l-stable method for differential-algebraic equations of multibody … · 2019. 1. 8. · order...

Research ArticleL-Stable Method for Differential-Algebraic Equations ofMultibody System Dynamics

Bowen Li1 Jieyu Ding 12 and Yanan Li1

1School of Mathematics and Statistics Qingdao University Qingdao 266071 China2Center for Computational Mechanics and Engineering Simulation Qingdao University Qingdao 266071 China

Correspondence should be addressed to Jieyu Ding djyqdueducn

Received 8 January 2019 Revised 22 March 2019 Accepted 28 April 2019 Published 29 May 2019

Academic Editor Samuel N Jator

Copyright copy 2019 Bowen Li et alThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

An L-stable method over time intervals for differential-algebraic equations (DAEs) of multibody system dynamics is presentedin this paper The solution format is established based on equidistant nodes and nonequidistant nodes such as Chebyshev nodesand Legendre nodes Based on Ehlersquos theorem and conjecture the unknown matrix and vector in the L-stable solution formula areobtained by comparison with Pade approximation Newton iterationmethod is used during the solution process Taking the planartwo-link manipulator system as an example the results of L-stable method presented are compared for different number of nodesin the time interval and the step size in the simulation and also compared with the classic Runge-Kutta method A-stable methodRadau IA Radau IIA and Lobatto IIIC methods The results show that the method has the advantages of good stability and highprecision and is suitable for multibody system dynamics simulation under long-term conditions

1 Introduction

The stability of the numerical solution of differential equa-tions is one of the important indicators to judge the qualityof a numerical method In the 1960s after the innovationand development of Dahlquist Widlund Gear and otherscholars the stability theory was deeply developed Dahlquist[1] proposed a linear equation 1199101015840 = 120583119910 as a model equationand proposed A-stable that is |119910119903| 997888rarr 0 119903 997888rarr infin and allℎ gt 0 Subsequently Axellson [2] determines the 119886119894119896 of the119910119894119896 = 119910119899119903minus1 + ℎ sum119899119896=1 119886119894119896119891(119910119896119903) by the quadrature coefficientsat the zeros of 119901119899 minus 119901119899minus1 or 119901119899 minus 119901119899minus2 constructing a classof A-stable methods where 119901119899 is the Legendre polynomialWidlund [3] proposed A(120572)-stable that is if all solutionsof 120572119896119909119899+119896 + + 1205720119909119899 = ℎ(120573119896119891119899+119896 + + 1205730119891119899) tendto zero as 119899 997888rarr infin when the method is applied witha fixed ℎ to any differential equation of the form 1199101015840 =120583119910where 120583 is a complex constant which lies in the set119878120572 = 119911 | arg(minus119911)| lt 120572 119911 = 0 Gear [4] proposedstiffness stable in order to generalize the A-stable conceptto nonlinear problems In 1975 Dahlquist [5] proposed theG-stable concept of single-method linear multistep method

Butcher [6 7] studied the nonlinear stability of Runge-Kuttamethod by using nonlinear equations 1199101015840 = 119891(119910) as modelequations proposed B-stable and algebraic stability of theRunge-Kutta method and established the algebraic stabilitycriteria for general linear methods Li Shoufu [8] establishedthe (kpq) stability theory of the general linear methodand the weak algebraic stability criterion of (kpq) For avery rigid differentialalgebraic equations it is hoped thatthe rigid component can be attenuated as soon as possibleand the stable domain contains the left half complex planeonly A-stable is not enough Ehle [9] proposed L-stableHairer [10] pointed out that the L-stable numerical methodwill introduce damping effect but it is very effective forsuppressing high-frequency oscillation and some scholarshave proposed implicit single-step and multistep methodsbased on L-stable for solving rigid equations but thesemethods are based on ordinary differential equations L-stable methods for DAEs are relatively rare

The typical form of multibody system dynamic equationsis DAEs which consist of ordinary differential equations andalgebraic constraint equations The numerical solution is amatter of concern in the study ofmultibody systemdynamics

HindawiMathematical Problems in EngineeringVolume 2019 Article ID 9283461 11 pageshttpsdoiorg10115520199283461

2 Mathematical Problems in Engineering

At present most of the numerical solutions of the DAEsmainly include Direct integration method and Newmarkmethod Although the calculation process is relatively simplethe stability is poor In the long-term simulation processthe error accumulation is more serious and it needs tobe corrected by combining constrained projection [11] Inthis paper the general form of L-stable method for multi-body system dynamics DAEs is established Based on theequidistant nodes Chebyshev nodes and Legendre nodesthe concrete construction process is given and the planar two-link manipulator system verified and compared

2 DAEs of Multibody System Dynamics

TheDAEs of index-3 in multibody systems dynamics are

M (q) q +ΦTq (q 119905) 120582 minus F (q q 119905) = 0

Φ (q 119905) = 0(1)

where q = [1199021(119905) 1199022(119905) 119902119899(119905)]T are generalized coordi-nates of the system q = dqd119905 are generalized speed ofthe system q = d2qd1199052 are generalized acceleration of thesystem 120582 = [1205821 1205822 120582119898]T are the Lagrange multipliervectors M isin 119877119899times119899 is a symmetric positive definite massmatrix F isin 119877119899 are generalized force vectors and Φ =[Φ1 Φ2 Φ119898]T are 119898 complete displacement constraints

For the constraint equation Φ the first derivative and thesecond derivative are respectively obtained for the time tand the DAEs form of the index-2 and the index-1 can beobtained by the combination of (1)

M (q) q +ΦTq (q 119905)120582 minus F (qq 119905) = 0

Φq (q 119905) q +Φ119905 (q 119905) = 0(2)


Φq (q 119905) q + [Φq (q 119905) q]q q + 2Φq119905 (q 119905) q +Φ119905119905 (q 119905)= 0

(3)

When the initial values q0 = q(0) q0 = q(0) are knownthe Runge-Kutta method can be used to directly solve theindex-1 equation (3) however if the selected step size islarge the accuracy will be reduced and the total energywill increase with time [12] smaller step will lead to lowercomputation efficiency and poor stability and the index-3equation (1) and the index-2 equation (2) cannot be solveddirectly using the Runge-Kutta methodThe L-stable methodis constructed below and can be used to solve the multibodysystem dynamics DAEs of index-1 -2 and -3

