Research ArticleA Structural Pounding Formulation Using SystematicModal Truncation

Jianye Shi , Franz Bamer , and BerndMarkert

Institute of General Mechanics, RWTH Aachen University, Templergraben 64, 52062 Aachen, Germany

Correspondence should be addressed to Franz Bamer; bamer@iam.rwth-aachen.de

Received 28 July 2017; Revised 8 March 2018; Accepted 25 March 2018; Published 7 May 2018

Academic Editor: Naveed Ahmad

Copyright © 2018 Jianye Shi et al.This is an open access article distributed under the Creative CommonsAttribution License, whichpermits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The evaluation of the response function of the structural pounding problem is generally time-consuming if high-order systemsare applied. The well-known modal truncation strategy is outstandingly efficient for a single linear ground-accelerated structure.However, for the analysis of the structural pounding problem, the classical modal truncation technique turns out to be ineffectiveas additional higher frequency motion due to possible contact impact occurs. This makes the determination of how many modesshould be taken into account in order to obtain a required level of accuracy more difficult. Therefore, in this paper, a systematicallycontrolled modal truncation strategy adapted to the seismic pounding formulation under consideration of high nonlinearity andnonsmoothness of contact problems is introduced. A comparative study of the classical and the controlled modal truncationtechnique is presented and a comparison with the commercial software package ABAQUS© is provided. It is shown that thecomputational accuracy is significantly improved when applying the new systematically controlled modal truncation strategy.

1. Introduction

It has been observed that the motion of buildings due toearthquake excitation may not cause as much damage tobuildings as the result of possible pounding between them[1–4]. Pounding occurs if the separation distance betweenadjacent buildings is not sufficient in order to compensate fortheir relative movements. It is dynamically complex and evenchaotic behavior that occurs if the eigenfrequencies of thesingle buildings differ considerably [5]. Therefore, numericalsimulations of the dynamic pounding problem are necessaryin order to understand the characteristics of structures basedonwhich guidelines for structural design can be provided [6–8].

Due to the high number of degrees of freedom and thenonlinearity as well as nonsmoothness of the contact prob-lem, the evaluation of the multiple structure pounding istime-consuming [9, 10]. As a result, the evaluation of the re-sponse to a whole frequency range can be computationallydisproportionately expensive. Thus, low-order models arenecessary that guarantee a certain level of accuracy.

Some studies have investigated model order reductiontechniques based on modal truncation strategies and the

proper orthogonal decomposition for structures with non-linear materials. However, they applied those strategies onlyto dynamical single body systems without considerationof multibody pounding; see [11–17]. Only limited researchstudies have been conducted considering model reductiontechniques for structural pounding problems. In the work ofZucca [18], another model order reduction technique basedon dynamic substructuring technique is applied to the forcedresponse of structures with contact interfaces. In the work ofGeradin and Rixen [19], a nodeless dual-superelement is for-mulated to reduce the degrees of freedom of the unilateralcontact problems. Most of the researches focus on the mod-eling andmitigation of the constitutive pounding law [1, 3, 8];few of them attempt to apply model reduction techniques forthe pounding problems.

The aim of this paper is to provide a low-order repre-sentation of structural pounding formulation by applying theclassical and a systematically controlled modal truncationtechnique considering dynamic contact impact interaction.The theoretical principles and mathematical formulation aswell as the solution methods in time domain are introducedin Section 2. A representative academic model, an ensem-ble of three adjacent frame structures, is demonstrated in

HindawiShock and VibrationVolume 2018, Article ID 6378085, 15 pageshttps://doi.org/10.1155/2018/6378085

2 Shock and Vibration

Section 2.2. In Section 3, the numerical investigations of thedynamic model are presented. In order to verify the in-house software, a comparison with the commercial softwareABAQUS© for the solution is given. Convergence studies andfrequency response analyses as well as error estimations arecarried out. The full and the reduced solutions are comparedconcerning efficiency and error of the displacements, accel-eration, and contact forces.The effectiveness of the improvedmethod is verified by performing a frequency response anal-ysis applying different number of considered modes forthe reduced computation. Additionally, the proposed con-trolled modal truncation strategy is examined on three realearthquake records. Conclusions and outlook of the demon-strated methods are presented in Section 4.

2. Mechanical Formulation

2.1. Initial Boundary Value Problem (IBVP) forMultiple Pounding

2.1.1. Strong Form. The problem statement associated withthe kinematic linear frictionless contact problem for thethree demonstrated deformable bodies is summarized in thefollowing equations. The IBVP for a dynamic system withfinite deformation must be satisfied for each body (cf. [20]):

divT(𝑖) + b(𝑖)0 = 𝜌(𝑖)u(𝑖) in Ω(𝑖)𝑡 × [0, 𝑇] ,u(𝑖) = u(𝑖) on 𝛾(𝑖)𝑢 × [0, 𝑇] ,

T(𝑖)n(𝑖) = t(𝑖) on 𝛾(𝑖)𝜎 × [0, 𝑇] ,u(𝑖) (𝑡0) = u(𝑖)0 in Ω(𝑖)0 ,u(𝑖) (𝑡0) = k(𝑖)0 in Ω(𝑖)0 .

