Fixed Base FRF Using Boundary Measurements as References: Analytical Derivation

Abstract

Modal parameters extracted from test articles with known boundary conditions are useful for model correlation and updating but are often not obtained because of the time and cost associated with moving and installing the test article on a rigid or isolated seismic mass and performing a separate modal test. It would be advantageous if frequency response functions from known boundary conditions, such as fixed base frequency response functions, could be obtained while the structure undergoes testing on fixtures that contain compliance or dynamics in the frequency range of interest, such as base shake tables.

This paper proposes a method that uses response measurements at the structure’s boundary to mathematically constrain the test article to produce equivalent fixed base frequency response functions. Fixed base frequency response functions are obtained by using the boundary measurements as additional references to mathematically constrain these degrees of freedom in the frequency range of interest.

This paper provides analytical justification for using the proposed methodology, discusses assumptions associated with boundary deformation, and finally presents an analytical example using a lumped parameter system.

27.1 Introduction

There has been much interest in extracting modal parameters from structures mounted on shake tables either by performing a conventional modal test while a structure is mounted to a shake table or by extracting modal parameters directly from shake table testing. In some cases the modal parameters extracted from these tests are nearly equivalent to modal parameters extracted from fixed base modal tests, especially when the inertia of the shake table is much larger than the structure being tested or when the shake table is effectively constrained to moving in one degree of freedom (the driving direction).

However, in many cases such as when large satellites are mounted to shake tables, the inertia of the satellite interacts with the shake table, resulting in motion at the base (such as rotation) in the extracted modes. Dynamics of the shake table can also be observed in the test data, and separating the effects of shake table dynamics from structure modes is oftentimes impossible.

Methods have been proposed to account for rigid body base motion by using rigid body motion of the shake table as a reference for calculating frequency response functions [1–5]. Using a coordinate transformation from absolute to relative motion, the resulting transfer functions are equivalent to fixed base acceleration/force FRF. The challenges with this method are that (1) some modes of a structure may not be observable from a rigid body base excitation, and (2) the method does not account for the potential flexible motion of the shake table.

One proposed method accounts for flexible motion of the shake table by introducing constraints based on measured deformations of the table [6]. When this is done, forces applied directly to the structure can be transformed into fixed base FRF.

This paper, in contrast, proposes using test-measured deformations at the structure-table interface along with forces applied directly to the structure as independent references when calculating frequency response functions. With this approach, the frequency response functions associated with the forces applied to the structure will be associated with fixed base FRF.

The key to implementing the proposed method is that there must be at least one independent excitation source per independent operating shape of the table-structure interface. This can be accomplished by exciting the table directly at several locations in such a way that these operating shapes are observable. Furthermore, this paper will also discuss the parameters for which it is appropriate to use base transmissibility measurements to extract fixed base modal parameters.

27.2 Derivation

Assume that a structural system can be discretized and modeled as a series of linear equations such that

$$ \left[ M \right]\left\{ {\ddot{x}} \right\} + \left[ C \right]\left\{ {\dot{x}} \right\} + K\left\{ x \right\} = \left\{ f \right\}. $$

(27.1)

Assume the structural system can then separated into interior degrees of freedom “I” and boundary degrees of freedom “B.” Boundary degrees of freedom include all degrees of freedom that are connected to anything outside the model, such as fixed or flexible supports. Interior degrees of freedom include all degrees of freedom on the structure that are not connected to anything outside. Elements that connect the model to the outside are not included so that the model can undergo a static rigid body deformation without generating any internal or external forces. The model is defined as:

$$ \left[ {\begin{array}{lll} {{M_{{II}}}} & {{M_{{IB}}}} \\{{M_{{BI}}}} & {{M_{{BB}}}} \\\end{array} } \right]\left\{ {\begin{array}{lll} {{{\ddot{x}}_I}} \\{{{\ddot{x}}_B}} \\\end{array} } \right\} + \left[ {\begin{array}{lll} {{C_{{II}}}} & {{C_{{IB}}}} \\{{C_{{BI}}}} & {{C_{{BB}}}} \\\end{array} } \right]\left\{ {\begin{array}{lll} {{{\dot{x}}_I}} \\{{{\dot{x}}_B}} \\\end{array} } \right\} + \left[ {\begin{array}{lll} {{K_{{II}}}} & {{K_{{IB}}}} \\{{K_{{BI}}}} & {{K_{{BB}}}} \\\end{array} } \right]\left\{ {\begin{array}{lll} {{x_I}} \\{{x_B}} \\\end{array} } \right\} = \left\{ {\begin{array}{lll} {{f_I}} \\{{f_B}} \\\end{array} } \right\} $$

