. ⇒ i i+1 2 , f f (9) MXi = −F2  F2 f f applicableforτi < t ≤ τ  Xei(t) = − 2M t − τi + ci t − τi + di ei

with the initial conditions:

f f f . f . f xei τi = xi τi = aiτi + bi, xei τi = xi τi = ai . f f f f . f (10) Xei τi = Xi τi = ciτi + di, Xei τi = Xi τi = ci

Therefore, the solution to Equation (9) is given by

. F1 f 2 f f f xei(t) = 2m (t − τi ) + xi τi t − τi + xi τi . (11) F1 f 2 f f f Xei(t) = − 2M (t − τi ) + Xi τi t − τi + Xi τi

f f f s where τei is the solution to xei τei − Xei τei = L + δi+1 r . . . 2 . f f f f F1 F2 − x τ − X τ + (xi τ − Xi τ ) + 2 + δxi f f i i i i i i m M τ = τ + + (12) ei i F1 F2 F1 F2 m + M m + M

f f where, δxi = Li+1 − (xi(τi ) − Xi(τi )). Appl. Sci. 2021, 11, x FOR PEER REVIEW 6 of 12

with the initial conditions: 𝑥(𝜏 )=𝑥(𝜏 )=𝑎𝜏 +𝑏,𝑥(𝜏 )=𝑥(𝜏 )=𝑎 (10) 𝑋(𝜏 )=𝑋(𝜏 )=𝑐𝜏 +𝑑,𝑋(𝜏 )=𝑋(𝜏 )=𝑐 Therefore, the solution to Equation (9) is given by

𝐹 𝑥 (𝑡) = (𝑡 − 𝜏 ) +𝑥 (𝜏 )(𝑡 − 𝜏 )+𝑥(𝜏) 2𝑚 (1 𝐹 𝑋 (𝑡) = − (𝑡 − 𝜏 ) +𝑋 (𝜏)(𝑡 − 𝜏)+𝑋(𝜏) 2𝑀

where 𝜏̃ is the solution to 𝑥(𝜏̃ )−𝑋(𝜏̃ )=𝐿+𝛿

𝐹 𝐹 (𝑥 (𝜏 )−𝑋 (𝜏 )) +2 + 𝛿𝑥 −(𝑥 (𝜏 )−𝑋 (𝜏 )) + 𝑚 𝑀 (122 𝜏̃ =𝜏 + + 𝐹 𝐹 𝐹 𝐹 + + ) 𝑚 𝑀 𝑚 𝑀 Appl. Sci. 2021, 11, 166 δττxL=−(( xff ) − X ( )) 6 of 12 where, i i+1 ii ii .

3.3.3.3. Approximate Approximate Response Response Modeling Modeling of ofSimplified Simpliﬁed AWES AWES InIn the the previous previous section, section, the the computation computation wa wass executed executed whenever whenever any any elements elements failed, failed, andand the the failure failure time time and and criteria criteria were were calculat calculateded at each at eachstep. For step. a very For long a very tether, long how- tether, ever,however, this requires this requires expensive expensive computational computational efforts. efforts. This This section section describes describes the theapproxi- approx- mateimate modeling modeling of an of AWES an AWES following following the hypothesis the hypothesis that thatthe mechanical the mechanical work work of an ofap- an proximateapproximate response response model model should should be close be closeto one to of one the of original the original models, models, as shown as shown in Fig- in ureFigure 5. 5.

Kd F1

Stage III

K1 Stage II Stage I

δ δ δ 3 1 2 FigureFigure 5. 5.ApproximationApproximation technique technique for for tether tether response response of ofan an AWES. AWES.

ForFor an an approximate approximate response response model, model, the the parame parametersters at ateach each point, point, such such asas stiffness, stiffness, force,force, and and elongation, elongation, should should be be obtained. obtained. The The stiffnesses stiffnesses of ofthe the tether tether at atthe the first ﬁrst stage stage (I) (I) andand the the last last stage stage (III) (III) are are given given by: by: 1 1 = 𝐾 = , 𝐾 = = K1 1 1 1 n−1 n−2 1 , K3 2 n n−1 (13)(13) + + + + + + + kcts ks kcps kp kcpp kctp kcts ks kcps

Fromthe approximate force-displacement relationship, the corresponding load values are expressed as: F1 = K1δ1, F2 = K2 δ2 − (n + 1 − 1) ls − lp , F3 = K3 δ3 − (n + 1 − 1) ls − lp (14)

Next, for the numerical computation, the responses are considered in detail for each stage. Response at stage I The equations of motion can be obtained according to the Euler–Lagrange equation ∂ ∂T ∂V t . + q = Qi. Equation (15) expresses the kinetic and potential energy and the ﬁnal ∂ ∂qi ∂ i equations are expressed in Equation (16) in matrix form.