(1)

The boundary Γ(𝑖) of each body is divided into three disjointparts, shown in Figure 1:

Γ(𝑖) = Γ(𝑖)𝑢 ∪ Γ(𝑖)𝜎 ∪ Γ(𝑖)𝑐 ,Γ(𝑖)𝑢 ∩ Γ(𝑖)𝜎 ∩ Γ(𝑖)𝑐 = ⌀, (2)

where the Dirichlet, Neumann, and contact boundary con-ditions are defined on Γ(𝑖)𝑢 , Γ(𝑖)𝜎 , Γ(𝑖)𝑐 , respectively. It should bepointed out that one body can have more contact surfaces;see Figure 1(b). Conventionally, one defines slave and mastersurfaces for the contact problem with superscripts (1) and (2),respectively; see [20, 21]. Generally, the definition of masterand slave is arbitrary. In the demonstrated formulation,Figure 1(b) plays a role as master with respect to both Figures1(a) and 1(c). This convention will be kept to describe theacademic test example and contact constraints in Sections 2.2and 2.3.

Taking small deformation into consideration, the lin-earized Green-Lagrangian strain can be expressed as

E(𝑖) = 12 (Grad u(𝑖) + Grad𝑇u(𝑖)) . (3)

Linear-elastic material is introduced; therefore, the constitu-tive relation between strain and displacement under smalldeformation is given as

T(𝑖) = C(𝑖)E(𝑖), (4)

where T is the Cauchy stress tensor and the tensor C isthe forth-order elasticity tensor. Karush-Kuhn-Tucker (KKT)conditions are applied to describe the contact constraints in astrong sense, which are in case of normal frictionless contact[21]

𝑔𝑛 ≥ 0,𝑝𝑛 ≤ 0,

𝑝𝑛𝑔𝑛 = 0on 𝛾(𝑖)𝑐 × [0, 𝑇] ,

t𝜏 = 0 on 𝛾(𝑖)𝑐 × [0, 𝑇] .

(5)

The scalar magnitude 𝑔𝑛 is the gap function between twoparticipants, which is determined by using the closest pointprojection.Themagnitude𝑝𝑛 is the contact traction acting onthe slave surfaces and the vector t𝜏 is the tangential tractiondescribing the friction and other viscous tractions. In thispaper, friction is neglected.

2.1.2. Weak Form. Based on the strong form of contactproblems formultiple deformable bodies, the weak statementcan be achieved in a variational sense using the principle ofvirtual displacement. Bymultiplying the virtual displacementas test functions with the local momentum balance in (1), onegets the weak form of the pounding system with frictionlesscontact

𝛿Wkin (u(𝑖), 𝛿u(𝑖)) + 𝛿Wint (u(𝑖), 𝛿u(𝑖))− 𝛿Wext (u(𝑖), 𝛿u(𝑖)) − 𝛿Wcon (u, 𝛿u) = 0, (6)

where 𝛿Wkin stands for the virtual work of the inertia columnforces, 𝛿Wint is the virtual work of the internal stresses, 𝛿Wextis the virtual work of the external forces, and 𝛿Wcon stands forthe virtual contact work. They are given as

𝛿Wkin = 3∑1

[∫Ω(𝑖)0

𝜌(𝑖)0 u(𝑖) ⋅ 𝛿u(𝑖)d𝑉0] ,

𝛿Wint = 3∑1

[∫Ω(𝑖)0

T(𝑖) ⋅ 𝛿E(𝑖)d𝑉0] ,

𝛿Wext = 3∑1

[∫Ω(𝑖)0

t(𝑖) ⋅ 𝛿u(𝑖)d𝑉0 + ∫Γ(𝑖)𝜎

b(𝑖)0 ⋅ 𝛿u(𝑖)dΓ(𝑖)] ,

𝛿Wcon = 3∑1

[∫Γ(𝑖)𝑐

t(𝑖)𝑁 ⋅ 𝛿u(𝑖)dΓ(𝑖)] .

(7)

It should be pointed out that the weak form is also valid foreach separate body Ω(𝑖). There are many possible ways to

Shock and Vibration 3

Table 1: Cross sections columns ([m]), cross sections beams ([m]), Young’s modulus of the columns (𝐸𝑐 [N/m2]), and Young’s modulus ofthe beams (𝐸𝑏 [N/m2]) of the frame system (FS).

FS Columns 𝐸𝑐 Beams 𝐸𝑏1 Hollow 2.1𝑒11 Full 3.5𝑒100.2 × 0.2, 𝑡 = 0.01 0.3 × 4.02 Hollow 2.1𝑒11 Full 2.1𝑒110.3 × 0.3, 𝑡 = 0.03 0.3 × 0.33 Hollow 2.1𝑒11 Full 3.5𝑒100.2 × 0.2, 𝑡 = 0.02 0.3 × 4.0

Γ(1)

Γ(2)

Γ(3)

Ω(1)0

Ω(2)0

Ω(3)0

Ω(1)t

Ω(2)t

Ω(3)t

Γ(1)c

Γ(1)u

Γ(2)u

Γ(3)u

X(1)

X(3)

u(1)(X(1) , t)

u(3)(X(3) , t)

(1)

(2)

(3)

x(1)

(1)u

(2)u

(3)u

c (x(1))

c (x(3))

(1)c

(2)c

(3)c

nc(x(1))

nc(x(3))

c (x(1))

c (x(3))

e3e2

e1

X(2)a

X(2)b

Γ(2)c

Γ(3)c

u(2)a (X(2) , t)

u(2)b (X(2) , t)

x(2)a

x(2)b

x(3)

(a)

(b)

(c)

Figure 1: Kinematics of a multiple pounding problem.

contribute to the contact virtual work such as Lagrange mul-tiplier method, augmented Lagrange multiplier method, ormortar method [20, 21]. In this work, the contact constraintsare regularized by the penalty method, which is introducedin Section 2.3.

