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Chapter 7

CHAPTER 7

CONCLUSIONS

Normal Modes Analysis

Both top and back plate modes (0, 0) showed good correlation with tested frequencies by Rossing [9]. The remaining top plate modes were fairly close in visual appearance and polarity, though frequency discrepancies were at a maximum of 35.7% difference for mode (1, 1). Frequencies of back plate modes other than the mode (0, 0) showed much discrepancy with a maximum of 109% difference at mode (0, 4). It appeared that the back plate FE model was much less stiff than the actual guitar, though the mode shapes were very similar in appearance and polarity, even the higher modes. Since the sensitivity analysis that was to follow relied weakly on correlation of higher modes and more on relative differences that various features and parameters exhibited on mode (0, 0), the current model was deemed adequate.

Sensitivity Analysis of Top Plate Frequency of Mode (0, 0)

Influence by Feature Inclusion

As expected, the x-braces had the most effect on stiffening the top plate mode (0, 0) with a 39.5% increase in frequency when added followed by the bass bars with a 24.3% frequency increase. After assembly and during the tuning process of the top plate by the luthier, the braces could be thinned to maintain stiffness but decrease mass thus increasing natural frequency, or, conversely, removing height to decrease the moment of inertia from a taller brace thus decreasing natural frequency.

Influence by Parameter Change

It appeared that parameter changes had less affect on the frequency than did the feature inclusions, but remained significant enough influence to render the top plate “out of tune” if ignored and unaccounted for with other adjustments. An interesting observation could be made that if the material obtained by the luthier was different from expected values of stiffness or density and the astute luthier realizes this prior to the building process, the luthier could easily accommodate for the foreseen frequency change by adjusting the thickness of the top plate cut, saving much time in the final tuning process.

Influence of Arching

Arching was seen to again have a less influence on the top plate natural frequency than the various feature inclusions, more on par with influences by the parameter adjustments. However, arching is again is an adjustment that can be utilized by the luthier to account for material properties that deviate from the expected values. By increasing or decreasing arching, the model showed that a 7-8% change in top plate natural frequency was achievable.

Sensitivity Analysis of Back Plate Frequency of Mode (0, 0)

Influence by Feature Inclusion

Similar observations were made here as were made for the top plate; the braces had the most influence with 68.8% increase to the mode (0, 0) frequency. Same suggestions for the final tuning process apply here as in the top plate analysis; adjust the braces.

Influence by Parameter Change

As stated in the top plate section, parameter changes have less influence on frequency than feature inclusion, but it appears that an unexpected material property can be offset by a thickness adjustment.

Influence by Arching

Arching has a similar affect on frequency of the back plate mode (0, 0) as it does in the top plate, thus adding some credibility to the results, though since both top and back plate used similar modeling techniques, error would appear in both cases.

Sensitivity Analysis of Top Plate MEW of Mode (0, 0)

Influence by Feature Inclusion

If the luthier was looking to adjust the modal effective weight of mode (0, 0) of the top plate, it appeared there was no need to look any further than the bass bars. They appeared to have a whopping 750.5% increase on the modal effective weight of the mode. Actual testing needs to be performed in order to verify this result as Richardson [8] does not comment on specific influence of features on this measurement. Model 11.7 shows that once the model is completely assembled, removal of the bass bars still showed the same influence on the modal effective weight. Since Richardson discusses the potential correlation between a lower modal effective weight and the strings ability to drive top plate, and therefore increasing sound intensity, the bass bars could possibly be manipulated to achieve the desired goal.

Influence by Parameter Change

As with frequency, the results here show that a thickness adjustment can be made given an unexpected material property.

Sensitivity Analysis of Back Plate MEW of Mode (0, 0)

Influence by Feature Inclusion

Though no mention was made by Richardson [8] of how modal effective weight of the back plate influences sound, it was analyzed here for completeness. The reinforcing strip appeared to have the most influence with an observed 244.3% increase; however, as will be pointed out in the following section, the model of the strip was much stiffer than actual due to material orientation and assumption of a continuous beam.

Influence by Parameter Change

Adjustment of back plate thickness appeared to be able to more than compensate for unexpected material properties with approximately a 75% swing on the modal effective weight of mode (0, 0).

Modeling Errors in FEM

The following is a list of modeling errors and areas of improvement for the next iteration of the Martin D-28 finite element model.

Neck

The heel of the actual neck is more saddle shaped than the more abrupt change of the model which would have an increased stiffness in the neck but no influence in the analysis performed here.

Reinforcing Strip

After the results showed that the reinforcing strip had a large influence on the modal effective weight of the back plate mode (0, 0), the blueprint was revisited for potential errors. The grain of the actual reinforcing strip runs perpendicular to the length of the beam, whereas the model has the grain parallel with the length. Also, the actual strip is discontinuous at the braces, but modeled as a continuous beam. Both of these issues will have an influence on the stiffness of the reinforcing strip, so results related to this feature should be observed with caution or disregarded.

Braces

Most beams do not follow the 10:1 length to depth ratio recommended for accuracy. They more accurately could have been modeled using solid elements but with a tremendous increase in modeling time and degree of freedom count.

Material Properties

Possible the most influential source of error may be the material properties used in the model versus the actual material properties of the guitar tested. Since Rossing [9] did not publish the properties of the tested guitar, material properties were taken from book values [4]. Because of large fluctuations within each species of wood, the material properties of each piece of wood used should be tested prior to assembly and directly incorporated into the FEM.

Accuracy of Blueprint

It is highly probable that differences exist between the published blueprint obtained for the guitar [5] and the actual production model of the guitar tested by Rossing [9]. Martin is obviously trying to maintain a lucrative business and publishing exact blueprints may not be in their best interests. For accuracy in future iterations, procedures performed by Elejabarrieta [2] and colleagues on a classical guitar should be implemented for the steel-string guitar. The FEM and actual guitar were built concurrently with properties used in the model exactly correlating with the measured properties from the actual model.