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`1. REPORT DATE (DD-MM-YYYY)
`2. REPORT DATE
`22-11-2002
`Technical
`4. TITLE AND SUBTITLE
`
`3. DATES COVERED (From - To)
`0101-1992 to 31-12-2003
`5a. CONTRACT NUMBER
`
`Generalized-a time integration solutions for
`hanging chain dynamics
`
`l>. AUTHOR(S)
`Jason I. Gobat
`Mark A. Grosenbaugh
`Michael S. Triantofyllou
`
`5b. GRANT NUMBER
`N00014-92-J-1269
`5c. PROGRAM ELEMENT NUMBER
`
`5d. PROJECT NUMBER
`
`5e. TASK NUMBER
`
`5f. WORK UNIT NUMBER
`
`7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES)
`
`Woods Hole Oceanograph.ic Insitution
`Woods Hole, MA 02543
`
`9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES)
`
`Office OL Naval Research
`Env.ironerai ntal Sciences Directorate
`Arlington, VA 22217-5660
`
`8. PERFORMING ORGANIZATION
`REPORT NUMBER
`
`10. SPONSOR/MONITOR'S ACRONYM(S)
`
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`
`12. DISTRIBUTION AVAILABILITY STATEMENT "
`
`APPROVED FOR PUBLIC RELEASE - DISTRIBUTION IS UNLIMITED
`
`13. SUPPLEMENTARY NOTES " ' "
`In citiing the report in abibliog..:aphy, the reference given
`Journal of Engineering Mechanics, 128(6) : 677-687
`14. ABSTRACT
`
`should read
`
`bstract: In this paper, we study numerically the two- and three-dimensional nonlinear dynamic response of a chain hanging under its
`own weight. Previous authors have employed the box method, a finite-difference scheme popular in cable dynamics problems, for this
`purpose. The box method has significant stability problems, however, and thus is not well suited to this highly nonlinear problem. We
`i lustrate these stability problems and propose a new time integration procedure based on the generalized-a method. The new method
`exhibits superior stability properties compared to the box method and other algorithms such as backward differences and trapezoidal rule.
`Of four time integration methods tested, the generalized-a algorithm was the only method that produced a stable solution for the
`three-dimensional whirling motions of a hanging chain driven by harmonic linear horizontal motion at the top.
`
`15. SUBJECT TERMS
`
`nonlinear responses, finite differences, cables, numerical models,
`dynamics
`
`16. SECURITY CLASSIFICATION OF:
`a. REPORT
`b. ABSTRACT
`C. THIS PAGE
`UL
`UL
`UL
`
`17. LIMITATION OF
`ABSTRACT
`
`18. NUMBER
`OF PAGES
`
`UL
`
`11
`
`19a. NAME OF RESPONSIBLE PERSON
`Mark Grosenbaugh
`19b. TELE PONE NUMBER {Include area code)
`508.289.2607
`Standard Form 298 (Rev. 8-98)
`Prescribed by ANSI-Std Z39-18
`
`PGS v. WESTERNGECO (IPR2014-00688)
`WESTERNGECO Exhibit 2049, pg. 1
`
`

