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An inexpensive instrument to measure the dynamic response of standing trees to wind loading
Agricultural and Forest Meteorology 132 (2005) 78–83 www.elsevier.com/locate/agrformet
An inexpensive instrument to measure the dynamic response of standing trees to wind loading John R. Moore a,*, Barry A. Gardiner b, Guy R.A. Blackburn b,c, Andrew Brickman d, Douglas A. Maguire a a
Department of Forest Resources, Oregon State University, Corvallis, OR 97331, USA Forestry Research, Northern Research Station, Roslin, Midlothian EH25 9SY, Scotland, UK c Forth Valley Primary Care, Larbert, Stirlingshire FK5 4SD, Scotland, UK d Department of Civil, Construction and Environmental Engineering, Oregon State University, Corvallis, OR 97331, USA b
Received 13 July 2004; received in revised form 20 July 2005; accepted 20 July 2005
Abstract A relatively inexpensive caliper type strain-gauge transducer was developed for measuring the dynamic behavior of tree stems and branches subjected to wind loading. Laboratory and field tests showed that the voltage signal produced by the transducer was linearly proportional to the strain it experiences, and that in turn this strain was linearly proportional to the displacement of the branches tested. The transducer was also found to be suitable for measuring the dynamic oscillations of trees. # 2005 Elsevier B.V. All rights reserved. Keywords: Wind loading; Strain-gauge transducer; Dynamic oscillations
1. Introduction Damage from wind is a problem in both natural and managed forests in many regions of the world (Quine and Gardiner, 1991), with effects ranging from foliage loss and crown breakage through to stem breakage and complete uprooting of trees. In addition, stresses * Corresponding author at: New Zealand Forest Research Institute Ltd., P.O. Box 29-237, Christchurch, New Zealand. Tel.: +64 3 364 2949; fax: +64 3 364 2812. E-mail address: [email protected] (J.R. Moore).
resulting from wind loading can induce physiological responses in trees that can include changes in stem and crown form, and the formation of reaction wood (Telewski, 1995). A number of authors (e.g., Mayer, 1987; Gardiner, 1994; Baker, 1997) have shown that trees behave as vibrating systems and that their response depends on the interrelationship between the frequency of applied wind loading and their own natural frequency. Studies by Scannell (1984) and Kerzenmacher and Gardiner (1998) hypothesized that the interaction between the vibration modes of different elements of a tree (i.e.,
0168-1923/$ – see front matter # 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.agrformet.2005.07.007
J.R. Moore et al. / Agricultural and Forest Meteorology 132 (2005) 78–83
main stem, first order branches, second order branches, etc.) was also important in determining the overall behavior of trees subjected to wind loading, particularly in the dissipation of energy. However, this hypothesis has largely been untested because of difficulties in collecting data. Previous studies that measured the natural frequencies of trees have used a variety of sensors including rotary or linear potentiometers connected to trees by thin wires (e.g. Milne, 1991), accelerometers (e.g., White et al., 1976), bi-axial tilt sensors (Flesch and Wilson, 1999), and prism-based systems (Hassinen et al., 1998). All of these sensors have proved satisfactory in providing data on the oscillation frequency of the main stem of a tree, with the possible exception of the errors that accumulate during the double integration of accelerometer data. However, their suitability for measuring the oscillation behavior of branches is questionable. Rotary potentiometers would be unsuitable for the task due to the difficulty in mounting them and preventing the wires from being obstructed by other branches. The cost of deploying an array of tilt sensors or prisms would be very high, and the resulting measurements would need to be separated into the component due to bending of the branch and the component due to the rigid body motion of the branch. A sensor that is capable of measuring the oscillations of both the stem and branches of a tree subjected to wind loading would help advance the understanding of tree biomechanics. Such a sensor should be relatively simple to manufacture and sufficiently inexpensive that arrays of them can be deployed to measure the oscillations of the stem of a tree and a number of branches simultaneously. This paper describes the development of a strain-gauge transducer that attempts to meet these requirements.
