A receptance-based method for frequency assignment via coupling of subsystems

In the field of receptance-based inverse structural modification, the forms of modifications are often limited to rank-one modifications or single-DoF (degree of freedom) oscillators. These modifications are easy to implement and have been studied thoroughly in the past. They are able to produce satisfactory results for dynamic property assignments when the number of modifications is adequate and the desired dynamic properties are not far away from the original ones. Liu et al. [1] achieved an eigenstructure assignment for a 5-DoF system using additional multiple single-DoF oscillators. However, there are situations in which modifications are flexible bodies and have multiple DoFs in order to bring greater dynamical influences on the structure of interest; the connection for modifications might also couple with multiple DoFs of the structure. The overhanging mass modification on a helicopter tailcone presented by Mottershead et al. [2] was a good example. Apparently, this type of structural modification problems can be studied in the fashion of forward structural modification and using system matrices. Park and Park [3] studied the receptance-based structural modification of coupling of substructures using eigenvalue sensitivity and eigenvalue reanalysis. Still, studying the problem in the inverse structural modification fashion is beneficial and would be a more straightforward approach for engineers. Hence, this paper aims to address the inverse structural modification problem of using modifications with additional DoFs based on receptances.

There have not been many studies focusing on the aforementioned structural modification problem. Related publications are reviewed as follows. Ozguven [4] proposed a general method to determine the receptances of a modified structure using receptances of the original structure and the dynamic stiffness matrix of the modification. The modification can be local or has additional DoFs. This idea was extended by Hang et al. [5] to study the forward structural modification of distributed systems. Their study incorporated the condensation procedures to circumvent the lack of rotational FRFs in experimental measurements. However, a condensation procedure inherently induces approximation error and thus the usable frequency range can be limited [6]. Structural modifications with additional DoFs, involving beam and plate, on a 3D structure were studied by Hang et al. [7] using a similar method. Wang and Zhu [8] predicted the response of a modified system using the subsystems’ FRFs, response of the original system, and the dynamic stiffness matrix of the modifications. The response prediction for six structural modification scenarios without additional DoFs was presented. The FRF-based calculation of the nonlinear FRFs of a linear system with local nonlinear modifications was later studied by Kalaycıoğlu and Özgüven [9] using the describing function method. The nonlinearity in a modification can be caused by, for instance, the clearances, friction, and nonlinear stiffness. Moreover, the decoupling problem for nonlinear systems was recently studied by Kalaycıoğlu and Özgüven [10].

Another type of method that can determine the receptances of a modified structure is substructure coupling. There are mainly two approaches to a substructure coupling problem [11]: (i) modal substructuring or component mode synthesis (CMS) and (ii) FRF-based substructuring (FBS). The Craig-Bampton method [12], proposed in 1968, is the most well-known CMS technique. For instance, Lindberg et al. [13] combined the Craig-Bampton method with an undeformed coupling interface approach to describe the coupling of soft and stiff substructures, which could represent the rubber bushings in a vehicle’s suspension system. Although CMS was proposed earlier than FBS, FBS has recently attracted more attention than CMS since FBS is able to use measured FRFs directly, and CMS techniques encounter a number of difficulties in their implementation in practice, for instance, a sufficient number of modes of the substructures has to be considered in order to accurately approximate the motion of the coupled system and the issue of modal truncation has to be dealt with. FBS-based methods are widely applied in the noise, vibration, and harshness (NVH) problems in vehicles [14], and the receptance coupling method proposed by Jetmundsen et al. [15] is possibly the most common and widely implemented FBS method in the frequency domain. A well-known application of the method, which is the so-called receptance coupling substructure analysis (RCSA), has been applied to the prediction of chatter-free cutting conditions in milling machines [16, 17] and the identification of joint parameters [18, 19].

Another FBS-based method was proposed by De Klerk et al. [20] as the Lagrange multiplier frequency-based substructuring method (LMFBS). The coupled system could be assembled in a dual coupling manner in which the Lagrange multipliers were introduced to satisfy the force equilibrium between subsystems and solved with the associated displacement compatibility equations. Both Jetmundsen’s method and LMFBS can yield the same results; however, in LMFBS, some FRFs at the interface DoFs can be ignored so as to avoid using inaccessible or noisy FRFs and improve the results. In addition, LMFBS can be extended to the substructure decoupling problem. A detailed discussion on this issue was presented by Voormeeren and Rixen [21]. The influence of the uncertainties in the measurements of vibration time histories of substructures on interpreted FRFs was studied by Voormeeren et al. [22]. Statistical moment method was used to quantify the propagated uncertainties in the coupled system’s FRFs. It was also shown that the small uncertainties in substructures’ FRFs could result in large errors in the FRFs of lightly damped coupled systems.

