A Dual Approach to Substructure Decoupling Techniques

Proceedings of the IMAC-XXVIII February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc.

A Dual Approach to Substructure Decoupling Techniques

S.N. Voormeeren and D.J. Rixen Delft University of Technology, Faculty of Mechanical, Maritime and Materials Engineering Department of Precision and Microsystem Engineering, section Engineering Dynamics Mekelweg 2, 2628CD, Delft, The Netherlands [email protected]

ABSTRACT In recent years, the structural dynamic community showed a renewed interest in dynamic substructuring (i.e. component coupling) techniques, especially in an experimental context. In this context the problem of propagation of uncertainty due to measurement errors was also investigated. In this paper the reverse problem is addressed: the decoupling (or identification) of a substructure from an assembled system. This problem arises when substructures cannot be measured separately but only when coupled to neighboring substructures, a situation regularly encountered in practice. Using a dual approach to substructure (dis)assembly, three substructure decoupling techniques will be derived in a unified way. Moreover, their accuracy due to measurement errors will be investigated by performing an uncertainty propagation analysis. The techniques are applied to a simulated experiment.

NOMENCLATURE Y Z u f g λ B L C E ?+ ?S ?s σ(?) ∆?

1 1.1

– – – – – – – – – – – – – – –

Receptance FRF matrix Dynamic stiffness FRF matrix Response vector External force vector Connection force vector Lagrange multiplier General Boolean matrix Boolean localization matrix Compatibility Boolean matrix Equilibrium Boolean matrix Generalized (pseudo) inverse Belonging to subsystem S Belonging to set of DoF s Standard deviation Confidence interval

INTRODUCTION Substructure Decoupling

Dynamic substructuring (DS) techniques have been well established over the past decades. These techniques consist in constructing the structural dynamic model of a large and complex system by assembling the dynamic models of its simpler components (also called subsystems or substructures). In recent years, the structural dynamic community showed a renewed interest in these substructure coupling techniques, especially in the context of experimental applications [6, 17].

One of the issues that has been investigated in this respect was the effect of random measurement errors on the accuracy of the coupled system description [8, 20]. However, sometimes one has to consider the reverse problem, namely how a substructure model can be found from the assembled system. This is a relevant issue for subsystems that cannot be measured separately, but only when coupled to their neighboring substructure(s). This can for example be the case for very delicate subsystems or substructures in operational conditions [12]. To illustrate the problem at hand, consider the subsystems A and B shown in figure 1 (a); when assembled they form system AB. In a dynamic substructuring analysis, the dynamics of assembly AB are obtained by coupling the dynamic models of A and B. In substructure decoupling, the reverse problem is solved. In this case, it is assumed that the dynamic models of the assembled system AB and the substructure A are known (e.g. from measurements). Based on this information, the aim is to find the dynamics of subsystem B as a “stand alone” component, that is, completely decoupled from subsystem A. Practical applications of substructure decoupling can be imagined in structural monitoring and vibration control techniques, where monitoring and controlling of individual (critical) components in an assembly can be very valuable. However, as outlined in [4], quite a number of challenges remain in the practical implementation of decoupling techniques. One important issue is the sensitivity of decoupling techniques to small (measurement) errors, especially around the antiresonances of the known component [18].1 This paper aims at writing the different decoupling techniques in a common framework in order to understand how they differ and how they are related, and if possible deriving more robust decoupling techniques using a so called “dual” (dis)assembly approach. Substructure coupling: ua uc u ua

u cA

ucB

b

ub

= AB

(unknown)

f i ∈ fb

+ A (known)

B (known)

i o

Substructure decoupling:

gc

= B (unknown)

A

– AB

uo ∈ ub

(known)

(a)

B

A (known) (b)

Figure 1: Substructure coupling (dynamic substructuring) and substructure decoupling (a); Finding the uncoupled response of subsystem B (b).

1.2

Problem Description

To illustrate the problem of substructure decoupling more thoroughly, let us consider the situation depicted in figure 1 (b). Suppose one is interested in the uncoupled dynamics of component B. To this end, the dynamics of the assembled system AB and the component A are assumed to be known in the form of frequency response functions or in short FRFs (assuming the systems are linear and time invariant). For the sake of illustration, suppose that we want to obtain the response uo of component B at DoF o, while it is excited by a force fi at DoF i, without the influence of neighboring subsystem A. Both DoF are internal to subsystem B and hence are part of the DoF set ub . The situation is depicted in figure 1 (b). In general, the decoupling problem can now be described as follows: 1 Note that in substructure decoupling the sensitivity is highest around the anti-resonance frequencies of the known subsystem(s), while in substructure coupling (DS) the sensitivities are found to be highest around the resonance frequencies of the subsystems [20].

• The force fi excites the system AB at DoF i internal to component B. • As a result, the assembled system AB shows a response uAB • Now take only the part of the response of AB associated to component B and realize that in addition to the excitation force fi , subsystem B in the assembly AB is also subjected to (connection) forces of component A. • Additional forces opposing these connection forces should thus be applied to the assembly AB in order to let B behave without “feeling” the influence of A. • Using the FRFs of uncoupled system A, one can determine these interface connection forces loads by imposing to the uncoupled model the coupled responses of subsystem A. Summarizing, one can now formulate the decoupling problem as finding the behavior of substructure B as part of the assembled system AB when additional forces are applied at the interface such that substructure B experiences no connection forces from subsystem A. Hence, substructure B behaves as if it were decoupled from A. The decoupling problem can be expressed in terms of equations by starting with the FRF description of the assembled system AB: ¡ ¢ uAB = Y AB f AB − g AB  AB  Ã" " # # " #! AB AB Yaa Yac Yab ua fa 0 (1) AB AB AB   uc fc − g c = Yca Ycc Ycb AB ub fb 0 Yba YbcAB YbbAB and the FRF matrix of subsystem A is assumed to be known too: ¡ ¢ uA = Y A f A + g A · A ¸ · A ¸ µ· A ¸ · ¸¶ A 0 ua Yaa Yac fa = + A gcA uA Yca YccA fcA c

