# Patent

## Title
Inverse method to estimate the properties of a flexural beam and the corresponding boundary parameters

## Abstract

A system and method is used for estimating the properties of a flexural beam. The beam is shaken transverse to its longitudinal axis. Seven frequency domain transfer functions of displacement are measured at spaced apart locations along the beam. The seven transfer functions are combined to yield closed form values of the flexural wavenumber in propagation coefficients at any test frequency.

## Background
STATEMENT OF GOVERNMENT INTEREST
The invention described herein may be manufactured and used by or for the Government of the United States of America for governmental purposes without the payment of any royalties thereon or therefore.


BACKGROUND OF THE INVENTION
(1) Field of the Invention
This invention relates to the field of structural properties, and in particular to the determination of the complex flexural wavenumber, corresponding wave propagation coefficients, and boundary condition parameters of a beam subjected to transverse motion.
(2) Description of the Prior Art
By way of example of the state of the art, reference is made to the following papers, which are incorporated herein by reference. Not all of these references may be deemed to be relevant prior art.
D. M. Norris, Jr., and W. C. Young, “Complex Modulus Measurements by Longitudinal Vibration Testing,” Experimental Mechanics, Volume 10, 1970, pp. 93-96.
W. M. Madigosky and G. F. Lee, “Improved Resonance Technique for Materials Characterization,” Journal of the Acoustical Society of America, Volume 73, Number 4, 1983, pp. 1374-1377.
S. L. Garrett, “Resonant Acoustic Determination of Elastic Moduli,” Journal of the Acoustical Society of America, Volume 88, Number 1, 1990, pp. 210-220.
I. Jimeno-Fernandez, H. Uberall, W. M. Madigosky, and R. B. Fiorito, “Resonance Decomposition for the Vibratory Response of a Viscoelastic Rod,” Journal of the Acoustical Society of America, Volume 91, Number 4, Part 1, April 1992, pp. 2030-2033.
G. F. Lee and B. Hartmann, “Material Characterizing System,” U.S. Pat. No. 5,363,701, Nov. 15, 1994.
G. W. Rhodes, A. Migliori, and R. D. Dixon, “Method for Resonant Measurement,” U.S. Pat. No. 5,495,763, Mar. 5, 1996.
R. F. Gibson and E. O. Ayorinde, “Method and Apparatus for Non-Destructive Measurement of Elastic Properties of Structural Materials,” U.S. Pat. No. 5,533,399, Jul. 9, 1996.
B. J. Dobson, “A Straight-Line Technique for Extracting Modal Properties From Frequency Response Data,” Mechanical Systems and Signal Processing, Volume 1, 1987, pp. 29-40.
C. Minas and D. J. Inman, “Matching Finite Element Models to Modal Data,” Journal of Vibration and Acoustics, Volume 112, Number 1, 1990, pp. 84-92,
T. Pritz, “Transfer Function Method for Investigating the Complex Modulus of Acoustic Materials: Rod-Like Specimen,” Journal of Sound and Vibration, Volume 81, 1982, pp. 359-376.
W. M. Madigosky and G. F. Lee, “Instrument for Measuring Dynamic Viscoelastic Properties,” U.S. Pat. No. 4,352,292, Oct. 5, 1982.
W. M. Madigosky and G. F. Lee, “Method for Measuring Material Characteristics,” U.S. Pat. No. 4,418,573, Dec. 6, 1983.
W. Madigosky, “In Situ Dynamic Material Property Measurement System,” U.S. Pat. No. 5,365,457, Nov. 15, 1994.
J. G. McDaniel, P. Dupont, and L. Salvino, “A Wave Approach to Estimating Frequency-Dependent Damping Under Transient Loading” Journal of Sound and Vibration, Volume 231(2), 2000, pp. 433-449.
J. Linjama and T. Lahti, “Measurement of Bending wave reflection and Impedance in a Beam by the Structural Intensity Technique” Journal of Sound and Vibration, Volume 161(2), 1993, pp. 317-331.
L. Koss and D. Karczub, “Euler Beam Bending Wave Solution Predictions of dynamic Strain Using Frequency Response Functions ” Journal of Sound and Vibration, Volume 184(2), 1995, pp. 229-244.
Measuring the flexural properties of beams is important because these parameters significantly contribute to the static and dynamic response of structures. In the past, resonant techniques have been used to identify and measure longitudinal properties. These methods are based on comparing the measured eigenvalues of a structure to predicted eigenvalues from a model of the same structure. The model of the structure must have well-defined (typically closed form) eigenvalues for this method to work. Additionally, resonant techniques only allow measurements at natural frequencies.
Comparison of analytical models to measured frequency response functions is another method used to estimate stiffness and loss parameters of a structure. When the analytical model agrees with one or more frequency response functions, the parameters used to calculate the analytical model are considered accurate. If the analytical model is formulated using a numerical method, a comparison of the model to the data can be difficult due to the dispersion properties of the materials.
Another method to measure stiffness and loss is to deform the material and measure the resistance to the indentation. This method can physically damage the specimen if the deformation causes the sample to enter the plastic region of deformation.
SUMMARY OF THE INVENTION
Accordingly, one objective of the present invention is to measure flexural wavenumbers.
Another objective of the present invention is to measure flexural wave propagation coefficients.
A further objective of the present invention is to measure Young's modulus when the beam is undergoing transverse motion.
Yet another objective of the present invention is to measure the boundary stiffness and dampening values when the beam is vibrated transversely.
The foregoing objects are attained by the method and system of the present invention. The present invention features a method of determining structural properties of a flexural beam mounted to a base. The method comprises securing a plurality of accelerometers spaced approximately equidistant from each other along a length of a beam. One accelerometer can be secured to the base. An input is provided to the beam. Seven frequency domain transfer functions of displacement are measured from the accelerometers secured to the beam. The flexural wavenumber is estimated from the seven frequency domain transfer functions.
The seven frequency domain transfer functions of displacement include the following equations: 


  
    
      
        T
        
          -
          3
        
      
      =
      
        
          
            
              U
              
                -
                3
              
            
            ⁡
            
              (
              
                
                  
                    -
                    3
                  
                  ⁢
                  α
                
                ,
                ω
              
              )
            
          
          
            
              V
              0
            
            ⁡
            
              (
              ω
              )
            
          
        
        =
        
          
            A
            ⁢
            
               
            
            ⁢
            
              cos
              ⁡
              
                (
                
                  3
                  ⁢
                  αδ
                
                )
              
            
          
          -
          
            B
            ⁢
            
               
            
            ⁢
            
              sin
              ⁡
              
                (
                
                  3
                  ⁢
                  αδ
                
                )
              
            
          
          +
          
            C
            ⁢
            
               
            
            ⁢
            
              cosh
              ⁡
              
                (
                
                  3
                  ⁢
                  αδ
                
                )
              
            
          
          -
          
            D
            ⁢
            
               
            
            ⁢
            
              sinh
              ⁡
              
                (
                
                  3
                  ⁢
                  αδ
                
                )
              
            
          
        
      
    
    ,
    
      

    
    ⁢
    
      
        T
        
          -
          2
        
      
      =
      
        
          
            
              U
              
                -
                2
              
            
            ⁡
            
              (
              
                
                  
                    -
                    2
                  
                  ⁢
                  α
                
                ,
                ω
              
              )
            
          
          
            
              V
              0
            
            ⁡
            
              (
              ω
              )
            
          
        
        =
        
          
            A
            ⁢
            
               
            
            ⁢
            
              cos
              ⁡
              
                (
                
                  2
                  ⁢
                  αδ
                
                )
              
            
          
          -
          
            B
            ⁢
            
               
            
            ⁢
            
              sin
              ⁡
              
                (
                
                  2
                  ⁢
                  αδ
                
                )
              
            
          
          +
          
            C
            ⁢
            
               
            
            ⁢
            
              cosh
              ⁡
              
                (
                
                  2
                  ⁢
                  αδ
                
                )
              
            
          
          -
          
            D
            ⁢
            
               
            
            ⁢
            
              sinh
              ⁡
              
                (
                
                  2
                  ⁢
                  αδ
                
                )
              
            
          
        
      
    
    ,
    
      

    
    ⁢
    
      
        T
        
          -
          1
        
      
      =
      
        
          
            
              U
              
                -
                1
              
            
            ⁡
            
              (
              
                
                  -
                  α
                
                ,
                ω
              
              )
            
          
          
            
              V
              0
            
            ⁡
            
              (
              ω
              )
            
          
        
        =
        
          
            A
            ⁢
            
               
            
            ⁢
            
              cos
              ⁡
              
                (
                αδ
                )
              
            
          
          -
          
            B
            ⁢
            
               
            
            ⁢
            
              sin
              ⁡
              
                (
                αδ
                )
              
            
          
          +
          
            C
            ⁢
            
               
            
            ⁢
            
              cosh
              ⁡
              
                (
                αδ
                )
              
            
          
          -
          
            D
            ⁢
            
               
            
            ⁢
            
              sinh
              ⁡
              
                (
                αδ
                )
              
            
          
        
      
    
    ,
    
      

    
    ⁢
    
      
        T
        0
      
      =
      
        
          
            
              U
              0
            
            ⁡
            
              (
              
                0
                ,
                ω
              
              )
            
          
          
            
              V
              0
            
            ⁡
            
              (
              ω
              )
            
          
        
        =
        
          A
          +
          C
        
      
    
    ,
    
      

    
    ⁢
    
      
        T
        1
      
      =
      
        
          
            
              U
              1
            
            ⁡
            
              (
              
                δ
                ,
                ω
              
              )
            
          
          
            
              V
              0
            
            ⁡
            
              (
              ω
              )
            
          
        
        =
        
          
            A
            ⁢
            
               
            
            ⁢
            
              cos
              ⁡
              
                (
                αδ
                )
              
            
          
          +
          
            B
            ⁢
            
               
            
            ⁢
            
              sin
              ⁡
              
                (
                αδ
                )
              
            
          
          +
          
            C
            ⁢
            
               
            
            ⁢
            
              cosh
              ⁡
              
                (
                αδ
                )
              
            
          
          +
          
            D
            ⁢
            
               
            
            ⁢
            
              sinh
              ⁡
              
                (
                αδ
                )
              
            
          
        
      
    
    ,
    
      

    
    ⁢
    
      
        T
        2
      
      =
      
        
          
            
              U
              2
            
            ⁡
            
              (
              
                
                  2
                  ⁢
                  δ
                
                ,
                ω
              
              )
            
          
          
            
              V
              0
            
            ⁡
            
              (
              ω
              )
            
          
        
        =
        
          
            A
            ⁢
            
               
            
            ⁢
            
              cos
              ⁡
              
                (
                
                  2
                  ⁢
                  αδ
                
                )
              
            
          
          -
          
            B
            ⁢
            
               
            
            ⁢
            
              sin
              ⁡
              
                (
                
                  2
                  ⁢
                  αδ
                
                )
              
            
          
          +
          
            C
            ⁢
            
               
            
            ⁢
            
              cosh
              ⁡
              
                (
                
                  2
                  ⁢
                  αδ
                
                )
              
            
          
          +
          
            D
            ⁢
            
               
            
            ⁢
            
              sinh
              ⁡
              
                (
                
                  2
                  ⁢
                  αδ
                
                )
              
            
          
        
      
    
    ,
    
      

    
    ⁢
    and
  




  
    
      
        T
        3
      
      =
      
        
          
            
              U
              3
            
            ⁡
            
              (
              
                
                  3
                  ⁢
                  δ
                
                ,
                ω
              
              )
            
          
          
            
              V
              0
            
            ⁡
            
              (
              ω
              )
            
          
        
        =
        
          
            A
            ⁢
            
               
            
            ⁢
            
              cos
              ⁡
              
                (
                
                  3
                  ⁢
                  αδ
                
                )
              
            
          
          +
          
            B
            ⁢
            
               
            
            ⁢
            
              sin
              ⁡
              
                (
                
                  3
                  ⁢
                  αδ
                
                )
              
            
          
          +
          
            C
            ⁢
            
               
            
            ⁢
            
              cosh
              ⁡
              
                (
                
                  3
                  ⁢
                  αδ
                
                )
              
            
          
          +
          
            D
            ⁢
            
               
            
            ⁢
            
              sinh
              ⁡
              
                (
                
                  3
                  ⁢
                  αδ
                
                )
              
            
          
        
      
    
    ,
    
      

    
    ⁢
    and
  



The flexural wavenumber is determined using the following equations: 



  
    Re
    ⁡
    
      (
      α
      )
    
  
  =
  
    {
    
      
        
          
            
              
                
                ⁢
                
                  
                    
                      1
                      
                        2
                        ⁢
                        δ
                      
                    
                    ⁢
                    Arc
                    ⁢
                    
                       
                    
                    ⁢
                    
                      cos
                      ⁡
                      
                        (
                        s
                        )
                      
