Estimating Rotational Frequency Response Function Using Mode Expansion and Frequency Response Function Synthesis

– The rotational frequency response function (RFRF) plays a crucial role in increasing the accuracy of the calculated results of the frequency-based substructuring method. However, RFRFs are often omitted due to the difficulties in the measurement process and limitations of the equipment. This paper presents a scheme of estimating the rotational FRF of an irregular plate structure using the FE model reduction and expansion method. The reduced FE model was introduced using the improved reduction system (IRS) and expanded to the experimental modal model (EMA model) using the system reduction and the expansion (SEREP) method. The FRF expanded method was then employed to derive the translational and rotational FRFs from the expanded EMA model. The accuracy of the expanded FRFs was evaluated with the EMA model of the irregular plate. It was found that the translational and rotational FRFs estimated from the proposed scheme were in good agreement with the EMA counterparts. Furthermore, the patterns of the estimated RFRFs were well correlated with the EMA RFRFs. This work shows that the proposed scheme may offer an attractive alternative way of accurately determining the RFRs of complex structures or structural components.


INTRODUCTION
Structural dynamic responses of a built-up structure are highly dependent on the properties of structural components and joints [1].The responses are often investigated and evaluated using FRFs due to overwhelming data [2][3][4].FRFs can be obtained from the structure analytically and experimentally [5,6].However, obtaining experimentally rotational FRFs that are crucial in substructuring methods is perceived to be very challenging and time-consuming in comparison with that of translational FRFs [7][8][9].There has been substantial research regarding measurement techniques of rotational FRFs.For instance, previous works showed that two sensitivity-matched and spatially separated accelerometers could be employed to estimate the rotational acceleration [10,11].Furthermore, the rotational FRF (RFRF) could be indirectly obtained using an attachment of T-block [8,12].The FRFs measured at the T-block was then decoupled from the system to obtain the rotational FRFof the test structure [4].Another alternate method used to estimate RFRs is using X-block attachment with three translational accelerometers.The X-block method was employed for structural modification of a helicopter rear tail [13].
Recently, the piezoelectric rotational accelerometer was introduced and has gained much attention in the dynamic substructuring field.Piezoelectric rotational accelerometers offer direct measurement of RFRFs [14].The comparison of the piezoelectric rotational accelerometer with the previous T-block method was carried out in [15].The comparison revealed that the results obtained from piezoelectric rotational accelerometers were far more accurate and reliable.However, there is an obvious drawback of a direct rotational sensor that seemed to be a highly-priced accelerometer compared to a classical translational accelerometer.The use of piezoelectric rotational accelerometers on a simple beam for the frequency-based substructuring (FBS) method was demonstrated in [16].However, it was reported that the measurements of rotational FRFs were only applicable for force excitation responses, resulting in the coupling matrix of the structure becomes an incomplete coupling matrix.Therefore, the analytical modal model was used to expand the moment excitation FRFs to acquire a full FRF matrix of the beam [17].
There are a number of attempts to indirectly obtain the moment excitation or torque [8,12].The authors introduced and attached the T-beam structure to an end of a shaft, and the torques were excited by forces excitation.The accelerations were measured by using linear accelerometers at a certain location of the T-beam.Using the FE model of the T-beam attachment, the moment between the torque to the angle of twist of the shaft was derived based on the measured linear acceleration to the excitation force.This paper proposes a scheme of estimating the RFRF of an irregular plate structure using the mode reduction, mode expansion method (MEM) and FRF synthesis.The FE model of the irregular plate (full FE model) was developed and then reduced according to the experimental model using the Guyan reduction and improved reduction system (IRS).Later, the reduced FE model is expanded to the EMA model using the system reduction and the ABSTRACT -The rotational frequency response function (RFRF) plays a crucial role in increasing the accuracy of the calculated results of the frequency-based substructuring method.However, RFRFs are often omitted due to the difficulties in the measurement process and limitations of the equipment.This paper presents a scheme of estimating the rotational FRF of an irregular plate structure using the FE model reduction and expansion method.The reduced FE model was introduced using the improved reduction system (IRS) and expanded to the experimental modal model (EMA model) using the system reduction and the expansion (SEREP) method.The FRF expanded method was then employed to derive the translational and rotational FRFs from the expanded EMA model.The accuracy of the expanded FRFs was evaluated with the EMA model of the irregular plate.It was found that the translational and rotational FRFs estimated from the proposed scheme were in good agreement with the EMA counterparts.Furthermore, the patterns of the estimated RFRFs were well correlated with the EMA RFRFs.This work shows that the proposed scheme may offer an attractive alternative way of accurately determining the RFRs of complex structures or structural components.journal.ump.edu.my/ijame◄ expansion (SEREP) method to estimate all the RFRFs.For validation purposes, the expanded FRFs were compared with the EMA counterparts.

