Shape Optimization of Rectangular Plates for Vibration-Based Fatigue Testing

When components are subjected to high frequency vibrations, care must be taken so as to ensure they will not fail prematurely due to high cycle fatigue [1]. To avoid these premature failures, a fatigue life model should be used during the design stage of the component. Many fatigue life models have been developed over the past century, including the classical Paris’ law, and its derivatives, such as the Walker relation [2–4]. In more recent times, there has been increased interest in describing fatigue life in multi-axial stress states by using critical plane, enclosed surface, integral, and material structure-based models [5,6]. Common to all fatigue life models, however, is the need for empirically measured constants. Often these constants are measured using a servo-hydraulic machine that puts the test specimen in pure axial stress. Consequently, the resulting empirical constants are most generally applicable for components in an axial stress state. However, in many high-performance applications, such as gas turbine engines, the stresses realized are not uniaxial, but instead fully reversed bending [7,8].

In fatigue tests performed on a typical uniaxial servo-hydraulic machine, the stress is essentially constant throughout the gage area of the specimen. Conversely, in bending there is a stress gradient through the thickness of the specimen. Due to this gradient, the fatigue life of a part in bending can often be an order of magnitude greater than its uniaxial life, even if the maximum stress is identical [9,10]. Consequently, using axial fatigue data for components in bending typically results in safe, albeit overdesigned, parts [11]. For high-performance applications, this may not be an acceptable performance loss, in which case bending fatigue data should be used. Notably, however, bending fatigue data is somewhat rare in literature, and many common sources of fatigue data do not tabulate data for bending [12]. One method of quickly measuring bending fatigue data uses the resonant modes of a rectangular plate. This type of testing is commonly referred to as vibration-based fatigue.

There are several varieties of vibration-based fatigue testing in modern usage, at least two of which exploit the resonant properties of flat plates [13]. For low frequency testing, the simpler of the two methods involves cantilevering a rectangular plate with lateral notches to a shaker. The shaker then applies a transverse base excitation to the plate at a frequency corresponding to its first bending mode. This causes a large, fully reversed bending stress to develop in the region between the lateral notches of the plate. Strain in this fatigue region is monitored until crack initiation and subsequent failure of the plate, allowing for determination of the typical fatigue properties [14,15]. By varying the size of plate used, the resonant frequency of the first bending mode can be controlled. In practice, this type of testing has allowed for resonant frequencies ranging from 100 Hz to 800 Hz [16–18].

For high cycle fatigue testing, it is desirable for the resonant frequency of the plate to be as high as possible, so that the test finishes quickly. In this scenario 800 Hz may not be sufficient, in which case it is necessary to use higher order resonant modes that occur at higher frequencies. One such mode is the so-called “two-stripe” mode of a cantilevered rectangular plate; its name owing to the presence of two nodal lines running lengthwise down the plate [19]. In comparison with aforementioned methodologies that use the first bending mode, this resonant mode exists at a substantially higher frequency and shifts the fatigue zone from near the clamped region to the center of the free edge opposite the clamp [7].

Typically, tests using the two-stripe mode utilize a 4.5 × 4.5 × 0.125 in. (114.3 × 114.3 × 3.175 mm) square plate. With these dimensions the two-stripe mode has its resonant frequency in the vicinity of 1600 Hz [20]. Utilizing this plate geometry, the fatigue tests utilizing the two-stripe mode have been shown to produce results consistent with more traditional methods [7,20]. Further confidence in this method has been built via additional tests for additively manufactured nickel alloy 718 and Ti-6Al-4V [21,22]. However, in some situations, the commonly used 4.5 × 4.5 in. (114.3 × 114.3 mm) plate may not be the most optimal candidate; for example, if a test frequency significantly different than 1600 Hz is necessary.

Initial work in the optimization of the geometry of plates used in this variety of fatigue testing was done in 2002 by George [19] and Seidt [23]. However, due to computational limitations of the era in which this research was conducted, the results presented were limited in scope. Notably, the length, thickness, and width along the clamped edge were fixed; only the width of the free edge was allowed to vary. In addition, the mesh used in the finite element analysis (FEA) simulations was relatively coarse by modern standards, leading to higher uncertainty in the presented results.