(27.2)

where “M,” “C,” and “K” are associated with elements in the mass, damping, and stiffness matrices, respectively. The subscripts “I” and “B” are associated with the interior and boundary degrees of freedom, respectively. Finally the letters “x” and “f” are associated with displacements and applied loads, respectively. The frequency domain version of this equation is written as

$$ \left( {{s^2}\left[ {\begin{array}{lll} {{M_{{II}}}} & {{M_{{IB}}}} \\{{M_{{BI}}}} & {{M_{{BB}}}} \\\end{array} } \right] + s\left[ {\begin{array}{lll} {{C_{{II}}}} & {{C_{{IB}}}} \\{{C_{{BI}}}} & {{C_{{BB}}}} \\\end{array} } \right] + \left[ {\begin{array}{lll} {{K_{{II}}}} & {{K_{{IB}}}} \\{{K_{{BI}}}} & {{K_{{BB}}}} \\\end{array} } \right]} \right)\left\{ {\begin{array}{lll} {{x_I}} \\{{x_B}} \\\end{array} } \right\} = \left\{ {\begin{array}{lll} {{f_I}} \\{{f_B}} \\\end{array} } \right\} $$

(27.3)

where s = iω. By substituting

$$ \left\{ {\begin{array}{lll} {{x_I}} \\{{x_B}} \\\end{array} } \right\} = \left( {\frac{1}{{{s^2}}}} \right)\left\{ {\begin{array}{lll} {{{\ddot{x}}_I}} \\{{{\ddot{x}}_B}} \\\end{array} } \right\} $$

(27.4)

the equation of motion can be written as

$$ \left( {\left[ {\begin{array}{lll} {{M_{{II}}}} & {{M_{{IB}}}} \\{{M_{{BI}}}} & {{M_{{BB}}}} \\\end{array} } \right] + \frac{1}{s}\left[ {\begin{array}{lll} {{C_{{II}}}} & {{C_{{IB}}}} \\{{C_{{BI}}}} & {{C_{{BB}}}} \\\end{array} } \right] + \frac{1}{{{s^2}}}\left[ {\begin{array}{lll} {{K_{{II}}}} & {{K_{{IB}}}} \\{{K_{{BI}}}} & {{K_{{BB}}}} \\\end{array} } \right]} \right)\left\{ {\begin{array}{lll} {{{\ddot{x}}_I}} \\{{{\ddot{x}}_B}} \\\end{array} } \right\} = \left\{ {\begin{array}{lll} {{f_I}} \\{{f_B}} \\\end{array} } \right\}. $$

(27.5)

If the degrees of freedom on the boundary can be measured, then they are known and can be moved to the right hand side of (27.5) to produce:

$$ \left( {\left[ {{M_{{II}}}} \right] + \frac{1}{s}\left[ {{C_{{II}}}} \right] + \frac{1}{{{s^2}}}\left[ {{K_{{II}}}} \right]} \right)\left\{ {{{\ddot{x}}_I}} \right\} = \left\{ {{f_I}} \right\} - \left( {\left[ {{M_{{IB}}}} \right] + \frac{1}{s}\left[ {{C_{{IB}}}} \right] + \frac{1}{{{s^2}}}\left[ {{K_{{IB}}}} \right]} \right)\left\{ {{{\ddot{x}}_B}} \right\}. $$

(27.6)

Therefore, if the base accelerations are used as independent references, then the resulting $[\left\{ {{{\ddot{x}}_I}} \right\}/\left\{ {{f_I}} \right\}]$ frequency response function matrix is equivalent to fixing the boundary interface.

The key is that all known degrees of freedom in $\left\{ {{f_I}} \right\}$ and $\left\{ {{{\ddot{x}}_B}} \right\}$must be independently measured or identically equal to zero. Applied loads on the structure are either measured or set equal to zero (by not applying a load at that degree of freedom). Setting a base acceleration degree of freedom identically equal to zero is nearly impossible. However, the boundary degrees of freedom can be represented as a combination of generalized coordinates associated with a few operating deflected shapes (ODS) such as rigid body shapes or perhaps shapes extracted from singular value decomposition of the measured degrees of freedom on the boundary. The level to which the reduced set of shapes captures the actual deformations of the boundary determines how well the fixed base FRF $[\left\{ {{{\ddot{x}}_I}} \right\}/\left\{ {{f_I}} \right\}]$ can be measured.