DESCRIPTION OF CASE STUDY
The proposed scheme to estimate the rotational FRF was demonstrated on an irregular shape plate structure, as presented in Figure 1.The irregular plate is made of steel and 350 mm in length [18].The rotational FRFs were determined at two different points: reference point 1 and reference point 2. The reference points, which are also the connection points, were investigated mainly for dynamic substructuring or structural modification purposes.In this work, only the z-axis and rotational FRF in the x-axis were investigated due to the limitation of the rotational accelerometer as discussed in the Piezoelectric Rotational Accelerometer Section.The mode of interest in this work was from 0 Hz and 2000 Hz.The dynamic behaviour in terms of mode shapes and natural frequencies of the irregular plate were obtained from experimental modal analysis (EMA) [19].On EMA, the irregular plate was discretised into 30 measurements points (EMA modal model), as presented in Figure 2. The measurement points decided to reduce the possibility of losing any mode and obtaining a suitable mesh for animating the modes of interest [20].The configuration of the EMA modal model used in developing the reduced FE model of the irregular plate is vital to the mode expansion.

Finite Element Modelling
The purpose of using the FE method, model reduction and expansion method in this study was to estimate the rotational modal vectors that are difficult to obtain experimentally.Prior to performing the model reduction and expansion method, the full FE model of the irregular plate was developed, as presented in Figure 3.A total of 2850 elements and 3030 nodes were required in the development of the FE model.2D shell elements that provide both translational and rotational modal vectors were used as the element profile.The standard material properties for steel used in this study are the modulus of elasticity of 210000 MPa, density of 7,850 kg/m 3 and Poisson's ratio of 0.3.The dynamic behaviour of the irregular plate was calculated by using Block Lanzos normal modes solution in free-free boundary conditions.The calculated dynamic behaviour was compared with the EMA counterparts for validation purposes [21].The natural frequencies and mode shapes were measured using impact hammer testing.The EMA setup is illustrated in Figure 4.The setup was designed as to that of the FE model.The EMA was performed by hanging the plate with two soft springs to imitate a free-free configuration [22].A single 10 mV/g uniaxial accelerometer was used to measure the dynamic data, and a 21.7mV/N impact hammer was used for the excitation.Siemens LMS data acquisition was used to obtain the dynamic data.The frequency range was set between 0 Hz to 2000 Hz with a step size of 1 Hz.To measure the natural frequencies and mode shape accurately, the excitation was performed 10 times to attain a high number of averaging.Meanwhile, the responses from the excitation were analysed using Siemens LMS Test.lab.

EXPERIMENTAL RESULTS
The accuracy of the developed FE model for the irregular plate was evaluated by comparing the FE natural frequencies and mode shapes with the EMA counterparts.The FE mode shapes were quantified by applying the modal assurance criterion (MAC).A detailed explanation of the MAC analysis is available in [23].The formulation of MAC used in this study is as follows.
where,   and   are finite element and EMA modal vectors.The MAC value that is approaching the value of 1 conveys the similarity between the FE and EMA mode shapes.Table 1  The comparison showed that the FE natural frequencies were in good agreement with the EMA counterparts with a total error of 4.08 per cent, which is within the acceptable level of accuracy.All the MAC values tabulated in Table 1 were above 0.90, indicating a high correlation between the FE mode shapes and EMA counterparts.A comparison of FE and EMA mode shapes is tabulated in Figure 5

Model Reduction Method
It is important to note that the issue of missed and swapped modes should be carefully addressed before a model reduction is carried out.In this work, two model reduction methods which are, the Guyan reduction method and the Improved Reduction System (IRS), were performed on the full FE model of the irregular plate to reduce the degrees of freedom (DOF) to that of the EMA modal model.The modes of a structure can be derived from the fundamental equation of motion (EOM).The linear equation of motion can be described by the following form: where M, C and K are the matrices of mass, damping and stiffness, respectively, and f is the load vector.The vectors ̈, u̇ and u denote the displacement, velocity, acceleration of the system.The general approach of the model reduction method is to approximate the state vector by using the transformation; where  and   is the transformation matrix and a reduced state vector, respectively.By applying the transformation in Eq.(3) into Eq.( 1), the reduced system can be described as; journal.ump.edu.my/ijame◄   ̈+  ̇+    =   (4) where   ,   and   are the reduced order of mass, damping and stiffness matrices.In this work, two condensation reduction methods were investigated for the FE model reduction of the irregular plate.The first one was the Guyan reduction method [24].The other one was the Improved Reduction System (IRS) [25], which its formulation is solely based on the   and   from the Guyan reduction method