2. Methods and materials 2.1. Principle of the strain gauge transducer As an alternative to direct measurement of tree deflection under an applied load, the strain in the outer fibers of the tree can be measured. From Hook’s Law and elementary beam theory (Gere and Timoshenko, 1984), it can be shown that the strain at the base of a
79
beam of uniform circular cross-section is linearly proportional to the displacement, i.e., e¼
3Dd 2L2
(1)
where d is the diameter of the beam (m), L the length (m), and D is the end deflection (m). Within the range of linear-elastic material behavior, the strain in a tree stem or branch should also be linearly proportional to displacement. Therefore, the dynamic properties of trees (i.e., natural frequency and damping ratio) can be determined from strain measurements as these will be linearly proportional to displacement. Similarly, spectral analysis of time series of strain data can be used to investigate the response of trees to dynamic wind loading. 2.2. Transducer design and construction While Ennos (1994) has successfully used shortterm strain gauge installations to measure the surface strain on a range of tropical tree species under conditions of artificial loading, longer term installation of such gauges has proven to be problematic because of the poor quality of the bond between the gauge and the underlying wood (Blackburn, 1997). Instead of mounting the strain gauges directly to the wood, an alternative is to use a strain transducer. Here the strain gauge or gauges are mounted on another substrate which is then attached to the tree. In his thesis, Blackburn (1997) developed a strain transducer for use on tree stems. His design used two stiff lever arms to concentrate the bending in a small area of thinner metal at the hinge (Fig. 1). The thickness of the two arms compared to the flexibility of the hinge means the bending of the arms is negligible and therefore the exact thickness of the arms is unimportant. The precise thickness of the cut separating the two arms, or its exact position, is unimportant as long as it is approximately central and separates the arms by sufficient distance to allow the transducer to function in compression as well as tension. Blackburn (1997) used a single active gauge; the other gauge was used to provide temperature compensation and was placed in a different plane to the active gauge so that it would not be subjected to any strain under normal loading conditions.
80
J.R. Moore et al. / Agricultural and Forest Meteorology 132 (2005) 78–83
Fig. 1. Photograph of the strain gauge transducer with ‘‘long’’ arms attached. The total length over which strain is measured is 120 mm.
The same basic transducer design was used by Moore (2002) with some modifications. Firstly, the 10 mm diameter hole at the hinge end of the transducer was increased in size to 19 mm (Fig. 2). This allowed the second gauge (previously the passive gauge) to be mounted on the inside of this hole, on the opposite side of the hinge to the other gauge. Such an arrangement with two active gauges has a greater sensitivity than that used by Blackburn (1997). One requirement for a successful transducer is that it must not stiffen or locally reinforce the area where measurements are being taken. The thickness (4.5 mm) of the hinge on the original transducer used by Blackburn (1997) was small relative to the diameter of tree stems tested (300–400 mm), but was quite significant relative to the diameter of a branch
(30–40 mm). Therefore, a second more compliant transducer with a hinge thickness of 2 mm was constructed for use on branches. Simple calculations show that the stiffness (i.e., product of modulus of elasticity, E, and second area moment of inertia, I) of branches 30–40 mm in diameter is at least thirty times greater that the stiffness of the transducer. Therefore, it is unlikely that the transducer will act to stiffen branches of this size to any significant extent. Both transducer designs were made using aluminum with two 350 V fully encapsulated strain gauges (CEA-13-062UW-350, Measurements Group Inc., Rayleigh, NC) mounted on them. These gauges were self-temperature compensating for aluminum and had a gauge factor of 2.135. The gauges were mounted on the calipers as per the instructions provided by Measurements Group Inc. A single length of four core shielded data cable (Belden Inc., Missouri) was attached to a pair of transducers. All four strain gauges had a common excitation input but had separate outputs. Because the transducers were intended for use in a hostile environment, the strain gauges were coated with polyurethane varnish (M-Coat A, Measurements Group Inc.) to provide electrical resistance (i.e., prevent short circuits). The data cable was secured in place using cable strain reliefs to prevent load being transmitted to the gauges which could result in a strain being measured or the gauge tearing. Further mechanical protection and waterproofing was provided by coating the whole hinge region with a non-corrosive flowable silicon rubber (RTV 3140, Dow Corning Corporation, Michigan). For a given strain in the substrate (i.e., the tree stem or branch), the displacement of the transducer, and hence the voltage signal produced, depends on the distance over which it measures; the larger this distance the greater will be the signal produced by the transducer. Therefore, flat aluminum arms were added to the transducer to increase the distance over which it measured strain from 38 to 120 mm in the case of the stem transducers and to 57 mm in the case of the branch gauges. Blackburn (1997) noted, however, that the addition of these arms has a detrimental effect on the thermal stability of the transducer. Because of the short duration over which dynamic measurements are made (approximately 15 min in general), thermallyinduced strains should be negligible. A cap screw was also added to the branch transducers to limit the
J.R. Moore et al. / Agricultural and Forest Meteorology 132 (2005) 78–83
81
Fig. 2. Schematic diagram of the strain gauge transducer. All dimensions are in mm and the thickness of the transducer is 9.5 mm.