`

`20021212 029
`DISTRIBUTION STATEMENT A
`Approved for Public Release
`Distribution Unlimited
`Generalized-« Time Integration Solutions
`for Hanging Chain Dynamics
`Jason I. Gobat1; Mark A. Grosenbaugh2; and Michael S. Triantafyllou3
`
`Abstract: In this paper, we study numerically the two- and three-dimensional nonlinear dynamic response of a chain hanging under its
`own weight. Previous authors have employed the box method, a finite-difference scheme popular in cable dynamics problems, for this
`purpose. The box method has significant stability problems, however, and thus is not well suited to this highly nonlinear problem. We
`illustrate these stability problems and propose a new time integration procedure based on the generalized-a method. The new method
`exhibits superior stability properties compared to the box method and other algorithms such as backward differences and trapezoidal rule.
`Of four time integration methods tested, the generalized-a algorithm was the only method that produced a stable solution for the
`three-dimensional whirling motions of a hanging chain driven by harmonic linear horizontal motion at the top.
`
`DOI: 10.1061/(ASCE)0733-9399(2002)128:6(677)
`
`CE Database keywords: Nonlinear responses; Finite differences; Cables; Numerical models; Dynamics.
`
`Introduction
`
`The dynamics of a chain hanging under its own weight is a classic
`problem in mechanics. Two of the more interesting aspects of the
`problem are the simultaneous presence of both high- and low-
`tension regimes in the chain and the unstable nature of large am-
`plitude motions. Triantafyllou and Howell (1993) and Howell and
`Triantafyllou (1993) considered both of these phenomena using a
`combination of analytic, numerical, and experimental results.
`They observed that the stability of the response in a harmonically
`driven system is strongly dependent on the frequency and ampli-
`tude of the excitation.
`The numerical model that they employed was based on a
`finite-difference scheme known as the box method. This method
`was first applied to a cable dynamics problem by Ablow and
`Schechter (1983). Because the box method is an implicit scheme,
`box method solutions for the classical cable dynamics equations
`are singular when the tension goes to zero anywhere on the cable.
`Howell and Triantafyllou (1993) removed this singularity by add-
`ing bending stiffness to the governing equations, thus providing a
`mechanism to propagate energy in the presence of zero tension
`(Burgess 1993). For small values of artificial bending stiffness
`
`'Postdoctoral Invest., Dept. of Physical Oceanography, Woods Hole
`Oceanographic Institution, Mail Stop No. 29, Woods Hole, MA 02543.
`E-mail: jgobat@whoi.edu
`2Associate Scientist, Dept. of Applied Ocean Physics and Engineer-
`ing, Woods Hole Oceanographic Institution, Mail Stop No. 7, Woods
`Hole, MA 02543. E-mail: mgrosenbaugh@whoi.edu
`Professor, Dept. of Ocean Engineering, Massachusetts Institute of
`Technology, 77 Massachusetts Ave., Cambridge, MA 02139. E-mail:
`mistetri@deslab.mit.edu FAX 617-258-9389.
`Note. Associate Editor: James L. Beck. Discussion open until Novem-
`ber 1, 2002. Separate discussions must be submitted for individual pa-
`pers. To extend the closing date by one month, a written request must be
`filed with the ASCE Managing Editor. The manuscript for this paper was
`submitted for review and possible publication on October 18, 2000; ap-
`proved on November 29, 2001. This paper is part of the Journal of
`Engineering Mechanics, Vol. 128, No. 6, June 1, 2002. ©ASCE, ISSN
`0733-9399/2002/6-677-687/$8.00+$.50 per page.
`
`this modification stabilized the numerical solution with no loss of
`accuracy compared to experimental results.
`The box method is popular because it is second-order accurate
`in both space and time and is relatively easy to implement. Be-
`cause the box method preserves the frequency content of the so-
`lution across all frequencies, however, it has the disadvantage of
`relatively poor stability in its temporal discretization. In a nonlin-
`ear problem, spurious high-frequency content can cause numeri-
`cal instabilities, and thus, it is desirous that a temporal integration
`scheme should be numerically dissipative at high frequencies.
`Koh et al. (1999) addressed this shortcoming of the box method
`by replacing the box method's temporal integration scheme with
`backward differences. They preserved the box method's straight-
`forward and easy to implement spatial discretization. Backward
`differences have also been used by Chatjigeorgiou and Mavrakos
`(1999) and Chiou and Leonard (1991) in conjunction with spatial
`discretizations based on collocation and direct integration, respec-
`tively. The scheme is only first-order accurate, but is very stable
`because it has strong numerical dissipation at high frequencies.
`Another temporal integration scheme that has been used in
`cable dynamics applications is the generalized trapezoidal rule
`(Sun et al. 1994). This scheme offers controllable numerical dis-
`sipation, but is second-order accurate only in its least dissipative
`form. Thomas (1993) compared three historically popular algo-
`rithms from the structural dynamics community, Newmark,
`Houbolt, and Wilson-0, for use in mooring dynamics problems.
`His conclusion was that Houbolt was the best choice of the three.
`Other authors, however, have noted that Houbolt has an undesir-
`able amount of low-frequency dissipation (Chung and Hulbert
`1994; Hughes 1987).
`Turning to the more recent structural dynamics literature,
`Gobat and Grosenbaugh (2001) proposed replacing the box meth-
`od's temporal integration with the generalized-a method devel-
`oped for the second-order structural dynamics problem by Chung
`and Hulbert (1993). This algorithm has the advantages of control-
`lable numerical dissipation, second-order accuracy, and straight-
`forward adaptation to the first-order nonlinear cable dynamics
`problem. Through appropriate choices of parameters, the method
`can also reproduce the spectral properties of several other algo-
`
`JOURNAL OF ENGINEERING MECHANICS / JUNE 2002 / 677
`
`PGS v. WESTERNGECO (IPR2014-00688)
`WESTERNGECO Exhibit 2049, pg. 2
`
`