(2)

Here u? is the substructure vector of degrees of freedom (DoF), Y ? the substructure receptance matrix and f ? the external force vector. The vector g ? represents the additional disconnection forces (with non-zero entries only at the interface DoF) felt from the coupling/decoupling of neighboring components. The subscripts a, b and c denote “internal to subsystem A”, “internal to subsystem B” and “coupling”, respectively; the superscripts A, B and AB denote the two subsystems and the assembled system. The explicit frequency dependency is omitted for clarity. To find the decoupled response ub of subsystem B to a force fb , we can start by writing the third equation in eq. (1) as (since fa = 0): ub = YbbAB fb − YbcAB gc . In this equation only the connection forces gc are unknown. To find an expression for these forces, use can be made of the compatibility condition. This condition states that the displacements at the interface DoF that need to be coupled (or decoupled) must be compatible, i.e.: uc = uA c

(3)

An expression for the interface forces can now be found by using the response of the interface DoF from eqs. (1) and (2): AB YccA gcA = uA = YcbAB fb − YccAB gc c = uc

Next, we will express that applying the forces predicted by the local problem on A to the interface of AB must leave the interface DoF of B as if they would be free, or in other words that the connection forces between the substructures are in equilibrium: gc + gcA = 0 Hence, the connection forces can now be expressed as: ¡ ¢−1 AB gc = YccAB − YccA Ycb fb .

(4)

The above expression can be plugged into the expression for ub . The resulting uncoupled frequency response functions Ybb of subsystem B, in terms of the FRFs of A and AB, then become: ³ ¡ ¢−1 AB ´ ub = YbbAB − YbcAB YccAB − YccA Ycb fb = YbbB fb (5) Using this approach, the complete decoupled FRF matrix of subsystem B can be reconstructed. There is however another, more systematic, way of obtaining this decoupled FRF, namely starting from a so called “dual” formulation. This approach will be discussed in the next section. 1.3

A Dual Formulation

In the above discussion, the decoupled receptance matrix of component B was found in a rather “ad hoc” manner, using the receptance matrices of A and AB (eqs. (2) and (1)) employing the interface compatibility and equilibrium conditions. A more systematic approach can be taken when starting from a dynamic stiffness representation of the subsystems by describing system AB as 

AB Zaa AB  Zca AB Zba

AB Zac AB Zcc AB Zbc

Z AB uAB " # ua  uc ub

AB Zab AB Zcb AB Zbb

= =

f AB − g AB " # " # fa 0 fc − gc fb 0

(6)

and subsystem A as ·

A Zaa A Zca

A Zac A Zcc

¸·

Z A uA ¸ uA a uA c

= f A + gA · A ¸ · ¸ 0 fa = + . gcA fcA

(7)

Here u? is the same DoF vector as before, Z ? the dynamic stiffness matrices, f ? the external force vector and g ? again the vector of connection forces (with non-zero entries only at the interface DoF). The compatibility and equilibrium conditions can be expressed as before, i.e. as in eqs. (3) and (4). However, two Boolean matrices can be introduced to write these conditions more compactly and allow a more systematic description of the problem. The first is a signed Boolean matrix B, operating on the substructure interface degrees of freedom. Using the partitioning of DoF as written in eqs. (6) and (7), this matrix can be: £ ¤ 0 −I ] B = B AB B A = [ 0 I 0 The B matrix has a number of rows equal to the total number of connections defined between the substructures, while the number of columns equals the total number of DoF of the substructures to be (dis)assembled. Using this Boolean matrix, the compatibility condition can be conveniently expressed as · ¸ £ ¤ uAB Bu = B AB B A = uc − uA (8) c = 0. uA The second Boolean matrix L localizes the interface DoF of the substructures in the global set of DoF and is similar to the localization matrices used in the assembly of individual elements in finite element models. In this case, the L matrix is   I 0 0 0 · AB ¸  0 0 0 I  L   L= =  0 I 0 0 , LA  0 0 I 0  0 0 0 I so that the equilibrium condition can be stated as:  LT g =

h LAB

T

LA

T

i·

g AB gA

¸

   =  

0 0 0 0 0 gc + gcA

    =0  

(9)

Interestingly, L actually represents the nullspace of B or vice versa (see [7]): ½ L = null (B) ¡ ¢ B T = null LT

(10)

