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· Fundamental & Applied Research
Partial Modal Analysis for Health Assessment of Living TreesJoakim Axmon, Dr. Maria Hansson and Prof. Leif Sörnmo
Rot in living trees cause substantial losses for the forestry industry. The common practice when evaluating forest stands for, e.g., purchase, is assessment based on visual signs. In this paper a new non-destructive assessment method based on the impact excitation method is proposed. The trunk of a living tree is excited by the impact of a hammer, and the vibrations are measured by accelerometers. Resonance frequencies, circumferential mode shapes and propagation velocity of a surface wave are analysed. A function describing the expected frequency for a sound tree is derived, and used in a detector whose performance is evaluated for 93 trees of species Norway spruce. The partial mode shape is used to ensure that the corresponding resonance frequencies are compared to each other. It is found that the detector is successful and outperforms assessments by skilled experts in forestry.
Keywords:Modal analysis, impact method, decay, tree, detector
Norway spruce (Picea abies) is, like most trees, susceptible to fungi, whose colonization of the trunk leads to rot. The fungus may enter the host tree via roots that are in contact with root systems of already infected trees, or via spores entering the trunk through injuries. The wood structure decays when the fungus converts constituents of the cell walls into nutritious carbons. The uninjured sapwood (the outer wood of the trunk) in a living tree is generally more resistant to decay than the heartwood (the inner wood). Hence, for butt rot caused by a fungus entering the host via the root system, there are often no external signs unveiling the decay, while the heartwood may already have been transformed into a moist and swampy structure (Wagener and Davidson 1954). During the progress of decay, the mechanical properties of the wood are altered. The modules of elasticity and shear, and the density, decrease. However, the rate differs: the density decreases at a smaller rate than do the modules (Wilcox 1978). Thus, wave propagation velocities, which are dependent on the square root of the ratio of the modulus to the density, are lower for decayed wood than for sound.
Despite methods developed during the last decades (Ouis 1999), visual tree assessment still seems to be the common practice when evaluating forest areas. In a Swedish study (Vollbrecht and Agestam 1995), five experts in forestry individually assessed 372 spruces visually. It was concluded that their assessment was only slightly better than random.
The majority of vibration methods used for detection of decay in wood are developed for application to logs, utility poles or construction timer (Ouis 1999). Only a few methods are intended for detection of decay in living trees. Among the impact excitation methods, time-of-flight measurement of stress waves (e.g. Mattheck and Bethge 1993) seems to be the dominating approach. Only recently has more advanced signal processing been used: Responses from trunks of living Norway spruce to impacts of hammer blows were measured using an array of accelerometers distributed along the circumference, and resonance frequencies and mode shapes were determined (Axmon and Hansson 1999). Furthermore, the propagation velocity of what appears to be a surface wave travelling along the circumference has been estimated (Axmon and Hansson 2000). Lawday and Hodges (2000) examined the response of a single ash tree (Fraxinus excelsior), using a modal hammer and an accelerometer, and studied the measured signals using the short-time Fourier transform.
|Fig 1: (a) Distribution of accelerometers and a screw around the cross section, and (b) the initial response from a sound tree with circumference 0.90 m to a hammer blow applied between sensor 12 and 1.|
The aim of this paper is to show that the joint information on modes and propagation velocity can be used to discriminate sound trees from decayed trees. The circumferential mode shapes are used to enable a comparison of resonance frequencies on a tree-to-tree basis. Then, based on a subset of sound trees, a prediction function yielding the expected resonance frequency of a certain mode, given the circumference and the propagation velocity, is formed. This prediction function is used in a detector, whose performance is evaluated for 93 trees.
