Comparison of Experimental Shaker Table Results with Classical FEA Frequency Response Analysis vs. Mechanical Event Simulation

ABSTRACT

The increasing use of time domain Finite Element Analysis (FEA) solution packages allows designers and engineers to analyze models experiencing static, dynamic and inertial based loads. This time-based event simulation tool can predict motion at discrete time intervals through the coupling of the stiffness matrices and the dynamic motion equations. One of the consequences from the simulation is that the results can be analyzed using methods such as Fast Fourier Transform (FFT) to provide vibration frequency response for various input loads.

In this paper, we compare experimental shaker table results with both classical FEA frequency response analysis and the corresponding mechanical event simulation of these experiments. Our work utilized the ALGOR mechanical event simulation tool (MES). Even though MES is geared towards directly simulating the physics of events, it can be used to predict mathematically abstract results such as frequency response spectra. Our comparison includes fundamental frequencies and transfer functions in response to impact, forced frequency vibration and random vibration spectral densities.

1. INTRODUCTION

The validation of hardware to vibration standards is a difficult and sometimes elusive task. As with any hardware requirements, the validation process must be laid out in full along with the requirements during the early phase of customer negotiation. The criticality of vibration requirements to the design must be made known to the engineering team and the testing requirements must be fed back to the management structure to fully understand the constraints being placed on the schedule. Failure to do so can result in redesign and retesting efforts due to last minute surprises late in the project schedule. These vibration problems are usually in areas of the design where quick fixes and retest options may not exist.

2. DESIGNING WITH VIBRATION CRITERIA

Starting any design effort with vibration criteria in mind does provide general guidelines to the engineering team. Because vibration is affected and influenced by all aspects of the mechanical design, these guidelines cover everything from boundary constraints to material selection to physical geometric configurations. Previous experience within the organization and published literature are invaluable aids as they document someone else’s successes and failures. This knowledge from the design of previous products or similar products often results in design constraints that are limited by the same design constraints from the earlier experience.

In new product development, however, there is often a challenge in that there is not necessarily the knowledge base from which to draw. New product design can be a dangerous area for this very reason. A well-planned design may still lack proper consideration in some area of the product for the required vibration environment. A complex system contains many areas for potential surprises. Even a good product design ends up putting a deliberate reliance on extensive post-development vibration testing. This can result in a test, redesign, test cycle that can lead to chasing a problem. This cycle ends only by discovering a solution (requiring further test validation) or in product specification relief from the customer based on what you now know the design can take from your testing.

3. TESTING TO VIBRATION CRITERIA

When it is required, actual validation relies on experimental test results. These tests can replicate the vibration requirements through methods such as (i) the use of artificial testers such as shaker table or impact type fixtures or (ii) actual field measurements over road test courses, flight tests or field test trials. Both of these scenarios can be very costly to schedule and support. This is especially true when the testing is used as a design tool and not solely a qualification tool.

Artificial testers are mechanical devices that attempt to replicate perceived field vibration environments. The structure under test is placed on the device and the vibration test is monitored through any of a variety of vibration feedback devices. System-level operational tests can then be performed as the structure is subjected to the required excitation. These tests can include a fixed frequency test, random vibration test, impact test or even natural frequency sweep test. The size of the system tested is limited only by cost and physical capability of the available fixtures.

Actual field trials and measurements are conducted through extensive monitoring of equipment while it is being used in the stipulated application environment. An equipment vehicle road test, such as the Munsen road test course, is a prescribed course laid out over which the test vehicle is driven. It contains specific types of bumps and obstacles at specific intervals and must be driven over at specific rates of speed. Other types of field trials can result in long duration testing in the service environment on ground, air or sea-based vehicles. Test unit monitoring can be continuous or at prescribed intervals. Proof of the design is survival through the life of the testing.

Failure in any of these testing program areas can be costly in terms of schedule and manpower. Field-testing relies heavily on proper data collection. Without proper instrumentation, a design failure can not always yield clues as to the magnitude of the vibration loads to which the unit was subjected and the source of those loads. Thus, the expense of the test is often lost without gaining much useful data as to the root cause of the failure. Perhaps the more painful part of a failure is that the only way to validate a design fix is to go back and again conduct the testing phase that was just completed.

Although testing will always be required for absolute assurance of product specification compliance, there is a great cost and schedule benefit for FEA tools to do this type of analysis. These software applications have demonstrated their capability well beyond that which anyone needs to comment. They have enabled engineers to save countless hours by eliminating poor design features and converging on a better design through rigorous analysis long before a prototype part is produced. In any complex system, if one series of a design, test and redesign cycle can be eliminated, the tool has more than proved its value in manpower cost savings alone. If the total price of testing a product is analyzed, more of these tools would be made available as additional cost savings to an organization.

