Structural dynamics simulation of a swinging fairground attraction using a novelimplementation of reduced order modeling

J.B. Blaauw

University of Twente, Drienerlolaan 5, Enschede

Abstract

This research combines the strengths of modal order reduction, Simscape Multibody and Ansys to come up with a novelmethod to determine transient stresses in a fairground attraction. Currently, many of these structures are designedusing static load cases. This study shows that these load cases can overestimate the stresses greatly in highly dynamicstructures. Also, dynamic effects like vibrations within the structure are not considered, possibly resulting in underesti-mates of the stresses in other places. This method is therefore not suitable for structures like fairground rides. On theother hand, full transient structural analyses are computationally expensive to run. The method outlined in this paperobtains a similar accuracy compared to a full transient study, but by using only a fraction of the computational time.This also enables the model to be used in real time applications of determining the dynamic behavior of structures.Furthermore, as this model is built up within Simulink, the powerful tools within the electrical and control domainprovided by Simulink are easy to apply on this structural model. A fairground attraction is used as a practical scenarioto apply this method on. The paper concludes by extensively comparing the results of the new method with the loadcases and full transient structural analysis.

Keywords: Modal order reduction, Simscape Multibody, Finite-element import method, Transient structuraldynamics, Load cases, Fairground attraction, Flexible multibody dynamics

1. Introduction

Accurate and easy modeling of dynamic structures en-ables designers to effectively optimize the geometry andweight of a design. These aspects are of great importanceconsidering fairground attractions, since large swingingmotions and rotating masses induce high dynamic load-ings. These motions, in combination with possible vibra-tions occurring within the structure, means material fa-tigue is an important consideration. On the other hand,these attractions must be easy to transport and assembleto travel from fairground to fairground. Therefore, it isessential for these attractions to be light weight. The as-sessment between strength and weight calls for an accuratemodeling technique to get to a design both lightweight andstrong.

Classical linear FEM is often used to determine thedeformation and stresses of such attractions. For exam-ple, one could model the structural components of movingparts as a rigid multibody system. Constraint forces overtime can be determined at the joints between bodies fromwhich load cases can be derived. Although giving a gen-eral idea about the dynamic response and internal loadsin this structure’s bodies, more complex dynamic behav-ior like the intricate influence on deformation of one bodyon other bodies cannot be derived with this quasi-staticapproach.

Conversely, looking at moving bodies that are notrigidly connected to the fixed world, for example the seat-ing structure of the ride, the movements are large and geo-metrically nonlinear. They therefore cannot be simulatedusing linear FEM. Also, as will be shown, the current stateof the art using quasi static load cases is not appropriateto accurately describe the stresses in these bodies, due tothem not being connected to the rigid world. Besides this,these load cases are also unable to predict vibrations andresulting metal fatigue within these bodies.

A different approach would be to perform a full flex-ible analysis using nonlinear FEM. Here, each body getsmeshed and solved using nonlinear FEM at the same time.This enables to see how bending and dynamic behavior ofbodies influence the structure and constraint forces. Themain disadvantage of this method is that this nonlinearmodel is computationally expensive to run. Also, solvingfor a full mesh for each time step results in a lot of degreesof freedom to solve for and a nonlinear FEM model is hardto reduce.

In an effort to solve aforementioned shortcomings, thispaper will describe a novel method capable of solving thesegeometrically nonlinear structures whilst keeping compu-tational costs low. This is done by combining the strengthsof the existing software packages Simscape Multibody,Matlab and Ansys. Using this method, one can quicklyset up a study evaluating the dynamic behavior of these

Preprint submitted to Engineering Structures November 27, 2020

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Figure 1: Layout of a KGM Afterburner. The main beam (2) swingsbackwards and forwards while the seats (1), rotates simultaneouslyalong the axis of the main beam. The frame (3) always remainsstationary. The red, green, and blue arrows indicate the location ofthe local x, y, and z axis, respectively

complex moving structures and quickly analyze transientstresses in the body. This method extends on the methoddescribed in [1], which uses modal order reduction to eval-uate flexible behavior of bodies within Simscape Multi-body. Although the rigid body motions are nonlinear, thismethod uses a local frame fixed to each body to deter-mine the flexible behavior using linear FEM. With thenew approach provided in this paper, tools have been setup to quickly import an existing CAD design into SimscapeMultibody. Modal superposition is implemented by com-bining Ansys and Matlab to determine the mode shapes.This enables the transient stresses within a body to bedetermined quickly. The complete procedure to set up amodel like this, starting with a CAD-assembly to simu-lating flexible behavior to getting deformation and stressdata, will be discussed extensively. Results will be com-pared to other methods in order to validate its accuracyand shortcomings.

As a practical load case, a fairground attraction calledthe Afterburner manufactured by KMG will be used. Thisattraction consists out of a swinging pendulum driven froma hinge at the top with a rotating frame with seats con-nected to the bottom end, see Figure 1. This attractionis also representative for many other rides that are pen-dulum like and consists out of beams. The main focus ofthis research is to get a fast and accurate flexible modelfor the main beam of this structure, as this body has acomplex but repetitive dynamic loading in multiple direc-tions which are hard to determine using traditional staticmethods or hand calculations.