27.3 Experimentally Measured Guyan Shapes

Since the base acceleration is treated as an independent reference from the applied forces on the structure, $\left\{ {{f_I}} \right\}$, then the frequency response function matrix relating base accelerations to interior accelerations is

$$ \left\{ {{{\ddot{x}}_I}} \right\} = - {\left( {\left[ {{M_{{II}}}} \right] + \frac{1}{s}\left[ {{C_{{II}}}} \right] + \frac{1}{{{s^2}}}\left[ {{K_{{II}}}} \right]} \right)^{{ - 1}}}\left( {\left[ {{M_{{IB}}}} \right] + \frac{1}{s}\left[ {{C_{{IB}}}} \right] + \frac{1}{{{s^2}}}\left[ {{K_{{IB}}}} \right]} \right)\left\{ {{{\ddot{x}}_B}} \right\}. $$

(27.7)

As the frequency, s, approaches 0, the equation of motion reduces to

$$ \left\{ {{{\ddot{x}}_I}} \right\} = - {\left[ {{K_{{II}}}} \right]^{{ - 1}}}\left[ {{K_{{IB}}}} \right]\left\{ {{{\ddot{x}}_B}} \right\}. $$

(27.8)

Note that the matrix $- {\left[ {{K_{{II}}}} \right]^{{ - 1}}}\left[ {{K_{{IB}}}} \right]$ are the Guyan shapes $\left[ {{\it \Psi_{{IB}}}} \right]$used in Craig-Bampton model reduction [6]. Therefore, if the assumed operating shapes used to describe the boundary motion in both test and in a Craig-Bampton reduction are the same, then the entire Craig-Bampton model reduction can potentially be experimentally determined. In practice, experimental measurement of the Guyan shapes for Craig-Bampton model reduction is feasible when the interface is reduced to a single node with six independent degrees of freedom (six rigid body shapes).

27.4 Relative Motion Transmissibility

Assume that the motion of the boundary can be reduced to a finite number of degrees of freedom and that these degrees of freedom can be used as unique references. The matrix $\left[ {{\it \Psi_{{IB}}}} \right] = - {\left[ {{K_{{II}}}} \right]^{{ - 1}}}\left[ {{K_{{IB}}}} \right]$ can be experimentally measured and then used in a coordinate transformation to define the relative displacement as

$$ \left\{ {\Delta x} \right\} = \left\{ x \right\} - \left[ T \right]\left\{ {{x_B}} \right\},{\hbox{or}}\,\,\left\{ {\begin{array}{lll} {\Delta {x_I}} \\{\Delta {x_B}} \\\end{array} } \right\} = \left\{ {\begin{array}{lll} {\Delta {x_I}} \\0 \\\end{array} } \right\} = \left\{ {\begin{array}{lll} {{x_I}} \\{{x_B}} \\\end{array} } \right\} - \left[ {\begin{array}{lll} {{\it \Psi_{{IB}}}} \\I \\\end{array} } \right]\left\{ {{x_B}} \right\}. $$

(27.9)

This equation can be re-ordered to define a coordinate transformation matrix from original coordinates to relative coordinates.

$$ \left\{ {\begin{array}{lll} {{x_I}} \\{{x_B}} \\\end{array} } \right\} = \left[ {\begin{array}{lll} I & {{\it \Psi_{{IB}}}} \\0 & I \\\end{array} } \right]\left\{ {\begin{array}{lll} {\Delta x} \\{{x_B}} \\\end{array} } \right\}. $$

(27.10)

The coordinate transformation can be applied to (27.5) and then the first row can be re-ordered to obtain

$${\begin{array}{lll}\left[ {{M_{{II}}}} \right]\left\{ {\Delta \ddot{x}} \right\} + \left[ {{C_{{II}}}} \right]\left\{ {\Delta \dot{x}} \right\} + \left[ {{K_{{II}}}} \right]\left\{ {\Delta x} \right\} = \left\{ {{f_I}} \right\} - \left[ {{M_{{II}}\it \Psi_{{IB}}} + {}{M_{{IB}}}} \right]\left\{ {{{\ddot{x}}_B}} \right\} - \left[ {{C_{{II}}\it \Psi_{{IB}}} + {}}{C_{{IB}}} \right]\left\{ {{{\dot{x}}_B}} \right\} - \left[ {{K_{{II}}\it \Psi_{{IB}}} + {}{K_{{IB}}}} \right]\left\{ {{x_B}} \right\}\end{array}} $$

