Identification method of nonlinear structural impact excitation loads based on hybrid regularization strategy

Abstract

The present invention discloses a method for identifying impact excitation loads of nonlinear structures based on a hybrid regularization strategy, which is specifically implemented according to the following steps: first, obtaining a transfer matrix between the impact excitation position and the vibration response measurement point position; measuring the vibration response of the nonlinear mechanical structure under the impact excitation load to be identified; constructing a sparse characterization model; solving the constructed sparse characterization model using a convex optimization algorithm to obtain the expansion force acting on the nonlinear mechanical structure; extracting the impact excitation load and the nonlinear force peak information from the expansion force; constructing a nonlinear response transfer equation based on the obtained impact excitation load, solving the nonlinear response transfer equation using an l2 norm regularization method, and obtaining the slow-changing trend information of the nonlinear force. The present invention solves the problem of low recognition accuracy and unstable results caused by directly using the l1 norm sparse regularization method in the prior art.

Description

Technical Field

[0001] The invention belongs to the technical field of nonlinear structural load identification and vibration analysis, and relates to a nonlinear structural impact excitation load identification method based on a hybrid regularization strategy. Background Art

[0002] With the improvement of the performance requirements and structural complexity of mechanical equipment, especially for advanced rail transportation equipment, marine engineering equipment, aerospace equipment and other fields, the nonlinear factors in mechanical equipment are receiving more and more attention. In addition, impact excitation load is a kind of dynamic load commonly found in mechanical equipment. The impact excitation load acting on the key structure of mechanical equipment is a key indicator for optimizing the design, reliability analysis, vibration control and state evaluation of the structure. Without knowing the impact excitation load on the structure, the research on the above problems is somewhat blind. However, considering the limitations of sensor measurement technology and installation space, it is difficult to directly measure the impact excitation load on the structure through suitable sensors for most mechanical structures. Therefore, it is of great academic significance and engineering value to study the load identification method of nonlinear mechanical structures under impact excitation load. The research on load identification of mechanical structures mainly focuses on linear structures, that is, various regularization methods are used to solve the ill-posed problem of the transfer equation established in the linear framework, so as to obtain the load to be obtained by inverting the structural vibration response that is easy to measure.

[0003] At present, some scholars have gradually extended the traditional l2 norm regularization method to the identification of simple harmonic excitation loads of nonlinear structures. However, when the excitation load acts on a nonlinear structure, the nonlinear factors contained in the structure will generate nonlinear forces under the action of the nonlinear structural vibration response, so that the excitation load and the nonlinear force can be regarded as an extension force acting on the nonlinear structure. When the excitation load of the nonlinear structure is an impact excitation load, although the impact excitation load is sparse, the nonlinear force caused by the structural vibration response and nonlinear factors is usually not sparse, that is, the extension force acting on the nonlinear structure is not sparse. Therefore, it is impossible to directly use the l1 norm sparse regularization method to realize the load identification of nonlinear structures under impact excitation loads. Summary of the invention

[0004] The purpose of the present invention is to provide a nonlinear structural impact excitation load identification method based on a hybrid regularization strategy, which solves the problems of low identification accuracy and unstable results caused by directly using the l1 norm sparse regularization method in the prior art.

[0005] The technical solution adopted by the present invention is a nonlinear structural impact excitation load identification method based on a hybrid regularization strategy, which is specifically implemented according to the following steps:

[0006] Step 1, obtaining a transfer matrix between an impact excitation position and a vibration response measurement point position in a nonlinear mechanical structure;

[0007] Step 2, measuring the vibration response of the nonlinear mechanical structure under the impact excitation load to be identified;

[0008] Step 3, construct a sparse representation model;

[0009] Step 4, using a convex optimization algorithm to solve the sparse representation model constructed in step 3 to obtain the expansion force acting on the nonlinear mechanical structure;

[0010] Step 5, extracting the impact excitation load and nonlinear force peak information from the extension force;

[0011] Step 6: construct a nonlinear response transfer equation based on the impact excitation load obtained in step 5, solve the nonlinear response transfer equation through the l2 norm regularization method, and obtain the slow-changing trend information of the nonlinear force.

[0012] The present invention is also characterized in that

[0013] Step 1 is implemented according to the following steps:

[0014] Step 1.1, obtain the coefficient matrix A of the state space model between the impact excitation position and the vibration response measurement point position d , B d , C d and D d .

