Nonlinear Dynamics

, Volume 96, Issue 2, pp 1115–1131 | Cite as

Transient dynamic analysis of cracked structures with multiple contact pairs using generalized HSNC

  • Meng-Hsuan Tien
  • Kiran D’SouzaEmail author
Original Paper


The development of efficient computational methods for cracked structures is critical in the fields of civil, mechanical, and aerospace engineering since the influence of cracks on structural dynamics can play an important role in design, prognosis, and health monitoring. The nonlinearity caused by the intermittent contact on the crack surfaces typically excludes the use of fast linear methods, without which the computation of the dynamics of complex cracked structures becomes very challenging. In this paper, an efficient computational scheme for predicting both the transient and steady-state responses of cracked structures with multiple contact pairs is introduced. The new algorithm is referred to as the generalized hybrid symbolic–numeric computational (HSNC) method. The generalized HSNC method extends the original HSNC method, which was recently developed for bilinear systems, to general piecewise-linear nonlinear systems. This work also combines the HSNC with the \(\hbox {X}-\hbox {X}_{r}\) method, a reduced-order modeling technique for cracked structures, to efficiently predict the dynamics of complex structures with cracks. The generalized HSNC approach is based on the idea that the nonlinear response of a cracked structure with multiple contact pairs can be obtained by combining linear responses of the system in each of its linear states. These linear responses can be symbolically expressed as functions of the initial conditions at starting time points in each time range where the system behaves linearly. The transition time where the system switches from one linear state to another is found using a nonlinear optimization solver with the initial values provided by an incremental search process. The method is able to individually track status of each contact pair; therefore, it can be used to predict the dynamics of the system when the crack surfaces are not completely open or closed. Moreover, both the transient and steady-state responses of complex cracked structures under various forcing conditions can be captured by the new method. The generalized HSNC method provides a flexible computational framework that is several orders of magnitude faster than traditional numerical integration methods. The dynamics of a spring–mass system and cantilever beam models that contain one crack and multiple cracks are investigated using the proposed method.


Piecewise-linear nonlinearity Cracked structure Reduced-order modeling Hybrid symbolic–numeric computation 


Compliance with ethical standards

Conflict of interest

The authors declared that they have no conflict of interest.


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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Department of Mechanical and Aerospace EngineeringThe Ohio State UniversityColumbusUSA

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