`

`rithms including the box method, backward differences, and trap-
`ezoidal rule. This latter property makes it a particularly conve-
`nient choice for the type of comparative study undertaken herein.
`The analyses of the box and generalized-a methods from
`Gobat and Grosenbaugh (2001) are summarized below. The per-
`formance of the new algorithm is studied by comparison to ana-
`lytic and experimental results for the free and forced response of
`the hanging chain. Throughout the analyses, comparisons are also
`made to trapezoidal rule and backward difference solutions.
`
`Analysis of Box Method
`
`The governing equations for a cable or chain can be written as a
`system of partial differential equations of the form (Howell 1992)
`
`dY dY
`M- + K-+F(Y,,,r) = 0
`
`(1)
`
`where Y= vector of N-dependent variables, M and K
`= coefficient matrices, and F= force vector. The independent
`variables are s, the Lagrangian coordinate measuring length along
`the unstretched cable, and t, time. Howell and Triantafyllou
`(1993) used the box method to discretize Eq. (1). In the box
`method the discrete equations are written using what look like
`traditional backward differences in both space and time, but be-
`cause the discretization is applied on the half-grid points with
`spatial and temporal averaging of adjacent grid points, the method
`is second-order accurate. The result is a four-point average cen-
`tered around the half-grid point.
`The stability of the box method can be analyzed by consider-
`ing an equivalent linear, single degree-of-freedom system in se-
`midiscrete form. This approach separates the spatial and temporal
`discretizations into distinct procedures. For each of the n -1 spa-
`tial half-grid points between the n nodes a set of N discrete equa-
`tions is assembled. Combining these N(n -1) equations with N
`equations describing the boundary conditions yields the semidis-
`crete equation of motion for all of the dependent variables at all
`of the nodes as (Gobat and Grosenbaugh 2001)
`
`MY+KY+F=0
`
`(2)
`
`The tilde over the matrices signifies that these are now dis-
`cretized, assembled quantities. The single degree-of-freedom, lin-
`ear, homogeneous analog of Eq. (2) is
`
`y + (Dy = 0
`
`(3)
`
`Applying the box method's temporal discretization to Eq. (3)
`yields
`
`yi+yi-1 + ia(yi+yi~l) = 0
`
`where
`
`3)' + y'-i = 2
`
`Ar
`
`Rearranging Eq. (5) gives the recursion relationships
`
`(4)
`
`(5)
`
`y' = 2
`
`-y'-1
`
`yi-yi-l
`Ar
`Ar y'yW'+r'i+r1
`Substituting each of the recursion relationships separately into Eq.
`(4), we can write equations for y' and y1 in matrix form as
`
`(6)
`
`(7)
`
`678 / JOURNAL OF ENGINEERING MECHANICS / JUNE 2002
`
`(8)
`
`.-,i —1
`
`-1
`
`2-coAr
`2 + wAr
`-4
`2 + wAr
`The 2X2 matrix on the right-hand side of Eq. (8) is the am-
`plification matrix. Spectral radius p of this matrix, defined as
`p = max(|X,|,|\2|) (9)
`governs the growth or decay of the solution from one time step to
`the next (Hughes 1987). \12= eigenvalues of the amplification
`matrix. For p=£ 1, the solution will remain steady or decay and is
`said to be stable. For p> 1, the solution will grow and is said to
`be unstable. For the box method,
`2-coAr
`\,= , Aj (10)
`2 + wAr
`
`\2=-l (ID
`and the spectral radius is unity (and the scheme is stable) for all
`values of w and Ar.
`In spite of this unconditional stability, however, the box
`method has three significant problems. The first problem is illus-
`trated by considering the update equation for y' written in the
`form
`
`y-
`
`2-wAr
`2 + ioAr
`
`(12)
`
`As (oAf goes to infinity this becomes
`
`y'=-y'
`This is the phenomenon known as Crank-Nicholson noise,
`whereby the high-frequency components of the solution oscillate
`with every time step. A second, related, problem is that the spec-
`tral radius is constant at unity. An artifact of the spatial discreti-
`zation process is that at some point the high-frequency (or equiva-
`lently, high-spatial wave-number) components of the solution are
`not well resolved and the numerical solution is inaccurate. For
`this reason it is desirous to have numerical dissipation in a
`scheme such that the spectral radius is less than unity for increas-
`ing values of o) Ar. The box method has no numerical dissipation.
`Finally, Hughes (1977) cites a problem with averaging schemes in
`general as applied to nonlinear problems. For the nonlinear single
`degree-of-freedom case, Eq. (4) can be written as
`y'" + y,'-1 + (üy + (ü,'"V"1 = 0 (14)
`The update equation for y', Eq. (12), becomes
`'2-co'_1Ar\
`2 + w'Ar
`and the stability becomes conditional as parameter o> changes
`with time. The practice suggested by Hughes (1977) for avoiding
`this problem is to use an averaged value of to, i.e.,
`w' + w'"
`
`(13)
`
`(15)
`
`f+y i-i +
`
`(y' + y'-I) = 0
`
`(16)
`
`Generalized-a Method
`
`Given the stability problems associated with the box method,
`Gobat and Grosenbaugh (2001) proposed replacing the temporal
`integration with Chung and Hulbert's (1993) generalized-a
`
`PGS v. WESTERNGECO (IPR2014-00688)
`WESTERNGECO Exhibit 2049, pg. 3
`
`

