## 1 Introduction  
In recent years, with the improvement of ship power and the widespread use of large-scale and complex shafting systems, the vibration problem of shafting systems becomes more and more serious. The torsional vibration of a shafting system is one of the main causes of shafting fracture and other faults, which has attracted wide attention. The dynamic performance of torsional vibration is one of the main aspects to guarantee the productivity and safety of machines and structures, and thus has great economic and social significance. However, there are many situations in which the structural design cannot meet the desired structural dynamic characteristics so that some accidents happen in serious cases, especially shaft breakage. Structural modifications (SMs) are known to be a very important technique to improve the dynamic characteristics of a structure in the form of fundamental structural parameters (such as mass, stiffness, and damping) to avoid resonance or the creation of a node on the system at a certain frequency. Additionally, SMs are also a procedure aimed at identifying the changes needed and determining the values of the physical parameters of a structure to achieve desirable dynamic characteristics (usually modal properties such as natural frequencies, mode shapes,and frequency response). Several theories and many algorithms have been established to solve SMs as a direct problem and as an inverse problem. Early studies about direct structural modification are mainly based on small modifications, localized modifications, and modal approximation [1]. The first useful method for structural modification was formulated and broaden by many researchers [2], which was based on a matrix perturbation approach to get an approximate solution for the modal properties of the modified structure. However, the limitation of the perturbation approach is that the modification should be small in order to get an accurate solution. Other methods such as the analytical approach of structural modification by adding different mass and stiffness attachments [3; 4; 5], the Rayleigh quotient method [6], and the sensitivity analysis method [7; 8] were applied as useful tools for structural modification. The inverse problem, being more complex, tries to identify, in the frame of a given set of possible locations, the most appropriate modifications required to obtain the desired dynamic behavior [9]. One major objective of doing that is to keep the frequency of excitation away from a natural frequency so that a large structural response can be prevented. The methods for inverse structural modifications are based on physical model parameters [10; 11; 12] (such as the mass matrix, stiffness matrix, and damping matrix) and experimental modal analysis [13; 14; 15; 16] (e.g., natural frequencies and modal shapes derived from modal testing or directly by the frequency response function) by either passive procedures, active procedures, or combining the above two procedures in a hybrid approach. There are large numbers of published papers in the literature describing the study of inverse problems, and therefore only a brief introduction to some major ones is given as follows. [PERSON] et al. [17] studied the problem of natural frequency assignment for mass-chain systems with inertias to determine whether an arbitrary set of positive numbers can be assigned as the natural frequencies of a chain of \(n\) masses. The solution and solvability condition for the inverse problem of linear undamped gyroscopic systems were presented by [PERSON] and [PERSON] [18], and the results were applied to model updating. A geared rotor-bearing system was designed and tested by [PERSON] et al. [19] to validate the method for solving the inverse structural modification using the measured receptances. [PERSON] et al. presented the strategy for the torsional vibration control of shaft systems based on the measured torsional receptances of the system [20].