Investigation on the Use of Various Decoupling Approaches

David Cloutier, Dr. Peter Avitabile Structural Dynamics and Acoustic Systems Laboratory

University of Massachusetts Lowell One University Avenue

Lowell, Massachusetts 01854

ABSTRACT

Substructuring methods allow for the development of system models from component information. Often times however, system response needs to be improved through the modification of one or more component representations. Decoupling the component is necessary in order to accomplish any additional design improvements. Decoupling can be performed different ways. Several approaches are considered for the evaluation of component decoupling from the system. Impedance and Mobility techniques are compared to an alternate force decoupling approach. Several models are studied to better understand the strengths and weaknesses of each of the techniques often employed. Several cases are studied and shown in the paper.

INTRODUCTION Frequency Based Substructuring has been used as a valuable tool for many years. This modeling approach using frequency response functions [1- 4] has received much attention in the development of system models. Many researchers have provided alternate approaches for the development of system models. In recent years however, many [5-13] have directed their efforts towards the disassembly or decoupling of system models to identify component characteristics. There are many reasons to obtain component information from a system representation. Often, an individual component may need to be redesigned to meet specific requirements to satisfy system response characteristics. The system response characteristics of the assembled system identify the overall characteristics but do not necessarily directly provide a clear identification of the individual component contributions to the total system response. Being able to identify specific component characteristics that provide the proper system level response is of critical importance when trying to redesign or retrofit components into a system model. Clearly, this information can provide extremely beneficial information if available. For this reason, many have attempted to identify component information from system response characteristics. When system response characteristics along with one of the component response characteristics are known, the remaining component information can be extracted. There are several approaches to finding the unknown component information. Impedance and Mobility modeling approaches have been employed by several researchers [5- 8]. While these approaches lead to the same equation when only response functions at the connections are used, the use of internal DOF in the formulation results in slightly different expressions. Both of these have provided possible mechanisms for obtaining the component characteristics for the unknown component which is part of a system model. An alternate approach was considered in which the constraint forces at the connections are used to estimate the unknown component. In this paper, the Impedance and Mobility approaches are reviewed along with an alternate force constraint approach. These are all compared to each other and strengths and drawbacks are discussed.

Proceedings of the IMAC-XXVIIIFebruary 1–4, 2010, Jacksonville, Florida USA

©2010 Society for Experimental Mechanics Inc.

THEORY

In general, two components A and B can be used to form a system representation AB. The two components are coupled at connection point (c), that results in a coupling force (fAB ) to tie the components together. An external force (fe) may be applied to either of the components. With this definition, the system is characteristically shown in Figure 1; also note that an equivalent representation of the component may be shown with only one component provided that the system is maintained in dynamic equilibrium using the connection force. Additionally, there may be interior points (i) on either component A or B. For the work presented in this paper, the general system coupling equations are presented first, followed by the three separate decoupling approaches investigated.

Figure 1. General System Description and Nomenclature.

General System Modeling Equations

Previous decoupling techniques include Impedance and Mobility [5] in which the frequency response functions (FRFs) of the connection DOF on the unknown component B are calculated from the complete known system AB and a known component A. By partitioning the FRF matrices into the connection DOF (c) and internal DOF (i), the known system can be written as,

{ }{ }

[ ] [ ][ ] [ ]

{ }{ }

AB ABAB AB

cc cic cAB ABAB AB

ic iii i

H Hu f

H Hu f

⎡ ⎤⎧ ⎫ ⎧ ⎫⎪ ⎪ ⎪ ⎪⎢ ⎥=⎨ ⎬ ⎨ ⎬⎢ ⎥⎪ ⎪ ⎪ ⎪⎩ ⎭ ⎩ ⎭⎣ ⎦

(1)

and for the known component A,

{ }{ }

[ ] [ ][ ] [ ]

{ }{ }

A AA A

cc cic cA AA A

ic iii i

H Hu f

H Hu f

⎡ ⎤⎧ ⎫ ⎧ ⎫⎪ ⎪ ⎪ ⎪⎢ ⎥=⎨ ⎬ ⎨ ⎬⎢ ⎥⎪ ⎪ ⎪ ⎪⎩ ⎭ ⎩ ⎭⎣ ⎦

(2)

The coupling conditions at connection DOF are:

{ } { } { }AB A B

c c cu u u= = (3)

{ } { } { }AB A B

c c cf f f= + (4)

and at internal DOFs on component B:

{ } { }AB B

i iu u= (5)

{ } { }AB B

i if f= (6)