. 2 . 2 T = 1 mx + 1 MX 2 1 2 1 (15) 1 2 1 2 V = 2 K1δ = 2 K1(x1 − X1 − L) ( .. ) m 0 x1 K −K x K L .. + 1 1 1 = 1 (16) 0 M X1 −K1 K1 X1 −K1L Next, the forced vibration response of the tether at the ﬁrst stage can be obtained as follows. 1 K1 L x1(t) = qm(t) + qM(t) = A1(t − τ0) + B1 + C1cosω1(t − τ0) + D1sinω1(t − τ0) + m 2 ω1 (17) m m m 1 K1 L X1(t) = qm(t) − qM(t) = A1(t − τ0) + B1 − C1cosω1(t − τ0) − D1sinω1(t − τ0) − 2 M M M M ω1 Appl. Sci. 2021, 11, x FOR PEER REVIEW 7 of 12

Fromthe approximate force-displacement relationship, the corresponding load values are expressed as: (144 𝐹 =𝐾𝛿 , 𝐹 =𝐾 𝛿 − (𝑛+1−1)𝑙 −𝑙 , 𝐹 =𝐾(𝛿 − (𝑛 + 1 − 1)(𝑙 −𝑙 )) ) Next, for the numerical computation, the responses are considered in detail for each stage. Response at stage I The equations of motion can be obtained according to the Euler–Lagrange equation + =𝑄. Equation (15) expresses the kinetic and potential energy and the final equations are expressed in Equation (16) in matrix form. 1 1 𝑇= 𝑚𝑥 + 𝑀𝑋 2 2 (15) 1 1 𝑉= 𝐾 𝛿 = 𝐾 (𝑥 −𝑋 −𝐿) 2 2

𝑚0𝑥 𝐾 −𝐾 𝑥 𝐾 𝐿 + = (16) 0𝑀𝑋 −𝐾 𝐾 𝑋 −𝐾𝐿 Next, the forced vibration response of the tether at the first stage can be obtained as follows.

1 𝐾𝐿 𝑥(𝑡) = 𝑞 (𝑡) + 𝑞 (𝑡) = 𝐴(𝑡 − 𝜏)+𝐵 +𝐶𝑐𝑜𝑠𝜔(𝑡 − 𝜏)+𝐷𝑠𝑖𝑛𝜔(𝑡 − 𝜏)+ 𝑚 𝜔 (17) 𝑋(𝑡) = 𝑞 (𝑡) − 𝑞(𝑡) = 𝐴(𝑡 − 𝜏)+𝐵 − 𝐶𝑐𝑜𝑠𝜔(𝑡 − 𝜏)− 𝐷𝑠𝑖𝑛𝜔(𝑡 − 𝜏)− Appl. Sci. 2021, 11, 166 7 of 12 where 𝜔 = 1 + By applying2 K1 the initialm conditions 𝑥(𝜏 ) =𝑥 , 𝑥 (𝜏 ) =𝑣 ,𝑋(𝜏)=𝑋, 𝑋 (𝜏 )=𝑉 to where ω1 = m 1 + M . Equation (17), we can obtain the coefficients. . . By applying the initial conditions x(τ0) = x0, x(τ0) = v0, X(τ0) = X0, X(τ0) = V0 to 𝑚𝑣 +𝑀𝑉Equation𝑚𝑥 (17), +𝑀𝑋 we can obtain𝐾 the𝐿(𝑚+𝑀 coefﬁcients.) −𝑚𝑀𝜔 (𝑥 −𝑋) 𝑀(𝑣 −𝑉) 𝐴 = , 𝐵 = , 𝐶 =− 2 , 𝐷 = (18) mv0 + MV0 mx0 + MX0 K1L(m + M) − mMω(x0 − X0) M(v0 − V0) = 𝑚+𝑀 = 𝑚+𝑀 = − 𝑚(𝑚+𝑀)𝜔 =(𝑚 + 𝑀)𝜔 A1 , B1 , C1 2 , D1 (18) m + M m + M m(m + M)ω (m + M)ω1 Response at stage II 1 TheResponse tether atresponse stage II at stage II is shown in Figure 6. The tether response at stage II is shown in Figure6.