(27.11)

Taking this into the frequency domain leads to

$$ {\begin{array}{lll}\left( {\left[ {{M_{{II}}}} \right] + \frac{1}{s}\left[ {{C_{{II}}}} \right] + \frac{1}{{{s^2}}}\left[ {{K_{{II}}}} \right]} \right)\left\{ {\Delta {{\ddot{x}}_I}} \right\} = \left\{ {{f_I}} \right\} - \left( {\left[ {{M_{{II}}\it \Psi_{{IB}}} + {}{M_{{IB}}}} \right] + \frac{1}{s}\left[ {{C_{{II}}\it \Psi_{{IB}}} + {}{C_{{IB}}}} \right] + \frac{1}{{{s^2}}}\left[ {{K_{{II}}\it \Psi_{{IB}}} + {}{K_{{IB}}}} \right]} \right)\left\{ {{{\ddot{x}}_B}} \right\}\end{array}} $$

(27.12)

The terms associated with the stiffness matrix, $\left[ {{K_{{II}}\it \Psi_{{IB}}} + {}{K_{{IB}}}} \right]$, are equal to $[0]$ since $\left[ {{\it \Psi_{{IB}}}} \right] = - {\left[ {{K_{{II}}}} \right]^{{ - 1}}}\left[ {{K_{{IB}}}} \right]$, and therefore the final equation becomes

$$ \left( {\left[ {{M_{{II}}}} \right] + \frac{1}{s}\left[ {{C_{{II}}}} \right] + \frac{1}{{{s^2}}}\left[ {{K_{{II}}}} \right]} \right)\left\{ {\Delta {{\ddot{x}}_I}} \right\} = \left\{ {{f_I}} \right\} - \left( {\left[ {{M_{{II}}\it \Psi_{{IB}}} + {}{M_{{IB}}}} \right] + \frac{1}{s}\left[ {{C_{{II}}\it \Psi_{{IB}}} + {}{C_{{IB}}}} \right]} \right)\left\{ {{{\ddot{x}}_B}} \right\}. $$

(27.13)

The forcing term associated with the mass matrix is constant with respect to frequency, but the forcing term associated with the damping matrix is proportional to 1/s. If the damping matrix, $\left[ C \right]$ is directly proportional to the stiffness matrix $\left[ K \right]$, then the term associated with the damping matrix is identically zero and the resulting $[\left\{ {\Delta {{\ddot{x}}_I}} \right\}/\left\{ {{{\ddot{x}}_B}} \right\}]$ matrix will resemble acceleration/force $[\left\{ {{{\ddot{x}}_I}} \right\}/\left\{ {{f_I}} \right\}]$ frequency response functions and will be suitable for modal analysis. Also, since the term associated with the damping matrix is proportional to 1/s, at some frequency the inertia forces will be much greater than the damping forces. Therefore, at higher frequencies the frequency response function $[\left\{ {\Delta {{\ddot{x}}_I}} \right\}/\left\{ {{{\ddot{x}}_B}} \right\}]$ will resemble a fixed base acceleration/force $[\left\{ {{{\ddot{x}}_I}} \right\}/\left\{ {{f_I}} \right\}]$ FRF and will also be suitable for modal extraction.

Note that the $[\left\{ {\Delta {{\ddot{x}}_I}} \right\}/\left\{ {{f_I}} \right\}]$ frequency response function matrix will always be suitable for extracting fixed base modes.