[0015] Step 1.2: Assemble the expansion force f according to the coefficient matrix obtained in step 1.1 ex (t) and the overall transfer matrix T between the vibration response measurement points n , expansion force f ex (t) includes the impact excitation load f(t) and the nonlinear force f nl (t), and from the overall transfer matrix T n Extract the transfer matrix T between the impact excitation position and the vibration response measurement point rf , and the transfer matrix T between the nonlinear force position and the vibration response measurement point rnl ;

[0016] In step 1.2, the extension force f ex (t) and the overall transfer matrix T between the vibration response measurement points n The assembly rules are as follows:

[0017]

[0018] Where N represents the total number of sampling points of the vibration response, and the transfer matrix T rfAccording to the spatial position of the impact excitation load f(t) in the extension force, the overall transfer matrix T is n Extracted from the transfer matrix T rnl According to the nonlinear force f nl (t) The spatial position of the expansion force is calculated column by column from the overall transfer matrix T n Extracted from

[0019] Impact excitation load f(t) and nonlinear force f nl The spatial position of (t) in the extension force is as follows:

[0020] f ex (t)={[f(0) f nl (0)] T [f(1) f nl (1)] T ...[f(N-1) f nl (N-1)] T} T (2)

[0021] In formula (2), the superscript T represents the transposition operation.

[0022] In step 2, the position of the vibration response measurement point is consistent with the position of the vibration response measurement point in step 1, and the vibration response of the nonlinear mechanical structure is measured by a vibration sensor.

[0023] In step 3, the sparse representation model is:

[0024]

[0025]

[0026] Where G(α) is the objective function, Y represents the vibration response of the measured nonlinear mechanical structure, λ is the sparse regularization parameter, and φ i (t) represents the cubic B-spline function, α i represents the sparse representation coefficient, Φ=[φ1,φ2,...,φ N ] represents the dictionary matrix composed of cubic B-spline functions, α=[α1,α2,...,α N ] T Represents the sparse representation coefficient to be solved.

[0027] In step 4, the convex optimization algorithm used is a two-step iterative threshold algorithm. The sparse representation model in step 3 is solved to obtain the sparse representation coefficient α, and then the relationship f is used to calculate the sparse representation coefficient α. ex (t) = Φα to obtain the expansion force f ex (t).

[0028] In step 5, according to the expansion force f identified in step 4 ex (t), combined with the impact excitation load and the spatial position of the nonlinear force in the extension force, the impact excitation load f(t) and the nonlinear force are obtained. The obtained impact excitation load f(t) is used as the identification result of the impact excitation load of the nonlinear mechanical structure, and the obtained nonlinear force As the identification result reflecting the nonlinear force peak information.

[0029] Step 6 is implemented according to the following steps:

[0030] Step 6.1: Extract the nonlinear response Y caused only by the nonlinear force from the measured vibration response of the nonlinear mechanical structure nl ;

[0031] Nonlinear response Y nl The extraction formula is:

[0032] Y nl =YT rf f nl (t) (5);

[0033] Step 6.2, determine the non-zero interval [j0,j s ];

[0034] In step 6.2, the nonlinear response Y nl The amplitude of the non-zero amplitude is determined in the nonlinear response position interval [j0,j s ];

[0035] Step 6.3, according to the non-zero interval [j0,j s ] for nonlinear response Y nl and the transfer matrix T rnl To make reductions;

[0036] Step 6.4, construct the nonlinear response transfer equation;

[0037] The constructed nonlinear force response transfer equation is:

[0038]

[0039] Among them, f nl_re (t) is the reduced nonlinear force, f nl_re (t) only includes the original nonlinear force f nl The j0th value of (t) to j s value.

[0040] Step 6.5, use the l2 norm Tikhonov regularization method to solve and obtain the nonlinear force slowly varying trend information.

[0041] In step 6.3, based on the non-zero interval [j0,j s ], only the nonlinear response Y is retained nl The j0th value to jth value s The values ​​are reduced to the nonlinear response vector Y nl_re , only the transfer matrix T between the nonlinear force and the vibration response measurement point is retained rnl The j0th column value to j s The column values ​​are reduced to the transfer matrix

[0042] In step 6.5, the nonlinear response transfer equation constructed in step 6.4 is solved using the l2 norm Tikhonov regularization method to obtain the reduced nonlinear force identification result Then according to the non-zero interval [j0,j s ] The complete nonlinear force is obtained by adding 0 elements As the identification result reflecting the nonlinear force slow-changing trend information, the identification result reflecting the nonlinear force peak information obtained in step 5 Combined with the impact excitation load, it fully reflects the complete information of the nonlinear force generated by the nonlinear mechanical structure under the impact excitation load.