3 Construction of the DAEs L-Stable Format

31 General Form of L-Stable Method In 1972 H A Wattsand L F Shampine [13] proposed a block implicit one-stepmethod for solving ordinary differential equations The rdiscrete moments are made into a block vector the output

values of rmoments are obtained from the input values of onemoment and it is proved that the method is A-stable In thismethod the time step in each block vector is equidistant Thisarticle improves on this method the block vector is modifiedto correspond to the point in the single-step interval and inthe iterative solution process a single-step interval [119905119896 119905119896+1]and a time step ℎ = 119905119896+1 minus 119905119896 are set selecting 119905119896 lt 1198881 lt1198882 lt lt 119888119903 = 119905119896+1 and constructing DAEs solution formatwhere 119888119894(119894 = 1 2 119903) can be an equidistant node or anonequidistant node

Taking the multibody system dynamics DAEs of index-1 equation (3) as an example after the order reduction thefollowing first-order differential equations can be obtained

q = z

M (q) z +ΦTq (q 119905)120582 minus F (z q 119905) = 0

Φq (q 119905) z + [Φq (q 119905) z]q z + 2Φq119905 (q 119905) z +Φ119905119905 (q 119905)= 0

(4)

Let y = [qT zT 120582T]T isin 1198772119899+119898 y119894 = y(119888119894) y119894 = y(119888119894) 119894 =1 2 119903 Y = [yT1 yT2 sdot sdot sdot yT119903 ]T Y = [yT1 yT2 sdot sdot sdot yT119903 ]T theconstruction solution formula is as follows

Y = e otimes y119896 + ℎd otimes y119896 + ℎB otimes Y (5)

where e = [1 1 sdot sdot sdot 1]T isin 119877119903 d isin 119877119903 B isin 119877119903times119903 y119896 =y(119905119896) y119896 = y(119905119896) and otimes is Kronecker

Bring (5) into the test equation

y1015840 (119905) = 120572y (119905)y (1199050) = y0

Re (120572) lt 0 (6)

and get stability function119877 (120582) = eT119903 (I119903 minus 120582B)minus1 (e + 120582d) (7)

where 120582 = 120572ℎ I119903 is the r-order unit matrix and e119903 =[0 sdot sdot sdot 0 1] isin 119877119903 For the stability function using thePade approximation we can construct L-stabilized solutionformula

Theorem 1 (119864ℎ119897119890 theorem and conjecture) For (119896 119895) minus 119875119886119889119890approximation if 119896 le 119895 le 119896 + 2 then this Pade approximationis A-stable if 119896 lt 119895 le 119896 + 2 then this Pade approximation isL-stable [14]

Perform a Taylor expansion on y119894 119894 = 1 2 119903 at 119905119896let the coefficient of the y(119903+1)

119896= y(119903+1)(119905119896) term be a pending

constant and omit the following high-order terms one can get

y119894 asymp y119896 + ℎ119894y119896 + ℎ21198942 y119896 + sdot sdot sdot + ℎ119903119894119903 y(119903)119896 + 120583119894(119903 + 1)y(119903+1)119896 119894 = 1 2 sdot sdot sdot 119903 (8)

Mathematical Problems in Engineering 3

where 120583119894 119894 = 1 2 sdot sdot sdot 119903 is pending constant and ℎ119894 = 119888119894minus119905119896 119894 =1 2 sdot sdot sdot 119903 Bringing (8) into (5) one can get the following

[B d] = [[[[[[[[ℎ1 ℎ21 sdot sdot sdot ℎ1199031 1205831ℎ2 ℎ22 sdot sdot sdot ℎ1199032 1205832ℎ119903 ℎ2119903 sdot sdot sdot ℎ119903119903 120583119903

]]]]]]]]sdot [[[[[[[[[[[

1 2ℎ1 3ℎ21 sdot sdot sdot (119903 + 1) ℎ11990311 2ℎ2 3ℎ22 sdot sdot sdot (119903 + 1) ℎ11990321 2ℎ119903 3ℎ2119903 sdot sdot sdot (119903 + 1) ℎ1199031199031 0 0 sdot sdot sdot 0]]]]]]]]]]]

minus1 (9)

From (9) Bd which are expressed using the undeterminedconstant can be obtained bringing these into the stabilityfunction (7) 13e value of the undetermined constant 120583119894 119894 =1 2 sdot sdot sdot 119903 can be obtained by comparisonwith the correspond-ing Pade approximation form so that the value of B d can beobtained bringing into formula (5) a specific solution formula

32 L-Stable Format Based on Equidistant Nodes For theequidistant node 119888119894(119894 = 1 2 119903) there are 119888119894 = 119905119896 + (119894119903)ℎ 119894 =1 2 119903 For example when 119903 = 3 ℎ = 1 the equidistantnode is 119888119894 = 119905119896 + 1198943 119894 = 1 2 3 at this time ℎ119894 = 1198943 119894 = 1 2 3and (9) becomes as follows

[B d] = [[[[[13 19 127 120583123 49 827 12058321 1 1 1205833

]]]]] sdot [[[[[[[[1 23 13 4271 43 43 32271 2 3 41 0 0 0

]]]]]]]]minus1

(10)

Calculate Bd substitute stability function (7) and compareit with (2 3) minus 119875119886119889119890 approximation formula one can solveundetermined constants 1205831 = 145 1205832 = 845 1205833 = 1 andbringing them into the expression B d one can obtain

B = 13 (107120 minus 37120 3401715 815 minus 11598 98 38 ) d = 13 ( 411202538 )

(11)

Bringing them into (5) can be used to obtain a solutionformula The same method can be used to construct L-stablesolution formats for other r values

33 L-Stable Format Based on Nonequidistant Nodes Whenthe node 119888119894(119894 = 1 2 119903) of the interval is the zero point ofthe Chebyshev polynomial within the interval [119905119896 119905119896+1] onegets the following119888119894 = 119905119896 + ℎ2 (1 + cos( 2119894 minus 12 (119903 minus 1)120587)) 119894 = 1 2 119903 minus 1 (12)

Take 119903 = 3 ℎ = 1 as an example get 1198881 = 119905119896 + 12 minus radic241198882 = 119905119896 + 12 + radic241198883 = 1 and bring it into (9) one can get