The subsequent text is a continuation of the work originally contributed by Seidt and George. Whereas their prior work focused on only a few sizes and geometries of plate, the present work expands the scope to several tens of thousands of geometric combinations of rectangular plates (width x length x thickness) ranging from 3 × 3 × 0.0625 in. (76.2 × 76.2 × 1.588 mm) up to 8 × 8 × 0.250 in. (203.2 × 203.2 × 6.35 mm) For each of these combinations the resonant frequency, and several indicators of fatigue performance, are identified. Using this data several promising geometries are identified and experimentally validated. Additional recommendations are given for how to quickly select plate dimensions when planning a test based around the operating frequencies of the test equipment.

The goal of the present work is to optimize the dimensions of a rectangular plate to achieve the best possible fatigue performance when driven at the two-stripe mode. Hence, a careful overview of the properties of the two-stripe mode is warranted. The normalized displacement and corresponding stress contours for the two-stripe mode are shown in Fig. 1 for a 4.5 × 4.5 × 0.125 in. (114.3 × 114.3 × 3.175 mm) plate. The plots were achieved via a modal analysis in ansys using a 33 × 33 × 6 element mesh with quadratic volume elements. In each of these contours the top edge is the clamped edge whereas the bottom edge is the free edge. Hereafter, when referencing the dimensions of the plate, width will refer to the dimension along the clamped edge which is shown horizontally in Fig. 1. Conversely, length will refer to the vertical dimension of Fig. 1.

In Fig. 1(a), the out-of-plane displacement contours for the two-stripe mode are given. The predominant feature shown by these contours is the presence of a V-shaped free edge. As shown in Fig. 1(b), this causes a large uniaxial stress to develop at the free edge. By monitoring damage accumulation in this region, data points for a S-N curve can be captured. Ideally, the uniaxial stress at the free edge of the plate is the largest stress in the plate, as this would ensure reliable failure in the expected region for the placement of gauges. However, as shown in Fig. 1(c), a second large stress exists at the clamped edge of the plate. For some rectangular geometries (though not including squares), this stress can be larger than the free edge stress and is disqualifying for use in fatigue testing. Finally, the Von Mises stress distribution is shown in Fig. 1(d). These contours reconfirm that the two largest stresses are those at the free edge and clamped edge.

In this work, the geometry of the rectangular plate is optimized via a parametric approach. Modal analyses are conducted using ansys apdl to determine the resonant frequencies and normalized stress distributions in a base-excited rectangular plate. The geometry is initialized and meshed using a APDL script. For consistency, the number of elements is fixed at 6534 between all geometries. Material properties corresponding to that of 6061-T6 aluminum are used. A value of 10,000 ksi (68.95 GPa) is used for the modulus of elasticity, a value of 0.33 is used for Poisson’s ratio, and a value of 0.0975 lb/in³ (2699 kg/m³) is used for the density [24]. The width and length of the plates are varied incrementally from 3 to 8 in. (76.2 to 203.2 mm) in 0.0625 in. (1.588 mm) increments, whereas the thickness is varied from 0.0625 to 0.25 in. (1.588 to 6.25 mm) in 0.0625 in (1.588 mm) increments. The stress field and deflections of the plate are output for the first ten resonant modes. In total, data for 52,488 modal simulations were captured.

A sample of the first five resonant modes for two different plate geometries is shown in Fig. 2. From these data, it is readily seen that the location of the two-stripe mode in the sequence of modes is a function of the geometry. In the case of a 4.5 × 3.5 × 0.125 in. (114.3 × 88.9 × 3.175 mm) plate, the two-stripe mode is the third mode whereas in a 4.5 × 5.5 × 0.125 in. (114.3 × 139.7 × 3.175 mm) plate it is the fifth mode. Thus, for each modal simulation, it is necessary to identify which of the first ten modes most resembles the two-stripe mode. For this study, three criteria are used to identify the two-stripe mode:

1. The concavity of the displacement at the free edge of the plate should be of constant sign, as shown in Fig. 3. This criterion serves to exclude torsional modes such as the second and fourth modes of Fig. 2(b). It also ensures that the plate has its characteristic V-shaped free edge. In practice plates rarely meet this condition along their entire width. Using an iterative approach, it was found that restricting this criterion to the interior 80% of the free edge allows for the successful identification of the two-stripe mode in most geometries while generating few false-positives.

