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Full Paper Proc. of Int. Conf. on Advances in Robotic, Mechanical Engineering and Design 2011

The Effect of the Reduction of Coordinates on the Identified Joint Dynamic Properties P.Soleimanian1, A.Tizfahm 2, M.H.Sadeghi 2 1

Tabriz University, Tabriz, Iran Email: soleimanianp@yahoo.com 2 Amirkabir University of Technology, Tehran, Iran Email: {a.mechanic1355@gmail.com, morteza@tabrizu.ac.ir} Abstract— In this study the effect of the coordinate reduction on the joint identification results in a complex structure (connected by an elastic joint) is tackled via a combined experimental-numerical approach. A theoretical model of a joint is established from the measured frequency- response functions (FRF) and the model of the joint is considered as a coupled dynamic stiffness matrix, which generally six degrees of freedom (DOFs) in each node. The effect of coordinate reduction on the identification results is demonstrated numerically. The considered numerical model is the FE models of substructures and assembly structure were updated by minimizing the error between the calculated translational FRFs from the measurement data, and the calculated ones based upon the FE model. The approach was validated by an experimental Cross-Beam structure. Finally, a reconstructedsubstructure-synthesis method, including the identified joint parameters the assembly dynamic response is reconstructed.

They extracted the joint’s parameters from the experimental data and established a theoretical model of a joint. For linear joint models, an identification approach was developed by Liu based on Ren’s method [4]. He concluded that the RDOFs have an important role in the identification process; it was not clear that how the RDOFs were included in the identification process. For obtaining the necessary information about RDOFs, Yang et al. [5] introduced the model of joint as a coupled stiffness matrix Instead of just a set of translational and rotational springs [6,7]. Some other researchers used substructure FRFs and joint-dependent FRFs of the whole structure, to identify joint properties [8,9]. The approach in this paper is a numerical-experimental technique, which is based on the method introduced by [10], and improved for identifying the dynamic properties of a real bolted joint in complex structures in a wide frequency range. To show the effect of coordinate reduction on the identification results, a theoretical model of a joint is established from the measured FRFs and the model of the joint is considered as a coupled dynamic stiffness matrix, which generally six DOFs in each node.

Index Terms—Coordinate Reduction, Joint Identification, Model Updating, Substructure-Synthesis, Rotational DOFs

I. INTRODUCTION Most of the assembly structures consisting of some substructures connected by joints play a significant role in dynamic properties of these systems. Correct identification of the joints properties representing the physical qualities involved in the joint region is critical in modeling of any assembly structure. The main purpose of joint identification is to estimate the parameters of the joint that minimize the difference between the measured response characteristics of the assembly and those predicted analytically [1]. Identifying the joint parameters in most of the structures requires rotational degrees of freedom (RDOFs) measurements, which is a very difficult task. As a result, the use of FE model in combination with experimental modal analysis (EMA) seems to be a rational choice. FE model updating has become a viable approach to increase the correlation between the dynamic response of a structure and the predictions from a model. In model updating, parameters of the model are adjusted so as to reduce a penalty function based on residuals between a measurement set and the corresponding model predictions [2]. The target of this approach is the jointidentification technique for linear systems involving several rigid and flexible joints developed by Ren and Beards [3].

II. JOINT IDENTIFICATION THEORY In general, each node of a structure has six DOFs or six coordinates and at each coordinate external force and moment can be present. The generalized displacement vector x and generalized force vector f can be defined as

Where superscript i represents particular node of the structure. Notations x, y and z represent translational DOFs, while  x ,  y and  z are rotational DOFs. The system of equations is adjusted in a way that, there is no limit in the number of substructures, joints and the corresponding DOFs. The proposed method comprises three subsystems as: all substructures system, the joint system, and the whole assembly system. These systems are shown in Fig. 1 [3]. In the substructure system two coordinates were considered: The joint coordinates and the internal coordinates, denoted by the subscripts b and a, respectively.

In the assembly structure, two coordinates were considered

________________________________________ 1

- Corresponding author. Tel: 00 98(0) 411 -33 924 74; Fa x: 0098 (0)411 -33541 53.