`

`Table 1. Algorithms Included in Generalized-a Method
`Algorithm
`
`a*
`
`Box method
`Backward differences
`Generalized trapezoidal
`Comwell and Malkus
`WBZ-a
`
`til]
`l y-a
`2 + a
`
`1st order problem
`
`2nd order problem
`
`Ablow and Schechter (1983)
`Koh et al. (1999)
`Sun et al. (1994)
`Cornwell and Malkus (1992)
`
`Newmark (1959)
`Hilber et al. (1977)
`Wood et al. (1981)
`
`method. The generalized-a method is a reasonably complete fam-
`ily of algorithms that is second-order accurate, has controllable
`numerical dissipation, and offers a clear approach to coefficient
`averaging for the nonlinear problem. Following Chung and Hul-
`bert's development of the generalized-a method for second-order
`equations, semidiscrete Eq. (2) becomes
`
`(l-aJMY' + anMi-1 + (l-at)KYi + aiKY'-1
`
`+ (l-ai)F'' + atF'-1 = 0 (17)
`The difference equation is the same as for the generalized trap-
`ezoidal rule (Hughes 1987),
`
`Y' = Y;-1 + Af[(l--y)Y''-I + 7Y'']
`
`(18)
`
`The three parameter family of algorithms given by Eqs. (17) and
`(18) defines the generalized-a method for the first-order semidis-
`crete problem. The method is second-order accurate if
`
`«m-at+7=2-
`
`(19)
`
`From the eigenvalues of the amplification matrix, the stability
`requirement is
`
`:2 7^2
`
`(20)
`
`Requiring second-order accuracy according to Eq. (19) and forc-
`ing the eigenvalues of the amplification matrix to be equal as
`oi&t—»o° to prevent bifurcation, yields formulas for a* and am as
`a function of X°° only
`
`X°°-l
`
`3X°°+1
`:2XC0-2
`
`(21)
`
`This yields a second-order accurate algorithm in which the only
`parameter is the eigenvalue (or spectral radius) at infinity.
`Algorithms that can be obtained through various choices of
`ak, a.m, y, and X°° are listed in Table 1. Spectral radii of some of
`these algorithms are shown in Fig. 1. Note that taking X°°
`e[0,l) as the basis for the spectral radius results in a different set
`of algorithms than \" e [ -1,0]. For p°°= 1 the only option is the
`negative eigenvalue and this results in the box method. A nondis-
`sipative algorithm with X°°= +1 cannot be achieved.
`In applying the generalized-a method to the nonlinear problem
`we must choose the time point at which we will evaluate M, K,
`and F. A natural choice, consistent with the practice suggested by
`Hughes (1977) for nonlinear first-order problems and exemplified
`by Eq. (16), is provided by the temporal averaging of terms that is
`already a part of the method. At time step i Eq. (17) becomes
`
`1.0
`
`0.9
`
`0.8
`
`0.7
`
`CO
`
`Q.
`3 0.6
`CO 0.5
`
`CO
`o
`©
`Q.
`CO
`
`0.4
`
`0.3
`
`0.2
`
`0.1
`
`0.0
`10
`
`10 10"' 10" 10' 10'
`
`Fig. 1. Spectral radii of generalized-a family algorithms
`
`JOURNAL OF ENGINEERING MECHANICS / JUNE 2002 / 679
`
`PGS v. WESTERNGECO (IPR2014-00688)
`WESTERNGECO Exhibit 2049, pg. 4
`
`