The outline of this paper is as follows: Before explain-ing the new method, the currently used method using loadcases will be discussed in section 2. The new method basedon [1] will be shortly explained in section 3. Next, the

modal order reduction strategy used will be discussed insection 4. After this, the newly developed method willbe extensively discussed in section 5. This method willbe optimized by validating the results to a full transientstudy using a cantilever beam in section 6. Applying thismethod on the Afterburner, the results will be shown anddiscussed in section 7. The paper will end by recapturingthe conclusions in section 8.

2. Industrial state of the art

The current state of the art uses load cases to deter-mine the strength of individual components within such aconstruction. A commonly used method to set up theseload cases starts with a rigid multibody analysis. The con-straint forces and moments over time between bodies arerecorded. For each time step, loads are combined into datapoints. For a 3D model, these data points consist out of 3forces and 3 moments for each interface. The data pointscan be plotted for each time step in a 6-dimensional graph.Now, a 6-dimensional box can be drawn enveloping eachdata point. The corners of this box, 64 of them, will con-tain combinations of the minimum and maximum forcesand moments that occurred during the simulation. Thesecombinations, multiplied with the required safety factorsenforced by the regulations, result in load cases which inturn are solved for within a flexible FEM modal in a staticway. Note that these combinations are not occurring inreal live but are commonly (much) fiercer. The thoughtis that if a structure can withstand these corner points ofthe envelope, it can withstand all combinations inside thisenvelope. As will be shown later in this paper, this leadsto rather conservative designs. Using methods that moreclosely resemble the occurring stresses within a structurewill often lead to large material and weight savings com-pared to using aforementioned load cases.

A package that can be used to perform such a rigidmultibody simulation is Simscape Multibody. This is anextension package on Matlab Simulink. Rigid multibodystructures can easily be set up in a method comparableto how Simulink works. The main differences are thatSimulink blocks here correspond to bodies or joints insteadof (transfer) functions, and connections between blocksdo not correspond to numerical data, but rigid connec-tions between bodies of the structure. Tools exist thatcan import assemblies from CAD programs directly intoSimscape as rigid multibody models. A strong advantageof Simscape Multibody is that these models can use thepowerful tools of Simulink to be applied on rigid multi-body structures. It is, for example, easy to set up multi-disciplinary studies, as control systems and electrical com-ponents can be included without much effort. Also, theuser interface will be easy to understand when one hassome experience with Simulink.

Modeling the Afterburner, and many similar(fairground) attractions, is straightforward. Looking atFigure 1, only 2 actuated joints are present: one enabling

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the pendulum motion located between the top of the mainbeam (2) and the rigid frame (3), and one enabling therotation of the seat located between the main beam (2)and seat structure (1). A simplified view of this model inSimulink can be seen in Figure 2.

The sweeping motion of the beam is controlled by aflip-flop controller, whilst the rotating motion of the seatis kinetically driven to ramp up to a constant speed. Thestudy starts from a static situation, and slowly ramps upto a sweep angle of 120◦. The seats speed up to a speedof 15 rpm.

The constraint forces and moments acting on both thetop and bottom of the beam are logged. These pointscan be processed into load cases as described in the be-ginning of this section. However, two sets of load casescan be extracted from this data: one where the load casesare applied on the top of the beam using the constraintforces and moments from this top frame, and one with theload cases applied on the bottom of the beam using theconstraint forces and moments of the bottom frame. Asboth sets represent the same structural problem, both setsshould result in similar stresses. However, as will be shownin subsection 7.1, this is not the case, again indicating thatusing load cases is not an accurate method of determiningthe stresses within a dynamic structure.

3. New method: Flexible bodies in SimscapeMultibody

As discussed in the previous section, Simscape Multi-body is initially only intended for rigid multibody struc-tures. Recently, however, a new method has been proposedby (S. Miller et al. 2017) enabling this software to also beused for flexible multibody structures. This is done by su-perimposing flexible behavior up on the previously definedrigid bodies. The most promising method discussed is the’finite-element import method’. Following this method,the previously rigid bodies are replaced by bodies includ-ing ’Deflection Joints’ at each interface, see Figure 3. Thedeflection joints measure the motion (displacement, veloc-ity, and acceleration) between the interface of a body andthe body itself and calculates the reaction force on thesejoints according to the linear equation of motion:

Mu + Du + Ku = F (1)

Here M, D and K are the local mass, damping andstiffness matrices for the flexible behavior of all the deflec-tion joints of the body, respectively. These matrices repre-sent the mass, damping and stiffness properties that eachjoint feel when moved relative to its undeformed position.u is the vector containing displacements and rotation ofeach frame relative to its undeformed position. The direc-tion of these displacements is expressed in a local framerigidly connected to the body, from here on called bodyfixed frame. The flexible behavior of a body is describedin the local frame. This means that the displacements of

this equation of motion remains small for stiff parts, whichmeans that the assumed linear nature of this system doesnot introduce large errors due to non-linear flexible behav-ior. For a 3-dimensional body with 6 degrees of freedomper frame and 3 frames, u has 18 entries. This also impliesthat the matrices of Equation 1 in this case must be of size18× 18. F is the vector with forces and moments appliedto the joints that occur because of the inertia, dampingand stiffness of the body.