4. CLASSICAL FEA VIBRATION ANALYSIS

Standard FEA packages contain modules for many vibration analysis functions. Some of the more common modules include fundamental frequency analysis and mode shapes, forced frequency response, random vibration response and shock response. The use of these modules is limited only by the quality of information put into them in the form of the model. Assumptions about boundary conditions, dynamic loads and component interactions can be difficult determinations to make. In all but a few cases, the types of models available to solve are more limited in scope than the options available under a static analysis module. Most static FEA packages contain some sort of gap or contact element, spring element, variable stiffness element or stress/strain relationships that will result in varying stiffness dependent on loading or displacement. Vibration dynamic packages do not have these options.

The classical linear vibration analysis relies on calculating the fundamental frequencies first. These frequencies are based on the material (stiffness and mass contributors) as well as the boundary constraints applied to the system. They do not look at external stimuli. Fundamental frequencies are the solution to the homogeneous differential equations and have nothing to do with the externally applied loads. Once the fundamental response is known, linear vibration theory can then look at the applied forcing function. This function is in the form of any of the aforementioned vibration tasks; forced frequency response, random response or impact response. The solver uses the homogeneous solution calculated from the fundamental frequency analysis and calculates a particular solution to mate with the specific driving function. This is the very reason that classical vibration solvers lack the options available in static solvers. Any displacement causing a change in stiffness from a stress/strain curve or gap elements is not known prior to the fundamental frequency analysis being done.

5. MES Motion-based Analysis

The advent of simulation tools has placed a new capability in the hands of the design team. These event simulation tools allow for the problem to be modeled as it is going to occur in a real-time environment. Each snapshot in time carries with it the inertial data that affects the next time increment. What this means is that the very method by which the required vibration results are obtained can now fundamentally shift away from the abstract methods of classical analysis and move more toward a real-time analysis.

Consider the following example of an MES problem run with the ALGOR MES Software tool.

Figure 1: Sketch showing a flat plate vibration model with all four sides rigidly constrained.

A flat square plate is modeled with rigid constraints all around its edges. A time-based load is applied in the middle of the plate deflecting the plate upward for a certain period of time. Gravity is always applied during this analysis. After a certain time, the applied load is released and the plate is allowed to move under its own inertial loads. The result is plate movement in a positive and negative direction.

Figure 2: Deflection time domain curve shown against the applied loading curve

From classical theory, we know that the motion of the plate is governed by the fundamental frequency and mode shape. There are no more externally applied loads other than gravity. From Roark and Young [1] for our specific example, we expect to see a theoretical fundamental frequency predicted from equation (1).

As a secondary verification analysis, the problem was solved with the modal analysis solver in ALGOR. The result for the first fundamental frequency was 108.3 Hz.

If we capture the time domain displacement results of the plate from the analysis and perform an FFT on those plate displacements, the resulting curve shows a large spike at the fundamental resonant frequency.

Figure 3: FFT analysis of the time domain displacements showing resonant peak at 101.5 Hz.

Figure 3 has confirmed that the plate has moved per our expectations. This was done without the mathematically abstract calculation of the fundamental frequencies; it is based solely on the plate geometry and boundary constraints and resulting time domain displacements.

6. Modeling Difficulties

Although almost any combination of element size, order and time step will result in motion, we have already shown that there is an expected motion and shape that the real plate will physically undergo. Therefore, the problem comes down to validation of the model against expected results: a constant challenge for any FEA model.

Our simple plate example has shown us that there is a definite need for higher-order elements. For 1600 quadrilateral elements with 4 nodes per element, the FFT predicted a fundamental frequency of 195.3 Hz. For 1600 quadrilateral elements with 8 nodes per element (4 midside nodes added), the FFT predicted a fundamental frequency of 101.5 Hz. These compare to the theoretical solution of 108.2 Hz. Even a smaller mesh density of 800 quadrilateral elements with 8 nodes per element (4 midside nodes added) resulted in a FFT predicted fundamental frequency of 106.4 Hz.

Element size played a more significant role in the predicted displacement results. The higher-order problems stated above each had calculated an FFT fundamental frequency in the immediate area of predicted value. However, the smaller mesh size of 1600 elements showed a maximum displacement of 0.0118 inches at the time of the load release while the 800 element mesh showed a displacement of 0.0020 inches at the same instant in time - a rather large discrepancy. Peak to peak vibration varied as well with amplitudes from 0.008 inches to -0.011 inches for the dense mesh and 0.0021 inches to -0.0028 inches for the more coarse mesh.

A static model was built with a very fine mesh and a static load was applied to the middle of the plate. The value was set to the same value the ramped load was at the time when the plate was released. Although there were no dynamic effects included in the static model, the relatively small deformation should be able to prove some indication as to the validity of either of the two different mesh densities in the previous MES models. The result of this model indicated a maximum deflection of 0.0110 inches, thus confirming that the smaller MES mesh density is certainly in the area of what we would expect. So, although the FFT fundamental frequency is in good agreement for each of the two higher-order element MES mesh densities, the peak motion of the plates necessitates the finer model.

For our smaller MES mesh density model, hard disk storage space was approximately 1.2 gigabytes for 2 seconds of calculated motion. The size requirements are on the order of 1 static problem for each time step in the analysis. For our example, that would be 2000 static problems of the same size with additional problems being thrown into account for any automatic time step reduction that is required by convergence for any given time step. While computation power is not a focus of this paper, it should be noted that with any FEA model, the proper balance must be obtained between accuracy and available hardware resources.

