Component-Based Transfer Path Analysis and Hybrid Substructuring at high frequencies

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IN

DEGREE PROJECT INDUSTRIAL ENGINEERING AND MANAGEMENT,

SECOND CYCLE, 30 CREDITS ,

STOCKHOLM SWEDEN 2020

Component-Based Transfer Path

Analysis and Hybrid

Substructuring at high frequencies

A treatise on error modelling in Transfer Path

Analysis

HARIKRISHNAN VENUGOPAL

KTH ROYAL INSTITUTE OF TECHNOLOGY

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Sammanfattning

För närvarande ser vi ett ökat intresse för felmodellering inom området modal provn-ing och analys. Flera fel som uppstår under testserier modelleras analytiskt eller nu-meriskt och propageras effektivt till olika systemkopplings- och gränssnittsreduktion-srutiner. Denna studie syftar till att hantera mänskliga fel, som positionsmätningsfel och orienteringsmätfel, och slumpmässiga brusbaserade fel i de uppmätta frekvensre-sponsfunktionerna (FRF) till den gränssnittsreduktionsalgoritm, som kallas ”Virtual Point Transformation” (VPT), och senare till en substrukturskopplingsmetod, som kallas FBS (Frequency-Based Substructuring). Dessa metoder utgör hörnstenen för ”Transfer Path Analsysis” (TPA). Dessutom har vanliga felkällor som sensormass-belastningseffekter och felorientering av sensorer undersökts. Slutligen har en ny metod för att beräkna sensorns positioner och riktningar, efter att mätning gjorts, baserat på systemets stelkroppsegenskaper och de applicerade krafterna.

Felpropageringen estimerades med en beräkningseffektiv, momentmetod av första ordningen och validerades senare med Monte-Carlo-simuleringar. Resultaten visar att orienteringsmätfelet är den mest signifikanta felkällan följt av FRF-fel och po-sitionsmätningsfel. Massbelastningseffekten kompenseras med hjälp av ”Structural Modification Using Response Functions” (SMURF) -metoden och sensorjusterin-gen korrigeras med hjälp av koordinatomvandling. Sensorpositionerna och posi-tioner och orientering beräknas exakt från stelkroppsegenskaperna och de applicer-ade krafterna; individuellt med matrisalgebra och samtidigt med en optimerings-baserad icke-linjär minsta kvadratlösare.

Nyckelord: Dynamisk substrukturering, Feluppskattning, Överföringsanalys, Osäk-erhetspropagering

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Abstract

The field of modal testing and analysis is currently facing a surge of interest in error modelling. Several errors which occur during testing campaigns are modelled analytically or numerically and propagated to various system coupling and inter-face reduction routines effectively. This study aims to propagate human errors, like position measurement errors and orientation measurement errors, and random noise-based errors in the measured Frequency Response Functions(FRFs) to the in-terface reduction algorithm called Virtual Point Transformation(VPT) and later to a substructure coupling method called Frequency-Based Substructuring(FBS). These methods form the cornerstone for Transfer Path Analsysis (TPA). Furthermore, com-mon sources of error like sensor mass loading effect and sensor misalignment have also been investigated. Lastly, a new method to calculate the sensor positions and orientations after a measurement has been devised based on rigid body properties of the system and from the applied force characteristics.

The error propagation was performed using a computationally efficient, moment method of the first order and later validated using Monte-Carlo simulations. The re-sults show that the orientation measurement error is the most significant followed by FRF error and position measurement error. The mass loading effect is compensated using the Structural Modification Using Response Functions (SMURF) method and the sensor misalignment is corrected using coordinate transformation. The sensor positions and orientations are accurately estimated from rigid body properties and applied force characteristics; individually using matrix algebra and simultaneously using an optimization-based non-linear least squares solver.

Keywords: Dynamic Substructuring, Error modelling, Transfer Path Analysis, Un-certainty Propagation

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Acknowledgements

I would like to express my gratitude to my supervisor Fabio Bianciardi from Siemens for giving me this great opportunity and for patiently supporting me throughout this thesis. His experience and knowledge in modal testing has been instrumental. I also thank Ulf Sellgren from KTH, for his helpful guidance and support in this thesis. Furthermore, he has been influential in helping me discover my interest in structural dynamics through his courses.

Last but not least, I would like to thank all my friends and family who have been supportive and kind with me throughout this endeavour. They have kept me going even through the toughest of times.

Harikrishnan Venugopal Stockholm, October 1, 2020

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Nomenclature

Notations

B Boolean Localization matrix f External force matrix

g Interface force matrix

G Transformation matrix for forces H Frequency Response Function I Moment of Inertia

m Virtual Point force matrix M Mass

Oi Direction cosine on the ith axis q Virtual Point displacement matrix

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DC Direction Cosine DOF Degree Of Freedom

FBS/LM-FBS Frequency Based Substructuring FE Finite Element

FRF Frequency Response Function IRF Impulse Response Functions NVH Noise Vibration and Harshness TPA Transfer Path Analysis

VP Virtual Point

VPT Virtual Point Transformation

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3.1 Levels for design of experiment . . . 31 4.1 Model Update results . . . 59 4.2 Results for sensor position estimation using rigid body properties . 80 4.3 Results for sensor position orientation using rigid body properties . 81 4.4 LMA-based estimation of sensor position from rigid body properties 81 4.5 LMA-based estimation of sensor orientation from rigid body

prop-erties . . . 81

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Chapter 1 Introduction

1.1 Background

A principal problem in Noise Vibration and Harshness(NVH) engineering is the characterization of paths through which vibration travels from one part of the car to another. The passenger feels vibrations and hears sounds that originate from sources that are hard to assess viz. the powertrain and the air surrounding the car. Analyzing these paths can give valuable insights upon which the overall comfort of the vehicle can be improved.

Transfer Path Analysis(TPA) is a method to identify and quantify the vibration that is transferred from one point of a system to another. A main advantage of doing this is that both the exciting force and individual path contributions to the response can be found. Also, the interface forces from an active sub-system(like an engine) can be measured and assembled into a simulated body. The coupling of sub-systems are done with the help of a set of methods, commonly referred to as dynamic substructuring (or hybrid substructuring). Thus it also helps in modular development of a system.

Dynamic substructuring typically requires a lot of measurements of the system and measurement errors affect the system coupling as well the TPA results. The effect of such errors are to be understood for defining the quality of the results obtained.

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1.2 Purpose

In the early 2000s electrical engines entered the vehicle market and pioneered a revolution in all related engineering services. In NVH engineering this meant that the frequency of interest shifted from the low frequency of the combustion engine to higher frequencies( >4kHz) of the electric motors. Damping these high frequency responses is of paramount importance due to the reasoning that sound is perceived to be louder at higher frequencies than at lower[2]. It is seen in figure 1.1 that a sound of 80 dB has 20 phon(unit of loudness) at 32 Hz but has 80 phon at 4kHz .