Impedance Approach

For the Impedance based approach, subtracting the inverse of (2) by the inverse of (1) and introducing the conditions (3), (4), (5) and (6), with some algebraic manipulation one can obtain,

[ ][ ][ ]

[ ][ ]

ABB cccccc AB

cc ic

HNH

N H

+ ⎡ ⎤⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦ ⎣ ⎦

(7)

with,

[ ] [ ] [ ] [ ] [ ] [ ]1 1AB A AB A

cc cc cc cc ci icN I H H H H

− −

= − − (8)

[ ] [ ] [ ] [ ] [ ]1 1AB A AB A

ic ic cc ii icN H H H H

− −

= − −

If only connection DOF are measured, then (7) simplifies to,

[ ] [ ] [ ] [ ]( ) [ ]1 1

B AB A AB

cc cc cc cc ccH I H H H

− −

= − (9)

Mobility Approach

For the Mobility based approach, subtracting (2) from (1) and introducing the coupling conditions (3), (4), (5) and (6), with some algebraic manipulation one can obtain,

[ ] [ ] [ ] [ ] [ ][ ][ ]

[ ] [ ] [ ][ ] [ ] [ ]

A AB AB AB AB AB cc cc ci cicc cc ci cc ci A AB AB A

ic ic ii ii

H H H HH H H I 0

H H H H

++⎛ ⎞⎡ ⎤ ⎡ ⎤−⎜ ⎟⎡ ⎤ ⎢ ⎥ ⎢ ⎥⎡ ⎤= −⎣ ⎦⎜ ⎟⎣ ⎦ ⎢ ⎥ ⎢ ⎥−⎜ ⎟⎣ ⎦ ⎣ ⎦⎝ ⎠

(10)

If only connection DOF are measured, then (10) simplifies to,

[ ] [ ] [ ] [ ] [ ]( ) 1B AB A 1 AB

cc cc cc cc ccH H I H H

−−= − (11)

Both approaches obtain the same solution when only connection DOFs are used, as seen by comparing (9) to (11).

Constraint Force Approach

Consider a complete known system AB assembled from known subcomponent A and unknown subcomponent B. The response, {x}c, of the system at the connection DOF (c) with a force, {f}e, at some arbitrary external DOF, (e) can be written as,

{ } [ ] { }AB AB

c ce ex H f= (12)

An alternate way to write the displacement of system AB would be to use the FRFs of the known component A and assuming a constraint force, {f}c. This will provide the necessary dynamic force that the unknown component B exerts on component A and is given as,

{ } [ ] [ ]{ }{ }

AB A A ec ce cc AB

c

fx H H

f

⎧ ⎫⎪ ⎪⎡ ⎤= ⎨ ⎬⎣ ⎦ ⎪ ⎪⎩ ⎭ (13)

Rewriting (13) to solve for the constraint force gives,

{ } [ ] [ ] { } { }1AB A A AB

c cc ce e cf H H f x

−

⎡ ⎤= −⎣ ⎦ (14)

From the Frequency Based Substructuring derivation [2], the constraint force between two components can be calculated by,

{ } [ ] [ ]( ) [ ] { }1AB A B A

c cc cc ce ef H H H f

−

= + (15)

Rewriting (15) to solve for the unknown component B,

[ ] [ ] { } { } [ ]1B A AB A

cc ce e c ccH H f f H

−

= − (16)

This equation holds true for a single connection. For two connections, (15) can be written as,

{ }{ }

[ ] [ ][ ] [ ]

[ ] [ ][ ] [ ]

[ ][ ]

{ }1AB A A B B A

c1 c1c1 c1c2 c1c1 c1c2 c1ceeAB A A B B A

c2 c2c1 c2c2 c2c1 c2c2 c2ce

f H H H H Hf

f H H H H H

−⎡ ⎤⎧ ⎫ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎪ ⎪ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥= +⎨ ⎬ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎪ ⎪⎩ ⎭ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦⎣ ⎦

(17)

Assuming [ ]B

c1c1H was previously calculated with the system having a single connection, one can find the FRF between

the two connections can be found by writing (17) into two individual equations and rearranging so that,

[ ] [ ] { } [ ] { } [ ] { } [ ] { }( ){ }1B A A AB A AB B AB AB

c1c2 c1ce e c1c1 c1 c1c2 c2 c1c1 c1 c2H H f H f H f H f f

−

= − − − (18)

Assuming reciprocity,

[ ] [ ]B B

c2c1 c1c2H H= (19)

Rewriting (17) again and using (18) and (19) gives,

[ ] [ ] { } [ ] { } [ ] { } [ ] { }( ){ }1B A A AB A AB B AB AB

c2c2 c2ce e c2c1 c1 c2c2 c2 c2c1 c1 c2H H f H f H f H f f

−

= − − − (20)

APPLICATION System Description

To study the robustness of each technique, a simple mass-spring model was created in MATLAB [14]. This model consists of two individual components A and B combined using rigid connections; extension of these equations to consider flexible connections could also be considered. Table 1 and Table 2 list the physical and modal properties of component A and component B, respectively.

