arXiv:1006.1722, Classical Physics, Cornell University Librar

S. Giurdanella, A.C. Sparavigna

English

PORTO Publications Open Repository TOrino

Self frequency-locking of a chain of oscillators 
 
S. Giurdanella and A.C. Sparavigna 
Dipartimento di Fisica, Politecnico di Torino 
C.so Duca degli Abruzzi 24, Torino, Italy 
 
 
Abstract 
The paper studies the vibrational modes of a slightly damped uniform chain, with n masses coupled by 
elastic forces. It will be shown that, for certain lengths of the chain, that is for certain values of n,  the 
damping of one of the masses at a specific position in the chain is able to constrain the vibration of the 
system to oscillate at a specific frequency. The damped mass turns out to be a node of the chain, 
subdividing it in two parts. This node can be considered as the synchronization element of the two 
subchains. As a consequence the oscillating system of n-masses is self-locking to the synchronized 
frequency of its subchains.   
 
PACS: 43.20.Tb, 42.60.Fc 
 
The dynamics of chains with harmonic oscillators is intuitive and quite easy to solve. For this reason, 
these chains remain the preferred models to use in solid state physics to describe the vibrational  
properties of crystalline lattices and their quantization in phonons. Modelling with harmonic chains is 
also a rather old research approach. As reported by Brillouin,  harmonic chains were the subject of  
Isaac Newton's researches on the speed of sound [1,2]. Anharmonic models instead are not so simple or 
so old: it was about 1954,  that Enrico Fermi proposed to study numerically a chain of particles linked 
by springs with weak nonlinearity. The so-called Fermi-Pasta-Ulam-Tsingou problem marked the 
beginning of studies on nonlinear physics, solitons and chaos [3-5]. 
Besides modelling the phononic systems, we can assume the harmonic chains as suitable models for  
metamaterials, that is, for those artificial compounds engineered to gain their properties from the 
structure rather than from the composition. Most of metamaterials are obtained from a periodic 
arrangement of few components.  If we consider the one-dimensional case for instance,  we 
immediately recognize that a chain of masses and springs is the simplest  metamaterial we can imagine. 
In fact the dispersion of frequencies is coming from the structure of the chain, not only from the 
stiffness of the springs. Moreover, the dynamics governing a mechanical chain with masses, springs 
and dampers, can  be immediately transferred to modelling the transmission lines (see Fig.1), built by 
means of electric circuits, with lumped capacitive and inductive components [6]. 
For all these reasons, harmonic chains are still deserving new investigations. In this paper, we propose 
a discussion on the vibrational modes of chains, starting from the results shown in Ref. [2]. The data 
reported in that paper are confined to a low number of masses. The model of the chain was a uniform 
one, made up of n equal m masses and n equal linear springs with stiffness k (see Fig.1, upper part). A 
damping contribution was attached, as in Fig.1, to a single mass of the chain. Solving the dynamical 
matrix of the system, the complex eigenvalues corresponding to the damped oscillations can be 
determined. The real part of the eigenvalue gives the life-time of oscillation, the imaginary part its 
proper  angular frequency. 
In Ref.2, the number of masses was small (n=5).  We extended the calculation to an arbitrary number 
of masses, observing that some undamped oscillating modes survive in chains with certain lengths, 
when the damping contribution is attached to a mass at a specific position.  The consequence is that the 
frequency of the chain is self-locking to  specific values. These values are determined by the length of 
the chain and the position of the damper. That is, we will see that, for chains with a given length, the 
damping of one of the masses at some specific positions in the chain constrains the vibration of the 
system to self-lock at specific frequencies. Moreover, the damped mass turns out to be a node of the 
chain. For this reason, the chain can be viewed as subdivided in two parts, that is, as if it were 
composed by two connected oscillating chains, which are synchronized to the same frequency. The 
oscillating system is self-locking to the synchronized frequency of its subchains. The damped mass is 
the synchornization element. Instead of synchronizing groups of coupled oscillators with sparse 
connections, as recently proposed [7], the synchronization can be achieved by means of a dissipative 
element in the chain. Since transmission lines are described by the same equations of mechanical 
chains, the presently discussed model could be proposed for their synchronization. 
Models with dampers can be interesting for other engineering applications. In civil engineering for 
instance, a damper is an effective control device commonly attached to a vibrating primary system for 
suppressing undesirable vibrations induced by winds and earthquake loads [8-10].  The study of nodes 
and of the self-locking at specific frequencies in vibrating models is fundamental for the quality of 
many civil structures.    
The model from Ref.2 is the mechanical one in Fig.1. Masses are on a horizontal frictionless surface.  
The chain starts with a mass connected to a wall and ends with a mass, connected only with the 