Figure 1.1: Equal loudness curves[1]

But it was found that TPA methods failed to give accurate results in these fre-quencies. The response measured and the same calculated from TPA is found to be contrasting. This is assumed to occur mainly due to simplifying assumptions or errors which are common in the measurements carried out for TPA[5]. It has been identified that the majority of the errors occur in the substructuring phase of TPA. The purpose of this thesis is to model the errors in dynamic structuring and also to compensate for some of the more commonly occurring errors during measurements.

1.3 Objective

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3 1.4. Delimitations

The main deliverables of the thesis are as follows:

1. Model and analyse rigidity-based errors in virtual point transformation using an FE model.

2. Find the ideal interface connection method within the FE model to approxi-mate results from dynamic substructuring.

3. Model the uncertainty in dynamic substructuring and virtual point transfor-mation due to position measurement error, orientation measurement error and random FRF errors.

4. Model and suggest compensation methods for common sources of error during measurements.

5. Formulate a technique to estimate sensor positions and orientations from the rigid body properties of a system.

1.4 Delimitations

• Transfer Path Analysis is a field of vast scope and multiple methods within can be explored. Due to time constraints, the scope of the thesis is limited to modelling the errors in dynamic substructuring. This can be later used for different methods within TPA.

• In this thesis, the individual sub-systems will be integrated together by a substructuring metod called Frequency Based Substructuring (FBS/LM-FBS). Other coupling methods are not considered since the FBS/LM-FBS method is already well established.

• Experimental results needed for the model update have been provided be-forehand, with no scope of performing a new experiment. Thus it has to be assumed that the results obtained are of high quality.

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Frame of Reference

To understand the challenges at hand, a methodical literature review is required. Several topics in Structural Dynamics, Transfer Path Analysis, Substructuring and Model Updation are presented here.

2.1 Introduction to Structural Dynamics

Structural Dynamics is the science explaining the response of a structure to dynamic loading. Such loads can be from people, waves, wind, machines etc. The response is measured using sensors in terms of displacement, velocity or acceleration.

2.1.1 A general framework for dynamic analysis

In general, dynamic analysis can be performed in five domains: physical, time, frequency, modal and state-space. Each of these domains are equivalent in nature. The domains and their conversion methods are shown in figure 2.1 .

• Physical Domain: In the physical domain, the system is characterized by physical properties of a dynamic system viz. mass, stiffness and damping. Other than for analytical cases, the mass, stiffness and damping matrices are obtained from a numerical FE model. However, this becomes computationally expensive for geometries with complex features that require a lot of DOFs to model.

• Modal Domain: In modal domain, the response of a structure is described in terms of its eigenmodes. In this method, a linear combination of global eigenmode shapes and quasi-static modes are used to represent the response of a system; the local dynamics are ignored to reduce the complexity of the

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5 2.1. Introduction to Structural Dynamics

Figure 2.1: Illustration of domains for dynamic simulations with methods for con-version between domains. (Diagram by Mvanderseijs / CC BY-SA)

problem. Accordingly, this method is also referred to as modal reduction, since it approximates the response by selecting modes that have the highest contribution.

• Frequency Domain: In Frequency domain, the system parameters like mass, stiffness and damping are not explicitly available. It assumes a time-invariant (system parameters remain constant), steady-state (transient responses are damped out) and linearized system. This method is useful from an experi-mental aspect as one usually measures the frequency response.

• Time Domain: In many problems, the system behaves non-linearly and have transient responses resembling Dirac delta functions(function with a sharp peak). Also sometimes, the modes are too close together to separate them realistically. In such conditions, using the time domain is more useful than the frequency domain. Time domain solutions use impulse response function (IRF) matrices to find the response. Several approaches are discussed in [3, 20, 21] • State-Space Domain: State-space representation contains input,output and

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matrices (called as ABCD-quadruples), where A being the state matrix, B the input matrix, C the output matrix and D the ’feed-forward’ matrix. The ma-trices are characteristic of a system and can algebrize its differential equations [3, 22, 23].

2.1.2 Frequency Response Functions

The response of a structure at a point due to an exciting force at another is char-acterized by functions of frequency involved.They are called Frequency Response Functions(FRFs)[3, 6, 7]. The responses can be displacements, velocity or accelera-tion. This relation is expressed in equation 2.1 below.

Un= Hnm.Fm (2.1)

Where Un represents the response of the system at degree of freedom n due to a force at m. The FRF connects the force and response together. An example of an FRF connecting the acceleration at a point to force at another is given in figure 2.2 below. The peaks in the FRF represent resonant conditions wherein the system’s response to excitation is maximum[3].

Figure 2.2: A sample Frequency Response Function

2.2 Dynamic Substructuring

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7 2.2. Dynamic Substructuring

then coupled to obtain the dynamics of the complete system. This is done by re-ducing the coupling problem to consider the dynamics only at the interfaces of the substructures; implemented by enforcing the equilibrium and compatibility bound-ary conditions.

The origin of dynamic substructuring lies in the mathematical technique of do-main decomposition . Here,a differential equation with constraints(also called as a boundary value problem) is splitted into smaller equations and solved iteratively. Additional equations are created to ensure continuity of the final solution. The technique was developed in 1890 by a mathematician, Herman Shwartz who devel-oped the method as a means to solve continuous sub-domains[12]. However, since a closed-form solution is rarely available, it led to the development of methods that discretized the domain and solved it in an approximate manner. Notable ones are the Rayleigh-Ritz method, Boundary Element Method(BEM) and Finite Element Method (FEM). In structural dynamics, the first techniques of substructuring were developed using mode shapes of the independent substructures and were collectively referred to as Component Mode Synthesis(CMS) [3, 7, 14]. Some of the classical methods in CMS are Craig-Bampton method [13, 3], Dual Craig-Bamption method [15, 3], MacNeal’s method[16, 3] and Rubin’s method [17].

Later in the 1980s, as sensor technology and signal processing techniques devel-oped, the experimental dynamics community started to embrace substructuring more. This lead to the formulation of Frequency-Based Substructuring(FBS) [18] by Jetmundsen et al. and the Structural Modification Using experimental frequency Response Functions(SMURF)[19] by Crowley et al. Even though both methods have identical concepts, the former gained popularity with the community; mostly because SMURF was aimed as a troubleshooting method to account for effect of extra masses (like sensors) on the measurement while FBS was taylored specifiaclly for substructuring.

Dynamic substructuring has proven to be extremely useful to product design and development. It offers the following advantages:

• Substructuring allows for modularity in design. Thus different teams can work independently on different components and then assemble them to predict the behaviour of the assembly. As a result, changes can be still made in the design phase.

• Substructuring allows for efficient sharing of information.Thus, low-complexity models can be shared and assembled easily; this significantly speeds up the design process.

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from a FE model.

• The modulraity in design offered by substructuring helps in giving attention to specific components within a design. For example, some components will have a recognized non-linear behaviour while other will require only the linear treatment. Besides, it will help in understanding the dynamic behaviour of each component better.