When the matrices are derived, these can be importedinto the flexible body model within Simscape. If this isdone, the original rigid body block can simply be replacedwith the suitable flexible body block. For the Afterburner,this implies replacing the ’Beam Rigid (2)’ block from Fig-ure 2 with a block built up like Figure 3. The top andbottom boundary frames of the beam are connected to’F1’ and ’F3’ from Figure 3. Frame ’F2’ is the body fixedframe which is placed in the center of mass; it has no ex-ternal connections. No further changes must be made tothe Simscape Multibody study, making the process easy ifa rigid body model of the structure is already on hand.

4. Modal order reduction

As discussed in the previous section, the mass, damp-ing, and stiffness matrices needs to be reduced to the rightformat before they can be implemented in the flexible Sim-scape model. This section will first explain how to de-rive these matrices for the boundary frames using Craig-Bampton modal order reduction. After this, these matri-ces will be extended to also include internal modes. Sec-tion 5 will continue with explaining how implement thesematrices into the flexible multibody model.

4.1. Craig-Bampton boundary modes

The mass, damping and stiffness matrices should bederived for a unit displacement in each degree of freedom(3 translations and 3 rotations) for each boundary node.To achieve this, Craig-Bampton model order reduction isused [2]. To do so, the stiffness matrix is first sorted suchthat boundary and internal nodes are grouped in the fol-lowing partitioned form:

KFEM =

[Kbb Kib

Kbi Kii

](2)

The indices i and b correspond the internal and bound-ary nodes, respectively. The boundary nodes consist outof the previously defined boundary frames and the bodyfixed frame. MFEM is partitioned in the same way. TheCraig-Bampton boundary modes can be set up:

ΦCB =

[I

K−1ii Kib

](3)

Here, I is an identity matrix. Kii and Kib are thequadrants of KFEM. Pre and post multiplying MFEM and

3

Actuatedseatsjoint

ActuatedtopjointRigid

Transform1

Positionout

Seatrotationcontroller

RigidTransform

BeamRigid(2)

SeatsRigid(1)

LogConstraintForcesMomentsBottom

LogConstraintForcesMomentsTop

SpeedinTorqueout

Sweepcontroller

RigidFrame(3)

Worldsettingsandorigin

Figure 2: Simplified view of the rigid Simscape Multibody model of the Afterburner. The gray lines are rigid connections between bodies andjoints. The body of interest is ”Beam Rigid (2)”, which has blocks placed on either end to log the constraint forces and moments acting onthis body. The numbering of each body corresponds to the numbering in Figure 1

Figure 3: Overview of a flexible body using the finite-import method.Image from [1]

KFEM give the matrices in the correct form to be used inthe Simscape model:

KCB = ΦTCBKFEMΦCB

MCB = ΦTCBMFEMΦCB

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The [ ]T indicates the transposed of the matrix. Theresulting matrices are square and symmetric and have asize of 6 times the number of boundary frames, that is,3 translations and 3 rotations for each frame. The or-der in which these frames are sorted corresponds with theorder defined in Kbb. In the case of the Afterburner, 3frames are defined, resulting in 18 by 18 matrices. Thisalso means that the number of degrees of freedom has re-duced significantly, enabling faster simulation times. TheCraig-Bampton modes corresponding to the top and bot-tom can be seen in Figure 4 and Figure 5 as an example.

4.2. Internal modes

Craig-Bampton modes only define displacement of theboundary frames. In practice however, when these pointsare fully constrained, parts of the beam can still vibrate.Examples of such internal modes for the beam can befound in Figure 6 and Figure 7. Depending on the ge-ometry and loads on the body, these modes can form asignificant part of the total response of the structure. Nottaking these modes in to account, may therefore reduce theaccuracy of the model. To solve this, the Craig-Bamptonmatrices can be extended to also include these internalmodes. To do this, first the internal modes must be calcu-lated using the standard equation for eigen modes, butwith the boundary frames fixed by removing its corre-sponding rows and columns from the mass and stiffness

4

Figure 4: 6 Craig-Bampton modes corresponding to the top bound-ary frame. The color indicates the displacement relative to the un-deformed model. Axis units are in meters

Figure 5: 6 Craig-Bampton modes corresponding to the bottomboundary frame. The color indicates the displacement relative tothe undeformed model. Axis units are in meters

Figure 6: First 4 internal modes of the part of the beam below thecenter of mass. The color indicates the displacement relative to theundeformed model. Axis units are in meters

Figure 7: First 4 internal modes of the part of the beam above thecenter of mass. The color indicates the displacement relative to theundeformed model. Axis units are in meters

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matrix: (MFEM,fixed − ω2

kKFEM,fixed

)Φint,k = 0 (5)

where k corresponds to internal mode k.Although it is possible to consider all internal modes,

it is generally not needed for accuracy. Normally, only thefew smallest internal modes cause for most of the flexi-ble behavior. For example, according to [3], only internalmodes with frequencies up to 4 times the highest frequencypresent in the constraint forces have to be taken into ac-count; internal modes with frequencies higher than this donot have a significant effect on the total flexible behavior.Therefore, only the eigenvectors in Φint corresponding tothese few modes are taken, here called Φn. This can beappended to Equation 3 like:

Φ =

[I 0

K−1ii Kib Φn

](6)

Pre- and post-multiplying in accordance with Equa-tion 4 results in the mass and stiffness matrix includingthe internal modes. For each internal mode, 1 extra rowand column is added to the matrices. For example, a 3Dbody containing 2 boundary frames, 1 body fixed frameand 2 internal modes will have 2× 6 + 1× 6 + 2 = 20 rowsand columns in its matrices.