2. The x-component of axial stress along the midline of the plate should be strictly increasing (or decreasing), as shown in Fig. 4. This serves to exclude certain bending modes that can become sinusoidal along the length of the plate. An example of this type of mode can be seen in the third mode of Fig. 2(b). Once again, this criterion often fails near the clamped edge and free edge of the plate, thus it is restricted to only the interior 70% of the plate. Once again, this restricted region was found via an iterative approach.

3. The maximum and minimum deflections must both occur at, or near, the free edge of the plate. Conditions (1) and (2) are typically sufficient for identifying the two-stripe mode. Certain bending modes, however, can pass criteria (1) and (2) while having a very low stress at the free edge. The existence of large deflections near the free edge ensures that the strains are also large. Modes that fail this criterion are illustrated by the first mode of Fig. 2(a).

These three criteria were found to be sufficient in identifying the two-stripe mode in 79.6% of plates. Furthermore, these criteria resulted in few false-positives; less than 0.1% of plates had more than a single mode shape pass these criteria. A visual representation of geometries in which the two-stripe mode was successfully identified is shown in Fig. 5(a). In this plot, the thickness of the plate is fixed at 0.125 in. (3.175 mm) and the horizontal and vertical axes correspond to the width and length of the plate, respectively. Color is used to represent the position of the two-stripe mode in the sequence of modes.

The two-stripe mode can be readily identified in the remaining 20.4% of geometries via simple interpolation. For each geometry in which the two-stripe mode was identified, the corresponding resonant frequency is also known. From these data, the resonant frequency of the two-stripe mode for each of the unidentifiable geometries can be interpolated. Upon identifying the approximate resonant frequency of these geometries, the two-stripe mode is identified as the one with frequency closest to that of the interpolated frequency. The results of interpolation on Fig. 5(a) are shown in Fig. 5(b). In general, it is seen that the two-stripe mode ranges from the third mode for wide, short plates, up to the ninth mode for long, thin plates.

Not only is the location of the two-stripe mode in the sequence known for all tested geometries, but the resonant frequency of the two-stripe mode is also known for each of those geometries. For a given material, the frequency of the two-stripe mode is a function of width, length, and thickness. Frequency contours for a plate thickness of 0.125 in. (3.175 mm) have been super-imposed on Fig. 5(b) and are shown in Fig. 6. This data shows that the frequency of the two-stripe mode tends to decrease with increasing plate sizes. It also shows that the frequency of the two-stripe mode is more sensitive to changes in the width of the plate than the length of the plate. Several jump discontinuities occur at the interfaces between the 3rd/4th, 5th/6th, and 8th/9th modes.

Although the two-stripe mode exists for all tested geometries, there is no guarantee that each of those geometries are suitable for fatigue testing. Two additional criteria must be satisfied to ensure a given geometry is suitable for fatigue testing:

1. The ratio of maximum stress at the free edge and maximum stress anywhere else in the plate must be greater than unity. This ensures that failure due to fatigue will reliably occur at the free edge of the plate, where it can be monitored by appropriate sensors [7,19]. For the two-stripe mode, the second largest stress occurs at the clamped edge of the plate and in a direction perpendicular to the stress at the free edge. This property can be seen in Figs. 1(b) and 1(c). Hence, forcing the ratio of these two stresses to be greater than one ensures that failure occurs in the desired location. Geometric combinations that fail to pass this criterion are denoted by the shaded regions in Fig. 7(a).

2. The frequency of the two-stripe mode must be sufficiently isolated from nearby resonant modes. When multiple resonant modes are close together in the frequency domain, the mode shapes can exhibit properties of their neighbors [7,25]. Hence, it is necessary to ensure that the two-stripe mode is sufficiently isolated from neighboring modes in the frequency domain. This is achieved by requiring the frequency of the two-stripe mode to be separated from its neighboring modes by at least 10%. Geometries that do not pass this criterion are once again denoted by the shaded regions in Fig. 7(b).

The results of enforcing both criteria simultaneously are shown in Fig. 7(c). Any geometry that falls within an unshaded region of this figure meets the minimum requirements for fatigue testing. There are two general trends that can be observed from this data. First, geometries fail the stress ratio criterion whenever the width is greater than the length, as indicated by the lower shaded region in Fig. 7(a). Second, all other geometries that fail the exclusion criteria are centered on lines of changing mode number, which by nature often occur when two neighboring modes converge in frequency, and thus fail based on frequency isolation as indicated in Fig. 7(b). In total, application of these two exclusion criteria reduce the number of usable geometries by over 70%, as shown by the shaded region in Fig. 7(c).

