Cantilever Beam Static and Dynamic Response Comparison with Mid-Point Bending for Thin MDF Composite Panels

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1 Cantiever Beam Static and Dynamic Response Comparison with Mid-Point Bending for Thin MDF Composite Panes John F. Hunt, a, * Houjiang Zhang, b Zhiren Guo, b and Feng Fu c A new cantiever beam apparatus has been deveoped to measure static and vibrationa properties of sma and thin sampes of wood or composite panes. The apparatus appies a known dispacement to a cantiever beam, measures its static oad, then reeases it into its natura first mode of transverse vibration. Free vibrationa tip dispacements as a function of time were recorded. This paper compares the test resuts from the cantiever beam static bending and vibration with standard midpoint simpy supported bending sampes. Medium density fiberboard panes were obtained from four different commercia sources. Comparisons were made using a set of fiberboard panes with thicknesses of 8.1, 4.5,.7, and.6 mm and nomina densities of 700, 770, 780, and 80 kg/m, respectivey. Cantiever beam static moduus and dynamic moduus of easticity ineary correated we but were consistenty higher than standard mid-point bending moduus of easticity having inear correations of 1.1:1 and 1.6:1, respectivey. The higher strain rates of both the static and vibrating cantiever beam coud be the primary reason for the sighty higher dynamic moduus vaues. The og decrement of the dispacement was aso used to cacuate the damping ratio for the cantiever beam. As expected, damping ratio had a sighty decreasing sope as density increased. This paper discusses the new apparatus and initia resuts. Keywords: Cantiever beam; Damping ratio; Dynamic moduus; Free-vibration; Static bending; Wood composite panes Contact information: a: USDA Forest Products Laboratory, Madison WI, 576 USA; b: Schoo of Technoogy, Beijing Forestry University, Beijing 10008, CHINA; c: Chinese Academy of Forestry, Wanshou Shan, Beijing, CHINA; *Corresponding author: jfhunt@fs.fed.us INTRODUCTION Evauation of wood and wood composites properties through vibrationa methods has been used with good success for severa decades (Mosemi 1967; Ross and Peerin 1994; Iic 00). In the iterature, most studies have focused on nondestructive testing using either ongitudina stress-wave or simpy supported transverse beam vibration frequency response techniques (Ross et a. 1991; Schad et a. 1995; Murphy 1997; Ross et a. 005; Hu 008). These studies have shown that nondestructive vibrationa properties correate we with bending and tensie modui as we as being abe to obtain damping coefficients. Much of this work focused on arge structura members. As new and ighter weight composite products are being deveoped with increasing demands on performance, there is a need for better anaysis and anaytica toos to quicky differentiate products or to describe enhanced performance characteristics. Research using transverse free-vibration for wood composites has aso been used and has shown simiar benefits for determining E, G, and damping vaues (Haines et a. 1996; Yoshihara Hunt et a. BioResources 8(1),