`

`(1 - a JM''_a'»Y,'+ amM'-a'"Y'-l + (1 - a^Kr^Y''
`
`+ aArK/-a*Y''-, + (l-at)F + atF"'-1 = 0
`
`(22)
`
`where the averaged coefficient matrices are defined as
`
`M''-a" = (l-aJMf + amM''_1
`
`(23)
`
`(24)
`
`K''-at=(l-ai)K,' + atK''-1
`This scheme has been implemented in a computer program for
`two- and three-dimensional simulations of cable dynamics (Gobat
`and Grosenbaugh 2000). At each time step, Eq. (22) is solved
`using a Newton-Raphson procedure. The solution from the pre-
`vious time step (or the static solution at the initial time step)
`serves as the initial guess in the nonlinear iterations. Because of
`this, the ultimate success of the solution is dependent on both the
`stability of the time integration and on the ability of the nonlinear
`solver to converge on a solution at time step i given an initial
`guess based on the solution at time step i— 1. To improve conver-
`gence the program implements an adaptive time stepping scheme
`whereby the time step (the distance between the guess at i -1 and
`the solution at i) is reduced by factors of 10 at any spots where
`the solver is not successful. A practical limit of four orders of
`magnitude below the base-line time step is set to prevent the
`solution from proceeding in the face of a physical or numerical
`instability unrelated to the nonlinear solution procedure (e.g.,
`Crank-Nicholson noise).
`All of the numerical solutions that follow were obtained using
`this program. Thus, the box method, trapezoidal rule, and back-
`ward difference results, while spectrally equivalent to previous
`implementations, may be more stable than previous solutions be-
`cause of the coefficient averaging scheme in Eq. (22). For clarity,
`spectrally equivalent historical names are retained in discussions
`of comparative algorithm performance that follow.
`
`Application to Hanging Chain Problem
`
`The performance of the different algorithms that can be imple-
`mented with the generalized-a family is studied by considering
`the free and forced response of the hanging chain shown in Fig. 2.
`In the free-response problem, we apply a small initial displace-
`ment to the chain and then at time f = 0, release it. The dynamic
`response of the chain for t >0 can be calculated analytically for
`the small motions that result. In the forced response problem we
`impose a sinusoidally varying horizontal displacement to the top
`of the chain and analyze the forced response. This latter problem
`was studied both numerically and experimentally by Howell and
`Triantafyllou (1993).
`
`Free Response to Initial Displacement
`
`For small motions and an inextensible chain, the equation of mo-
`tion is
`
`m
`
`dq
`
`d
`Is
`where m = mass per length of the chain, q = transverse displace-
`ment of the chain, g = acceleration due to gravity, and 5
`= independent coordinate along the chain with s = 0 at the free
`end. Assuming a harmonic solution of the form
`
`(25)
`
`q(s,t) = q{s)[A coswf+B sin oaf]
`
`(26)
`
`the mode shapes; q(s), are (Triantafyllou et al. 1986)
`
`680 / JOURNAL OF ENGINEERING MECHANICS / JUNE 2002
`
`Q(0
`
`Fig. 2. Definitions for hanging chains problems
`
`q(s) = CiJ0\2a^lS-j+c2Y0\^2<* yg
`
`■£
`
`(27)
`
`where J0 and YQ= zero-order Bessel functions of the first and
`second kind, respectively. The requirement that the solution be
`finite at s = 0 leads to the elimination of the Y0 term and the
`requirement that q(L) = 0 leads to the natural frequencies, o>.
`They are given by the roots of
`
`Jn\ 2<>>
`
`(28)
`
`$-
`The complete response is given as the sum of the response in all
`modes:
`
`q(s,t)=^ J0 2con\/- [Ancoswf + ß„sinw/] (29)
`n = 1 \ o /
`
`The coefficients A„ and B„ are determined from the initial
`displacement, q0(s), and velocity, q0(s). Given qQ(s) = 0, we can
`immediately determine that B„ = 0. To determine A„ we first write
`
`q{sfi)-- = 2 AnJ0 2w„y- =4o(* 0
`
`(30)
`
`Multiplying both sides by J0(2d>n\[s/g), integrating from s = 0 to
`s = L, and making use of the fact that
`
`j Jo\2i*n^S-jJ0\[2umy]S-jds = 0 fox n^m (31)
`
`yields the following equation for A„ :
`
`j\0(s)J0[2<*nyJ-\
`ds
`An = ; ^—
`
`J0\ 2lD„
`
`(32)
`
`PGS v. WESTERNGECO (IPR2014-00688)
`WESTERNGECO Exhibit 2049, pg. 5
`
`