FigureFigure 6. 6.ForceForce and and displacement displacement plot plot for for the the first ﬁrst and and second second stages. stages.

InIn Figure Figure 6,6 ,

1 F − F = = − 2 1 = ( − ) − K2 2 N N−1 , Kd , F2 K2 δ2 N ls lp (19) + + δ2 − δ1 Kcts Ks Kcss

Since the response of the second stage (II) is inelastic, the virtual work resulting from nonconservative force needs to be plugged into the Lagrange equation.

→ → → → δW = Fext i × δ(∆) = −[F1 − Kd(∆ − δ1)] i × δ(∆) = −[F1 − Kd(x − X − L − δ1)] i × (δx2 − δX2) i (20) = −[F1 − Kd(x2 − X2 − L − δ1)]δx2 + [F1 − Kd(x2 − X2 − L − δ1)]δX2 = Q1δx2 + Q2δX2

∂ ∂T ∂V Based on the rule of Euler–Lagrange’s equation, t . + q = Qi, the equations of ∂ ∂qi ∂ i motion are given by:

( .. ) m 0 x2 K −K x −F − K (L + δ ) .. − d d 2 = 1 d 1 (21) 0 M X2 −Kd Kd X2 F1 + Kd(L + δ1)

Next, the forced vibration response can be obtained as follows:

1 1 x2(t) = qm(t) + qM(t) = A2 t − τf 1 + B2 + C2Exp ω2 t − τf 1 + D2Exp −ω2 t − τf 1 2 [F1 + Kd(δ1 + L)] m ω2 (22) m m m m 1 1 X2(t) = qm(t) − qM(t) = A2 t − τf 1 + B2 − C2Exp ω2 t − τf 1 − D2Exp −ω2 t − τf 1 − 2 [F1 + Kd(δ1 + L)] M M M M m ω2

2 Kd m where ω2 = m 1 + M (14). Appl. Sci. 2021, 11, 166 8 of 12

. By applying the initial conditions x2 τf 1 = x f 1, x2 τf 1 = v f 1, X2 τf 1 = . X f 1, X2 τf 1 = Vf 1 to Equation (22), we can obtain the coefﬁcients:

mv f 1+MVf 1 mx f 1+MX f 1 A2 = , B2 = , " h m+M m+M i # (m + M)F1 + mMω2 −ω2 x f 1 − X f 1 − v f 1 − Vf 1 2 C2 = − / m(m + M)ω1 +(m + M)Kd(L + δ1) " h i # (m + M)F1 + mMω2 −ω2 x f 1 − X f 1 + v f 1 − Vf 1 2 D2 = − / m(m + M)ω1 +(m + M)Kd(L + δ1) (23) Tether status check at stage II The tether failure can be checked according to the work–energy theorem.

−W = (Total Area) = A126 + A2346 − A345 ∗ ∗ (24) = 1 F δ + R δ [F − K (δ − δ )]dδ − R δ [F∗ + K (δ − δ∗)]dδ 2 1 1 δ1 1 d 1 d 1

∗ where δ∗ = x (t∗) − X (t∗) − L, F∗ = F − K (δ∗ − δ ), andd = F + δ∗ 2 2 1 d 1 K1 The kinetic energy variation can be expressed as:

∗ 1 2 2 1 2 2 ∆T = T − T = m v∗ + V∗ − m v + V (25) 0 2 2 0 0 In cases in which the kinetic energy variation of the moving ﬂying object (m) is higher than the amount of work produced by the external force and elongation (δ2), all the main elements will fail. Response at the second stage (II) when the failure of primary elements occurs The corresponding response at the second stage when some of the primary elements fail is as follows:

x3(t) = qm(t)+ qM(t) = A3 t − τf 2 + B3 + C3cosω3 t − τf 2 + D3sinω3 t − τf 2 1 ∗ ∗ + 2 [−F + K1(δ + L)] mω3 m (26) X3(t) = qm(t)− M qM(t) m m = A3 t − τf 2 + B3 − M C3cosω3 t − τf 2 − M D3sinω3(t m 1 ∗ ∗ −τf 2 − 2 [−F + K1(δ + L)] M mω3