2 011). These generay use fexibe supports hed at the node points for free vibration. Vibration is initiated by tapping, and the resuting frequency is measured by a piezo materia or microphone apparatus at one end of the beam. There is no direct correation for dispacement vs. time measurement with most of these investigations. Other equipment such as the dynamic mechanica anayzers (DMA) use sma sampes that are vibrated at a known frequency, and the vibrationa response of the sampes are used to measure fundamenta properties of the particuar beam (Keey et a. 1987; Menard 008; Jiang et a. 008). This test method is very usefu for exporing specific characteristics or the infuence of individua parameters that can be differentiated as the sampes vibrate. specimen size. DMA sampes are significanty smaer and thinner than representative, as-produced, commercia sampes that woud be cut from typica composite panes for bending tests. According to ASTM D 107 (ASTM 006), bending or tensie test specimens need to be 50.8 mm wide for thickness beow 6 mm and 76. mm wide for a other thicknesses. Specimen engths shoud be nominay 4 times their thickness. This size of sampe coud not be used within currenty manufactured DMA vibrationa test equipment. Testing as-produced composite sampes requires arger fixtures to measure the vibrationa properties. The USDA Forest Products Laboratory deveoped a dynamic cantiever beam vibration (CBV) apparatus to test thin to moderatey thick as-produced wood-fiber composite materias (Turk et a. 008). Based on the initia apparatus, the authors are working cooperativey to deveop an improved cantiever beam test apparatus that measures both static bending and vibrationa properties using a one test set-up. The new cantiever beam apparatus has a oad ce attached at the oading point. It is possibe then, during the preoading phase, to measure static oad appied at a given deformation to obtain a static bending vaue. Then once reeased into its free vibration mode, direct dispacement measurements are obtained, and the data can then be used to determine frequency. This apparatus has the advantage of obtaining both static and dynamic properties from the same specimen having the same test conditions with the same boundary conditions. This new apparatus reduces many test variabes, resuting in improved comparisons between static and dynamic responses of a specimen. The authors understand that most vibration theory treats the static and dynamic modui as equivaent (Harris and Pierso 00); however, there are differences observed in the comparison of static bending and dynamic vibration data for simpy supported beams (Ross et a. 1991). The goa of our research was to deveop an apparatus to measure both static bending and transverse vibrationa properties of cantiever beams for thin composite materia anayses that uses the same specimen and test set-up for improved comparisons. This paper discusses the equations used and preiminary test resuts from the new apparatus using both the static cantiever beam moduus of easticity (SMOE) and dynamic cantiever beam vibration moduus of easticity tests (DMOE). Comparisons were made with moduus of easticity for standard simpy supported beam (BMOE) tests. This work is part of a continuing research program to deveop the cantiever beam vibration apparatus for improved testing and evauation. Cantiever Beam Bending Equations For static bending of a cantiever beam, as shown in Fig. 1, the equation that describes defection is as foows, Hunt et a. BioResources 8(1),

3 y P E s I (1) where P is static oad (N), y is dispacement of static oad point (m), is uncamped or E s is static moduus of easticity (SMOE, Pa), and I is area moment of inertia of the beam cross section (m 4 ). To cacuate the static MOE, we can rewrite Equation (1) in terms of known beam dimensions as foows, P 4 P SMOE E () s yi ybt where b is base width of the beam (m), and t is thickness of the beam (m). Therefore, given a measured dispacement (y) at a oad (P), the SMOE can be determined. Both Equations (1) and () do not incude shear deformation terms. It is assumed that the ratios of beams ength to thickness (:t) are so sma that shear effects can be negected. ASTM standards suggest a span ength to thickness ratio of 4. For our cantiever beam, ½ the ength of a fu span woud resut in a ratio of /t of 1. Our specimens were much onger. P y Fig. 1. Static bending of a cantiever beam Cantiever Beam Vibration Equations The frequency of the first mode of free vibration of a cantiever beam is given by Equation () (Harris and Pierso 00), n1 f E m d u I () where n1 is frequency of the first natura mode of vibration (radians sec 1 ), f is the detected frequency of the first natura mode of vibration (Hz), is ength of the cantiever beam (m), E d is dynamic moduus of easticity (Pa), I is area moment of inertia of the beam cross section (m 4 ), and m u is mass per-unit ength (kg m 1 ). Equation () can be rearranged and written in terms of known vaues to provide the dynamic moduus of easticity (DMOE), DMOE m I u ( f ) M L 1 bt ( f ) (4) where M is mass of the specimen (kg), L is tota ength of the specimen (m), b is base width of the specimen (m), and t is thickness of the specimen (m). Hunt et a. BioResources 8(1),