5. From CAD design to transient stresses

To perform a flexible multibody simulation in Sim-scape, many properties of the bodies must be processedand imported, like the matrices discussed before and thedimensions of the body. This process is quite involved, asit requires multiple different programs to work together.This study automates this process to a large extend. Forthis, Matlab is used as the programming environment be-cause of its close integration with Simulink and Simscape.The resulting process is graphically represented in Fig-ure 8. Upcoming part of the paper will discuss each pro-cess block of this figure extensively. It is assumed that aCAD design and rigid multibody Simscape study are al-ready available.

5.1. Ansys Workbench

To start, a mesh should be created from the CAD de-sign of each body that needs to be modeled flexibly. Fromthis the mass and stiffness matrices MFEM and KFEM

can be derived for these bodies. Also, boundary framesmust be defined to connect other bodies to in the Sim-scape model. These frames must be rigidly connected tothe nearby nodes in the mesh to prevent rigid body modesof these frames. It should be ensured that the nodes cor-responding to these frames have both translational as ro-tational degrees of freedom, as commonly nodes within aFEM mesh do not have rotational degrees of freedom. Fur-thermore, because the flexible behavior is expressed in a

local, body fixed frame, the position of this frame mustbe chosen. For accuracy, it is best practice to place thisframe in the center of mass and connect this frame rigidlyto the FEM mesh [4]. Also, this frame should have bothtranslational and rotational degrees of freedom.

For the main beam of the Afterburner, the first frameis placed at the joint near the top of the beam where itconnects to the static frame as can be seen in Figure 1.The second frame is placed at the bottom of the beam,where it connects to the seat structure. Furthermore, aframe is placed in the center of mass, connected to theperimeter of the hollow beam.

For this study, Ansys Workbench is used because of theeasy to use user interface, making the process more acces-sible to less experienced people. However, this method isnot limited to Ansys Workbench as other FEM programscan also be used, as long as it is possible to obtain the massand stiffness matrices and the nodal locations. The bound-ary frames are placed at so-called ’remote points’ that arerigidly connected to nearby nodes. The main disadvantageof using Ansys Workbench is the limited options to selectonly a few nodes to connect these remote points to, whichcan result in large parts of the body becoming rigid. OtherFEM programs might offer more options to prevent this.Next, using an APDL command snippet, MFEM, KFEM

and the locations of each node is exported into text files.Depending on the size of the matrices, these files can be-come large and take long to process. A trade off must bemade between the size of these matrices and accuracy ofthe mesh.

5.2. Data preparation

Now the full mass and stiffness matrices are known,they can be reduced in the same method as discussed insection 4. This makes the matrices much easier to han-dle, as their size is greatly reduced. Now, a few moresteps are required to prepare the data to be imported intoSimscape, which will be explained in upcoming sections.First, a damping matrix must be derived from the reducedmass and stiffness matrices. Then, these matrices must beconverted into a state space formulation which can thenbe imported into Simscape. Furthermore, the mass of thebody must be divided over its boundary and body fixedframes. Multiple methods of dividing the mass will bediscussed in the final subsection.

5.2.1. Damping

Damping is added for this model to mimic the dampingproperties of the real-live construction. This has the addedbenefit to prevent energy build op in the flexible bodies,causing the system to become unstable over time.

In the current model, Equation 1, viscous dampingcan be added straightforwardly. Two common dampingmodels are often used: proportional damping and modaldamping [3]. Choosing the damping model and ratios isnot straightforward, as this not only depends on mate-rial properties, but also on the geometry of the body. In

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Material propertiesInterface framesAnsys Workbench

Data preparation

Damping constant# internal modesMass division

Simscape

Mass & stiffness matricesNodal locationsLocation interface frames

Data structure for flexiblebody in Simscape

Simscape rigid multibody model

Setup APDL script

Ansys APDL

APDL Script for Craig-Bampton modes

Modal superposition

Participation factor perCraig-Bampton mode

Stresses per Craig-Bampton mode

Stress data over time for each node

CAD-design

Figure 8: Schematic representation of the workflow

this case, modal damping with a constant damping ratio of0.05 is chosen to provide predictable and realistic dampingbehavior for this study. When performing another study,other models and damping ratios may be needed to accu-rately model the structure.

5.2.2. State Space

The flexible model within Simscape Multibody requiresthe mass, damping and stiffness matrices to be convertedinto a state-space system. In accordance with [1], this isexecuted as follows:

x = Ax + Bu (7)

y = Cx + Du (8)

where

A =

[0 I

−KCB,ii

MCB,ii− LCB,ii

MCB,ii

]

B =

[0 0 0

−KCB,ib

MCB,ii− LCB,ib

MCB,ii−MCB,ib

MCB,ii

]

C =

−KCB,bi −MCB,biKCB,ii

MCB,ii

−LCB,bi −MCB,biLCB,ii

MCB,ii

I

D =

[a b c0 0 0

]

7

where

a = −KCB,bb −MCB,biKCB,ib

MCB,ii

b = −LCB,bb −MCB,biLCB,ib

MCB,ii

c = −MCB,bb −MCB,biMCB,ib

MCB,ii

This system uses u, u and u combined into a vector asinput (u) and outputs the reaction forces in each degree offreedom (y). Internal modes are handled inside the state-space system and are part of the state vector (x). AsSimulink does not output the state vector in its State-space block, the C matrix is appended with the identitymatrix and the D matrix is appended with zeros to matchthe number of rows of the C matrix. In this way, theexcitation of internal modes and its derivatives are addedto the output vector y and can be used for verification.