Although Fig. 7(c) provides a set of geometries suitable for fatigue testing, it does not inform which of those geometries perform best for fatigue testing. It does, however, suggest two possible candidates: square plates and rectangular plates with a length-to-width ratio of 1.37. Because the frequency level curves have a steep slope, square plates will attain any given frequency with the minimum possible mass, which can allow for greater excitation on a force-limited shaker. However, as seen in Fig. 8(a), square plates have a low stress ratio, only slightly exceeding the minimum requirement of 1. Conversely, plates with a length-to-width ratio of 1.37 can attain a stress ratio exceeding 1.8. In addition, these plates exist in a large geometric space and are relatively impervious to tolerancing issues in their production. Because the minimum mass plates and maximum stress ratio plates exist on nonintersecting curves, it is impossible to produce a plate that simultaneously minimizes the mass and maximizes the stress ratio. As shown in Fig. 8(b), plates with a 1.37 length-to-width ratio have the additional property of achieving the maximum possible frequency isolation. Because each mode has two neighboring frequencies, the plot shows the isolation of the nearest of the two frequencies (i.e., the minimum isolation). It is also worth noting plates with a length-to-width ratio of 2.5 can have stress ratios exceeding 2.4. However, these plates exist in a very small geometric region and are likely to be hard to precisely machine in practice.

In Figs. 5–8, the thickness of the plate is fixed at 0.125 in. (3.175 mm) Beginning with Fig. 9, the thickness is allowed to vary while restricting the resonant frequency to narrow bands. By plotting all of the data between two fixed frequency contours for several different thicknesses, the response to plate thickness can be visualized. This is shown in Fig. 9. In this figure, each of the bands represents a different thickness of plate ranging from 0.0625 in. (1.588 mm) on the left to 0.25 in. (6.35 mm) on the right. The frequency range occupied by each band is the same; the left boundary of each band corresponds to 1575 Hz whereas the right boundary corresponds to 1425 Hz. This plot is similar to Fig. 8, as both square plates and plates with a length-to-width ratio of 1.37 are suitable for fatigue testing at all simulated thicknesses.

In Figs. 6 and 7, the resonant frequency of the plate was taken as the dependent variable. However, in practice it is likely that the desired resonant frequency is simply the maximum that a given shaker system can operate at. Hence, it is worthwhile to formulate the dimensions of the optimal plate in terms of the resonant frequency and plate thickness. Shown in Fig. 10 is a set of level curves corresponding to the plate width that maximizes the stress ratio and frequency isolation for a given resonant frequency and plate thickness. The corresponding length of plate is found by multiplying the width by 1.37.

The FEA results presented prior are validated in two respects. First, it is ensured that resonant modes occur at their expected frequencies, and that those frequencies are sufficiently isolated. Second, the stress ratio is verified by measuring the strain at the free and clamped edges of select plates at a fixed excitation level. Tests are performed on two sets of 6061-T6 aluminum plates ranging from 0.0625 to 0.196 in. (1.588 to 4.978 mm) thick. The first set of plates has equal length and width whereas the second set has a length-to-width ratio of 1.37.

Tests were performed on a Data Physics V617 shaker unit. The V617 shaker can apply a sinusoidally varying force in excess of 1000 lb. at frequencies up to 2400 Hz. A clamping block made of 304 stainless steel is mounted to the shaker via three 5/16 in. bolts torqued to 22 ft-lb (29.8 N-m). When used in conjunction with this clamping block, the shaker can supply base excitations exceeding 40 #x2019;s to the plates. Plates are secured to the clamping block via four 1/2 in. SAE Grade 5 bolts torqued to 90 ft-lb (122 N-m). This results in approximately 38,400 lb (170.8 kN) of preload and a minimum compressive pressure of 3000 psi (20.7 MPa) in the clamped region of the plates. The clamp includes two alignment posts to ensure consistent alignment of the plates in the clamp. The clamping configuration is shown in Fig. 11.