displacement of the bending arms and thus prevent plastic deformation of the hinge. 2.3. Transducer testing Before the transducers were deployed operationally, two sets of tests were conducted on them. The first was to determine the relationship between the change in output voltage from the transducers and the change in strain. Eight transducers (four stem and four branch) were randomly selected and each mounted in a micrometer. Because of the aperture size of the micrometer, the transducers were tested without their arms. The transducers were connected to a CR21X datalogger (Campbell Scientific, Logan, UT) in a ‘‘full-bridge’’ arrangement with two 11 kV balancing resistors. The change in output voltage was recorded at various levels of caliper displacement. The second series of tests aimed to determine the feasibility of attaching the gauges to actual branches and to test whether the relationship between displacement and strain was linear. The tests were conducted on two branches from a single Douglas-fir (Pseudotsuga menziesii Mirb. Franco) tree growing on the Oregon State University campus. The first branch was 3.50 m long and 87 mm in diameter at its proximal
end. Corresponding dimensions for the second branch were 3.00 m and 73 mm. Two transducers were attached to each branch; one gauge was attached to the top surface of the branch and the other to the side of the branch. The transducers were located as close as possible to the proximal end of each branch and held in place with 15 mm self-tapping wood screws. Masses of up to 14 kg were suspended from a point at a known distance from the proximal end of the two branches. The resulting deflection and transducer output were recorded during the loading and unloading of these masses. Damped free vibration tests were also conducted on the two branches with a transducer scanning frequency of 10 Hz.
3. Results and discussion There was some variation in the relationship between transducer displacement and change in voltage, however it appeared to be approximately linear (Fig. 3). In addition, the relationship appeared to be similar for stem and branch transducers, despite the latter being more compliant. The relationship between transducer displacement and output voltage was quantified and the assumption of linearity tested by
82
J.R. Moore et al. / Agricultural and Forest Meteorology 132 (2005) 78–83
Fig. 3. Relationship between displacement and output voltage for eight randomly selected strain gauge transducers. The line was fitted to the data using ordinary least squares regression.
Fig. 4. Relationship between displacement and measured strain for the two test branches. Lines were fitted to the data points using ordinary least squares regression.
fitting a linear model to the data from the micrometer tests using ordinary least squares regression
Results from the second series of tests showed that the gauges were able to record the bending-induced strain satisfactorily. Furthermore, the measured strain was linearly proportional to the branch displacement (Fig. 4), as expected for a linear-elastic system. In both cases, over 90% of the variation in measured strain was accounted for branch displacement. The dynamic tests also yielded satisfactory results as the gauges were able to respond sufficiently rapidly to record the transient response of the branches. For branch displacements of a typical magnitude expected due to wind loading, the transducers produced a strong signal (Fig. 5). When the branch was displaced in the vertical direction, the measured strain in the top surface was
Dvoltage ¼ b0 þ b1 Ddisplacement
(2)
Estimates of the model parameters b0 and b1 are given in Table 1. There was a strong linear relationship between transducer displacement and Dvoltage (r2 = 0.88), with a 1 mm transducer displacement resulting in a 1.78 mV voltage change. (In an earlier series of tests, Blackburn (1997) found that applied strain accounted for 99.3% of the variation in transducer output. These earlier tests displaced the transducers by mounting them in a milling machine and moving the bed of the machine. This is a more accurate method of displacing the transducer by a known amount than was used in the tests reported here.) With the arms added to the transducers, strains of 4685 me and 9862 me in the stem and branch transducers, respectively were required to cause a 1 mV voltage change. This difference reflects the different lengths of the arms that were added to the calipers. Table 1 Parameter estimates for the relationship between transducer displacement and voltage change Parameter
Estimate
Standard error
t-value
Pr(>jtj)
b0 b1
0.1800 1.7789
0.0444 0.0548
4.0547 32.4442
0.0001 <0.0001
Fig. 5. Time history of bending induced strain on the top and side surfaces of a branch under damped free vibration.