[0043] In step 1.1, the vibration response measurement point positions include nonlinear positions, and the number of vibration response measurement points is equal to the sum of the number of nonlinear factors and the number of impact excitations. The coefficient matrix A of the state space model is d , B d , C d and D d It is obtained by combining vibration testing at low excitation levels with subspace identification methods, or by establishing a finite element model of the basic linear structure after removing nonlinear factors from the nonlinear mechanical structure.

[0044] The beneficial effects of the present invention are as follows: the present invention makes full use of the vibration characteristics of the nonlinear mechanical structure, combines the l1 norm sparse regularization method with the l2 norm Tikhonov regularization method, and solves the problems of low load identification accuracy and unstable results of the nonlinear mechanical structure under the impact excitation load; it can not only realize the identification of the impact excitation load of the nonlinear mechanical structure, but also accurately obtain the peak value information and slow-changing trend information of the nonlinear force, providing a basis for monitoring the state of the nonlinear factors of the structure; the model is reduced through the non-zero amplitude information of the nonlinear vibration response, which greatly reduces the scale of the transfer matrix inverse operation and has the advantage of high calculation efficiency; the present invention has good applicability and is not restricted by the nonlinear position, nonlinear type and impact excitation position. BRIEF DESCRIPTION OF THE DRAWINGS

[0045] Figure 1It is a flow chart of a method for identifying nonlinear structural impact excitation loads based on a hybrid regularization strategy of the present invention;

[0046] Figure 2 is a schematic diagram of the impact load identification result of a nonlinear cantilever beam system with a gap in one embodiment of the present invention;

[0047] Figure 3 is a schematic diagram of nonlinear force peak information identification results of a nonlinear cantilever beam system with a gap in one embodiment of the present invention;

[0048] Figure 4 It is a schematic diagram of the identification result of the nonlinear force slowly varying trend information of a nonlinear cantilever beam system with a gap in one embodiment of the present invention. DETAILED DESCRIPTION

[0049] The present invention is described in detail below with reference to the accompanying drawings and specific embodiments.

[0050] The flowchart of the method for identifying nonlinear structural impact excitation loads based on the hybrid regularization strategy of the present invention is as follows: Figure 1 As shown, the specific implementation steps are as follows:

[0051] Step 1, obtaining a transfer matrix between an impact excitation position and a vibration response measurement point position in a nonlinear mechanical structure;

[0052] Step 1 is implemented according to the following steps:

[0053] Step 1.1, obtain the coefficient matrix A of the state space model between the impact excitation position and the vibration response measurement point position d , B d , C d and D d .

[0054] In step 1.1, the vibration response measurement point positions include nonlinear positions, and the number of vibration response measurement points is equal to the sum of the number of nonlinear factors and the number of impact excitations. The coefficient matrix A of the state space model is d , B d , C d and D d It is obtained by combining vibration testing at low excitation levels with subspace identification methods, or by establishing a finite element model of the basic linear structure after removing nonlinear factors from the nonlinear mechanical structure.

[0055] Step 1.2: Assemble the expansion force f according to the coefficient matrix obtained in step 1.1 ex (t) and the overall transfer matrix T between the vibration response measurement points n , expansion force f ex (t) includes the impact excitation load f(t) and the nonlinear force f nl(t), and from the overall transfer matrix T n Extract the transfer matrix T between the impact excitation position and the vibration response measurement point rf , and the transfer matrix T between the nonlinear force position and the vibration response measurement point rnl ;

[0056] In step 1.2, the extension force f ex (t) and the overall transfer matrix T between the vibration response measurement points n The assembly rules are as follows:

[0057]

[0058] Where N represents the total number of sampling points of the vibration response, and the transfer matrix T rf According to the spatial position of the impact excitation load f(t) in the extension force, the overall transfer matrix T is n Extracted from the transfer matrix T rnl According to the nonlinear force f nl (t) The spatial position of the expansion force is calculated column by column from the overall transfer matrix T n Extracted from

[0059] Impact excitation load f(t) and nonlinear force f nl The spatial position of (t) in the extension force is as follows:

[0060] f ex (t)={[f(0) f nl (0)] T [f(1) f nl (1)] T ...[f(N-1) f nl (N-1)] T} T (2)

[0061] In formula (2), the superscript T represents the transposition operation.