The shaker is controlled via a Data Physics Abacus 901 signal controller used in conjunction with a Dytran 3056D5 T accelerometer. The accelerometer is mounted to the top of the clamping block via a stud torqued to 24 in-lb (2.71 N-m). A Polytec OFV-353 laser vibrometer is mounted 18 in. (45.72 cm) above the plate and is used to measure the velocity of the plate. Velocity is measured at the center of the free edge and is positioned between 0.1 and 0.2 in. (2.54 and 5.08 mm) from the edge. Data from the vibrometer are used to locate the resonant frequency of the plate.

Before application of the strain gages, the resonant frequencies and damping properties of the plates were measured. This testing was a two-step process. In the first step, the two-stripe mode and its nearest neighbors were located via a wide, low amplitude sine sweep at a rate of 0.1 octaves/min. A typical result for this type of sweep is shown in Fig. 12(a). This data allows for the quantification of the frequency isolation of the two-stripe mode.

In the second step, the frequency response of the two-stripe mode was captured in higher resolution. A second sine sweep was run in a region of ±10 Hz around the resonant frequency with a lower sweep rate of 0.1 Hz/s. A modal curve fit was then applied to the resonant mode to determine the resonant frequency and damping ratio, as shown in Fig. 12(b) [27]. An additional five sine sweeps were done in the same manner. Reported values for the resonant frequency and damping ratio are the average of these six sweeps. Measurement of the resonant frequency and damping ratio before application of the strain gages allows for the quantification of the impact of the gages on the resonant frequency and response of the plate.

The strains at the free and clamped edges of the plate are measured using Micro Measurements CEA-06-062UW-350 strain gages wired in a quarter bridge configuration using a Vishay 2310B amplifier. Lead wires from the gage at the free edge of the plate are routed to the clamped edge while approximately following the nodal lines of the plate. These wires are then connected to larger wires at the clamped edge via a bondable solder terminal. This configuration ensures the contribution of the gages and wires to the stiffness of the plate is minimized. A typical example of this wiring configuration is shown in Fig. 13(a). The corresponding nodal lines for this plate geometry are shown in Fig. 13(b).

Upon application of the strain gages, the resonant and damping properties of the plate were measured by repeating the curve fit demonstrated in Fig. 12. The strain was then measured by utilizing the resonance search and dwell functionality of DataPhysics SignalStar shaker controller software. Using this functionality, the plate was driven at its resonant mode with base excitations ranging from 1 to10 g’s in 1 g increments. From 1 to 9 g’s the plate was held at each excitation level for 1 min. At the last step, the plate was held at 10 g’s for 2 min. A time history of the strain amplitude during a typical test is shown in Fig. 14(a). Subsequent reported data are for the last 60 s of the test. A typical time history for the last 60 s of a test is shown in Fig. 14(b).

In total, 30 plates of various geometries were machined. Fifteen of these plates were square and the remaining 15 had a length-to-width ratio of 1.37 (hereafter referred to as “rectangular”). For both geometries, five plates were selected to target a resonant frequency of 1000 Hz, five to target a frequency of 1500 Hz, and five to target 2000 Hz. For each target frequency, the plates were cut from sheets of 6061-T6 aluminum at five different thicknesses. Equation (1) was used to select plate width based on the thickness and frequency target. Due to manufacturing tolerances, however, the actual dimensions of the plates received were slightly different from what was requested. The as-manufactured dimensions are shown in Table 1 for each of the 30 plates. Because slight changes to the plate geometry can result in large changes to the resonant frequency, the expected frequencies for the as-manufactured plates, calculated using Eq. (1), are provided in Table 1. These dimensionally corrected frequency values will be used for comparison against the measured frequencies. The measured frequency and damping ratio were obtained via the curve fit demonstrated in Fig. 12(b).

For all tested plates, the observed error in the resonant frequencies, relative to the simulation results, ranged from 0.05% to 1.74%. For the rectangular plates, the lowest error consistently occurred in the 0.063 in. (1.588 mm) thick plates, whereas the highest error occurred in the 0.160 in. (4.064 mm) plates. This trend was reversed in the square plates; the 0.063 in. plates had the largest errors and the 0.160 in. plates had among the lowest error. On average, the resonant frequencies of the rectangular plates were biased upwards from the simulated values by 0.57%. The square plates were biased upward by 0.75%. Values of the damping ratio for each plate are also given in Table 1. These ratios ranged from 0.0305% up to 0.1289% and tended to decrease with increasing resonant frequency. The rectangular plates exhibited lower damping ratios than the square plates.