Although the substructure coupling problem and the problem of parameter identification are close to the idea of structural modification, only few papers have reported the direct integration of them. Ram [23] determined the receptance at the connection interface of a coupled system (mass-spring-damper system) which consisted of two subsystems connected via a spring, a dashpot, or a mass. The result could be directly extended to a structural modification problem through solving the value of the basic connecting element, which was later implemented by Birchfield et al. [24] on a coupled simulated rotor model; however, the practical applications in both papers were restricted since only the basic elements were considered but the connecting elements can be rather complex in practice. Kyprianou et al. [25] assigned natural frequencies and antiresonance frequencies of a L-shaped structure through a straight beam. Their approach is receptance based, and thus, the numerical models of the structure are not required. The assignment was achieved by finding a solution to the multivariate polynomial equations in the design variables including the breadth, depth, and thickness of the added beam. Belotti and Richiedei [26] proposed an inverse structure modification method for eigenstructure assignment using the addition of auxiliary spring-mass systems whose topology was defined in advance. The method was based on homotopy optimization, which is an effective approach to solving non-convex problems via replacing the original objective function with a homotopy map. Although the system matrices were required in the proposed method, it showed great performances and flexibilities in the assignment problem.

In this paper, receptance coupling technique is implemented to characterize the receptance of a coupled system in terms of the receptances of the subsystems. The results can be used to assign natural frequencies for the coupled system via modifying the subsystem. The frequency assignment problem is cast as an optimization problem with an additional term penalizing large modifications. A number of numerical examples, which involve two stationary rotor systems and a joint, are given to demonstrate the validity of the frequency assignment. It is shown that the bending natural frequency and the torsional natural frequency of the coupled system can be simultaneously or separately assigned. Moreover, the problem of measuring the rotational receptances (in bending) is also dealt with in this paper. The method proposed by Tsai et al. [27] is extended to produce high-quality rotational receptance estimations. The proposed rotational receptance estimation technique can facilitate the implementation of the frequency assignment method to achieve bending and/or torsional natural frequency assignment in practice.

In this section, the classic receptance coupling technique proposed in [15] is first briefly introduced, and then it will be extended to predict the receptance functions of a coupled system consisting of three subsystems based on the receptances from the subsystems.

When applying receptance coupling technique to a coupled system, the conditions of displacement compatibility and force equilibrium must be satisfied at the connection interface; moreover, it is assumed that the subsystems are rigidly connected, i.e., there are no additional degrees of freedom (DoFs) at the connection interface. The classical receptance coupling technique considers two subsystems A and B rigidly connected through several DoFs (denoted as t) as shown in Fig. 1. The conditions at the interface can be written aswhich indicates that the force vector of the coupled system, \(\mathbf{f}_t \), at the interface is the summation of the force vectors of the subsystems, \(\hbox {f}_t^{\mathrm{A}} \) and \(\hbox {f}_t^{\mathrm{B}} \), and that the resultant displacement vector, \(\mathbf{u}_t \), is equal to the displacement vectors \(\hbox {u}_t^{\mathrm{A}} \) and \(\hbox {u}_t^{\mathrm{B}} \), of each subsystem. If the internal DoFs of Subsystems A and B of interest are denoted as i and j, the displacement vectors on the two subsystems can be represented as

