Journal of Experimental Botany, Vol. 48, No. 310, pp. 1125-1132, May 1997
Journal of
Experimental
Botany
Measurements of the natural frequencies of trees
C.J. Baker1
Department of Civil Engineering, University of Nottingham, Nottingham NG72RD, UK
Received 20 December 1996; Accepted 21 January 1997
Abstract
Recent theoretical work has suggested that tree natural frequency is an important indication of tree
stability in windy conditions. This paper reports natural
frequency measurements on a large number of urban
trees using a remote laser-based method. The majority
of the measurements were made on healthy lime trees,
and showed a consistent variation of natural frequency
with tree size, with out-of-leaf, winter frequencies
being substantially greater than in-leaf, summer frequencies. These measurements also revealed that
different tree geometries can produce significant
changes in natural frequency. A number of measurements were made on diseased lime trees (crown dieback, rot, etc.) and these displayed natural frequencies
that were very different from those for the healthy
trees.
Key words: Trees, stability, natural frequencies.
Introduction
In a recent paper the author (Baker, 1995) describes a
dynamic analysis of the behaviour of trees and cereal
crops in wind. This was based on the investigation of a
two mass system, one representing the root ball, and the
other representing the tree canopy or cereal ear, with the
masses being connected by a weightless stem. This model
was used to predict failure wind speeds for a variety of
plants. Whilst one should be wary about placing too
much reliance on the results of such a simple model, none
the less the failure wind speeds that were predicted were
broadly in agreement with observed values. Perhaps the
most important point to emerge was the importance of
the natural frequency of the plant/root system in the
dynamic process. It was found that most of the model
parameters, canopy mass, root inertia, stem stiffness, root
plate resistance, etc. were of importance in the process
1
only in so far as they affected the natural frequency, and
that the predicted failure wind speeds and associated
return periods were particularly sensitive to changes in
this natural frequency, the lower the natural frequency,
the lower the failure wind speed. It was concluded that a
knowledge of the natural frequency of trees and cereals
was likely to be of considerable use in the determination
of plant stability in high winds.
Now, for trees, the data for natural frequency are very
sparse. Milne (1991) reports measurements on a number
of Sitka spruce, and Gardiner (1989, 1995) and Peltola
(1995) also reported measurements on similar trees.
Roodbaraky et al. (1994) give details of measurements
made on a small number of broad-leaved trees of different
species that indicated considerable variations in natural
frequency between the in-leaf and out-of-leaf cases.
Clearly, it would be desirable to obtain more data for a
variety of tree species. The experiments described in this
paper set out to do this. A rapid laser-based measurement
was used that did not require on-tree instrumentation
and allowed a large number of trees to be surveyed in a
relatively short time. Measurements were carried out
mainly on lime trees spanning the size range, in both
healthy and diseased states, but the natural frequencies
of a number of specimens from other species were also
measured.
In this paper, details of the method and sample results
will be given. Only an outline analysis of the data will be
presented. These data are being used to validate a theoretical dynamic modelling of urban trees in a doctoral study
(Saunderson, unpublished results) and a more detailed
analysis of the data in the light of this work will be
published in due course.
The measurements
The natural frequencies of the sample trees were obtained
by measuring the power spectrum of tree velocity using
To whom correspondence should be addressed: Fax: + 44 115 951 3898. E-mail: Chnstopher.Baker©Nottingham.ac.uk
© Oxford University Press 1997
1126
Baker
a tripod mounted Laser Doppler Interferometer. This
method has been used successfully in the past for measuring the natural frequencies of building structures and
components and is known to provide natural frequency
data that are consistent with other methods of measurement (e.g. measuring the time period of free oscillations).
A beam from a 15 mW helium neon laser was shone at a
suitable point on the specimen being investigated. The
reflected light was collected and compared with a reference
beam in an interferometer, to produce interference fringes.
