Damage can be identified using generalized flexibility matrix based methods, by using the first natural frequency and the corresponding mode shape. However, the first mode is not always appropriate to be used in damage detection. The contact interface of rodfastenedrotor may be partially separated under bending moment which decreases the flexural stiffness of the rotor. The bending moment on the interface varies as rotating speed changes, so that the first and secondmodal parameters obtained are corresponding to different damage scenarios. In this paper, a structural damage detection method requiring single nonfirst mode is proposed. Firstly, the system is updated via restricting the first few mode shapes. The mass matrix, stiffness matrix, and modal parameters of the updated system are derived. Then, the generalized flexibility matrix of the updated system is obtained, and its changes and sensitivity to damage are derived. The changes and sensitivity are used to calculate the location and severity of damage. Finally, this method is tested through numerical means on a cantilever beam and a rodfastenedrotor with different damage scenarios when only the second mode is available. The results indicate that the proposed method can effectively identify single, double, and multiple damage using single nonfirst mode.
Damage in a structure produces variations in its geometric and physical properties, which can result in changes in its natural frequencies and mode shapes. In the last years, several researchers have developed many damage detection methods based on dynamic parameters. Fan and Qiao [
Although the generalized flexibility matrix based damage detection approach can precisely detect the location and severity of damage by using only the first natural frequency and the corresponding mode shape, there are still many limitations in these methods. One limitation lies in the damage detection of rodfastenedrotor of heavy duty gas turbine. The flexural stiffness of the interface decreases when some zones of the contact interface are separated with bending moment on the rotor [
In this paper, a structural damage detection method based on changes in the flexibility matrix only using single natural frequency and the corresponding mode shape is presented. Firstly, restricted by the first several mode shapes, the system is updated. The flexibility matrix of updated system can be obtained by using nonfirstmodal parameters of original system. Then, sensitivity of flexibility of the updated system to damage is derived. Taking advantage of generalized flexibility matrix, which can considerably reduce the error caused by truncating higherorder modal parameters, the location and severity of the damage are calculated. Finally, two numerical examples for a cantilever beam and a rodfastenedrotor are used to illustrate the effectiveness of the proposed method, when only the second natural frequency and the corresponding mode shape are available.
The differential equation governing the free vibration of a linear, undamped structural system can be expressed as
Leftmultiplying (
The complete mode shapes are difficult to obtain, particularly when a limited number of sensors are available. However, incomplete mode shape data can be expanded to complete mode shapes by mode shape expansion technique. The expansion method in [
In this method, only the decrease in structure stiffness due to damage is considered. Changes in mass property are ignored. The damage parameters are denoted by
According to the definition of flexibility and stiffness matrix, they satisfy the following relationship:
Postmultiplying (
In order to reduce the error result from truncating higherorder modes, generalized flexibility matrix
The generalized flexibility matrix for the updated system can also be approximately determined by using its first frequency
In order to verify the effectiveness of the proposed method, two numerical examples are considered. The first numerical example is a cantilever beam, and the second one is a rodfastenedrotor considering partial separation of interface.
A twodimensional cantilever beam with a rectangular section, as shown in Figure
A cantilever beam.
When the first mode is unavailable, location and severity of damage can be obtained by using the second natural frequency and the corresponding mode shape with the proposed method in this paper. Parameter
The results are also compared with the results obtained by using the method in [
Figure
Damage detection by only using the secondmodal parameters for damage case 1: (a) and (b) are the results by using the proposed method in this paper and the method in [
Similarly, Figure
Damage detection by only using the secondmodal parameters for damage case 2: (a) and (b) are the results by using the proposed method in this paper and the method in [
To consider effect of the measured noise of frequency and mode shape on accuracy of the proposed method, 1% and 5% random noise are added in the frequency and mode shape for damage detection, respectively [
Damage detection by only using the secondmodal parameters for damage (a) case 1 and (b) case 2, when 1% and 5% random noise are added in the frequencies and mode shapes, respectively.
To investigate effect of using incomplete mode shapes on accuracy of the proposed method, 80 and 60 nodal displacements of the complete second mode shape are used to detect the damage. The noise effect is also considered at the same time. Figure
Damage detection by using the second natural frequency and 80 nodal displacements of the corresponding mode shape: (a) case 1; (b) case 2.
Damage detection by using the second natural frequency and 60 nodal displacements of the corresponding mode shape: (a) case 1; (b) case 2.
The rodfastenedrotors are commonly used in heavy duty gas turbines and aero engines. As shown in Figure
Schematic diagram of heavy duty gas turbine and node dividing.
Detection of partial separation of contact interface for a rodfastenedrotor in heavy duty gas turbine is investigated in this paper. The beam elements are used to model the rodfastenedrotor. The total number of elements is 37, as shown in Figure
Location and extent of multiple damage of rodfastenedrotor.
Element number  10  11  12  13 


Damage extent  5.2%  16.2%  11.9%  2.2% 
Bending moment on each interface of the rodfastenedrotor at the second critical speed.
The location and extent of damage are detected with the method proposed in this paper and the result is shown in Figure
Damage detection by only using the secondmodal parameter of the rodfastenedrotor.
From the results obtained above, the proposed method can effectively identify single, double, and multiple damage without the first frequency and the corresponding mode shape for numerical examples.
The firstmodal parameter of system plays the most important role in its flexibility matrix; the original flexibility method cannot be used to detect damage when the firstmodal parameter is unavailable. A damage detection method based on flexibility change by using single nonfirst mode is presented. The system is updated via restricting the first few mode shapes. The mass matrix, stiffness matrix, and modal parameters of the updated system are obtained. Then sensitivity of the new flexibility matrix to damage is derived. The damage extent of each element can be calculated by solving a linear equation. This method is tested through numerical means on a cantilever beam and a rodfastenedrotor with different damage scenarios, when only the second natural frequency and mode shape are available. The results of the numerical examples indicate that the proposed method can effectively identify single, double, and multiple damage.
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work was supported by the Natural Science Foundation of China (no. 51275385) and Major State Basic Research Development Program of China (no. 2011CB706505).