2.2.1 Substructure coupling

Coupling along interfaces

A complex superstructure divided into substructures have to be coupled such that two interface conditions must be always satisfied:

• Compatibility: The displacements at the interface of the substructure must be equal and in same direction.

• Equilibrium: The forces at the interfaces must be equal and opposite. A graphical representation of the coupling conditions is shown below in figure 2.3

Figure 2.3: Illustration of boundary conditions for coupling of two substructures Let us consider the two substructures depicted in figure 2.3.Here 1 and 4 are internal DOFs and 2 and 3 are the interface DOFs of substructure A and B respectively. The displacement and force matrix related to these nodes can be defined as seen in equation 2.2 below. Matrix u represents the displacments, f represents external forces and g represents interface forces.

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Firstly, the compatibility conditions will be discussed followed by force equilib-rium.There are two ways of representing the coupling conditions viz. using local substructure coordinates or using global coordinates of the resulting assembly. Com-patibility conditions in substructure, local coordinates are shown in equation 2.3 below.

B.u = 0 ; B = [0 I −I 0 ] (2.3) Compatibility conditions in global, assembly coordinates are shown below in equa-tion 2.4. Here q represents displacement in global set of coordinates.

u = Lq ; L =     I 0 0 0 I 0 0 0 I 0 0 I     and q =     q1 q2/3 q2/3 q4     (2.4) Both B and L are referred to as Boolean Localization matrices; it has identity ma-trices for performing necessary Boolean operations on other mama-trices. Comparing equation 2.3 and 2.4 we get the important relation that B and L make each other’s null space, i.e. one can be obtained from the other.

B.u = B.Lq = 0 ∀q _{∵ L = null(B); ⇒ B.L = 0} (2.5) The force equilibrium is based on Newton’s third law of motion, stating that every force should have an equal and opposite reaction. This relation is represented in local coordinates using the signed Boolean matrix B.

g = −BT.λ (2.6)

Where λ represents the magnitude of the interface forces. It should be noted that on expanding the matrices, interface forces exist only on the collocated DOFs. Similarly, in the global coordinates we use the Boolean matrix L as shown in equation 2.7.

LT.g = 0 (2.7)

From equations 2.6 and 2.7, we can see that for any magnitude of the interface forces, the equilibrium is automatically satisfied. This can be easily shown using the fact that B and L are null-spaces of each other.

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clarity of operation and popularity within the community. Similar to the matri-ces of displacements and formatri-ces, the matrix of frequency response functions can be assembled. ZA=Z A 11 Z12A Z₂₁A Z₂₂A ZB =Z B 33 Z34B Z₄₃B Z₄₄B ; HA=H A 11 H12A H₂₁A H₂₂A HB =H B 33 Y34B H₄₃B H₄₄B (2.9) Both the impedance (Z ) and admittance(H ) form of the FRF matrix is shown above. However,for multiple substructures, the matrix operations are much easier if the FRF matrices are in block-diagonal form. The FRF matrix for the substructures A and B in figure 2.3 can be written as follows.

Z =     ZA 11 Z12A Z₂₁A Z₂₂A 0 0 0 0 0 0 0 0 ZB 33 Z34B ZB 43 Z44B     ; H =     HA 11 H12A H₂₁A H₂₂A 0 0 0 0 0 0 0 0 HB 33 Y34B HB 43 H44B     (2.10) These uncoupled FRF matrices in equation 2.10 can be used with the displacement and force matrices to obtain the corresponding corresponding dynamic equation of the linear system. For a system with mass matrix M, stiffness matrix K and damping matrix C, we have:

Zu = f + g where Z = −M ω2+ (iω) C + K (2.11)

u = H(f + g) (2.12)

It should be noted that in equation 2.11 and 2.12 the interface forces and the local displacements are unknown. Thus the number of unknowns exceed the number of equations. The necessary additional equations are obtained by enforcing interface conditions. The two standard methods to do this are discussed next.

Primal Assembly

In primal assembly, the unknown displacements u, also known as primal unknowns are used to define the interface problem. By defining the local displacements u in the global assembled coordinates q, the compatibility conditions are satisfied a priori.In particular, by using a suitable Boolean Localization matrix L. The force equilibrium equations in 2.6 are also added. From equations 2.4 and 2.11, we get:

ZLq = f + g (2.13)

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Pre-multiplying equation 2.13 with LT, the interface forces drop out and as a result we get equal number of unknowns and equations.

ˆ

Zq = ˆf ; with Z = Lˆ TZL and ˆf = LTf (2.15) Here ˆZ represents the impedance matrix in global coordinates and ˆf represents the external nodal forces in global coordinates. The inverse of the global impedance will give the global admittance matrix. Lastly, it should be noted that the interface forces are not obtained as a part of the calculations. They can be obtained once the problem is converted back to local coordinates.

g = Zu − f =ZL LTZL −1

LT − If (2.16) Dual Assembly

Similar to the primal assembly problem, the interface forces or the Dual unknowns are used to define the interface problem. In the Primal assembly problem, the interface force equilibrium is automatically satisfied instead of the compatibility. Therefore,the interface compatibility equations are added seperately to create a de-terminate system of equations.From equations 2.3 and 2.11 we get:

Z.u + BT.λ = f (2.17)

B.u = 0 (2.18)

Equation 2.17 can be rewritten as:

u = H. f − BT.λ where H = Z−1 (2.19) Applying compatibility condition(eq.2.18) to equation 2.19 we get:

BH f − BTλ = 0 (2.20) Expanding 2.20 and rearranging we get the relation connecting the interface forces to the external ones.

λ = (BHBT)−1.BHf = (BHBT)−1.Bu (2.21) The λ in equation 2.21 represents the intensity of the interface forces. Observe that in equation 2.21 , Bu shows the interface incompatibility due to external forces f and (BHBT₎−1 _{is the assembled system impedance at the interface. The remaining} unknowns, the local displacements are obtained from equations 2.17 and 2.21.

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From equation 2.22 we can also obtain the assembled admittance. Hass = H − HBT BHBT

−1

BH (2.23)

This equation forms the basis of the LM-FBS procedure explained next.

2.2.2 Lagrange-Multiplier Frequency Based Substructuring

Lagrange-Multiplier Frequency Based Substructuring (LM-FBS) [3, 6, 7] is a proce-dure used to couple structures by measuring individual sub-system FRFs. It uses the dual assembly coupling procedure, which was discussed earlier . There are four major steps involved in the LM-FBS procedure:

1. The substructure FRF matrices must be obtained with both interface and internal DOFs.

2. The substructure displacements,forces and FRF matrices must be represented in a global manner. The displacement and force matrix are formed by ar-ranging the local substructure displacement and forces respectively along a column. The global FRF matrix is formed by arranginf the substructure FRFs as a block diagonal matrix.

u =    u(1) ... u(n)    f =    f(1) ... f(n)    H =    H(1) . . . 0 ... ... ... 0 . . . H(n)    (2.24)

3. The system connectivity is to be specified using the signed Boolean localization matrix B; it connects interface local DOFs u with signs to differentiate each other. Otherwise, the global Boolean localization matrix L can be made for global DOFs q and using the null-space relation B can be obtained.