Small movements of the specimen in the usually low-level
ambient winds, produced movements in the interference
fringes across a pair of photodetectors, which a signal
conditioning unit then translated into a voltage signal
that was proportional to tree velocity in the direction of
the incident beam. From the resulting velocity time series,
velocity spectra were calculated in the range 0-10 Hz at
0.04 Hz intervals using conventional Fast Fourier
Transform techniques. This method is very sensitive to
small tree movements and a reliable spectrum could be
obtained using a 4 min sample. The natural frequencies
of the tree/root system can then be determined from the
frequencies of the peaks in the spectrum. It was found
during the course of the measurements that a low frequency peak was always present, due to the mean velocity
component. This has also been observed in previous
building measurements. Thus only spectral peaks above
a frequency of 0.20 Hz should be considered to be of
relevance in what follows.
The measurements were carried out on trees on or
around the Nottingham University campus on a total of
62 trees. Of these 27 were healthy limes (Tilia x europea)
covering the complete size range, 13 were limes showing
obvious signs of disease—crown dieback etc., 10 were
healthy trees of a variety of other species and 12 were
diseased trees of a variety of species. Measurements
were made in July 1995 when all the trees were fully
in-leaf and the ground was very dry, and again in late
November 1995 when most of the trees (all the limes, but
not all the other species) had lost their leaves and the
ground conditions were moist, but far from saturated.
The trees that will be considered in this paper are tabu-
lated in Tables 1 and 2—Table 1 for the healthy limes
and Table 2 for the diseased limes. Note that the assessment of tree health was made by an experienced aboriculturalist. The trees of species other than lime will not be
considered in what follows, but the results can be obtained
from the author if required. Each table shows the tree
number, breast height diameter, tree height, summer
natural frequency n,, and winter natural frequency nw.
These latter parameters will be discussed in what follows.
In addition, Table 1 gives an indication of spectrum type
(see Results) and Table 2 gives brief details of the health
of the tree.
Results
Assessment of the method
Roodbaraky et al. (1994) made measurements of the
natural frequency of a small London plane tree (Platanus
acerifolia) in two ways—by measuring the period of
natural oscillations of the tree when deflected and
released, and from the power spectrum of tree displacement in winds. This tree was also measured in the present
experiments. The results of Roodbaraky et al. (1994)
(actually measured in 1992/93) gave the following values.
Tree displacement and release
«, = 0.42Hz
^ = 0.75-0.87 Hz
Tree displacement spectra
n, = 0.25-0.43 Hz
rc>=0.80Hz
The spectra of velocity measured in the experiments
reported here are shown in Fig. 1. Note that, as with all
the spectra presented in this paper the values of the
ordinates are totally arbitrary. The peaks in these figures
correspond to values of n, and nw of 0.25 and 0.62 Hz,
respectively. These two sets of values are in reasonably
close agreement (particularly when the period of three
years between measurements are considered) and give a
degree of confidence in the experimental technique.
A further question that arises is how important is the
precise position of the target on the tree. One tree, a
sycamore (Acer pseudoplatanus) (tree 50), was measured
lTre»f55:Ptan»:WMar|
08
i
IV
!
s
\ 1
/!\W Ml
Fr»qu«nqr(Hi)
(a)
Fig. 1. Measured spectra for plane tree (number 55); (a) summer (b) winter.
1
•
(b)
;
•
'
!
Natural frequencies of trees 1127
repeatedly with the laser beam targeted at different points
on the tree. The measured spectra are shown in Fig. 2.
Note that the slight low frequency peak is below 0.2 Hz
and thus likely to be spurious. It can be seen that there
is little variation in the position of the main peak of the
spectra at 0.4 Hz. Even for the trunk measurement (test
57) an expansion of the >>-axis shows a well-defined peak
at this frequency. One can thus conclude that the precise
position of the target is unimportant. All the other
experiments reported in this section were carried out with
the laser beam targeted on the trunk of the tree below
the canopy.
Measurements on the healthy limes
/
/\
2
11
1
CLi
ir
/
• Tatf57
• T«*»
L
if
• TatU
-»Ta*t1
\\
14
OJ
Fraqunytfd
Fig. 2. Effect of target position on measured
(number 50), summer conditions.
spectra—sycamore
Figure 3 shows the variation of tree height with the
diameter at breast height (dbh) for the healthy lime trees.