2.2. An Academic Multiple Pounding Benchmark Example. Asystem of two three-story frames and one higher four-storyadjacent frame is selected to represent the academic pound-ing benchmark example. The selected frames are modeledby column-beam elements. The frames are arranged with aninitial gap of 𝑔0,1 = 𝑔0,2 = 0.2m. The widths and the heights

of the frames are 𝑙1 = 6.0m, 𝑙2 = 5.0m, and 𝑙3 = 5.0m andℎ1 = 3.0m, ℎ2 = 3.2m, and ℎ3 = 3.4m. This is depicted inFigure 2. Different cross sections as well as different stiffnessparameters are chosen for the columns and beams.Themeas-urements and the material parameters of the cross sectionsare presented in Table 1.

The structures are discretized by two node structuralbeam/truss elements with three degrees of freedom per node,that is, axial and vertical local displacements and rotations (intotal 978 degrees of freedom).

In order to keep the convention of indices for slave andmaster surfaces in Section 2.1, the contact domain between

4 Shock and Vibration

rep. node

ℎ1

ℎ3

ℎ2

ℎ2

ℎ2

ℎ2

ℎ3

ℎ3

ℎ1

ℎ1 1 2 3

l1 l1

l2 l2

l3 l3

Γ(1)c1,2

Γ(1)c2,3

Γ(2)c1,2

Γ(2)c2,3

g0,2g0,1 xg

Figure 2: Dynamic system of adjacent structures; frame height measures ℎ1, ℎ2, and ℎ3; frame width measures 𝑙1, 𝑙2, and 𝑙3; initial gap 𝑔0,1;initial gap 𝑔0,2; representative output node (rep. node); contact surfaces Γ(1)𝑐1,2 , Γ(2)𝑐1,2 , Γ(1)𝑐2,3 , and Γ(2)𝑐2,3 .

frameS1 and frameS2 will be marked as Γ(1)𝑐1,2 (slave side) andΓ(2)𝑐1,2 (master side) and, furthermore, Γ(1)𝑐2,3 (slave side) and Γ(2)𝑐2,3(master side) for the contact domain between frame S2 andframe S3, rather than Γ(1)𝑐 , Γ(2)𝑐 , and Γ(3)𝑐 ; see Figure 2. Ana priori master-slave separation is possible in this academiccase.

2.3. Method of Contact Constraint Enforcement. Poundingbetween two adjacent objects is usuallymodeled by introduc-ing the impact elements, which are widely used to describethe contact force if collision occurs (cf. [1, 3, 8, 22–24]).Mostly pounding elements with a linear or a nonlinear springdamper system, following the Hertz law, are applied (cf.[2, 25–29]). In this paper, a pounding element with a linearspring is chosen to describe the impact forces in bothhorizontal and vertical directions by interpolating the nodetractions. This is equivalent to the penalty method, using thepenalty parameter for the stiffness of the spring.This methodenjoys the particular advantage of removing the constraintsexplicitly from the variational formulation; see [21]. Thecontact constraints, regularized by the penalty method, arebriefly introduced.

As mentioned in Section 2.1, the slave side is labeled byindex (1) and themaster side by index (2), respectively. Basedon the node-to-segment formulation, the gap function men-tioned in Section 2.1 is defined by solving the problem of theminimumdistance between the slave point x(1)𝑃 and the oppo-site point on the master surface 𝛾2𝑐 of the discretized system;see Figure 3. The possible element of the master surface canbe defined through the equation

x(2) (𝜉) = x(2)1 + (x(2)2 − x(2)1 ) 𝜉. (8)

The gap function is then defined as

𝑔𝑛 = (x(1)𝑃 − (1 − 𝜉∗) x(2)1 − 𝜉∗x(2)2 ) ⋅ n(2), (9)

where 𝜉∗ is the parameter of the intersection point coordi-nate. The contact traction can be described as the productof the penalty parameter 𝜀𝑛 and the gap function 𝑔𝑛. Thenonconform meshes between two bodies are combined inthis sense. Inserting the gap function into the weak form (7),the virtual contact work can be formulated as

𝛿Wcon = 3∑1

[∫Γ(𝑖)𝑐

𝜀(𝑖)𝑛 ⟨𝑔𝑛⟩ 𝛿𝑔𝑛dΓ(𝑖)] , (10)

where the bracket ⟨⋅⟩ is the Macaulay bracket, which means

⟨𝑥⟩ fl {{{𝑥, if 𝑥 ≥ 0,0, if 𝑥 < 0. (11)

Evaluation of the volume integration of the individual termin (7) delivers the explicit formulation of the set of equationsof motion describing the discretized dynamic system

Mu + Cu + Ku + Fc (u) = F (𝑡) , (12)

where M, C, and K are the global mass and damping andstiffness matrices, respectively. Rayleigh-Damping is chosenfor the damping matrix C, which is proportional to the massand stiffness matrix:

C = 𝜇M + 𝜆K, (13)

Shock and Vibration 5

x(1)P

(1)c

x(2)2

x(2)1

n(2)

gn

(2)c

Figure 3: Illustration of the contact zone and the gap function.

where the second and the fifthmodes are used to evaluate thecoefficients 𝜇 and 𝜆:

𝜇 = 2𝜉 𝜔2𝜔5𝜔2 + 𝜔5 ,𝜆 = 2𝜉𝜔2 + 𝜔5 ,

(14)

with a damping ratio 𝜉 = 4%, 𝜔2 = 1.796 rad/s, and 𝜔5 =7.085 rad/s. The contact force is then directly expressed asa function of the displacement field by the penalty method,which couples all degrees of freedom of all frames.The initialconditions are assumed to be zero for both the displacementsand velocities.