4. Apply LM-FBS formulation (see eq.2.23) for each frequency line in the required range.

2.3 Transfer Path Analysis

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13 2.3. Transfer Path Analysis

the response can be defined by both force and the path, such an analysis can help in identifying critical regions of improvement.

2.3.1 Component-based Transfer Path Analysis

The classical method of TPA uses an assembled system to characterize the vibration paths. This has been a major drawback, because if the passive structure is modified or replaced it can no longer predict the dynamic behaviour of the system since the forces present in the interface are contact forces; which are dependent on the complete assembly.

This issue is resolved by independently assessing the forces of the active sub-system at the interfaces. Several techniques have been developed in this regard. They are collectively referred to as Component-Based TPA. The methods to find the equivalent force at the interface include attaching the active source to a rigid support or leaving the interface free to vibrate or by assembling to a reference structure whose transfer functions are known[3, 5, 6]

2.3.2 The Equivalent force concept

The equivalent force from the active sub-system to the passive one is determined by the measurement of FRF and responses at the interfaces. Usually for estimation of equivalent force, a reference passive structure is used with its FRFs known. Here the Dual assembly formulation will be used here to estimate the equivalent force. In figure 2.4 one can find an assembled system with source at point 1 in A and a receiver body B with response points at 4. Both parts are connected together at point 2 in A and point 3 in B.

Figure 2.4: Illustration of Equivalent Force Concept

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represent interface forces and f1 is the source at an interior point in A.     uA 1 uA₂ uB₃ uB 4     =     HA 11 H12A 0 0 H₂₁A H₂₂A 0 0 0 0 H₃₃B H₃₄B 0 0 HA 43 H44B     .     f1 g₂A gB₃ 0     (2.25) The sub-systems are coupled by enforcing compatibility and equilibrium at the inter-face. This is done with the help of boolean operators. A matrix of identity functions referred to as the Boolean Localization Matrix(B) is generated for this purpose.

B =0 1 −1 0 (2.26)

Applying compatibility and equilibrium using operator 2.26 we get:

δ = B.u = 0 ⇒ uA₂ = uB₃ (2.27) g = −BT.λ ⇒ −gA₂ = gB₃ = λ (2.28) Here the equation 2.27 represents the incompatibility and equation 2.28 represents the force equilibrium at the interface. λ represents the interface force that keeps the sub-systems together and δ is the incompatibility in the interface which is ideally zero.

Applying equation 2.27 and 2.28 to equation 2.25 we get:

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15 2.3. Transfer Path Analysis

Thus from equation 2.33 one can see that the equivalent force is only dependent on the characteristics of the source body A. This makes it a particularly useful concept to relate a source from a measured body to any other structure, even a Finite Element one. A closer look into equation 2.32 will reveal that this is in fact equivalent to the result from 2.22. This ties the equivalent force calculation to substructuring and the subsequent errors found within.

2.3.3 An application of TPA

Now we have effectively characterized the force applied by the source and have transferred it to the interface. Next, the system is coupled using the LM-FBS formulation and the equivalent force concept is used to predict the response. Rather than a direct substitution, the dual interface problem is explained specific to this case for better understanding. Assume in figure 2.4 that an and external force equal to the equivalent force acts on the sub-system A. This will invoke a response as shown in equation 2.34.     uA₁ uA₂ uB 3 uB₄     =     H₁₁A H₁₂A 0 0 H₂₁A H₂₂A 0 0 0 0 HB 33 H34B 0 0 H₄₃A H₄₄B     .     0 feq 0 0     ≡ u = H.f (2.34)

Since in reality A is connected to B, this response will cause an incompatibility in the interface.

δ = B.u = uA₂ − uB

3 (2.35)

Applying equation 2.35 to equation 2.34 we get:

δ =HA 11 H22A −H B 33 −H B 34 .     0 feq 0 0     (2.36)

In order to prevent this incompatibility interface forces must exist. From the dual interface problem(see equation 2.21) we can see that the interface forces can be expressed in terms of the external forces. This is shown in equation 2.37 below.

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Now that the interface forces are defined, we can find the coupled response at all the points in the system.

    uA₁ uA₂ uB 3 uB₄     =     H₁₁A H₁₂A 0 0 H₂₁A H₂₂A 0 0 0 0 HB 33 H34B 0 0 H₄₃A H₄₄B     .     0 −λ λ 0     (2.38)

Referring back to the dual assembly problem in equation 2.23 we can assemble the FRF matrix using all the previous computations:

u = H_assAB.f (2.39) H_assAB =         H₁₁A H₁₂A 0 0 HA 21 H22A 0 0 0 0 HB 33 H34B 0 0 H₄₃B H₄₄B     −     H₁₂A HA 22 −HB 33 −HB 43     H₂₂A + H₃₃B−1 HA 21 H22A H33A H1134     (2.40) The relation connecting the equivalent force and the receiver points can be derived from the equation 2.40.

u4 = H42B.λ = H B 43. H A 22+ H B 33 −1 .H₂₂A.feq (2.41) This should be same as the response produced in the system when it is assembled and thus it provides the opportunity to relate FRFs in equation 2.41 to FRF of the assembled system.

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17 2.4. Interface modelling using Virtual Point Transformation

2.4 Interface modelling using Virtual Point

Trans-formation

In most cases the experimentally measured responses are closer to the actual response when compared to a numerically simulated one. But experimental methods are prone to many simplifications that affect its accuracy, mostly due to lack of proper measurement apparatus. One such prominent example is to neglect the rotational degrees of freedom at the interface, which leads to spurious results most of the time [5, 8, 24]. This is mostly because the interface DOF is either inaccessible or sensors for measuring 6-DOF are unavailable. Accordingly, to counter this shortcoming, a number of interface modelling methods are proposed [8, 25, 26, 27]. One such method which considers the interface rigid is explained next.

In the real world, the interface connections are mostly surface and line connections, but for estimation of FRF and equivalent forces, a point is easier to model. Virtual Point transformation can be used for this purpose [3, 7, 8]. It can also be used to in-troduce rotational degrees of freedom through multiple translational measurements. One main assumption of this method is that it assumes that the interfaces are rigid.

2.4.1 Interface displacement reduction

Multiple sensors are connected around a joint of interest as shown in figure 2.5. The first step is to select a point which is to be the virtual point.