[0062] Step 2, measuring the vibration response of the nonlinear mechanical structure under the impact excitation load to be identified;

[0063] In step 2, the position of the vibration response measurement point is consistent with the position of the vibration response measurement point in step 1, and the vibration response of the nonlinear mechanical structure is measured by a vibration sensor.

[0064] Step 3, construct a sparse representation model for identification of extension forces of nonlinear mechanical structures;

[0065] In step 3, the sparse representation model is:

[0066]

[0067]

[0068] Where G(α) is the objective function, Y represents the vibration response of the measured nonlinear mechanical structure, λ is the sparse regularization parameter, and φ i (t) represents the cubic B-spline function, α i represents the sparse representation coefficient, Φ=[φ1,φ2,...,φ N ] represents the dictionary matrix composed of cubic B-spline functions, α=[α1,α2,...,α N ] T Represents the sparse representation coefficient to be solved.

[0069] Step 4, using a convex optimization algorithm to solve the sparse representation model constructed in step 3 to obtain the expansion force acting on the nonlinear mechanical structure;

[0070] In step 4, the convex optimization algorithm used is a two-step iterative threshold algorithm. The sparse representation model in step 3 is solved to obtain the sparse representation coefficient α, and then the relationship f is used to calculate the sparse representation coefficient α. ex (t) = Φα to obtain the expansion force f ex (t).

[0071] Step 5, extracting the impact excitation load and nonlinear force peak information from the extension force;

[0072] In step 5, according to the expansion force f identified in step 4 ex (t), combined with the impact excitation load and the spatial position of the nonlinear force in the extension force, the impact excitation load f(t) and the nonlinear force are obtained. The obtained impact excitation load f(t) is used as the identification result of the impact excitation load of the nonlinear mechanical structure, and the obtained nonlinear force As the identification result reflecting the nonlinear force peak information.

[0073] Step 6: construct a nonlinear response transfer equation based on the impact excitation load obtained in step 5, solve the nonlinear response transfer equation through the l2 norm regularization method, and obtain the slow-changing trend information of the nonlinear force.

[0074] Step 6 is implemented according to the following steps:

[0075] Step 6.1: Extract the nonlinear response Y caused only by the nonlinear force from the measured vibration response of the nonlinear mechanical structure nl ;

[0076] Nonlinear response Y nl The extraction formula is:

[0077] Y nl =YT rff nl (t) (5);

[0078] Step 6.2, determine the non-zero interval [j0,j s ];

[0079] Through the nonlinear response Y nl The amplitude of the non-zero amplitude is determined in the nonlinear response position interval [j0,j s ].

[0080] Step 6.3, according to the non-zero interval [j0,j s ] for nonlinear response Y nl and the transfer matrix T rnl To make reductions;

[0081] According to the non-zero interval [j0,j s ], only the nonlinear response Y is retained nl The j0th value to jth value s The values ​​are reduced to the nonlinear response vector Y nl_re , only the transfer matrix T between the nonlinear force and the vibration response measurement point is retained rnl The j0th column value to j s The column values ​​are reduced to the transfer matrix

[0082] Step 6.4, construct the nonlinear response transfer equation;

[0083] The constructed nonlinear force response transmission equation is:

[0084]

[0085] Among them, f nl_re (t) is the reduced nonlinear force, f nl_re (t) only includes the original nonlinear force f nl The j0th value of (t) to j s value.

[0086] Step 6.5, using the l2 norm Tikhonov regularization method to solve and obtain the nonlinear force slowly varying trend information;

[0087] In step 6.5, the nonlinear response transfer equation constructed in step 6.4 is solved using the l2 norm Tikhonov regularization method to obtain the reduced nonlinear force identification result Then according to the non-zero interval [j0,j s ] The complete nonlinear force is obtained by adding 0 elements As the identification result reflecting the nonlinear force slow-changing trend information, the identification result reflecting the nonlinear force peak information obtained in step 5 Combined with the impact excitation load, it fully reflects the complete information of the nonlinear force generated by the nonlinear mechanical structure under the impact excitation load.