A. Material and Measurement Setup
A total of 94 trees (Norway spruce) from three forest stands in Southern Sweden have been examined. For each specimen, the circumference and two perpendicular diametric measures were measured and filed. Remarks were made on factors that might influence the outcome of the experiments, e.g., strange geometry of the cross section, resin exudation, the occurrence of unusually many dead branches, and injuries. Twelve accelerometers (range ±51 m/s2, bandwidth 11 kHz at 10% tolerance) were positioned equidistantly around the cross sections (1.2 m above ground), and a screw onto which the impact excitations were applied was firmly attached to the sapwood at a position midway between two consecutive sensors, Fig.1(a). The role of the screw is to bypass the bark, thereby saving it from mechanical injuries, but also to avoid attenuation of the exciting force. Blows by hand force of a hammer (weight 0.2 kg) served as exciting force. The responses of the cross sections in terms of the radial components of the vibrations were sampled from the sensor outputs (rate Fs=100 kHz) and stored for post-processing. The initial part of a response from a sound tree with circumference 0.90 m is shown in Fig.1(b). The signals are transient with duration of 20-30·10-3 s.
After the set of experiments, the trees were felled and the status of each tree was determined by studying the fell cut at ground level. The trees were classified into five groups:
|Fig 2: (a) Dimensions for an annular cylinder, and (b) circumferential mode shapes.|
Furthermore, trees with remarks on, e.g., triangularly shaped cross sections, were distinguished. The number of trees in each class is 43, 23, 7, 7 and 14, respectively, with the figure within brackets yielding the number of trees with remarks. According to Habermehl (1982), it is expected that the mechanical properties of trees in Decay I are somewhat affected, and for trees in Decay I and II, impaired.
B. Modal Analysis of Isotropic Cylinders
Some results for cylinders of homogeneous, isotropic and linearly elastic (HILE) materials are needed for the further development. In computational modal analysis of HILE cylinders based on Ritz' method, the displacement of a tiny volume element in the cylinder wall is described by
where r,q , and z is the radial, tangential, and longitudinal coordinate, respectively, t is the continuous time index, W is a temporal angular frequency, y is an arbitrary temporal phase, and m is the mode number of the circumferential mode shape, Fig.2. The functions Ur(r,z), Uq (r,z) and Uz(r,z) are scaling factors dependent on radial and longitudinal position, generally modelled by power series in the two coordinates (Leissa and So 1995, Verma et al. 1987). Somewhat simplified, these functions determine the longitudinal component, and m the circumferential component of the full three-dimensional mode shape.
In the Ritz analysis, the kinetic and potential energies T and V, respectively, are calculated over the body, with the coefficients of the power series serving as a generalized coordinate system. The principle of conservation of energy is invoked, stating that for a conservative system undergoing harmonic oscillations around an equilibrium the total energy, i.e., the sum of potential and kinetic energies, is constant and the kinetic energy reaches its maximum when the potential energy reaches its minimum, and vice versa. An energy function L=Vmax-Tmax is formed (the subscripts refer to the maximum quantity of energy during a vibratory cycle), and minimized with respect to the unknown coefficients of the power series. This results in a generalized eigenvalue problem, from which the eigenvectors yield the coefficients for the power series, and the associated eigenvalues the resonance frequencies.
The circumferential mode number is assigned prior to the analysis; that the assumption made on the displacement is correct has been verified for metal cylinders (Singal et al. 1987).
Calculations based on Ritz' method have been performed for a large number of cases of height-to-outer radius ratio H/RO and inner radius-to-outer radius RI/RO, Fig.2(a), (Axmon 2000). The results reported are given as non-dimensional frequencies, the square root of the eigenvalues obtained from the analysis. The non-dimensional frequency l 1/2 is related to the dimensional frequency in Hz by
where G is the modulus of shear and r is the density. The modulus of shear is related to the modulus of elasticity via the Poisson's ratio. In Fig.3, results for the lowest resonance frequency for each circumferential mode shape m=2, 3 and 4 are shown, with H/RO ranging from 2 to 20 and RI/RO ranging from 0 (solid cylinder) to 0.9. The results are based on a Poisson's ratio of 0.3 (applicable for many metals), and free-free end conditions. The following observations are made. When H/RO exceeds 6, an increment of H/RO seems to have no substantial influence on the frequencies. Additionally, the longitudinal component of the mode shape does not seem to have any nodes when H/RO>6 (Axmon 2000), implying that all cross sections of the cylinder vibrate in an in-phase motion. Furthermore, for an increasing ratio of RI/RO, i.e., introducing a hole of growing dimension in the cylinder, the resonance frequency of the lowest mode (m=2) is altered at an earlier stage than m=3, and m=3 earlier than m=4. The results show that for cylinders with sufficiently large ratio H/RO, the frequency may be considered dependent only on the ratio RI/RO, and the best indicator for this ratio is the mode m=2.