E-mail address: soleimanianp@yahoo.com

© 2011 AMAE DOI: 02.ARMED.2011.01.20

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Full Paper Proc. of Int. Conf. on Advances in Robotic, Mechanical Engineering and Design 2011

f c1  x1c   2   Z j  2   Z j  xc    fc  (5) f c  x c  The internal coordinates before and after assembling of joint do not change, hence:

 fa    fn 

Figure 1.

and  x a    x n  (6) Considering no external forces and moment acting at the connection coordinates, the compatibility and equilibrium conditions can be written:

the arrangement system in the joint identification process

 f b    f c    0

and

 x   x    x 

(7) From system (4) the responses at the connection coordinates can be expressed:

as the internal and joint coordinates which are denoted by the subscripts n and j, respectively. The relationships between the displacement and the force vectors of the substructure system, which includes two substructures, can be expressed as:

j

b

c

 xb   H ba  f a   H bb  f b 

(8)

By substituting (5) into (8) and considering (7): Z j  x c   - f b   Z j H ba  f a   Z j H bb  f b 

(9)

Rearranging (9):

 fb   (I  Z j H bb )-1 - Z j H ba  fa 

(10)

Substituting (10) in (4) and using (6) and (7) yields: x n    x j  H aa  H ab (I  Z j H bb ) 1 Z j H ba  1  H ba  H bb (I  Z j H bb ) Z j H ba

Where H is the receptance matrix, superscripts 1 and 2 represent particular substructure, while x and f are generalized displacement and force vector, respectively.

And for assembly system:

H ab (I  Z j H bb ) 1  f n    H bb (I  Z j H bb ) 1  f j 

(11) By comparing (11) and (4):

For simplicity the (2), (3) can be rewritten as:

In this equation:

H aa  H nn  H ab (I  Z j H bb ) 1 Z j H ba

(12)

H ba  H jn  H bb (I  Z j H bb ) 1 Z j H ba

(13)

H nj  H ab (I  Z j H bb ) 1

(14)

H jj  H bb (I  Z j H bb ) 1

(15)

For practical applications, (12) can not be used directly in the process of identification and an improved form of this equation should be presented. First pre-multiplying (12) by matrix H ab  , then pre-multiplying by ( I  Z j H bb ) leads to: 

(I  Z j H bb )H ab (H aa  H nn )  Z j H ba

(16)

With rearranging (16): 



H ab (H aa  H nn )  Z j (H ba  H bb H ab (H aa  H nn )) (17)

And finally 

H aa  H nn  H ab Z j (H bb H ab (H aa  H nn )  H ba )

(18)

Where the superscript ‘+’ denotes the pseudo-inverse [11] of the matrix. Equation (18) is a direct solution and basis for the identification of the dynamic stiffness matrix of the joint system Z j . Equations (18) generally has a form

The characteristic of the joint system can be expressed in the

A ( M N ) Z j

form of the dynamic stiffness matrix Z j :

( NN )

B ( N  L )  C( M  L)

(19)

Where A , B and C are the coefficient matrices; and M , N and L represent the size of the corresponding matrices. © 2011 AMAE DOI: 02.ARMED.2011.01.20

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Full Paper Proc. of Int. Conf. on Advances in Robotic, Mechanical Engineering and Design 2011 If

120/V accelerometer as shown in Fig. 3.

and also matrices A and B are non-singular, (19) becomes determined or over-determined. Hence Z j is solved M, L  N

uniquely [3, 10].

The least squares solution of (19) is: Z j  A C B

(20)

Where the superscript ‘+’ denotes the pseudo-inverse [11] of the matrix, as mentioned before. Also (19) can be written as a system of linear equations, as: E( )z ( )  g( )

(21)

Where z is a frequency dependent ( N

2

1

) vector whose

elements are constructed from Z j . E is the coefficient matrix constructed from matrices A and B , and g is a coefficient vector constructed from the matrix C [4]. To convert the vector z into a frequency-independent vector y , a linear transformation introduced as follows: z ( )  T f y

(22)

Where Tf is a transformation matrix and has the following form

Figure 2. The plot of real assembly

2

    Tf  I - 2 I i I iI  0 0  

(23)

In this equation  0 is a reference angular frequency which is usually equal to the maximum measured frequency [4].

Vector y is formed in terms of the mass, stiffness and damping matrices (including viscous damping c and structural damping d ) as follows:





y  k  02 m  0 c d Substitution of (22) into (21) yields:

(24)

E( )T f y  g ( )

(25)

The least-square (average) solution of (25) will be:  n T y    E( ) E( )  i1



  

Figure 3. Experimental test setup for FRF measurement

1

n

 E( )  g( )  

  i 1

T



(26)

IV. FINITE ELEMENT MODEL UPDATING For accurate estimation of unmeasured FRFs, the updated model of the substructures and assembly structure was created. The linearized eigen-sensitivity method was employed for model updating [12, 13]. The updated FE model of the substructures as well as the joint was made by using three dimensional parametric Euler-Bernoulli beam elements having two-node/12-DOF in each element. The updated results for substructures and assembly structure are presented in Fig. 4 and Fig. 5 respectively. Damping matrix of the substructures and assembly structure were extracted from modal parameters, estimated from the measured data.