2.4. Modal Truncation Strategies. The global set of equationsof motion (12) is solved numerically. Dependent on the num-ber of degrees of freedom and the performance of the contactsearch algorithms, this is generally time-consuming. Thus,the simulation of the system response for the whole rangeof frequencies is even much more challenging. Therefore,solution strategies applying the classical and a systematicallycontrolled modal truncation technique are introduced.

2.4.1. Classical Modal Truncation. The undamped homoge-neous set of equations of motion Mu + Ku = 0 leads afterinserting the solution ansatz function u = Φ𝑒−𝑖𝜔𝑡 to theeigenvalue problem [K − 𝜔2𝑖M]Φ𝑖 = 0. The solution of theeigenvalue problem leads then to the spectral matrix Ω =diag[𝜔2𝑖 ], with the eigenfrequencies𝜔𝑖 in ascending order andthe corresponding eigenmatrix Φ = [Φ1,Φ2, . . . ,Φ𝑛] (cf.[12]). The transformation into the reduced space is physicallymotivated by the fact that, dependent on the frequency rangeof the excitation force, the whole deformation energy of thevibrating system is associated with a low number of vibrationmodes. In case of seismic ground excitation, it can a prioribe assumed that the whole deformation energy is stored inthe first couple of low frequency modes of vibration (cf.[13]). Based on modal analysis, the transformation matrix

can be extracted from the sorted modal matrix according tocorrespondent eigenfrequencies of the full system with 𝑚 ≪𝑛, which means

Φ = [[Φ1,Φ2, . . . ,Φ𝑚⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

Φ𝑟

, Φ𝑚+1,Φ𝑚+2, . . . ,Φ𝑛⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟truncated modes

] . (15)

The physical coordinate u is transformed into the reducedspace z by

u = Φ𝑟z. (16)

Additionally, left transformation byΦ𝑇𝑟 leads to

Φ𝑇𝑟MΦ𝑟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

m∈R𝑚×𝑚z +Φ𝑇𝑟CΦ𝑟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

c∈R𝑚×𝑚z +Φ𝑇𝑟KΦ𝑟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

k∈R𝑚×𝑚z + Φ𝑇𝑟 Fc⏟⏟⏟⏟⏟⏟⏟⏟⏟

fc∈R𝑚×1

= Φ𝑇𝑟 F (𝑡)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟f(𝑡)∈R𝑚×1

. (17)

Concerning the multiple structural pounding problem, thematricesM and K are block-wise diagonal and only the con-tact force is responsible for possible interaction. The solutionof the eigenvalue problem including these matrices provideseigenvalues, which can be sorted in increasing order, but itis not possible to assign these eigenvalues to the framesystemwhich they belong to without visible check.Therefore,concerning themultiple pounding problem, this is an uncon-trolled modal truncation strategy.Themotivation is to dividethe whole mechanical problem into subsystem coupled by acontact term and investigate each subsystem separately.

2.4.2. Controlled Modal Truncation. The observation of theafore presented classical uncontrolled modal truncationmethod is that the systemmodes do not necessarily cover theessential modes of all bodies within the overall number ofconsidered modes. Applying a systematic modal truncationstrategy, the system modes are picked in a controlled way,which means that the main modes of each body and theunessential modes are truncated. In other words, the trunca-tion will be carried out for each separate body. Therefore, thesystem is decoupled into subsystems and for each subsystemthe following set of equations of motion must hold:

M(𝑖)u(𝑖) + C(𝑖)u(𝑖) + K(𝑖)u(𝑖) + Fc(𝑖) (u) = F(𝑖) (𝑡) . (18)

For each body, the modal truncation technique is now sepa-rately executed with𝑚(𝑖) ≪ 𝑛(𝑖); that is,

u(𝑖) = Φr(𝑖)z(𝑖). (19)

The set of equations of motion of one body is rewritten by leftmultiplication of the reduced modal matrix as

Φ(𝑖)𝑟

𝑇M(𝑖)Φ(𝑖)𝑟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟m(𝑖)

z(𝑖) +Φ(𝑖)𝑟 𝑇C(𝑖)Φ(𝑖)𝑟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟c(𝑖)

z(𝑖) +Φ(𝑖)𝑟 𝑇K(𝑖)Φ(𝑖)𝑟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟k(𝑖)

z(𝑖)

+Φ(𝑖)𝑟 𝑇Fc(𝑖)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟fc(𝑖)

= Φ(𝑖)𝑟 𝑇F(𝑖) (𝑡)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟f(𝑖)(𝑡)

. (20)

6 Shock and Vibration

Assembling all of the controlled modal truncation matrices,one obtains one controlled modal truncation matrix

u = [[[[

u(1)

u(2)

u(3)

]]]]= [[[[

Φ(1)𝑟 0 0

0 Φ(2)𝑟 0

0 0 Φ(3)𝑟

]]]][[[[

z(1)

z(2)

z(3)

]]]]= Φ𝑟z, (21)

and the controlled truncated set of equations of motion:

Φ𝑇

𝑟MΦ𝑟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟m

z + Φ𝑇𝑟CΦ𝑟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟c

z + Φ𝑇𝑟KΦ𝑟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟k

z + Φ𝑇𝑟 Fc⏟⏟⏟⏟⏟⏟⏟⏟⏟fc

= Φ𝑇𝑟 F (𝑡)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟f(𝑡)

. (22)

Through this transformation, the global movement of thesystem is captured and, therefore, a more reliable low-ordersystem is ensured.