In a primal approach, the interface conditions are satisfied by introducing one unique set of interface DoF. This way, the interface compatibility is a priori satisfied and as a result the interface connection forces are eliminated, thereby satisfying the equilibrium condition as well.2 This approach was implicitly taken in the previous section. In a dual approach however, the equilibrium condition is chosen to be satisfied a priori. This is obtained by choosing the interface forces in the form g = BT λ Here, λ are Lagrange multipliers, corresponding physically to the interface force intensities. By choosing the interface forces in this form, they act in opposite directions for any pair of dual interface degrees of freedom, due to the construction of Boolean matrix B. The equilibrium condition in eq. (9) thus becomes LT g = LT B T λ = 0. Because LT is in the nullspace of B T , see eq. (10), this condition is always satisfied. The compatibility condition as shown in eq. (8) should however be taken into account explicitly as an additional equation. The decoupling problem can therefore be formulated in a dual way as:  T  Z AB uAB + B AB λ = f AB   T Z A uA − B A λ = f A    AB AB B u + B A uA = 0 One can transform this to matrix vector notation as:  AB     T Z 0 B AB uAB f AB T  0 Z A −B A   uA  =  f A  λ 0 B AB B A 0 However, this system of equations is non-symmetric. By multiplying the second equation by minus 1, the system becomes symmetric:  AB    T  Z 0 B AB uAB f AB T  0 (11) −Z A B A   uA  =  −f A  AB A λ 0 B B 0 This last relation clearly shows that the decoupling of a subsystem from a total system is equivalent to a dual assembly of a negative dynamic stiffness for the substructure that one wants to subtract (here substructure B). The actual uncoupled FRFs of B can now be found by eliminating the Lagrange multipliers. At first, start by writing explicitly the substructure DoF as: ³ ´ −1 T uAB = Z AB f AB − B AB λ (12) uA = −Z A

−1

³

´ T −f A − B A λ

Substitution in the compatibility condition and solving for λ gives: ³ ´−1 −1 T −1 T −1 λ = B AB Z AB B AB − B A Z A B A B AB Z AB f AB Here it is assumed that f A = 0; since this component is “subtracted” there is no excitation at its DoF. Substitution in the expression for uAB (eq. (12)) gives the decoupled responses: µ ¶ ³ ´−1 AB AB −1 AB −1 AB T AB AB −1 AB T A A−1 AT AB AB −1 u = Z −Z B B Z B −B Z B B Z f AB 2 More

details on primal and dual substructure assembly can be found in [7].

It should now be realized that the dynamic stiffness matrices are the inverse of the receptance matrices, i.e Z AB

−1

= Y AB

and

ZA

−1

= Y A.

Hence one can write: µ ¶ ³ ´−1 T T T uAB = Y AB − Y AB B AB B AB Y AB B AB − B A Y A B A B AB Y AB f AB

(13)

A clear physical interpretation can be given to the above expression. To this end, let us express the above equation as: T

uAB = Y AB f AB − Y AB B AB Zint uint where

³ ´−1 T Zint = B AB Y AB B AB + B A Y A B A uint = B AB Y AB f AB

This form of the decoupling problem can now be interpreted as follows: • The term Y AB f AB represents the response of the assembled system AB to an external excitation f AB ; • This also leads to interface displacements uint ; • However, these interface displacements are due to the combined stiffness of A and B. Therefore, a corrected interface stiffness Zint must be calculated to eliminate the influence of substructure A; • The adjusted interface stiffness times the interface displacements, Zint uint leads to a correction force at the interface DoF; T

• This force correction is spread to the other subsystem DoF by multiplication by Y AB B AB . Next, we can insert the expressions for the receptance matrices of systems A and AB in (13) and calculate the products with the Boolean matrices. This gives:   AB  " " #  AB # AB AB Yaa Yac Yab Yac ua fa ¡ ¢ £ ¤ −1 AB A AB AB AB AB AB AB AB  fc uc =  Yca (14) Ycc Ycb  −  Ycc  Ycc − Ycc Yca Ycc Ycb AB ub fb Yba YbcAB YbbAB YbcAB If one now tries to obtain the same transfer function as before, i.e. the response ub to an excitation force fb , one can start by noting that again fa = 0 but also fc = 0 since the interface connection forces are present in the Lagrange multipliers and no external excitation is assumed at the interface DoF. Insertion in the previous equation and extracting the third row then gives the expression for the uncoupled FRFs Ybb of B: ³ ¡ ¢−1 AB ´ ub = YbbAB − YbcAB YccA − YccAB Ycb fb (15) Evidently, the expression for YbbB is exactly equal to the one found earlier in eq. (5). So, starting from a dual formulation in terms of dynamic stiffness FRFs, the same decoupled (receptance) FRFs can be obtained as starting from a receptance representation of the systems. Finally, note that the responses ua in eq. (14) correspond to the responses of the internal DoF of A assembled system AB when additional interface forces are applied and are generally not of interest. 1.4

Paper Outline

The idea outlined above will be further expanded in the next section. From the dual formulation, three methods for substructure decoupling will be derived. Section 3 thereafter presents an uncertainty analysis of these techniques in order to investigate the sensitivity of the methods to small (measurement) errors in the FRFs. The results of a case study are presented in section 4, while the paper is ended by some conclusions and recommendations in section 5. Note that all the discussions in this paper consider the case of decoupling two substructures for the sake of illustration. Nevertheless it is straightforward to generalize the approach for any number of substructures that need to be coupled and/or decoupled.

fi ∈ fb

λ

uo ∈ ub

i o Z AB

f i ∈ fb

λ ZA

uo ∈ ub

i o ZB

Figure 2: Substructure decoupling from a dual perspective.