5.2.3. Mass division

In the Simscape model, the boundary frames are notrigidly connected to the rest of the body and do not haveany inertia, as is shown in Figure 3. In Simulink, this re-sults in unstable system behavior because a force is put ona massless object resulting in high accelerations. To solvethis problem, some inertia must be added to the externalnodes. Two methods are compared:

First, as common in classical flexible multibody dy-namics, only mass is put on the body fixed frame. To pre-vent unstable behavior, in this application it means themass on the external frames should be as low as possiblewithout causing high simulation times or instability. Todo so, a small portion of the total inertia of the body issubtracted from the body fixed frame and added to eachboundary frame. An assessment must be made to find theoptimum portion of mass put on the boundary frames.

Second, the inertia of the body can be equally dividedover each body such that each boundary frame reflects itsadjoining part of the body. This process has many sim-ilarities with mass lumping when using a lumped systemapproach. There is, however, no consensus on how the to-tal inertia of the system must be divided over the threeframes [3, 5, 6]. An advantage is that the inertia proper-ties of the deformed beam are more accurately reflected inthe total multibody system [1].

Multiple methods of mass lumping have been com-pared. However, many either result in negative inertiaswhich are not supported by Simscape or require shapefunctions of the body which, due to the complex geom-etry, are not available in this case. A commonly usedmethod is the HRZ lumping method, because of its sim-plicity and relatively good accuracy [7]. Here, the trans-lational mass on the main diagonal of the Craig-Bamptonmatrix is summed. Because cross terms are neglected, thissum does not correspond exactly to the real mass of the

body. This difference in mass has been compensated forby multiplying the mass on each frame with the same con-stant in order to keep the total mass correct. The rota-tional inertias on the main diagonal are also multiplied bythe same constant. All off-diagonal terms are set to zero.This method normally results in a good mass division, butrotational inertia is less accurate. Generally, this does notimpose large errors because the rotational inertias onlyhave a minor influence on the flexible behavior of a body[3].

Both methods are being compared to each other in sub-section 7.2.

5.3. Simscape

The only change that needs to be made in Simscape isreplacing the rigid body block with a flexible body block.This block builds upon the work done by MathWorks [1]to automatically import the right data setup in the pre-vious step from the Matlab Workspace. What is new isthat this block also stores its deformation data, consistingout of participation factors of each Craig-Bampton modeand each internal mode for every time step, in the MatlabWorkspace so it can later be used for further processing.

For this study, the ’daessc’ solver provided with Sim-scape Multibody is used in its default settings, as thisproved to be a fast solver with accurate results. Opti-mizing the solver settings might improve simulation speedor accuracy and stability, but the default settings seem towork quick and reliable with this system.

5.4. Setup APDL script

This model uses modal superposition to determine thestresses inside a body. This process will be discussed moreextensively in subsection 5.6. The problem is that for eachCraig-Bampton and internal mode a static structural anal-ysis must be performed to determine the stresses occurringduring each mode. As setting up a study manually foreach mode is laborious, this Matlab script automates thisprocess by setting up an Ansys APDL file containing thedeformations for each boundary node during each mode.This file can simply be imported into Ansys APDL whereall studies automatically run successively, and all data isstored in text files.

A possible disadvantage of this method is that, basedon the FEM software used, this method is unable to workwith internal modes. The culprit here is that some FEMprograms do not determine stresses when performing amodal study, making modal superposition impossible.Switching to other FEM programs might offer a solution.In this research, it is assumed only the Craig-Bamptonmodes result in stresses; stresses due to internal modesare neglected. As the natural frequency of the internalmodes is much higher than the occurring frequencies inthe construction, this is not a large problem for this case.However, depending on the geometry of the body, this as-sumption might have a large effect on the results.

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5.5. Ansys APDL

The APDL script set up at the previous section is im-ported and run in Ansys APDL. This will perform thestatic structural study for each Craig-Bampton mode suc-cessively for each boundary frame defined insubsection 5.1.

For example, the first mode consists of all frames beingfixed at their original position, only with the first framedisplaced 1 meter in the x direction. The deformations,normal stresses and shear stresses at each node are cal-culated and exported into a text file. Then, this processis repeated until each Craig-Bampton mode is calculated.The body fixed frame is fixed during each study, as thelocal frame used during the study in Simscape multibodyis rigidly connected to this frame. A total of 6 studies foreach boundary frame are performed. In case of the After-burner, having 2 boundary frames, this thus results in 12studies.

5.6. Combining data: Modal superposition

Commonly, the deformation data derived in the previ-ous section is used to perform a transient structural anal-ysis. This process will be explained in detail in section 6.Performing a transient analysis, however, means perform-ing the time integration and solution convergence twice;once during the simulation in Simscape and once in theFEM package. Also, determining the stress for a com-plete mesh is computationally expensive. In Simscape, thepossible deformation shapes are limited by the number ofCraig-Bampton and internal modes considered. Having tosolve for the full FEM mesh at each time step is thereforeof no use, as the accuracy is determined by the numberof deformation modes considered, not by the number ofnodes in the FEM mesh.

