Several structures are completely or partially manufactured in a factory and then transported to the final situation where they are going to be located. An accurate methodology to check the serviceability of the structure in the factory, previously to the transportation and final assembly, will diminish significantly the costs of validation of its dynamic behavior. The structural dynamic modification (SDM) can be used to predict the modal parameters of a supported structure from the experimental modal parameters corresponding to the same structure but tested in a configuration easy to reproduce in a factory, such as the freefree condition. However, the accuracy obtained with this technique depends on how well the boundary conditions modelled with the SDM replicate the real support conditions. In this paper the SDM theory is used to predict the modal parameters of a pinnedfree beam, a cantilever beam, and a 3D steel beam structure from the experimental natural frequencies and mode shapes of the same structures tested in freefree configuration. The predictions provided by the SDM theory are validated by operational modal testing on the supported structures. It is shown how the aforementioned boundary conditions can be modelled with the SDM, and the accuracy provided by the technique is investigated.
Several types of structures are usually assembled in a factory or other manufacturing site and then transported to the construction site where the structure is to be located. The vibration serviceability state limit is one of the criteria that must be considered in the dynamic design of these structures in order to avoid resonances, which could cause structural damages or affect the comfort of the users [
The analytical and numerical predictions can be validated by static tests (loading tests) or modal tests performed once the structure is fully assembled. This methodology requires moving personnel and equipment to the place where the structure is located in order to perform the tests which increases significantly the financing costs of the project if the structure, once in operation, needs some kind of reinforcements or repairs to be undertaken in order to fulfil the standards [
The structural dynamic modification theory (SDM) [
In this paper, a methodology to predict the modal parameters of a structure, using the modal parameters corresponding to modal tests performed in the factory and the SDM theory, is presented. The technique consists of two steps. Firstly, the structure is tested using boundary conditions easy to reproduce in the factory/lab, such as the freefree condition (i.e., hanging the structure on two bridge cranes by means of flexible elements). In the second step, the natural frequencies and mode shapes of the structure with the new boundary conditions are predicted with SDM. In this case, the SDM consists of introducing the boundary conditions expected for the structure in operation and no mass or stiffness changes have to be considered. This means that the accuracy provided by the methodology only depends on how well the boundary conditions are modelled and the test setup and the identification techniques used in the modal tests.
The results provided by the methodology can be used to validate the structural design comparing the predictions obtained with the SDM with those provided by a numerical or analytical model.
In this paper, it is explained how to apply the methodology and the accuracy achieved with the SMD in simple structures with pinned and clamped boundary conditions is investigated. The modal parameters of a pinnedfree beam, a cantilever beam, and a 3D beam structure are predicted using the SDM, and the modal parameters corresponding to the same structures are tested in freefree boundary conditions. The predicted modal parameters are validated with those obtained by operational modal tests performed on the supported structures.
The equation of motion of a multiple degree of freedom system with mass matrix
In this case, free vibration with proportional damping Equation (
If the dynamic system given by Equation (
Using modal decomposition, the responses can be written as a linear combination of the mode shape vectors by means of [
Premultiplication of Equation (
Taking into account that the eigenvectors of the unperturbed system satisfy the orthonormal conditions, the following are obtained:
A twometer long steel cantilever beam with a rectangular hollow section was used in the investigation. The mechanical and material properties considered for the beam are shown in Table
Mechanical and material properties of the beam.
Section (mm)  Young’s modulus (Pa)  Inertia (strong axis, 
Inertia (weak axis, 
Mass per unit length (kg/m) 

RHS 80 × 40 × 4  2.1 × 10^{11}  64.8  21.5  13.9 
The beam was initially tested in freefree configuration using OMA. In order to reproduce the freefree conditions, the beam was initially suspended using two springs as it is shown in Figure
Beam in freefree configuration.
Test setup for the freefree configuration.
The experimental modal parameters were identified from the experimental responses with the frequencydomain decomposition (FDD) technique [
Singular value decomposition for the beam under freefree conditions from 0 to 750 Hz.
Thirteen modes were identified in the range of 0–1066 Hz. Six modes correspond to the strong axis (
Experimental and numerical natural frequencies for the freefree configuration.
Mode  Mode shape  Natural frequency (Hz)  

Weak axis ( 
Strong axis (  
Numerical  Experimental  Error (%)  Numerical  Experimental  Error (%)  
1 