The isolation of the two-stripe mode in the frequency domain was also measured for all 30 plates (per criterion 2 in Sec. 2.2). The rectangular plates were expected to be isolated from lower frequency modes by 20.18% and from higher frequency modes by 33.55%. For the rectangular plates, measured values for the lower frequency isolation ranged from 20.81% to 21.72%, well exceeding the 10% criterion outlined in Sec. 2.2. The measured upper isolation was slightly lower than expected and ranged from 29.96% to 31.85%. For the square plates the simulated lower frequency isolation is 12.35% and the upper frequency isolation is 22.26%. The experimentally measured values ranged from 9.93% to 11.53% for the lower frequency isolation and from 23.20% to 24.20% for the upper frequency isolation. In all cases, the data weres tightly clustered, thus values of the frequency isolation are omitted from Table 1.

Data for the plates upon application of the strain gages is given in Table 2. Of the 30 plates, gages were applied to the 18 plates with thicknesses 0.090, 0.125, and 0.160 in. (2.286, 3.175, and 4.064 mm). Upon application of the strain gages and lead wires, 15 of the 18 plates experienced a subtle drop in resonant frequency. For the rectangular plates, these drops ranged from −0.01% to −0.45% with the average change being −0.26%. The square plates exhibited an average shift of −0.09%. For the majority of plates, application of the gages did not substantially affect the damping ratio.

The strains measured at the free edge of the plates are given in Table 2. As a point of reference, strains estimated using the damping ratios prior to application of the gages are also included in Table 2. These strains were calculated via a harmonic analysis in ansys utilizing the damping ratio’s provided in Table 1. For the rectangular plates, the free edge strains were consistent across the frequency range for both the 0.090 in. and 0.125 in. thick plates. The 0.090 in. plates achieved strains ranging from 581.7–583.6 μɛ and the 0.125 in. plates achieved 697.8–725.9 μɛ. In contrast, the strains in the 0.160 plates ranged from 706.1 μɛ to 842.5 μɛ. The experimentally measured strains in the rectangular plates agreed with the simulated values to within 10% in most cases with the largest deviation being the 160–1500-R plate at 13.52%.

Unlike the rectangular plates, the square plates exhibited a substantial variation in the free edge strain at all thicknesses. The strain in the 0.090 in. thick plates ranged from 561.2 to 702.8 μɛ. The strain in these plates decreased linearly with increasing resonant frequency. Conversely, the 0.125 and 0.160 in. thick plates achieved their largest strains at a resonant frequency of 1500 Hz, with strains at 1000 Hz and 2000 Hz being substantially lower. The strains measured in the square plates also differed from the simulated values more than the rectangular plates. On average the simulated strain values were 16.39% higher than the measured values, with a maximum of 28.56%.

Of the 18 plates with gages at the free edge, six of them had an additional gage added near the clamped edge to measure the strain ratio (per criterion 1 in Sec. 2.2). The strains at the clamped edges of the plates are given in Table 3. In all cases, the strain measured at the clamped edge was consistently lower than expected. For the rectangular plates, the strain ratio ranged from 3.08 to 3.60 instead of its expected value of 2.25. Similarly, the simulated ratio for the square plates was 1.50, but the measured strain ratio ranged from 1.99 to 2.36. The larger strain ratios tended to correspond to the smaller plates. A simulated strain ratio value that corrects for the effects of gage size and location on the plate is also included in Table 3. For more information on how this value was calculated, see Sec. 4.1. These compensated values tend to agree better with the measured ratio but are still not fully consistent.

The experimentally measured resonant frequencies given in Table 1 show good agreement with values calculated using Eqs. (1) and (2), with an average error of 0.57% and 0.75%, respectively. Among the rectangular plates, the 160-1000-R, 160-1500-R, and 160-2000-R geometries consistently had the largest error relative to Eq. (1). Similarly, among the square plates, the 063-1000-S, 063-1500-S, and 063-2000-S geometries consistently had the largest error relative to Eq. (2). Because the errors are consistently large for a specific plate thickness, it is not likely they were caused by inaccurate measurement of the plate width and length. Consequently, the most likely source of this error stems from the resolution of the instrument used to measure the plate thickness. The instrument used to measure the thickness has a resolution of 0.0005 in. (0.0127 mm) and is rated to an accuracy of ±0.001 in. (0.0254 mm) Thus, the actual thickness of the 063-1000-S, 063-1500-S, and 063-2000-S geometries may range from 0.061 to 0.063 in. (1.549–1.600 mm). Similarly, the thickness of the 160-1000-R, 160-1500-R, and 160-2000-R geometries may range from 0.155 to 0.157 in. (3.937–3.988 mm). If a value of 0.063 in. is taken for the square plates the maximum error decreases to 0.25%. If a value of 0.157 is taken for the rectangular plates, the maximum decreases to 0.76%. Thus, the error of the plates is largely explained by inaccuracies in the measurement of the thickness of the plate.