                    
                  
                  +
                  
                    
                      n
                      ⁢
                      
                         
                      
                      ⁢
                      π
                    
                    
                      2
                      ⁢
                      δ
                    
                  
                
              
            
            
              
                n
                ⁢
                
                   
                
                ⁢
                even
              
            
          
          
            
              
                
                ⁢
                
                  
                    
                      1
                      
                        2
                        ⁢
                        δ
                      
                    
                    ⁢
                    Arc
                    ⁢
                    
                       
                    
                    ⁢
                    
                      cos
                      ⁡
                      
                        (
                        
                          -
                          s
                        
                        )
                      
                    
                  
                  +
                  
                    
                      n
                      ⁢
                      
                         
                      
                      ⁢
                      π
                    
                    
                      2
                      ⁢
                      δ
                    
                  
                
              
            
            
              
                n
                ⁢
                
                   
                
                ⁢
                odd
              
            
          
        
        ⁢
        
           
        
        ⁢
        where
        ⁢
        
          

        
        ⁢
        s
      
      =
      
        
          
            [
            
              Re
              ⁡
              
                (
                φ
                )
              
            
            ]
          
          2
        
        +
        
          
            [
            
              Im
              ⁡
              
                (
                φ
                )
              
            
            ]
          
          2
        
        -
        
          
            
              
                {
                
                  
                    
                      [
                      
                        Re
                        ⁡
                        
                          (
                          φ
                          )
                        
                      
                      ]
                    
                    2
                  
                  +
                  
                    
                      [
                      
                        Im
                        ⁡
                        
                          (
                          φ
                          )
                        
                      
                      ]
                    
                    2
                  
                
                }
              
              2
            
            -
            
              {
              
                
                  
                    2
                    ⁡
                    
                      [
                      
                        Re
                        ⁡
                        
                          (
                          φ
                          )
                        
                      
                      ]
                    
                  
                  2
                
                +
                
                  
                    2
                    ⁡
                    
                      [
                      
                        Im
                        ⁡
                        
                          (
                          φ
                          )
                        
                      
                      ]
                    
                  
                  2
                
                -
                1
              
              }
            
          
        
      
    
  




and said imaginary part comprises: 



  
    Im
    ⁡
    
      (
      α
      )
    
  
  =
  
    
      1
      δ
    
    ⁢
    
      log
      e
    
    ⁢
    
      
        {
        
          
            
              Re
              ⁡
              
                (
                φ
                )
              
            
            
              cos
              ⁡
              
                [
                
                  
                    Re
                    ⁡
                    
                      (
                      α
                      )
                    
                  
                  ⁢
                  δ
                
                ]
              
            
          
          -
          
            
              Im
              ⁡
              
                (
                φ
                )
              
            
            
              sin
              ⁡
              
                [
                
                  
                    Re
                    ⁡
                    
                      (
                      α
                      )
                    
                  
                  ⁢
                  δ
                
                ]
              
            
          
        
        }
      
      .
    
  




Using the flexural wavenumber and various equations disclosed within the present invention, the complex valued modulus of elasticity can be determined at each frequency, as well as the wave property coefficient, and the boundary parameters.
Thus, this invention has the advantages that all measurements can be calculated at every frequency that a transfer function measurement is made. They do not depend on system resonance's or curve fitting to transfer functions. The calculation from transfer function measurement to calculation of all system parameters is exact, i.e., no errors are introduced during this process. Furthermore, the measurements can be calculated without adverse consequences to the tested beam.



BRIEF DESCRIPTION OF THE DRAWINGS
These and other features and advantages of the present invention will be better understood in view of the following description of the invention taken together with the drawings wherein:
FIG. 1 is a schematic block diagram of a conventional testing system including two springs and two dashpots attached to a shaker table;
FIG. 2 is a schematic block diagram of a conventional testing system including one spring and one dashpot attached to a shaker table;
FIG. 3 is a schematic block diagram of a conventional testing system including two springs and two dashpots, one of which is attached to a shaker table;
FIG. 4 is a schematic block diagram of a conventional testing system wherein the beam is attached directly to a shaker table;
FIG. 5A is a graph of the magnitude of a typical transfer function of a beam;
FIG. 5B is a graph of the phase angle of a typical transfer function of a beam;
FIG. 6 is a graph of the function s versus frequency;
FIG. 7A is a graph of the real part of a flexural wavenumber versus frequency;
FIG. 7B is a graph of the imaginary part of a flexural wavenumber versus frequency;
FIGS. 8-11 are graphs of the wave propagation coefficients versus frequency;
FIG. 12 is a graph of the real and imaginary parts of the Young's Modulus versus frequency;
FIG. 13 is a graph of the boundary conditions of the system shown in FIG. 1 versus frequency; and
FIG. 14 is a graph of the boundary conditions of the system shown in FIG. 2 versus frequency.



DESCRIPTION OF THE PREFERRED EMBODIMENT
The method and system, according to the present invention, is used to develop and measure complex flexural wavenumbers and the corresponding wave propagation coefficients of a beam undergoing transverse motion. An inverse method has been developed using seven transfer function measurements. These seven transfer function measurements are combined to yield closed form values of flexural wavenumber and wave propagation coefficients at any given test frequency. Finally, Young's modulus, spring stiffnesses, dashpot damping values, and boundary condition parameters, among other parameters, are calculated from the flexural wavenumber and wave propagation coefficients.
According to an exemplary test configuration 10, FIG. 1, a shaker table 12 initiates transverse motion 14 into a beam 16. The beam 16 is connected to the shaker table 12 with a spring 18 and dashpot 20 at each end 22. FIG. 1 represents a double translational spring and damper input configuration. Other test configurations are also possible, including the shaker table 12 inputting energy into only one end 22 of the beam 16 with the other end terminated to ground 24 directly, as shown in FIG. 2, or terminated to ground 24 with a spring 18 and dashpot 20, as shown in FIG. 3, or terminated to ground 24 and the shaker 12 directly, as shown in FIG. 4. FIG. 2 represents a single translational spring and damper input configuration with the other end pinned. FIG. 3 represents a single translational spring and damper input configuration with the other end having a translational spring and damper. FIG. 4 represents a single pin input configuration with the other end pinned. These approaches are intended for use when a beam 16 is to undergo motion in the transverse direction 14. This system typically arises in cars, ships, aircraft, bridges, buildings and other common structures.
In any of the embodiments shown in FIGS. 1-4 sensors 26 such as accelerators are positioned equally along beam 22. As discussed above, a minimum of seven such sensors 22 are required. Optionally, a reference sensor 28 can be joined to shaker table 12 to read the input motion 14. The input motion 14 can also be read directly from the shaker table 12 controls.
For simplicity, the present invention will be described as it relates to the derivation of the linear equations of motion of the system with a spring 18 and dashpot 20 boundary condition at each end 22, but this is for exemplary purposes only, and is not intended to be a limitation.
The system model of the beam is the Bernoulli-Euler beam equation written as 



  
    
      
        
          EI
          =
          
            
              
                
                  
                    ∂
                    4
                  
                  ⁢
                  
                    u
                    ⁡
                    
                      (
                      
                        x
                        ,
                        t
                      
                      )
                    
                  
                
                
                  ∂
                  
                    x
                    4
                  
                
              
              +
              
                ρ
                ⁢
                
                   
                
                ⁢
                
                  A
                  b
                
                ⁢
                
                  
                    
                      ∂
                      2
                    
                    ⁢
                    
                      u
                      ⁡
                      
                        (
                        
                          x
                          ,
                          t
                        
                        )
                      
                    
                  
                  
                    ∂
                    
                      t
                      2
                    
                  
                
              
            
            =
            0
          
        
        ,
      
    
    
      
        (
        1
        )
      
    
  




where x is the distance along the length of the beam in meters, t is time in seconds, u is the displacement of the beam in the (transverse) y direction in meters, E is the unknown frequency-dependent, complex Young's modulus (N/m2), I is the moment of inertia (m4), ρ is the density (kg/m3), and Ab is the cross-sectional area of the beam (m2). Implicit in equation (1) is the assumption that plane sections remain planar during bending (or transverse motion). Additionally, Young's modulus, the moment of inertia, the density, and the cross sectional area are constant across the entire length of the beam. The displacement is modeled as a steady state response and is expressed as

u(x,t)=U(x,ω)exp(iωt),  (2)

where ω is the frequency of excitation (rad/s), U(x,ω) is the temporal Fourier transform of the transverse displacement, and i is the square root of −1. The temporal solution to equation (1), derived using equation (2) and written in terms of trigonometric functions, is

U(x,ω)=A(ω)cos[α(ω)x]+B(ω)sin[α(ω)x]+C(ω)cosh[α(ω)x]+D(ω)sinh[α(ω)x]′  (3)

where A(ω), B(ω), C(ω), and D(ω) are wave propagation coefficients and α(ω) is the flexural wavenumber given by 



  
    
      
        
          α
          ⁡
          
            (
            ω
            )
          
        
        =
        
          
            
              [
              
                
                  ω
                  2
                
                
                  (
                  
                    
                      EI
                      /
                      ρ
                    
                    ⁢
                    
                       
                    
                    ⁢
                    
                      A
                      b
                    
                  
                  )
                
              
              ]
            
            
              1
              /
              4
            
          
          .
        
      
    
    
      
        (
        4
        )
      
    
  




For brevity, the ω dependence is omitted from the wave propagation coefficients and the flexural wavenumber during the remainder of the disclosure and α(ω) is references as α. Note that equations (3) and (4) are independent of boundary conditions, and the inverse model developed in the next section does not need boundary condition specifications. Boundary conditions are chosen, however, to show that the boundary parameters can be estimated and to run a realistic simulation.
One of the most typical test configurations is the beam mounted to shock mounts on each end that are attached to a shaker table that generates a vibrational input, as shown in FIG. 1. Using the middle of the beam as the coordinate system origin, these boundary conditions are modeled as 



  
    
      
        
          
            
              
                ∂
                2
              
              ⁢
              
                u
                ⁡
                
                  (
                  
                    
                      
                        -
                        L
                      
                      /
                      2
                    
                    ,
                    t
                  
                  )
                
              
            
            
              ∂
              
                x
                2
              
            
          
          =
          0
        
        ,
      
    
    
      
        (
        5
        )
      
    
  
  
    
      
        
          
            
              -
              EI
            
            ⁢
            
              
                
                  ∂
                  2
                
                ⁢
                
                  u
                  ⁡
                  
                    (
                    
                      
                        
                          -
                          L
                        
                        /
                        2
                      
                      ,
                      t
                    
                    )
                  
                
              
              
                ∂
                
                  x
                  3
                
              
            
          
          =
          
            
              
                k
                1
              
              ⁡
              
                [
                
                  
                    u
                    ⁡
                    
                      (
                      
                        
                          
                            -
                            L
                          
                          /
                          2
                        
                        ,
                        t
                      
                      )
                    
                  
                  -
                  
                    v
                    ⁡
                    
                      (
                      t
                      )
                    
                  
                
                ]
              
            
            +
            
              
                c
                1
              
              ⁡
              
                [
                
                  
                    
                      ∂
                      
                        u
                        ⁡
                        
                          (
                          
                            
                              
                                -
                                L
                              
                              /
                              2
                            
                            ,
                            t
                          
                          )
                        
                      
                    
                    
                      ∂
                      t
                    
                  
                  -
                  
                    
                      ∂
                      
                        v
                        ⁡
                        
                          (
                          t
                          )
                        
                      
                    
                    
                      ∂
                      t
                    
                  
                
                ]
              
            
          
        
        ,
      
    
    
      
        (
        6
        )
      
    
  
  
    
      
        
          
            
              
                ∂
                2
              
              ⁢
              
                u
                ⁡
                
                  (
                  
                    
                      L
                      /
                      2
                    
                    ,
                    t
                  
                  )
                
              
            
            
              ∂
              
                x
                2
              
            
          
          =
          0
        
        ,
      
    
    
      
        (
        7
        )
      
    
  
  
    
      
        
          
            
              -
              EI
            
            ⁢
            
              
                
                  ∂
                  3
                
                ⁢
                
                  u
                  ⁡
                  
                    (
                    
                      
                        L
                        /
                        2
                      
                      ,
                      t
                    
                    )
                  
                
              
              
                ∂
                
                  x
                  3
                
              
            
          
          =
          
            
              
                k
                2
              
              ⁡
              
                [
                
                  
                    u
                    ⁡
                    
                      (
                      
                        
                          L
                          /
                          2
                        
                        ,
                        t
                      
                      )
                    
                  
                  -
                  
                    v
                    ⁡
                    
                      (
                      t
                      )
                    
                  
                
                ]
              
            
            +
            
              
                c
                2
              
              ⁡
              
                [
                
                  
                    
                      ∂
                      
                        u
                        ⁡
                        
                          (
                          
                            
                              L
                              /
                              2
                            
                            ,
                            t
                          
                          )
                        
                      
                    
                    
                      ∂
                      t
                    
                  
                  -
                  
                    
                      ∂
                      
                        v
                        ⁡
                        
                          (
                          t
                          )
                        
                      
                    
                    
                      ∂
                      t
                    
                  
                
                ]
              
            
          
        
        ,
      
    
    
      
        (
        8
        )
      
    
  




where

ν(t)=V0(ω)exp(iωt),  (9)

which is the input into the system from the shaker table.