[B d] = [[[[ℎ1 ℎ21 ℎ31 1205831ℎ2 ℎ22 ℎ32 12058321 1 1 1205833]]]] sdot [[[[[[

1 2ℎ1 3ℎ21 4ℎ311 2ℎ2 3ℎ22 4ℎ321 2 3 41 0 0 0]]]]]]minus1

(13)

where ℎ1 = 12 minus radic24 ℎ2 = 12 + radic24Bring B d into the stability function (7) andmake it equal

to the (2 3) minus 119875119886119889119890 approximation to get 1205831 = (10 minus 5radic2)321205832 = (10 + 5radic2)32 and 1205833 = 1 and get the correspondingBd

B = (11 minus 2radic224 5 minus 8radic224 7radic2 minus 1485 + 8radic224 11 + 2radic224 minus7radic2 + 14823 23 minus16 ) d = (radic2 minus 748 minusradic2 + 748 minus16)119879

(14)

and bringing these into (5) one can get the formulaWhen the node 119888119894(119894 = 1 2 119903) is the zero point of the

Legendre polynomial take 119903 = 3 ℎ = 1 as an example get1198881 = (12)(minusradic33 + 1) 1198882 = (12)(radic33 + 1) 1198883 = 1 get1205831 = (3 minus radic3)12 1205832 = (3 + radic3)12 1205833 = 1 and get thecorresponding Bd

B = ( 27 + radic372 9 minus 17radic372 3 + 5radic3729 + 17radic372 27 minus radic372 3 minus 5radic37212 12 0 )d = (minusradic3 + 372 radic3 minus 372 0)119879

(15)

and bringing these into (5) can be used to obtain a solutionformula The L-stability solution format for nonequidistantnodes of other r values can be constructed using the samemethod

4 Numerical Example

Planar two-link manipulator is shown in Figure 1 the lengthof the link is 1198711 1198712 where 1198711 = 1119898 1198712 = radic3119898 and the


O

Y

X

(x1 y1)2

(x2 y2)

L1 m1

L2 m21

Figure 1 Planar two-link manipulator

minus3 minus2 minus1 0 1 2 3x

minus3

minus2

minus1

0

1

2

3

y

L1 (solid) L2( dotted)

Figure 2 Displacement trajectory at the end of the connecting rod

quality is 1198981 1198982 where 1198981 = 1119896119892 1198982 = 2119896119892 The cornersare denoted by 1205791 and 1205792 and at the initial time 1205791 = 1205873 1205792 =minus1205876 (1199091 1199101)(1199092 1199102) represent the coordinate position ofthe connecting rod centroid

Based on the Lagrange function planar two-link dynam-ics model is established where q = (1199091 1199101 1205791 1199092 1199102 1205792)119879M = diag([1198981 1198981 1198681 1198982 1198982 1198682]) is a generalized massmatrix 1198681 = (112)119898111987121 and 1198682 = (112)119898211987122 arethe moments of inertia of the connecting rod and F =[0 minus1198981119892 0 0 minus1198982119892 0]119879 is a generalized force vector

Here using the (2 4) minus 119875119886119889119890 we take r=4 as an exampleto divide the cells into four equal parts for the solution andthe step size h=001 and the time history 10s are selected andprogrammed using MATLABThe Newton iteration methodis used for calculation and the maximum allowable error is10minus3Thedisplacement trajectory at the end of the connectingrod can be obtained as shown inFigure 2 the starting positionof the connecting rod end is indicated by a circle symbol andthe ending position is represented by a starThe displacementtime course of the connecting rod end is shown in Figure 3

0 1 2 3 4 5 6 7 8 9 10t (s)

minus1

minus05

0

05

1

L1

x1y1

0 1 2 3 4 5 6 7 8 9 10t (s)

minus4

minus2

0

2

4

L2x2y2

Figure 3 Displacement time history of the end of the rod

The total energy kinetic energy and potential energy of thesystem are shown in Figure 4 Displacement constraintsspeed-level constraints and acceleration-level constraints areshown in Figure 5

It can be seen from Figures 2ndash5 that the L-stable methodgiven in this paper can maintain good stability duringthe simulation process and the total energy error and theconstraint error of each level are small which is suitablefor multibody system dynamics simulation The numericalverification and comparison of the method are carried out indifferent situations(1) The simulation time is t=10s the number of identicalequidistant nodes (r=4) is taken in each time interval and theresults of different step sizes are compared in Table 1 Whenthe step size is reduced the time spent on program operationis increasing but the maximum energy error (max |120576119867|)is decreasing at the same time displacement constraintsspeed-level constraints and acceleration-level constraints arebetter when the step size is smaller Therefore in the caseof a short simulation time choosing a smaller step size willimprove the accuracy of the solution and accuracy bettermaintains displacement constraints speed-level constraintsand acceleration-level constraints(2) The simulation time is t=10s and the step size ish=001 selecting the number of different equidistant nodes ineach time interval for comparison (r=34) in Table 2 Whenr=4 although the time spent onprogramoperation is increas-ing the maximum energy error displacement constraintsspeed-level constraints and acceleration-level constraints areless than r=3 therefore an increase in the number of nodescan improve the accuracy


0 1 2 3 4 5 6 7 8 9 10t (s)

1274345

127435

1274355

127436

tota

l ene

rgy

(Nm

)

0 1 2 3 4 5 6 7 8 9 10t (s)

minus50

0

50

100

ener

gies

(Nm

)

total energykinetic energypotential energy

Figure 4 Total energy kinetic energy and potential energy time history of the system

0 1 2 3 4 5 6 7 8 9 10t (s)

minus1

0

1

0 1 2 3 4 5 6 7 8 9 10t (s)

minus5

0

5

0 1 2 3 4 5 6 7 8 9 10t (s)

minus1

0

1

Ｎ

ＮＮ

times10minus6

times10minus7

times10minus11

Figure 5 Time history of displacement and speed-level and acceleration-level constraints

Table 1 Results of L-stable method under different steps

Runtimes max |120576H| displacement constraints speed-level constraints acceleration-level constraintsh=001 58906 70007e-05 68459e-07 37480e-07 56595e-12h=0005 108750 88078e-07 19154e-09 66686e-09 19846e-12h=0001 426406 55074e-11 76766e-13 41800e-13 14211e-13

Table 2 Results of the L-stable method under different nodes

Runtimes max |120576H| displacement constraints speed-level constraints acceleration-level constraintsr=3 16875 00021 58212e-05 39267e-04 62629e-12r=4 58906 70007e-05 68459e-07 37480e-07 56595e-12


Table 3 Comparison of L-stable method and Explicit Runge-Kutta method at the same step size h=001