Upon application of the strain gages, 15 of the 18 plates experienced a downward shift in their resonant frequency. These shifts were small however, with the maximum amounting to only a −0.45% change. Conversely, there was no consistent trend with the damping ratio of the plates. In ten of the plates, the damping ratio increased, in seven of the plates it decreased, and one plate had no change. These changes were relatively small however, with the largest decrease being −9.25% and the largest increase being 8.02%. The discrepancy can likely be explained by small changes in the clamped edge boundary condition between tests. For example, as the four clamp bolts wear due to repeated torque cycles, the pressure distribution in the clamped region may change. It was also observed that fretting at the plate-clamp interface was causing a build up of metallic particles in that region. This would have also slightly affected the boundary condition over time. Because the differences in resonant frequency and damping ratio before and after application of the gages are relatively small, it is concluded that the addition of the strain gage did not substantially alter the mode shape of the plates.

The strains measured at the free edge for the rectangular plates are given in Table 2. The measured strains for the 0.090 in. and 0.125 in. thick plates were quite consistent, ranging from 581.7–583.6 μɛ and 697.8–725.9 μɛ, respectively. The three 0.160 in. thick plates had substantially more variation in the measured strain, achieving 779.6 μɛ, 842.5 μɛ, and 706.1 μɛ for the 160-1000-R, 160-1500-R, and 160-2000-R plates, respectively. For all three plates the measured strains were close to the simulated values, thus the discrepancy in the strains is likely not caused by improper mounting of the strain gages. To resolve these outliers, a second 160-2000-R plate was machined and tested in the same manner as previous plates. However, no appreciable change in the results was observed on the new plate. Thus, the low strain in the 160-2000-R plate seems to be a feature of the geometry, and not an outlier caused by a defect in the plate.

A plot of the strain amplitude for all 18 measured plates as a function of thickness is shown in Fig. 15. It can be seen that, although there is a large spread in the strains, there is still a very clear trend upward as the plate thickness increases. It is worth noting that, typically, thicker plates would increase the mass of the system, thereby decreasing the maximum excitation the shaker can supply and offsetting any potential gains due to the increase in the plate thickness. However, the mass of a typical shaker armature and clamping block will often exceed the mass of the plate by an order of magnitude. Thus, changes to the size of the plate will not typically have a substantial impact on the maximum excitation a shaker system can supply. Consequently, on lower force shaker systems it is recommended that thicker plates be used if the increase in thickness does not raise the resonant frequency beyond the range of the instrumentation.

The strain measured at the clamped edge of the plates was substantially lower than expected for all six plates tested. For the rectangular plates, simulation data indicate that the strain ratio between the free edge and clamped edge should be near 2.25. However, measurements showed this ratio to range from 3.06 to 3.60. This discrepancy is partially explained by the physical size of the gage and the area over which it measures the strain. As shown in Fig. 16, the strain near the clamped edge decreases rapidly with distance from the edge. At a distance of 3% of the total plate length from the clamped edge the strain decreases to 78% of its maximum. At 6% distance, the strain decreases to 56% of its maximum at the edge. For the 125-1500-R plate, the area of the gage spanned 0.11 to 0.21 in. (2.794 to 5.334 mm) from the clamped edge. Based on simulation data, using the average strain over this area results in a strain ratio of 2.80. This process was repeated for the other five plates with gages mounted at the clamped edge. The results are given in Table 3. The remaining discrepancy is likely caused by the boundary condition not being perfectly rigid.