If the goal is to find the receptance function \({\mathbf{H}}_{ij} \) of the coupled system, the relationship between \(\mathbf{u}_i^{\mathrm{A}} \) and \(\mathbf{f}_j \) or \(\mathbf{u}_j^{\mathrm{B}} \) and \(\mathbf{f}_i \) needs to be established when the reciprocity principle holds. Through Eqs. (1) and (2), it can be shown thatandSubstituting Eq. (3) into Eq. (4) results in\(\mathbf{u}_i^{\mathrm{A}}\) in Eq. (4) can be expressed explicitly in terms of forces \(\mathbf{f}_i ,\mathbf{f}_j ,\hbox {and } \mathbf{f}_t \) by substituting \(\mathbf{f}_t^{\mathrm{B}} \) using Eq. (5), which leads toIt is clear that \({\mathbf{H}}_{ij} \) can be obtained by assuming forces other than \(\mathbf{f}_j \) are zero, that is,Equation (7) can be further improved to require fewer matrix inversions by matrix operations, which results inIt indicates that the cross receptance of the coupled system can be estimated by the receptance functions of the subsystems; moreover, the point receptances of the coupled system can also be obtained, for instance, substituting \(\mathbf{f}_t^{\mathrm{B}} \) in Eq. (5) into \(\mathbf{u}_j^{\mathrm{B}} ={\mathbf{H}}_{jj}^{\mathrm{B}} \mathbf{f}_j +{\mathbf{H}}_{jt}^{\mathrm{B}} \mathbf{f}_t^{\mathrm{B}} \) givesFor the purpose of completeness, the generalized formula for receptance coupling [15], which includes all the DoFs of the coupled system, is well known as

The frequency assignment through coupling of subsystems is studied through numerical simulations in this section. The technique used for the frequency assignment is mainly based on Eq. (20). It should be noted that not every natural frequency can be assigned to the coupled structure as the structural modifications usually do not modify the whole structure but only a few design parameters. Moreover, the design parameters are usually confined by their design constraints. Therefore, the problem is herein treated as a multivariable optimization problem with inequality constraints.

In the following discussion, the coupled system under consideration consists of three subsystems, and it is assumed that one of the subsystems is modifiable. In practice, this kind of assembly scenario can be easily seen. If the modifiable subsystem is the subsystem connected to the other two subsystems and the receptances of the two subsystems at the interface are known, the receptance of the coupled system at the connection interface can be represented as a function of the design variable x and Laplace variable s shown as below:

The importance of including the rotational receptances has been demonstrated in Figs. 5 and 6. However, it is currently not possible to directly measure high-quality rotational receptances in practice due to the lack of instruments for moment excitation and measurement. As a result, indirect measurement techniques have been proposed. The method recently proposed by Tsai et al. [27] (Method 2 in the article) for estimating torsional receptances using a T-block attachment has shown better results than those reported in the literature. Generally speaking, torsional receptance is a rotational receptance but it is about the longitudinal (axial) direction; thus, in theory, the method can be applied to the measurements of rotational receptances in bending by changing the T-block’s orientation. The setup is illustrated in Fig. 12. In that case, the imparted translational force can yield moment excitation to the parent structure at the rotational DoF of interest in the x–z plane. Additionally, since only the receptances at the connection DoFs of the subsystems are required for the frequency assignment of the coupled system, the results obtained from the indirect measurement technique can produce the necessary information for the assignment problem.

The receptances in the x–z plane pertaining to \(\theta _y \) that cannot be conventionally measured are \(h_{x, \theta _y } , h_{\theta _y , x} \), and \( h_{\theta _y , \theta _y } \). In fact, \(h_{\theta _y , x} \) can be directly measured using rotational accelerometers and an impact hammer, but the rotational accelerometers require a flat mounting surface for optimal measurement performance. This suggests that the structure of interest might have to be machined or altered. For a shaft or a rotor, this kind of alteration might not be favorable as it could introduce asymmetry and increase costs; for that reason, \(h_{\theta _y , x} \) is treated as unmeasurable and will later be indirectly measured.

As an example, the receptances of Subsystem A alone at the interface (DoFs at node 8 in Fig. 4) are to be measured using the setup in Fig. 12. Subsystem A and the assembled structure are assumed to be lightly proportionally damped, and the damping of the T-block is excluded in the estimation process. The selections of the DoFs for response are DoFs 4, 8, and 21, and that for excitation are DoFs 4, 18, and 19 (see Fig. 12). The receptances \( h_{\theta _y , \theta _y } \) and \(h_{x, \theta _y } \) of Subsystem A at the connection end are estimated. Figure 13 gives the corresponding estimation results using Eq. (22) in [27] with noise level equal to 0% and 2.5%, respectively. It can be seen that the estimated FRFs agree well with the exact ones when noise is not present (see Fig. 13a). However, Fig. 13b suggests that the estimation is quite sensitive to noise, and the results are not satisfactory even when the rotational information (DoF 8) is included in the estimation process. It is also found that the noticeable fluctuation around 500 Hz is associated with a bending natural frequency of the assembled system (Subsystem A plus T-block, 496.84 Hz to be exact).