previous one.  A damping friction can be added to each mass. The authors studied the case with a 
single damped mass and with all the masses damped, giving all the eigenvalues for chains till n=5.  The 
aim of that paper was the proposal of an approximate analytic estimation for eigenvalues. 
Tab.2 in Ref.2 is showing the values in the case with n=4 and p ranging from 1 to 4. In the case that the 
damping effect is placed on the third mass, one of the complex conjugated pair of eigenvalues is purely 
imaginary. This means that the corresponding mode survives on long times, when all the others modes 
have been damped by their real part different from zero. The authors did not discuss the presence of 
this undamped mode and the reason for its existence. We decided then to investigate the presence of 
pure oscillating modes for longer chains, in the case of a single damped mass (the case with all the 
masses damped have no undamped modes). 
We use the same equations as proposed in Ref.2 (here reported for the reader convenience in the 
Appendix): first of all we searched existing undamped modes when the number n of masses increases. 
The n conjugated pairs λ=d±iw eigenvalues were obtained in two ways, by means of a MATLAB® 
program and with the QR algorithm for real Hessenberg matrices as in Ref.11, for Fortran 
programming. For each n, we supposed a single damped mass with its position p spanning from 1 to n. 
We found that for n=4, 7, 10, 12, 13, 16, 17, 19, 22, 24, 25, 27, 28 .... these modes exist, that is, there 
are purely imaginary eigenvalues (d=0). Fig.2  reports on the left, the behaviour of the real (x-axis) and 
of the imaginary (y-axis) of the eigenvalues, for the corresponding choices of n and p as reported in the 
figure, and for the specified physical parameters.  Note that the real part of the eigenvalues decreases in 
magnitude when the number of masses increases, this is due to the fact that we have damped only one 
mass in the chain and its effect is reduced increasing its length. 
The right part of the figure shows the behaviour for a specific chain with n=19 and p=18, which has 
eigenvalues with a null real part, and that of the two following chains n=20 and n=21, which do not 
possess undamped modes. There are eigenvalues λ=d±iw having a quite small real part, d, definitely 
different from zero. We see modes with a null real part also for other positions of the damped mass, 
different from p=n−1, as reported in Table I  for chains with few masses.  
Let us start discussing more deeply the case of a chain with 4 masses (m1,m2,m3,m4), the third (m3) 
damped. Fig.3.a reports the behaviour of the third (m3) and fourth masses (m4) as a function of time, for 
m=1 kg, k=1 N/m, c=0.01 s-1. The position  x3  is damped to zero. Image 3.b reports the behaviour of  
the positions (x2,x4) of the oscillating masses  (m2,m4), with the time as parameter. The initial point of 
the motion is at the right upper corner marked by a red dot, whereas the limit of the motion for 
asymptotic times is described by the grey diagonal line. Fig.3.c and 3.d report the phase space diagrams 
for masses m4 and m3 respectively, where 3.d has a clear limit-cycle, which is observed also for m2. In 
the case of 3.d, the limit is the central point of the image, coherently with the behaviour in 3.a. 
The evaluation of the corresponding dynamical matrix is easily to make: the pure imaginary  
eigenvalue has magnitude w=√(k/m). The corresponding eigenvector is parallel to (1,1,0,-1).  This 
means that we can subdivide the chain and the corresponding dynamical problems in two parts, one 
concerning an oscillating system composed by the first two masses (m1,m2) and the other by the last 
mass m4, as exemplified  in Fig.4.  The eigenvalues of the first oscillator have magnitudes  w=1.000 
and  1.732, the second has w=1.000. It is then possible for the two parts (A and B) of the chain to 
oscillate with the same frequency, w=1.000, and with a phase suitable to let mass 3 at its equilibrium 
position. That is, m3 becomes a node for the longitudinal oscillations of the chain.  
As we can see from Table I, the same behaviour is observed for longer  chains. In this case there is the 
possibility to have part B composed by more than one single mass: this fact increases the number of 
possible frequencies. Table II displays all the modes of the two sub-chains A and B. The common 
values are marked with the bold character. Fig.5 is visualising the same result as in Table II, with the 
comparison of dispersion of the chain without dampers and dispersions of the two sub-chains. The 
figure and the table show the synchronized frequencies. From these results, it is clear that the 
synchronization of the two parts of the chain is possible only for specific values of n and p. The 
following Fig.6 is reporting the undamped modes appearing in the case of very long chains. From such 
charts, one has the possibility to decide where a damper must be placed to have the oscillation of the 
system  to self-lock at one of the synchronization frequency.   
Let us conclude remarking that we are dealing with a free-end chain: this mechanical system has its 
analogous in a transmission line with the open-end termination. It is possible then that the behaviour 
displayed by our model could have interesting for applications in devices including transmission lines, 
such as for many mechanical structures. 
 
