ZHOU Shuo, Lü Xiao-huan, WANG Xiao-xue. On the Construction of Stiffness Matrices With 3 Vector Pairs for Beam Vibration Systems[J]. Applied Mathematics and Mechanics, 2015, 36(3): 303-314. doi: 10.3879/j.issn.1000-0887.2015.03.008
 Citation: ZHOU Shuo, Lü Xiao-huan, WANG Xiao-xue. On the Construction of Stiffness Matrices With 3 Vector Pairs for Beam Vibration Systems[J]. Applied Mathematics and Mechanics, 2015, 36(3): 303-314.

# On the Construction of Stiffness Matrices With 3 Vector Pairs for Beam Vibration Systems

##### doi: 10.3879/j.issn.1000-0887.2015.03.008
Funds:  The National Natural Science Foundation of China（11072085）
• Rev Recd Date: 2014-12-14
• Publish Date: 2015-03-15
• The stiffness matrix of the discrete vibrating beam model is a real symmetric 5-diagonal matrix, so the inverse problem of the vibrating beam is substantially an inverse eigenvalue problem of the real symmetric 5-diagonal matrix. The existence and uniqueness conditions for the solution to the inverse problem of the real symmetric 5-diagonal matrix vector pair were given by means of the vector pairs and the Moore-Penrose generalized inverse, and the existence and uniqueness conditions for the solution to the inverse problem of the bi-symmetric 5-diagonal matrix vector pair were discussed in combination with the partitioned matrices. Then the inverse eigenvalue problem of the real symmetric 5-diagonal matrix, of which the sub-diagonal elements were negative and the rest elements were positive, was calculated. Since the data required for the construction of discrete beam models are available through measurements, the presented method is well suited to modal analysis, or analysis and design of system structures. Furthermore, the related numerical algorithms and numerical experiments illustrate the effectiveness of the method.
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