Inserting equation (3) into equation (5), (6), (7), (8), and (9) yields the solution to the wave propagation coefficients. Inserting these back into equation (3) is the displacement of the system, and is sometimes called the forward solution. The wave coefficient A is 



  
    
      
        
          A
          =
          
            
              A
              T
            
            
              A
              B
            
          
        
        ,
      
    
    
      
        (
        10
        )
      
    
  




where 



  
    
      
        
          A
          T
        
        =
        
          
            
              [
              
                
                  (
                  
                    
                      k
                      1
                    
                    +
                    
                      i
                      ⁢
                      
                         
                      
                      ⁢
                      ω
                      ⁢
                      
                         
                      
                      ⁢
                      
                        c
                        1
                      
                    
                  
                  )
                
                -
                
                  (
                  
                    
                      k
                      2
                    
                    +
                    
                      i
                      ⁢
                      
                         
                      
                      ⁢
                      ω
                      ⁢
                      
                         
                      
                      ⁢
                      
                        c
                        2
                      
                    
                  
                  )
                
              
              ]
            
            ⁢
            
              (
              
                EI
                ⁢
                
                   
                
                ⁢
                
                  α
                  3
                
              
              )
            
            ⁢
            
              cos
              ⁡
              
                (
                
                  α
                  ⁢
                  
                    L
                    2
                  
                
                )
              
            
            ⁢
            
              cosh
              ⁡
              
                (
                
                  α
                  ⁢
                  
                    L
                    2
                  
                
                )
              
            
            ⁢
            
              sinh
              ⁡
              
                (
                
                  α
                  ⁢
                  
                    L
                    2
                  
                
                )
              
            
          
          -
          
             
            
              
                
                  
                    [
                    
                      
                        (
                        
                          
                            k
                            1
                          
                          +
                          
                            i
                            ⁢
                            
                               
                            
                            ⁢
                            ω
                            ⁢
                            
                               
                            
                            ⁢
                            
                              c
                              1
                            
                          
                        
                        )
                      
                      -
                      
                        (
                        
                          
                            k
                            2
                          
                          +
                          
                            i
                            ⁢
                            
                               
                            
                            ⁢
                            ω
                            ⁢
                            
                               
                            
                            ⁢
                            
                              c
                              2
                            
                          
                        
                        )
                      
                    
                    ]
                  
                  ⁢
                  
                    (
                    
                      EI
                      ⁢
                      
                         
                      
                      ⁢
                      
                        α
                        3
                      
                    
                    )
                  
                  ⁢
                  
                    sin
                    ⁡
                    
                      (
                      
                        α
                        ⁢
                        
                          L
                          2
                        
                      
                      )
                    
                  
                  ⁢
                  
                    
                      cosh
                      2
                    
                    ⁡
                    
                      (
                      
                        α
                        ⁢
                        
                          L
                          2
                        
                      
                      )
                    
                  
                
                -
                
                  4
                  ⁢
                  
                    (
                    
                      
                        k
                        1
                      
                      +
                      
                        i
                        ⁢
                        
                           
                        
                        ⁢
                        ω
                        ⁢
                        
                           
                        
                        ⁢
                        
                          c
                          1
                        
                      
                    
                    )
                  
                  ⁢
                  
                    (
                    
                      
                        k
                        2
                      
                      +
                      
                        i
                        ⁢
                        
                           
                        
                        ⁢
                        ω
                        ⁢
                        
                           
                        
                        ⁢
                        
                          c
                          2
                        
                      
                    
                    )
                  
                  ⁢
                  
                    sin
                    ⁡
                    
                      (
                      
                        α
                        ⁢
                        
                          L
                          2
                        
                      
                      )
                    
                  
                  ⁢
                  
                    cosh
                    ⁡
                    
                      (
                      
                        α
                        ⁢
                        
                          L
                          2
                        
                      
                      )
                    
                  
                  ⁢
                  
                    sinh
                    ⁡
                    
                      (
                      
                        α
                        ⁢
                        
                          L
                          2
                        
                      
                      )
                    
                  
                
              
              ,
              and
            
          
        
      
    
    
      
        (
        11
        )
      
    
  
  
    
      
        
          A
          B
        
        =
        
          
            2
            ⁢
            
              
                (
                
                  EI
                  ⁢
                  
                     
                  
                  ⁢
                  
                    α
                    3
                  
                
                )
              
              2
            
            ⁢
            
              
                sin
                2
              
              ⁡
              
                (
                
                  α
                  ⁢
                  
                    L
                    2
                  
                
                )
              
            
            ⁢
            
              cosh
              ⁡
              
                (
                
                  α
                  ⁢
                  
                    L
                    2
                  
                
                )
              
            
          
          -
          
            
              2
              ⁡
              
                [
                
                  
                    (
                    
                      
                        k
                        1
                      
                      +
                      
                        i
                        ⁢
                        
                           
                        
                        ⁢
                        ω
                        ⁢
                        
                           
                        
                        ⁢
                        
                          c
                          1
                        
                      
                    
                    )
                  
                  -
                  
                    (
                    
                      
                        k
                        2
                      
                      +
                      
                        i
                        ⁢
                        
                           
                        
                        ⁢
                        ω
                        ⁢
                        
                           
                        
                        ⁢
                        
                          c
                          2
                        
                      
                    
                    )
                  
                
                ]
              
            
            ⁢
            
              (
              
                EI
                ⁢
                
                   
                
                ⁢
                
                  α
                  3
                
              
              )
            
            ⁢
            
              
                sin
                2
              
              ⁡
              
                (
                
                  α
                  ⁢
                  
                    L
                    2
                  
                
                )
              
            
            ⁢
            
              cosh
              ⁡
              
                (
                
                  α
                  ⁢
                  
                    L
                    2
                  
                
                )
              
            
            ⁢
            
              sinh
              ⁡
              
                (
                
                  α
                  ⁢
                  
                    L
                    2
                  
                
                )
              
            
          
          -
          
            2
            ⁢
            
              
                (
                
                  EI
                  ⁢
                  
                     
                  
                  ⁢
                  
                    α
                    3
                  
                
                )
              
              2
            
            ⁢
            
              
                cos
                2
              
              ⁡
              
                (
                
                  α
                  ⁢
                  
                    L
                    2
                  
                
                )
              
            
            ⁢
            
              
                sinh
                2
              
              ⁡
              
                (
                
                  α
                  ⁢
                  
                    L
                    2
                  
                
                )
              
            
          
          -
          
            
              2
              ⁡
              
                [
                
                  
                    (
                    
                      
                        k
                        1
                      
                      +
                      
                        i
                        ⁢
                        
                           
                        
                        ⁢
                        ω
                        ⁢
                        
                           
                        
                        ⁢
                        
                          c
                          1
                        
                      
                    
                    )
                  
                  -
                  
                    (
                    
                      
                        k
                        2
                      
                      +
                      
                        i
                        ⁢
                        
                           
                        
                        ⁢
                        ω
                        ⁢
                        
                           
                        
                        ⁢
                        
                          c
                          2
                        
                      
                    
                    )
                  
                
                ]
              
            
            ⁢
            
              (
              
                EI
                ⁢
                
                   
                
                ⁢
                
                  α
                  3
                
              
              )
            
            ⁢
            
              cos
              ⁡
              
                (
                
                  α
                  ⁢
                  
                    L
                    2
                  
                
                )
              
            
            ⁢
            
              sin
              ⁡
              
                (
                
                  α
                  ⁢
                  
                    L
                    2
                  
                
                )
              
            
            ⁢
            
              
                sinh
                2
              
              ⁡
              
                (
                
                  α
                  ⁢
                  
                    L
                    2
                  
                
                )
              
            
          
          +
          
            
              2
              ⁡
              
                [
                
                  
                    (
                    
                      
                        k
                        1
                      
                      +
                      
                        i
                        ⁢
                        
                           
                        
                        ⁢
                        ω
                        ⁢
                        
                           
                        
                        ⁢
                        
                          c
                          1
                        
                      
                    
                    )
                  
                  -
                  
                    (
                    
                      
                        k
                        2
                      
                      ⁢
                      i
                      ⁢
                      
                         
                      
                      ⁢
                      ω
                      ⁢
                      
                         
                      
                      ⁢
                      
                        c
                        2
                      
                    
                    )
                  
                
                ]
              
            
            ⁢
            
              (
              
                EI
                ⁢
                
                   
                
                ⁢
                
                  α
                  3
                
              
              )
            
            ⁢
            
              
                cos
                2
              
              ⁡
              
                (
                
                  α
                  ⁢
                  
                    L
                    2
                  
                
                )
              
            
            ⁢
            
              cosh
              ⁡
              
                (
                
                  α
                  ⁢
                  
                    L
                    2
                  
                
                )
              
            
            ⁢
            
              sinh
              ⁡
              
                (
                
                  α
                  ⁢
                  
                    L
                    2
                  
                
                )
              
            
          
          -
          
            
              2
              ⁡
              
                [
                
                  
                    (
                    
                      
                        k
                        1
                      
                      +
                      
                        i
                        ⁢
                        
                           
                        
                        ⁢
                        ω
                        ⁢
                        
                           
                        
                        ⁢
                        
                          c
                          1
                        
                      
                    
                    )
                  
                  -
                  
                    (
                    
                      
                        k
                        2
                      
                      +
                      
                        i
                        ⁢
                        
                           
                        
                        ⁢
                        ω
                        ⁢
                        
                           
                        
                        ⁢
                        
                          c
                          2
                        
                      
                    
                    )
                  
                
                ]
              
            
            ⁢
            
              (
              
                EI
                ⁢
                
                   
                
                ⁢
                
                  α
                  3
                
              
              )
            
            ⁢
            
              cos
              ⁡
              
                (
                
                  α
                  ⁢
                  
                    L
                    2
                  
                
                )
              
            
            ⁢
            
              sin
              ⁡
              
                (
                
                  α
                  ⁢
                  
                    L
                    2
                  
                
                )
              
            
            ⁢
            
              
                cosh
                2
              
              ⁡
              
                (
                
                  α
                  ⁢
                  
                    L
                    2
                  
                
                )
              
            
          
          -
          
            8
            ⁢
            
              (
              
                
                  k
                  1
                
                +
                
                  i
                  ⁢
                  
                     
                  
                  ⁢
                  ω
                  ⁢
                  
                     
                  
                  ⁢
                  
                    c
                    1
                  
                
              
              )
            
            ⁢
            
              (
              
                
                  k
                  2
                
                +
                
                  i
                  ⁢
                  
                     
                  
                  ⁢
                  ω
                  ⁢
                  
                     
                  
                  ⁢
                  
                    c
                    2
                  
                
              
              )
            
            ⁢
            
              cos
              ⁡
              
                (
                
                  α
                  ⁢
                  
                    L
                    2
                  
                
                )
              
            
            ⁢
            
              sin
              ⁡
              
                (
                
                  α
                  ⁢
                  
                    L
                    2
                  
                
                )
              
            
            ⁢
            
              cosh
              ⁡
              
                (
                
                  α
                  ⁢
                  
                    L
                    2
                  
                
                )
              
            
            ⁢
            
              sinh
              ⁡
              
                (
                
                  α
                  ⁢
                  
                    L
                    2
                  
                
                )
              
            
          
        
      
    
    
      
        (
        12
        )
      
    
  




The wave coefficient B is 



  
    
      
        
          B
          =
          
            
              B
              T
            
            
              B
              B
            
          
        
        ,
      
    
    
      
        (
        13
        )
      
    
  




where 



  
    
      
        
          B
          T
        
        =
        
          
            
              -
              
                [
                
                  
                    (
                    
                      
                        k
                        1
                      
                      +
                      
                        i
                        ⁢
                        
                           
                        
                        ⁢
                        ω
                        ⁢
                        
                           
                        
                        ⁢
                        
                          c
                          1
                        
                      
                    
                    )
                  
                  +
                  
                    (
                    
                      
                        k
                        2
                      
                      ⁢
                      i
                      ⁢
                      
                         
                      
                      ⁢
                      ω
                      ⁢
                      
                         
                      
                      ⁢
                      
                        c
                        2
                      
                    
                    )
                  
                
                ]
              
            
            ⁢
            
              (
              
                EI
                ⁢
                
                   
                
                ⁢
                
                  α
                  3
                
              
              )
            
            ⁢
            
              sin
              ⁡
              
                (
                
                  α
                  ⁢
                  
                    L
                    2
                  
                
                )
              