A more effective way to simplify the stress computa-tions is using modal superposition of the stresses. Fromthe Simscape study, it is already known how much eachmode is excited at each time step. From the previous sub-section, the stresses occurring with each Craig-Bampton orinternal mode are known. Now, multiplying the stressesper mode by its flexible coordinate η and adding the resultsof the different modes together results in a good approxi-mation of the total stresses at each point in the mesh:

σ =n∑

i=1

ηiσi (9)

Here, σ being the tensor with normal and shear stressesin each direction for each point in the mesh, n the numberof modes taken into account, ηi the participation factor ofmode i and σi the normal and shear stresses at each nodefor mode i. This function can be repeated for each timestep to obtain the stresses over time.

The modal superposition is executed within Matlab, asit makes it easy to use the data gathered previously fromSimscape. To start, Matlab imports the displacement datafrom Simscape and the stress per Craig-Bampton-mode

from the text files. For each timestep, the normal andshear stresses are multiplied by the flexible coordinate ofeach Craig-Bampton mode and added to the result of theother Craig-Bampton mode shapes, like Equation 9. Thisresults in the normal and shear stresses and displacementsin each local direction at each time step.

With this information, it is possible to retrieve thestresses and displacement of each node at each time step.This data can for example be used for constructive evalua-tion or fatigue studies. The Von Mises equivalent stress isalso calculated from the individual stress directions. Thisis later used in this study for comparison with Ansys Work-bench.

6. Validation with Ansys Workbench

Using modal superposition to describe the stresses anddeformation of the body requires some assumptions tobe made. The main assumption is that the deflection ofthe body can be fully described by a combination of itsCraig-Bampton modes of the boundary frames. To val-idate whether these assumptions and simplifications areaccurate, the results are compared to a second method fordetermining the stresses.

This method starts with the same deformation data ofthe boundary frames as used with the method explainedin the previous section. The deformation of the boundaryframes with respect to the body fixed frame is imposedon the same boundary frames in Ansys Workbench. Thebody fixed frame is kept stationary, as Ansys Workbenchdoes not support rigid body movements. It is assumed thateffects of the rigid body motion, like the shifting directionof gravity, is already present in the deformation data of theboundary frames and should therefore not be compensatedfor.

Analytically, the main difference of this method com-pared to Simscape, is that the position and stresses ofeach node are not prescribed based on the Craig-Bamptonmodes as the model is solved for a full mesh. This meansthat the deflection of each node within the body is not lim-ited to the deflection described by a limited combinationof modes but is free to take on any position. From thisview, this method offers a more accurate representation ofthe real world. On the contrary, this method is computa-tionally more expensive than modal superposition.

To import the data from Simscape into Ansys, it shouldbe converted to a data structure compatible with AnsysWorkbench. For this reason, two important assumptionsmust be made:

First, Ansys Workbench is unable to work with largedata structures. As Simscape Multibody uses small timesteps to come to an accurate and stable result, the amountof time steps is rather large. To decrease the number ofsteps, the deformation data is resampled using a lowersample frequency. For this, first a fast Fourier transform ofthe deformation has been made to analyze the frequency

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content of the deformation data. From this, it can beseen that there are barely any vibrations present in theAfterburner above 10 Hz. The data should therefore beresampled at at least the Nyquist frequency of 20 Hz. Toaccount for some margin of error, the data is resampled at40 Hz.

Secondly, the rotation convention between SimscapeMultibody and Ansys Workbench is different. SimscapeMultibody can export rotations as either axis-angle,quaternions or 3d rotation matrices. It is also possibleto get the rotation of the boundary frames by requestingthe participation factors of each rotational Craig-Bamptonmode. The rotation order of this method is the extrin-sic x-y-z Tait-Bryan angles. Ansys Workbench on theother hand uses the intrinsic Z-X-Y Euler Angle sequence.Transforming between those conventions can be complexas the solution is not always unique.

Generally, however, rotations due to deformation aresmall. In the case of the Afterburner, the rotation in anydirection is at most 0.013 radians. Therefore, it is assumedrotations are linear, making rotation order unimportant.This removes the need for converting one convention intothe other as the rotational data can directly be taken fromthe Craig-Bampton participation factors and applied inAnsys Workbench.

To import, the translation and rotation data is con-verted into a xml-file used by Ansys Workbench to importdisplacement data for a transient analysis. This results in6 separate files per boundary frame, which can be assignedto each degree of freedom of each boundary frame. Afterthis, the simulation can be run. The resulting data will becompared in the next section.

7. Results and discussion

This section is divided into three subsections. First,the Afterburner will be simulated using the current stateof the art as discussed in section 2. This method usesthe rigid multibody Simscape model to determine the con-straint forces and moments on the main beam. Static loadcases are determined from these constraint forces and mo-ments. The results between the two sets of load cases, withthe forces either applied at the top or bottom boundary,will be compared and discussed.

Now a baseline is set using the rigid multibody model,the results can be compared to the flexible multibodymodel of the Afterburner. First, however, some model set-tings and the model accuracy will be tested on the beamwhen rigidly fixed at its top. Using this setup, the in-fluence of the mass division and internal modes on theaccuracy and simulation time is compared to a transientstructural analysis. The optimal settings for simulatingthe Afterburner will be derived using the results.