2

A DUAL FRAMEWORK FOR SUBSTRUCTURE DECOUPLING

Based on the dual formulation of the previous section, one can now go one step further in by realizing that the total system AB itself can be written as a dual assembly of systems A and B, so:      T ZA 0 BA uAB fAAB A   T  AB   AB  Z AB =  Z B BB   0   uB  =  fB  0 BA BB 0 λAB This can be inserted in eq. (11) and writing (from now on it will be assumed that f A = 0):    AB   T T  uA ZA 0 BA 0 CA fAAB T  0    Z B BB 0 0    uAB   fBAB  B   BA BB   AB    0 0 0   λ = 0        0 ˜ A −C AT   uA   0  0 0 −Z 0 λ CA 0 0 −C A 0

(16)

Here the Boolean matrix B AB has been replaced by [C A 0 0] (zeros have been added at the Lagrange multipliers λAB ) and B A = −C A , in order to emphasize that the Boolean matrix used for the decoupling of substructures is not necessarily the same as the one used (implicitly) for coupling. Note that the dynamic stiffness matrix of the separate ˜ A , while the dynamic stiffness of A in the total system AB is denoted by Z A . Taking component A now is denoted by Z the above expression as a starting point, three approaches to the decoupling problem can be formulated. These will be discussed next. 2.1

Standard Decoupling

The easiest way of solving the decoupling problem of eq. (16) is by choosing C A = B A . This means that compatibility and equilibrium are only required at the interface DoF between substructure A and assembled system AB. Basically, in this approach the connection forces between the substructures (that are used to eliminate the influence of A in the response of AB) are determined using the minimum information needed, namely only the responses on the interface DoF of subsystem A. We will refer to this approach as “standard decoupling”. The Boolean matrix can in this case be expressed as: C A = [0ca

Icc ]

The subscripts c and a respectively refer to the number of interface and internal DoF of A. Consequently, the Boolean matrix acting on AB is found as: C AB = [0ca

Icc

0cb ]

The decoupled FRFs of B can be found by inserting this choice for the Boolean matrix in eq. (16). For the sake of simplicity, the assembled system AB is expressed simply as Z AB . This leads to the description of the decoupling problem in the form of eq. (11), i.e.:  AB    T  Z 0 C AB uAB f AB T  0 (17) −Z A C A   uA  =  0  AB 0 λ C CA 0

By eliminating the Lagrange multipliers λ and solving for the degrees of freedom uB , the decoupled FRFs of B are found. Not surprisingly, since the same Boolean matrix is used, the result will be exactly the same as found in the previous section in eq. (14). In practice however, when dealing with measured FRF matrices, measurement errors are inevitable. As a result the separately measured FRFs of component A are slightly different from the ones (implicitly) measured when system A is ˜ A ≈ Z A . Exactly these small discrepancies between the measured part of AB. Hence, eq. (16) is slightly modified since Z components can cause the standard decoupling method to fail on a practical problem, as they amplify to large errors on the decoupled FRFs [2].

2.2

Decoupling with Additional Internal DoF

Compared to the standard approach to decoupling as described above, a somewhat more clever approach can be taken. The idea is very simple: in addition to using the information at the interface DoF to determine the connection forces, one might also use the knowledge of the internal DoF of subsystem A. As their responses are also (partly) due to the connection forces, this can be valuable additional information for determining those forces. This is believed to reduce the ˜ A ≈ Z A and therefore enhances the estimation of the decoupled FRFs. This approach problems described above when Z has been termed “extended interface” decoupling in [5]. In this approach (some of) the internal DoF are also taken into account in the compatibility and equilibrium conditions. In the limit case where all internal DoF of subsystem A are used, the Boolean matrix for component A becomes identity: · ¸ Iaa 0 A C = 0 Icc Consequently, the Boolean matrix acting on AB is found as: · ¸ Iaa 0 0 C AB = 0 Icc 0 The decoupling problem is now found by inserting these Boolean matrices in eq. (17). We can proceed by eliminating the Lagrange multipliers and solving for the uncoupled responses, as in section 1.3. This gives the following result: "

ua uc ub

#

 = Y AB



AB Yaa AB  − Yca AB Yba

 AB µ· AB Yac Yaa AB  Ycc AB Yca YbcAB

AB Yac YccAB

¸

· −

A Yaa A Yca

A Yac YccA

¸¶−1 ·

AB Yaa AB Yca

AB Yac YccAB

AB Yab YcbAB

 ¸ " fa #  fc fb (18)

Indeed, the complete FRF matrix Y A is now used to find the uncoupled responses of B. Solving the decoupling problem with additional internal DoF as outlined above also raises the question what internal DoF should be taken into account. An answer to this question is not very easy to find. Of course, when performing decoupling using measured data one can use some indicators (e.g. coherence functions) to filter out badly measured FRFs, but currently no general criteria exist. ˜ A ≈ Z A . In case Z ˜ A = Z A , the Furthermore, it should be noted that the above approach only works when indeed Z problem becomes singular at all frequencies, as was already observed in [5]. Physically, this is due to the fact that additional Lagrange multipliers are introduced to satisfy equilibrium at the internal DoF. However, due to the compatibility condition being enforced exactly at all DoF, these Lagrange multipliers become redundant when the FRFs of substructure A are measured as being the same in AB and in A alone. In [5] it is proposed to overcome the singularity by applying truncated SVD techniques to perform the inversion. 2.3

Non-Collocated Compatibility and Equilibrium Conditions

The decoupling methodology can be further generalized by realizing that a certain freedom exists in the choice of DoF on which to enforce compatibility and equilibrium. In other words, it is not required to enforce compatibility and equilibrium on

the same (number of) DoF. Physically this translates to applying forces at some set of DoF in order to satisfy equilibrium at another set of DoF. As long as the controllability properties of these DoF sets are correct, this is perfectly possible. This idea can be translated into equations by taking different Boolean matrices for the compatibility and equilibrium conditions, as:  AB    T  Z 0 E AB uAB f AB T  0 (19) −Z A E A   uA  =  0  AB A 0 λ C C 0 Here E ? are the Boolean matrices governing the equilibrium while C ? are the matrices enforcing compatibility. Solving the above equation for the uncoupled responses as before then gives µ ¶ ³ ´+ T T T uAB = Y AB − Y AB E AB C AB Y AB E AB − C A Y A E A C AB Y AB f AB , (20) ³ ´ T T where + denotes the (Moore-Penrose) pseudo-inverse, since the term C AB Y AB E AB − C A Y A E A is now no longer necessarily a square matrix. The trick is now to make a smart choice for both interface conditions such that the influence of possible errors in measured FRFs of A and AB are minimized. Such a minimization may be obtained by choosing the Boolean matrices for the enforcing the compatibility as · ¸ · ¸ Iaa 0 Iaa 0 0 CA = and C AB = , 0 Icc 0 Icc 0 while for the Booleans that govern the equilibrium condition we choose: E A = [ 0ca

Icc ]

and

E AB = [ 0ca

Icc

0cb ] .