0.5  — 

0.5  — 
2 


0.6  — 

0.6  — 
3 

73.78  71.4  3.33  129.73  125.96  2.99 
4 

203.57  195.14  4.32  357.62  342.49  4.42 
5 

398.77  374.50  6.48  701.10  653.75  7.24 
6 

659.17  600.67  9.74  1159.09  1047.2  10.68 
7 

984.89  868.21  13.44  1731.83  —  — 
Moreover, although the numerical model is not needed in the proposed methodology, a finite element model was assembled using the program LISA 8.0 [
The SDM theory was applied to predict the natural frequencies and mode shapes of the beam in the pinnedfree configuration with respect to both strong and weak axis.
The methodology followed to predict the modal parameters will be described in detail for the weak axis. The first six experimental mode shapes (scaled to the maximum component equal to unity) of the freefree beam (weak axis) were considered in the predictions (Table
Analytical model of the (a) freefree and (b) pinnedfree configurations.
In this paper, a lumped mass matrix was assembled in order to scale the mode shapes. This model is expected to provide good results because the mass is uniformly distributed and the mass of the beam is known (it was previously weighted).
The analytical natural frequencies and the mode shapes were predicted using the eigenvalue problem given by Equation (
To predict the pinnedfree case using SDM, the analytical model presented in Figure
Analytical natural frequencies obtained for different spring stiffness
First component of the predicted mode shapes for different values of
Therefore, the following stiffness change matrix was used in the analytical predictions:
In order to reproduce experimentally the pinned boundary condition with respect to the weak axis, the beam was welded to a plate as it is shown in Figure
Detail of the weld bead along the long side (a) and short side (b).
The same methodology was followed with respect to the strong axis. The experimental modes were identified by OMA, and the natural frequencies estimated with the FDD technique are shown in Table
Natural frequencies of the pinnedfree beam with respect to the strong axis (
Mode  Natural frequency (Hz)  Error (%)  

Experimental  Predicted (Equation ( 
Numerical  Experimental − predicted  Experimental − numerical  
1  0.64  0  0  —  — 
2  89.01  90.64  89.40  1.80  0.44 
3  279.51  286.53  289.72  2.45  4.65 
4  568.4  570.78  604.51  0.42  6.35 
5  939.52  913.62  1033.84  2.83  10.04 
Natural frequencies of the pinnedfree beam with respect to the weak axis (
Mode  Natural frequency (Hz)  Error (%)  

Experimental  Predicted (Equation ( 
Numerical  Experimental − predicted  Experimental − numerical  
1  5.72  0  0  —  — 
2  52.73  50.42  50.84  4.39  3.72 
3  161.88  163.38  164.77  0.92  1.75 
4  327.26  335.64  343.785  2.56  4.81 
5  541.98  558.61  587.94  3.07  7.82 
6  792.48  829.24  897.333  4.64  11.68 
From Tables
The modal assurance criterion (MAC) between the experimental and the analytical mode shapes predicted with Equation (
MAC between the experimental and the analytical mode shapes.
Mode  Strong axis ( 
Weak axis (  

1  2  3  4  5  1  2  3  4  5  6  
1  1.00  0.06  0.06  0.06  0.06  1.00  0.08  0.07  0.06  0.10  0.07 
2  0.05  1.00  0.10  0.09  0.10  0.05  1.00  0.07  0.13  0.07  0.08 
3  0.05  0.05  0.99  0.10  0.06  0.06  0.09  1.00  0.05  0.14  0.10 
4  0.03  0.12  0.05  0.93  0.14  0.06  0.09  0.09  0.97  0.08  0.12 
5  0.07  0.07  0.10  0.10  0.80  0.07  0.08  0.08  0.16  0.97  0.09 
6  —  —  —  —  —  0.05  0.05  0.11  0.06  0.18  0.93 
A similar procedure was followed to predict the modal parameters of the fixedfree configuration (cantilever). The beam was fixed by welding two steel plates (B) 6 mm thick to both beam and horizontal plate (A) (Figure
Details of the fixed support of the free configuration.
The analytical model shown in Figure
Analytical model for the fixedfree configuration.
The fixed boundary conditions for the analytical model were simulated by adding two springs
The magnitude of the springs stiffnesses was established using the same procedure as that followed for the pinnedfree configuration. The evolution of the natural frequencies for different values of
Evolution of the natural frequencies for different
Evolution of the natural frequencies for different values of (a)
Evolution of the first component of the mode shapes for different values of
The predicted natural frequencies together with the experimental ones identified by operational modal analysis using the FDD technique are shown in Table
Experimental and predicted natural frequencies for the fixedfree configuration.
Weak axis ( 
Strong axis (  

Mode  Natural frequencies (Hz)  Error  Natural frequencies (Hz)  Error  
Experimental frequency  Frequency predicted (Equation ( 
Experimental − predicted (%)  Experimental frequency  Frequency predicted (Equation ( 
Experimental − predicted (%)  
1  10.93  9.55  12.60  15.617  15.67  0.33 
2  67.69  78.09  15.36  103.25  117.87  14.16 
3  186.88  185.90  0.53  296.6  305.81  3.11 
4  362.91  377.11  3.91  573.72  611.66  6.61 
5  570.67  1075.59  >100.00  1016.21  1064.90  >100.00 
6  825.29  2291.05  >100.00  —  2280.30  >100.00 
The MAC between the predicted and the experimental mode shapes is shown in Table
MAC between experimental and predicted mode shapes.
MAC  