27.5 Rigid Body Boundary Motion

A special case of relative motion transmissibility occurs when the boundary degrees of freedom are assumed to be rigid body degrees of freedom. Assume that boundary motion operating shapes are rigid body deformations. In this case, the boundary degrees of freedom are rigid body degrees of freedom. In this case, the displacement can be represented as

$$ \left\{ x \right\} = \left\{ {\Delta x} \right\} + \left[ R \right]\left\{ r \right\} = \left[ {\begin{array}{lll} I & R \\\end{array} } \right]\left\{ {\begin{array}{lll} {\Delta x} \\r \\\end{array} } \right\} $$

(27.14)

where the boundary degrees of freedom are generalized coordinates $\left\{ r \right\}$ associated with rigid body deformation of the entire structural system $\left[ R \right]$. Substituting this into (27.1) yields

$$ \left[ {\begin{array}{lll} M & {MR} \\{{R^T}M} & {{R^T}MR} \\\end{array} } \right]\left\{ {\begin{array} {lll}{\Delta \ddot{x}} \\{\ddot{r}} \\\end{array} } \right\} + \left[ {\begin{array}{lll} C & {CR} \\{{R^T}C} & {{R^T}CR} \\\end{array} } \right]\left\{ {\begin{array}{lll} {\Delta \dot{x}} \\{\dot{r}} \\\end{array} } \right\} + \left[ {\begin{array}{lll} K & {KR} \\{{R^T}K} & {{R^T}KR} \\\end{array} } \right]\left\{ {\begin{array}{lll} {\Delta x} \\r \\\end{array} } \right\} = \left\{ {\begin{array}{lll} f \\{{R^T}f} \\\end{array} } \right\} $$

(27.15)

Taking the first row of (27.15) yields

$$ \left[ M \right]\left\{ {\Delta \ddot{x}} \right\} + \left[ C \right]\left\{ {\Delta \dot{x}} \right\} + \left[ K \right]\left\{ {\Delta x} \right\} = \left\{ f \right\} - \left( {\left[ {MR} \right]\left\{ {\ddot{r}} \right\} + \left[ {CR} \right]\left\{ {\dot{r}} \right\} + \left[ {KR} \right]\left\{ r \right\}} \right). $$

(27.16)

A rigid body deformation should produce no internal forces proportional to displacement nor velocity, and therefore

$$ \left[ {CR} \right] = \left[ {KR} \right] = \left[ 0 \right]. $$

(27.17)

The final equation becomes

$$ \left[ M \right]\left\{ {\Delta \ddot{x}} \right\} + \left[ C \right]\left\{ {\Delta \dot{x}} \right\} + \left[ K \right]\left\{ {\Delta x} \right\} = \left\{ f \right\} - \left[ {MR} \right]\left\{ {\ddot{r}} \right\}. $$

(27.18)

In this particular case, the frequency response function $[\left\{ {\Delta {{\ddot{x}}_I}} \right\}/\left\{ {{{\ddot{r}}_B}} \right\}]$ will resemble a fixed base $[\left\{ {\Delta {{\ddot{x}}_I}} \right\}/\left\{ {{f_I}} \right\}]$ FRF and can be used for fixed base modal extraction.

In summary, subtracting the static component of the $[\left\{ {{{\ddot{x}}_I}} \right\}/\left\{ {{{\ddot{x}}_B}} \right\}]$ leads to a relative motion frequency response function matrix $[\left\{ {\Delta {{\ddot{x}}_I}} \right\}/\left\{ {{{\ddot{x}}_B}} \right\}]$. The $[\left\{ {\Delta {{\ddot{x}}_I}} \right\}/\left\{ {{{\ddot{x}}_B}} \right\}]$ matrix can be used for modal extraction if the base motion is rigid, if the damping matrix is proportional to the stiffness matrix, or if the damping forces are much less than the inertia forces as is the case with low levels of damping. However, note that $[\left\{ {{{\ddot{x}}_I}} \right\}/\left\{ {{f_I}} \right\}]$ or $[\left\{ {\Delta {{\ddot{x}}_I}} \right\}/\left\{ {{f_I}} \right\}]$ FRF matrices will always be valid for fixed base modal analysis.

27.6 Numerical Example

A numerical example is used to prove the concept. In this example, shown in Fig. 27.1, a structural system is represented by eight degrees of freedom (DOF) that are marked with the letter “X.” Elements connecting degrees of freedom 1 through 6 represent a structure mounted on a shake table. Elements connecting DOF 4 through 8 represent a shake table and its connection to ground. DOF 1 through 3 and 4 through 6 represent interior and boundary degrees of freedom on the structure, respectively. DOF 7 represents the interface between the shake table and ground, and DOF 8 represents ground and was assumed to be fixed in this example. A lumped parameter mass matrix was used. The mass element M_S was used for DOF 1 through 6. DOF 7 was represented by M_B, which was 10 times the value of M_S. Spring elements K_S represent elements on the structure and also interface between the shake table and ground. The spring element K_G represents the stiffness of the shake table to ground connection and was assumed to be 10 times K_S. The dashpot values for each of the damping elements were assigned by random numbers, and then the entire damping matrix was scaled so that the first fixed base mode had approximately 2% damping.