Runtimes max |120576H| displacement constraints speed-level constraints acceleration-level constraintsL 58906 70007e-05 68459e-07 37480e-07 56595e-12ERK4 03906 02628 00087 00021 99476e-14

0 1 2 3 4 5 6 7 8 9 10t (s)

127

1275

128

1285

129

1295

13

1305

tota

l ene

rgy

(Nm

)

L-Stability block formatRunge-Kutta method

Figure 6 L-stable method and fourth-order Explicit Runge-Kutta method for total system time history (10s)

(3) The simulation time is t=10s and the step size ish=001 We compare L-stable method when r=4 with fourth-order Explicit Runge-Kutta method (ERK4) in Table 3 Inthe L-stable method (L) for solving differential-algebraicequations the maximum energy error the displacementconstraints and the speed-level constraints are obviouslysmaller than those in the fourth-order Explicit Runge-Kuttamethod although the acceleration-level constraints are notas good as those in the fourth-order Explicit Runge-Kuttamethod the total energy of the L-stable method keeps stablewith time and the total energy of the fourth-order ExplicitRunge-Kutta method has been increasing rapidly with timeas in Figures 6 and 7(4) The simulation time is t=10s and the step sizeis h=001 Under the same number of nodes (r=4) thedifferential-algebraic equations of index-1 index-2 andindex-3 are solved in L-stable method in Table 4 The timetaken for the differential-algebraic equations of the L-stabilityblock format to solve index-1 index-2 and index-3 is roughlythe same and the maximum energy errors for index-1 andindex-2 are significantly smaller than those for index-3 Thedisplacement constraints for index-3 are the smallest thespeed-level constraints for index-2 are the smallest and theacceleration- level constraints for index-1 are the smallest Bycomprehensive comparison the overall accuracy of index-1 ishigher

(5) The step size is h=001 under the same step sizethe L-stable method of index-3 selects equidistant nodesto be compared with nonequidistant nodes (Chebyshevnodes Legendre nodes) in Table 5 By comparison it isfound that when the L-stable method selects nonequidis-tant nodes the maximum energy error is smaller thanequidistant nodes and Chebyshev nodes have higher overallaccuracy

5 Accuracy Experiments and Analysis

The differential-algebraic equation of index-3 is used to solvethe planar two-link model in time interval 119905 isin [0 1] takingthe result of fourth-order Explicit Runge-Kutta method ℎ =0000001 as its approximate exact solution and calculate thelogarithmic relationship between the overall error of q andthe step size of the L-stable method at t=1s r=4 as shown inFigure 8 the logarithm of the overall error and step size foreach component of q is shown in Figures 9 and 10 when r=3the logarithmic relation curve of the overall error is shown inFigure 11 the logarithm of the overall error and step size foreach component is shown in Figures 12 and 13 The formulalog 119890 log ℎ is used to calculate the accuracy When r=4 theL-stable method overall error can guarantee the fourth-orderaccuracy When r=3 the L-stable method overall error can


0 10 20 30 40 50 60 70 80 90 100t (s)

10

15

20

25

30

35

40

45

tota

l ene

rgy

(Nm

)



Table 4 Comparison of differential-algebraic equations for L-stable method solving indices-123

Runtimes max |120576H| displacement constraints speed-level constraints acceleration-level constraintsindex -1 58906 70007e-05 68459e-07 37480e-07 56595e-12index -2 54219 32163e-05 43949e-09 17764e-14 00704index -3 56094 00011 27756e-15 59405e-04 09854

Table 5 Comparison of the results of the L-stable method taking uniform and nonuniform nodes

Runtimes max |120576H| displacement constraints speed-level constraints acceleration-level constraintsequidistant nodes 56094 00011 27756e-15 59405e-04 09854Chebyshev nodes 52188 43245e-05 29976e-15 18094e-04 06880Legendre nodes 51094 25327e-04 28866e-15 29047e-04 07560

guarantee the second-order accuracy ie the accuracy of r isgreater

6 Stability Analysis

According to Theorem 1 one can construct A-stable methodand get matrix B and d

B = 14 ((((

20812520 minus109420 1912520 minus 4350406245 415 245 minus 190377280 111140 167280 minus 315606445 815 6445 1445))))

d = 14 (18475040 2990 179560 1445)119879(16)

Taking the planar two-link manipulator as an examplewith the simulation time being t=10s and the step size beingh=001 the differential-algebraic equations for the index-1-2 and -3 are obtained in Table 6 It can be seen that theresults of solving the index-1 -2 are basically the same as thoseobtained by the L-stable method but when solving the index-3 there is a singularity that cannot be solved It can be seenthat the L-stable method is also helpful for solving high-indexproblems

The L-stable method in this paper can be writteninto the corresponding Runge-Kutta method [15] Tak-ing the (1 3) minus 119875119886119889119890 approximation as an example the


minus8 minus75 minus7 minus65 minus6 minus55 minus5 minus45log h

minus30

minus28

minus26

minus24

minus22

minus20

minus18

minus16

minus14

minus12

log

Figure 8 The overall error of q and the logarithm of h

Table 6 Comparison of DAEs for L-stable and A-stable method solving indices-123

index-1 Runtimes max |120576H| displacement constraints speed-level constraints acceleration-level constraintsA-stable 62188 38045e-05 23863e-07 20615e-07 56648e-12L-stable 58906 70007e-05 68459e-07 37480e-07 56595e-12index-2 Runtimes max |120576H| displacement constraints speed-level constraints acceleration-level constraintsA-stable 54688 11145e-05 16059e-09 17729e-14 00214L-stable 54219 32163e-05 43949e-09 17764e-14 00704index-3 Runtimes max |120576H| displacement constraints speed-level constraints acceleration-level constraintsA-stable SingularL-stable 56094 00011 27756e-15 59405e-04 09854

Table 7 Comparison of L-stable method (r=3) with Radau IA Radau IIA and Lobatto IIIC methods

Runtimes max |120576H| displacement constraints speed-level constraints acceleration-level constraintsL-stable 16875 00021 58212e-05 39267e-04 62629e-12Radau IA 149063 14075 01803 01087 98100Radau IIA 135625 01035 21396 82918e-04 98100Lobatto IIIC 139375 01038 21396 00010 98100

corresponding fourth-order Runge-Kutta coefficient table isas follows

0 0 0 0 013 172 4372 minus2972 1823 112 1936 136 1361 18 38 38 1818 38 38 18(17)