Figure 2.5: Virtual Point Transformation for a joint [8]

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in the global axis (upper case subscripts) are found from local displacements(lower case subscripts) as seen in equation 2.43.

  uk X uk Y uk_Z  =   ek x,X uky,X ukz,X ek

x,Y eky,Y ekz,Y ek x,Z eky,Z ekz,Z  .   uk x uk y uk z  = Ek.uk (2.43) Now the displacements of the virtual point represented as qv _{(for virtual point v)} can be related to the global displacements as shown in equation 2.44 below.

  uk_X uk Y uk Z  =   1 0 0 0 rk_Z −rk Y 0 1 0 −rk Z 0 rkX 0 0 1 rk Y −rYk 0  .         qv X q_Yv qv Z qv θx q_θyv qv θz         +   µk_X µk Y µk Z   (2.44) u = Rq + µ (2.45)

The matrix R, called sensor interface displacement modes matrix(IDM), transforms the sensor displacements in the local frame to 6 virtual point displacements and rotations.Here, r represents the vector of the sensor position w.r.t the virtual point .Matrix µ represents the residual displacements that are not represented in the virtual point. For example, if the virtual point is taken so that only five DOFs are to be evaluated, then the residual will have the remaining displacements. Residual matrix also includes displacements that do not obey the rigidness criterion [8]. The residual matrix µ is found to be orthogonal to the matrix R, hence RT_.µ=0_{. So} equation 2.45 can be pre-multiplied by RT as seen in equation 2.46.

RTu = RT(Rq + µ) = (RTR)q (2.46) ⇒ q = T u; T = (RT_R)−1

.RT (2.47)

The matrixT in equation 2.47 is the standard Moore-Penrose inverse of the displac-ment IDM matrix R. Matrix q defines the virtual point displacedisplac-ments. It should be noted that the residual displacements are absent from this formulation; thus q has only the rigid body interface displacements.

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19 2.4. Interface modelling using Virtual Point Transformation R =    R(1,1) _{. . .} _R(1,nk) ... ... ... R(nv,1) _{. . . R}(nv,nk)   ; u =    u(1,1) ... u(1,nk)    (2.48)

It is suggested that at least twice the number of measurements as the virtual point DOFs are required for a good estimate. i.e 12 sets of displacements for 6 DOF virtual point. Also it is suggested to use at least three 3-D sensors in different positions. Otherwise, if only two 3-D sensors are used, it is impossible to track the rotation in the axis between the sensors. Inverting the virtual point forces and moments with the virtual responses will give the virtual FRF[8].

2.4.2 Interface force reduction

A similar approach can be followed to find the virtual point forces and moments. By definition the local frame sensor displacements are uniquely described by the virtual point displacements (see eq.2.44). However, this is not the case with forces and moments. For example, the same moment can be created at the virtual point by different pairs of forces and therefore forces are not uniquely defined. On the contrary, the reverse holds true i.e if the position and direction of the forces are known, one can calculate the resultant moments from them. This implies that the interface force reduction is performed inverse to the displacement reduction.

Analogous to the displacement transformation matrix, we have force IDM matrix G to transform the local frame forces to the virtual point. Referring to figure 2.5, this relation can be stated as follows:

        mv_x mv y mv z mv θx mv_θ_y mv_θ_z         =         1 0 0 0 1 0 0 0 1 0 −rk Z rYk −rk Z 0 −rXk −rk Y rXk 0         .   fk x f_yk f_zk   (2.49) mv = Gvk.fk (2.50)

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G =    G(1,1) _{. . .} _G(1,nk) ... ... ... G(nv,1) _{. . . G}(nv,nk)   ; f =    f(1,1) ... f(1,nk)    (2.51)

In case of additional excitations fh_{, the force IDM matrix can be changed, provided} their position and orientation are known. The IDM matrix for these forces is ob-tained by changing the vector rk and fk in equation 2.49 to rh and fh respectively (see fig.2.5). Further,the rules for stacking can be applied by adding row-wise to the force matrix and the column-wise to the IDM matrix. However, if excitations exist only on the sensor faces, then G = RT_.

Now if we invert 2.50, we get the the local frame forces f. But from equation 2.49, we can see that the number of equations are greater than the number of unknowns. So a least squares solution is suggested by using the Moore-Penrose inverse. It should be noted that the forces thus computed will only load the interface in a rigid manner.

f = T_fTm ; T_fT = GT. G.GT−1 (2.52)

2.4.3 Virtual Point Receptance

From the definitions of virtual point displacement and forces, one can construct a receptance matrix for the virtual point; this directly transforms the local forces and displacments to the virtual point reference. In the frequency domain we can write the dynamic equations follows:

u = Huf.f (2.53)

By substituting 2.47 and 2.52 in 2.53 we get:

q = TuH TT_f .m (2.54) ⇒ q = Hqm.m ; Hqm = TuH TT_f (2.55) Here Hqm is the virtual receptance matrix connecting the virtual displacements to virtual forces. Finally, it can be observed that the reduced interface has lesser DOFs than the DOFs measured by the sensor. This ’weakening’ of the interface is beneficial for coupling.

2.4.4 Special case: Inclusion of sensor orientation

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21 2.4. Interface modelling using Virtual Point Transformation

seen in figure 2.6. This change in orientation has to be included within the IDM matrices. Assume that a sensor P is an inclined position then the sensor IDM matrix(R)is modified as follows:

R =A B (2.56)

A = 



OkxP OkyP OkzP OlxP OlyP OlzP OmxP OmyP OmzP   (2.57) B =   rP Y.OkzP − rZPOkyP rZP.OkxP − rXP.OkzP rXP.OkyP − rYPOkxP r_YP.OlzP − rZPOlyP rZP.OlxP − rXP.OlzP rXP.OlyP − rYPOlxP rP_Y.OmzP − rP

ZOmyP rZP.OmxP − rXP.OmzP rPX.OmyP − rYPOmxP 

 (2.58)

Figure 2.6: Tilted axes of sensor P

Several additional orientation terms are added to the matrix. From the figure above, we can see that the axes of the sensor are k, l and m. Each of these measuring directions(unit vectors) can be represented as direction cosines on the global axis (x, y and z). For example, unit vector along k can be represented as OikxPˆi + OkyP.ˆj + OkzP.kˆ where ˆi, ˆj and ˆk are unit vectors along x, y and z axes as shown in figure 2.6. The subscripts for, say OkxP, can be understood as the direction cosine of local sensor axis k along the global axis x for sensor P . The stacking laws for multiple sensors and multiple virtual points remain the same as before. A similar change can also be made for force IDM matrix in case of an inclined applied force.

2.4.5 Quality of Virtual Point Transformation

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larger than the virtual point DOFs. Therefore, the effect of the residuals of the least-square methods can be studied. In previous sections it was explained that only rigid body motions are ’filtered’ to the virtual point and that the elastic behaviour is represented in the residual vector. Hence, such a study can also help to identify regions with elastic behaviour; which therefore cannot be used for measurements for VPT.

Specific Sensor Consistency

Sensor consistency compares the measured responses to the responses predicted by the virtual point and is evaluated per load channel. This function was originally proposed as an indicator of the rigidity of the interface [47].