This choice for the Boolean matrices corresponds to enforcing compatibility at all DoF of A (interface and internal) while the equilibrium condition is satisfied by introducing Lagrange multipliers only at the interface DoF. Hence, we have a + c compatibility conditions and only c Lagrange multipliers to solve. As before we can insert these Boolean matrices in eq. (19), eliminate the Lagrange multipliers and solve for the uncoupled responses. This gives:   AB   " #  AB AB AB µ· AB ¸ · A ¸¶+ · AB ¸ " fa # Yaa Yac Yab Yac ua AB AB Y Y Y Y Y AB ac ac aa ac ab  fc uc =  Yca − YccAB YcbAB  −  YccAB  AB YccAB YccA Yca YccAB YcbAB AB AB AB AB ub fb Yba Ybc Ybb Ybc The result is indeed decoupling from an overdetermined set of equations, sometimes called the “mobility approach” in literature [4, 18, 5]. It is easy to see that in this method the decoupling is performed by solving for the interface forces in a least squares sense. Using the non-collocated compatibility and equilibrium conditions, many variations can be imagined. Note however that in order to end up with a solvable set of equations, it should always hold that rank (C ? ) ≥ rank (E ? ) ≥ c, where c is the number of interface (coupling ) DoF. Another interesting possibility is to choose for the compatibility Boolean matrices CA = [ 0

Icc ]

C AB = [ 0

,

Icc

0 ]

and for the equilibrium matrices E A = [ Ica

0cc ]

,

E AB = [ Ica

0cc

0cb ]

This corresponds to applying forces at the internal DoF of A such that the interface compatibility condition can still be satisfied. This requires that the interface DoF of A are controllable from inputs at its internal DoF. Using the above choice for the Boolean matrices, one obtains the following expression for the uncoupled responses:   AB  " " #  AB # AB AB Yaa Yac Yab Yaa ua fa ¡ ¢ £ ¤ −1 AB A AB AB  AB uc =  Yca Yca − Yca YccAB YcbAB  −  Yca Yca YccAB YcbAB  fc AB AB AB AB ub fb Yba Ybc Ybb Yba In this way, the inversion of (possibly sensitive) driving point FRFs on the interface can be avoided. Moreover, it should be noted that knowledge of the driving point FRFs on the interface is not required in case fc = 0 and one is not interested in uc (i.e. only in the internal FRFs of B). This can be beneficial in practical applications, where driving point FRFs of the interface DoF may be hard to obtain.

3

UNCERTAINTY ANALYSIS

Although from the previous sections it is evident that the theory of substructure decoupling is rather straightforward, the practical application of decoupling techniques remains troublesome. This is a problem experienced as well in the field of substructure coupling techniques (dynamic substructuring) [6]. In particular, decoupled substructure models seem to be very sensitive to errors in the (experimental) description of the other subsystem(s) and assembled system. From a general point of view, two kinds of errors can be made in (sub)system measurements and modeling. The first type, called bias errors, are systematic errors which lead to measured values being systematically offset. Careful design of an experiment and/or model creation allows one to reduce the influence of those errors. The second kind of error, which is addressed here, is more random of nature. Random errors are fluctuations that can be evaluated through statistical or interval analysis. When performing dynamic measurements, all sorts of random errors (“measurement noise”) can be encountered, such as round-off errors in A/D conversion, sensor noise, varying impact locations from a hammer test, etc. From a broader perspective, one can also count uncertain parameter values (e.g. geometric dimensions or material properties) as a random error, which is especially important when coupling and decoupling a mix of numerical models and measurements. In contrast to systematic errors, random errors are generally due to factors that cannot be controlled. Therefore they introduce uncertainty, following the definition of uncertainty of Hazelrigg [10]: “In an experiment, when the sample space contains more than one element with non-zero probability, there is uncertainty.” An important question is how these uncertainties in the assembled and subsystem descriptions propagate, and possibly amplify, in the substructure decoupling process. In dynamic substructuring, common belief is that small errors in a subsystem interface description can lead to significant discrepancies in the coupled system’s representation, due to the numerical conditioning of the subsystems’ interface flexibility matrix [1, 13, 3, 16]. Since this matrix needs to be inverted in the coupling process, ill conditioning could severely magnify the small errors in subsystems. Similar problems for decoupling techniques are described in [18, 4, 5]. In this paper an uncertainty propagation analysis will be performed to quantify this effect for the decoupling techniques, allowing a comparison of the sensitivity of the three methods to small errors. Such an uncertainty analysis can be a valuable tool in the light of model verification and validation. The remainder of this section is organized as follows. First, the general theory of uncertainty propagation is outlined in section 3.1, thereafter these methods will be applied to the decoupling techniques in 3.2.