In the final subsection, the flexible response of thebeam of the Afterburner will be simulated following themethod described in section 5, using the optimal settings

and the flexible multibody model. The constraint forceson the beam will be compared for the rigid and flexiblemultibody model. After this, transient studies using An-sys and modal superposition are compared.

During this section, all directions mentioned are in thelocal body fixed frame of the beam, see Figure 1. Ingraphs, the x-direction is marked red, the y-direction ismarked green and the z-direction is marked blue.

7.1. Industrial state of the art: Load cases

The constraint forces and moments on the beam arerecorded for both the top and bottom boundary frame us-ing a rigid multibody simulation. For one period of thesweeping motion, the results of the rigid multibody studycan be found in Figure 9 and Figure 10. Note how the mo-ments in transverse direction due to the gyroscopic effectsof the rotating seats are larger than the actuating torque.

As noted in section 2, the current state of the art is touse the maximum and minimum values of these forces overtime to set up load cases. In this case there are 6 loadings(3 forces and 3 moments) making up for 64 load cases.To evaluate these load cases, the beam is rigidly fixed toone boundary frame whilst a load case is put on the freeboundary frame. This is done for both the top and thebottom boundary. For each node, the maximum occurringVon Mises stress during all 64 load cases is recorded. Theseresults are displayed in Figure 11 and Figure 12.

Examining these figures, it is clear that the location ofwhere the beam is clamped has a great effect on the results.The maximum stress is different for both cases, with theload case applied at the bottom boundary frame result-ing in lower stresses. Furthermore, the location where themaximum stress occurs is different for both cases, as it islocated near the clamped end as expected for a cantileverbeam. All in all, the results are significantly different whenclamping the top frame compared to clamping the bottomframe, where in practice, both studies should yield thesame results because the same rigid multibody problem issolved. Clearly, this method, which is the current stateof the art, is not suitable to determine the stresses in thiscase where the beam is not rigidly connected at either end.

7.2. Model optimization

Before applying the novel method on the Afterburner,some values can be optimized to improve accuracy anddecrease computational costs. These studies also give anindication about the accuracy of the flexible model usedin Simscape. In this paper, attention will be paid to twoimportant factors: the weight distribution and the internalmodes.

For the optimization studies, the beam is rigidly con-nected at its top side. Two studies have been executed:one static and one dynamic. For the static study, a con-stant loading of −100 kN is put in the x direction. Thedisplacement of the end in the x direction is comparedbetween the results of the modal order reduced model of

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Figure 9: Constraint forces and moments on the top hinge duringone period for the rigid model. The x direction is red, y directiongreen and the z direction blue

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Figure 10: Constraint forces and moments on the bottom hingeduring one period for the rigid model. The x direction is red, ydirection green and the z direction blue

Figure 11: Maximum stress per node during all 64 load cases appliedat the top frame. Maximum equivalent stress is 1.53 × 108 Pa

Figure 12: Maximum stress per node during all 64 load cases appliedat the bottom frame. Maximum equivalent stress is 1.28 × 108 Pa

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Table 1: Effect of internal modes on the frequency response

Nr ofinternalmodes

Frequencydifference(%)

Staticdifference(%)

SimulationtimeSimscape(s)

0 16.77 0.27 2.22 16.77 0.27 2.24 16.73 0.27 3.040 16.73 0.27 10.0

Simscape and a fully meshed model in Ansys Workbench.For the dynamic study, the main vibration frequency inthis direction is compared by giving the beam a short im-pulse on the free end and letting it vibrate freely. Again,the Simscape model is compared to the Ansys Workbenchmodel.

To start, the influence of internal modes is evaluated.For this study, HRZ-lumping is used for the mass division.The results for different amounts of internal modes canbe found in Table 1. The frequency difference is definedas |ωwb−ωsc

ωwb|, with ωwb the lowest vibration mode in Ansys

Workbench and ωsc the lowest vibration mode in SimscapeMultibody.

It is striking that increasing the internal modes barelyhas any effect on the frequency response and static dis-placement. For the static displacement, this behavior isexpected, as internal modes do not influence the staticdisplacement of the boundary frames. A possible explana-tion for the dynamic behavior can be found in [3], whichstates that internal modes with frequencies more than 4times higher than the main excitation frequency do nothave a significant influence on the dynamic behavior ofthe system. Looking at Figure 6, the lowest internal modehas a frequency of 72.24 Hz, whereas the main frequencyis more than 4 times lower with 8.80 Hz, fulfilling this ruleof thumb. Increasing the number of internal modes doesincrease the simulation time of the Simscape Model, al-though many internal modes have to be considered for thesimulation time to increase significantly.

Secondly, the two methods to divide mass over theframes discussed section 5.2.3 are compared with the re-sults of Ansys Workbench. The results can be seen in Ta-ble 2. The mass division has no influence on the static dis-placement and is therefore not shown in the table. Clearly,putting the most mass on the mid frame results in more ac-curate dynamic results than HRZ lumping. For this case,the results seem to stabilize at a mass fraction of 0.01 %.Above this percentage the results barely change, but sim-ulation time increases. Therefore, a mass fraction of 0.01% will be used for the final simulation.