Fig. 27.1

Example lumped parameter model

Forces were applied to degrees of freedom 1 through 6 and are shown here as red arrows. Forces at degrees of freedom 1 through 3 represent forces $\left\{ {{f_I}} \right\}$ applied to the structure. Forces at degrees of freedom 4 through 6 represent shakers attached directly to the shake table. There must be at least the same number of shakers attached to the shake table as there are independent degrees of freedom of the table in the frequency range of interest.

Frequency response functions were generated that represent a modal test of the entire structural system. These frequency response functions associated with DOF 1, 2, and 3 as references and responses are represented by a summary complex mode indicator function (CMIF) plot in Fig. 27.2 below. Note that there are at least five modes in the frequency range of interest even though there are only three structure degrees of freedom (1 through 3) for the same structure with a fixed base where degrees of freedom 4 through 6 are held fixed. These extra modes are due to the dynamics associated with the table (DOF 4 though 6) and the interaction between the shake table and ground (DOF 7).

Fig. 27.2

Complex mode indicator functions for baseline-measured base acceleration/applied force FRF for DOF 1, 2, and 3 before fixed base correction is applied

A coordinate transformation was then applied to the frequency response functions based on the following relationship.

$$ \begin{array}{lll} \left\{ {\begin{array}{lll} {f_1^f} \\{f_2^f} \\{f_3^f} \begin{array}{lll} {{\ddot{x}}_4} \\{{\ddot{x}}_5} \hfill \\{{\ddot{x}}_6}\end{array} \end{array}} \right\} = \left[ \begin{array}{lll} \begin{array}{llllllllllll} 1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 1 & 0 & 0 & 0 \\{{{\ddot{x}}_4}/f_1^{\,*}} & {{{\ddot{x}}_4}/f_2^{\,*}} & {{{\ddot{x}}_4}/f_3^{\,*}} & {{{\ddot{x}}_4}/f_4^{\,*}} & {{{\ddot{x}}_4}/f_5^{\,*}} & {{{\ddot{x}}_4}/f_6^{\,*}} \\\end{array} \hfill \\\begin{array}{llllllllllll} {{{\ddot{x}}_5}/f_1^{\,*}} & {{{\ddot{x}}_5}/f_2^{\,*}} & {{{\ddot{x}}_5}/f_3^{\,*}} & {{{\ddot{x}}_5}/f_4^{\,*}} & {{{\ddot{x}}_5}/f_5^{\,*}} & {{{\ddot{x}}_5}/f_6^{\,*}} \\{{{\ddot{x}}_6}/f_1^{\,*}} & {{{\ddot{x}}_6}/f_2^{\,*}} & {{{\ddot{x}}_6}/f_3^{\,*}} & {{{\ddot{x}}_6}/f_4^{\,*}} & {{{\ddot{x}}_6}/f_5^{\,*}} & {{{\ddot{x}}_6}/f_6^{\,*}} \\\end{array} \\ \end{array} \right]\left\{ {\begin{array}{lll} {f_1^{\,*}} \\{f_2^{\,*}} \\{f_3^{\,*}} \\\begin{array}{llllll} f_4^{*} \\f_5^{*} f_6^{*} \\\end{array} \\\end{array}} \right\}\end{array} $$

(27.19)

Or

$$ \left\{ {f^f} \right\} = {\left[ T \right]^{{ - 1}}}\left\{ {{f^{\,*}}} \right\}\;\;{\hbox{and}}\;{\hbox{therefore}}\;\;\left\{ {f^{\,*}} \right\} = \left[ T \right]\left\{ {{f^f}} \right\} $$

(27.20)

where the superscript “f” refers to fixed base degrees of freedom (corrected fixed base forces and base accelerations) and the superscript “*” refers to forces applied to the baseline uncorrected system. The accelerations due to the baseline system FRF can be defined as

$$ \left\{ {{a_I}} \right\} = \left[ H \right]\left\{ {{{\,f}^{\,*}}} \right\} $$

(27.21)

where $\left\{ {{a_I}} \right\}$ are the accelerations of DOFs 1, 2, and 3.