2 (m+M)K1 where ω3 = mM . . . By applying the initial conditions x3 τf 2 = x f 2, x3 τf 2 = v f 2, X3 τf 2 = X f 2, X3 τf 2 = Vf 2 to Equation (26), we can obtain the coefﬁcients:

∗ ∗ ∗ ∗ mv2 +MV2 mx2 +MX2 A3 = m+M , B3 = m+M (m+M)F∗+mMω 2(x∗−X∗)−K (δ∗+L)(m+M) M(v∗−V∗ (27) = 3 2 2 1 = 2 2 C3 2 , D3 (m+M)ω m(m+M)ω3 3 The ﬁnal time is given by: 1 C3 τf 3 = τf 2 + π − ArcSin(0) − ArcTan (28) ω3 D3

Response at the last stage (III) Appl. Sci. 2021, 11, 166 9 of 12

The corresponding response at the last stage when some of primary elements fail is as follows:

x3(t) = qm(t)+ qM(t) = A3 t − τf 2 + B3 + C3cosω3 t − τf 2 + D3sinω3 t − τf 2 1 + 2 [−F2 + K3(δ2 + L)] mω3 m (29) X3(t) = qm(t)− M qM(t) m m = A3 t − τf 2 + B3 − M C3cosω3 t − τf 2 − M D3sinω3(t m 1 −τf 2 − 2 [−F2 + K3(δ2 + L)] M mω3

2 (m+M)K3 where ω3 = mM . . . By applying the initial conditions x3 τf 2 = x f 2, x3 τf 2 = v f 2, X3 τf 2 = X f 2, X3 τf 2 = Vf 2 to Equation (29), we can obtain the coefﬁcients:

mv f 2+MVf 2 mx f 2+MX f 2 A3 = m+M , B3 = m+M , [(m+M)F +mMω 2(x −X )−K (δ +L)(m+M)] m(v −V ) (30) = 2 3 f 2 f 2 5 2 = 2 2 C3 2 , D3 (m+M)ω m(m+M)ω3 3

4. Numerical Results Figures7 and8 show the comparison plots of the approximate and the full response of the tether structure. The higher the initial velocities were, the closer the approximate

Appl. Sci. 2021, 11, x FOR PEER REVIEWresponse was to the full response result. Also, as shown in Figures9–12, the approximate10 of 13 solutions demonstrated very good correlation to the full response results as the number of Appl. Sci. 2021, 11, x FOR PEER REVIEW 10 of 13 elements, that is, the length of tether, increased from N = 15 to N = 1000. The beneﬁt of the approximate solution concerns the computational cost. For N = 1000, the computation time was just 5 s while the full analytical results took about 10 min. Even for N = 10,000 time was just 5 s while the full analytical results took about 10 min. Even for N = 10,000 timeelements, was justthe 5approximate s while the fullmodel analytical took 10 results s while took the about fully 10analytical min. Even model for Ntook = 10,000 more elements,elements,than an thehour. the approximate approximate model modeltook took 1010 ss whilewhile thethe fullyfully analyticalanalytical model model took took more more than anthan hour. an hour.

(a) v = 7 m/s (b) v = 13 m/s (c) v = 14 m/s (a) v = 7 m/s (b) v = 13 m/s (c) v = 14 m/s

Figure 7. Forced responses at various initial velocities (v = 7 m/s, 13 m/s, 14 m/s). Number of segments = 15, red line: Figure 7. Forced responses at various initial velocities (v = 7 m/s, 13 m/s, 14 m/s). Number of seg- approximate solution, blue line: exact solution. Figurements =7. 15, Forced red line: responses approximate at various solution, initial blue velocities line: exact (v = 7solution. m/s, 13 m/s, 14 m/s). Number of segments = 15, red line: approximate solution, blue line: exact solution.