3.1

Theory of Uncertainty Propagation – Moment Methods

The study of uncertainty propagation comprises the determination of a function’s uncertainty based on the uncertainties of its input variables. Different approaches exist to investigate the uncertainty propagation from a number of inputs to an output [11]. In the uncertainty propagation analysis in this paper the so called ‘moment’ method will be used, an efficient, sensitivity based method for uncertainty analysis. The approach taken here is analog to [20, 8]. For the sake of compactness, the theory will be summarized in this section, details can be found in the aforementioned papers. In statistics, the ‘moments’ are properties that characterize a variable’s probability distribution. The most common statistical moments are the following four ‘central’ moments (taken about the mean): 1. The first moment corresponds to the mean of the distribution. 2. The second moment represents the variance. 3. The third moment is the skewness, expressing the symmetry of a distribution. 4. The fourth moment is the kurtosis, describing the distribution’s ‘peakiness’. In many cases however, not all four (central) moments are required to characterize a probability distribution. A normal or Gaussian probability distribution for example is completely defined by its first two moments. A Gaussian distribution is often a good approximation of the random external influences on a measurement, and will therefore be used in the uncertainty propagation method derived here. Indeed, in [19] it was found that the random influences on a real-life structural dynamic measurement closely obey this distribution.

T

Assume now a set of n stochastic input variables xi assembled in vector x , [x1 · · · xn ] , which have known mean values x ¯i and standard deviations σ(xi ). Let g be a function of the variables in x. The moments of the function g(x) can then be calculated from a truncated Taylor series expansion about the mean value of the input variables. Hence, the approach taken here is a special type of sensitivity analysis. In this paper it is assumed that the input uncertainties are small and Gaussian distributed, and that all functions considered are continuous and can be linearized around the mean value ¯ In that case a first order Taylor series expansion suffices to obtain approximations for the first of the input variables x. and second moments. These approximations are usually called first order, first moment (FOFM) and first order, second moment (FOSM) approximations, respectively. Approximating the function g(x) by a first order Taylor series around the mean values of the input variables gives the first moment of the function g(x) as: ¯ E[g(x)] = g(x).

(21)

Using some statistics [9] and assuming that the variables are uncorrelated, one finds the variance (the second moment) of the function as: ¶2 n µ X ∂g Var[g(x)] = Var[xi ]. (22) ∂xi i=1 The zero correlation assumption might be somewhat crude, as it is not expected that random errors on measured structural dynamic signals (e.g. forces and accelerations) will be fully uncorrelated. However, this assumption is still made as it considerably simplifies the subsequent analysis. Note that this assumption can be regarded valid as long as the noise on the measured mechanical signals is dominated by uncorrelated random influences (e.g. sensor resolution). When the noise on the signals is ‘mechanical’ of nature (e.g. vibrations from the environment, fluctuations in applied excitation, etc.), the errors on the measured signals will be highly correlated due to the physical structure. Hence, the excitation should have a deterministic character, the resulting practical implications have been addressed in [8]. In practice it is convenient to express the second moment in terms of the standard deviation, which has the same unit as the function itself. Since the standard deviation is the positive square root of the variance and the standard deviation in the input variables was defined as σxi , the standard deviation of the function g(x) is found to be v u n µ ¶2 uX ∂g σ(g(x)) = t σ(xi ) . (23) ∂xi i=1 This last equation forms the starting point for the uncertainty propagation derivation for the decoupling techniques, although it first needs to be generalized in case the function g is a vector/matrix function. The derivative of a matrix with respect to any of its entries may be written as ∂G , Pij , ∂Gij

(24)

where matrix Pij is a ‘Boolean’ type of matrix with the same size as matrix G and the elements of Pij are all zero except for entry (i, j), which equals one. With this definition one can now express equation (23) in matrix form as v  v  u ¾2  u n X m ½ n X m uX uX  ∂G u u 2 σ(G) = t σ(Gij ) =t (25) {Pij σ(Gij )} ,     ∂Gij i=1 j=1 i=1 j=1 where G has dimension n-by-m. Here the curly-bracket notation {· · ·}, indicates that the square and square root operations must be performed elementwise. Note that expression (25) thus simply states that the standard deviation of a matrix is the sum of the standard deviations of its entries. Another helpful result needed for the upcoming uncertainty propagation analysis is the derivative of the inverse matrix G−1 to its elements Gij [14]: ∂G−1 ∂G −1 = −G−1 G = −G−1 Pij G−1 . ∂Gij ∂Gij

(26)

With some matrix algebra and the definition of the Moore-Penrose pseudoinverse it can be easily shown that it also holds that: ∂G+ ∂G + = −G+ G = −G+ Pij G+ . ∂Gij ∂Gij

(27)