[0088] In the nonlinear structure impact excitation load identification method based on the hybrid regularization strategy of the present invention, the state space model coefficient matrix of the nonlinear mechanical structure is obtained by combining vibration testing at a low excitation level with a subspace identification method or by establishing a finite element model of the basic linear structure after removing the nonlinear factors of the nonlinear mechanical structure, and the transfer matrix between the excitation position and the vibration response measurement point position is assembled, and the transfer matrix between the impact excitation load and the vibration response measurement point, as well as the transfer matrix between the nonlinear force position and the vibration response measurement point are extracted.

[0089] The vibration response of the nonlinear mechanical structure under the impact excitation load to be identified is measured by using sensors, a sparse characterization model is constructed, and the extension force is obtained through a two-step iterative threshold algorithm, from which the impact excitation load and nonlinear force are extracted. The extracted impact excitation load is used as the identification result of the impact load of the nonlinear mechanical structure, and the extracted nonlinear force is used as the identification result reflecting the peak information of the nonlinear force. The obtained impact excitation load is used to calculate the nonlinear response caused only by the nonlinear force, and the nonlinear response transfer equation is constructed in combination with the nonzero interval of the nonlinear response. The identification result reflecting the slow-changing trend information of the nonlinear force is obtained through the Tikhonov regularization method.

[0090] Example

[0091] The nonlinear structure impact excitation load identification method based on the hybrid regularization strategy of the present invention is used for simulation verification. Taking the cantilever beam model with gap nonlinearity as an example, the impact excitation load amplitude and loading time are 30N and 0.2s, the cantilever beam length is l=1m, and the impact excitation load action position is l f =0.2m, gap nonlinear action position l c =0.8m. Through the nonlinear structure impact excitation load identification method based on the hybrid regularization strategy of the present invention, the impact excitation load identification result of the nonlinear structure is as follows: Figure 2 As shown in FIG. 1 , the impact excitation load identification value approaches 0 in the impact non-loading area and is close to the theoretical value of the impact excitation load in the impact loading area. Therefore, the present invention can accurately realize the identification of the impact excitation load of the nonlinear mechanical structure. The peak information identification result of the gap nonlinear force is shown in FIG. Figure 3 As shown, from Figure 3 It can be seen from the local enlarged diagram that the peak value of the nonlinear force identification value is close to the corresponding theoretical value, but there is a difference in the part where the nonlinear force curve changes slowly. The slow-changing trend information of the gap nonlinear force, such as Figure 4 As shown, Figure 4The results given show that in the part where the nonlinear force curve changes slowly, the identification results are in good agreement with the theoretical results. Figure 3 And reflect the nonlinear force slowly changing trend information Figure 4 To achieve accurate observation of complete information of nonlinear force.

[0092] The present invention makes full use of the vibration characteristics of nonlinear mechanical structures, combines the L1 sparse regularization method with the L2 regularization method, and identifies the impact excitation load and nonlinear force of the nonlinear mechanical structure under the impact excitation load. The identification result is accurate, which can not only realize the identification of the impact excitation load of the nonlinear mechanical structure, but also accurately obtain the peak information and slow-changing trend information of the nonlinear force, which is helpful to understand the vibration generation and transmission mechanism of the nonlinear mechanical structure, and lays a foundation for the optimal design and state monitoring of the mechanical structure.