M3M1 M2K1 K2 K3

A

M3M1 M2K1 K2 K3

A

Figure 2. Physical Representation of Component A.

Table 1. Properties of Component A.

M4 M5 M6K4 K5 K6

B

M4 M5 M6K4 K5 K6M4 M5 M6K4 K5 K6

B

Figure 3. Physical Representation of Component B.

Table 2. Properties of Component B.

FRFs of each component were synthesized from the modal properties listed in Table 1 and Table 2. Using Frequency Based Substructuring [2], the two components were combined using rigid connections. System AB-1 was created using a rigid connection between DOF 3 on component A and DOF 4 on component B, represented by Figure 4. The natural frequencies and damping are listed in Table 3.

M3 M4 M5 M6K4 K5 K6

B

M1 M2K1 K2 K3

A

M3 M4 M5 M6K K K

B

M4 M5 M6K K K

M4 M5 M6K K K

B

M1 M2K1 K2 K3

A

M1 M2K1 K2 K3

A

M3 M4 M5 M6K4 K5 K6

B

M4 M5 M6K4 K5 K6M4 M5 M6K4 K5 K6

B

M1 M2K1 K2 K3

A

M1 M2K1 K2 K3

A

M3 M4 M5 M6K K K

B

M4 M5 M6K K K

M4 M5 M6K K K

B

M1 M2K1 K2 K3

A

M1 M2K1 K2 K3

A

Figure 4. Physical Representation of System AB-1.

Table 3. Modal Properties of System AB-1.

System AB-2 was created using rigid connections between DOF 1 on component A to DOF 6 on component B and also between DOF 3 on component A to DOF 4 on component B. Figure 5 displays a representation of this system, and Table 4 lists the natural frequencies and damping of this system.

M3 M4 M5 M6K4 K5 K6

B

M1 M2K1 K2 K3

A

M3 M4 M5 M6K K K

B

M4 M5 M6K K K

M4 M5 M6K K K

B

M1 M2K1 K2 K3

A

M1 M2K1 K2 K3

A

M3 M4 M5 M6K4 K5 K6

B

M4 M5 M6K4 K5 K6M4 M5 M6K4 K5 K6

B

M1 M2K1 K2 K3

A

M1 M2K1 K2 K3

A

M3 M4 M5 M6K K K

B

M4 M5 M6K K K

M4 M5 M6K K K

B

M1 M2K1 K2 K3

A

M1 M2K1 K2 K3

A

Figure 5. Physical Representation of System AB-2.

Table 4. Modal Properties of System AB-2.

Perturbation of Frequency Response Functions

Response analysis was performed in LMS CADA-X Forced Response Simulation module [15] to simulate real measurements. A uniform random force was applied to each of the systems and responses were computed. All FRF manipulation was performed using the LMS CADA-X Matrix Toolbox [16]. Two cases were evaluated. One with FRFs to study the decoupling approaches without any noise on the functions and a second case with noise. A one percent random noise was applied to the force spectrum and frequency response functions were computed for all DOFs. All decoupling approaches were first computed with the applied noise resulting in noisy estimates of the unknown component. All techniques produced similar results when calculated directly with noise, but will not be presented in this paper in order to keep the results brief. These cases will be presented in future work [13] along with experimental studies currently in progress. Modal parameter estimation was performed to smooth the FRFs, as typically done with measurements. Table 5 lists the frequencies and damping obtained from modal parameter estimation. New FRFs were synthesized from this modal data for use in the decoupling computations. Figures 6, 7 and 8 show the effect of the applied noise and synthesized FRFs from modal parameters.

Table 5. Reference Model Modal Characteristics

Compared to Model with Noise.

Figure 6. Drive Point FRF of Connection DOF 3 on

Component A.

Figure 7. Drive Point FRF of Connection DOF 3 on

System AB-1.

Figure 8. Drive Point FRF of Connection DOF 3 on

System AB-2.