In (26) values of the mass, stiffness and damping matrices are constant and are valid for the entire frequency range.

III. EXPERIMENTAL CASE STUDY The test assembly is a Cross-Beam laboratory-scale structure, consisting of two steel beams (as substructures 1 and 2) with identical lengths and cross sections (800,30  10mm) , which were connected (asymmetrically) together by an elastic joint, as shown in Fig. 2. The joint consisted of a nut, bolt, rubber and steel washers. (U1z  U9 z )and (D1z  D9 z ) are measured points of substructures 1 and 2 respectively. A B&K 4809 shaker powered by a B&K 2706 amplifier was used to random excitation of the system. The applied force was measured by a force transducer connected to a charge amplifier (B&K 8200+2647a) and the response was measured by a DJB A/ © 2011 AMAE DOI: 02.ARMED.2011.01.20

V. SOLVING OF FULL AND REDUCED SYSYTEMS In general case, all the internal coordinates and all the joint coordinates are included in the identification process.

Although this is acceptable for simple theoretical cases, it is rather impossible from practical point of view. Every 8

Full Paper Proc. of Int. Conf. on Advances in Robotic, Mechanical Engineering and Design 2011 VI. NUMERICAL CASE STUDY

discretization of continuous structures or numerical models is always leading to the coordinate reduction. The identified

Numerical FE model is an updated model of a assembly Cross-Beam that consist of updated substructures and joint.

Each substructure has 126 coordinates and assembly has 252 coordinates (Fig. 6). The geometry and material properties of substructures and joint are given in Table 1.

joint dynamic stiffness matrix Z j is full only in case, when all the joint coordinates are included in the identification process.

Full joint dynamic stiffness matrix can be written in terms of the stiffness, mass and damping matrices as follows: Z j ( )  K   2 M  i C

(27)

Figure 4. Comparison of FRFs for upper substructure (U9 z

/ U1z )

Figure 6. 3-dimensional FE model with 252 coordinates TABLE I. G EOMETRY AND MATERIAL PROPERTIES

/ U1z ) Where K, M and C represent the stiffness, mass and the viscous damping matrices, respectively. In this case, not only the direct solution, but also the average solution is possible.

When joint coordinates is reduced in the identification Figure 5. Comparison of FRFs for assembly structure (U9 z

process, the reduced joint dynamic stiffness matrix Z rj is identified. Equation (26) can be written in a form Z rj ( )  K r ( )   2 M r ( )  i Cr ( )

Knowing matrices K, M, C and using equation (28)

1 (29) K   2 M  i C receptance matrices of each substructure and an assembly were calculated. Selected frequency range was 0–3.2 kHz. For simplicity, receptances are denoted with regard to the response coordinate and the excitation coordinate.

H ( ) 

Where the reduced matrices K , M and C are frequencydependent. Hence it follows that the average solution (26) cannot be valid any more. When the number of joint coordinates is reduced, only one solution for is reasonable, i.e. the direct solution (18) at each individual frequency point.

Reduction of the internal coordinates does not change the r

r

r

Therefore, the notation H ( z1,m5) represents the receptance.

size of the identified joint dynamic stiffness matrix Z j , and

H ( z1,m 5) 

accordingly should not have big influence on the results of the identification. The internal coordinates can then be chosen more freely and only those can be selected that can easily be measured. However, it is essential that the number of internal coordinates is greater than or at least equal to the

(30)

6.1. Identification on a full system When all the coordinates are included in the identification process either a direct solution or an average solution of system (18) can be found. Both should give the same results (Fig. 7 and Fig. 8).

number of joint coordinates( N a  N b ). © 2011 AMAE DOI: 02.ARMED.2011.01. 20

z1( ) m5( )

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Full Paper Proc. of Int. Conf. on Advances in Robotic, Mechanical Engineering and Design 2011

Figure 7. Comparison of real dynamic stiffness

(z j ) ( f

zj 1 , z j1 )

Figure 10. Real dynamic stiffness

Figure 8. Comparison of real dynamic stiffness

(z j ) ( m

Figure 11. Comparison of original and reconstructed assembly receptance

( H (z ,f ) ) 6

z6

6.3. Reduction of internal coordinates In this case, all the joint coordinates, but only few internal coordinates were included in the identification process. The vector of selected joint coordinates is:

6.2. Reduction of joint coordinates In this case, all the internal coordinates, but only the



x c  x11 , y11 , z11 , x11 , y11 , z11 , x 22 , y 22 , z 22 , x 22 , y 22 , z 22

translational joint coordinates ( x11 , y11 , z11 , x22 , y 22 , z 22 ) were included in the identification process. The direct solution for an element of the joint dynamic stiffness matrix is shown in Fig. 10. The result of the identification is not aquadratic function and, therefore, the proper average solution cannot be found. However, the direct solution is fully acceptable and can be used for the prediction of the assembly response.

