9 On Localised Modes in Bio-inspired Hierarchically Organised …
161
that acts on the mass m N−1 is equal to zero and this mass does not oscillate. The
frequency of vibration corresponds to the natural frequency of the masses from the
highest order of hierarchy and is equal to
√
k N /m N . The number of the localised
modes n 1 that appear at this frequency corresponds to the number of the pairs of
masses of the highest hierarchy: n 1 = 1 group* 2
N masses/2 in each pair = 2
N−1 .
Group 2: This group includes the case when the masses from the penultimate and
ultimate order of hierarchy oscillate (Fig. 9.8b). One mass m N−1 and two masses m N
attached to it move in one direction, while the other mass m N−1 and two masses m N
attached to it move in the opposite direction. Thus, the spring forces that act on the
mass m N−2 cancel out and this mass stays at rest. This case is characterised by two
modal shapes and two natural frequencies. The overall number of this type of modal
shapes n 2 is equal to: n 2 = 2 groups * 2
N−2 modal shape = 2
N−1 .
Group 3: There are oscillations of the masses from the three highest orders of
hierarchy (note that one existing group is presented in Fig. 9.8c). There are two sets
of such masses, numbered by I and II in Fig. 9.8c. They move in the identical way
but in the opposite directions, yielding a zero-valued resulting force acting on the
mass m N−3 , so that this mass stays at rest. This case is characterised by three modal
shapes and three modal frequencies. The overall number of this type of modal shapes
n 3 is defined by: n 3 = 3 groups * 2
N−3 modal shape.
Group 4: This class contains the modes when all masses besides the main one m
(Fig. 9.8d) oscillate. There are two sets of such masses. One set contains N masses
and is characterised by N modal shapes. The overall number of these modal shapes
n N is: n N = N groups * 2
N−N modal shape = N.
The following Theorem can now be formulated.
Theorem [8] A chain consisting of the main spring-mass system to which N hierarchically organised parallel systems of spring-masses is attached (Fig. 9.7) is characterised by localised modes whose number is defined by the difference between the
number of degrees of freedom and the number of masses in the hierarchy including
the main mass, i.e.
N
i=1
i · 2
N−i
=
N+1
i=1
2
N+1−i
− (N + 1)
(9.12)
This Theorem is given here without the proof, which can be found in [8]. Instead
of giving the proof, the claim of the theorem is compared with the results presented
in Sects. 9.2 and 9.3.
So, when N = 1, only one localised mode exists: the masses m 1 oscillate, but the
main mass does not. This coincides with the case obtained in Sect. 9.2, illustrated in
Fig. 9.3.
When N = 2, Theorem implies that there are four localised modes. Two modes
are localised in the highest-order of hierarchy and two modes are localised in the
penultimate and ultimate order of hierarchy and corresponds to different frequencies.
This agrees with the findings from Sect. 9.3, presented in Fig. 9.6.
161
that acts on the mass m N−1 is equal to zero and this mass does not oscillate. The
frequency of vibration corresponds to the natural frequency of the masses from the
highest order of hierarchy and is equal to
√
k N /m N . The number of the localised
modes n 1 that appear at this frequency corresponds to the number of the pairs of
masses of the highest hierarchy: n 1 = 1 group* 2
N masses/2 in each pair = 2
N−1 .
Group 2: This group includes the case when the masses from the penultimate and
ultimate order of hierarchy oscillate (Fig. 9.8b). One mass m N−1 and two masses m N
attached to it move in one direction, while the other mass m N−1 and two masses m N
attached to it move in the opposite direction. Thus, the spring forces that act on the
mass m N−2 cancel out and this mass stays at rest. This case is characterised by two
modal shapes and two natural frequencies. The overall number of this type of modal
shapes n 2 is equal to: n 2 = 2 groups * 2
N−2 modal shape = 2
N−1 .
Group 3: There are oscillations of the masses from the three highest orders of
hierarchy (note that one existing group is presented in Fig. 9.8c). There are two sets
of such masses, numbered by I and II in Fig. 9.8c. They move in the identical way
but in the opposite directions, yielding a zero-valued resulting force acting on the
mass m N−3 , so that this mass stays at rest. This case is characterised by three modal
shapes and three modal frequencies. The overall number of this type of modal shapes
n 3 is defined by: n 3 = 3 groups * 2
N−3 modal shape.
Group 4: This class contains the modes when all masses besides the main one m
(Fig. 9.8d) oscillate. There are two sets of such masses. One set contains N masses
and is characterised by N modal shapes. The overall number of these modal shapes
n N is: n N = N groups * 2
N−N modal shape = N.
The following Theorem can now be formulated.
Theorem [8] A chain consisting of the main spring-mass system to which N hierarchically organised parallel systems of spring-masses is attached (Fig. 9.7) is characterised by localised modes whose number is defined by the difference between the
number of degrees of freedom and the number of masses in the hierarchy including
the main mass, i.e.
N
i=1
i · 2
N−i
=
N+1
i=1
2
N+1−i
− (N + 1)
(9.12)
This Theorem is given here without the proof, which can be found in [8]. Instead
of giving the proof, the claim of the theorem is compared with the results presented
in Sects. 9.2 and 9.3.
So, when N = 1, only one localised mode exists: the masses m 1 oscillate, but the
main mass does not. This coincides with the case obtained in Sect. 9.2, illustrated in
Fig. 9.3.
When N = 2, Theorem implies that there are four localised modes. Two modes
are localised in the highest-order of hierarchy and two modes are localised in the
penultimate and ultimate order of hierarchy and corresponds to different frequencies.
This agrees with the findings from Sect. 9.3, presented in Fig. 9.6.