Fig 3: Non-dimensional frequencies for the lowest resonance frequency of the circumferential mode shape |
(a) m=2, (b) m=3, and (c) m=4, for varying thickness of the cylinder wall and height-to-outer radius.
Calculations for solid cylinders show that the differences between free-free and fixed-free end conditions are small for the lowest resonance frequency of each circumferential mode shape m=2, 3 and 4 (Leissa and So 1995).
C. Signal Model and Signal Analysis
Wood is non-homogeneous and best modelled orthotropic due to the different properties of the sapwood and the heartwood, the orientation of the fibres and the formation of annual growth rings (Schniewind 1989). However, it has been observed for freshly cut spruce logs undergoing continuous sinusoidal excitation in the radial direction that although a full three-dimensional mode shape is not always excited, circumferential components arise in the vicinity of the cross section where the exciting force is applied (Skatter 1996, Skatter and Dyrseth 1997). Thus, our signal model of the measured response is derived from the HILE case. Wood is considered linearly elastic for a certain range of the exciting force (Schniewind 1989). It has been established that the impacts by the hammer blows by hand force are within the linear region: the effect of varying exciting force is limited to a common scaling of the signals measured around the cross section (Axmon 2000).
The sampled signal yk(n) from the output of sensor k, positioned at angle q k taken from the point of excitation, is modelled by a sum of exponentially damped sinusoids,
where n is the sample time index and h k(n) is measurement noise. The parameters Ai, a i, fi and y i correspond to the amplitude, damping, normalized frequency (f=F/Fs) and temporal phase, respectively, and mi and Fi to the circumferential mode number and the spatial orientation, respectively, for the ith resonance frequency. The mode number determines the number of sinusoids along the circumference, and the spatial orientation determines where the nodes and antinodes occur. It is assumed that I modes are excited due to the impact of the hammer. The signal model is based on ur(r,q ,z,t) in Eq.(1), which is extended by inclusion of exponential damping and a possibly non-zero spatial orientation. All sensors are ideally positioned at the same height z and at the same radial distance r=RO from the centre of the trunk. Thus Ur(r,z) becomes a frequency-dependent scaling common to all sensor outputs. Eq.(1) describes displacement, whereas the measured signals describe acceleration; the difference is an additional frequency-dependent scaling and a phase shift by p.
The resonance frequencies, damping factors, temporal phases, and weights by means of Wi,k=Aicos(miqk+fi) are estimated using Prony's method (Kumaresan and Tufts 1982), a high-resolution frequency analysis method based on linear prediction. It is capable of resolving closely spaced frequency components, but the estimates of damping factors and amplitudes are highly influenced by the presence of noise. With the sensors equidistantly distributed along the circumference, i.e., q k=(2k-1)p/12, the parameters Ai, mi and f i are estimated from the set of weights Wi,k, k=1...12, using the discrete Fourier transform. The analysis is performed on a signal segment whose onset is selected such that the texture of the signal matrix is well defined; for the response in Fig.1(b) this is after approximately 4´ 10-3 s. That the mode shapes follow the presumed model is shown in Fig.4, where the weights Wi,k, k=1...12, are shown for the four modes i=1...4 identified from the signal in Fig.1(b).