2.5. Time Integration. The nonlinear dynamic system issolved using the central difference integration scheme (cf.[12]). Considering second-order Taylor forward and back-wards expansion, the approximation of the velocity isobtained by the sum of the two expansions, and the approxi-mation of the acceleration is obtained by the difference of thetwo expansions.Thus, the approximations of the velocity andacceleration by the central difference method are describedby

u𝑘 = u𝑘+1 − u𝑘−12Δ𝑡 ,u𝑘 = u𝑘−1 − 2u𝑘 + u𝑘+1Δ𝑡2 .

(23)

Inserting the approximations in (12) leads to an explicitformulation for the displacement at the subsequent time step:

( 1Δ𝑡2M + 12Δ𝑡C) u𝑘+1= F𝑘 − (K − 2Δ𝑡2M) u𝑘 − ( 1Δ𝑡2M + 12Δ𝑡C) u𝑘−1+ Fc𝑘.

(24)

As presented in (24), the displacement vector at time instant𝑡𝑘+1 is evaluated based on the already known displacementvectors at time instants 𝑡𝑘−1 and 𝑡𝑘. Therefore, an additionalvalue for u−1 must be calculated before the start of the inte-gration loop. This is realized by the second-order backwardsTaylor extrapolation:

u−1 = u0 − u0Δ𝑡 + u0Δ𝑡22 . (25)

The initial acceleration u0 is obtained by inserting the initialconditions u0 and u0 into the global equation of motion(12).The coefficient matrix ((1/Δ𝑡2)M+ (1/2Δ𝑡)C) is positivedefinite and can be LU factorized a priori, which acceleratesthe solution procedure. The big advantage of explicit timeintegration schemes is that for each step no iteration isrequired. However, the disadvantage is that relatively small

time steps must be chosen in order to ensure convergence. Inotherwords, if the chosen time stepΔ𝑡 exceeds a critical value,the solution function increases exponentially to infinity. Thecritical time step of the central difference integration schemedepends on the highest eigenfrequency of the dynamicsystem:

Δ𝑡 ≤ 𝑡crit = 2𝜔max, (26)

where 𝜔max defines the highest eigenfrequency of thedynamic system. Applying the frequency based controlledmodal truncation strategy, the highest eigenfrequency of thelow-order system is significantly reduced. Thus, the criticaltime step is considerably enlarged. Applying this strategy, amuch higher iteration time step can be chosen, which leadsto a considerably smaller number of integration loops to beprocessed and a significant decrease of the total computa-tional time.

The new strategy, presented in this paper, is a remarkablyefficient explicit time integration scheme, as it is able to over-come the general downside of these types of methods, that is,computational instability. Implicit time integration schemes,as, for example, the Newmark method (cf. [12]), require, onthe one hand, an iterative procedure within one time integra-tion step, and, therefore, more computational effort withinthis integration step. On the other hand, in case of the rightchoice of certain parameters, they are unconditionally stable.Choosing an arbitrary large time step, generally, a convergedsolution function can be observed. However, applying verylarge time steps, a solution function is obtained, which con-verges to the static solution. This is caused by the numericaldamping effect. Thus, in order to ensure response functionswith sufficient accuracy, additional calculations can ensure asufficiently accurate solution. In summary, a smaller numberof integration steps must be applied, but the computationaleffort and storage per integration step are considerably highercompared to explicit time integration schemes.

Within one test calculation, the standard central differ-ence scheme is compared with the new reduced integra-tion strategy and, additionally, with the implicit Newmarkmethod, using the commercial software package ABAQUS©.Through this approach, not only is the overall correctness ofthe implemented in-house code verified but also the com-putational efficiency is tested. The comparison of the threemethods can be found in Section 3.5. A more detailed dis-cussion on the advantages and disadvantages of the differenttime integration methods for the dynamic Signorini problemis given in the study of Doyen et al. [30].

3. Comparison of Methods

3.1. SystemModal Analysis. Theuncontrolled classical modesof vibration of the whole system are calculated. The first 10modes are depicted in Figure 4. It is observed that, due todifferent stiffness and characteristics of frames, the systemmodes alter from one frame to another frame in an irregularpattern. The disadvantage of the classical modal truncationtechnique is obvious: if one takes, for example, the first 6

Shock and Vibration 7

Mode 1 (1 = 1.514 rad/s).

Mode 5 (5 = 7.085 rad/s). Mode 6 (6 = 9.850 rad/s).

Mode 2 (2 = 1.796 rad/s). Mode 3 (3 = 5.198 rad/s). Mode 4 (4 = 5.967 rad/s).

Mode 7 (7 = 11.728 rad/s). Mode 8 (8 = 18.909 rad/s).

Mode 9 (9 = 25.197 rad/s). Mode 10 (10 = 35.716 rad/s).

Figure 4: Presentation of the first 10 system modes.

modes and cuts the rest modes, the distribution ofmodes willbe [3, 1, 2] concerning the substructures [S1(red),S2(green),S3(blue)]. A consistent distribution is therefore not achieved,which leads to an unsatisfactory low-order model.

The idea of the controlled modal truncation is motivatedby the above introduced disorder of system modes. The aimof the systematic modal truncation is to decouple the wholenonlinear contact problem and to keep the important modesof each frame by executing the modal truncation for eachframe subsystem separately. The choice of the modes can beempirically selected according to the number of stories of theframes or according to the frequency range of excitation. Forinstance, if the first ten modes are used and the rest of themodes are truncated, one can choose threemodes for the firstframe, four modes for the second frame, and three modesfor the third frame. This is depicted in Figure 5. A consistentdistribution of modes over the subsystem is achieved and asufficient number of deformation patterns within the contactarea is ensured in order to describe global contact behavior.These twomethods are compared in detail applying harmonicand transient seismic excitations in the following sections.