 
Appendix 
The dynamical equations for a linear, discrete mechanical system with n degrees of freedom is given by 
 0=++ )t()t()t( qKqCqM &&&     (A1) 
where M, C and K are n×n mass-, damping- and stiffness matrices, respectively, and q(t) is the n-
dimensional coordinate vector. Let us assume 
 te~ λqq =     (A2) 
leads to the eigenvalue problem  
 0=++ ][ 2 qKCM ~λλ     (A3) 
Where λ and q~  denote an eigenvalue and the corresponding eigenvector, respectively. The case that we 
discussed in this paper has a viscous damper at the p-th mass. The motion of the oscillator is governed 
by Eq.A1 with matrices in the form: 
 
















−
−
−−
−
===
110
12
121
012
),(diag),(diag
...
.........kcm ip KCM δ ,       (A4) 
 
We follow as in Ref.2 the eigenvalue problem formulation from Müller and Schiehlen, Shabana  and 
Newland (see Ref.12 for details). The state vector y(t) be defined as the (2n ×  1) vector: 
 
                                             





=
q
q
y
&
                (A5) 
 
The (2n ×  2n) system matrix is given as: 
 






−−
= −−
CMKM
I0
A 11          (A6) 
 
The physical parameters are m, k, and c, damping coefficient.  A MATLAB® script  has been 
implemented to solve numerically the corresponding eigenvalues and eigenvectors problem. The 
dimension of matrix, the position p of damping, mass, stiffness and damping coefficient were the 
parameters to use in the numerical program. At the end of calculation, real and imaginary parts of 
eigenvalues were plotted by means of a dedicated MATLAB® tool, or a tabular vision was considered. 
A Fortran programming was also used, with the QR algorithm for real Hessenberg matrices [11], to 
increase the speed of calculations. 
 
 
References 
1. L. Brillouin, Wave Propagation in Periodic Structures, 2nd Edn. Dover, New York, 1953. 
2. M. Gürgöze and A. Özer, On the slightly damped uniform n-mass oscillator, Computers & 
Structures, Vol.5, pp.797-803, 1996. 
3. T. Dauxois,  Fermi, Pasta, Ulam and a mysterious lady. Physics Today, Vol.61, pp.55-57, 2008.  
4. J. Ford, The Fermi-Pasta-Ulam problem: Paradox turns discovery, Phys. Rep., Vol.213, pp.271-310, 
1992. 
5. T.P. Weissert, The Genesis of Simulation in Dynamics: Pursuing the Fermi-Pasta-Ulam Problem, 
Springer, 1997.  
6. E. Weber and F. Nebeker, The Evolution of Electrical Engineering, IEEE Press, Piscataway, New 
Jersey USA, 1994.   
7. Z. Zheng, X. Feng, B. Ao and M.C. Cross, Synchronization of groups of coupled oscillators with 
sparse connections, Europhysics Letters,Vol.87, pp.50006-1-50006-5, 2009. 
8. H. Frahm, Device for damping vibration of bodies, U.S. Patent No.989-958, 1911. 
9. R.J. McNamara, Tuned mass dampers for buildings, Journal of the Structure Division, ASCE, 
Vol.103 pp.1985–98, 1977. 
10. P.D. Cha and C. Pierre, Imposing nodes to the normal modes of a linear elastic structure, Journal of 
Sound and Vibration, Vol.219, pp. 669-687, 1999. 
11. Numerical Recipes in Fortran 77, Second Edition, 1992, http://www.nrbook.com/a/bookfpdf.php. 
12. M. Gürgöze, On various eigenvalue problem formulations for viscously damped linear mechanical 
systems International, Journal of Mechanical Engineering Education, Vol.33, pp.235-243,  2005. 
 