J.R. Moore et al. / Agricultural and Forest Meteorology 132 (2005) 78–83
much greater than the strain in the side of the branch (Fig. 5). While there were some small strains recorded by the gauge mounted on the side of the branch (possibly due to some motion of the branch in the horizontal direction), the transducers appear relatively insensitive to motion perpendicular to their axis of sensitivity. While these instruments have not been tested against other displacement transducers, the principles upon which they are designed and calibrated are well established in the fields of civil and mechanical engineering. Based on the results of the calibration tests reported here, the instruments are capable of measuring displacement to within 6% of the true value at the 95% confidence level. The output from the gauges in turn is linearly related to the deflection of the branch or stem to which they are attached. Provided displacements are not excessive (i.e., beyond the linear elastic range of the gauge), then the gauges will provide an unbiased estimate of displacement.
Acknowledgements The authors would like to thank Milo Clauson and Dick Holbo (Oregon State University), Dave Brooks and John Strachan (Forest Research) and Martin Hill (Holtech Associates) for their guidance during the development and testing of the transducers. Partial funding was provided by the Department of Forest Resources at Oregon State University. Comments from an anonymous reviewer on an earlier version of the manuscript are also appreciated.
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References Baker, C.J., 1997. Measurement of the natural frequencies of trees. J. Exp. Bot. 48 (310), 1125–1132. Blackburn, G.R.A., 1997. The growth and mechanical response of trees to wind loading. Ph.D. Thesis. University of Manchester, Manchester, 216 pp. Ennos, A.R., 1994. The biomechanics of root anchorage. Biomimetics 2, 129–137. Flesch, T.K., Wilson, J.D., 1999. Wind and remnant tree sway in forest cutblocks. II. Relating measured tree sway to wind statistics. Agric. For. Meteorol. 93, 243–258. Gardiner, B.A., 1994. Wind and wind forces in a plantation spruce forest. Boundary-Layer Meteorol. 67, 161–186. Gere, J.M., Timoshenko, S.P., 1984. Mechanics of Materials, 2nd ed. PWS Engineering, Boston, p. 768. Hassinen, A., Lemettinen, M., Peltola, H., Kellomaki, S., Gardiner, B., 1998. A prism-based system for monitoring the swaying of trees under wind loading. Agric. For. Meteorol. 90, 187–194. Kerzenmacher, T., Gardiner, B.A., 1998. A mathematical model to describe the dynamic response of a spruce tree to the wind. Trees 12, 385–394. Mayer, H., 1987. Wind-induced tree sways. Trees 1, 195–206. Milne, R., 1991. Dynamics of swaying Picea sitchensis. Tree Physiol. 9, 383–399. Moore, J.R., 2002. Mechanical behavior of coniferous trees subjected to wind loading. Ph.D. Thesis. Oregon State University, Corvallis, 211 pp. Quine, C.P., Gardiner, B.A., 1991. Storm damage to forests: a major abiotic threat with global occurrence. In: Proceedings of the Tenth World Forestry Conference, Paris, pp. 339–345. Scannell, B., 1984. Quantification of the interactive motions of the atmospheric surface layer and a conifer canopy. Ph.D. Thesis. Cranfield Institute of Technology, Bedford, UK. Telewski, F.W., 1995. Wind-induced physiological and developmental responses in trees. In: Coutts, M.P., Grace, J. (Eds.), Wind and Trees. Cambridge University Press, pp. 237–263. White, R.G., White, M.F., Mayhead, G.J., 1976. Measurement of the motion of trees in two dimensions. Technical Report 86, Institute of Sound and Vibration Research, University of Southampton, 22 pp.
An inexpensive instrument to measure the dynamic response of standing trees to wind loading John R. Moore a,*, Barry A. Gardiner b, Guy R.A. Blackburn b,c, Andrew Brickman d, Douglas A. Maguire a a
Department of Forest Resources, Oregon State University, Corvallis, OR 97331, USA Forestry Research, Northern Research Station, Roslin, Midlothian EH25 9SY, Scotland, UK c Forth Valley Primary Care, Larbert, Stirlingshire FK5 4SD, Scotland, UK d Department of Civil, Construction and Environmental Engineering, Oregon State University, Corvallis, OR 97331, USA b
Received 13 July 2004; received in revised form 20 July 2005; accepted 20 July 2005
Abstract A relatively inexpensive caliper type strain-gauge transducer was developed for measuring the dynamic behavior of tree stems and branches subjected to wind loading. Laboratory and field tests showed that the voltage signal produced by the transducer was linearly proportional to the strain it experiences, and that in turn this strain was linearly proportional to the displacement of the branches tested. The transducer was also found to be suitable for measuring the dynamic oscillations of trees. # 2005 Elsevier B.V. All rights reserved. Keywords: Wind loading; Strain-gauge transducer; Dynamic oscillations
1. Introduction Damage from wind is a problem in both natural and managed forests in many regions of the world (Quine and Gardiner, 1991), with effects ranging from foliage loss and crown breakage through to stem breakage and complete uprooting of trees. In addition, stresses * Corresponding author at: New Zealand Forest Research Institute Ltd., P.O. Box 29-237, Christchurch, New Zealand. Tel.: +64 3 364 2949; fax: +64 3 364 2812. E-mail address: [email protected] (J.R. Moore).