4 Equation (4) is an ideaized equation of vibration that negects the effects of shear force and rotary motion in the specimen. Harris and Pierso (00) cacuated that if the specimen size was made such that the radius of gyration divided by the free ength was ess than 0.0 (dimensioness), then the frequency correction factor approaches 1.0. As the correction factor approaches 1.0, shear and rotary effects coud be considered negigibe for cantiever-free vibration (Eq. (5)). This works out for a beam having a ratio of free ength to thickness (:t) greater than 14.5 (Eq. (6)). Then for thin composites from 1- to 10-mm thick to negect any effects of shear, the ength for the test sampe ength shoud be from 14.5 to 145 mm, respectivey. Radius_of_ Gyration Free_Lengt h I/Area bt (1 bt ) t t 0.0 (5) Free Length Thickness t (6) Free vibration of a cantiever beam appears as a damped sine wave, as shown in Fig.. The damping component or the interna friction during the vibration impacts the resonant frequency so that it is ess than the natura resonant frequency without damping. The ogarithmic decrement of vibrationa decay ( ) is a measure of interna friction and can be expressed in the form (for free vibrations) of Equation (7), n A A n n 1 n 1 1 n A A 1 n 1 f f r (7) where is the ogarithmic decrement of vibrationa decay, A 1 is the first ampitude of the damped sine wave seected, A n is the nth ampitude of the damped sine wave seected, A n+1 is the (n + 1) th ampitude of the damped sine wave seected, f is the natura resonant frequency without the damping, and f r is the resonant frequency tested, damping ratio. A 1 A n t Fig.. Damped sine wave for free vibration of a beam From Equation (7), we can cacuate the damping ratio ( ) using the ogarithmic decrement of vibrationa decay ( ) in Equation (8): Hunt et a. BioResources 8(1),

5 4 (8) Based on Equations (7) and (8), we can cacuate natura resonant frequency (f) from the measured resonant frequency (f r ), as shown in Equation (9). f 1 f r (9) The natura resonant frequency (f) can be substituted into Equation (4) to cacuate the DMOE. Simpy Supported Beam Equations The standard test method used to obtain bending MOE (BMOE) for composite panes is outined in ASTM D107 (ASTM 006). Equation (10) is used to determine BMOE based on the oad/defection ) curve for a simpy supported beam with a constant cross section. The method suggests obtaining the inear ratio of from 10% to 40% maximum oad. bending _ MOE P s y 4 bt where BMOE is bending moduus of easticity (Pa), y is mid-point defection (m), P is mid-point oad (N), s is span, simpy supported beam ength (m), b is base width of the specimen (m), and t is thickness of the specimen (m). (10) EXPERIMENTAL Materias Five sets of commercia medium density fiberboards (MDF) having four different fiber types, processing, thicknesses, and densities (Tabe 1) were tested. The materias were obtained from a oca retai outet, so the specific fiber and resin types or other manufacturing characteristics were not avaiabe for this test. One MDF pane was tested at two engths of 40 mm and 0 mm (Sets M), tota ength (L), width (b), and thickness (t) were measured prior to testing. The MDF specimen size, number, and average density are isted in Tabe 1. The /t ratios for the specimens ranged from 61 to 111, which was 4 to 7 times greater than the vaue of 14.5 that has been suggested as a minimum for incuding shear effects; thus, we assumed shear effects were negigibe. Cantiever Beam Apparatus and Test Methods The cantiever beam vibration (CBV) apparatus consisted of a support base, a beam ength bracket, a specimen camp, a aser sensor, a primary dispacement mechanism, and a oad ce ocated within the dispacement mechanism (Fig. ). Hunt et a. BioResources 8(1),