3.2

Uncertainty Analysis of Decoupling Techniques

Now the theory of uncertainty propagation using moment methods has been outlined, the decoupling methods discussed in section 2 can be analyzed. The assembled and subsystem FRFs serve as input for the decoupling techniques, it is therefore assumed that the uncertainty on these FRFs is known. How to obtain the uncertainty on individual FRFs from measured excitations and responses is described in [20]. Furthermore, the uncertainty on the FRFs is assumed to be specified in terms of confidence intervals instead of in terms of standard deviations. These intervals are denoted by ∆? and are specified here at the 95% level. Describing uncertainty using confidence intervals allows to express the uncertainty on the mean FRFs and includes the influence of repeated measurements [20].3 Note that the theory from the previous section is equally valid for confidence intervals. Basically, the uncertainty analysis is to a large extent similar as the one performed in [20], since the set of equations that form the starting point are largely similar. Here, we will perform the uncertainty analysis on the most general form of the dual decoupling problem, i.e. the “non-collocated” approach in eq. (20). From this equation, the decoupled FRFs of B can be expressed as: ³ ´+ T T T Y dec = Y AB − Y AB E AB C AB Y AB E AB − C A Y A E A C AB Y AB (28) Since the “standard” and “extended interface” methodologies only differ from the “non-collocated” in the choice of the Boolean matrices, the uncertainty analysis is also valid for these approaches. In order to calculate the confidence interval of the decoupling equation, one should first realize that the decoupled FRFs are a function of the FRFs in Y AB and Y A : ¡ ¢ Y dec = Y dec Y AB , Y A Therefore, the derivative of Y dec to the FRFs in these matrices is needed. To find the derivative, it should be noted that there are no ‘duplicate’ FRFs in the matrices Y AB and Y A . This means that, for example, when deriving Y dec to an FRF in Y A , the first term in eq. (28) will be zero. The derivative is now found as: AB A AB ∂Y dec ∂Y AB ∂Y AB AB T AB ∂Y AB T A ∂Y AT AB ∂Y = − E D + D C E D − D C E D − D C , (29) 2 1 2 1 2 1 ∂Yij ∂YijAB ∂YijAB ∂YijAB ∂YijA ∂YijAB

where use was made of the product rule on the second term in equation (28) and D1 and D2 are defined as ³ ´+ T T T ∆ D1 = Y AB E AB C AB Y AB E AB − C A Y A E A ³ ´+ T T ∆ D2 = C AB Y AB E AB − C A Y A E A C AB Y AB

(30)

By combining the above and equation (25), one finds the confidence interval on the decoupled FRFs of subsystem B

∆Y¯

dec

=

v( u u P P n dec ∂Y t j

¯ AB AB ∆Yij ∂Yij

Pij −

Pij E AB T D2

i

=

v( u u P P ©¡ t i

j

o2

+

P P n ∂Y dec k

+

l

A ∂Ykl

∆Y¯klA

o2

D1 C AB Pij E AB T D2

)

−

D1 C AB Pij

¢

ª2 ∆Y¯ijAB

+

P P ©¡ k

l

D1 C A Pkl E AT D2

¢

ª2 ∆Y¯klA

Note again that the uncertainties on the receptance matrices of AB and A (∆Y¯ijAB and ∆Y¯klA ) are assumed to be known. Furthermore, note that the square and square root operations must be performed elementwise as indicated by the curly-bracket notation. 3 This can be interpreted as follows: after 100 measurements the average FRF magnitude at a certain frequency is found, for example, as Y = 10.0, while √ the standard deviation is 0.2.√One can therefore say with 95% confidence that the true mean FRF magnitude is between 10 − 1.96 · 0.2 100 = 9.61 and 10 + 1.96 · 0.2 100 = 10.39. The factor 1.96 originates from the fact that the confidence intervals is stated at the 95% level, which corresponds to 1.96 times the standard deviation. Now suppose one would take another 300 measurements, giving a total of 400 measurements. Even though the average and standard deviation might √ not have changed, the 95%√confidence interval has now narrowed so that the true mean magnitude is somewhere between 10 − 1.96 · 0.2 400 = 9.80 and 10 + 1.96 · 0.2 400 = 10.20.

)

4

CASE STUDY

In this section the results of a case study are presented. The simple problem used for this study is shown in figure 3 and consists of two lightly damped mass-spring-damper systems (modal damping < 1%). Subsystem A has 7 degrees of freedom, subsystem B possesses 4 DoF and the systems are coupled at a 2 DoF interface.

Subsystem A uA

kA

uA mA

mA

kA

uA

uA

uA

kA

kA

mA

uB

uA

kA

mA kA

Subsystem B

Interface mB

mA uA

kA

mA

uB

kB

uB

uB

mB

kB

mB

kB

mB

mA

Figure 3: Simple system used for case study.

The remainder of this section is organized as follows. The next section presents the application of the decoupling techniques to the simple decoupling problem of figure 3. Thereafter, section 4.2 shows some results from the uncertainty analysis.

4.1

Calculation of Decoupled FRFs

˜A ≈ ZA To resemble a practical situation, the FRFs of A in AB and the FRFs of A separately are not exactly equal, i.e. Z as in eq. (16). The disturbance on the FRFs of A was a normally distributed random error on the amplitude with a 95% confidence interval of ±5% and a normally distributed random error on the phase with a 95% confidence interval of ±2o . This was done to resemble real-life measurement situations, where external disturbances result in both amplitude and phase uncertainty. Note that a different random error was taken for each frequency; in total 200 frequency points were evaluated. The results of the decoupling using the three methods are shown in figure 4 for an arbitrary FRF of B; Decoupled FRF YB1B4 - Magnitude 30

True Standard Extended Non-collocated

20 10 0

3 2 Phase [rad]

Magnitude [dB]

Decoupled FRF YB1B4 - Phase 4

-10 -20

1 0 -1

-30

-2

-40

-3

-50

0

10

20 30 Frequency [Hz]

40

50

-4

0

10

20 30 Frequency [Hz]

40

50

Figure 4: Decoupled FRF of subsystem B obtained with three decoupling techniques vs. true FRF of B.

the “true” FRF is also shown. The following compatibility and equilibrium conditions were chosen: • “Standard” method: compatibility and equilibrium only at the two interface DoF.

• “Extended interface” method: compatibility and equilibrium at all DoF of A, so at seven DoF. • “Non-collocated” method: compatibility at all (seven) DoF of A, equilibrium only at the (two) interface DoF. As one can see, the three decoupling methods seem to perform identical for this system. All three methods show some spurious peak around 6 Hz, but seem to resemble the true FRF quite nicely at the other frequencies. Note however that due to the dB-scale the results may seem better then they are and using the FRFs for subsequent manipulations (e.g. for a modal identification or a substructuring analysis) might give problems. To zoom in on the actual error, an uncertainty analysis has been performed. This will be discussed next.