Consider a virtual point i and a load case fj at a point j. The sensor displacements for this load case is denoted as ui,j. Now we compare the responses after the IDM projection (eui,j) and the original responses (ui,j).

ui,j = Hi,j fj (2.59) e

ui,j = RTuHi,j fj (2.60) Then the specific sensor consistency is estimated using the formulation for spectral coherence and is shown in equation below:

ρ_u_i,j = (ui,j + uei,j) u ∗ i,j + ue

∗ i,j

2 ui,j . u∗i,j+eui,j . eu

∗ i,j

(2.61)

The equation above is evaluated for each frequency line in a range of frequencies. Also, if specific sensor consistencies are avreraged over all the load cases, we get the average sensor consistency, which will be used to assess the performance of the VPT in this thesis. The sensor consistecy becomes low when the interface lacks rigidity or when the positions and orientations of the sensors are reported incorrectly. Specific Impact Consistency

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23 2.5. Model Updation for Numerical simulation

This metric is used to identify if there is any difference between the response obtained due to a single force and the response obtained after reduction of that force to the virtual point.

yi,j = wjT.Hi,j (2.62) e

yi,j = wTjHi,j G Tf (2.63) As done previously, we define the specific impact consistency as a variant of the coherence function. This formulation is depicted below:

ρfi,j = (yi,j+ eyi,j) y ∗ i,j + ye ∗ i,j 2 yi,j . y∗i,j +yei,j . ey

∗ i,j

(2.64)

Impact consistency is more of a quality check for the measurements than the sensor consistency. The specific impact consistency can be low when impact positions and directions are reported wrongly or if the impact itself is faulty (e.g. double pulse, low energy) or if the sensor response channels become overloaded due to the impact [7]. If no weighting function is used then from eq.2.64, the average impact consistency is obtained.

2.5 Model Updation for Numerical simulation

In many a cases, the experimental model is far superior to the numerical one in terms of accuracy. Other than the imperfections due to discretization, there arises errors in the FE model due to lack of information regarding the material properties of the test bench components(like Modulus of Elasticity and Poisson’s ratio). This error can be reduced by optimizing the FE model to correlate with some of the estimated modal parameters from experiments (like resonance frequencies). Nonetheless, the updated parameters should also be physically significant. Model updating has been extensively used in the industry for structural damage identification and structural health monitoring [9, 28, 29, 30].

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of the nodes post updating, something which was lacking in the direct methods. It-erative methods either use eigendata (like eigen-frequencies or eigen-vectors) or FRF data. Collins et al.[33] used natural frequencies and mode shapes, Arora et al.[34] used both resonant and anti-resonant frequencies and Lin and Ewins [35] formulated a model update method based on measured FRFs. Iterative methods which include damping has also been proposed by Arora et al. [36].

Due to the advantages of the iterative method and its popularity, it will be used for model updating in this thesis. All iterative methods use optimization algorithms for reaching the optimum. An optimization problem has four components:

• Design Variable: Design variables are those whose values are to be optimized so that the numerical model correlate with the experimental one. For instance, design variables can be mass matrix and stiffness matrix of a FE model whose values upon change results in simulation outputs that agree with the mea-sured outputs. Design variables are also given starting values to initiate the optimization problem.

• Objective: The objective function is a function whose value has to be maxi-mized or minimaxi-mized to optimize the problem. In the case of model updating, objective function connects the discrepancies between the experiment and sim-ulation to the design variables. Consider the case of an FRF-based model up-date. Here the objective function can be FRF-correlation criteria like FRAC, FDAC, etc whose values have to be maximized to get the accurate model. In cases where no apparent relation between the objective function and design variables exist, Response Surface methods can be used to formulate one [38]. • Constraints: The optimization algorithm treats the model update problem

as a mere mathematical one, devoid of physical meaning. Thus more often than once, an unconstrained problem leads the optimization to converge on a spurious solution for the design variables. Hence, acceptable limits are kept on the values of design variables. For example, consider the case of a model update problem with the elastic modulus as a design variable. Usually there is a good guess of the material used; therefore elastic modulus should be constrained within the limits possible for that material.

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25 2.6. Modelling of uncertainty in dynamics

2.6 Modelling of uncertainty in dynamics

In the previous sections, different analytical techniques for interface reduction and substructure coupling are discussed. However, the input parameters for these calcu-lations are prone to error, thereby affecting the accuracy of the results. For example, consider the measurement of sensor position and orientation in the case of calcula-tion of virtual point properties. Hence, it is necessary to characterize the results, not as absolute values, but within a range defined by a confidence interval.

Uncertainty quantification in dynamics is generally a newly-explored topic.Voormeeren et al. used the moment method, both first and second order method, to calculate amplitude and phase uncertainty of a coupled FRF and verified them with a Monte-Carlo simulation [48]. In this case, random errors in the FRFs were considered as sources of uncertainty. Further development on this method came from Trainotti et al.[49], when the moment method was extended to VPT. Here, the moment method was also studied in detail for validity of its statistical assumptions. Accordingly, the method was also extended to classical and component-based TPA[50, 51].

2.6.1 Theory of uncertainty propagation

Uncertainty thrives in all processes in nature. An experiment performed multiple times with the same conditions often give different results. Therefore, the results are presented along with statistical errors. If the errors are large, it can be because of an unrecognized factor which acts as an input or because the input conditions are not maintained to the required degree of precision or because the experiment is sensitive to small changes in the input conditions which cannot be avoided. In many a case, all these factors are conjoined.

In many cases, the relation between the input variables and output is not defined. Accordingly,an approximate model is created. Here we explore the mathematical technique to approximately quantify the error in the output, provided the errors in the input are available.

Consider a non-linear function f(x) defined by a single variable x . The function is represented graphically in figure 2.7.

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Figure 2.7: Error propagation of a one dimensional non-linear function below. f (x) = f (µx) + ∂f ∂x x=µx .σx (2.65)

Thus the mean and standard deviation of f(x) are:

µy = f (µx) (2.66) σy = ∂f ∂x x=µx .σx (2.67)

Equations 2.66 and 2.67 cover the basic framework behind uncertainty propagation. But there are some aspects that affect the accuracy of such an approximation:

• Usually the population mean (actual mean of x)is unknown and only mean of a sample is available. So it is assumed that the sample mean is close to the population mean.

• The function f(x) might be extremely non-linear near µx. Henceforth, the first-order linear approximation is no longer sufficient. It is suggested that the method be used, provided f(x) is almost linear within one standard deviation of the mean [11]. This is also referred to as the small error assumption. • Similar to the population mean, the population variance is usually not known.

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Chapter 3 Implementaion

The first step is to make an accurate FE model of the TPA test setup in Siemens. This includes meshing of its CAD model followed by a model updating procedure with the experimentally obtained FRF measurements. Once a satisfactory model is obtained, both interface reduction and substructure coupling methods are tested on it. The results of these tests give valuable insights on these methods and also on FE modeling in general.