FE Model Reduction using the Guyan Reduction Method
In the condensation-based reduction method, the degrees of freedoms (DOFs) of the irregular plate are separated into masters, , and slaves, .It is worth noting that the Guyan Reduction method only provides the transformation matrix, T based on the K matrix.The slave DOFs are condensed in the reduction process resulting in the reduction of the state vector that contains only the master DoFs.The stiffness matrix in Eq. ( 2) can be partition as; assuming that there is no load acting on the slaves DoFs, with   = 0, the equation yields; from which   can be identified as; Neglecting the inertia from the M matrix results in the Guyan reduction transformation matrix,   in the form of; and the   is; The same transformation procedure is applied on M matrix to obtain the   in the form of; The master DOFs were acquired from the 30 measurement points of the EMA modal model of the irregular plate.The reduced model of the irregular plate is shown in Figure 4.The natural frequencies and mode shapes of the reduced model of the irregular plate were calculated using the normal modes solution.They were then compared with those of the full FE model for validation purposes.Note that a normal mode solution is a mathematical procedure of determining a set of normal coordinates describing the collective motion of the system.Table 3 presents the calculated MAC values and the comparison of the natural frequencies between the full FE model and the reduced FE model using Guyan reduction method.From Table 3, the 6 th and 7 th modes recorded a low correlation as compared with the natural frequencies between the full FE and Guyan FE model.The 6 th mode with 6.60 per cent of percentage error was the largest contributor to the total percentage error of 18.75 per cent.Furthermore, it was found that the natural frequencies obtained from the Guyan FE journal.ump.edu.my/ijame◄ model were slightly higher compared with the full FE ones.The higher discrepancies obtained are because the Guyan reduction method only provides the transformation matrix T, which is based on the K matrix and the   , as stated in Eq. (11).Furthermore, there are no inertia terms as pseudo-static forces to the transformation matrix T in the   .A detailed discussion of results comparison between the Guyan FE and full FE model is available in [26].

FE Model Reduction using IRS
The previous section showed that the Guyan reduction method is incapable of accurately predicting the natural frequencies and mode shapes.The inaccuracy in the prediction can be clearly seen in the 6 th and 7 th modes, which are the largest contributor to the total percentage error and the lowest MAC values calculated.The IRS method introduced by [25], which consider of the inertia in term of pseudo-static forces to the transformation matrix, is adopted to improve the Guyan reduction method.The detailed information on the derivation of the Guyan reduction and IRS methods is available in [26].The transformation matrix for IRS,   can be written as; or by inserting Eq. ( 12) into Eq.( 5), the reduced stiffness and mass matrices are: the transformation,   , reduced stiffness,   and mass   matrices introduced in the previous section were used to calculate the   ,   and   of the irregular plate.The dynamic behaviour of the IRS reduced model of the irregular plate was calculated by using the same solution with the Full FE.Table 5 presents the MAC values and comparison of the natural frequencies between the full FE model and IRS FE model.

FRF Synthesis Method
The FRFs of the expanded model of the plate is derived via the FRF synthesis method [28].For this method, the expanded model's synthesised FRF matrix Hexp (ω k ) and mode shapes  are expressed in the form of: where N represents the number of modes, {}  represents the ith expanded mode shapes,    represents ith natural frequency and   represents the i-th modal damping ratio.The expanded model's synthesised FRFs are compared with the EMA counterparts and presented in the next subsection.Note that only the z-axis FRF was used in the comparisons because all modes of the irregular plate mostly are presented in the z-direction.

Validation of FRF between Expanded and EMA model
Figure 7(a) to 7(c) present the direct comparisons of the translational FRF between the expanded and EMA models.The FRFs were obtained at reference points 1 and 2. The cross axis FRF between the two pints were also compared for reciprocity check.From Figure 7(a) to 7(c), all the resonance peaks calculated from the expanded model showed a strong correlation with those of EMA, particularly the peaks at connection point 1.On the other hand, the anti-resonance peaks of the expanded model were slightly mismatched, particularly at the frequency range between 1000-2000Hz.journal.ump.edu.my/ijame◄

Validation of Rotational FRF
Rotational FRFs (RFRFs) are less frequently considered in the experimental modal analysis compared to the translational FRFs.Nevertheless, RFRFs have an important role in particular structural dynamics applications, for instance, structural modifications, substructuring, FE model updating and model validation.Even though the RFRFs represents 75 per cent of all the FRF matrix, they are often excluded due to the difficulties in measuring them.In this case study, the RFRFs derived from the expanded test model were evaluated with the EMA FRF.The rotational FRFs was measured using a piezoelectric rotational accelerometer.A brief presentation and technical specifications of the piezoelectric rotational accelerometer are given in this section as they are used in the rotational FRFs validation.