|Fig 4: Mode shapes (lines) identified from the Prony weights (·); (a) m=2 found at 780 Hz, (b) m=3 at 1136 Hz, (c) m=4 at 1408 Hz, and (d) m=5 at 1627 Hz. Sound tree, circumference 0.90 m.|
As can be noted in Fig.1(b), there seems to be a surface wave propagating from the point of excitation in both directions along the circumference. The problem is considered as two mirrored plane waves impinging on a uniform linear array of sensors, and the propagation velocity is estimated using a technique based on Delay-and-Sum beamforming (Johnson and Dudgeon 1993), with the summation replaced by singular value decomposition. This allows a robust criterion to be defined for when the sensor outputs are aligned to each other. To reduce interference, the wave heading in one of the directions is suppressed by spatiotemporal filtering. It has been shown that the algorithm is capable of producing estimates with only small errors even when the sensor locations are randomised in the vicinity of the presumed positions (Axmon 2000).
The results from the modal analysis of the trees in the database are shown in Fig.5(a), where the resonance frequency of the mode m=2 is plotted versus the circumference. The mode m =2 was found for 93 of 94 trees, m =3 for 70, m =4 for 38 and m =5 for 16. For the tree in Sound I with the least dimensions, no modes were detected. It is obvious that the resonance frequencies of the sound trees, despite the varying dimensions, are approximately proportional to the reciprocal of the circumference. Proportionality would by expected for HILE cylinders of substantial ratio H/RO and with similar ratios of RI/RO. The spread is, however, large. One of the assumptions made is that the loss of elasticity due to decay may be modelled as introducing a hole in the trunk of a sound tree; many of the decayed trees possess lower frequencies than sound trees of matching dimensions, but there is a considerable overlap between the groups.
|Fig 5: Results from analysis. (a) Resonance frequencies for the circumferential mode m=2, and (b) propagation velocities of the surface wave. Based on 93 trees (65 sound, 28 decayed)..|
The results concerning the propagation velocity are shown in Fig.5(b), from which it can be noted that many of the decayed trees exhibit velocities below the average of sound trees. The velocities of sound trees are almost uniformly distributed over the interval 315-395 m/s. There is, however, a tendency of decreasing velocity for increasing circumference.
D. Regression Analysis and Detector
Denoting the resonance frequency, the propagation velocity and the circumference of the pth tree in the database by Fp, vp and Cp, respectively, and calculating weights corresponding to non-dimensional frequency,
it is expected for sound trees that if they behave like HILE cylinders, will not depend on the circumference or the propagation velocity, although it may have a spread since estimates are used in Eq.(4). However, plotting the weights for Sound I and II versus the circumference and the velocity, Fig.6(a), it is obvious that the weights are dependent upon these parameters. The linear regression for Sound I shows that the weights seem to increase with increasing circumference, and to decrease with increasing velocity.
The frequency of the pth of 42 trees in Sound I is modelled by a random variable,
where G(Cp,vp,a0,a1,a2) yields the expected frequency of a sound tree, given the explanatory variables Cp and vp and the parameters a0, a1 and a2 subjected to estimation, and where ep is a residual assumed to be normally distributed and with zero mean. The regression model is
The parameters a0, a1 and a2 are estimated for the trees in Sound I using non-linear regression based on Gauss-Newton's method (Rawlings 1988). The estimates are â0=2.32, â1= -1.00´ 10-1 m-1 and â2=3.23´10-4 sm-1.
When the weights in Eq.(4) are adjusted,
almost all dependency on the circumference and the velocity is accounted for, Fig.6(b).
|Fig 6: Non-dimensional frequencies (a) prior to and (b) after adjustment for dependency on the explanatory variables circumference and velocity. Linear regression (line) for Sound .|
The detector is based on a threshold, D , for the residual between the estimated frequency of the tree under test and predicted frequency of a sound tree with matching circumference and propagation velocity. Since it has been observed that all sound trees studied have propagation velocities above 310 m/s, an additional threshold vlow is applied to the velocity. Hence, the pth tree in the database is considered sound only when the two conditions
are met. The residuals are shown in Fig.7(a), where the vertical line show a threshold for the velocity of vlow=310 m/s. The residuals for the sound trees are close to normally distributed.