3.1 Finite element modelling and updating

3.1.1 Test setup description

The TPA test setup seen in figure 3.1a is used in Siemens for demonstrating var-ious TPA methods. The source and receiver parts meet each other, separated by aluminium blocks. Therefore, from a substructuring point of view, each half of the aluminium block is assigned to each substructure. This split is seen in figure 3.1b. We have a passive substructure with three connection points viz. Subconn1, Sub-conn2 and Subconn3(see fig.3.2b). Referring to 2.4, all the connection regions are reduced to a point in the middle of the connecting surface.

The three aluminium connector arms of the passive substructure are attached to a common base made of steel. The active side comprises of a wiper electromotor with three connection points to match the substructure(see fig.3.3). Its interior design is unknown. The passive substructure is assembled to the active substructure using a bolt. Specifically, from each connection of the passive substructure to the corresponding threaded holes in the active substructure.The bolt acts as a stiffener for the interface. Moreover, the aluminum blocks which are the connection interfaces

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not only give stiffness to the interface (useful for interface reduction) but also provide an ideal surface for measurements.

(a) TPA demonstrator setup (b) Substructure interface (indicated in blue)

Figure 3.1: TPA demonstrator setup with passive structure and active source.

(a) Passive substructure with connection re-gions

(b) Assembling components of passive sub-structure

Figure 3.2: Passive substructure

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29 3.1. Finite element modelling and updating

3.1.2 FE modelling

The first step to accomplish in FE modelling is to obtain adequate meshing of the model. Complex geometric features which may cause problems during meshing also must be smoothed out. For the passive substructure, a 2D mesh of 8-node quad elements are used on the surfaces. They act as a seeding mesh for the 3D elements. The meshing software uses the location of the nodes in the 2D mesh to create the 3D mesh. Moreover, this method of meshing was chosen because local skewing of elements was obeserved for normal 3D meshing technique. The element size for the aluminium arms and the blocks are 2.2 mm. This is also the case for the bolt and washers. The steel base and is assigned an element size of 5mm. The final meshed models are depicted in figure 3.4.The modal correction, meshing and simulation is done in Siemens Simcenter 3D.

(a) Meshed active substructure (b) Meshed passive substructure

Figure 3.4: Meshed models

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3.1.3 Model Update

In order for accurate modelling of the TPA setup, a model update on the material properties is necessary. Model update on each substructure is done separately with the FRFs estimated. In this thesis, the response surface methodology for model updating is followed. This is because the RSA method requires fewer simulations and is hence efficient. An initial guess of the materials is made for each component. Afterwards, the experimental results and the one obtained from the simulations are used to obtain an objective function for the optimization. The material properties act as the design variables. Limits for the design variables form the constraints of the optimization problem. They are usually set based on engineering experience. For the model update case in this thesis ,we have the following settings:

• Objective: The objective function used here is called Frequency response As-surance Criterion or FRAC [42, 43]. The FRAC is estimated for a simulated FRF and the corresponding FRF obtained experimentally ,for each frequency line. The FRAC value varies from zero to one. One meaning complete corre-lation and zero implying a lack of correcorre-lation. Thus the aim is to maximise the FRAC value. The objective function is shown below:

FRACij = P ωHij(ω) ˆHij ∗ (ω) 2 P ωHij(ω)Hij∗ (ω) P ωHˆij(ω) ˆHij ∗ (ω) (3.1) Here Yij refers to the FRF from i to j, estimated from simulation. Similarly, ˆHij is the corresponding FRF obtained from experiment. The * symbol indicates complex conjugate.

• Design variable: The modulus of elasticity of the aluminium and steel are taken as the design variables. All components except the standard ones like bolts and washers are considered.

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31 3.1. Finite element modelling and updating

Table 3.1: Levels for design of experiment Levels

Material Property -1(Lower lim.) 0 1(Upper lim.) EAl(N/mm2) 68000 70500 73000 ESteel(N/mm2) 190000 202500 215000

• Algorithm: Since the objective function (eq. 3.1) does not contain any mate-rial properties, the first step is to derive such a relation. This can be done using a design of experiments(DOE) setup with different levels of material properties as seen in 3.1. Here, the Central Composite Design(CCD) metodology is used for the DOE, specifically the face-centered CCD wherein the star points of the CCD are on the center of each face(α = ±1)[46]. CCD is relatively simple to execute and can also form a quadratic model. This is advantageous for an initial study. The DOE for the modulus of elasticity is shown in table 3.5a be-low. At first, the DOE with the elastic moduli of both materials are prepared and a linear regression model of the FRAC function is created with the elastic moduli as its variables. Thus a response surface of FRAC can be created(see fig.3.5b) . Now the optimization can be performed much easily to obtain the optimum values of elastic moduli. Naturally question arises whether the initial regression model is accurate after the initial optimization. For this matter, a DOE with a tighter limit can be made around the optimized point. Nonethe-less, this is not recommended if the improvements are negligible around the optimized point.

(a) A DOE setup of the Face-centered CCD

type (b) Response surface of FRAC

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From manual inspection of the FRFs it was clear that there was a difference in the location of peaks between the simulated and measured FRFs. By def-inition, a shift in the resonant frequency is easily achievable by changing the elastic modulus when compared to the damping ratio. Hence the model up-date based on elastic modulus alone. Moreover, it also reduces the chance of the optimization falling into a local optima. Such a division of optimization was also beneficial considering the time constraints.

3.2 Modelling for VPT and FBS

Now that the material properties for the model are decided, both interface reduction and substructure coupling are tested on the FE model for their functionality in the realm of simulation. The first step is to check the viability of the VPT model for interface reduction. Accordingly, this simplifying of the interface allows for an easy substructure coupling via FBS, which will be followed suit.

Simcenter 3D uses NASTRAN-based codes for executing simulations. The codes for dynamic simulation are separated into different analysis packages depending on the underlying theory. For this analysis, SOL 111-Modal Frequency Response analysis is used. This method reduces the problem size based on the fact that not all the modes of a structure is needed for calculating its response. The frequency response can be calculated by summing up individual modal responses.

3.2.1 Study on interface rigidity

It was mentioned in section 2.4 that the VPT filters out the flexible motion and assigns only the rigid body motion to the virtual point. Therefore a question arises of how the flexibility of the regions of measurement affect the VPT. This issue will be tackled in this section. The experiment is shown in figure 3.6.

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33 3.2. Modelling for VPT and FBS

In this study, we use the connection point Subconn1 of the passive substructure. The virtual point(as seen in figure) is selected at the midpoint of the sliced part of the bolt. As per considerations of the VPT, an over-determination of two is suggested. Therefore, four positions are taken in the vicinity of the virtual point and and triaxial sensors are mounted on them; thereby we get 12 local frame DOFs for 6 virtual point DOFs. Subsequently, these four sensors are kept in four different position setups as shown in the figure above and virtual point transformation is performed for all sets of positions. This is done keeping in mind that, as we move further away from the virtual point more and more flexible behaviour is observed. Finally, a distinction is made on the measuring region based on its relative rigidity: the block part is significantly more rigid than the neck part. Such a distinction can help to understand the results more intuitively. For a comparison, we consider a virtual point FRF and the same FRF evaluated directly at the virtual point.