In the complete model of the Afterburner, the seatsare fixed to the now free end of the beam. As the massof the seats is substantially higher than the mass of thebeam, the influence of mass division within the beam be-comes smaller. With this mass added to the end of the

Table 2: Effect of mass division on the frequency response

Mass fractionon boundaryframes (%)

Frequencydifference(%)

Simulation timeSimscape(s)

HRZ lumping 16.77 2.21 2.72 2.00.1 4.33 2.00.01 4.46 2.00.001 4.50 2.20.000001 4.50 2.6

beam and using HRZ lumping, the error in the vibrationalfrequency is only 1.55 % whereas placing most mass onthe mid frame, the error decreases to 0.02 %. Still, usingthis last simulation method results in the best distinctionbetween accuracy and simulation time.

7.3. New method: Transient and modal analysis

The methods outlined in section 5 and section 6 shouldgive better insights in the stresses occurring in the beamover time. First, comparing the constraint forces and mo-ments between the rigid and flexible multibody model,Figure 13 and Figure 14, it is clear some vibrations oc-cur within the flexible model. However, as the system isquite stiff, the differences are small compared to the rigidstudy. Therefore, setting up load cases using the data ofthe flexible study instead of the rigid study will result insimilar inaccurate results and does not provide additionalbenefits.

The deflections of the boundary frames of the beamfollowing from the flexible multibody model can be seenin Figure 15 and Figure 16. This data is used for both thetransient analysis as the modal analysis.

To compare both the results from modal superpositionas well as the transient study in Ansys, for both stud-ies the equivalent von Mises stress at the node which un-dergoes the highest stresses during the simulation is com-pared. This is the same node for both studies. In Figure 17and Figure 18, the stresses calculated using both methodsare plotted. Comparing the stresses, the resemblance isstriking. Only minor differences between the two methodsare visible. The absolute mean difference between the twomethods is 0.11 %. Although the differences are small, themodal superposition only takes 1/52 of the computationaltime and the results file takes only 1/21 of the storagespace, making it more attractive from a computationalstandpoint.

For this case, the simulation runs faster than real time.This enables many possibilities in real time applicationsand real time control problems. Setting up such a controlloop is also quite straightforward, as Simscape Multibodyworks seamlessly with Simulink, offering many tools andsupports multiple platforms to work with.

Comparing the maximum stresses during the transientanalysis (95.41 MPa) with the maximum stresses during

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Figure 13: Constraint forces and moments on the top hinge duringone period for the rigid and flexible model. The x direction is red,y direction green and the z direction blue

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Figure 14: Constraint forces and moments on the bottom hingeduring one period for the rigid and flexible model. The x directionis red, y direction green and the z direction blue

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Figure 15: Linear deflection of the top and bottom boundary framewith respect to the body fixed frame

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Figure 16: Rotational deflection of the top and bottom boundaryframe with respect to the body fixed frame

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Figure 17: Comparison of the transient and modal analysis

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Modal Superposition Matlab

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Figure 18: Zoomed in comparison of the transient and modalanalysis

Figure 19: Maximum stress per node during the transient analysis us-ing modal superposition. Maximum equivalent stress is 0.95 × 108 Pa

the quasi-static load cases (153 MPa and 129 MPa), it isclear that quasi static load cases overestimates the stressessignificantly. This effect can clearly be seen by comparingFigure 19 with Figure 11 and Figure 12. Using a tran-sient analysis instead of the commonly used quasi-staticmethod, this might lead to reducing the weight of a struc-ture, which is very beneficial for a movable fairground at-traction.

8. Conclusions

This paper has successfully developed a novel methodof determining transient stresses inside a multibody struc-ture. This is done by combining existing methods likemodal superposition and programs like Matlab, SimscapeMultibody and Ansys Workbench. It is shown that thecurrent state of the art to determine stresses within abody, using quasi static load cases, often overestimatesthe stresses greatly. Also, the location of the maximumstresses is not always realistic. The method discussed inthis paper helps solving these problems.

Comparing the new method with a full mesh transientstudy, the stresses are in great resemblance. However, aftersome manual setup, the new method is computationallymuch lighter without losing accuracy. This enables real-time simulations of complex structures and the possibilityto include flexible behavior of bodies in control problems.

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References

[1] S. Miller, T. Soares, Y. V. Weddingen, J. Wendlandt, Model-ing flexible bodies with simscape multibody software, Technicalpaper, MathWorks (2017).

[2] R. R. Craig, M. C. C. Bampton, Coupling of substructures fordynamic analyses., AIAA Journal 6 (7) (1968) 1313–1319.

[3] R. D. Cook, D. S. Malkus, M. E. Plesha, R. J. Witt, Conceptsand Applications of Finite Element Analysis, 4th Edition, JohnWiley & Sons. Inc., 2001.

[4] A. Cardona, Superelements modelling in flexible multibody dy-namics, Multibody System Dynamic 4 (2000) 245–266.

[5] C. A. Felippa, Introduction to finite element methods, Boulder,Colorado, USA (2001).

[6] K. H. Yang, Chapter 8 - modal and transient dynamic analysis,in: K. H. Yang (Ed.), Basic Finite Element Method as Appliedto Injury Biomechanics, Academic Press, 2018, pp. 309 – 382.

[7] E. Hinton, T. Rock, O. C. Zienkiewicz, A note on mass lumpingand related processes in the finite element method, EarthquakeEngineering & Structural Dynamics 4 (3) (1976) 245–249.

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