            
            ⁢
            
              cosh
              ⁡
              
                (
                
                  α
                  ⁢
                  
                    L
                    2
                  
                
                )
              
            
            ⁢
            
              sinh
              ⁡
              
                (
                
                  α
                  ⁢
                  
                    L
                    2
                  
                
                )
              
            
          
          -
          
             
            
              
                
                  [
                  
                    
                      (
                      
                        
                          k
                          1
                        
                        +
                        
                          i
                          ⁢
                          
                             
                          
                          ⁢
                          ω
                          ⁢
                          
                             
                          
                          ⁢
                          
                            c
                            1
                          
                        
                      
                      )
                    
                    +
                    
                      (
                      
                        
                          k
                          2
                        
                        +
                        
                          i
                          ⁢
                          
                             
                          
                          ⁢
                          ω
                          ⁢
                          
                             
                          
                          ⁢
                          
                            c
                            2
                          
                        
                      
                      )
                    
                  
                  ]
                
                ⁢
                
                  (
                  
                    EI
                    ⁢
                    
                       
                    
                    ⁢
                    
                      α
                      3
                    
                  
                  )
                
                ⁢
                
                  cos
                  ⁡
                  
                    (
                    
                      α
                      ⁢
                      
                        L
                        2
                      
                    
                    )
                  
                
                ⁢
                
                  
                    sinh
                    2
                  
                  ⁡
                  
                    (
                    
                      α
                      ⁢
                      
                        L
                        2
                      
                    
                    )
                  
                
              
              ,
              and
            
          
        
      
    
    
      
        (
        14
        )
      
    
  
  
    
      
        
          B
          B
        
        =
        
          
            2
            ⁢
            
              
                (
                
                  EI
                  ⁢
                  
                     
                  
                  ⁢
                  
                    α
                    3
                  
                
                )
              
              2
            
            ⁢
            
              
                cos
                2
              
              ⁡
              
                (
                
                  α
                  ⁢
                  
                    L
                    2
                  
                
                )
              
            
            ⁢
            
              
                sinh
                2
              
              ⁡
              
                (
                
                  α
                  ⁢
                  
                    L
                    2
                  
                
                )
              
            
          
          -
          
            
              2
              ⁡
              
                [
                
                  
                    (
                    
                      
                        k
                        1
                      
                      +
                      
                        i
                        ⁢
                        
                           
                        
                        ⁢
                        ω
                        ⁢
                        
                           
                        
                        ⁢
                        
                          c
                          1
                        
                      
                    
                    )
                  
                  -
                  
                    (
                    
                      
                        k
                        2
                      
                      +
                      
                        i
                        ⁢
                        
                           
                        
                        ⁢
                        ω
                        ⁢
                        
                           
                        
                        ⁢
                        
                          c
                          2
                        
                      
                    
                    )
                  
                
                ]
              
            
            ⁢
            
              (
              
                EI
                ⁢
                
                   
                
                ⁢
                
                  α
                  3
                
              
              )
            
            ⁢
            
              
                cos
                2
              
              ⁡
              
                (
                
                  α
                  ⁢
                  
                    L
                    2
                  
                
                )
              
            
            ⁢
            
              cosh
              ⁡
              
                (
                
                  α
                  ⁢
                  
                    L
                    2
                  
                
                )
              
            
            ⁢
            
              sinh
              ⁡
              
                (
                
                  α
                  ⁢
                  
                    L
                    2
                  
                
                )
              
            
          
          -
          
            2
            ⁢
            
              
                (
                
                  EI
                  ⁢
                  
                     
                  
                  ⁢
                  
                    α
                    3
                  
                
                )
              
              2
            
            ⁢
            
              
                sin
                2
              
              ⁡
              
                (
                
                  α
                  ⁢
                  
                    L
                    2
                  
                
                )
              
            
            ⁢
            
              
                cosh
                2
              
              ⁡
              
                (
                
                  α
                  ⁢
                  
                    L
                    2
                  
                
                )
              
            
          
          +
          
            
              2
              ⁡
              
                [
                
                  
                    (
                    
                      
                        k
                        1
                      
                      +
                      
                        i
                        ⁢
                        
                           
                        
                        ⁢
                        ω
                        ⁢
                        
                           
                        
                        ⁢
                        
                          c
                          1
                        
                      
                    
                    )
                  
                  -
                  
                    (
                    
                      
                        k
                        2
                      
                      +
                      
                        i
                        ⁢
                        
                           
                        
                        ⁢
                        ω
                        ⁢
                        
                           
                        
                        ⁢
                        
                          c
                          2
                        
                      
                    
                    )
                  
                
                ]
              
            
            ⁢
            
              (
              
                EI
                ⁢
                
                   
                
                ⁢
                
                  α
                  3
                
              
              )
            
            ⁢
            
              cos
              ⁡
              
                (
                
                  α
                  ⁢
                  
                    L
                    2
                  
                
                )
              
            
            ⁢
            
              sin
              ⁡
              
                (
                
                  α
                  ⁢
                  
                    L
                    2
                  
                
                )
              
            
            ⁢
            
              
                cosh
                2
              
              ⁡
              
                (
                
                  α
                  ⁢
                  
                    L
                    2
                  
                
                )
              
            
          
          +
          
            
              2
              ⁡
              
                [
                
                  
                    (
                    
                      
                        k
                        1
                      
                      +
                      
                        i
                        ⁢
                        
                           
                        
                        ⁢
                        ω
                        ⁢
                        
                           
                        
                        ⁢
                        
                          c
                          1
                        
                      
                    
                    )
                  
                  -
                  
                    (
                    
                      
                        k
                        2
                      
                      +
                      
                        i
                        ⁢
                        
                           
                        
                        ⁢
                        ω
                        ⁢
                        
                           
                        
                        ⁢
                        
                          c
                          2
                        
                      
                    
                    )
                  
                
                ]
              
            
            ⁢
            
              (
              
                EI
                ⁢
                
                   
                
                ⁢
                
                  α
                  3
                
              
              )
            
            ⁢
            
              
                sin
                2
              
              ⁡
              
                (
                
                  α
                  ⁢
                  
                    L
                    2
                  
                
                )
              
            
            ⁢
            
              cosh
              ⁡
              
                (
                
                  α
                  ⁢
                  
                    L
                    2
                  
                
                )
              
            
            ⁢
            
              sinh
              ⁡
              
                (
                
                  α
                  ⁢
                  
                    L
                    2
                  
                
                )
              
            
          
          +
          
            
              2
              ⁡
              
                [
                
                  
                    (
                    
                      
                        k
                        1
                      
                      +
                      
                        i
                        ⁢
                        
                           
                        
                        ⁢
                        ω
                        ⁢
                        
                           
                        
                        ⁢
                        
                          c
                          1
                        
                      
                    
                    )
                  
                  -
                  
                    (
                    
                      
                        k
                        2
                      
                      +
                      
                        i
                        ⁢
                        
                           
                        
                        ⁢
                        ω
                        ⁢
                        
                           
                        
                        ⁢
                        
                          c
                          2
                        
                      
                    
                    )
                  
                
                ]
              
            
            ⁢
            
              (
              
                EI
                ⁢
                
                   
                
                ⁢
                
                  α
                  3
                
              
              )
            
            ⁢
            
              cos
              ⁡
              
                (
                
                  α
                  ⁢
                  
                    L
                    2
                  
                
                )
              
            
            ⁢
            
              sin
              ⁡
              
                (
                
                  α
                  ⁢
                  
                    L
                    2
                  
                
                )
              
            
            ⁢
            
              
                sinh
                2
              
              ⁡
              
                (
                
                  α
                  ⁢
                  
                    L
                    2
                  
                
                )
              
            
          
          +
          
            8
            ⁢
            
              (
              
                
                  k
                  1
                
                +
                
                  i
                  ⁢
                  
                     
                  
                  ⁢
                  ω
                  ⁢
                  
                     
                  
                  ⁢
                  
                    c
                    1
                  
                
              
              )
            
            ⁢
            
              (
              
                
                  k
                  2
                
                +
                
                  i
                  ⁢
                  
                     
                  
                  ⁢
                  ω
                  ⁢
                  
                     
                  
                  ⁢
                  
                    c
                    2
                  
                
              
              )
            
            ⁢
            
              cos
              ⁡
              
                (
                
                  α
                  ⁢
                  
                    L
                    2
                  
                
                )
              
            
            ⁢
            
              sin
              ⁡
              
                (
                
                  α
                  ⁢
                  
                    L
                    2
                  
                
                )
              
            
            ⁢
            
              cosh
              ⁡
              
                (
                
                  α
                  ⁢
                  
                    L
                    2
                  
                
                )
              
            
            ⁢
            
              sinh
              ⁡
              
                (
                
                  α
                  ⁢
                  
                    L
                    2
                  
                
                )
              
            
          
        
      
    
    
      
        (
        15
        )
      
    
  




The wave coefficient C is 



  
    
      
        
          C
          =
          
            
              C
              T
            
            
              A
              B
            
          
        
        ,
      
    
    
      
        (
        16
        )
      
    
  




where 



  
    
      
        
          C
          T
        
        =
        
          
            
              [
              
                
                  (
                  
                    
                      k
                      1
                    
                    +
                    
                      i
                      ⁢
                      
                         
                      
                      ⁢
                      ω
                      ⁢
                      
                         
                      
                      ⁢
                      
                        c
                        1
                      
                    
                  
                  )
                
                -
                
                  (
                  
                    
                      k
                      2
                    
                    +
                    
                      i
                      ⁢
                      
                         
                      
                      ⁢
                      ω
                      ⁢
                      
                         
                      
                      ⁢
                      
                        c
                        2
                      
                    
                  
                  )
                
              
              ]
            
            ⁢
            
              (
              
                EI
                ⁢
                
                   
                
                ⁢
                
                  α
                  3
                
              
              )
            
            ⁢
            
              
                cos
                2
              
              ⁡
              
                (
                
                  α
                  ⁢
                  
                    L
                    2
                  
                
                )
              
            
            ⁢
            
              sinh
              ⁡
              
                (
                
                  α
                  ⁢
                  
                    L
                    2
                  
                
                )
              
            
          
          -
          
            
               
              
                
                  
                    [
                    
                      
                        (
                        
                          
                            k
                            1
                          
                          +
                          
                            i
                            ⁢
                            
                               
                            
                            ⁢
                            ω
                            ⁢
                            
                               
                            
                            ⁢
                            
                              c
                              1
                            
                          
                        
                        )
                      
                      -
                      
                        (
                        
                          
                            k
                            2
                          
                          +
                          
                            i
                            ⁢
                            
                               
                            
                            ⁢
                            ω
                            ⁢
                            
                               
                            
                            ⁢
                            
                              c
                              2
                            
                          
                        
                        )
                      
                    
                    ]
                  
                  ⁢
                  
                    (
                    
                      EI
                      ⁢
                      
                         
                      
                      ⁢
                      
                        α
                        3
                      
                    
                    )
                  
                  ⁢
                  
                    cos
                    ⁡
                    
                      (
                      
                        α
                        ⁢
                        
                          L
                          2
                        
                      
                      )
                    
                  
                  ⁢
                  
                    sin
                    ⁡
                    
                      (
                      
                        α
                        ⁢
                        
                          L
                          2
                        
                      
                      )
                    
                  
                  ⁢
                  
                    cosh
                    ⁡
                    
                      (
                      
                        α
                        ⁢
                        
                          L
                          2
                        
                      
                      )
                    
                  
                
                -
                
                  4
                  ⁢
                  
                    (
                    
                      
                        k
                        1
                      
                      +
                      
                        i
                        ⁢
                        
                           
                        
                        ⁢
                        ω
                        ⁢
                        
                           
                        
                        ⁢
                        
                          c
                          1
                        
                      
                    
                    )
                  
                  ⁢
                  
                    (
                    
                      
                        k
                        2
                      
                      +
                      
                        i
                        ⁢
                        
                           
                        
                        ⁢
                        ω
                        ⁢
                        
                           
                        
                        ⁢
                        
                          c
                          2
                        
                      
                    
                    )
                  
                
              
              ]
            
            ⁢
            
              cos
              ⁡
              
                (
                
                  α
                  ⁢
                  
                    L
                    2
                  
                
                )
              
            
            ⁢
            
              sin
              ⁡
              
                (
                
                  α
                  ⁢
                  
                    L
                    2
                  
                
                )
              
            
            ⁢
            
              
                sinh
                ⁡
                
                  (
                  
                    α
                    ⁢
                    
                      L
                      2
                    
                  
                  )
                
              
              .
            