resulting from wind loading can induce physiological responses in trees that can include changes in stem and crown form, and the formation of reaction wood (Telewski, 1995). A number of authors (e.g., Mayer, 1987; Gardiner, 1994; Baker, 1997) have shown that trees behave as vibrating systems and that their response depends on the interrelationship between the frequency of applied wind loading and their own natural frequency. Studies by Scannell (1984) and Kerzenmacher and Gardiner (1998) hypothesized that the interaction between the vibration modes of different elements of a tree (i.e.,
0168-1923/$ – see front matter # 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.agrformet.2005.07.007
J.R. Moore et al. / Agricultural and Forest Meteorology 132 (2005) 78–83
main stem, first order branches, second order branches, etc.) was also important in determining the overall behavior of trees subjected to wind loading, particularly in the dissipation of energy. However, this hypothesis has largely been untested because of difficulties in collecting data. Previous studies that measured the natural frequencies of trees have used a variety of sensors including rotary or linear potentiometers connected to trees by thin wires (e.g. Milne, 1991), accelerometers (e.g., White et al., 1976), bi-axial tilt sensors (Flesch and Wilson, 1999), and prism-based systems (Hassinen et al., 1998). All of these sensors have proved satisfactory in providing data on the oscillation frequency of the main stem of a tree, with the possible exception of the errors that accumulate during the double integration of accelerometer data. However, their suitability for measuring the oscillation behavior of branches is questionable. Rotary potentiometers would be unsuitable for the task due to the difficulty in mounting them and preventing the wires from being obstructed by other branches. The cost of deploying an array of tilt sensors or prisms would be very high, and the resulting measurements would need to be separated into the component due to bending of the branch and the component due to the rigid body motion of the branch. A sensor that is capable of measuring the oscillations of both the stem and branches of a tree subjected to wind loading would help advance the understanding of tree biomechanics. Such a sensor should be relatively simple to manufacture and sufficiently inexpensive that arrays of them can be deployed to measure the oscillations of the stem of a tree and a number of branches simultaneously. This paper describes the development of a strain-gauge transducer that attempts to meet these requirements.
2. Methods and materials 2.1. Principle of the strain gauge transducer As an alternative to direct measurement of tree deflection under an applied load, the strain in the outer fibers of the tree can be measured. From Hook’s Law and elementary beam theory (Gere and Timoshenko, 1984), it can be shown that the strain at the base of a
79
beam of uniform circular cross-section is linearly proportional to the displacement, i.e., e¼
3Dd 2L2
(1)
where d is the diameter of the beam (m), L the length (m), and D is the end deflection (m). Within the range of linear-elastic material behavior, the strain in a tree stem or branch should also be linearly proportional to displacement. Therefore, the dynamic properties of trees (i.e., natural frequency and damping ratio) can be determined from strain measurements as these will be linearly proportional to displacement. Similarly, spectral analysis of time series of strain data can be used to investigate the response of trees to dynamic wind loading. 2.2. Transducer design and construction While Ennos (1994) has successfully used shortterm strain gauge installations to measure the surface strain on a range of tropical tree species under conditions of artificial loading, longer term installation of such gauges has proven to be problematic because of the poor quality of the bond between the gauge and the underlying wood (Blackburn, 1997). Instead of mounting the strain gauges directly to the wood, an alternative is to use a strain transducer. Here the strain gauge or gauges are mounted on another substrate which is then attached to the tree. In his thesis, Blackburn (1997) developed a strain transducer for use on tree stems. His design used two stiff lever arms to concentrate the bending in a small area of thinner metal at the hinge (Fig. 1). The thickness of the two arms compared to the flexibility of the hinge means the bending of the arms is negligible and therefore the exact thickness of the arms is unimportant. The precise thickness of the cut separating the two arms, or its exact position, is unimportant as long as it is approximately central and separates the arms by sufficient distance to allow the transducer to function in compression as well as tension. Blackburn (1997) used a single active gauge; the other gauge was used to provide temperature compensation and was placed in a different plane to the active gauge so that it would not be subjected to any strain under normal loading conditions.