(a) v = 7 m/s (b) v = 13 m/s (c) v = 14 m/s (a) v = 7 m/s (b) v = 13 m/s (c) v = 14 m/s Figure 8. Elongations at variousFigure initial 8. Elongations velocities at (v various = 7 m/s, initial 13 velocities m/s, 14 (v m/s).= 7 m/s, Number13 m/s, 14 ofm/s). segments Number =of 15,segments red line: = approximate solution, blue line:Figure15, exact red 8.line: solution. Elongations approximate at various solution, initial blue velocities line: exact (v solution. = 7 m/s, 13 m/s, 14 m/s). Number of segments = 15, red line: approximate solution, blue line: exact solution.

(a) v = 20 m/s (b) v = 34 m/s (c) v = 37 m/s (a) v = 20 m/s (b) v = 34 m/s (c) v = 37 m/s Figure 9. Forced responses at various initial velocities (v = 20 m/s, 34 m/s, 37 m/s). Number of seg- Figurements =9. 100, Forced red responsesline: approximate at various so lution,initial velocitiesblue line: (vexact = 20 solution. m/s, 34 m/s, 37 m/s). Number of segments = 100, red line: approximate solution, blue line: exact solution. Appl. Sci. 2021, 11, x FOR PEER REVIEW 10 of 13

time was just 5 s while the full analytical results took about 10 min. Even for N = 10,000 elements, the approximate model took 10 s while the fully analytical model took more than an hour.

(a) v = 7 m/s (b) v = 13 m/s (c) v = 14 m/s

Figure 7. Forced responses at various initial velocities (v = 7 m/s, 13 m/s, 14 m/s). Number of segments = 15, red line: approximate solution, blue line: exact solution.

Appl. Sci. 2021 , 11 , 166 (a) v = 7 m/s (b) v = 13 m/s (c) v = 14 m/s 10 of 12

Figure 8. Elongations at various initial velocities (v = 7 m/s, 13 m/s, 14 m/s). Number of segments = 15, red line: approximate solution, blue line: exact solution.

Appl.Appl. Sci.Sci. 20212021,, 1111,, xx FORFOR PEERPEER REVIEWREVIEW 1111 ofof 1313 (a) v = 20 m/s (b) v = 34 m/s (c) v = 37 m/s

Figure 9. Forced responsesFigure at various 9. Forced initial responses velocities (vat =various 20 m/s, initial 34 m/s, velocities 37 m/s). (v Number = 20 m/s, of segments34 m/s, 37 = 100,m/s). red Number line: of seg- approximate solution, bluements line: = exact 100, solution.red line: approximate solution, blue line: exact solution.

((aa)) vv == 2020 m/sm/s ((bb)) vv == 3434 m/sm/s ((cc)) vv == 3737 m/sm/s

Figure 10. Elongations atFigureFigure various 10.10. initial ElongationsElongations velocities atat (v variousvarious = 20 m/s, initialinitial 34 m/s, velocitiesvelocities 37 m/s). (v(v == Number 2020 m/s,m/s, 34 of34 segmentsm/s,m/s, 3737 m/s).m/s). = 100, NumberNumber red line: ofof seg-seg- approximate solution, bluementsments line: == exact 100,100, solution. redred line:line: approximateapproximate sosolution,lution, blueblue line:line: exactexact solution.solution.

((aa)) vv == 7070 m/sm/s ((bb)) vv == 113113 m/sm/s ((cc)) vv == 114114 m/sm/s

Figure 11. Forced responsesFigureFigure at various 11.11. ForcedForced initial responsesresponses velocities atat (v variousvarious = 70 m/s, initialinitial 113 velocitiesvelocities m/s, 114 m/s). (v(v == 7070 Number m/s,m/s, 113113 of m/s,m/s, segments 114114 m/s).m/s). = 1000, NumberNumber ofof red line: approximate solution,segmentssegments blue == line: 1000,1000, exact redred solution. line:line: approximateapproximate solution,solution, blueblue line:line: exactexact solution.solution.