          
        
      
    
    
      
        (
        17
        )
      
    
  




The wave coefficient D is 



  
    
      
        
          D
          =
          
            
              D
              T
            
            
              B
              B
            
          
        
        ,
      
    
    
      
        (
        18
        )
      
    
  




where 



  
    
      
        
          D
          T
        
        =
        
          
            
              -
              
                [
                
                  
                    (
                    
                      
                        k
                        1
                      
                      +
                      
                        i
                        ⁢
                        
                           
                        
                        ⁢
                        ω
                        ⁢
                        
                           
                        
                        ⁢
                        
                          c
                          1
                        
                      
                    
                    )
                  
                  +
                  
                    (
                    
                      
                        k
                        2
                      
                      +
                      
                        i
                        ⁢
                        
                           
                        
                        ⁢
                        ω
                        ⁢
                        
                           
                        
                        ⁢
                        
                          c
                          2
                        
                      
                    
                    )
                  
                
                ]
              
            
            ⁢
            
              (
              
                EI
                ⁢
                
                   
                
                ⁢
                
                  α
                  3
                
              
              )
            
            ⁢
            
              
                sin
                2
              
              ⁡
              
                (
                
                  α
                  ⁢
                  
                    L
                    2
                  
                
                )
              
            
            ⁢
            
              cosh
              ⁡
              
                (
                
                  α
                  ⁢
                  
                    L
                    2
                  
                
                )
              
            
          
          -
          
             
            
              
                [
                
                  
                    (
                    
                      
                        k
                        1
                      
                      +
                      
                        i
                        ⁢
                        
                           
                        
                        ⁢
                        ω
                        ⁢
                        
                           
                        
                        ⁢
                        
                          c
                          1
                        
                      
                    
                    )
                  
                  +
                  
                    (
                    
                      
                        k
                        2
                      
                      +
                      
                        i
                        ⁢
                        
                           
                        
                        ⁢
                        ω
                        ⁢
                        
                           
                        
                        ⁢
                        
                          c
                          2
                        
                      
                    
                    )
                  
                
                ]
              
              ⁢
              
                (
                
                  EI
                  ⁢
                  
                     
                  
                  ⁢
                  
                    α
                    3
                  
                
                )
              
              ⁢
              
                cos
                ⁡
                
                  (
                  
                    α
                    ⁢
                    
                      L
                      2
                    
                  
                  )
                
              
              ⁢
              
                sin
                ⁡
                
                  (
                  
                    α
                    ⁢
                    
                      L
                      2
                    
                  
                  )
                
              
              ⁢
              
                
                  sinh
                  ⁡
                  
                    (
                    
                      α
                      ⁢
                      
                        L
                        2
                      
                    
                    )
                  
                
                .
              
            
          
        
      
    
    
      
        (
        19
        )
      
    
  




These coefficients are used for the simulation below. If the beam model corresponds to FIGS. 2, 3, or 4, then the boundary conditions given in equations (5)-(8) change slightly as do the wave propagation coefficients.
Equation (3) has five unknowns and is nonlinear with respect to the unknown flexural wavenumber. It will be shown that using seven independent, equally spaced measurements, that the five unknowns can be estimated with closed form solutions. Furthermore, in the next section, it will be shown that the components that comprise the beams mounting system can also be estimated. Seven frequency domain transfer functions of displacement are now measured. These consist of the measurement at some location divided by a common measurement. Typically this would be an accelerometer at a measurement location and an accelerometer at the base of a shaker table. These seven measurements are set equal the theoretical expression given in equation (3) and are listed as 



  
    
      
        
          
            T
            
              -
              3
            
          
          =
          
            
              
                
                  U
                  
                    -
                    3
                  
                
                ⁡
                
                  (
                  
                    
                      
                        -
                        3
                      
                      ⁢
                      
                         
                      
                      ⁢
                      δ
                    
                    ,
                    ω
                  
                  )
                
              
              
                
                  V
                  0
                
                ⁡
                
                  (
                  ω
                  )
                
              
            
            =
            
              
                A
                ⁢
                
                   
                
                ⁢
                
                  cos
                  ⁡
                  
                    (
                    
                      3
                      ⁢
                      
                         
                      
                      ⁢
                      α
                      ⁢
                      
                         
                      
                      ⁢
                      δ
                    
                    )
                  
                
              
              -
              
                B
                ⁢
                
                   
                
                ⁢
                
                  sin
                  ⁡
                  
                    (
                    
                      3
                      ⁢
                      
                         
                      
                      ⁢
                      α
                      ⁢
                      
                         
                      
                      ⁢
                      δ
                    
                    )
                  
                
              
              +
              
                C
                ⁢
                
                   
                
                ⁢
                
                  cosh
                  ⁡
                  
                    (
                    
                      3
                      ⁢
                      
                         
                      
                      ⁢
                      α
                      ⁢
                      
                         
                      
                      ⁢
                      δ
                    
                    )
                  
                
              
              -
              
                D
                ⁢
                
                   
                
                ⁢
                
                  sinh
                  ⁡
                  
                    (
                    
                      3
                      ⁢
                      
                         
                      
                      ⁢
                      αδ
                    
                    )
                  
                
              
            
          
        
        ,
      
    
    
      
        (
        20
        )
      
    
  
  
    
      
        
          
            T
            
              -
              2
            
          
          =
          
            
              
                
                  U
                  
                    -
                    2
                  
                
                ⁡
                
                  (
                  
                    
                      
                        -
                        2
                      
                      ⁢
                      
                         
                      
                      ⁢
                      δ
                    
                    ,
                    ω
                  
                  )
                
              
              
                
                  V
                  0
                
                ⁡
                
                  (
                  ω
                  )
                
              
            
            =
            
              
                A
                ⁢
                
                   
                
                ⁢
                
                  cos
                  ⁡
                  
                    (
                    
                      2
                      ⁢
                      
                         
                      
                      ⁢
                      α
                      ⁢
                      
                         
                      
                      ⁢
                      δ
                    
                    )
                  
                
              
              -
              
                B
                ⁢
                
                   
                
                ⁢
                
                  sin
                  ⁡
                  
                    (
                    
                      2
                      ⁢
                      
                         
                      
                      ⁢
                      α
                      ⁢
                      
                         
                      
                      ⁢
                      δ
                    
                    )
                  
                
              
              +
              
                C
                ⁢
                
                   
                
                ⁢
                
                  cosh
                  ⁡
                  
                    (
                    
                      2
                      ⁢
                      
                         
                      
                      ⁢
                      α
                      ⁢
                      
                         
                      
                      ⁢
                      δ
                    
                    )
                  
                
              
              -
              
                D
                ⁢
                
                   
                
                ⁢
                
                  sinh
                  ⁡
                  
                    (
                    
                      2
                      ⁢
                      
                         
                      
                      ⁢
                      αδ
                    
                    )
                  
                
              
            
          
        
        ,
      
    
    
      
        (
        21
        )
      
    
  
  
    
      
        
          
            T
            
              -
              1
            
          
          =
          
            
              
                
                  U
                  
                    -
                    1
                  
                
                ⁡
                
                  (
                  
                    
                      -
                      
                         
                      
                      ⁢
                      δ
                    
                    ,
                    ω
                  
                  )
                
              
              
                
                  V
                  0
                
                ⁡
                
                  (
                  ω
                  )
                
              
            
            =
            
              
                A
                ⁢
                
                   
                
                ⁢
                
                  cos
                  ⁡
                  
                    (
                    
                      α
                      ⁢
                      
                         
                      
                      ⁢
                      δ
                    
                    )
                  
                
              
              -
              
                B
                ⁢
                
                   
                
                ⁢
                
                  sin
                  ⁡
                  
                    (
                    
                      α
                      ⁢
                      
                         
                      
                      ⁢
                      δ
                    
                    )
                  
                
              
              +
              
                C
                ⁢
                
                   
                
                ⁢
                
                  cosh
                  ⁡
                  
                    (
                    
                      α
                      ⁢
                      
                         
                      
                      ⁢
                      δ
                    
                    )
                  
                
              
              -
              
                D
                ⁢
                
                   
                
                ⁢
                
                  sinh
                  ⁡
                  
                    (
                    αδ
                    )
                  
                
              
            
          
        
        ,
      
    
    
      
        (
        22
        )
      
    
  
  
    
      
        
          
            T
            0
          
          =
          
            
              
                
                  U
                  0
                
                ⁡
                
                  (
                  
                    0
                    ,
                    ω
                  
                  )
                
              
              
                
                  V
                  0
                
                ⁡
                
                  (
                  ω
                  )
                
              
            
            =
            
              A
              +
              C
            
          
        
        ,
      
    
    
      
        (
        23
        )
      
    
  
  
    
      
        
          
            T
            1
          
          =
          
            
              
                
                  
                    U
                    1
                  
                  ⁡
                  
                    (
                    
                      δ
                      ,
                      ω
                    
                    )
                  
                
                
                  
                    V
                    0
                  
                  ⁡
                  
                    (
                    ω
                    )
                  
                
              
              ⁢
              A
              ⁢
              
                 
              
              ⁢
              
                cos
                ⁡
                
                  (
                  αδ
                  )
                
              
            
            +
            
              B
              ⁢
              
                 
              
              ⁢
              
                sin
                ⁡
                
                  (
                  
                    α
                    ⁢
                    
                       
                    
                    ⁢
                    δ
                  
                  )
                
              
            
            +
            
              C
              ⁢
              
                 
              
              ⁢
              
                cosh
                ⁡
                
                  (
                  αδ
                  )
                
              
            
            +
            
              D
              ⁢
              
                 
              
              ⁢
              
                sinh
                ⁡
                
                  (
                  
                    α
                    ⁢
                    
                       
                    
                    ⁢
                    δ
                  
                  )
                
              
            
          
        
        ,
      
    
    
      
        (
        24
        )
      
    
  
  
    
      
        
          
            T
            2
          
          =
          
            
              
                
                  
                    U
                    2
                  
                  ⁡
                  
                    (
                    
                      
                        2
                        ⁢
                        δ
                      
                      ,
                      ω
                    
                    )
                  
                
                
                  
                    V
                    0
                  
                  ⁡
                  
                    (
                    ω
                    )
                  
                
              
              ⁢
              A
              ⁢
              
                 
              
              ⁢
              
                cos
                ⁡
                
                  (
                  
                    2
                    ⁢
                    αδ
                  
                  )
                
              
            
            +
            
              B
              ⁢
              
                 
              
              ⁢
              
                sin
                ⁡
                
                  (
                  
                    2
                    ⁢
                    α
                    ⁢
                    
                       
                    
                    ⁢
                    δ
                  
                  )
                
              
            
            +
            
              C
              ⁢
              
                 
              
              ⁢
              
                cosh
                ⁡
                
                  (
                  
                    2
                    ⁢
                    αδ
                  
                  )
                
              
            
            +
            
              D
              ⁢
              
                 
              
              ⁢
              
                sinh
                ⁡
                
                  (
                  
                    2
                    ⁢
                    α
                    ⁢
                    
                       
                    
                    ⁢
                    δ
                  
                  )
                
              
            
          
        
        ,
        
          

        
        ⁢
        and
      
    
    
      
        (
        25
        )
      
    
  
  
    
      
        
          
            T
            3
          
          =
          
            
              
                
                  
                    U
                    3
                  
                  ⁡
                  
                    (
                    
                      
                        3
                        ⁢
                        δ
                      
                      ,
                      ω
                    
                    )
                  
                
                
                  
                    V
                    0
                  
                  ⁡
                  
                    (
                    ω
                    )
                  
                
              
              ⁢
              A
              ⁢
              
                 
              
              ⁢
              
                cos
                ⁡
                
                  (
                  
                    3
                    ⁢
                    αδ
                  
                  )
                
              
            
            +
            
              B
              ⁢
              
                 
              
              ⁢
              
                sin
                ⁡
                
                  (
                  
                    3
                    ⁢
                    α
                    ⁢
                    
                       
                    
                    ⁢
                    δ
                  
                  )
                
              
            
            +
            
              C
              ⁢
              
                 
              
              ⁢
              
                cosh
                ⁡
                
                  (
                  
                    3
                    ⁢
                    αδ
                  
                  )
                
              
            
            +
            
              D
              ⁢
              
                 
              
              ⁢
              
                sinh
                ⁡
                
                  (
                  
                    3
                    ⁢
                    α
                    ⁢
                    
                       
                    
                    ⁢
                    δ
                  
                  )
                
              
            
          
        
        ,
      
    
    
      
        (
        26
        )
      
    
  




where δ is the sensor to sensor separation distance (m) and V0(ω) is the reference measurement. Note that the units of the transfer functions given in equations (20)-(26) are dimensionless.