There can be seen to be a high level of correlation between
these parameters. In what follows the variation of tree
natural frequency will be presented as a function of breast
height diameter only, this being assumed to be an
adequate descriptor of tree size. It can be argued that it
would be more sensible to plot the data against
dbh/(trunk height) 2 as this is a better measure of the
natural frequency of a tapered cantilever (Gardiner,
1989). Now, whilst this is true of forest trees where the
trunk height is well defined, the same cannot be said for
the trees measured here, where the precise equivalent
trunk heights were difficult to determine. However, an
analysis of this type is reported, using the present data,
in the work of Saunderson (unpublished results).
20
18 -
16
o
o o
o
o o°
•
14
o
12
10
8
6
10
20
30
Fig. 3. Variation of tree height with dbh for healthy lime trees.
40
50
dbh (cm)
60
70
80
90
1128
Baker
An examination of the measured spectra revealed that
there were three distinct types. These are shown in Fig. 4.
Type I spectra (the most common) had a low frequency
peak at between 0.3-0.6 Hz in the summer, and
0.5-1.5 Hz in the winter (Fig. 4a), the precise figures
being dependent upon tree size. In general, only one peak
(other than the spurious low frequency peak) could be
seen, but on some spectra low level, higher frequency,
peaks could be detected. Type II spectra showed no
significant peaks for one or both of the summer or the
winter conditions (Fig. 4b). Type III spectra are similar
to type I spectra, but have a natural frequency range
significantly below the type I range for similar diameter
trees (Fig. 4c). These three different types corresponded
to three different geometries of tree (Fig. 5). Type I
(Fig. 5a) corresponds to the normal geometry of the limes
that were measured, with main branches coming from the
trunk at angles of around 20-30 ° to the vertical. Type II
(Fig. 5b) corresponded to a 'bushy' type of tree with a
well-developed crown with the main branches much closer
to the horizontal. These are generally the smaller, younger
trees. Type III (Fig. 5c) corresponded to trees within
avenues, with steeply inclined main branches at angles of
less than 10° to the vertical.
The variation of n, and nw with dbh from Table 1 for
type I spectra are shown on Figs 6 and 7. It can be seen
(a)
(b)
(c)
Fig. 4. Different types of spectra for healthy lime trees, (a) Type I (number 23), (b) type II (number 30), (c) type III (number 8); summer
conditions on left, winter conditions on right.
Natural frequencies of trees
\
Fig. 5. TypKal trees with different spectrum types, (a) Type I (number 23); (b) type II (number 30); (c) type III (number 8).
1129
1130
Baker
Table 1. Data for healthy lime trees
Tree
number
dbh (cm)
Height
(m)
Hei]
n,(Hz)
13
12
11
22
30
10
8
9
32
35
33
26
31
27
21
29
34
3
18
28
25
20
24
23
2
19
4
7.0
9.9
13.4
29.2
30.2
34.7
36.9
38.5
56.0
56.3
56.3
58.3
59.5
60.5
61.1
61.1
61.1
63.7
640
65.4
66.2
64.4
71.0
72.3
73.2
75.1
82.8
4.7
5.9
6.2
9.8
8.3
13.8
14.1
14.1
14.1
14.1
14.5
15 1
14.5
14.1
15.6
13.7
14.6
13.1
15.7
16.5
15.6
18.3
19.5
18.7
18.2
14.6
16.5
—
0.50
0.53
0.53
—
0.31
0 25
0.20
0.53
—
0.42
0.50
—
—
0.50
—
0.42
0.39
0.59
0.48
0 48
0.40
0.31
0.31
0.35
0.42
0.39
Table 2. Data for diseased lime tress (asterisk indicates values
of frequency are statistically from a different population to those
of Table 1, spectral type I)
Spectrum
type
1.42
1.10
1.02
0.93
0.47
0.42
0.42
0.75
0.82
0.74
0.70
0.85
0.79
0.79
0.82
0.70
0.62
0.70
0.66
0.65
0.55
0.42
0.51
0.43
0.65
0.55
I/I
I/I
II
II
III
I/I
I
I
I/I
[I/I
1
I/I
Tree
number
dbh (cm)
Height (m)
45
31.2
9.5
46
40.1
98
38
42.0
11.0
42
45.2
13 1
37
45.8
11.6
47
46.2
11.0
39
47.7
12.3
48
47.7
10.7
40
47 7
12.3
41
48.4
12.3
43
509
10 1
49
52.5
12.6
44
55.1
12.0
n,(Hz)
n w (Hz)
0.59*
0.98
Condition
Crown
dieback
0.65*
Crown
0 68*
dieback
1.35*
1 40*
Extensive
dieback
0.91*
0.82
Crown
dieback
1.25*
Dieback,
0.91*
pruned
0.68*
0.90
Crown
dieback
0.91*
1.21*
Extensive
dieback
0.90
Dieback and
0.70*
basal rot
0.70*
0.89
Dieback and
basal fungi
0.91*
1.05*
Extensive
dieback
Type III Type III Crown
dieback
1.05*
Dieback and
0.85*
basal rot
Crown
Type III 1.05*
dieback
1 U
1 4
-
-
1.2
-
-
1
-
-
0.8
-
-
06
o
.