4.2

Uncertainty Analysis

˜ A ≈ Z A , with the same error on A as mentioned above, in order to avoid The uncertainty analysis is performed with Z conditioning problems with the “extended interface” method. Additionally, confidence intervals have been assumed on the FRFs of AB and A of the same size as before, i.e. a 95% confidence interval of ±5% on the amplitude and ±2o on the phase. With this data the uncertainty analysis as outlined in the previous section has been performed. The result is shown in figure 5. The same FRF of B is considered as before and the relative confidence intervals on the magnitude (the confidence interval on the magnitude divided by the magnitude of the FRF) and the confidence intervals on the phase are plotted for the three methods. As can be seen, the “standard” and “extended interface” methods perform identical, they Error on decoupled FRF YB1B4 - Magnitude Standard Extended Non-collocated

Relative error | ∆Y / Y| [%]

1200 1000 800 600 400 200 0

0

10

20 30 Frequency [Hz]

40

50

Error on decoupled FRF YB1B4 - Phase 3.5 Phase error ∠(Y + ∆Y) - ∠ Y [rad]

1400

3 2.5 2 1.5 1 0.5 0

0

10

20 30 Frequency [Hz]

40

50

Figure 5: Comparison of the uncertainty of the three decoupling techniques on the magnitude (relative) and phase of an decoupled FRF.

produce exactly the same uncertainty on the decoupled FRF. An unambiguous reason for this finding cannot yet be given. Furthermore, this is in contrast to the findings in [5], where a clear difference was reported between the performance of the standard method and the extended interface method, in favor of the latter. This is believed to be due to the fact that in this paper a truncated SVD inversion is used instead of a normal inverse. This implicitly leads to an approach similar as for the non-collocated method, since some (the smallest) singular values are thrown away which physically corresponds to forcing some Lagrange multipliers to zero and thus implicitly gives an overdetermined system. Note that the standard and extended interface methods yield relative confidence intervals on the amplitude of decoupled FRFs of more then 1000% at some frequencies. This is an error amplification of more than a factor 200. More or less the same holds for the phase angle, where the input error of ±2o is amplified to phase errors of 2.5 radians or approximately ±150o . Note that in the region between 20 and 30 Hz the phase error is wrongly shown due to an “unwrapping” error. The “non-collocated” method on the other hand seems to generally give much smaller confidence intervals on the decoupled FRF. At some frequencies, the method performs up to ten times better (i.e. it produces relative errors that are ten times smaller) then the other two methods. Still it gives pretty large confidence intervals at some frequencies. The effect of these large confidence intervals is shown in figure 6. Here the average decoupled FRF, obtained without noise,

is shown with the confidence interval shown as a grey band. The likelihood that a decoupled FRF is in this band is 95%. One can see that for the standard and extended interface decoupling techniques at certain frequencies the intervals become so large, that “spurious peaks” can appear [15]. This can lead to problems when, for example, a modal identification is performed on the decoupled subsystem. The non-collocated decoupling technique suffers from this problem to a lesser extent.

30

20

20

20

10

10

10

0 -10 -20 -30

0 -10 -20 -30

Average FRF

-40 -50

Magnitude [dB]

30

Magnitude [dB]

Magnitude [dB]

Decoupled FRF YB1B4 with Confidence Interval 30

0

20 40 Frequency [Hz]

-10 -20 -30

Average FRF

-40

CI - 'Standard'

0

Average FRF

-40

CI - 'Non-collacted'

CI - 'Extended Int.'

60

-50

0

20 40 Frequency [Hz]

60

-50

0

20 40 Frequency [Hz]

60

Figure 6: Uncoupled subsystem FRF with its confidence interval (CI) shown as a band of possible outcomes.

5

CONCLUSIONS & RECOMMENDATIONS

In this paper a dual framework was presented for substructure decoupling techniques. From this framework, three different types of decoupling methods were derived using different Boolean matrices for the compatibility and equilibrium conditions imposed on the structures that need to be decoupled. A “standard” decoupling method was found when equilibrium and compatibility were only at the interface DoF; an “extended interface” method was found when both conditions were also enforced at internal DoF of the known subsystem. A third method was proposed in which equilibrium and compatibility were chosen at non-collocated DoF, such that an the decoupled FRFs are determined in a least squares sense. Furthermore, this method allows to compute the internal responses of the decoupled subsystem without knowledge of the driving point FRFs on the interface. The three methods were subjected to a uncertainty analysis based on a sensitivity method for finding statistical moments in order to assess their accuracy when small, random errors are present in the dynamic systems’ descriptions. Subsequently, both approaches were applied to a numerical decoupling problem. This case study showed that the standard and extended interface decoupling methods perform identical in terms of uncertainty propagation. Both methods can greatly amplify input errors. The non-collocated method generally performs much better (up to a factor 10), this method thus seems preferable over the other two methods. Given the possible improvements in accuracy of the decoupled FRFs, it might therefore be worthwhile measuring additional internal DoF in a practical decoupling problem. However, in order to make decoupling techniques truly robust, further research has to be performed. For example, research is needed in order to establish criteria for selecting the additional (internal) DoF that need to be taken into account. One could for instance think of choosing optimal compatibility and equilibrium conditions at each frequency. Further research is also required to investigate the influence of errors on the phase and amplitude of the input FRFs, to tell the effect of both types of errors on the decoupled FRF.

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