Equation (22) is now subtracted from equation (24), equation (21) is subtracted from equation (25), and equation (20) is subtracted from equation (26), yielding the following three equations: 



  
    
      
        
          
            
              B
              ⁢
              
                 
              
              ⁢
              sin
              ⁢
              
                (
                
                  α
                  ⁢
                  
                     
                  
                  ⁢
                  δ
                
                )
              
            
            +
            
              D
              ⁢
              
                 
              
              ⁢
              
                sinh
                ⁡
                
                  (
                  
                    α
                    ⁢
                    
                       
                    
                    ⁢
                    δ
                  
                  )
                
              
            
          
          =
          
            
              
                T
                1
              
              -
              
                T
                
                  -
                  1
                
              
            
            2
          
        
        ,
      
    
    
      
        (
        27
        )
      
    
  
  
    
      
        
          
            
              B
              ⁢
              
                 
              
              ⁢
              
                sin
                ⁡
                
                  (
                  
                    2
                    ⁢
                    α
                    ⁢
                    
                       
                    
                    ⁢
                    δ
                  
                  )
                
              
            
            +
            
              D
              ⁢
              
                 
              
              ⁢
              
                sinh
                ⁡
                
                  (
                  
                    2
                    ⁢
                    α
                    ⁢
                    
                       
                    
                    ⁢
                    δ
                  
                  )
                
              
            
          
          =
          
            
              
                T
                2
              
              -
              
                T
                
                  -
                  2
                
              
            
            2
          
        
        ,
        and
      
    
    
      
        (
        28
        )
      
    
  
  
    
      
        
          
            B
            ⁢
            
               
            
            ⁢
            
              sin
              ⁡
              
                (
                
                  3
                  ⁢
                  α
                  ⁢
                  
                     
                  
                  ⁢
                  δ
                
                )
              
            
          
          +
          
            D
            ⁢
            
               
            
            ⁢
            
              sinh
              ⁡
              
                (
                
                  3
                  ⁢
                  α
                  ⁢
                  
                     
                  
                  ⁢
                  δ
                
                )
              
            
          
        
        =
        
          
            
              
                T
                3
              
              -
              
                T
                
                  -
                  3
                
              
            
            2
          
          .
        
      
    
    
      
        (
        29
        )
      
    
  




Equations (27), (28), and (29) are now combined to give 



  
    
      
        
          
            
              cosh
              ⁡
              
                (
                αδ
                )
              
            
            ⁢
            
              cos
              ⁡
              
                (
                αδ
                )
              
            
          
          -
          
            
              [
              
                 
              
              ⁢
              
                
                  
                    T
                    2
                  
                  -
                  
                    T
                    
                      -
                      2
                    
                  
                
                
                  2
                  ⁢
                  
                    (
                    
                      
                        T
                        1
                      
                      -
                      
                        T
                        
                          -
                          1
                        
                      
                    
                    )
                  
                
              
              ]
            
            [
            
               
            
            ⁢
            
              
                cosh
                ⁡
                
                  (
                  αδ
                  )
                
              
              +
              
                cos
                ⁡
                
                  (
                  αδ
                  )
                
              
            
            ]
          
          +
          
            [
            
               
            
            ⁢
            
              
                
                  T
                  3
                
                -
                
                  T
                  
                    -
                    3
                  
                
                +
                
                  T
                  1
                
                -
                
                  T
                  
                    -
                    1
                  
                
              
              
                4
                ⁢
                
                  (
                  
                    
                      T
                      1
                    
                    -
                    
                      T
                      
                        -
                        1
                      
                    
                  
                  )
                
              
            
            ]
          
        
        =
        0.
      
    
    
      
        (
        30
        )
      
    
  




Equation (22) is now added to equation (24) and equation (21) is added to equation (25), yielding the following two equations: 



  
    
      
        
          
            
              A
              ⁢
              
                 
              
              ⁢
              
                cos
                ⁡
                
                  (
                  αδ
                  )
                
              
            
            +
            
              C
              ⁢
              
                 
              
              ⁢
              
                cosh
                ⁡
                
                  (
                  αδ
                  )
                
              
            
          
          =
          
            
              
                T
                1
              
              +
              
                T
                
                  -
                  1
                
              
            
            2
          
        
        ,
      
    
    
      
        (
        31
        )
      
    
  




and 



  
    
      
        
          
            A
            ⁢
            
               
            
            ⁢
            
              cos
              ⁡
              
                (
                
                  2
                  ⁢
                  αδ
                
                )
              
            
          
          +
          
            C
            ⁢
            
               
            
            ⁢
            
              cosh
              ⁡
              
                (
                
                  2
                  ⁢
                  αδ
                
                )
              
            
          
        
        =
        
          
            
              
                T
                2
              
              +
              
                T
                
                  -
                  2
                
              
            
            2
          
          .
        
      
    
    
      
        (
        32
        )
      
    
  




Equations (23), (31), and (32) are now combined to yield the following equation: 



  
    
      
        
          
            
              cosh
              ⁡
              
                (
                αδ
                )
              
            
            ⁢
            
              cos
              ⁡
              
                (
                αδ
                )
              
            
          
          -
          
            
              [
              
                 
              
              ⁢
              
                
                  
                    T
                    2
                  
                  -
                  
                    T
                    
                      -
                      2
                    
                  
                
                
                  2
                  ⁢
                  
                    T
                    0
                  
                
              
              ]
            
            [
            
               
            
            ⁢
            
              
                cosh
                ⁡
                
                  (
                  αδ
                  )
                
              
              +
              
                cos
                ⁡
                
                  (
                  αδ
                  )
                
              
            
            ]
          
          +
          
            [
            
               
            
            ⁢
            
              
                
                  T
                  2
                
                +
                
                  T
                  
                    -
                    2
                  
                
                +
                
                  2
                  ⁢
                  
                    T
                    0
                  
                
              
              
                
                  4
                  ⁢
                  
                    T
                    0
                  
                
                )
              
            
            ]
          
        
        =
        0.
      
    
    
      
        (
        33
        )
      
    
  




Equation (30) and (33) are now combined, and the result is a binomial expression with respect to the cosine function, and is written as

αcos2(αδ)+b cos(αδ)+c=0,  (34)

where

a=4T12−4T−12+4T−2T0−4T0T2,  (35)

b=2T−2T−1−2T−2T1+2T−1T0−2T0T1+2T−1T2−2T1T2+2T0T3−2T−3T0,  (36)

and

c=T−12−T12+T22−T−22+T−3T−1−T−1T3+T−3T1−T1T3+2T0T2−2T−2T0.  (37)

Equation (34) is now solved using 



  
    
      
        
          
            cos
            ⁡
            
              (
              αδ
              )
            
          
          =
          
            
              
                
                  -
                  b
                
                ±
                
                  
                    
                      b
                      2
                    
                    -
                    
                      4
                      ⁢
                      
                         
                      
                      ⁢
                      ac
                    
                  
                
              
              
                2
                ⁢
                a
              
            
            =
            φ
          
        
        ,
      
    
    
      
        (
        38
        )
      
    
  




where φ is typically a complex number. Equation (38) is two solutions to equation (34). One, however, will have an absolute value less than one and that is the root that is further manipulated. The inversion of equation (38) allows the complex flexural wavenumber α to be solved as a function of φ at every frequency in which a measurement is made. The solution to the real part of α is 



  
    
      
        
          Re
          ⁡
          
            (
            α
            )
          
        
        =
        
          {
          
            
              
                
                  
                    
                      
                        1
                        
                          2
                          ⁢
                          δ
                        
                      
                      ⁢
                      Arc
                      ⁢
                      
                         
                      
                      ⁢
                      
                        cos
                        ⁡
                        
                          (
                          s
                          )
                        
                      
                    
                    +
                    
                      
                        n
                        ⁢
                        
                           
                        
                        ⁢
                        π
                      
                      
                        2
                        ⁢
                        δ
                      
                    
                  
                
                
                  
                    n
                    ⁢
                    
                       
                    
                    ⁢
                    even
                  
                
              
              
                
                  
                    
                      
                        1
                        
                          2
                          ⁢
                          δ
                        
                      
                      ⁢
                      Arc
                      ⁢
                      
                         
                      
                      ⁢
                      
                        cos
                        ⁡
                        
                          (
                          
                            -
                            s
                          
                          )
                        
                      
                    
                    +
                    
                      
                        n
                        ⁢
                        
                           
                        
                        ⁢
                        π
                      
                      
                        2
                        ⁢
                        δ
                      
                    
                  
                
                
                  
                    n
                    ⁢
                    
                       
                    
                    ⁢
                    odd
                  
                
              
            
            ,
          
        
      
    
    
      
        (
        39
        )
      
    
  




where 



  
    
      
        
          s
          =
          
            
              
                [
                
                  Re
                  ⁡
                  
                    (
                    φ
                    )
                  
                
                ]
              
              2
            
            +
            
              
                [
                
                  Im
                  ⁡
                  
                    (
                    φ
                    )
                  
                
                ]
              
              2
            
            -
            
              
                
                  
                    {
                    
                      
                        
                          [
                          
                            Re
                            ⁡
                            
                              (
                              φ
                              )
                            
                          
                          ]
                        
                        2
                      
                      +
                      
                        
                          [
                          
                            Im
                            ⁡
                            
                              (
                              φ
                              )
                            
                          
                          ]
                        
                        2
                      
                    
                    }
                  
                  2
                
                -
                
                  {
                  
                    
                      
                        2
                        ⁡
                        
                          [
                          
                            Re
                            ⁡
                            
                              (
                              φ
                              )
                            
                          
                          ]
                        
                      
                      2
                    
                    -
                    
                      
                        2
                        ⁡
                        
                          [
                          
                            Im
                            ⁡
                            
                              (
                              φ
                              )
                            
                          
                          ]
                        
                      
                      2
                    
                    -
                    1
                  
                  }
                
              
            
          
        
        ,
      
    
    
      
        (
        40
        )
      
    
  




n is a non-negative integer and the capital A denotes the principal value of the inverse cosine function. The value of n is determined from the function s, which is a periodically varying cosine function with respect to frequency. At zero frequency, n is 0. Every time s cycles through π radians (180 degrees), n is increased by 1. When the solution to the real part of α is found, the solution to the imaginary part of α is then written as 



  
    
      
        
          Im
          ⁡
          
            (
            α
            )
          
        
        =
        
          
            1
            δ
          
          ⁢
          
            log
            e
          
          ⁢
          
            
              {
              
                
                  
                    Re
                    ⁡
                    
                      (
                      φ
                      )
                    
                  
                  
                    cos
                    ⁡
                    
                      [
                      
                        
                          Re
                          ⁡
                          
                            (
                            α
                            )
                          
                        
                        ⁢
                        δ
                      
                      ]
                    
                  
                
                -
                
                  
                    Im
                    ⁡
                    
                      (
                      φ
                      )
                    
                  
                  
                    sin
                    ⁡
                    
                      [
                      
                        
                          Re
                          ⁡
                          
                            (
                            α
                            )
                          
                        
                        ⁢
                        δ
                      
                      ]
                    
                  
                
              
              }
            
            .
          
        
      
    
    
      
        (
        41
        )
      
    
  




Once the real and imaginary parts of wavenumber α are known, the complex valued modulus of elasticity can be determined at each frequency with 



  
    
      
        
          E
          ⁡
          
            (
            ω
            )
          
        
        =
        
          
            
              Re
              ⁡
              
                [
                
                  E
                  ⁡
                  
                    (
                    ω
                    )
                  
                
                ]
              
            
            +
            
              i
              ⁢
              
                 
              
              ⁢
              
                Im
                ⁡
                
                  [
                  
                    E
                    ⁡
                    
                      (
                      ω
                      )
                    
                  
                  ]
                
              
            
          
          =
          
            
              
                ρ
                ⁢
                
                   
                
                ⁢
                
                  A
                  b
                
                ⁢
                
                  ω
                  2
                
              
              
                
                  I
                  ⁡
                  
                    [
                    
                      
                        Re
                        ⁡
                        
                          (
                          α
                          )
                        
                      
                      +
                      
                        i
                        ⁢
                        
                           
                        
                        ⁢
                        
                          Im
                          ⁡
                          
                            (
                            α
                            )
                          
                        
                      
                    
                    ]
                  
                
                4
              
            
            .
          
        
      
    
    
      
        (
        42
        )
      
    
  




assuming that the density, area, and moment of inertia of the beam are known. Equations (20)-(42) produce an estimate Young's modulus at every frequency in which a measurement is conducted.

