Combining the Fixed Base Correction Method with The Acoustic Modal Method

Author: Dale Schick with information from ATA Engineering and Northrop Grumman Corporation

Introduction

In our previous application note, infrasonic acoustic modal survey testing using the ARS Neutron™,  we described how experimental data from Acoustic Research Systems (ARS), ATA Engineering, and Northrop Grumman Corporation pushes the possibility of true Experimental Modal Analysis via acoustic excitation by driving the ARS Neutron™ to infrasonic frequencies. Though this enables new kinds of testing, it also presents a challenge in terms of fixturing. While traditional experimental modal testing optimizes the boundary condition of the Device Under Test (DUT) by affixing to isolation masses, The Acoustic Modal Method often occurs in facilities which are not equipped for modal.

This application note explores the work done by ATA Engineering known as the Fixed Base Correction (FBC) method. By well-instrumenting or modeling the fixturing, some careful analysis can help effectively remove the fixture from the measurements, leaving the DUT response in isolation for correlation to Finite Element Models (FEM’s.) In a practical sense, this increases the usefulness of in-situ testing, be it acoustic or through vibration, so that the data can be more useful.

Review of Fixed Base Correction

One way to address the boundary condition challenge is to mount the test article on a well-modeled support that can be inserted into the analysis model and incorporated in the model updating effort. Another way to address both the uncertain boundary condition and the excitation of all the target modes in a modal survey test is to supplement the acoustic test with conventional excitation sources, such as modal shakers and impact testing, and use the FBC technique to measure FRFs associated with the boundary fixed.

FBC exploits the fact that when using drive point accelerations as references, mode shapes associated with the resulting FRFs are associated with those same acceleration degrees of freedom (DOF) fixed. The method has been used to measure fixed base modes on a variety of dynamically active boundary conditions including six-DOF base shake systems, shaker tables, and static testing fixtures.

Figure 1: Typical Acoustic Qualification Test Set Up

Acoustic excitation is beneficial in exciting acoustically sensitive modes; and some acoustic devices like the Neutron™ exciters are capable of a wider bandwidth, especially in the lower frequencies, than is typical. Two major factors that contribute to this capability are:

1. High initial efficiency through acoustic impedance matching (horn loading) and drivers with very high field strength to coil resistance ratios, and
2. Low mechanical losses through extremely robust construction – any flexure in the structure of the loudspeaker will be somewhat parasitic to the output unless the construction is very rigid.

The second point provides the added benefit of allowing their use for low-frequency modal testing above 2 Hz.

As such, recent research has been performed to explore how modal parameters (shape, frequency, damping) can be extracted from either test-measured or frequency response function (FRF)-synthesized cross-spectrum matrices that were created using acoustic excitation. The test article in this research is a flight-like mass model of a spacecraft built by Northrop Grumman Corporation in Dulles, VA. The spacecraft was configured to emulate an in-situ modal test performed alongside a traditional DFA test, with the spacecraft situated on a transportation cart which was isolated from the floor by using four rubber isolators under floor-jacks.

Figure 2: Spacecraft Mass Model on a Transportation Cart

Combining Acoustic Excitation and Fixed Base Correction

The straightforward way to implement FBC with acoustic excitation is to run all the exciters simultaneously including both the numerous acoustic excitation sources and the modal shakers mounted to the base and then use both the acoustic sources and the base acceleration sources as references when calculating FRFs. The resulting FRFs, including the acoustic excitation sources, will be associated with those acceleration reference DOFs fixed.

There, however, may be future scenarios where a very large number of independent acoustic references, on the order of 80 or more, may be employed that could make calculating FRFs difficult and time consuming. As such, this section presents an alternative approach that estimates modes from the fixed base corrected output cross-spectral matrix using FRFs from a separate test where only the accelerations of the boundary are used as references to remove motion from the base. The approach follows a Master’s thesis by L.M. Elias that separated responses due to disturbance sources from disturbances due to dynamically active boundaries. The FRF equation using acoustic sources and boundary accelerations as references can be written as

                                                     (1)

where {𝐴} are measured responses such as accelerations, {𝐴_𝐵 } are the boundary accelerations and can be generalized coordinates such as constraint shapes, {𝑃} are the acoustic sources, [H_Ap] is the FRF matrix associated with the acoustic sources {P} activated while holding the base accelerations {𝐴_𝐵} equal to zero, and [H_AB] is the test-measured FRF matrix due to exciting the boundary directly while holding the acoustic sources equal to zero. In practice, it is straightforward to generate the matrix [H_AB] directly by exciting the base directly with conventional exciters while not turning on the acoustic sources. The matrix [H_Ap], however, cannot be measured if the base is not excited directly because it is not possible to hold the base accelerations equal to zero while the acoustic sources are activated. Instead, the matrix must be generated from test data where all of the conventional base exciters and each acoustic source are activated.