80
J.R. Moore et al. / Agricultural and Forest Meteorology 132 (2005) 78–83
Fig. 1. Photograph of the strain gauge transducer with ‘‘long’’ arms attached. The total length over which strain is measured is 120 mm.
The same basic transducer design was used by Moore (2002) with some modifications. Firstly, the 10 mm diameter hole at the hinge end of the transducer was increased in size to 19 mm (Fig. 2). This allowed the second gauge (previously the passive gauge) to be mounted on the inside of this hole, on the opposite side of the hinge to the other gauge. Such an arrangement with two active gauges has a greater sensitivity than that used by Blackburn (1997). One requirement for a successful transducer is that it must not stiffen or locally reinforce the area where measurements are being taken. The thickness (4.5 mm) of the hinge on the original transducer used by Blackburn (1997) was small relative to the diameter of tree stems tested (300–400 mm), but was quite significant relative to the diameter of a branch
(30–40 mm). Therefore, a second more compliant transducer with a hinge thickness of 2 mm was constructed for use on branches. Simple calculations show that the stiffness (i.e., product of modulus of elasticity, E, and second area moment of inertia, I) of branches 30–40 mm in diameter is at least thirty times greater that the stiffness of the transducer. Therefore, it is unlikely that the transducer will act to stiffen branches of this size to any significant extent. Both transducer designs were made using aluminum with two 350 V fully encapsulated strain gauges (CEA-13-062UW-350, Measurements Group Inc., Rayleigh, NC) mounted on them. These gauges were self-temperature compensating for aluminum and had a gauge factor of 2.135. The gauges were mounted on the calipers as per the instructions provided by Measurements Group Inc. A single length of four core shielded data cable (Belden Inc., Missouri) was attached to a pair of transducers. All four strain gauges had a common excitation input but had separate outputs. Because the transducers were intended for use in a hostile environment, the strain gauges were coated with polyurethane varnish (M-Coat A, Measurements Group Inc.) to provide electrical resistance (i.e., prevent short circuits). The data cable was secured in place using cable strain reliefs to prevent load being transmitted to the gauges which could result in a strain being measured or the gauge tearing. Further mechanical protection and waterproofing was provided by coating the whole hinge region with a non-corrosive flowable silicon rubber (RTV 3140, Dow Corning Corporation, Michigan). For a given strain in the substrate (i.e., the tree stem or branch), the displacement of the transducer, and hence the voltage signal produced, depends on the distance over which it measures; the larger this distance the greater will be the signal produced by the transducer. Therefore, flat aluminum arms were added to the transducer to increase the distance over which it measured strain from 38 to 120 mm in the case of the stem transducers and to 57 mm in the case of the branch gauges. Blackburn (1997) noted, however, that the addition of these arms has a detrimental effect on the thermal stability of the transducer. Because of the short duration over which dynamic measurements are made (approximately 15 min in general), thermallyinduced strains should be negligible. A cap screw was also added to the branch transducers to limit the
J.R. Moore et al. / Agricultural and Forest Meteorology 132 (2005) 78–83
81
Fig. 2. Schematic diagram of the strain gauge transducer. All dimensions are in mm and the thickness of the transducer is 9.5 mm.
displacement of the bending arms and thus prevent plastic deformation of the hinge. 2.3. Transducer testing Before the transducers were deployed operationally, two sets of tests were conducted on them. The first was to determine the relationship between the change in output voltage from the transducers and the change in strain. Eight transducers (four stem and four branch) were randomly selected and each mounted in a micrometer. Because of the aperture size of the micrometer, the transducers were tested without their arms. The transducers were connected to a CR21X datalogger (Campbell Scientific, Logan, UT) in a ‘‘full-bridge’’ arrangement with two 11 kV balancing resistors. The change in output voltage was recorded at various levels of caliper displacement. The second series of tests aimed to determine the feasibility of attaching the gauges to actual branches and to test whether the relationship between displacement and strain was linear. The tests were conducted on two branches from a single Douglas-fir (Pseudotsuga menziesii Mirb. Franco) tree growing on the Oregon State University campus. The first branch was 3.50 m long and 87 mm in diameter at its proximal
end. Corresponding dimensions for the second branch were 3.00 m and 73 mm. Two transducers were attached to each branch; one gauge was attached to the top surface of the branch and the other to the side of the branch. The transducers were located as close as possible to the proximal end of each branch and held in place with 15 mm self-tapping wood screws. Masses of up to 14 kg were suspended from a point at a known distance from the proximal end of the two branches. The resulting deflection and transducer output were recorded during the loading and unloading of these masses. Damped free vibration tests were also conducted on the two branches with a transducer scanning frequency of 10 Hz.