Additionally, combining equations (27) and (28) yields 



  
    
      
        B
        =
        
          
            
              2
              ⁢
              
                (
                
                  
                    T
                    1
                  
                  -
                  
                    T
                    
                      -
                      1
                    
                  
                
                )
              
              ⁢
              
                cosh
                ⁡
                
                  (
                  αδ
                  )
                
              
            
            -
            
              (
              
                
                  T
                  2
                
                -
                
                  T
                  
                    -
                    2
                  
                
              
              )
            
          
          
            4
            ⁢
            
               
            
            ⁢
            
              
                sin
                ⁡
                
                  (
                  αδ
                  )
                
              
              ⁡
              
                [
                
                  
                    cosh
                    ⁡
                    
                      (
                      αδ
                      )
                    
                  
                  -
                  
                    cos
                    ⁡
                    
                      (
                      αδ
                      )
                    
                  
                
                ]
              
            
          
        
      
    
    
      
        (
        43
        )
      
    
  




and 



  
    
      
        D
        =
        
          
            
              
                (
                
                  
                    T
                    2
                  
                  -
                  
                    T
                    
                      -
                      2
                    
                  
                
                )
              
              -
              
                2
                ⁢
                
                  (
                  
                    
                      T
                      1
                    
                    -
                    
                      T
                      
                        -
                        1
                      
                    
                  
                  )
                
                ⁢
                
                  cos
                  ⁡
                  
                    (
                    αδ
                    )
                  
                
              
            
            
              4
              ⁢
              
                 
              
              ⁢
              
                
                  sinh
                  ⁡
                  
                    (
                    αδ
                    )
                  
                
                ⁡
                
                  [
                  
                    
                      cosh
                      ⁡
                      
                        (
                        αδ
                        )
                      
                    
                    -
                    
                      cos
                      ⁡
                      
                        (
                        αδ
                        )
                      
                    
                  
                  ]
                
              
            
          
          .
        
      
    
    
      
        (
        44
        )
      
    
  




Combining equations (23) and (31) yields 



  
    
      
        A
        =
        
          
            
              2
              ⁢
              
                T
                0
              
              ⁢
              
                cosh
                ⁡
                
                  (
                  αδ
                  )
                
              
            
            -
            
              (
              
                
                  T
                  1
                
                +
                
                  T
                  
                    -
                    1
                  
                
              
              )
            
          
          
            2
            ⁡
            
              [
              
                
                  cosh
                  ⁡
                  
                    (
                    αδ
                    )
                  
                
                -
                
                  cos
                  ⁡
                  
                    (
                    αδ
                    )
                  
                
              
              ]
            
          
        
      
    
    
      
        (
        45
        )
      
    
  




and 



  
    
      
        C
        =
        
          
            
              
                (
                
                  
                    T
                    1
                  
                  +
                  
                    T
                    
                      -
                      1
                    
                  
                
                )
              
              -
              
                2
                ⁢
                
                  T
                  0
                
                ⁢
                
                  cos
                  ⁡
                  
                    (
                    αδ
                    )
                  
                
              
            
            
              2
              ⁡
              
                [
                
                  
                    cosh
                    ⁡
                    
                      (
                      αδ
                      )
                    
                  
                  -
                  
                    cos
                    ⁡
                    
                      (
                      αδ
                      )
                    
                  
                
                ]
              
            
          
          .
        
      
    
    
      
        (
        46
        )
      
    
  




Equations (43)-(46) are the estimates of the complex wave propagation coefficients. These are normally considered less important than the estimate of the flexural wavenumber. It will be shown, however, that these coefficients can be used to estimate the boundary condition parameters of the beam.

Inserting equations (2), (3), (4), and (9) into equation (6) and solving for the boundary parameters at x=−L/2 yields 



  
    
      
        
          k
          1
        
        =
        
          Re
          ⁢
          
            {
            
              
                
                  (
                  
                    EI
                    ⁢
                    
                       
                    
                    ⁢
                    
                      α
                      3
                    
                  
                  )
                
                ⁢
                
                  ⌊
                  
                    
                      A
                      ⁢
                      
                         
                      
                      ⁢
                      
                        sin
                        ⁡
                        
                          (
                          
                            α
                            ⁢
                            
                              L
                              2
                            
                          
                          )
                        
                      
                    
                    +
                    
                      B
                      ⁢
                      
                         
                      
                      ⁢
                      
                        cos
                        ⁡
                        
                          (
                          
                            α
                            ⁢
                            
                              L
                              2
                            
                          
                          )
                        
                      
                    
                    +
                    
                      C
                      ⁢
                      
                         
                      
                      ⁢
                      
                        sinh
                        ⁡
                        
                          (
                          
                            α
                            ⁢
                            
                              L
                              2
                            
                          
                          )
                        
                      
                    
                    -
                    
                      D
                      ⁢
                      
                         
                      
                      ⁢
                      
                        cosh
                        ⁡
                        
                          (
                          
                            α
                            ⁢
                            
                              L
                              2
                            
                          
                          )
                        
                      
                    
                  
                  ⌋
                
              
              
                [
                
                  
                    A
                    ⁢
                    
                       
                    
                    ⁢
                    
                      cos
                      ⁡
                      
                        (
                        
                          α
                          ⁢
                          
                            L
                            2
                          
                        
                        )
                      
                    
                  
                  -
                  
                    B
                    ⁢
                    
                       
                    
                    ⁢
                    
                      sin
                      ⁡
                      
                        (
                        
                          α
                          ⁢
                          
                            L
                            2
                          
                        
                        )
                      
                    
                  
                  +
                  
                    C
                    ⁢
                    
                       
                    
                    ⁢
                    
                      cosh
                      ⁡
                      
                        (
                        
                          α
                          ⁢
                          
                            L
                            2
                          
                        
                        )
                      
                    
                  
                  -
                  
                    D
                    ⁢
                    
                       
                    
                    ⁢
                    
                      sinh
                      ⁡
                      
                        (
                        
                          α
                          ⁢
                          
                            L
                            2
                          
                        
                        )
                      
                    
                  
                  -
                  1
                
                ]
              
            
            }
          
        
      
    
    
      
        (
        47
        )
      
    
  




and 



  
    
      
        
          c
          1
        
        =
        
          
            1
            ω
          
          ⁢
          Im
          ⁢
          
            
              {
              
                
                  
                    (
                    
                      EI
                      ⁢
                      
                         
                      
                      ⁢
                      
                        α
                        3
                      
                    
                    )
                  
                  [
                  
                    
                      A
                      ⁢
                      
                         
                      
                      ⁢
                      
                        sin
                        ⁡
                        
                          (
                          
                            α
                            ⁢
                            
                              L
                              2
                            
                          
                          )
                        
                      
                    
                    +
                    
                      B
                      ⁢
                      
                         
                      
                      ⁢
                      
                        cos
                        ⁡
                        
                          (
                          
                            α
                            ⁢
                            
                              L
                              2
                            
                          
                          )
                        
                      
                    
                    +
                    
                      C
                      ⁢
                      
                         
                      
                      ⁢
                      
                        sinh
                        ⁡
                        
                          (
                          
                            α
                            ⁢
                            
                              L
                              2
                            
                          
                          )
                        
                      
                    
                    -
                    
                      D
                      ⁢
                      
                         
                      
                      ⁢
                      
                        cosh
                        ⁡
                        
                          (
                          
                            α
                            ⁢
                            
                              L
                              2
                            
                          
                          )
                        
                      
                    
                  
                  ]
                
                
                  [
                  
                    
                      A
                      ⁢
                      
                         
                      
                      ⁢
                      
                        cos
                        ⁡
                        
                          (
                          
                            α
                            ⁢
                            
                              L
                              2
                            
                          
                          )
                        
                      
                    
                    -
                    
                      B
                      ⁢
                      
                         
                      
                      ⁢
                      
                        sin
                        ⁡
                        
                          (
                          
                            α
                            ⁢
                            
                              L
                              2
                            
                          
                          )
                        
                      
                    
                    +
                    
                      C
                      ⁢
                      
                         
                      
                      ⁢
                      
                        cosh
                        ⁡
                        
                          (
                          
                            α
                            ⁢
                            
                              L
                              2
                            
                          
                          )
                        
                      
                    
                    -
                    
                      D
                      ⁢
                      
                         
                      
                      ⁢
                      
                        sinh
                        ⁡
                        
                          (
                          
                            α
                            ⁢
                            
                              L
                              2
                            
                          
                          )
                        
                      
                    
                    -
                    1
                  
                  ]
                
              
              }
            
            .
          
        
      
    
    
      
        (
        48
        )
      
    
  




Similarly, inserting equations (2), (3), (4), and (9) into equation (8) and solving for the boundary parameters at x=L/2 yields 



  
    
      
        
          k
          2
        
        =
        
          Re
          ⁢
          
            {
            
              
                
                  (
                  
                    
                      -
                      EI
                    
                    ⁢
                    
                       
                    
                    ⁢
                    
                      α
                      3
                    
                  
                  )
                
                [
                
                  
                    A
                    ⁢
                    
                       
                    
                    ⁢
                    
                      sin
                      ⁡
                      
                        (
                        
                          α
                          ⁢
                          
                            L
                            2
                          
                        
                        )
                      
                    
                  
                  -
                  
                    B
                    ⁢
                    
                       
                    
                    ⁢
                    cos
                    ⁢
                    
                      (
                      
                        α
                        ⁢
                        
                          L
                          2
                        
                      
                      )
                    
                  
                  +
                  
                    C
                    ⁢
                    
                       
                    
                    ⁢
                    
                      sinh
                      ⁡
                      
                        (
                        
                          α
                          ⁢
                          
                            L
                            2
                          
                        
                        )
                      
                    
                  
                  +
                  
                    D
                    ⁢
                    
                       
                    
                    ⁢
                    
                      cosh
                      ⁡
                      
                        (
                        
                          α
                          ⁢
                          
                            L
                            2
                          
                        
                        )
                      
                    
                  
                
                ]
              
              
                [
                
                  
                    A
                    ⁢
                    
                       
                    
                    ⁢
                    
                      cos
                      ⁡
                      
                        (
                        
                          α
                          ⁢
                          
                            L
                            2
                          
                        
                        )
                      
                    
                  
                  +
                  
                    B
                    ⁢
                    
                       
                    
                    ⁢
                    
                      sin
                      ⁡
                      
                        (
                        
                          α
                          ⁢
                          
                            L
                            2
                          
                        
                        )
                      
                    
                  
                  +
                  
                    C
                    ⁢
                    
                       
                    
                    ⁢
                    
                      cosh
                      ⁡
                      
                        (
                        
                          α
                          ⁢
                          
                            L
                            2
                          
                        
                        )
                      
                    
                  
                  +
                  
                    D
                    ⁢
                    
                       
                    
                    ⁢
                    
                      sinh
                      ⁡
                      
                        (
                        
                          α
                          ⁢
                          
                            L
                            2
                          
                        
                        )
                      
                    
                  
                  -
                  1
                
                ]
              
            
            }
          
        
      
    
    
      
        (
        49
        )
      
    
  




and 



  
    
      
        
          c
          2
        
        =
        
          
            1
            ω
          
          ⁢
          Im
          ⁢
          
            
              {
              
                
                  
                    (
                    
                      
                        -
                        EI
                      
                      ⁢
                      
                         
                      
                      ⁢
                      
                        α
                        3
                      
                    
                    )
                  
                  [
                  
                    
                      A
                      ⁢
                      
                         
                      
                      ⁢
                      
                        sin
                        ⁡
                        
                          (
                          
                            α
                            ⁢
                            
                              L
                              2
                            
                          
                          )
                        
                      
                    
                    -
                    
                      B
                      ⁢
                      
                         
                      
                      ⁢
                      cos
                      ⁢
                      
                        (
                        
                          α
                          ⁢
                          
                            L
                            2
                          
                        
                        )
                      
                    
                    +
                    
                      C
                      ⁢
                      
                         
                      
                      ⁢
                      sinh
                      ⁢
                      
                        (
                        
                          α
                          ⁢
                          
                            L
                            2
                          
                        
                        )
                      
                    
                    +
                    
                      D
                      ⁢
                      
                         
                      
                      ⁢
                      
                        cosh
                        ⁡
                        
                          (
                          
                            α
                            ⁢
                            
                              L
                              2
                            
                          
                          )
                        
                      
                    
                  
                  ]
                
                
                  [
                  
                    
                      A
                      ⁢
                      
                         
                      
                      ⁢
                      
                        cos
                        ⁡
                        
                          (
                          
                            α
                            ⁢
                            
                              L
                              2
                            
                          
                          )
                        
                      
                    
                    +
                    
                      B
                      ⁢
                      
                         
                      
                      ⁢
                      
                        sin
                        ⁡
                        
                          (
                          
                            α
                            ⁢
                            
                              L
                              2
                            
                          
                          )
                        
                      
                    
                    +
                    
                      C
                      ⁢
                      
                         
                      
                      ⁢
                      
                        cosh
                        ⁡
                        
                          (
                          
                            α
                            ⁢
                            
                              L
                              2
                            
                          
                          )
                        
                      
                    
                    +
                    
                      D
                      ⁢
                      
                         
                      
                      ⁢
                      
                        sinh
                        ⁡
                        
                          (
                          
                            α
                            ⁢
                            
                              L
                              2
                            
                          
                          )
                        
                      
                    
                    -
                    1
                  
                  ]
                
              
              }
            
            .
          
        
      
    
    
      
        (
        50
        )
      
    
  




Thus, once the flexural wavenumber and wave coefficients are estimated, the properties of the springs and dashpots at the boundaries can be calculated.