The performance of the detector is evaluated by means of the epidemiological measures sensitivity and specificity. Here the sensitivity is the ratio of decayed trees detected as decayed to all decayed trees, and the specificity the ratio of sound trees detected as sound to all sound trees. As large figures as possible are desired; for non-overlapping groups, it is possible to achieve 100% jointly. However, when there is an overlap, there is a trade-off to be made.
A sensitivity and specificity graph for the proposed detector is shown in Fig.7(b) for a residual threshold D in the range of 1 to 120 Hz and a velocity threshold vlow=310 m/s. All 93 trees are included in the evaluation. A simultaneous sensitivity and specificity of approximately 74% is achieved for D =53 Hz. To increase the likelihood of that a tree detected as decayed actually is decayed, D >53 Hz should be used. Conversely, to increase the likelihood of that a tree detected as sound actually is sound, D <53 should be selected. However, in the former case, more of the decayed trees will be detected as sound, and in the latter, more of the sound trees will be detected as decayed. Hence, trade-offs are made.
|Fig 7: Detector. (a) Residuals between the estimated frequency and the predicted frequency. The sound trees are nearly normally distributed with a standard deviation of 48 Hz (dash-dotted lines). (b) Performance by means of sensitivity and specificity. Intersection: 74% at D=53 Hz. The results are based on 93 trees.|
Tossing a coin when determining the status of each tree, the sensitivity and the specificity would in the long run both reach 50%. Thus, the intersection at approximately 74% in Fig.7(b) is well above random assessment and shows that the detection is successful, although a higher score would be desirable.
The resonance frequency of the mode m=2 seems to be dependent on the circumferential dimension in a manner not predicted by the HILE cylinder model. Besides that wood is non-homogeneous and orthotropic, this could depend on the relative distribution of heartwood and sapwood; Mattheck and Bethge (1993) found size-dependent variations in stress wave propagation velocities and suggested that it depended on that larger trees have a higher percentage of heartwood in the cross section. The signal model is still applicable since the circumferential mode shapes detected are close to sinusoidal.
If the trunks were isotropic, a Rayleigh surface wave could be expected from the superficial excitation (Graff 1975). However, the trunks are not, and hence the type of the wave travelling along the circumference is yet not determined.
The coefficients of the prediction function, Eq.(6), are based on the trees in the Sound I group only. The residuals of the trees in both Sound I and Sound II are very close to normally distributed, which verifies the approach. It is possible that some trees in Sound II are infected at the examined cross section, since fungi may enter the tree via injuries. This is not noticed when determining the status from the fell cut. Similarly, trees in Decay I may be perfectly sound near the examined cross section.
The measurements were performed under similar conditions in three forest stands in a relatively small geographical region. It is possible that the coefficients are dependent on location and environmental conditions, and thus have to be tuned for different regions and climates.
Comparing the performance to the visual assessment reported by Vollbrecht and Agestam (1995), where five foresters assessed a large number of trees, it was found that requiring an agreement in the assessment, only 11% of the sound trees and 24% of the decayed trees were correctly identified by the experts. When instead using the majority's decision, i.e., when at least three of five foresters made the same assessment of a tree, 53% of the sound and 67% of the decayed trees are identified. These figures correspond to specificity and sensitivity, respectively. Hence, it is concluded that the proposed method is more reliable than visual tree assessment.
This work was financed by the Swedish Research Council. Additional support has kindly been provided by the Royal Physiographic Society in Lund. The authors would like to thank the colleagues at the Dept. of Electroscience and the personnel at SkogForsk, Sävar, for assistance with the measurements, and B. Bengtsson and Trolleholms Gods AB, respectively, for providing forest stands for measurements.
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