3. Results and discussion There was some variation in the relationship between transducer displacement and change in voltage, however it appeared to be approximately linear (Fig. 3). In addition, the relationship appeared to be similar for stem and branch transducers, despite the latter being more compliant. The relationship between transducer displacement and output voltage was quantified and the assumption of linearity tested by
82
J.R. Moore et al. / Agricultural and Forest Meteorology 132 (2005) 78–83
Fig. 3. Relationship between displacement and output voltage for eight randomly selected strain gauge transducers. The line was fitted to the data using ordinary least squares regression.
Fig. 4. Relationship between displacement and measured strain for the two test branches. Lines were fitted to the data points using ordinary least squares regression.
fitting a linear model to the data from the micrometer tests using ordinary least squares regression
Results from the second series of tests showed that the gauges were able to record the bending-induced strain satisfactorily. Furthermore, the measured strain was linearly proportional to the branch displacement (Fig. 4), as expected for a linear-elastic system. In both cases, over 90% of the variation in measured strain was accounted for branch displacement. The dynamic tests also yielded satisfactory results as the gauges were able to respond sufficiently rapidly to record the transient response of the branches. For branch displacements of a typical magnitude expected due to wind loading, the transducers produced a strong signal (Fig. 5). When the branch was displaced in the vertical direction, the measured strain in the top surface was
Dvoltage ¼ b0 þ b1 Ddisplacement
(2)
Estimates of the model parameters b0 and b1 are given in Table 1. There was a strong linear relationship between transducer displacement and Dvoltage (r2 = 0.88), with a 1 mm transducer displacement resulting in a 1.78 mV voltage change. (In an earlier series of tests, Blackburn (1997) found that applied strain accounted for 99.3% of the variation in transducer output. These earlier tests displaced the transducers by mounting them in a milling machine and moving the bed of the machine. This is a more accurate method of displacing the transducer by a known amount than was used in the tests reported here.) With the arms added to the transducers, strains of 4685 me and 9862 me in the stem and branch transducers, respectively were required to cause a 1 mV voltage change. This difference reflects the different lengths of the arms that were added to the calipers. Table 1 Parameter estimates for the relationship between transducer displacement and voltage change Parameter
Estimate
Standard error
t-value
Pr(>jtj)
b0 b1
0.1800 1.7789
0.0444 0.0548
4.0547 32.4442
0.0001 <0.0001
Fig. 5. Time history of bending induced strain on the top and side surfaces of a branch under damped free vibration.
J.R. Moore et al. / Agricultural and Forest Meteorology 132 (2005) 78–83
much greater than the strain in the side of the branch (Fig. 5). While there were some small strains recorded by the gauge mounted on the side of the branch (possibly due to some motion of the branch in the horizontal direction), the transducers appear relatively insensitive to motion perpendicular to their axis of sensitivity. While these instruments have not been tested against other displacement transducers, the principles upon which they are designed and calibrated are well established in the fields of civil and mechanical engineering. Based on the results of the calibration tests reported here, the instruments are capable of measuring displacement to within 6% of the true value at the 95% confidence level. The output from the gauges in turn is linearly related to the deflection of the branch or stem to which they are attached. Provided displacements are not excessive (i.e., beyond the linear elastic range of the gauge), then the gauges will provide an unbiased estimate of displacement.
Acknowledgements The authors would like to thank Milo Clauson and Dick Holbo (Oregon State University), Dave Brooks and John Strachan (Forest Research) and Martin Hill (Holtech Associates) for their guidance during the development and testing of the transducers. Partial funding was provided by the Department of Forest Resources at Oregon State University. Comments from an anonymous reviewer on an earlier version of the manuscript are also appreciated.
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