3.2. Sinusoidal Seismic Excitation. The level of accuracy withrespect to the number of considered modes for both theclassical and the controlled modal truncation technique isstudied under sinusoidal excitation. The results are focusedon the displacement, the velocity, and the acceleration as wellas the contact force of the horizontal degrees of freedom ofthe output node depicted in Figure 2.Theground accelerationis defined as ��𝑔 = 𝑎0 sin(]𝑡) with 𝑎0 = 2.5m/s2 and ] =4.0 rad/s. In order to examine the accuracy of the strategies,six, ten, and fifteen modes of vibration are chosen and com-pared. We obtain unsatisfactory results for all values ofinterest if only the first six modes (with a mode distributionof [2, 2, 2] concerning [S1,S2,S3] for the controlled modaltruncation strategy) are chosen; see Figure 6. The deviationof the acceleration and the contact force is considerably

large. Consideration of only six modes for both classical andsystematic modal truncation is obviously insufficient for areliable approximation of the response outputs.

The results improve significantly if ten ormoremodes aretaken into account. Regarding the classical modal truncation,if the first ten modes, illustrated in Figure 4, are chosen,the results improve considerably as shown in Figure 6 (redcurves). However, noticeable errors for the acceleration andthe contact forces are observed. Bymeans of controlledmodaltruncation technique (with a mode distribution of [3, 4, 3]concerning [S1,S2,S3]), which is depicted in Figure 5, thedisplacements and velocities are almost the same comparedto the full system. It can be concluded that the systematicallycontrolled modal truncation technique delivers much betterresults than the uncontrolled modal truncation for everyquantity of interest; see Figure 7.

If 15 system modes (with a mode distribution of [4, 7, 4]concerning [S1,S2,S3] for the controlled modal truncationstrategy) are taken into consideration, the results between thefull and the reduced systems agree almost; see Figure 8. Thecritical time step is smaller than the truncated systemwith tenmodes. The convergence study with respect to the number ofmodes is carried out in the next section. Error analysis of thetwo techniques is given in Section 3.4.

3.3. Frequency Response Analysis. The frequency responseanalysis is essential for a better understanding of systemcharacteristics in earthquake engineering. It is also the mostcommonly used method regarding the design of structuressubject to earthquakes [31]. The frequency response analysisinvestigates how themaximum response of the displacement,the velocity, and the acceleration as well as the contact forceof the structure behaviors subjected to harmonic excitationswith different frequencies. For the demonstrated model, thedriving angular frequencies vary from 0.1 rad/s to 5.0 rad/s.The results regarding themaximumdisplacements, the veloc-ities, and the acceleration as well as the impact forces versus

8 Shock and Vibration

Mode 2 of 1 (

Mode 2 of 3 (Mode 1 of 3 (

Mode 3 of 1 (

Mode 3 of 2 ( Mode 4 of 2 (

Mode 3 of 3 (

Mode 1 of 1 (1 = 1.514 rad/s). 2 = 5.967 rad/s).

2 = 7.085 rad/s).1 = 1.796 rad/s).

3 = 9.850 rad/s).

Mode 1 of 2 (1 = 5.198 rad/s). Mode 2 of 2 (2 = 18.909 rad/s). 3 = 36.495 rad/s).

3 = 11.728 rad/s).

4 = 54.292 rad/s).

Figure 5: Presentation of the controlled modes.

full systemuncon. MTcon. MT

−1

−0.5

0

0.5

uℎ

(m)

104 12 1462 80t (s)

(a) Displacement

full systemuncon. MTcon. MT

−2

0

2

104 12 1462 80t (s)

uℎ

(m/s

)

(b) Velocity

full systemuncon. MTcon. MT

−100

−50

0

104 12 1462 80t (s)

uℎ

(m/s

2)

(c) Acceleration

full systemuncon. MTcon. MT

·105

−8

−6

−4

−2

0

Fc

(N)

104 12 1462 80t (s)

(d) Contact force

Figure 6: Results for the displacement, the velocity, and the acceleration as well as the contact force with𝑚 = 6.

Shock and Vibration 9

104 12 1462 80t (s)

−0.5

0

0.5uℎ

(m)

full systemuncon. MTcon. MT

(a) Displacement

−1

0

1

2 6 80 10 12 144t (s)

full systemuncon. MTcon. MT

uℎ

(m/s

)

(b) Velocity

104 12 1462 80t (s)

−80

−60

−40

−20

0

full systemuncon. MTcon. MT

uℎ

(m/s

2)

(c) Acceleration

−4

−2

0

Fc

(N)

104 12 1462 80t (s)

·105

full systemuncon. MTcon. MT

(d) Contact force

Figure 7: Results for the displacement, the velocity, and the acceleration as well as the contact force with𝑚 = 10.

driving angular frequencies applying the first ten modes areshown in Figure 9. It is observed that the extremal values arereached near ] ≈ 2.3 rad/s. The numerical calculations of thefirst five eigenfrequencies are 1.514, 1.796, 5.198, 5.967, and7.085 rad/s. The extremal values are not located near thesefrequencies as complex pounding interaction influences theoverall system behavior.

It is observed that the controlled modal truncationtechnique approximates the results significantly better thanthe standard modal truncation technique. The maximumdisplacements, velocities, and contact forces for the wholerange of interest of the excitation frequencies are well approx-imated by the controlled modal truncation method. Thecomputational effort for each test frequency of the full systemis about 315 seconds. However, the truncated system needsonly 15 seconds. As the whole evaluation is based on therepresentative node, it is necessary to carry out a global errorestimation, considering all degrees of freedom.