o
04
0
o
-
"o"
£-
a
o
oo
"""
-•
02
n
20
40
60
80
100
dbh (cm)
Fig. 6. Natural frequency versus dbh for healthy limes with type I spectra, summer conditions (best fit line is (height) = 0 569-0.0021 (dbh) with
99% confidence limits about best fit line of 0.043 (1+2.6 x 10" 3 (dblv 58.4 f)12).
Natural frequencies of trees
80
1131
100
dbh (cm)
Fig. 7. Natural frequency versus dbh for healthy limes with type I spectra, winter conditions (best fit line is (height) = 1.332-0.0102 (dbh) with 99%
confidence limits about best fit line of 0.069 ( l + 2 . 3 6 x l O * 3 (dbh-55.1) 2 )" 2 ).
that there is a consistent variation with tree size, with a
decrease in natural frequency with dbh. Values of n^ are
consistently above those of n, as would be expected as
the trees are lighter without leaves, although this difference becomes smaller at higher values of dbh.
There were three trees with the type III spectra—
numbers 8, 9 and 10. The frequency values of these trees
all fall outside the 99% confidence limits for the type I
trees, with values considerably below the type I values.
Measurements on diseased limes
The data for the diseased limes are shown in Table 2. It
was found that, to within the 99% confidence level, the
results for all trees identified by an asterisk in that table
(i.e. all except nw for trees 40, 42, 45, 47, and 48) are
from a different population to the healthy limes, with
higher natural frequencies in both the summer and winter
conditions. The values for trees 40, 47 and 48 lie only
just within the 99% limits.
Discussion
Firstly, consider how these results relate to the results of
previous investigators. The good agreement with the
results of Roodbaraky el al. (1994) have already been
mentioned. The experiments of Gardiner (1989) defined
a standard Sitka Spruce of dbh 0.166m and natural
frequency of 0.33 Hz. Now whilst it is recognized that it
is not particularly meaningful to compare natural frequencies across species a comparison of this work with the
results of Fig. 6 (for the summer case with the trees
in-leaf), indicates that the values are broadly comparable
and give some confidence in the present method.
The data for the healthy limes with spectra type I show
that the natural frequency falls with tree size, with the
summer values (tree in-leaf) being significantly lower
than the winter values. At any one tree size there is a
fairly wide spread of values which may well reflect the
different nature of the rooting systems. The work of
Baker (1995) suggests that such a variation could result
in a significant variation in failure wind speed, for nominally identical trees with those specimens with low natural
frequencies being at considerably more at risk to
windthrow than those with the higher natural frequencies.
The turbulent energy within the oncoming wind decreases
rapidly over the range from 0.1-lHz. Specimens with low
natural frequency will thus react more strongly to fluctuations within the winds, with increased displacements,
base bending moments, etc. This may in part explain why
apparently identical adjacent trees may be affected very
differently in windstorms. It would have been of considerable interest to compare the root systems of the different
trees that were measured. This was not of course possible
within the constraints of the experiments.