Piezoelectric Rotational Accelerometer
Kistler 8840 piezoelectric direct rotational accelerometer, as presented in Figure 8, was used in measuring rotational FRFs.This sensor is quartz crystal-based and powered by a 20-30 VDC power supply.The technical specification of the rotational accelerometer can be found in [14,15] and presented in Table 9.In this study, only the force excitation rotational FRF was acquired as the moment-excite rotational FRF cannot be performed currently.Figure 9 presents the mounting configuration of the rotational accelerometer.The rotational accelerometer was mounted to the connection points of the irregular plate using an M5 Hex socket cap bolt facing the x-axis direction (rotational x-axis).In this work, only one rotational accelerometer was mounted at a time to minimise the mass loading effect on the measured data.The test structure was excited by using the same modal hammer used in the previous section.The frequency range was set from 0 Hz to 2000 Hz due to the limitation of the sensor according to the technical specification stated in Table 10.

CONCLUSION
A scheme enabling the estimation of experimental rotational FRF using the FE model reduction and expansion method is presented.The proposed scheme accurately estimates the rotational FRFs of the structure, particularly for lower modes (0 Hz to 1500 Hz) and produces lower quality rotational FRFs for higher modes of 1500 Hz to 2000 Hz.However, the higher modes, generally, are not really of concern in dynamic structural analysis.
The greatest wealth of the proposed scheme over experimental modal analysis for rotational FRF measurements, which is usually very challenging, time-consuming and definitely requires rotational accelerometers, lies in the use of the FE model reduction and expansion method, which eliminates the great constraints of experimental modal analysis, to estimate rotational FRFs accurately.

ACKNOWLEDGEMENT
The authors would like to express their gratitude to the Malaysian Ministry of Higher Education (MoHE) for providing financial support for this work through the Fundamental Research Grant Scheme FRGS600-IRMI/FRGS 5/3 (335/2019).

Figure 2 .
Figure 2. A total of 30 measurement points on the irregular plate.

Figure 3 .
Figure 3. FE model of the irregular plate.
. The comparison was carried out by mapping the mode shapes of the FE Computer LMS Frontend z-axis excitation Soft springs z x y journal.ump.edu.my/ijame◄ model onto the corresponding EMA mode shapes.From Figure 5, there were no missed or swapped modes identified between the FE and EMA model.

Figure 5 .
Figure 5. FE and EMA mode shapes comparison.

Figure 5 .
Figure 5. Reduced model of the irregular plate.

Figure 6 .
Figure 6.Expanded and EMA mode shapes comparison.

Figure 10 (Figure 10 .
Figure 10(a) to 10(c) present the direct comparisons of the rotational FRFs between the expanded and EMA models.The rotational FRFs were obtained at points 1z1rx, 2z2rx and 1z2rx of both models.The patterns and amplitudes of the resonance and the anti-resonance estimated from the expanded model were almost similar to those obtained from the EMA model, particularly in the frequency range from 0 Hz to 1500 Hz.The resulting outcomes show that the expanded model is capable of accurately estimating rotational FRFs for lower modes.However, at high amplitudes of response (1500 Hz to 2000 Hz), the rotational FRFs estimated from the expanded model were low quality and deviated from the EMA ones.

Table 1 .
presents the calculated MAC values, damping and comparison of natural frequencies between the EMA and FE model of the irregular plate.It is worth mentioning that the measured damping values are used for FRF synthesis purposes.Comparison of the natural frequencies between the EMA and FE model of the irregular plate.

Table 3 .
MAC values and comparison of the natural frequencies between the full FE and Guyan model.

Table 4 presents
MAC values calculated from the comparison between the full FE model and Guyan FE model.On MAC analysis, low MAC values with 0.73 and 0.73 were observed in the 6 th and 7 th modes, respectively.In contrast, the other four modes (1 st , 2 nd , 4 th and 5 th modes) recorded high MAC values with approximately 1.0.

Table 4 .
MAC values between full FE (column) and Guyan FE (row) mode shapes