3.4. Error Estimation. As shown in Section 3.1, the results aremainly dependent on howmanymodes are chosen.The root-mean-square deviation (RMSD) of the node output historiesis evaluated and plotted in Figure 10.The RMSD is defined as

RMSD (𝑢ℎ) = √∑𝑛𝑖=1 (𝑢ℎ,𝑖 − ��ℎ,𝑖)𝑛 , (27)

where ��ℎ,𝑖 is the reference solution of the full system and 𝑢ℎ,𝑖is a reduced solution at the time step 𝑖. The RMSD errorsof the standard method are larger than the controlled modaltruncation if the applied number of modes𝑚 is smaller than20. If more than 20 modes are taken into consideration, theRMSD of both methods are almost the same.

3.5. Comparison of the New Strategy with the Commercial Soft-ware Package ABAQUS©. In order to verify the correctnessand efficiency of the implemented in-house software and thenew strategy presented this paper, the full system response iscompared with the commercial software package ABAQUS©and the reduced system using the new reduction strategy.Applying ABAQUS©, the same discretization compared tothe in-house simulation is applied and the same penaltyparameter, that is, stiffness parameter of the linear poundingelement, is chosen within the contact module. The test iscarried out in two steps.

In the first step, a statical analysis is performed. A hori-zontal force is applied at the top leftnode of the first frame sys-tem, where the magnitude of the force is chosen to be largeenough in order to ensure contact between frames one andtwo and contact between frames two and three.The displace-ments of numerous representative nodes of the first, the sec-ond, and the last frame are evaluated by usingABAQUS©.The

10 Shock and Vibration

full systemuncon. MTcon. MT

−0.4

−0.2

0

0.2

uℎ

(m)

104 12 1462 80t (s)

(a) Displacement

full systemuncon. MTcon. MT

−1

0

1

104 12 1462 80t (s)

uℎ

(m/s

)

(b) Velocity

full systemuncon. MTcon. MT

−80

−60

−40

−20

0

104 12 1462 80t (s)

uℎ

(m/s

2)

(c) Acceleration

·105

full systemuncon. MTcon. MT

−4

−2

0

Fc

(N)

104 12 1462 80t (s)

(d) Contact force

Figure 8: Results for the displacement, the velocity, and the acceleration as well as the contact force with𝑚 = 15.

results of the commercial software package agree perfectlywith the in-house software tool. This verifies the correctnessof the stiffness matrix and the contact formulation of the in-house software.

In the second step, the dynamic response of the struc-tural system is evaluated, using the in-house implementa-tion and the commercial software package ABAQUS©. Thecomparison of the displacement of the representative node,using ABAQUS© and the in-house software, is presented inFigure 11 by the blue dashed and black lines, respectively.Additionally, the solution of the new proposed controlledmodal truncation strategy is depicted by the red dotted linein this figure.

In Figure 11, it can be seen that the in-house softwareagrees well with the result of the commercial software tool.Thus, the correctness of the calculation is verified. However,small deviations are observed, which are caused by the factthat, for the ABAQUS© simulation, the implicit Newmarkintegration is chosen, applying a time step of 0.001 seconds,and, for the in-house implementation, the explicit central dif-ference method is used. It is also observed that, for theimplicit integration scheme of the commercial software pack-age, a maximum time step of 0.001 seconds must be chosen;otherwise, numerical damping is considerably high, and

an inaccurate solution is observed. Detailed comments aboutthe advantages and disadvantages of the two methods arediscussed in Section 2.5. The computational cost of the in-house software, using an integration time step of 0.0001seconds for the explicit integration scheme, is comparablewith the one of commercial softwareABAQUS©.On the samemachine, the calculation time using the in-house tool is about315 seconds and the calculation time using the commercialsoftware is about 297 seconds. However, the calculation time,applying the novel strategy, is only 15 seconds. It is observedthat the new method shows an outstanding efficiency bypreserving a required level of accuracy. Thus, the new meth-od provides the possibility of evaluating high dimensionaldynamic multiple pounding systems, which demand gener-ally very high computational effort, using the standard inte-gration scheme.

3.6. Excitation under Different Earthquake Measurements.Regarding general problems in earthquake engineering,ground excitations are not harmonic. They are transient.Therefore, the presented methods are verified applying dif-ferent earthquake excitations. Two earthquake test samplesdepicted in Figure 12 are chosen.The calculation is performedtaking into account ten modes (with a mode distribution

Shock and Vibration 11

0 1 2 3 4 50

2

4

6

full systemuncon. MTcon. MT

uℎ

max

(m)

] (rad/s)

(a) Displacement

full systemuncon. MTcon. MT

5

10

1 432 50] (rad/s)

uℎＧ；Ｒ(Ｇ

/s)

(b) Velocity

full systemuncon. MTcon. MT

0

200

400

600

800

1 432 50] (rad/s)

uℎＧ；Ｒ(Ｇ

/s2)

(c) Acceleration

full systemuncon. MTcon. MT

0 1 2 3 4 5

0

2

4

6

Fℎ

max

(N)

·106

] (rad/s)

(d) Contact force

Figure 9: Frequency response of the maximum of the displacement, the velocity, and the acceleration as well as the contact force.

of [3, 4, 3] concerning [S1,S2,S3] for the controlled modaltruncation strategy). Data of the earthquake measurementsare presented in Table 2.

The numerical results of the representative node, pre-sented in Figures 13 and 14, reveal that the systematically con-trolled modal truncation technique provides reliable approx-imations of the system response. Taking into account thedisplacements and velocities compared to the full benchmarkresponse, a higher accuracy is achieved by the controlledmodal truncation technique. Concerning the accelerationand contact forces, especially at the contact time instants, thelow-order model must be improved by taking more modesfor both the classical and the controlled modal truncationmethods.