The columns of the fixed base FRF matrix are defined as the acceleration due to a unit input in $\left\{ {f^f} \right\}$. This relationship can be defined by multiplying (27.20) by (27.21) to obtain

$$ \left\{ {{a_I}} \right\} = \left[ H \right]\left[ T \right]\left\{ {{{\,f}^f}} \right\} = \left[ {{H^F}} \right]\left\{ {{{\,f}^{\,f}}} \right\}. $$

(27.22)

In practice, the base accelerations can be used directly as reference channels to calculate the fixed base FRF. When this coordinate transformation is applied, the fixed base FRF are obtained. The CMIF of the corrected fixed base FRF associated with applied forces to the structure are shown in Fig. 27.3 below. In this analysis example, the fixed base FRF from analysis are identical to the corrected fixed base FRF.

Fig. 27.3

Complex mode indicator functions for fixed base acceleration/applied force FRF for DOF 1, 2, and 3. Results for fixed base FRF and corrected fixed base FRF are identical

The fixed base FRF associated with base accelerations as references are presented in Fig. 27.4 below.

Fig. 27.4

Frequency response functions for the structure DOF 1, 2, and 3 using the base accelerations at DOF 4, 5, and 6 as references

The value of these FRF at 0 Hz is the Guyan shapes associated with Craig-Bampton model reduction. The Guyan shapes were then subtracted from these FRF to obtain $[\left\{ {\Delta {{\ddot{x}}_I}} \right\}/\left\{ {{{\ddot{x}}_B}} \right\}]$ FRF functions. The relative acceleration FRF are presented below in Fig. 27.5.

Fig. 27.5

Relative acceleration frequency response functions for the structure DOF 1, 2, and 3 using the base accelerations at DOF 4, 5, and 6 as references

Modal parameters were extracted using AFPoly [7] for different sets of frequency response functions in order to determine if FRF with base accelerations as references could be used to extract modal parameters. An example of the stability diagram for the corrected fixed base FRF using applied loads as references is presented in Fig. 27.6. There are three sets of stable poles—one for each fixed base structural mode. The stability diagram for the corrected fixed base FRF using base accelerations as references is presented in Fig. 27.7. In this case, there are three sets of stable poles for the three fixed base modes, and one extra set of poles for a very lightly excited resonance in the tertiary CMIF curve. This light resonance is likely due to the “1/s” term associated with damping in (27.13).

Fig. 27.6

Stability diagram for fixed base acceleration/applied force FRF for DOF 1, 2, and 3. There are three sets of stable poles—one for each fixed base mode

Fig. 27.7

Stability diagram for fixed base acceleration/base acceleration FRF for DOF 1, 2, and 3. There are four sets of stable poles—three sets are for the fixed base modes, and one set of stable poles is non-structural, as is likely associated with the “1/s” term associated with the damping matrix

Table 27.1 presents modal parameters from the eigenvalue analysis, the fixed based FRF, the structure accelerations divided by the base accelerations, and finally the relative acceleration FRF. For the cases with base acceleration as the reference, the extra set of stable poles was removed. The modal extraction results are identical to the level of precision used in the table for all four cases.

Table 27.1 Modal parameter results using different sets of FRF: baseline damping

In order to investigate a system with a higher overall damping level, the damping matrix was multiplied by a factor of 10 and the process was repeated. The modal extraction result for the different sets of FRF is presented in Table 27.2. Again, the extracted modal parameters are nearly identical as well. In this case, it appears the frequency-dependent “1/s” term associated with the damping matrix can be ignored as long as the pole associated with it is not selected. However, it is recommended that more test cases be performed in the future to better understand this effect.

Table 27.2 Modal parameter results using different sets of FRF: 10x baseline damping

27.7 Summary

This paper has presented a method for more accurate estimates of fixed base frequency response functions for structures tested on flexible supports such as shake tables. The key to implementation is to excite the boundary supports directly with at least as many independent sources (such as modal shakers) as there are independent boundary deformations in the frequency range of interest. The boundary accelerations, along with any applied forces on the structure, can then be used as independent references when calculating frequency response functions. The frequency response functions associated with applied loads or rigid body deformations of the shake table are mathematically guaranteed to be sufficient for calculating modal parameters. The same structural modal parameters were calculated using flexible base motion as reference degrees of freedom in the example in this paper whether using absolute acceleration or relative acceleration as responses.