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

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## Abstract

This paper presents a theoretical study of the frequency assignment problem of a coupled system via structural modification of one of its subsystems. It deals with the issue in which the available modifications are not simple; for example, they are not point masses, grounded springs, or spring-mass oscillators. The proposed technique is derived based on receptance coupling technique and formulated as an optimization problem. Only a few receptances at the connection ends of each subsystem are required in the structural modification process. The applicability of the technique is demonstrated on a simulated rotor system. The results show that both bending natural frequencies and torsional natural frequencies can be assigned using a modifiable joint, either separately or simultaneously. In addition, an extension is made on a previously proposed torsional receptance measurement technique to estimate the rotational receptance in bending. Numerical simulations suggest that the extended technique is able to produce accurate estimations and thus is appropriate for this frequency assignment problem of concern.

## 1 Introduction

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.

## 2 Receptance coupling technique

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.

*t*) as shown in Fig. 1. The conditions at the interface can be written as

*i*and

*j*, the displacement vectors on the two subsystems can be represented as

### 2.1 Receptance coupling on three subsystems

*i*is on Subsystem A and

*k*is on Subsystem C (Case 1), and the other is to find the receptances at the connection interface (Case 2), i.e., DoFs

*t*and

*p*. It will be shown that both cases only require partial information from subsystems to make the estimation.

*Case 1*

*l*denotes any internal DoFs in Subsystem D, the arrangement of DoFs in the coupled system can be illustrated by Fig. 3.

*l*is equal to DoFs

*i*in Fig. 2, \({\mathbf{H}}_{lp}^{\mathrm{D}} \) and \({\mathbf{H}}_{pp}^{\mathrm{D}} \) can be rewritten by the following equations:

*Case 2*

## 3 Numerical simulation

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.

**x**and Laplace variable

*s*shown as below:

### 3.1 Numerical model

*x*–

*z*plane. Each node contains three DoFs including one translational DoF in the

*x*direction, one rotational DoF about the

*y*-axis, and one torsional DoF about the

*z*-axis. The material properties of the shafts, the disks, and the bearing stiffness are given in Table 1, and the parameters of the systems are listed in Table 2. The shafts in Subsystems A and C are solid circular shafts and are mostly 2 cm in diameter except that a section in the middle of the shaft in Subsystem C is 3 cm in diameter. By default, the diameter of Subsystem B is 3 cm and the total length is 5 cm. The first four undamped natural frequencies of the corresponding assembled system are listed in Table 3.

System properties

Young’s modulus (GPa) | 210 | Transverse bearing stiffness (N/m) | \(10^{7}\) |
---|---|---|---|

Poisson ratio | 0.33 | Rotational bearing stiffness (Nm/rad) | 10 |

Density (\(\hbox {kg}/\hbox {m}^{3})\) | 7850 |

System parameters

Parameter | Value (cm) | Parameter | Value (cm) | Disks | (OD, ID, thickness) (cm) |
---|---|---|---|---|---|

\(l_{1,2} \) | 5 | \(l_{7,8} \) | 2 | Node 2 | (8, 2, 2) |

\(l_{2,3} \) | 5 | \(l_{8-10} \) | 5 (default) | Node 4 | (5, 2, 2) |

\(l_{3,4} \) | 5 | \(l_{10,11} \) | 2 | Node 6 | (5, 2, 2) |

\(l_{4,5} \) | 5 | \(l_{11,12} \) | 5 | Node13 | (6, 2, 2) |

\(l_{5,6} \) | 5 | \(l_{12,13} \) | 3 | ||

\(l_{6,7} \) | 5 |

Undamped natural frequencies of the original assembled system

Mode number | 1 | 2 | 3 (torsional) | 4 |
---|---|---|---|---|

| 587.5 | 617.2 | 701.1 | 774.3 |

### 3.2 Bending frequency assignment

*r*th desired frequency. In addition, the parameter \(\beta \) controls the trade-off between the cost of carrying out large modifications and the cost of the frequency assignment (the determinant function). A numerical example is given below to first demonstrate the bending natural frequency assignment of a coupled system by coupling of subsystems.

**x**that minimizes Eq. (22) given the desired frequency so that \({s}=\left( {2{\uppi }\times 575 } \right) {i}\). The constraints of the design variables in Subsystem B are

Natural frequencies of the modified coupled system

Mode number | 1 (achieved) | 2 (torsional) | 3 | 4 |
---|---|---|---|---|

| 574.96 | 586.38 | 599.82 | 634.39 |

### 3.3 Torsional frequency assignment

*i*(in Subsystem A) to DoF

*k*(in Subsystem C) of the coupled system. The denominator can be rearranged in the form of \(\hbox {det}\left( {{\mathbf{H}}_{\mathrm{AC}} \left( s \right) +{\mathbf{H}}_{\mathrm{B}} \left( s \right) } \right) \) with the same definitions of the two matrices. Thus, the objective function presented in the bending natural frequency assignment problem can also be applied to the torsional frequency assignment problem.

Natural frequencies of the modified coupled system

Mode number | 1 | 2 | 3 (torsional) | 4 |
---|---|---|---|---|

| 575.70 | 617.29 | 714.91 | 809.32 |

### 3.4 Two-frequency assignment

The two-frequency assignment problem is rather demanding under the current context as there are only two design variables. Since it has been demonstrated that either a bending natural frequency or a torsional natural frequency can be assigned using the same objective function, simultaneously assigning both frequencies to the coupled system is thus possible.

Natural frequencies of the modified coupled system

Mode number | 1 | 2 | 3 (torsional) | 4 |
---|---|---|---|---|

| 598.83 | 616.61 | 669.93 | 695.94 |

## 4 Estimation of rotational receptances

*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.

*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 as

**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 to

**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 as

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.

## 5 Conclusions

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.

## Notes

### Acknowledgements

The first author is grateful for the support of the UoL-NTHU Dual PhD Scholarship.

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