4. Conclusion and Outline

In this paper, the classical and a systematic modal truncationtechnique are studied, taking into account computationalcontact impact problem. Pounding is realized applying linearpounding elements within a node-to-surface contact formu-lation. Due to the pounding impact the classical modal trun-cation technique is not sufficient to approximate the system

response since the unconsidered higher frequency vibrationmodes play also an important role for the system response.The computational results show that the systematically con-trolled modal truncation is able to deliver a more reliableapproximation of the full systems, especially for the dis-placements and velocities. This is also verified applying realearthquake excitations. The systematically controlled modaltruncation technique makes the frequency response analysisfor complex structural systemswith a high number of degreesof freedommore feasible with consideration of the computa-tional effort. It should be pointed out that the stated methodis of course not limited to the seismic pounding problemonly,but to a much broader field of dynamical contact and impactproblems.

It must be added that the systematically controlled modalreduction technique is effective for materially linear systems.In the present paper, it is shown that it can also be efficientlyapplied for dynamic pounding problems. However, it is stillnecessary to increase the accuracy of the presented method,especially, with respect to the acceleration and the contactforce functions. The next step concerning future researchshould also concentratemore on local vibration near the con-tact area. Here, it should be more focused on substructuringtechniques, performed by, for example, the Craig-Bampton

12 Shock and Vibration

0 20 40 60 80 100n

uncon. MTcon. MT

0

0.1

0.2

0.3 u

ℎ(m

)

(a) Displacement

uncon. MTcon. MT

0.2

0.4

0.6

40 10060 80200n

uℎ

(m/s

)

(b) Velocity

uncon. MTcon. MT

2

4

6

8

40 10060 80200n

uℎ

(m/s

2)

(c) Acceleration

uncon. MTcon. MT

0 20 40 60 80 100

2

4

6

n

·104

F ℎ

(N)

(d) Contact force

Figure 10: Error analysis using RMSD of the displacement, the velocity, and the acceleration as well as the contact force.

full system

−0.5

0

0.5

uℎ

(m)

144 1210 16 1862 8 200t (s)

new strategy (m = 12)ABAQUS©

Figure 11: Comparison of the displacement between the in-house software and ABAQUS©.

method, where in this paper defined subsystems are againsubdivided into substructures.

In case of material nonlinearity, further extensions ofthe above-mentioned strategies are necessary. Generally, themethod of proper orthogonal decomposition was found tobe useful for creating model order reduced descriptions,which are able to provide accurate approximations of non-linear dynamic systems. The objective for upcoming studies,regarding the accurate low-order description of poundingproblems, is to combine the proposed strategywith theCraig-Bampton method (cf. [18]) and with the proper orthogonal

decomposition (POD), which has been presented in recentstudies [11, 12]. In particular, the frame subsystems, intro-duced in this paper, can be described by constraint modes,concerning the global motion of the structure, and internalmodes, describing the local behavior of the structure withinthe contact area. Then, the nonlinear system behavior canbe approximated using proper orthogonal modes, whichare evaluated from an information matrix (snapshot matrix[13]), which contains already existing solution vectors of thematerially nonlinear system and the constraint as well as theinternal modes of the frame subsystems.

Shock and Vibration 13

Table 2: Earthquake excitation list.

Event Year Location 𝑛𝑡 [–] 𝑇 [s] 𝑀 PGA [m/s2]Kobe 1995 Hanshin (Japan) 2001 20.000 7.2 6.81Bam 2003 Bam (Iran) 400 19.950 6.6 3.58𝑛𝑡: number of time steps, 𝑇: duration of the record,𝑀: moment magnitude, and PGA: peak ground acceleration.

−5

0

144 1210 16 1862 8 200t (s)

xg

(m/s

2)

(a) Kobe earthquake

−2

0

2

4

144 1210 16 1862 8 200t (s)

xg

(m/s

2)

(b) Bam earthquake

Figure 12: Acceleration time history of the Kobe and the Bam earthquake excitation.

0 2 4 6 8 10 12 14 16 18 20

uℎ

(m)

t (s)

full systemuncon. MTcon. MT

−0.2

0

0.2

(a) Displacement

full systemuncon. MTcon. MT

−1

0

1

144 1210 16 1862 8 200t (s)

uℎ

(m/s

)

(b) Velocity

full systemuncon. MTcon. MT

−60

−40

−20

0

144 1210 16 1862 8 200t (s)

uℎ

(m/s

2)

(c) Acceleration

full systemuncon. MTcon. MT

−4

−2

0

Fc

(N)

144 1210 16 1862 8 200t (s)

·105

(d) Contact force

Figure 13: Results for the displacement, the velocity, and the acceleration as well as the contact force under seismic excitation (Kobe).

14 Shock and Vibration

−0.4

−0.2

0

0.2

uℎ

(m)

144 1210 16 1862 8 200t (s)

full systemuncon. MTcon. MT

(a) Displacement

−0.5

0

0.5

1

144 1210 16 1862 8 200t (s)

full systemuncon. MTcon. MT

uℎ

(m/s

)

(b) Velocity

full systemuncon. MTcon. MT

−80

−60

−40

−20

0

144 1210 16 1862 8 200t (s)

uℎ

(m/s

2)

(c) Acceleration

·105

Fc

(N)

−4

−2

0

144 1210 16 1862 8 200t (s)

full systemuncon. MTcon. MT

(d) Contact force

Figure 14: Results for the displacement, the velocity, and the acceleration as well as the contact force under seismic excitation (Bam).

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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