A method of mode order determination for engineering structural modal identification

The procedures of the mode order determination method for the modal identification of engineering structures are as follows:

• Step 1:
• Sample and process the impulse response;
• The structural impulse responses yk are collected;
• The Hankel matrix and H(k−1) and H(k) are built by yk:
• H
• 
• (
• k
• )
• =
• (
• y
• k
• y
• k
• +
• 1
• …
• y
• k
• +
• cH
• -
• 1
• y
• k
• +
• 1
• y
• k
• +
• 2
• …
• y
• k
• +
• cH
• …
• …
• …
• …
• y
• k
• +
• rH
• -
• 1
• y
• k
• +
• rH
• …
• y
• k
• +
• rH
• +
• cH
• -
• 2
• )
• where k+i represents the k+i time point;
• the number from k to k+rH+cH−2 is the number of time points for the time history;
• H(k−1) can be obtained by replacing k with k−1;
• H(k−1) is then decomposed by singular matrix decomposition as follows:
• H(k−1)=UΓ2VT
• where Γ is singular value matrix;
• U and V are unitary matrix;
• Step 2:
• obtain the modal shape matrix
• The rank cH of singular matrix is assumed to be the structural mode order;
• Then the eigenvalue λj can be obtained by eigensystem realization algorithm;
• The relation equation between modal responses and structural responses is built using N eigenvalues, and the modal shape matrix Φj is obtained as follows:
• (
• Φ
• 1
• Φ
• 1
• …
• Φ
• N
• )
• =
• (
• y
• 1
• y
• 2
• …
• y
• p
• )
• 
• (
• λ
• 1
• λ
• 1
• 2
• …
• λ
• 1
• p
• λ
• 2
• λ
• 2
• 2
• …
• λ
• 2
• p
• ⋮
• ⋮
• ⋱
• ⋮
• λ
• N
• λ
• N
• 2
• …
• λ
• N
• p
• )
• +
• where “+” is the generalized inverse;
• p=rH+cH−1 is the pth time point;
• Step 3:
• Obtain the modal response for the jth mode and the square mean root of the modal response for the jth mode;
• The jth modal response Yp,j is expressed as follows:
• Yp,j=(y1,jy2,j . . . yp,j)=Φj(λjλj2 . . . λjp)
• The expression for the scalar, i.e. the square mean root of the jth modal response, is:
• εj=sqrt(Yp,jYp,jH)
• Step 4:
• Obtain the mode order;
• The jth MRCI is obtained by the summation of the scalars for all degree-of-freedom as follow:
• MRCI
• 
• (
• j
• )
• =
• ∑
• r
• =
• 1
• m
• 
• ɛ
• j
• 
• (
• r
• )
• where r denotes the number of degree-of-freedom;
• The relation map between mode order and MRCI is drawn, where the horizontal axis denotes the mode order and the vertical axis represents the normalized MRCI by divided by the maximization of MRCI;
• The mode order is determined by the obvious gap between two adjacent MRCI.