6 Tabe 1. Parameters and Data for Specimens Reevant to Dynamic Testing and Mid-point Bending Testing Bending settings Pane Thickness, t 8.1 (mm) 4.5 (mm).7 (mm).6 (mm).6 (mm) Specimen ID MDF MDF MDF.7 40 MDF.6 40 MDF.6 0 Span, s (mm) Length, L (mm) Width, b (mm) Nomina density (kg/m ) Defection rate (mm/min) Number of specimens for MOE test Number of specimens for MOR test Number of specimens for dynamic testing Length to thickness ratio, /t The specimens were inserted 50 mm into the camp and centered beneath a oading pate. The 50-mm grip ength was subtracted from the tota ength (L) to obtain the free beam ength (). The specimens were camped using a pate and screw assemby in which the screw was tightened to a constant torque to appy a constant pressure to secure the specimen. The specimen was hung verticay to minimize gravitationa effects during transverse vibration. On the free end of the specimen, a aser-dispacement -ine or zero-oad position. A dispacement hook (not shown) was connected to a oad ce and hooked to the end of the specimen to appy a consistent initia dispacement of 11.1 mm. At this initia dispacement, the oad was recorded, and the static moduus of easticity was cacuated. The hook was reeased from the end of the specimen reeasing the specimen to its free vibration state (first mode). The aser measured vibration dispacement of the beam as a function of time. Dispacement data were coected at a samping rate of 1,000 Hz. The software determined the frequency using Equations (7), (8), and (9), then using Equation (4) to cacuated the DMOE. A typica vibration response curve for specimen (.6 mm (t) 50 mm (b) 40 mm (L)) is shown in Fig. 4. A the specimen widths were nominay 50 mm as ASTM D107 standard specifies for specimens ess than 6 mm thick. For our series, the thickest panes were (8.1 mm), which woud have required a width of 76 mm. This apparatus was designed for a maximum width of 50 mm. In the future, wider camps may be necessary; however, we chose to keep a specimen widths at 50 mm for consistent testing. The cantiever beam was initiay dispaced to 11.1 mm and then reeased into its free vibration state. The 11.1 mm initia cantiever dispacement equates to approximatey 0% maximum stress as cacuated from moduus of rupture (MOR) (Eq. (11)) from the bending specimens. MOR Mt I (11) Hunt et a. BioResources 8(1),

7 where M is moment (N-m), and I is area moment of inertia (m 4 ). Beam vibration frequencies ranged from 8.7 to 1. Hz resuting in to 115 data points to describe each cyce within the vibration dispacement curve. B ra c k e t C a m p S p e c im e n L a s e r s e n s o r B a s e P rim a ry d is p a c e m e n t m e c h a n ism Fig.. Cantiever Beam Vibration tester shown with a specimen in position Votage (V) Time (s) Fig. 4. A typica cantiever beam free-vibration response For mid-point testing, the span for each of the four test series was set at 4 times the nomina thickness. The respective spans for each test series are isted in Tabe 1. The cross-head defection rates were set according to the ASTM test methods to provide consistent strain rates for each of the thicknesses. A of specimens were first tested using the cantiever beam vibration test, and then they were tested using the static mid-point bending test method. Approximatey haf of the specimens were tested to faiure to obtain maximum MOR. Figure 5 shows the mid-point bending test set-up. Hunt et a. BioResources 8(1),

8 Fig. 5. The mid-point bending test set-up RESULTS AND DISCUSSION Apparatus Repeatabiity To verify repeatabiity of the CBV apparatus, a random specimen (.6 mm (t) 50 mm (b) 40 mm (L)) was oaded and tested five consecutive times without removing it from the specimen grip or re-adjusting the positioning screws. The resuts show exceent repeatabiity, with a maximum variation in recorded frequency of 0.0 Hz (Tabe.). Simiar observations were made with other sampes evauated mutipe times. Aso, it can be seen that the DMOE was sighty higher than the SMOE. These differences wi be discussed in the next section. Tabe. Repeated Testing Resuts for a Singe Specimen Without Repositioning Initia Specimen ID Static MOE (GPa) Dynamic MOE (GPa) dispacement (mm) Resonant frequency f r (Hz) MDF MDF MDF MDF MDF Average Comparison of DMOE with SMOE Figure 6 shows five pots comparing DMOE with SMOE for each of the MDF types. The DMOE had a inear correation with SMOE for each of the board types with sopes ranging between 1.10 and The combined average inear correation sope was 1.1 with R vaue of 0.96 (Fig. 6(f)). This inear reationship spans the range of specimens having significanty different fiber types, processing, thicknesses, densities, and specimen ength, yet the reationship is very consistent. The resuts show that having the same test set-up, the same test conditions, and the same specimen provided very good correation between the two test methods. Hunt et a. BioResources 8(1),