The Radau IA Radau IIA and Lobatto IIIC methodsin the Implicit Runge-Kutta method are also L-stabilized[16] The above method is used to calculate the planar two-link manipulator dynamics model In order to compare theresults with the results of this paper the fourth-order methodis selected for calculation The step size is h=001 and thesimulation time is t=10s The comparison results are shownin Table 7

By comparison it is found that running time max-imum energy error displacement constraints speed-levelconstraints and acceleration-level constraints of the L-stablemethod constructed in this paper for solving the multibodysystemdynamicsmodel are better than those in theRadau IA



minus32

minus30

minus28

minus26

minus24

minus22

minus20

minus18

minus16

minus14

log ex1log ey1log e1

log

Figure 9 The overall error of 1199091 1199101 1205791and the logarithm of h



minus32

minus30

minus28

minus26

minus24

minus22

minus20

minus18

minus16

minus14

log

||e||

Figure 10 The overall error of 1199092 1199102 1205792 and the logarithm of h



minus18

minus17

minus16

minus15

minus14

minus13

minus12

minus11

minus10

minus9

minus8

log



minus20

minus18

minus16

minus14

minus12

minus10

minus8

log



Radau IIA and Lobatto IIIC methods of the Implicit Runge-Kutta although these methods are L-stable

7 Conclusion

This paper constructs L-stable method for solvingdifferential-algebraic equations of multibody system dynam-ics The numerical results show that under the equidistant

node when the same number of nodes is selected the smallerthe step size the higher the accuracy and when the step sizeis fixed the more the nodes the higher the accuracy Whencomparedwith the Explicit Runge-Kuttamethod the L-stablemethod better maintains the displacement constraints andspeed-level level constraints the maximum energy error issignificantly smaller than the Explicit Runge-Kutta methodand the total energy retention effect is better under long-term



minus22

minus20

minus18

minus16

minus14

minus12

minus10

minus8

log



simulation When compared with A-stable method L-stablemethod has no singularity in calculating high-index systemsof equations and each index keeps good results Whencompared with Radau IA Radau IIA and Lobatto IIICmethod L-stable method is more efficient and has betterperformance than the above methods The numerical resultsshow that the nonequidistant nodes (Chebyshev nodes andLegendre nodes) result accuracy is higher than equidistantnode The method is simple in mathematics easy to useand easy to program It is easy to implement in computersand worthy of research and promotion How to improve thecalculation efficiency while ensuring the numerical accuracyis the content that needs further study in the future

Data Availability

The data used to support the findings of this study areincluded within the article

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This study has been supported by the National NaturalScience Foundations of China under Grant no 1147214311772166

References

[1] G G Dahlquist ldquoA special stability problem for linearmultistepmethodsrdquo BIT Numerical Mathematics vol 3 pp 27ndash43 1963

[2] O Axelsson ldquoA class of A-stable methodsrdquo BIT Journal vol 9no 3 pp 185ndash199 1969

[3] O B Widlund ldquoA note on unconditionally stable linear mul-tistep methodsrdquo BIT Numerical Mathematics vol 7 pp 65ndash701967

[4] C W Gear 13e Automatic Integration of Stiff Ordinary Differ-ential Equations North Holland publishing Company Amster-dam Netherlands 1963

[5] G Dahlquist Error Analysis for a Class of Methods for Stiff Non-linear Initial Value Problems Springer-Verlag Berlin Germany1975

[6] J C Butcher ldquoA stability property of implicit Runge-Kuttamethodsrdquo BIT Journal vol 15 no 4 pp 358ndash361 1975

[7] K Burrage and J C Butcher ldquoStability criteria for implicitRunge-Kutta methodsrdquo SIAM Journal on Numerical Analysisvol 16 no 1 pp 46ndash57 1979

[8] S F Li ldquoNonlinear stability of general linear methodsrdquo Journalof Computational Mathematics vol 9 no 2 pp 97ndash104 1991

[9] B L Ehle ldquoA-stable methods and Pade Approximations to theexponentialrdquo SIAM Journal onMathematical Analysis vol 4 no5 pp 671ndash680 1973

[10] E Hairer and G Wanner Solving Ordinary Differential Equa-tions II stiff and differential-Algebraic problems Science PressBeijing China 2nd edition 2006

[11] J Y Ding and Z K Pan ldquoGeneralized-120572 projection methodfor differential-algebraic equations of multibody dynamicsrdquoEngineering Mechanics vol 30 no 4 pp 380ndash384 2013

[12] L P Wen C H Yang and H Y Wen ldquoStability and asymp-totic stability of multistep runge-kutta methods for nonlinearfunctional-integro-differential equationsrdquoNatural Science Jour-nal of Xiangtan University vol 40 no 1 pp 1ndash5 2018

[13] H A Watts and L F Shampine ldquoA-stable block implicit one-step methodsrdquo Bit Numerical Mathematics vol 12 no 2 pp252ndash266 1972

[14] Z D Yuan G Fei and D G Liu Numerical Solution of InitialValue Problems for Stiff Ordinary DifferentialEquations SciencePress Beijing China 2016

[15] Z Q Wu ldquoL-Stable methods for numerical solution of struc-tural dynamics equations and multibody dynamics equationsrdquoinChangsha Hunan National University of Defense Technologypp 1ndash132 2009

[16] F S Zhu ldquoA Class of Runge-Kutta methods with L-stability andB-stabilityrdquo Journal of Mathematics vol 21 no 2 pp 183ndash1882001

Hindawiwwwhindawicom Volume 2018

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Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

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Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

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Dierential EquationsInternational Journal of

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Submit your manuscripts atwwwhindawicom


At present most of the numerical solutions of the DAEsmainly include Direct integration method and Newmarkmethod Although the calculation process is relatively simplethe stability is poor In the long-term simulation processthe error accumulation is more serious and it needs tobe corrected by combining constrained projection [11] Inthis paper the general form of L-stable method for multi-body system dynamics DAEs is established Based on theequidistant nodes Chebyshev nodes and Legendre nodesthe concrete construction process is given and the planar two-link manipulator system verified and compared