Numerical simulations conducted to determine the effectiveness of this method use the following parameters to define a baseline problem: Re(E)=(3·1010+107f) N/m2, Im(E)=(3·109+106f) N/m2, ρ=5000 kg/m3, Ab=0.02 m2, I=6.67×10−5 m4, L=3 m, δ=0.5 m, k1=50000 N/m, c1=4000 N·s/m, k2=60000 N/m, and c2=5000 N·s/m where f is frequency in Hz. FIGS. 5A and 5B represent a typical transfer function of the beam displacement measured at x=0 m, which is the middle of the beam, divided by base displacement. The top plot, FIG. 5A, is the magnitude versus frequency and the bottom plot, FIG. 5B, is the phase angle versus frequency. This figure was constructed by inserting the above parameters into equations (3), (4), (10), (11), (12), (13), (14), (15), (16), (17), (18), and (19) and calculating the solution (a forward model).
FIG. 6 graphs the function s versus frequency. It was calculated by inserting the left-hand side of equations (20)-(26) into equations (34)-(40) and represents the first step of the inverse method calculations. FIGS. 7A and 7B represent the flexural wavenumber versus frequency. The top plot, FIG. 7A, is the real part and the bottom plot, FIG. 7B, is the imaginary part. The values created using equation (4) (the forward solution) are shown as solid lines and the values calculated (or estimated) using equations (34)-(41) (the inverse solution) are shown with x's and o's. Note that there is total agreement among the forward and inverse solutions. FIGS. 8-11 are the wave propagation coefficients A, B, C, and D versus frequency, respectively. The top plots are the magnitudes and the bottom plots are the phase angles. The values created using equation (10)-(19) (the forward solution) are shown as solid lines and the values calculated using equations (43)-(46) (the inverse solution) are shown with x's and o's. FIG. 12A and FIG. 12B graph the real and imaginary parts of Young's modulus versus frequency. The actual values are shown as solid lines and the values calculated using equation (42) are shown with x's and o's. FIG. 13 is the boundary condition parameters of mount one versus frequency. The top plot is the stiffness and the bottom plot is the damping. The actual values are shown as solid lines and the values calculated using equations (47) and (48) are shown with x's and o's. FIG. 14 is the boundary condition parameters of mount two versus frequency. The top plot, FIG. 14A, is the stiffness and the bottom plot, FIG. 14B, is the damping. The actual values are shown as solid lines and the values calculated using equations (49) and (50) are shown with x's and o's.
In light of the above, it is therefore understood that within the scope of the appended claims, the invention may be practiced otherwise than as specifically described.

## Claims

1. A method of determining structural properties of a flexural beam comprising the steps of:
securing at least seven accelerometers spaced approximately equidistant from each other along a length of said beam; 
providing a transverse vibrational input to said beam at a base; 
measuring seven frequency domain transfer functions of displacement from said secured accelerometers; and 
estimating a flexural wavenumber from said seven frequency domain transfer functions; 
wherein said seven frequency domain transfer functions comprise: 



  
    
      
        
          T
          
            -
            3
          
        
        =
        
        ⁢
        
          
            
              
                U
                
                  -
                  3
                
              
              ⁡
              
                (
                
                  
                    
                      -
                      3
                    
                    ⁢
                    
                       
                    
                    ⁢
                    δ
                  
                  ,
                  ω
                
                )
              
            
            
              
                V
                0
              
              ⁡
              
                (
                ω
                )
              
            
          
          =
          
            
              A
              ⁢
              
                 
              
              ⁢
              cos
              ⁢
              
                (
                
                  3
                  ⁢
                  α
                  ⁢
                  
                     
                  
                  ⁢
                  δ
                
                )
              
            
            -
          
        
      
    
  
  
    
      
        
        ⁢
        
          
            
              B
              ⁢
              
                 
              
              ⁢
              sin
              ⁢
              
                (
                
                  3
                  ⁢
                  αδ
                
                )
              
            
            +
            
              C
              ⁢
              
                 
              
              ⁢
              cosh
              ⁢
              
                (
                
                  3
                  ⁢
                  α
                  ⁢
                  
                     
                  
                  ⁢
                  δ
                
                )
              
            
            -
            
              D
              ⁢
              
                 
              
              ⁢
              sinh
              ⁢
              
                (
                
                  3
                  ⁢
                  α
                  ⁢
                  
                     
                  
                  ⁢
                  δ
                
                )
              
            
          
          ,
        
      
    
  
  
    
      
        
          T
          
            -
            2
          
        
        =
        
        ⁢
        
          
            
              
                U
                
                  -
                  2
                
              
              ⁡
              
                (
                
                  
                    
                      -
                      2
                    
                    ⁢
                    
                       
                    
                    ⁢
                    δ
                  
                  ,
                  ω
                
                )
              
            
            
              
                V
                0
              
              ⁡
              
                (
                ω
                )
              
            
          
          =
          
            
              A
              ⁢
              
                 
              
              ⁢
              cos
              ⁢
              
                (
                
                  2
                  ⁢
                  α
                  ⁢
                  
                     
                  
                  ⁢
                  δ
                
                )
              
            
            -
          
        
      
    
  
  
    
      
        
        ⁢
        
          
            
              B
              ⁢
              
                 
              
              ⁢
              sin
              ⁢
              
                (
                
                  2
                  ⁢
                  αδ
                
                )
              
            
            +
            
              C
              ⁢
              
                 
              
              ⁢
              
                cosh
                ⁡
                
                  (
                  
                    2
                    ⁢
                    α
                    ⁢
                    
                       
                    
                    ⁢
                    δ
                  
                  )
                
              
            
            -
            
              D
              ⁢
              
                 
              
              ⁢
              
                sinh
                ⁡
                
                  (
                  
                    2
                    ⁢
                    α
                    ⁢
                    
                       
                    
                    ⁢
                    δ
                  
                  )
                
              
            
          
          ,
        
      
    
  
  
    
      
        
          T
          
            -
            1
          
        
        =
        
        ⁢
        
          
            
              
                U
                
                  -
                  1
                
              
              ⁡
              
                (
                
                  
                    -
                    δ
                  
                  ,
                  ω
                
                )
              
            
            
              
                V
                0
              
              ⁡
              
                (
                ω
                )
              
            
          
          =
          
            
              A
              ⁢
              
                 
              
              ⁢
              cos
              ⁢
              
                (
                
                  α
                  ⁢
                  
                     
                  
                  ⁢
                  δ
                
                )
              
            
            -
          
        
      
    
  
  
    
      
        
        ⁢
        
          
            
              B
              ⁢
              
                 
              
              ⁢
              sin
              ⁢
              
                (
                αδ
                )
              
            
            +
            
              C
              ⁢
              
                 
              
              ⁢
              
                cosh
                ⁡
                
                  (
                  
                    α
                    ⁢
                    
                       
                    
                    ⁢
                    δ
                  
                  )
                
              
            
            -
            
              D
              ⁢
              
                 
              
              ⁢
              
                sinh
                ⁡
                
                  (
                  
                    α
                    ⁢
                    
                       
                    
                    ⁢
                    δ
                  
                  )
                
              
            
          
          ,
        
      
    
  
  
    
      
        
          
            T
            0
          
          =
          
          ⁢
          
            
              
                
                  U
                  0
                
                ⁡
                
                  (
                  
                    0
                    ,
                    ω
                  
                  )
                
              
              
                
                  V
                  0
                
                ⁡
                
                  (
                  ω
                  )
                
              
            
            =
            
              A
              +
              C
            
          
        
        ,
      
    
  
  
    
      
        
          
            T
            1
          
          =
          
          ⁢
          
            
              
                
                  U
                  1
                
                ⁡
                
                  (
                  
                    δ
                    ,
                    ω
                  
                  )
                
              
              
                
                  V
                  0
                
                ⁡
                
                  (
                  ω
                  )
                
              
            
            =
            
              
                A
                ⁢
                
                   
                
                ⁢
                
                  cos
                  ⁡
                  
                    (
                    
                      α
                      ⁢
                      
                         
                      
                      ⁢
                      δ
                    
                    )
                  
                
              
              +
              
                B
                ⁢
                
                   
                
                ⁢
                
                  sin
                  ⁡
                  
                    (
                    αδ
                    )
                  
                
              
              +
              
                C
                ⁢
                
                   
                
                ⁢
                
                  cosh
                  ⁡
                  
                    (
                    
                      α
                      ⁢
                      
                         
                      
                      ⁢
                      δ
                    
                    )
                  
                
              
              +
              
                D
                ⁢
                
                   
                
                ⁢
                
                  sinh
                  ⁡
                  
                    (
                    
                      α
                      ⁢
                      
                         
                      
                      ⁢
                      δ
                    
                    )
                  
                
              
            
          
        
        ,
      
    
  
  
    
      
        
          T
          2
        
        =
        
        ⁢
        
          
            
              
                U
                2
              
              ⁡
              
                (
                
                  
                    2
                    ⁢
                    
                       
                    
                    ⁢
                    δ
                  
                  ,
                  ω
                
                )
              
            
            
              
                V
                0
              
              ⁡
              
                (
                ω
                )
              
            
          
          =
          
            
              A
              ⁢
              
                 
              
              ⁢
              cos
              ⁢
              
                (
                
                  2
                  ⁢
                  α
                  ⁢
                  
                     
                  
                  ⁢
                  δ
                
                )
              
            
            +
          
        
      
    
  
  
    
      
        
        ⁢
        
          
            
              B
              ⁢
              
                 
              
              ⁢
              sin
              ⁢
              
                (
                
                  2
                  ⁢
                  αδ
                
                )
              
            
            +
            
              C
              ⁢
              
                 
              
              ⁢
              
                cosh
                ⁡
                
                  (
                  
                    2
                    ⁢
                    α
                    ⁢
                    
                       
                    
                    ⁢
                    δ
                  
                  )
                
              
            
            +
            
              D
              ⁢
              
                 
              
              ⁢
              
                sinh
                ⁡
                
                  (
                  
                    2
                    ⁢
                    α
                    ⁢
                    
                       
                    
                    ⁢
                    δ
                  
                  )
                
              
            
          
          ,
          and
        
      
    
  
  
    
      
        
          T
          3
        
        =
        
        ⁢
        
          
            
              
                U
                3
              
              ⁡
              
                (
                
                  
                    3
                    ⁢
                    
                       
                    
                    ⁢
                    δ
                  
                  ,
                  ω
                
                )
              
            
            
              
                V
                0
              
              ⁡
              
                (
                ω
                )
              
            
          
          =
          
            
              A
              ⁢
              
                 
              
              ⁢
              cos
              ⁢
              
                (
                
                  3
                  ⁢
                  α
                  ⁢
                  
                     
                  
                  ⁢
                  δ
                
                )
              
            
            +
          
        
      
    
  
  
    
      
        
        ⁢
        
          
            B
            ⁢
            
               
            
            ⁢
            sin
            ⁢
            
              (
              
                3
                ⁢
                αδ
              
              )
            
          
          +
          
            C
            ⁢
            
               
            
            ⁢
            
              cosh
              ⁡
              
                (
                
                  3
                  ⁢
                  α
                  ⁢
                  
                     
                  
                  ⁢
                  δ
                
                )
              
            
          
          +
          
            D
            ⁢
            
               
            
            ⁢
            
              
                sinh
                ⁡
                
                  (
                  
                    3
                    ⁢
                    α
                    ⁢
                    
                       
                    
                    ⁢
                    δ
                  
                  )
                
              
              .

2. The method of claim 1 further comprising the step of securing at least one accelerometer to said base.

3. The method of claim 1 further comprising securing said beam to a shaker table using a spring and a dashpot disposed at both a first and a second end of said beam.

4. The method of claim 3 further comprising the step of securing at least one accelerometer to said base.

5. The method of claim 1 wherein said base is a shaker table and further comprising:
securing a first end of said beam to the shaker table using a spring and a dashpot; and 
securing a second end of said beam to a fixed object by a pinned connection.

6. The method of claim 5, further comprising the step of securing at least one accelerometer to said base.

7. The method of claim 1 wherein said base is a shaker table and further comprising the steps of:
securing a first end of said beam to the shaker table using a spring and a dashpot; and 
securing a second end of said beam to a fixed object using a spring and a dashpot.

8. The method of claim 7 further comprising the step of securing at least one accelerometer to said base.

9. The method of claim 1 wherein said base is a shaker table and further comprising the steps of:
securing a first end of maid beam directly to the shaker table; and 
securing a second end of maid beam to a fixed object by a pinned connection.

10. The method of claim 9 further comprising the step of securing at least one accelerometer to said base.

11. The method of claim 1 further comprising the step of determining a complex valued modulus of elasticity at each frequency using said flexural wavenumber.

12. The method of claim 1 further comprising the step of determining wave property coefficient using said flexural wavenumber.