Mode  Weak axis ( 
Strong axis (  
1  2  3  4  5  6  1  2  3  4  5  6  
1  1.00  0.09  0.06  0.10  0.13  0.00  1.00  0.09  0.07  0.08  0.08  — 
2  0.07  1.00  0.08  0.12  0.09  0.00  0.05  0.96  0.20  0.06  0.09  — 
3  0.08  0.12  0.99  0.08  0.18  0.00  0.05  0.03  0.95  0.18  0.03  — 
4  0.08  0.08  0.09  0.92  0.10  0.00  0.09  0.15  0.06  0.81  0.49  — 
5  0.07  0.09  0.05  0.31  0.59  0.04  0.03  0.02  0.04  0.01  0.31  — 
6  0.07  0.10  0.09  0.07  0.67  0.10  —  —  —  —  —  — 
From Tables
The steel beam structure shown in Figure
Geometry of the structure (a) and the setup for the freefree configuration (b).
The structure was initially tested in freefree configuration using operational modal analysis. The structure was suspended using two springs. Twelve accelerometers in the vertical direction with a sensitivity of 100 mv/g were attached to the structure. The test setup is shown in Figure
The structure was excited with an impact hammer applying randomly hits both in time and in space. The responses were recorded with a sampling frequency of 2132 Hz during approximately 7 minutes. The experimental natural frequencies estimated with the frequencydomain decomposition technique are presented in Table
Experimental and numerical natural frequencies obtained for freefree configuration.
Mode  Natural frequencies (Hz)  Error (%)  

Experimental  Numerical  
1  23.15  23.65  2.16 
2  27.83  30.13  8.26 
3  46.24  48.03  3.88 
4  56.02  57.06  1.86 
5  70.58  71.07  1.84 
6  65.90  71.87  7.86 
7  102.70  105.03  2.27 
8  131.60  136.26  3.54 
9  136.10  144.90  6.47 
10  152.60  154.67  1.36 
Numerical frequencies and mode shapes for freefree configuration and pinnedsupported configuration.
Numerical frequencies and mode shapes  



Undeformed state  


Mode  Freefree configuration  Pinnedsupported configuration 


1 


2 


3 


4 


5 


6 


The SDM theory was applied to predict the natural frequencies and mode shapes of the structure with pinned supports in points A, B, C, and D (Figure
Details of the support.
Two solid rigid modes and six elastic modes of the freefree structure (Table
Experimental, numerical, and predicted natural frequencies for pinnedfree configuration.
Vertical axis  

Mode  Natural frequency (Hz)  Error (%)  
Experimental  Predicted (Equation ( 
Numerical  Experimental − predicted  Experimental − numerical  
1  25.25  25.74  24.02  −1.94  4.87 
2  33.57  33.88  31.12  −0.92  7.30 
3  80.68  77.42  78.22  4.04  3.05 
4  104.10  95.02  98.97  8.72  4.93 
5  127.52  132.33  131.47  −3.77  3.10 
6  187.28  187.42  189.65  −0.07  1.27 
Operational modal analysis was applied to the pinnedsupported structure using the test setup shown in Figure
Test setup used in the experimental tests.
Singular value decomposition (SVD) in the frequency range 0–200 Hz.
With respect to the mode shapes, the MAC between the experimental mode shapes and those predicted with the SDM theory is presented in Table
MAC between experimental and analytical mode shapes.
Mode  Vertical axis  

1  2  3  4  5  6  
1  1.00  0.03  0.00  0.00  0.01  0.00 
2  0.00  0.99  0.00  0.00  0.00  0.00 
3  0.00  0.06  1.00  0.00  0.00  0.13 
4  0.00  0.11  0.00  1.00  0.00  0.00 
5  0.01  0.00  0.00  0.00  0.99  0.00 
6  0.00  0.02  0.15  0.00  0.00  1.00 
Structural dynamic modification theory has been applied in this paper to predict the modal parameters of supported structures using the modal parameters of the same structures tested in freefree boundary conditions. This methodology could be used in a wide range of applications such as structures that are usually assembled in a factory and then transported to the site where the structure is to be located. A pinnedfree beam, a cantilever beam, and a 3D steel beam were used to validate the proposed methodology. The technique can be summarized in two steps: (1) to test the structure in freefree boundary conditions and estimation of the modal parameters using operational modal analysis and (2) to use the SDM theory to predict the modal parameters of the supported structures using Equation (
The technique has been validated by comparing the modal parameters predicted with SDM with those estimated with OMA. Accurate results have been obtained for pinnedsupported structures (error less than 10%), whereas the fixed boundary condition is more difficult to replicate, the error being less than 16% in the natural frequencies.
It has been demonstrated that the technique provides good results when the boundary conditions considered in Equation (
The data used to support the findings of this study are available from the corresponding author upon request.
The authors declare that they have no conflicts of interest.
The financing support given by the Spanish Ministry of Economy and Competitiveness through the project BIA201128380C0201 is gratefully appreciated.