`

`10'
`
`10'
`
`*
`
`•*»l 5.
`
`CD 10
`co
`c o
`8-10-'
`
`lio-2
`o
`"co
`<§ 10"3t
`
`■o
`o
`"510-5
`Q
`CO
`Q-10-%
`
`io"7t
`
`10
`
`zr.
`~^ analytic solution
`box method
`* trapezoidal aile
`* backward difference
`* V = -0.5
`° X~ = 0
`- X = 0.25
`
`8 ?o
`V o
`
`o o
`
`10 15 20 25
`non-dimensional frequency
`
`30
`
`35
`
`Fig. 3. Power spectra peaks of response of free end of chain for analytic solution and for six variants of generalized-a method with At
`= 0.01 s andn = 50
`
`1U
`
`- '1
`
`1
`
`1
`
`1
`
`0 analytic solution
`box method
`V
`trapezoidal rule ,
`+
`backward difference !
`
`10' [*
`I
`
`0)10°
`
`CO c o
`
`CO 10
`
`"|io-2 r
`o
`"co
`S10-3
`E
`■0
`
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`c
`0 10 5
`O
`CO
`0-10-6
`
`10-7
`
`in-8
`
` 1 . 1.
`
`]
`-:
`
`•
`~
`
`'■
`
`Ul Ol
`
`0 0 1
`II II II
`8 8 8
` K K
`
`a
`►
`X K
`
`0 0
`
`O 0
`
`7 _
`1
`V V
`
`* f
`4 °**»ft
`♦
`♦
`
`*>
`
`V
`
`V
`
`» 0 0 0
`V
`
`0
`
`V
`
`♦
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`0
`
`♦
`
`♦
`
`1
`
`1
`
`1
`
`1
`
`1
`10 15 20 25
`non-dimensional frequency
`
`♦
`
`♦
`
`1
`
`30
`
`35
`
`Fig. 4. Power spectra peaks of response of free end of chain for analytic solution and for six variants of generalized-a method with At
`= 0.001 sandn = 50
`
`JOURNAL OF ENGINEERING MECHANICS / JUNE 2002 / 681
`
`PGS v. WESTERNGECO (IPR2014-00688)
`WESTERNGECO Exhibit 2049, pg. 6
`
`

`

`10' r
`
`10' |r
`
`O analytic solution
`D n = 50
`$ n = 200
`V n = 400
`>■ n = 800
`
`0
`
`CO c o
`
`Q.
`C/3
`0 10" [r
`
`c g
`in
`Sio"1
`E
`T3
`I
`C o
`c10"2
`o
`Q
`
`0.
`
`10'3h
`
`10
`
`fa
`
`♦ D
`$ n
`
`•H,
`* f 8 u H
`
`10 15 20 25
`non-dimensional frequency
`
`30
`
`35
`
`Fig. 5. Power spectra peaks of response of free end of chain for analytic solution and for \"
`
`7, Af= 0.001 s, with n = 50, 200, 400, and 800
`
`0.3
`
`— 0.25
`E,
`
`ra 0.2
`
`c 1
`
`Box: n=100, At = 0.01
`\
`V{I
`
`Trapezoidal: n=100,At = 0.01 Backward Diff: n=100, At = 0.01
`
`I'
`
`'i-
`I-
`i!.:
`i
`i
`
`■
`
`n
`.■■V
`•* 'z
`/ /
`
`§ 0.15
`-c
`> 0.1
`
`3.43 s
`■-• 3.44 s
`- - 3.45 s
` 3.46 s
`
`0.1 0.2
`
`0 0.1 0.2 -0.1
`horiz coordinate (m)
`
`0.1
`
`X~ = -0.5: n=100, At = 0.01
`
`X°° = 0.0:n=100,At = 0.01
`
`-0.7: n=100, At = 0.01
`
`0.3
`
`0.25
`
`0.2
`
`0.15
`
`0.1
`
`0.2
`
`0.05
`0 0.1 0.2 -0.1
`horiz coordinate (m)
`
`Fig. 6. Snapshots of chain configuration near time of expected intersection for six algorithms that can be obtained from within generalized-a
`method
`
`682/ JOURNAL OF ENGINEERING MECHANICS/JUNE 2002
`
`PGS v. WESTERNGECO (IPR2014-00688)
`WESTERNGECO Exhibit 2049, pg. 7
`
`