((aa)) vv == 7070 m/sm/s ((bb)) vv == 113113 m/sm/s ((cc)) vv == 114114 m/sm/s

FigureFigure 12.12. ElongationsElongations atat variousvarious initialinitial velocitiesvelocities (v(v == 7070 m/s,m/s, 113113 m/s,m/s, 114114 m/s).m/s). NumberNumber ofof seg-seg- mentsments == 1000,1000, redred line:line: approximateapproximate sosolution,lution, blueblue line:line: exactexact solution.solution.

5.5. ConclusionsConclusions TheThe applicabilityapplicability andand benefitsbenefits ofof aa tailoredtailored tethertether forfor anan airborneairborne windwind energyenergy systemsystem (AWES)(AWES) werewere investigated.investigated. TheThe AWESAWES systemsystem withwith aa tethertether structurestructure waswas simplifiedsimplified asas aa two-degree-of-freedomtwo-degree-of-freedom systemsystem andand anan exactexact responseresponse waswas mathematicallyobtained.mathematicallyobtained. TheThe compositecomposite tethertether waswas provenproven toto showshow bebettertter enduranceendurance underunder impulsiveimpulsive loadingloading com-com- paredpared toto thethe conventionalconventional structurestructure sincesince moremore mechanicalmechanical workwork waswas requiredrequired toto reachreach completecomplete structurestructure failure,failure, thanksthanks toto thethe mumultipleltiple loadload pathspaths (primary(primary andand secondarysecondary ele-elements).ments). Furthermore,Furthermore, anan approximateapproximate responresponsese modelmodel waswas developed,developed, whichwhich resultedresulted inin Appl. Sci. 2021, 11, x FOR PEER REVIEW 11 of 13

(a) v = 20 m/s (b) v = 34 m/s (c) v = 37 m/s

Figure 10. Elongations at various initial velocities (v = 20 m/s, 34 m/s, 37 m/s). Number of segments = 100, red line: approximate solution, blue line: exact solution.

(a) v = 70 m/s (b) v = 113 m/s (c) v = 114 m/s

Appl. Sci. 2021, 11, 166 Figure 11. Forced responses at various initial velocities (v = 70 m/s, 113 m/s, 114 m/s). Number11 of 12 of segments = 1000, red line: approximate solution, blue line: exact solution.

(a) v = 70 m/s (b) v = 113 m/s (c) v = 114 m/s

Figure 12. Elongations atFigure various 12. initial Elongations velocities at (v various = 70 m/s, initial 113 m/s, velocities 114 m/s). (v = Number 70 m/s, of113 segments m/s, 114 = m/s). 1000, redNumber line: of seg- approximate solution, bluements line: = exact 1000, solution. red line: approximate solution, blue line: exact solution.

5. 5.Conclusions Conclusions TheThe applicability applicability and beneﬁtsbenefits of of a a tailored tailored tether tether for for an airbornean airborne wind wind energy energy system system (AWES)(AWES) were were investigated. investigated. The The AWES AWES system system with a tether structurestructure was was simpliﬁed simplified as a two-degree-of-freedomas a two-degree-of-freedom system system and and an anexact exact response response was was mathematicallyobtained. mathematicallyobtained. The compositeThe composite tether tether was proven was proven to show to show better better endurance endurance under under impulsive impulsive loading loading com- paredcompared to the to conventional the conventional structure structure since since more mechanicalmechanical work work was was required required to reach to reach complete structure failure, thanks to the multiple load paths (primary and secondary complete structure failure, thanks to the multiple load paths (primary and secondary ele- elements). Furthermore, an approximate response model was developed, which resulted ments).in a dramatic Furthermore, reduction an of approximate computational respon cost asse well model as showing was developed, good correlation which with resulted the in original results.

Author Contributions: K.H. compiled the literature review, conceptualized ideas, performed the numerical simulation, and wrote the paper, in addition to acquiring funding resources. The author has read and agreed to the published version of the manuscript. Funding: This study has been supported by Brain Pool Program through the National Research Foun- dation of Korea (NRF) funded by the Ministry of Science and ICT (grant number: 2019H1D3A2A02102093). Institutional Review Board Statement: Not applicable. Informed Con sent Statement: Not applicable. Data availability: The data that support the finding of this research are not publicly available due to confidentiality constraints. Acknowledgments: K.H. acknowledges the administrative and technical support of the Offshore Floating Wind Energy System Engineering Department of University of Ulsan. Conflicts of Interest: The author declares no conflict of interest.

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