The comparison of original and reconstructed assembly receptance is shown in Fig. 11. The original and the reconstructed receptance curves in Fig. 11 are not the exact fit, which is a consequence of the reduction of joint coordinates.

x a   N1, N 5, N13, N 21, N 24, N 28, N 35, N 37, N 41





Ni  xi , yi , zi , xi , yi , zi , i  1,5,13,21,24,28,35,37,41

The example of the identified joint dynamic stiffness (direct solution) is shown in Fig. 12.

The identified dynamic stiffness in Fig. 12 is equivalent to the dynamic stiffness in Fig. 7; therefore, also the predicted assembly response should be very similar to that in Fig. 9.

Figure 9. Comparison of original and reconstructed assembly

© 2011 AMAE DOI: 02.ARMED.2011.01. 20



While the vector of selected internal coordinates is:

Figure 12. Real dynamic stiffness

( H (z ,f ) ) 6

zj 1 , z j1 )

zj1 , j 1 )

Fig. 7 and Fig. 8 show the comparison of the direct and the average solution for the identified translational and rotational dynamic stiffness, respectively. Full compliance between the direct and the average results is obtained. From the identified joint parameters and the substructure FRFs, the assembly FRFs were reconstructed. Fig. 9 shows the comparison of assembly receptances. Very good agreement between the original and the reconstructed receptance curves is obtained, which confirms the accuracy of the identification.

receptance

(z j ) ( f

z6

10

(z j ) ( f

zj 1 , z j1 )

Full Paper Proc. of Int. Conf. on Advances in Robotic, Mechanical Engineering and Design 2011 [4] W. Liu, Structural Dynamic Analysis and Testing of Coupled Structures, PhD Thesis, Imperial College of Science, Technology and Medicine, London, 2000.

[5] T. Yang, H.S. Fan, S.C. Lin, Joint stiffness identification using FRF measurements, Computers and Structures 81 (2003) 25492556.

[6] K.T. Yang, Y.S. Park, Joint structural parameter identification using a subset of frequency response function measurement, Mechanical Systems and Signal Processing 7 (1993) 509-530.

[7] Y. Ren, C.F. Beards, On the nature of FRF joint identification technique, Proceedings of the 11th International Modal Analysis Conference, Florida, USA,(1993) 473-478.

[8] S.W. Hong, C.W. Lee, Identification of linearized joint structural parameters by combined use of measured and computed frequency responses, Mechanical Systems and Signal Processing 5 (1991) 267-277.

[9] J.H. Wang, C.M. Liou, Experimental identification of mechanical joint parameters, Journal of Sound and Vibration 143 (1991) 28-36.

[10] Y. Ren, The Analysis and Identification of Friction Joint Parameters in the Dynamic Response of Structures, PhD Thesis, Imperial College of Science, Technology and Medicine, London, 1992.

[11] N.M. Maia, J.M.M. Silva, Theoretical and Experimental Modal Analysis, Wiley, New York, 1997.

[12] Gan.Chen, FE Model Validation for Structural Dynamics, PhD Thesis, Imperial College of Science, Technology and Medicine, London, 2001.

[13] Sen.Huang, Dynamic Analysis of Assembly Structures with Nonlinearity, PhD Thesis, Imperial College of Science, Technology and Medicine, London, 2007.

CONCLUSIONS In this paper a theoretical model of a joint is established from the measured FRFs to show the effect of coordinate reduction on the identification results in complex structures and wide frequency range. The accuracy in results of model updating shows that, this theory is done on a real structure.

The direct and the average solution were done in this paper.

The average solution can only be used, when all the joint coordinates are included in the identification process. The obtained results show that, by reducing coordinates of joint, the original and the reconstructed receptance are not the exact fit, which is a consequence of the reduction of joint coordinates. In order to get a unique solution for the joint dynamic stiffness matrix, the number of internal coordinates has to be higher than a number of joint coordinates.

REFERENCES [1] R.A. Ibrahim, C.L. Pettit, Uncertainties and dynamic problems of bolted joints and other fasteners, Journal of Sound and Vibration 279 (2005) 857-936.

[2] M.I. Friswell, J.E. Mottershead, H.Ahmadian, Finite-element model updating using experimental test data: parameterization and regularization, Philosophical Transactions: Mathematical, Physical and Engineering Sciences, Vol. 359, No. 1778, Experimental Modal Analysis (2001) 169-186.

[3] Y. Ren, C.F. Beards, Identification of joint properties of a structure using FRF data, Journal of Sound and Vibration 186 (1995) 567-587.

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