`

`Box: n=100, At = 0.01
`
`Box: n=100, At = 0.001
`
`Box: n=100, At = 0.0001
`
`0.25
`
`0.2
`
`0.15
`
`0.1
`
`\v
`ifjt
`:'u
`
`— 0.25
`
`to
`
`>rdinate
`
`O0.15
`■c
`
`3.43 s
` 3.44 s
`' 0.1 3.45 s
`0-4D S
`
`n nc
`
`0.1
`
`0.2
`
`0.05
`0 0.1 0.2
`horiz coordinate (m)
`
`Box: n=200, At = 0.01
`
`Box n= =200, At = = 0.001
`
`Box: n=200, At = 0.0001
`
`/
`/' /
`//
`I
`A. J
`
`0.25
`
`•
`
`0.2
`
`■' \
`
`'i'l
`
`■'•'/
`
`1
`'Ml
`Aj J
`
`0.15
`
`0.1
`
`0.05
`
`0.1
`
`0.2
`
`i'l
`11
`
`Ml
`
`0.25
`
`0.2
`
`0.15
`
`0.1
`
`oi ro oi
`- P io
`o o
`coordinate (m)
`
`■c
`CD
`" 0.1
`
`0.05
`
`0.05
`0.1 0.2 0 0.1 0.2
`horiz coordinate (m)
`
`Fig. 7. Snapshots of chain configuration near time of expected intersection for box method with different spatial and temporal discretizations
`
`The analytic solution was computed for a chain released from
`an initial catenary configuration. For simplicity all of the model
`parameters (mass per length, gravity, length) were set to unity.
`The horizontal force applied at s = 0 to create the initial deflection
`was set to 0.001 N. All of the integrals for the analytic solution
`were computed using the trapezoidal rule with 10,000 panels. A
`400 s time series of the response at the free end was constructed
`using the first 20 modes of the analytic solution. The analytic
`result was sampled at 20 Hz to adequately capture the response up
`to mode 20. (The natural frequency for mode 20 is approxi-
`
`mately 5 Hz.)
`Analytic solutions were compared to numerical simulation re-
`sults for a chain released from the same initial configuration. For
`simulation results El was set to 10-6 Nm2 and EA to 109 N. This
`setting for El corresponds to the value of El* = EI/mgL3 used in
`Howell's (1992) comparison of experimental and simulation re-
`sults and in the simulations of the forced hanging chain problem
`that follow. The results from Howell demonstrated that this value
`is sufficient to stabilize the numerical solution in the presence of
`zero tension, but is small enough as to have a negligible effect on
`
`Trapezoidal: n=100,At = 0.01
`0.3
`
`Trapezoidal: n=200,At = 0.01
`
`Trapezoidal: n=100,At = 0.001
`
`— 0.25
`E,
`CD
`■& 0.2
`c
`'■£
`§0.15
`t
`CD
`* 0.1
`
`0.05
`
`•A'/
`■//A
`Ml \
`~A^J
`
`■
`
`■
`
`• 3.43 s
`•-■ 3.44 s
`- - 3.45 s
` 3.46 s
`
`0.1
`
`0.2
`
`0 0.1 0.2
`horiz coordinate (m)
`
`' \
`
`0.25
`
`0.2 0
`■''A
`
`0.15.
`
`.
`
`0.1
`
`0 05
`
`■
`
`0.2
`
`0.1
`
`Fig. 8. Snapshots of chain configuration near time of expected intersection for trapezoidal rule with different spatial and temporal discretizations
`
`JOURNAL OF ENGINEERING MECHANICS / JUNE 2002 / 683
`
`PGS v. WESTERNGECO (IPR2014-00688)
`WESTERNGECO Exhibit 2049, pg. 8
`
`