3.2.2 Effect of inconsistent points on VPT

In the previous section, we studied the influence of rigidity on virtual point trans-formation. However, in many situations, one may take measurements in a non-rigid region which will lower the overall consistency of the output. This effect is analysed in this section.

For this experiment we take four base point close to the virtual point. These points are in a region that exhibit excellent rigid behaviour (same points as position 1 in the previous section). Afterwards, an additional point is added to the base points; there are three additional points and they are added one by one to the base points and VPT is performed. The simulation setup is shown in figure 3.7.Once the VPT is performed on all the cases, one of reduced FRFs is taken for comparison.

(a) Base points (b) Additional points

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3.2.3 FE modelling of substructure coupling

In section 2.2, we discussed the basics of dynamic substructuring and analytical models are developed. The LM-FBS formulation directly deals with FRFs and therefore it is best suited for experimental results. However, in many cases it is also advantageous to have a numerically simulated model for comparison and for computationally efficient testing. Thus implementation of substructure coupling in a numerical model is of great demand. The aim will be to model the interfaces in an infinitely stiff manner as required by the FBS formulation (otherwise interface stiffness can also be added as a term in the base formula).

Figure 3.8: Virtual points for coupling

Several coupling techniques are implemented and later verified with coupling results calculated analytically. Driving point FRFs are measured at the virtual point of each substructure and then coupled using FBS to obtain the analytical solution. For simplicity, only Subconn1 is coupled. The measured virtual points are shown in figure 3.8. The same FRFs are acquired using the coupling techniques for assembly of the FEM model and then used for comparison. It should be noted that only the bolt tips are connected, the aluminium blocks are not coupled. This is because, ideally it should be a point-to-point connection (at the virtual points) so a close approximation would be to consider a surface as close to these points. The following coupling techniques are tried:

1. Regular Gluing: This is the most common method to assemble structures in Simcenter 3D. Gluing creates a separate face(also called pseudo face) to which the connecting faces of the structures are attached. It translates both forces and displacements across the interface. It is important to note that the elements of the gluing face has a finite value of stiffness which is calculated as the average of element stiffness at both the connecting faces.

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35 3.2. Modelling for VPT and FBS

the surface connect to a point in the middle. Rigid RBE2 elements are used for the mesh.

3. Spider-mid node: A spider mesh is used to connect one surface to a node created at the midpoint of the other surface. The mesh is made of RBE2 elements.

4. Face-Face: Nodes of one face is connected to the other using RBE2 elements. This ensures rigidity of the connecting surfaces. A 2D dependent mesh is cre-ated with one connecting face as the master which ensures that the connecting faces have the same meshing.

5. Universal connection: Universal connection is used to connect two points or a set of points or even elements automatically by elements specified by Simcenter 3D. One disadvantage of this method is that the connecting mesh is not explicitly visible. Here, mid node of one connecting face is meshed to the mid node of the other.

6. Mesh mating: Mesh mating method is used to create identical meshing for the connection surfaces.It creates a 2D mesh on the target surface identical to the source surface and then attaches them together using the gluing method.

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3.3 Uncertainty Propagation in VPT and FBS

In section 2.6 we introduced the basics of uncertainty modelling and also discussed its advancements on application relating to interface reduction and substructure coupling. This thesis will focus on a holistic approach to uncertainty modelling in dynamics by including more errors that realistically can occur during a measurement. Furthermore, the moment method will be used to model the uncertainty for better computational efficiency. Three sources of errors are considered here viz. sensor position measurement error, sensor orientation measurement error and FRF error. They will be explained in detail below.

3.3.1 Error in position measurement

Referring back to section 2.4, we can see that for VPT, the sensor position is to be measured. This is done manually and therefore human errors can creep into the measurement. A reasonable error estimate for position is decided as ±3 mm from its mean value. It should be noted that only the position measurement error is considered, not the positioning error itself. This is because positioning error can change the measured FRFs itself and is therefore very much dependent on the substructure in use, which is not the case for postion measurement error.

Here , we will model the error in position measurement to be used in uncertainty propagation to virtual point transformation and afterwards to frequency based sub-structuring. Errors in VPT is propagated to FBS through the reduced virtual point FRFs are used for coupling.

Let’s consider the sensor IDM matrix in equation 2.44. R =   1 0 0 0 rk Z −rkY 0 1 0 −r_Zk 0 r_Xk 0 0 1 rk Y −rYk 0  . (3.2) Here rk X, r k Y and r k

Z are the components of the position vector of sensor k with respect to the virtual point. As explained earlier, an error value of ±3 mm is chosen for each position of the sensor. Now, the moment method assumes that the input errors are distributed normally(see 2.6). Accordingly, standard deviation for each position component can be calculated from its absolute error, assuming a confidence interval and total number of trials. A confidence interval about the mean value shows the likelihood, with a degree of confidence, that the true value lies within that interval. Here, we assume that the position error is 3mm at 95% confidence(γ = 1.96) for 100 trials. This implies that:

0.003 = 1.96.σ√ x

Component-Based Transfer Path Analysis and Hybrid Substructuring at high frequencies

Component-Based Transfer Path

Analysis and Hybrid

Substructuring at high frequencies

A treatise on error modelling in Transfer Path

Analysis

HARIKRISHNAN VENUGOPAL

Sammanfattning

Abstract

Acknowledgements

Nomenclature

Notations

Contents

Chapter 1

Introduction

1.1

Background

1.2

Purpose

1.3

Objective

1.4

Delimitations

Frame of Reference

2.1

Introduction to Structural Dynamics

2.1.1

A general framework for dynamic analysis

2.1.2

Frequency Response Functions

2.2

Dynamic Substructuring

2.2.1

Substructure coupling

2.2.2

Lagrange-Multiplier Frequency Based Substructuring

2.3

Transfer Path Analysis

2.3.1

Component-based Transfer Path Analysis

2.3.2

The Equivalent force concept

2.3.3

An application of TPA

2.4

Interface modelling using Virtual Point

Trans-formation

2.4.1

Interface displacement reduction

2.4.2

Interface force reduction

2.4.3

Virtual Point Receptance

2.4.4

Special case: Inclusion of sensor orientation

2.4.5

Quality of Virtual Point Transformation

2.5

Model Updation for Numerical simulation

2.6

Modelling of uncertainty in dynamics

2.6.1

Theory of uncertainty propagation

Chapter 3

Implementaion

3.1

Finite element modelling and updating

3.1.1

Test setup description

3.1.2

FE modelling

3.1.3

Model Update

3.2

Modelling for VPT and FBS

3.2.1

Study on interface rigidity

3.2.2

Effect of inconsistent points on VPT

3.2.3