Type II spectra usually seem to occur when the trees
are in-leaf, with type I spectra when the trees are out-ofleaf (Table 1), although even in the latter case the peaks
might not be well defined (Fig. 4b). This type of spectrum
is of considerable interest and indicates that such trees
have no preferred resonant frequency at which the tree
1132
Baker
will react strongly to turbulence fluctuations in the wind.
This may in part be due to large aerodynamic damping
of oscillations by the trees when in-leaf, that obscures
any resonant peak. Since aerodynamic damping increase
with wind speed (Wood, 1995), this might suggest that
at higher wind speeds the effect of the natural frequency
value on the transfer of energy from the wind to the tree
might not be important.
The data for healthy trees with type III spectra indicate
much lower values of natural frequency than for those
trees with the type I spectra. These results suggest that
trees with such geometries, typical of trees grown closely
together around the edges of estates, would have a greater
risk of windthrow than type I trees. The lower natural
frequencies may result either from the above-ground
geometry or, perhaps more likely, from differences in
rooting behaviour.
A further interesting point to emerge from these measurements is the general lack of higher frequency harmonics. These harmonics might play a role in the process of
windthrow for some trees causing high stresses at points
other than the tree base (Wood, 1995; Guitard and
Castera, 1995), but these results suggest they are of little
significance.
The data for the diseased limes indicate natural frequencies much higher than for the healthy trees, probably
reflecting variations in trunk stiffness, wood density,
crown weight, etc. These variations are highly significant
in a statistical sense. Although the higher natural frequencies would suggest a reduced risk of windthrow, it is
likely that they will be accompanied by a decrease in
fracture strength of the timber, which may result in an
increased risk of wind damage, although perhaps not of
uprooting. Now Baker (1995) hypothesizes that natural
frequency is a good indicator of tree stability in high
winds. Whilst the present results do not confirm this,
none the less they do leave open this possibility.
The following conclusions can be drawn.
(a) The laser interferometer method has been shown to
be a useful method for measuring the natural frequency of trees.
(b) For healthy lime trees the natural frequency decreases
as dbh increases with the summer, in-leaf, values
being significantly below the winter, out-of-leaf
values.
(c) At any one dbh there can be a significant variation
in tree natural frequency. If the hypothesis of Baker
(1995) is correct this suggests a significant variation
in susceptibility to wind throw for nominally identical
trees which will probably result from variations in
root strength.
(d) Natural frequency seems to be a function of tree
spacing, with closely planted trees with near vertical
main branches and constrained rooting systems
having considerably lower values than those trees
which are more widely spaced.
(e) Limes showing obvious signs of disease have values
of natural frequency considerably above those for
healthy limes
Acknowledgements
This work was made possible through a Royal Society research
grant. The laser measurements were carried out by Mr Peter
Winney of P and P Engineering Ltd, and the experiments were
supervised by Dr Hilary Roodbaraky. Their help is gratefully
acknowledged.
References
Baker CJ. 1995. The development _of a theoretical model for
the windthrow of plants. Journal of Theoretical Biology
175, 355-72.
Gardiner BA. 1989. Mechanical characteristics of Sitka Spruce.
Forestry Commission Occasional Paper No. 24.
Gardiner BA. 1995. The interaction of wind and tree movement
in forest canopies. In: Coutts M, Grace J, eds. Wind and
trees. Cambridge University PTess, 41-59.
Guitard DGE, Castera P. 1995. Experimental analysis and
mechanical modelling of wind induced tree sways. In: Coutts
M, Grace J, eds. Wind and trees. Cambridge University
Press, 182-94.
Milne R. 1991. Dynamics of swaying Picea sitchenis. Tree
Physiology 9, 383-99.
Peltola H. 1995. Studies on the mechanism of wind induced
damage of Scots pine. PhD thesis, University of Joenssu,
Finland.
Roodbaraky HJ, Baker CJ, Dawson AR, Wright CJ. 1994.
Experimental observations of urban trees in high winds.
Journal of Wind Engineering and Industrial Aerodynamics
52, 171-84.
Wood CJ. 1995. Understanding wind forces on trees. In: Coutts
M, Grace J, eds. Wind and trees. Cambridge University
Press, 133-64.