7. Foundation for Shaker Table Comparisons

For us at SNC, the goal of tool utilization is to get to where we can simulate the entire test in a virtual environment. By using an MES tool programmed with actual test conditions, the design can be much closer to its final form by the time the first qualification vibration test is done. Any testing data or actual field data can further refine the model so that an accurate picture of the product can be retained for any other design effort. We have had an ongoing effort to correlate the MES solver against our measured shaker table results under a variety of different circumstances.

To do that, the real world tests have to be converted to a format that the ALGOR MES can understand. In the case of shaker table tests, this means modeling fixed frequency excitation, random vibration and shock testing as a time domain forcing function.

For fixed frequency excitation, this is not a difficult task. This forcing function usually takes the form of a prescribed gravity loading versus time curve.

Most of the vibration testing we do is in the form of random vibration profiles. This type of vibration best represents the operating environment of vehicle traffic, movement or random excitation. Random excitation refers to anywhere that the vibration excitation can be characterized as non-pattern oriented and non-predictable vibration. A typical resulting Acceleration Spectral Density (ASD) curve that we get may look like the following (see Figure 4):

Figure 4: Sample ASD curve showing frequency dependent amplitude.

To employ ALGOR’s MES, the curve must be transformed into a time domain signal that can be input into the analysis. Because we do not have the exact data that generated the ASD, we convert it using an approximate method.

The basic assumption is that by taking a very thin slice from the ASD curve, along with the frequency axis, you can calculate the root mean square (RMS) value of the vibration under that curve in the region bounded by your slice. This is given by the formula:

If your slice is appropriately small enough, then the approximate time domain vibration curve that gives you the same rms value as above is of the form:

This calculation is done at many intervals along the ASD curve until the entire curve has been broken into small frequency sections. All of the individual time domain gravity curves calculated from above are then superimposed to obtain a total time profile. This time domain profile is then entered into the MES as the gravity time domain curve. An example of the results of this calculation is shown below for a simple ASD curve that goes from 1 Hz to 100 Hz and has a constant amplitude of 0.01 g²/Hz.

Figure 5: Approximated time history curve from PSD curve.

It is always good practice to run a check calculation that converts the time domain data back into an ASD curve so that you can verify your choice for f and your MES time increment give an ASD curve within the acceptable limits. A general C program that breaks the curve up is capable of very good accuracy because the time increments and f can be very small.

Finally, we address the area of impact vibration. Here the MES tool is set up as an entire impact event. The virtual object is dropped or impacted as the event would occur in real life. This can be in the form of an impact hammer on a base plate or drop test of cargo on a floor from a certain height. If the event can be modeled, then there is no need to consider any kind of external applied loadings from the impact itself. The MES will include the dynamic impact and transfer the loads to the virtual object.

8. CONCLUSION

Simulation problems are ones in which we had very limited ability to solve directly just a few years ago. Vibration analysis was done through either the classical FEA tools or through load transfer estimation techniques. Classical tools lacked the versatility of some of the features available in static analysis solely because the displacement was not known until after the specific solution to the forcing function of the differential equation had been solved. This only occurred once the fundamental frequencies and mode shapes were calculated. A gap element and variable stiffness geometries were not easily solved, if at all. Load transfer techniques rely on personal experience and similar product comparison. The ability to reasonably set boundary constraints and estimate impact loads is crucial to the success of these traditional FEA models.

Because vibration validation is made through testing, it is our goal to become more proficient at using the MES tool to do these virtual motion tests. We want to strive to eliminate additional testing requirements caused by design failures and the need to retest. Once we have established a validation model correlated with measured results, the model then becomes a tool we can use for further design. It is not the goal to eliminate testing, but to augment it with more knowledge before the product enters the test bed.

We have shown that a level of self-validation can be achieved by comparing an FFT of the MES displacements with predicted classical vibration methods. Although the goal is not to calculate vibration frequencies through the MES, it is a natural consequence of the motion of the bodies. This verification establishes the mesh density and time steps needed for a successful MES model.

It has also been shown how the event simulation tool can solve stress and displacement problems for any of the vibration problems of forced response, random excitation and impact analysis. It can do this with element types and variable stiffness cases not easily obtained with classical FEA methods.

As is common within FEA, the challenge relates to the available computing power and electronic storage space. Our examples can easily consume days of time to solve and require gigabytes of space to store information. A good handle on the required FEA model parameters can be achieved through repeated model runs. However, because the MES is a complete FEA solution at every time step, the size of the problem can grow very large. Care must be taken to balance model parameters with available resources.

MES tools are enabling us to look at problems today in a way not previously accessible on the average designer’s desk. But, self-validation of a model is still the key to a successful analysis. This must be done before any attempt at an external correlation can be made. Once self-validation is complete and external correlation can be established, you then have a powerful model on which to base a design.

REFERENCES