 
Tables 
 
Number of masses, 
n 
Damped mass, 
p 
Eigenvalues (Im) 
w 
4 3 1.000 
7 3 1.000 
7 5 0.618, 1.618 
7 6 1.000 
10 3 1.000 
10. 6 1.000 
10 7 0.445, 1.247, 1.802 
10 9 1.000 
12 5 0.618, 1.618 
12 10 0.618, 1.618 
13 3 1.000 
13 6 1.000 
13 9 0.347, 1.000, 1.532, 1.879 
13 12 1.000 
Table I: Values of the undamped mode eigenvalues in the case of m=1 kg, k=1 N/m, c=0.01 s-1. 
 
 
 
 
n=7, p=5 
w: 0.618, 1.618 
 
 
part A (4 masses), w: 0.618, 1.176, 1.618, 1.902 
part B (2 masses), w: 0.618, 1.618 
 
 
n=10, p=7 
w: 0.445, 1.247, 1.802 
 
 
 part A (6 masses), w: 0.445, 0.868, 1.247, 1.563, 1.802, 1.950 
part B (3 masses), w: 0.445, 1.247, 1.802 
 
 
n=13, p= 9 
w: 0.347, 1.000, 1.532,  
1.879 
 
 
part A (8 masses), w: 0.347, 0.684, 1.000, 1.285, 1.532, 1.732, 
1.879, 1.969 
part B (4 masses),  w: 0.347, 1.000, 1.532, 1.879 
 
 
Table II: Values of the undamped mode eigenvalues in the case of m=1 kg, k=1 N/m, c=0.01 s-1, for 
three chains with the specified position of the damped mass and the two corresponding parts A and B 
of  them. The synchronized frequencies are marked by the bold character.  
Figures 
 
 
 
 
Fig.1: Chain of coupled oscillators with a single damped mass, in the upper part, in the upper part of 
the image. A transmission line with electric oscillating circuits in the lower part: the depicted case is 
that of open-end termination. The two systems are described by the same dynamical equations.  
  
 
 
 
 
 
Fig.2: Behaviour of  real (x-axis) and  imaginary (y-axis) parts of eigenvalues, for corresponding 
choices of n and p as reported in the box, and for the specified physical parameters. The right part of 
the figure shows the behaviour of a chain (n=19 and p=18, red crosses) with pure imaginary 
eigenvalues, and those of  two following chains n=20 and n=21, which have no undamped modes. 
Eigenvalues have a rather small real part, different from zero. 
 
 
 
 
 
 
 
Fig.3: This image concerns the chain with 4 masses (m1,m2,m3,m4), the third (m3) damped. Panel 3.a 
reports the behaviour of the third (m3) and fourth masses (m4) as a function of time, for m=1 kg, k=1 
N/m, c=0.01 s-1.  x3  is damped. 3.b reports the behaviour of  the two positions (x2,x4) of oscillating 
masses  (m2,m4), with time as parameter. The initial point of the motion is at the right upper corner 
marked with the red dot, whereas the motion limit for asymptotic times is coincident with central part 
of the grey diagonal line. 3.c and 3.d report the phase space diagrams for masses m4 and m3 
respectively. Note the limit-cycle in 3.c. For 3.d, the limit is the central point. 
 
 
 
 
 
 
 
 
Fig.4: When only the undamped mode survives, the third mass becomes a node of the chain.  This 
allows to view the chain as composed by two sub-chains (parts A and B). They oscillate synchronized 
at the same frequency and with a phase suitable to let mass 3 at its equilibrium position.  
 
 
 
 
 
 
Fig.5: The same as in Table II, with the comparison of the dispersion of the chain without dampers with 
n=13, and the dispersion of the two sub-chains with n=4 and n=8. The red crosses are the modes 
appearing when the mass at position 9 is damped. The physical parameters are m=1 kg, k=1 N/m, 
c=0.01 s-1. 
 
 
 
 
 
 
 
 
 
Fig.6: The red crosses are the values of the undamped modes appearing when the mass at position p is 
damped. The green points represent the dispersion of the chain without dampers. Physical parameters 
are m=1 kg, k=1 N/m, c=0.01 s-1. 
 
 
 
  
 
 
 


Self frequency-locking of a chain of oscillators

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