Crack initiation in the fatigue zone of a plate results in a net reduction in stiffness. Correspondingly, crack initiation is accompanied by a drop in resonant frequency and a change in mode shape. To model these changes, the stiffness of the six elements through the thickness of the plate at the location of largest stress are incrementally decreased. The normalized drop in resonant frequency as a function of stiffness reduction at the crack initiation site is shown in Fig. 17 for both a square and rectangular plate. The square plate exhibited a 0.25% drop in resonant frequency whereas the rectangular plate exhibited a drop of 0.38%. Consequently, it is expected that crack initiation will be slightly easier to detect on the rectangular plates as compared with the square plates. In both cases, the reduction in stiffness resulted in the location of maximum stress following the tip of the crack and receding slightly toward the clamped edge. This did not dramatically alter the mode shape, however, and the plates continued to satisfy the three identification criteria outlined in Sec. 2.1. Thus, it is not expected that crack initiation will affect the quality of data measured until the crack spans a substantial portion of the plate’s length (by which point the test would have ended).

As shown in Fig. 15, decreasing the plate thickness tends to decrease the achievable strain at any given excitation level. In addition, thin, or low frequency, plates will require larger deflections to achieve those strains. Consequently, geometric nonlinearities in the plate can exacerbate nonlinear hardening effects even at modest excitation levels. This behavior is shown in Fig. 18 for several different thicknesses of plate driven at a 15 g base excitation. All plates shown are substantially below fatigue-level strains; however, it is seen that the thinner plates exhibit a substantial amount of nonlinear hardening. Further increasing the excitation level can cause these plates to exhibit path dependence in their frequency response [28,29]. At extreme excitation levels, oscillators with this behavior can undergo period-doubling bifurcations or develop chaotic waveforms [30,31]. At this point, fatigue testing becomes impractical.

To investigate the phenomenon further, a large 8.00 × 10.96 × 0.063 in. (203.2 × 278.4 × 1.600 mm) plate was machined and tested. By Eq. (1), the plate is expected to have a resonant frequency of 230 Hz, although it should be noted that the dimensions of this plate exceed the range of geometries for which the equation is calibrated. Sine sweeps at an amplitude of 0.1 g’s in the vicinity of this mode confirm a resonant frequency of 232 Hz with a reasonably well-behaved frequency response function. However, upon increasing the excitation level to 5 g’s, severe nonlinear effects manifested themselves. The frequency response function of the plate at a 5 g excitation level is shown in Fig. 19. When sweeping downwards through the frequency range the plate encounters a period-doubling bifurcation at 231 Hz following a short spike in the amplitude. When sweeping upwards through the frequency range the plate encounters period-doubling bifurcations at 230.8 Hz and 231.9 Hz along with a sharp fall in the response at 233 Hz. When the effects shown in Fig. 19 are present, fatigue testing is essentially impossible. Thus, some caution should be used when using very thin plates at low frequencies.

In summary, this paper has analyzed the two-stripe resonant mode of a thin, rectangular plate as it pertains to vibration-based fatigue testing. Using a parametric approach in FEA software, it was determined that there exists two types of plates suitable for fatigue testing: square plates and plates with a length-to-width ratio of 1.37. Square plates represent the minimum mass geometry and may allow for slightly higher base excitations to be extracted from a shaker system. Conversely, plates with a 1.37 length-to-width ratio have a higher ratio of strain between the fatigue zone and the clamped edge. This may allow for fatigue tests to be conducted even in the presence of severe fretting effects at the clamped edge. Equations were developed that provide the necessary plate dimensions to achieve a given resonant frequency when using a specific plate thickness.

The results of the FEA analysis were validated using an electrodynamic shaker to find the resonant modes of 30 plates of varying geometries. It was found that the frequency of the plates agreed well with the simulated results. In addition, the strain in the fatigue zone was measured for 18 of the 30 plates. This data showed that thicker plates tended to achieve higher strains for a given excitation level. Finally, additional strain gages were mounted at the clamped edge of 6 of the 18 plates. These gages were used to find the ratio of strain between the free and clamped edges of the plate. It was shown that plates with a 1.37 length-to-width ratio achieve a substantially higher strain ratio than square plates, as predicted by the simulations.

This work was supported by a grant from ARCTOS Technology Solutions, LLC on behalf of the Air Force Research Laboratory (award # 142411-0000003-19-02-C13). The authors also wish to thank support from the Young Investigator Program at the Air Force Office of Scientific Research (award # FA9550-21-1-0437).

There are no conflicts of interest.

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request. The authors attest that all data for this study are included in the paper.