2 DAEs of Multibody System Dynamics

TheDAEs of index-3 in multibody systems dynamics are

M (q) q +ΦTq (q 119905) 120582 minus F (q q 119905) = 0

Φ (q 119905) = 0(1)

where q = [1199021(119905) 1199022(119905) 119902119899(119905)]T are generalized coordi-nates of the system q = dqd119905 are generalized speed ofthe system q = d2qd1199052 are generalized acceleration of thesystem 120582 = [1205821 1205822 120582119898]T are the Lagrange multipliervectors M isin 119877119899times119899 is a symmetric positive definite massmatrix F isin 119877119899 are generalized force vectors and Φ =[Φ1 Φ2 Φ119898]T are 119898 complete displacement constraints

For the constraint equation Φ the first derivative and thesecond derivative are respectively obtained for the time tand the DAEs form of the index-2 and the index-1 can beobtained by the combination of (1)


Φq (q 119905) q +Φ119905 (q 119905) = 0(2)


Φq (q 119905) q + [Φq (q 119905) q]q q + 2Φq119905 (q 119905) q +Φ119905119905 (q 119905)= 0

(3)

When the initial values q0 = q(0) q0 = q(0) are knownthe Runge-Kutta method can be used to directly solve theindex-1 equation (3) however if the selected step size islarge the accuracy will be reduced and the total energywill increase with time [12] smaller step will lead to lowercomputation efficiency and poor stability and the index-3equation (1) and the index-2 equation (2) cannot be solveddirectly using the Runge-Kutta methodThe L-stable methodis constructed below and can be used to solve the multibodysystem dynamics DAEs of index-1 -2 and -3

3 Construction of the DAEs L-Stable Format

31 General Form of L-Stable Method In 1972 H A Wattsand L F Shampine [13] proposed a block implicit one-stepmethod for solving ordinary differential equations The rdiscrete moments are made into a block vector the output

values of rmoments are obtained from the input values of onemoment and it is proved that the method is A-stable In thismethod the time step in each block vector is equidistant Thisarticle improves on this method the block vector is modifiedto correspond to the point in the single-step interval and inthe iterative solution process a single-step interval [119905119896 119905119896+1]and a time step ℎ = 119905119896+1 minus 119905119896 are set selecting 119905119896 lt 1198881 lt1198882 lt lt 119888119903 = 119905119896+1 and constructing DAEs solution formatwhere 119888119894(119894 = 1 2 119903) can be an equidistant node or anonequidistant node

Taking the multibody system dynamics DAEs of index-1 equation (3) as an example after the order reduction thefollowing first-order differential equations can be obtained

q = z

M (q) z +ΦTq (q 119905)120582 minus F (z q 119905) = 0

Φq (q 119905) z + [Φq (q 119905) z]q z + 2Φq119905 (q 119905) z +Φ119905119905 (q 119905)= 0

(4)

Let y = [qT zT 120582T]T isin 1198772119899+119898 y119894 = y(119888119894) y119894 = y(119888119894) 119894 =1 2 119903 Y = [yT1 yT2 sdot sdot sdot yT119903 ]T Y = [yT1 yT2 sdot sdot sdot yT119903 ]T theconstruction solution formula is as follows

Y = e otimes y119896 + ℎd otimes y119896 + ℎB otimes Y (5)

where e = [1 1 sdot sdot sdot 1]T isin 119877119903 d isin 119877119903 B isin 119877119903times119903 y119896 =y(119905119896) y119896 = y(119905119896) and otimes is Kronecker

Bring (5) into the test equation

y1015840 (119905) = 120572y (119905)y (1199050) = y0

Re (120572) lt 0 (6)

and get stability function119877 (120582) = eT119903 (I119903 minus 120582B)minus1 (e + 120582d) (7)

where 120582 = 120572ℎ I119903 is the r-order unit matrix and e119903 =[0 sdot sdot sdot 0 1] isin 119877119903 For the stability function using thePade approximation we can construct L-stabilized solutionformula

Theorem 1 (119864ℎ119897119890 theorem and conjecture) For (119896 119895) minus 119875119886119889119890approximation if 119896 le 119895 le 119896 + 2 then this Pade approximationis A-stable if 119896 lt 119895 le 119896 + 2 then this Pade approximation isL-stable [14]

Perform a Taylor expansion on y119894 119894 = 1 2 119903 at 119905119896let the coefficient of the y(119903+1)

119896= y(119903+1)(119905119896) term be a pending

constant and omit the following high-order terms one can get

y119894 asymp y119896 + ℎ119894y119896 + ℎ21198942 y119896 + sdot sdot sdot + ℎ119903119894119903 y(119903)119896 + 120583119894(119903 + 1)y(119903+1)119896 119894 = 1 2 sdot sdot sdot 119903 (8)




]]]]]]]]sdot [[[[[[[[[[[


minus1 (9)



[B d] = [[[[[13 19 127 120583123 49 827 12058321 1 1 1205833

]]]]] sdot [[[[[[[[1 23 13 4271 43 43 32271 2 3 41 0 0 0

]]]]]]]]minus1

(10)


B = 13 (107120 minus 37120 3401715 815 minus 11598 98 38 ) d = 13 ( 411202538 )

(11)




[B d] = [[[[ℎ1 ℎ21 ℎ31 1205831ℎ2 ℎ22 ℎ32 12058321 1 1 1205833]]]] sdot [[[[[[

1 2ℎ1 3ℎ21 4ℎ311 2ℎ2 3ℎ22 4ℎ321 2 3 41 0 0 0]]]]]]minus1

(13)




(14)




(15)


4 Numerical Example



O

Y

X

(x1 y1)2

(x2 y2)

L1 m1

L2 m21



minus3

minus2

minus1

0

1

2

3

y






0 1 2 3 4 5 6 7 8 9 10t (s)

minus1

minus05

0

05

1

L1

x1y1

0 1 2 3 4 5 6 7 8 9 10t (s)

minus4

minus2

0

2

4

L2x2y2





0 1 2 3 4 5 6 7 8 9 10t (s)

1274345

127435

1274355

127436

tota

l ene

rgy

(Nm

)

0 1 2 3 4 5 6 7 8 9 10t (s)

minus50

0

50

100

ener

gies

(Nm

)



0 1 2 3 4 5 6 7 8 9 10t (s)

minus1

0

1

0 1 2 3 4 5 6 7 8 9 10t (s)

minus5

0

5

0 1 2 3 4 5 6 7 8 9 10t (s)

minus1

0

1

Ｎ

ＮＮ

times10minus6

times10minus7

times10minus11









0 1 2 3 4 5 6 7 8 9 10t (s)