9 The sighty higher DMOE vaues coud be due to higher strain rates during beam vibration. According to ASTM D 107, approximatey mm/mm/min (ASTM 006). For panes approximatey 6 mm thick, ASTM D 107 suggests using -mm/min cross-head movement, and for panes 1 mm thick, a rate of 6 mm/min is suggested. From the specimens tested in vibration, the highest dispacement rates occurred each time the beam passed through the neutra point and sowed to zero when the beam reached maximum dispacements. A conservativey frequency woud be 4,400 to 1,00 times faster than the ASTM test method. Wood and wood composites are rate-dependent materias, and the higher the strain rates, the higher the MOE vaues obtained. The faster strain rate during vibration coud be the significant contributor to the higher MOE vaues. Further anaysis needs to be done to determine effects of strain rate on MOE for the CBV apparatus. The effect of higher strain rates is mentioned here, but the anaysis for this effect is beyond the scope of this paper and wi be addressed in ater research and artices. Comparison between DMOE, SMOE, and Mid-Point BMOE Figure 7 shows the reationships between DMOE and BMOE for each pane series. There was good inear reationship between DMOE and BMOE on panes MDF , MDF , and MDF.6 0. The best coefficient of determination was with MDF.6 0 at For specimens from MDF.7 40, the coefficient of determination was very ow with an R of This ow correation may be due to a sma data spread of a singe data set. When a the data were combined, the inear correation was 1.6 with the coefficient of determination of 0.91 (Fig. 7e). Simiar data were obtained (but not shown) from the static cantiever beam as compared with the standard midpoint bending test. The overa data comparison was 1.1 correation with a 0.9 coefficient of determination (Fig. 7e). Both DMOE and SMOE showed exceent overa correations with BMOE. However, the sighty higher DMOE and SMOE vaues coud be partiay due to higher strain rates during beam vibration, as described previousy, as we as higher strain rates due to the quick appication of the initia 11.1-mm dispacement to the tip of the beam. Both dispacement rates were faster than the - to 6-mm/min dispacement rate used for the midpoint bending test. Aso, we reaize that the camp on the one end of the cantiever has some infuence on the bending response of the beam, but we are unsure of the exact magnitude. A possibe infuence on the DMOE vaue is the effective ength determination,, of the uncamped portion of the beam. In Equation 4 the uncamped ength is quadruped, so if there were an infuence, then it might show up based on differences in the free engths used to cacuate DMOE. If there were an infuence of 1 mm beneath the camp that might add to the effective ength, then the cacuated DMOE woud be 0.8, 1.4, and. % higher for the 550 mm, 40 mm, and 0 mm ong specimens, respectivey. It woud require an effect under the camp of 10.7 mm to change the DMOE by 6 % for the 0 mm ong specimens. The 40 mm and 550 mm beams woud change by 8.6% and 15. % for a 10.7 mm increase in the effective ength, respectivey. The higher DMOE vaues over the BMOE as shown in Figure 7(e) are inear at about 6 % greater than the BMOE. For our set-up, we beieve the camp had ony a minima effect on the free ength. Hunt et a. BioResources 8(1),

10 (a) MDF (b) MDF (c) MDF.7 40 (d) MDF.6 40 (e) MDF.6 0 (f) Combined data Fig. 6. DMOE and SMOE reationship tested by the cantiever bending apparatus Hunt et a. BioResources 8(1),