To remove the requirement of calculating the matrix [H_Ap] directly, a first test that measures the matrix [H_AB] directly is performed by exciting the boundary directly while the acoustic sources are turned off, i.e., {P}={0}. Thus,

                                                         (2)

Equation (1) is expanded to {A}= [H_Ap]{P}+[H_AB]{A_B}, where [H_Ap]{P} and [H_AB]{A_B} are the response of the system due to the acoustic sources assuming the base accelerations are equal to zero and the response due to the base accelerations assuming the acoustic sources are equal to zero, respectively. This equation can be rewritten to solve for [H_Ap]{P} as

                                          (3)

where {𝐴̅} is the response of the acoustic inputs holding the base accelerations fixed. Since {𝐴} and {A_B} are measured and [H_AB] is known, the product {𝐴̅} can be calculated for each frame of data to calculate the output cross-spectrum [𝑆_𝐴̅𝐴̅] as

                                                                 (4)

The frames are then averaged together to calculate the total response cross-spectrum due to acoustic inputs holding the base motion equal to zero. The resulting cross-spectrum matrix is then used to estimate modal parameters using operational modal analysis techniques.

Experimental Example

The test article in figure 3, a satellite mass model provided by Northrop Grumman Corporation, is mounted to its transportation cart, which is a typical testing configuration for acoustic qualification testing. For this test, however, the transportation cart was supported on isolators used to regularize and isolate the transfer of loading from the cart to the ground. A total of 24 Neutron™ acoustic sources were used in an eight-stack configuration with three sources per stack. Control microphones placed in front of each source were used to set voltage levels for its corresponding source. ATA Engineering-owned modal shakers were used to excite the base of the test article as well, but this paper presents the results using impact testing on the base to generate the base FRFs.

A test display model is presented in figure 3. A total of 151 response accelerometers were mounted to the test article including twelve seismic accelerometers, which were mounted to observe the six rigid body modes of the transportation cart, the primary twist mode of the cart, and the three out-of-plane deformations (two roll and one vertical) of the center connection.

Impacts were performed using an un-instrumented mallet with a soft eraser bonded to the impact face to shape the frequency content of the impact to lower frequencies. Approximately five to eight impacts were performed near each of the twelve seismic accelerometers mounted on the transportation cart. The impacts were spaced greater than 8 seconds apart so that 8-second frames could be used to calculate FRFs. Ten transportation acceleration constraint shapes were used as references when calculating FRFs. The six rigid body modes of the transportation cart were supplemented with outer perimeter twist. Three additional constraint shapes were added: the of overall inner square rigid body RX, and RY, and Z motion out of phase with outer perimeter rigid body RX, RY, and RZ motion, respectively.

Figure 3: Test Article and Accelerometer Location

The cross-spectrum matrix of the acoustic sources while holding the base fixed is calculated using Equation (4). The first three singular values of both cross-spectrum matrices are presented in figure 4, along with the first three singular values of the FRF matrix associated with base accelerations as references. Note that the peaks in the fixed base corrected cross-spectrum matrix match those of the fixed base corrected FRFs for many frequencies, including the two primary modes. However, not all peaks are present in both sets because the FRFs associated with the base accelerations emphasize high effective mass modes while the cross-spectrum matrix from the acoustic sources emphasize acoustically sensitive modes.

Figure 4: First three singular values of Cross-Spectrum Matrix (LeftO and Base Acceleration Matrix (Right)

In order to verify fixity, mode shapes were extracted from the fixed base corrected cross-spectrum matrix using the OPOLY parameter estimation tool in ATA’s MATLAB-based IMAT™ software [11]. Three of these modes are presented in Figure 5, and the transportation cart does not move.

Figure 5: Example of Test-Measured Fixed Base Modes

As with almost every modal survey test that is conducted, some modes of interest may not be well excited because the location and orientation of the excitation source is not positioned properly. In these cases, it is oftentimes beneficial to excite the test article in a way that will emphasize a mode in question with whatever means is most expedient. For example, if the primary torsion mode of the test article were not well excited, perhaps the acoustic sources could be rotated so that they could provide both radial and rotational inputs instead of providing a purely radial input.

FBC + The Acoustic Modal Method

The excerpts in this application note verify that data from acoustic sources can be combined with conventional sources for modal survey tests and presents a method to process acoustic test data when the number of independent sources becomes very large.

If the boundary such as a transportation cart is well modeled, then the test results from the acoustic sources, either from calculated FRFs or from response cross-spectra, and the FRFs associated with conventional excitation techniques can be used directly to update an analysis model. In addition, FBC works with acoustic sources, which is important for cases when performing a modal survey test in a DFA test configuration is desirable and when well-modeled boundary conditions are not available. The additional benefit of combining FBC with acoustic excitation is that all high effective mass modes and acoustically sensitive modes are well excited and can be extracted.

In our next Application Note, we will shift gears and examine how Multiple Single-Input/Single-Output (M-SISO) drive signals can help to control Direct Field Acoustic (DFA) tests while removing test complexity.