127

1275

128

1285

129

1295

13

1305

tota

l ene

rgy

(Nm

)








0 10 20 30 40 50 60 70 80 90 100t (s)

10

15

20

25

30

35

40

45

tota

l ene

rgy

(Nm

)










B = 14 ((((


d = 14 (18475040 2990 179560 1445)119879(16)





minus30

minus28

minus26

minus24

minus22

minus20

minus18

minus16

minus14

minus12

log







0 0 0 0 013 172 4372 minus2972 1823 112 1936 136 1361 18 38 38 1818 38 38 18(17)





minus32

minus30

minus28

minus26

minus24

minus22

minus20

minus18

minus16

minus14


log




minus32

minus30

minus28

minus26

minus24

minus22

minus20

minus18

minus16

minus14

log

||e||




minus18

minus17

minus16

minus15

minus14

minus13

minus12

minus11

minus10

minus9

minus8

log



minus20

minus18

minus16

minus14

minus12

minus10

minus8

log




7 Conclusion





minus22

minus20

minus18

minus16

minus14

minus12

minus10

minus8

log




Data Availability




Acknowledgments


References
























Journal of












Journal of







Volume 2018






Volume 2018











]]]]]]]]sdot [[[[[[[[[[[


minus1 (9)



[B d] = [[[[[13 19 127 120583123 49 827 12058321 1 1 1205833

]]]]] sdot [[[[[[[[1 23 13 4271 43 43 32271 2 3 41 0 0 0

]]]]]]]]minus1

(10)


B = 13 (107120 minus 37120 3401715 815 minus 11598 98 38 ) d = 13 ( 411202538 )

(11)




[B d] = [[[[ℎ1 ℎ21 ℎ31 1205831ℎ2 ℎ22 ℎ32 12058321 1 1 1205833]]]] sdot [[[[[[

1 2ℎ1 3ℎ21 4ℎ311 2ℎ2 3ℎ22 4ℎ321 2 3 41 0 0 0]]]]]]minus1

(13)




(14)




(15)


4 Numerical Example



O

Y

X

(x1 y1)2

(x2 y2)

L1 m1

L2 m21



minus3

minus2

minus1

0

1

2

3

y






0 1 2 3 4 5 6 7 8 9 10t (s)

minus1

minus05

0

05

1

L1

x1y1

0 1 2 3 4 5 6 7 8 9 10t (s)

minus4

minus2

0

2

4

L2x2y2





0 1 2 3 4 5 6 7 8 9 10t (s)

1274345

127435

1274355

127436

tota

l ene

rgy

(Nm

)

0 1 2 3 4 5 6 7 8 9 10t (s)

minus50

0

50

100

ener

gies

(Nm

)



0 1 2 3 4 5 6 7 8 9 10t (s)

minus1

0

1

0 1 2 3 4 5 6 7 8 9 10t (s)

minus5

0

5

0 1 2 3 4 5 6 7 8 9 10t (s)

minus1

0

1

Ｎ

ＮＮ

times10minus6

times10minus7

times10minus11









0 1 2 3 4 5 6 7 8 9 10t (s)

127

1275

128

1285

129

1295

13

1305

tota

l ene

rgy

(Nm

)








0 10 20 30 40 50 60 70 80 90 100t (s)

10

15

20

25

30

35

40

45

tota

l ene

rgy

(Nm

)










B = 14 ((((


d = 14 (18475040 2990 179560 1445)119879(16)





minus30

minus28

minus26

minus24

minus22

minus20

minus18

minus16

minus14

minus12

log







0 0 0 0 013 172 4372 minus2972 1823 112 1936 136 1361 18 38 38 1818 38 38 18(17)





minus32

minus30

minus28

minus26

minus24

minus22

minus20

minus18

minus16

minus14


log




minus32

minus30

minus28

minus26

minus24

minus22

minus20

minus18

minus16

minus14

log

||e||




minus18

minus17

minus16

minus15

minus14

minus13

minus12

minus11

minus10

minus9

minus8

log



minus20

minus18

minus16

minus14

minus12

minus10

minus8

log




7 Conclusion





minus22

minus20

minus18

minus16

minus14

minus12

minus10

minus8

log




Data Availability




Acknowledgments


References
























Journal of












Journal of







Volume 2018






Volume 2018









O

Y

X

(x1 y1)2

(x2 y2)

L1 m1

L2 m21



minus3

minus2

minus1

0

1

2

3

y






0 1 2 3 4 5 6 7 8 9 10t (s)

minus1

minus05

0

05

1

L1

x1y1

0 1 2 3 4 5 6 7 8 9 10t (s)

minus4

minus2

0

2

4

L2x2y2





0 1 2 3 4 5 6 7 8 9 10t (s)

1274345

127435

1274355

127436

tota

l ene

rgy

(Nm

)

0 1 2 3 4 5 6 7 8 9 10t (s)

minus50

0

50

100

ener

gies

(Nm

)



0 1 2 3 4 5 6 7 8 9 10t (s)

minus1

0

1

0 1 2 3 4 5 6 7 8 9 10t (s)

minus5

0

5

0 1 2 3 4 5 6 7 8 9 10t (s)

minus1

0

1

Ｎ

ＮＮ

times10minus6

times10minus7

times10minus11









0 1 2 3 4 5 6 7 8 9 10t (s)

127

1275

128

1285

129

1295

13

1305

tota

l ene

rgy

(Nm

)








0 10 20 30 40 50 60 70 80 90 100t (s)

10

15

20

25

30

35

40

45

tota

l ene

rgy

(Nm

)










B = 14 ((((


d = 14 (18475040 2990 179560 1445)119879(16)





minus30

minus28

minus26

minus24

minus22

minus20

minus18

minus16

minus14

minus12

log







0 0 0 0 013 172 4372 minus2972 1823 112 1936 136 1361 18 38 38 1818 38 38 18(17)





minus32

minus30

minus28

minus26

minus24

minus22

minus20

minus18

minus16

minus14


log




minus32

minus30

minus28

minus26

minus24

minus22

minus20

minus18

minus16

minus14

log

||e||




minus18

minus17

minus16

minus15

minus14

minus13

minus12

minus11

minus10

minus9

minus8

log



minus20

minus18

minus16

minus14

minus12

minus10

minus8

log




7 Conclusion





minus22

minus20

minus18

minus16

minus14

minus12

minus10

minus8

log




Data Availability




Acknowledgments


References
























Journal of












Journal of







Volume 2018






Volume 2018









0 1 2 3 4 5 6 7 8 9 10t (s)