The results can be improved by measuring \(h_{x,x} \) and modifying the estimation equation. For the connection end, assuming that the motion in the z-axis direction is not coupled with the motion in the other directions, \({\mathbf{H}}_{\mathrm{cc}}^{\mathrm{A}} \) in Eq. (19) in [27] can be expressed explicitly and rewritten aswith the same definitions of matrices L and R as in [27]. Since the axial receptance \(h_{z,z} \) is not of interest, the last row and the last column in L and R can both be removed, leading towhere the remaining entries in L and R are arranged in the row vectors whose lengths are determined by the number of excitations. If \(h_{x,x} \) can be measured directly, then \(h_{x,\theta _y } \) and \(h_{\theta _y , \theta _y } \) can be estimated using the two equations in Eq. (25), respectively. The equations can be solved in a least-squares sense. For instance, the residual of the first equation can be written asNote that all the terms in Eq. (26) are complex for damped structures; thus, the real part and the imaginary part are solved separately. Considering the real part of each term is extracted, the residual squared can be expressed aswhere \(\mathbf{p}_1 ^{\mathrm{T}}=\left( {\mathbf{l}_1 ^{\mathrm{T}}-h_{x, x} \mathbf{r}_1 ^{\mathrm{T}}} \right) \). \(\partial \varepsilon /\partial h_{x, \theta _y } =0\) results in the following equation:The real part of \(h_{x,\theta _y } \) can now be obtained, and the imaginary part of \(h_{x,\theta _y } \) can be solved similarly. Once \(h_{x,\theta _y } \) is estimated, \(h_{\theta _y , \theta _y } \) can be estimated using the second equation in Eq. (25). If the principle of reciprocity holds, the equation can be written asin which \(h_{\theta _y ,\theta _y } \) can also be solved in the same fashion. Figure 14a gives the results using the modified equations. Compared with Fig. 13b, it is clear that the fluctuations due to the noise are greatly reduced. The noticeable fluctuation around the assembled system’s bending natural frequency disappears, and the variation in errors is reduced. However, the quality of the receptances is still not good enough for receptance-based structural modifications.

For this reason, the noise elimination technique proposed by Sanliturk and Cakar [28] is applied to filter out the noise in the FRFs. The number of singular values used to separate the meaningful data from noise is set to 20 for \(h_{\theta _y ,\theta _y } \) and 40 for \(h_{x,\theta _y } \), respectively. The corresponding filtered FRFs are given in Fig. 14b. It is clear that the noise is almost eliminated and that now the filtered FRFs match the exact FRFs quite well. It can be noted that the noise around the peaks of the FRF remains, which is because of the magnitude-dependent noise assumption. Fortunately, this type of noise may not be very common in reality. Therefore, it is believed that the proposed technique for rotational receptance estimation is able to produce high-quality FRFs for the structural modification problem.

The work presented in this paper deals with the problem of frequency assignment through coupling of subsystems. A technique that integrates receptance coupling method, structural modification, and optimization is presented. The receptance method is first extended to a coupled system of three subsystems, and a simple representation of the receptance functions at the interface of the coupled system is derived. The formulation shows that only the receptance functions of the subsystems at the connection coordinates are required to determine the receptance functions of the coupled system at the same coordinates. This relationship is further used as a basic formula for the objective function for structural modification.

The technique is applied to a lightly damped finite element model to verify its applicability. The model consists of three subsystems that are connected in series. It is assumed that the middle subsystem is the only undamped subsystem and is a circular beam whose diameter and length can be modified within certain ranges. Through modifying the undamped middle beam, the coupled system can thus possess some desired natural frequencies. The simulation results suggest that the current technique works well with the frequency assignment problems. Bending natural frequencies or torsional natural frequencies can be assigned to the coupled system either respectively or simultaneously. None of the simulation examples exceeds more than 1% of error between the desired natural frequencies and the achieved ones.

The rotational receptances are essential to the structural modification problem in the context. In the paper, an improvement is made to a known technique to improve the technique’s robustness on the rotational receptance estimation. The numerical simulation results also suggest that the technique is able to produce accurate estimations and is appropriate for the frequency assignment problem of concern.