11 (a) MDF (b) MDF (c) MDF.7 40 (d) MDF.6 0 (e) Overa data for DMOE vs. BMOE (f) Overa data for SMOE vs. BMOE Fig. 7. Reationship between DMOE, SMOE, and mid-point BMOE We know that the pressure from the camps decreases sighty the thickness of the beam, thus creating a thinner moment of inertia (I) for the beam at the insertion point that then shoud increase the defection for a given oad according to Equation (1) and as a resut woud decrease the SMOE. However, the SMOE was sti higher than the BMOE. The boards used in this study were reativey high in density, and the decrease in thickness woud be minima. There is a need to study the exact infuence of the camps, but we beieve that the effects are minima. Hunt et a. BioResources 8(1),

12 Another factor that may have infuenced the bending difference between the DMOE and BMOE may have come from the higher shear strain or defection that woud resut in a ower cacuated BMOE. The span was 4 times the thickness according to ASTM standards, but there may have been sufficient shear to ower the BMOE vaues. There is a need to further examine the comparison based on shear strain infuences of the mid-span, static cantiever bend test, and the cantiever beam vibration test. The authors aso understand that moisture content has a strong infuence on materia properties. For this test sequence, there was time between testing for the DMOE and BMOE. Moisture contro was not possibe for this test sequence. Therefore there may have been sight property differences (up or down) due to moisture content fuctuation that woud then have infuenced the mechanica properties obtained from either the DMOE or BMOE testing. However, for the DMOE and SMOE testing, there woud be no time difference because same specimen was used for both tests and woud have been tested at the same time with the same set-up, thus eiminating any moisture content infuences for their comparison. Damping Ratio and DMOE Reationship with Density The damping ratio ( ) for a specimens ranged between 0.06 and 0.1 (Fig. 8). As expected, damping decreased as density increased. Damping ratio reates to the ost energy as stress is transferred within the board. Since increased density generay impies improved bonding (between fibers), it suggests better fiber network connections and ower energy osses during vibration. The reationship between damping ratio and density was potted as a inear reationship. However, damping ratio (energy oss) is more compex than a inear reationship and is affected by many interacting parameters other than by average density aone, such as density distribution through the thickness. In Fig. 9, MDF sampes (circed) showed higher DMOE than the others, but the pot of the damping ratio vs. density (Fig. 8.) shows no significant differences compared with the other panes. The differences may be a combined effect from density profie, fiber aignment, fiber type, resin amount, or resin type for the MDF 4.5 series. Fig. 8. Reationship for Damping Ratio as a function of density Hunt et a. BioResources 8(1),

13 7 6 y = 0.014x R² = Density (kg/m ) Fig. 9. DMOE as a function of density. Circed data are sampes from MDF pane CONCLUSIONS As composites become more compex and are required to achieve improved performance, there is a need to measure and study performance differences such that the pane can be engineered for a particuar performance criterion. Additiona study wi be required to sort out the infuences of other pane parameters on the interaction between static and dynamic properties. 1. The cantiever beam apparatus provides an easy method to measure pre-oad and end dispacement of a fiberboard composite beam that can then be used to determine static beam mechanica properties. The SMOE of MDF beams correates very we with mid-point BMOE.. The cantiever beam apparatus provides an easy method to initiate a free vibration of a beam and measure end dispacement as a function of time. Tip dispacement vs. time can be used to determine frequency. Then from the physica properties of the beam, the DMOE of the beam can be determined. Overa DMOE of MDF beams correate very we with mid-point BMOE.. Damping ratio was shown to decrease as density increased. However, the reationship is more compex than a simpe inear correation with density. 4. The testing is nondestructive and highy repeatabe for determining SMOE, DMOE, and damping. With one test set-up, the CBV apparatus aows mutipe measurements that can provide more compex anayses and may provide better understanding of the composite pane than coud be obtained with just one static test. 5. The cantiever beam can be cut from as-produced composite pane pieces to determine vibrationa properties. 6. Additiona research is necessary to determine reasons for the difference between test methods for determining MOE. Hunt et a. BioResources 8(1),