`

`the accuracy of the simulation result (based on comparisons with
`experiment). The model results were insensitive to changes in this
`value of at least an order of magnitude.
`Because the primary distinction among the various algorithms
`contained within the generalized-a family is the amount of nu-
`merical dissipation, all results are compared in the frequency do-
`main. For each 400 s time series, power spectra of the response at
`the free end were computed using nonoverlapping 256 point fast
`Fourier transforms. For clarity, only the peaks of the spectra are
`plotted. This prevents clutter and allows for a comparison of the
`spectral roll off of each of the algorithms compared to the roll off
`from the analytic solution.
`Fig. 3 shows a comparison between the analytic solution and
`numerical solutions for six different parametrizations of the
`generalized-a method. At this time step, Ar = 0.01 s, most of the
`algorithms are accurate out to the fifth or sixth mode. The notable
`exception is the first-order accurate backward difference solution,
`which substantially underestimates the response even in the first
`mode. All of the algorithms show a marked fall off from the
`analytic solution at higher frequencies, with the solutions for X°°
`5*0 showing the most decay and the trapezoidal rule appearing to
`be the most accurate.
`In Fig. 1, the numerical damping of most algorithms increases
`with increasing o>Ar. The idea that we should see less numerical
`damping at a fixed frequency with a decrease in A r is illustrated
`in Fig. 4, which shows the same results comparison as in Fig. 3
`for a time step of Ar=0.001 s. At this time step most algorithms
`are accurate out to the tenth mode. Only backward differences,
`which due to its first-order accuracy is again a poor solution even
`at very low frequencies, and X°°=0 are worse than this.
`That the other algorithms, with their varying levels of dissipa-
`tion, have converged to the same solution suggests that the re-
`maining error is not due to numerical dissipation. Fig. 5 shows the
`comparison for four cases with X™ = — j and Af= 0.001 s, with a
`varying number of nodes. As the node density is increased, the
`numerical model is better able to resolve the mode shapes asso-
`ciated with the higher frequencies. At n = 800, the numerical so-
`lution is in agreement with the analytic solution over the full
`range of the analytically computed response.
`These results demonstrate that the ability of the model to ac-
`curately resolve high-frequency response is dependent on tempo-
`ral and spatial discretizations and on the numerical dissipation for
`a given algorithm. Given sufficient temporal and spatial resolu-
`tion, most of the algorithms appear ultimately capable of accu-
`rately calculating the free response of the swinging chain. Based
`on its better accuracy at the larger time step, the best choice of
`algorithm for this problem appears to be the trapezoidal rule.
`
`Two-Dimensional Forced Response to Imposed
`Motion
`
`The forced hanging chain problem that we consider was studied
`by Howell and Triantafyllou (1993). In this problem, a 1.75-m-
`long chain is suspended from an actuator which imposes a sinu-
`soidally varying horizontal linear displacement, Q(t), to the top
`of the chain (Fig. 2). In experiments, Howell and Triantafyllou
`observed that the free end of the chain intersects the chain above
`it at approximately 3.4 s.
`Fig. 6 shows the configuration of the lower portion of the
`chain from 3.43 to 3.46 s for six different numerical algorithms,
`all with n= 100 and Ar=0.01 s. The box method and trapezoidal
`rule both closely match the experimental result, with intersection
`occurring by the 3.43 s mark. For the other algorithms the inter-
`
`684/JOURNAL OF ENGINEERING MECHANICS/JUNE 2002
`
`section occurs later; the delay in the time of intersection is pro-
`portional to the amount of numerical dissipation in the algorithm.
`The backward differences solution is again the worst; the chain
`never intersects itself. Likewise for X.°° = 0, though it comes closer
`to doing so. For X°°= — 0.7, intersection actually happens at 3.47
`s and for X"= -0.5, at 3.50 s.
`The situation changes somewhat if we consider the effect of
`temporal and spatial discretization. Fig. 7 shows the same time
`points for versions of the box method with n= 100 or 200 and
`Af = 0.01, 0.001, and 0.0001 s. In this case we see that increasing
`the number of nodes does not significantly affect the solution,
`suggesting that n = 100 is adequate to accurately capture the re-
`sponse. An increase in temporal resolution, however, from Ar
`= 0.01 s to Ar=0.001 s, leads to a delay in the crossover to ap-
`proximately 3.46 s. The result at the even smaller Ar = 0.0001 s
`confirms that the solution has converged at these smaller time
`steps. Fig. 8 shows this same behavior for the trapezoidal rule.
`The only notable difference between trapezoidal rule and box
`method solutions is the better smoothness of the trapezoidal rule
`solutions at Ar=0.01 s.
`Similar results for X°° = -0.5 are shown in Fig. 9. In this case,
`the solution at Ar=0.001 s is slightly different than the solutions
`from the trapezoidal rule and the box method at the 3.46 s point.
`The solutions for Ar=0.0001 s are in good agreement with the
`converged solutions for Ar = 0.001 s in Figs. 7 and 8. A notable
`difference in the solutions for the various algorithms does appear
`between 3.5 and 4.0 s (i.e., following crossover). Both trapezoidal
`rule and box method solutions required significant adaptation of
`the time step to get through the collapse of the lower portion of
`the chain following the crossover. The enhanced stability of solu-
`tions with X°°= -0.5 allowed for a smooth numerical solution in
`this region, with no or very little adaptation. Without experimen-
`tal verification, however, we cannot say if the X°°= — 0.5 solution
`is accurate.
`At sufficiently small time steps and adequate spati