Claims

1. A nonlinear structural impact excitation load identification method based on a hybrid regularization strategy, characterized in that: Follow the steps below to implement it: Step 1, obtaining a transfer matrix between an impact excitation position and a vibration response measurement point position in a nonlinear mechanical structure; Step 2, measuring the vibration response of the nonlinear mechanical structure under the impact excitation load to be identified; Step 3, construct a sparse representation model; Step 4, using a convex optimization algorithm to solve the sparse representation model constructed in step 3 to obtain the expansion force acting on the nonlinear mechanical structure; Step 5, extracting the impact excitation load and nonlinear force peak information from the extension force; Step 6, constructing a nonlinear response transfer equation based on the impact excitation load obtained in step 5, solving the nonlinear response transfer equation by using the l2 norm regularization method to obtain the slow-changing trend information of the nonlinear force; The step 1 is specifically implemented according to the following steps: Step 1.1, obtain the coefficient matrix of the state space model between the impact excitation position and the vibration response measurement point position , , and ; Step 1.2: Assemble the expansion force according to the coefficient matrix obtained in step 1.1 The overall transfer matrix between the vibration response measurement points , expansion capability Including shock excitation loads and nonlinear forces , and from the overall transfer matrix Extract the transfer matrix between the impact excitation position and the vibration response measurement point , and the transfer matrix between the nonlinear force position and the vibration response measurement point ; In step 1.2, the expansion force The overall transfer matrix between the vibration response measurement points The assembly rules are as follows: （1） in, N Represents the total number of sampling points of the vibration response, the transfer matrix Based on impact excitation load The spatial positions in the expansion forces are obtained column by column from the overall transfer matrix The transfer matrix is ​​extracted from Based on nonlinear force The spatial positions in the expansion forces are obtained column by column from the overall transfer matrix Extracted from Impact excitation load and nonlinear forces The spatial position in the extension force is as follows: （2） In formula (2), the superscript T represents the transposition operation; In step 2, the position of the vibration response measuring point is consistent with the position of the vibration response measuring point in step 1, and the vibration response of the nonlinear mechanical structure is measured by a vibration sensor; In step 3, the sparse representation model is: （3） （4） in, is the objective function, represents the measured vibration response of a nonlinear mechanical structure, is the sparse regularization parameter, represents the cubic B-spline function, represents the sparse representation coefficient, represents the dictionary matrix composed of cubic B-spline functions, represents the sparse representation coefficient to be solved; In step 4, the convex optimization algorithm used is a two-step iterative threshold algorithm, and the sparse representation model in step 3 is solved to obtain the sparse representation coefficient , and then through the relationship Get expansion power ; In step 5, according to the expansion force identified in step 4 , combined with the impact excitation load and the spatial position of the nonlinear force in the extension force, the impact excitation load is obtained and nonlinear forces , the impact excitation load obtained As a result of the identification of the impact excitation load on the nonlinear mechanical structure, the nonlinear force As the identification result reflecting the peak information of nonlinear force; The step 6 is specifically implemented according to the following steps: Step 6.1: Extract the nonlinear response caused only by the nonlinear force from the measured vibration response of the nonlinear mechanical structure ; Nonlinear response The extraction formula is: （5）； Step 6.2, determine the non-zero interval of the nonlinear response [ j 0, j s ]; In step 6.2, the nonlinear response The amplitude of the non-zero amplitude is determined in the nonlinear response. j 0, j s ]; Step 6.3, according to the non-zero interval [ j 0, j s ] for nonlinear response and the transfer matrix To reduce; Step 6.4, construct the nonlinear response transfer equation; The constructed nonlinear force response transmission equation is: （6）； in, is the reduced nonlinear force, Only the original nonlinear forces are included No. j 0 values ​​to j s values; Step 6.5, using the l2 norm Tikhonov regularization method to solve and obtain the nonlinear force slowly varying trend information; In step 6.3, according to the non-zero interval [ j 0, j s ], only the nonlinear response is retained No. j 0 values ​​to j s The values ​​are reduced to the nonlinear response vector , only the transfer matrix between the nonlinear force and the vibration response measurement points is retained No. j 0 column value to j s The column values ​​are reduced to the transfer matrix ; In step 6.5, the nonlinear response transfer equation constructed in step 6.4 is solved using the l2 norm Tikhonov regularization method to obtain the reduced nonlinear force identification result , and then according to the non-zero interval [ j 0, j s ] The complete nonlinear force is obtained by adding 0 elements , As the identification result reflecting the nonlinear force slow-changing trend information, the identification result reflecting the nonlinear force peak information obtained in step 5 Combined with the impact excitation load, it fully reflects the complete information of the nonlinear force generated by the nonlinear mechanical structure under the impact excitation load.

2. The method for identifying nonlinear structural impact excitation loads based on a hybrid regularization strategy according to claim 1 is characterized in that: In step 1.1, the vibration response measurement point positions include nonlinear positions, the number of vibration response measurement points is equal to the sum of the number of nonlinear factors and the number of impact excitations, and the coefficient matrix of the state space model is , , and It is obtained by combining vibration testing at low excitation levels with subspace identification methods, or by establishing a finite element model of the basic linear structure after removing nonlinear factors from the nonlinear mechanical structure.

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