14 7. Additiona research is necessary to determine how other pane properties such as density profie through the thickness, fiber aignment, and fiber-type impact cantiever static and dynamic properties. Athough the CBV apparatus provides more information than the standard bending test method, both the USDA Forest Products Laboratory and Beijing Forestry University wi continue to cooperativey deveop the cantiever beam apparatus. Our research has shown that the apparatus has the potentia to provide quaitative and quantitative information that can be used to study and understand the fundamenta materia properties of as-produced thin fiber-based composites. Additiona research is necessary to determine effects of strain rate, camping ength, camping pressure, width of sampe, density profie, fiber ength, and fiber orientation on the static and dynamic modui vaues. ACKNOWLEDGMENTS The authors thank the 948 fund of Chinese State Forestry Administration for funding this research, grant #: The corresponding author thanks C. Turk and D. J. Marr from USDA Forest Products Laboratory for their initia CBV deveopment. REFERENCES CITED ASTM (006). Standard D 107. Evauating the properties of wood-base fiber and American Society for Testing and Materias, West Conshohocken, PA. Haines, D. W., Leban, J. M., and for spruce, fir and isotropic materias by the resonance fexure method with Wood Sci. and Tech. 0, 5-6. Harris, C. M., Pierso, A. G. (00). Shock and Vibration Handbook, 5th Ed. McGraw- Hi, New York, 00, 1568 pp. Hu, Y.C. (008). Nondestructive testing of mechanica parameters for wood-based materias Proceedings: 17th Word Conference on Nondestructive Testing, 5 8 Oct 008, Shanghai, China. Iic, J. (00). Dynamic MOE of 55 Hoz Roh Werkst. 61(), Jiang, J. L., Lu, J. X., and Yan, H. P. (008). Dynamic viscoeastic properties of wood treated by three drying methods measured at high-temperature rang J. Wood Fiber Sci. 40(1), Keey, S. S., Rias, T. G., and Gasser, W. G. (1987). Reaxation behavior of the amorphous components of wood J. Mater. Sci. (), Menard, K. P. (008). Dynamic Mechanica Anaysis: A Practica Introduction, CRC Press, Boca Raton, FL. Mosemi (1967). Dynam Forest Prod. J. 17(1), 5-. Murphy, J. F. (1997). Transverse vibration of a simpy supported beam with symmetric J. Test. Eva. 5(5), Hunt et a. BioResources 8(1),

15 Ross, R. J., Geske, E. A., Larson, G. L., and Murphy, J. F. (1991). Transverse vibration nondestructive te Res. Pap. FPL RP 50. USDA, Forest Products Laboratory, Madison, WI. Ross, R. J., and Peerin, R. F. (1994). Nondestructive testing for assessing wood members in structures: A FPL GTR 70. USDA, Forest Products Laboratory, Madison, WI. Ross, R. J., Zerbe, J. I., Wang, X., Green, D. W., and Peerin, R. F. (005). Stress wave nondestructive evauation of Dougas- Forest Prod. J. 55(), Schad, K. C., Kretschmann, D. E., McDonad, K. A., Ross, R. J., and Green, D. A. (1995). Stress wave techniques for determining quaity of dimensiona umber from FPL RN 065. USDA, Forest Products Laboratory, Madison, WI. Turk, C., Hunt, J. F., and Marr, D. J. (008). Cantiever-beam dynamic moduus for wood compo FPL RN 008. USDA, Forest Products Laboratory, Madison, WI. - pane quasi-isotropic medium- BioResources 6, Artice submitted: August 16, 01; Peer review competed: September 9, 01; Revised version received and accepted: November 1, 01; Pubished: November 1, 01. Hunt et a. BioResources 8(1),