Decoupling of System – 1 Connection

The first case considers system AB-1 in which there is a single rigid connection between DOF 3 and DOF 4. Due to the simplicity of the model, effects of truncation are of no concern for any of the cases. Figure 9 shows all techniques produce exact results when pure FRFs are used. Figure 10 shows the effect of noise applied to the data. The Constraint Force approach shows noise scattered throughout the entire FRF due to the noise being present in the constraint force calculation, equation (14). The Impedance and Mobility approaches gave a fairly accurate result except for a spurious peak occurring at approximately 60 Hz. Further investigation to this discrepancy is related to the shift in antiresonances in connection FRFs which are inverted in the calculation. Figure 11 displays an overlay of the FRFs used in the computation and the resultant FRF estimation of Component B. The problematic frequencies are highlighted, which are both located at antiresonances of system AB-1 and component A. (These antiresonances of the perturbed FRFs have a slight shift in frequency from the true FRFs, which is shown to produce these peaks during the manipulation of the FRF equations.) An advantage of both the Impedance and Mobility approaches is the ability to include additional internal DOFs which are present on both the system and known component. The estimation of the unknown component is significantly improved with the use of internal DOF, as shown in Figure 12. Using DOF 1 produced slightly more accurate results for both approaches. Including both DOFs in the computation produces the same results as the results using DOF 1. (Use of additional internal DOF is currently under investigation with larger models to allow for more general selection of DOF; on this limited size model, the use of additional DOF is limited.) For all cases studied thus far, the Mobility with internal DOF was shown to produce more accurate results.

Figure 9. FRF Estimation of Component B using Exact

FRFs: Reference (Black), Force Approach (Red), Impedance and Mobility (Blue).

Figure 10. FRF Estimation of Component B using

Perturbed FRFs: Reference (Black), Force Approach (Red), Impedance and Mobility (Blue).

Figure 11. Overlay of FRFs used in Calculation and

Perturbed Estimation of Component B (Top: Component A, Bottom: System AB-1).

Figure 12. FRF Estimation of Component B using

Internal DOF: Reference (Black), Impedance (Red), Mobility (Blue).

Decoupling of System – 2 Connections

The second case studied uses system AB-2, in which component A and B are connected with two rigid connections. As with the first case, all three methods produce exact results when pure FRFs are used, as shown in Figure 13. Impedance and Mobility approaches are much more efficient approaches than Force, as only a single calculation must be performed. As derived in the theory section, the Force approach needs a FRF on the unknown component initially found from the single connection case to find the additional connection FRFs. Figure 14 presents the estimation of the connection drive point FRFs on component B using perturbed FRFs. As previously mentioned, the Constraint Force approach requires the FRF of component B with a single connection; therefore any inaccuracies in this estimation will be amplified in successive calculations. Impedance and Mobility are again significantly improved with the use of an internal DOF, as shown in Figure 15. Mobility approach with internal DOF again produced the most accurate estimations.

Figure 13. FRF Estimation of Component B using Exact

FRFs: Reference (Black), Force Approach (Red), Impedance and Mobility (Blue).

Figure 14. FRF Estimation of Component B using

Perturbed FRFs: Reference (Black), Force Approach (Red), Impedance and Mobility (Blue).

Figure 15. Estimation of Component B using Internal DOF:

Reference (Black), Impedance (Red), Mobility (Blue). OBSERVATIONS

Current investigations of the Constraint Force technique were shown to be promising with a single connection. While this technique is not as efficient for multiple connections, current work is focused on computing the unknown component with multiple connections in a single calculation. A single matrix calculation would significantly reduce the noise amplified in successive calculations. Impedance and Mobility appeared to be advantageous when internal DOF are available, although slight shifts in antiresonances due to modal parameter estimation on the noisy FRFs were shown to produce spurious peaks when only connection DOF are used. These spurious peaks are not present when the noisy FRFs are directly used in the decoupling approaches, although the entire FRF computed has a significant amount of noise. Future studies will be focused on applying these techniques to a test fixture in which a variety of test issues will be presented.

CONCLUSIONS The decoupling problem was presented in which FRFs are of an unknown component are estimated from a known system and component. Impedance and Mobility techniques are investigated and compared to a Constraint Force-based decoupling approach. The effect of noise was studied for all techniques to replicate real measurements. The current Constraint Force approach was shown to amplify noise of the estimated component for multiple connections. Impedance and Mobility approaches were shown to produce spurious peaks in the estimation of the unknown component when only connection DOF FRFs are used. The use of internal DOF significantly improved the estimated FRF for these approaches.

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