1274345

127435

1274355

127436

tota

l ene

rgy

(Nm

)

0 1 2 3 4 5 6 7 8 9 10t (s)

minus50

0

50

100

ener

gies

(Nm

)



0 1 2 3 4 5 6 7 8 9 10t (s)

minus1

0

1

0 1 2 3 4 5 6 7 8 9 10t (s)

minus5

0

5

0 1 2 3 4 5 6 7 8 9 10t (s)

minus1

0

1

Ｎ

ＮＮ

times10minus6

times10minus7

times10minus11









0 1 2 3 4 5 6 7 8 9 10t (s)

127

1275

128

1285

129

1295

13

1305

tota

l ene

rgy

(Nm

)








0 10 20 30 40 50 60 70 80 90 100t (s)

10

15

20

25

30

35

40

45

tota

l ene

rgy

(Nm

)










B = 14 ((((


d = 14 (18475040 2990 179560 1445)119879(16)





minus30

minus28

minus26

minus24

minus22

minus20

minus18

minus16

minus14

minus12

log







0 0 0 0 013 172 4372 minus2972 1823 112 1936 136 1361 18 38 38 1818 38 38 18(17)





minus32

minus30

minus28

minus26

minus24

minus22

minus20

minus18

minus16

minus14


log




minus32

minus30

minus28

minus26

minus24

minus22

minus20

minus18

minus16

minus14

log

||e||




minus18

minus17

minus16

minus15

minus14

minus13

minus12

minus11

minus10

minus9

minus8

log



minus20

minus18

minus16

minus14

minus12

minus10

minus8

log




7 Conclusion





minus22

minus20

minus18

minus16

minus14

minus12

minus10

minus8

log




Data Availability




Acknowledgments


References
























Journal of












Journal of







Volume 2018






Volume 2018











0 1 2 3 4 5 6 7 8 9 10t (s)

127

1275

128

1285

129

1295

13

1305

tota

l ene

rgy

(Nm

)








0 10 20 30 40 50 60 70 80 90 100t (s)

10

15

20

25

30

35

40

45

tota

l ene

rgy

(Nm

)










B = 14 ((((


d = 14 (18475040 2990 179560 1445)119879(16)





minus30

minus28

minus26

minus24

minus22

minus20

minus18

minus16

minus14

minus12

log







0 0 0 0 013 172 4372 minus2972 1823 112 1936 136 1361 18 38 38 1818 38 38 18(17)





minus32

minus30

minus28

minus26

minus24

minus22

minus20

minus18

minus16

minus14


log




minus32

minus30

minus28

minus26

minus24

minus22

minus20

minus18

minus16

minus14

log

||e||




minus18

minus17

minus16

minus15

minus14

minus13

minus12

minus11

minus10

minus9

minus8

log



minus20

minus18

minus16

minus14

minus12

minus10

minus8

log




7 Conclusion





minus22

minus20

minus18

minus16

minus14

minus12

minus10

minus8

log




Data Availability




Acknowledgments


References
























Journal of












Journal of







Volume 2018






Volume 2018









0 10 20 30 40 50 60 70 80 90 100t (s)

10

15

20

25

30

35

40

45

tota

l ene

rgy

(Nm

)










B = 14 ((((


d = 14 (18475040 2990 179560 1445)119879(16)





minus30

minus28

minus26

minus24

minus22

minus20

minus18

minus16

minus14

minus12

log







0 0 0 0 013 172 4372 minus2972 1823 112 1936 136 1361 18 38 38 1818 38 38 18(17)





minus32

minus30

minus28

minus26

minus24

minus22

minus20

minus18

minus16

minus14


log




minus32

minus30

minus28

minus26

minus24

minus22

minus20

minus18

minus16

minus14

log

||e||




minus18

minus17

minus16

minus15

minus14

minus13

minus12

minus11

minus10

minus9

minus8

log



minus20

minus18

minus16

minus14

minus12

minus10

minus8

log




7 Conclusion





minus22

minus20

minus18

minus16

minus14

minus12

minus10

minus8

log




Data Availability




Acknowledgments


References
























Journal of












Journal of







Volume 2018






Volume 2018










minus30

minus28

minus26

minus24

minus22

minus20

minus18

minus16

minus14

minus12

log







0 0 0 0 013 172 4372 minus2972 1823 112 1936 136 1361 18 38 38 1818 38 38 18(17)





minus32

minus30

minus28

minus26

minus24

minus22

minus20

minus18

minus16

minus14


log




minus32

minus30

minus28

minus26

minus24

minus22

minus20

minus18

minus16

minus14

log

||e||




minus18

minus17

minus16

minus15

minus14

minus13

minus12

minus11

minus10

minus9

minus8

log



minus20

minus18

minus16

minus14

minus12

minus10

minus8

log




7 Conclusion





minus22

minus20

minus18

minus16

minus14

minus12

minus10

minus8

log




Data Availability




Acknowledgments


References
























Journal of












Journal of







Volume 2018






Volume 2018










minus32

minus30

minus28

minus26

minus24

minus22

minus20

minus18

minus16

minus14


log




minus32

minus30

minus28

minus26

minus24

minus22

minus20

minus18

minus16

minus14

log

||e||




minus18

minus17

minus16

minus15

minus14

minus13

minus12

minus11

minus10

minus9

minus8

log



minus20

minus18

minus16

minus14

minus12

minus10

minus8

log




7 Conclusion





minus22

minus20

minus18

minus16

minus14

minus12

minus10

minus8

log




Data Availability




Acknowledgments


References
























Journal of












Journal of







Volume 2018






Volume 2018










minus18

minus17

minus16

minus15

minus14

minus13

minus12

minus11

minus10

minus9

minus8

log



minus20

minus18

minus16

minus14

minus12

minus10

minus8

log




7 Conclusion





minus22

minus20

minus18

minus16

minus14

minus12

minus10

minus8

log




Data Availability




Acknowledgments


References
























Journal of












Journal of







Volume 2018






Volume 2018










minus22

minus20

minus18

minus16

minus14

minus12

minus10

minus8

log




Data Availability




Acknowledgments


References
























Journal of












Journal of







Volume 2018






Volume 2018















Journal of












Journal of







Volume 2018






Volume 2018








l-stable method for differential-algebraic equations of multibody … · 2019. 1. 8. · order...

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