- Email: [email protected]
Chains of oscillators with negative stiffness elements
Journal of Sound and Vibration 333 (2014) 6676–6687
Contents lists available at ScienceDirect
Journal of Sound and Vibration journal homepage: www.elsevier.com/locate/jsvi
Chains of oscillators with negative stiffness elements Elena Pasternak a, Arcady V. Dyskin b,n, Greg Sevel c,1 a
School of Mechanical and Chemical Engineering, The University of Western Australia, 35 Stirling Highway, Crawley WA 6009, Australia School of Civil, Environmental and Mining Engineering, The University of Western Australia, 35 Stirling Highway, Crawley WA 6009, Australia c 10 Nila Street, Wembley Downs 6019, WA, Australia b
a r t i c l e in f o
abstract
Article history: Received 10 November 2013 Received in revised form 26 June 2014 Accepted 29 June 2014 Handling Editor: L.N. Virgin Available online 6 August 2014
Negative stiffness is not allowed by thermodynamics and hence materials and systems whose global behaviour exhibits negative stiffness are unstable. However the stability is possible when these materials/systems are elements of a larger system sufficiently stiff to stabilise the negative stiffness elements. In order to investigate the effect of stabilisation we analyse oscillations in a chain of n linear oscillators (masses and springs connected in series) when some of the springs' stiffnesses can assume negative values. The ends of the chain are fixed. We formulated the necessary stability condition: only one spring in the chain can have negative stiffness. Furthermore, the value of negative stiffness cannot exceed a certain critical value that depends upon the (positive) stiffnesses of other springs. At the critical negative stiffness the system develops an eigenmode with vanishing frequency. In systems with viscous damping vanishing of an eigenfrequency does not yet lead to instability. Further increase in the value of negative stiffness leads to the appearance of aperiodic eigenmodes even with light damping. At the critical negative stiffness the low dissipative mode becomes non-dissipative, while for the high dissipative mode the damping coefficient becomes as twice as high as the damping coefficient of the system. A special element with controllable negative stiffness is suggested for designing hybrid materials whose stiffness and hence the dynamic behaviour is controlled by the magnitude of applied compressive force. & 2014 Elsevier Ltd. All rights reserved.
1. Introduction The force–displacement response of materials and structures, at least in their stable state is characterised by positive stiffness whereby an increase in displacement results in the corresponding increase in the reaction force. In this case the work done leads to increase in the system's potential energy. However, in some circumstances when the system achieves unstable state (e.g. post-buckling regime in columns or shells or post-peak softening stage of the loading of brittle materials and rocks) it exhibits negative stiffness (at least incrementally) such that the potential energy formally receives negative increments under loading. Of course, such a system is unstable: maintaining the stable state and further loading are only possible in the presence of a stabilising device, either a sufficiently stiff loading frame (e.g., [1–4]) or a matrix with conventional positive stiffness which the negative stiffness elements are immersed into [5,6]. This device (matrix) increases its own potential energy in the process and thus maintains the energy balance.
n
Corresponding author. E-mail addresses: [email protected] (E. Pasternak), [email protected] (A.V. Dyskin), [email protected] (G. Sevel). 1 Participated in this project while on employ at UWA.
http://dx.doi.org/10.1016/j.jsv.2014.06.045 0022-460X/& 2014 Elsevier Ltd. All rights reserved.
E. Pasternak et al. / Journal of Sound and Vibration 333 (2014) 6676–6687
6677
There are many examples of systems that exhibit negative stiffness as a structural or effective property. In particular discrete systems of elastic springs, arches, link models as well as link and lever models with elastic springs are known to exhibit negative stiffness in the unloading part of the loading curve (e.g., [9,11–18]). These also include tubes and columns pre-buckled to an S-shaped configuration [6,19], single foam cells [20], ‘wine-rack’ [21] and zig-zag [22] structures, platelike interlocking structures of cubic elements [3,4] and materials with phase transitions (e.g., [7,23,24]). The effect of apparent negative stiffness can be produced by rotating non-spherical grains [25] (This mechanism of negative stiffness is explained below in Section 5.). In life nature, the negative stiffness regime was observed in the mechanical response of a hair cell of the bullfrog's sacculus, which is believed to be responsible for amplification of its sensitivity [26,27]. Negative stiffness elements also provide the simplest models of the effect of unstable regions created by nonlinear springs with non-convex energy [28,29]. The behaviour of systems made of conventional elements that under some conditions exhibit negative stiffness can be (and has been) analysed by the conventional means of structural mechanics. Thus, Wang [30] modelled the effect of a negative stiffness inclusion within a positive stiffness matrix by considering a double hexagonal structure in which external hexagon is made of standard elements, while the internal hexagon is made of negative stiffness elements. He found that depending on the value of the negative stiffness and the loading conditions, there exist zones of increased effective (positive) stiffness followed by abrupt transition to negative effective stiffness. Continuum analysis of the behaviour of systems with negative stiffness elements and especially of materials with negative stiffness inclusions is commonly performed employing standard methods of mechanics of solids with the signs of the stiffnesses (moduli) reversed. Thus, Lakes [5,6] and Lakes et al. [7] used the Voigt–Reuss and Hashin–Strikman bounds by replacing the positive moduli with the corresponding negative ones to infer that the presence of negative stiffness inclusions makes the material stiffer. Lakes and Drugan [8] came to the same conclusion by considering a matrix with negative stiffness spherical inclusions in dilute concentrations. Lakes et al. [7] report the results of experiments with composite materials consisted of inclusions of vanadium dioxide (ferroelastic material with transformation from monoclinic to tetragonal at 67 1C) in a pure (99.99 per cent) tin matrix. Despite the dilute concentration of inclusions (1 per cent volumetric fraction) the composite is observed to produce considerable effect on the mechanical damping (the loss tangent) and the stiffness, namely drop and then rise of the compliance at 65.5–66 1C as well as rise and then considerable drop in damping in the same temperature interval and then a smaller peak in damping at 67 1C. Anomalous fluctuations with temperature in the Young's modulus and loss tangent at very low frequencies (0.1 Hz) as well as negative Poisson's ratio in tetragonal BaTiO3 after ageing under bending were further reported in [31]; however no extremely high stiffness was found. Furthermore, in the applications of systems with negative stiffness elements to achieve the sound attenuation and isolation it is understood that the negative stiffness elements are responsible for reducing the overall stiffness of the system and hence reducing the resonant frequency below the excitation thus avoiding resonances (e.g. [18]). The use of negative stiffness elements in increasing damping and energy dissipation was discussed in [9,32–34]. In general, as the presence of the negative stiffness in a composite phase formally violates the condition of positive definiteness of elastic energy, it was necessary to analyse in which cases the phase with positive definite energy could stabilise the composite. Drugan [35] proved that a cylinder or sphere made from a material whose tensor of elastic moduli is strongly elliptic (a certain range of negative Young's and bulk moduli is permitted) can be stabilised by a thin coating made of a conventional material as long as it is sufficiently stiff. Kochmann and Drugan [36] extended the analysis of the dynamic stability to any thickness of the coating in the case of cylindrical inclusion. In particular, for the cylinder in an infinite matrix a simple criterion of stability was obtained: μII/μI 4 1 λI/μI, where λ and μ are the Lame constants and the superscripts I and II refer to the material of the cylinder and the matrix respectively. It is interesting, that in the case of negative shear modulus, μI ¼ mμII, this criterion reduces to m4 1þλI/μII, that is the stability requires high values of negative modulus. Dyskin and Pasternak [10,25] considered effective elastic moduli of a composite consisting of a conventional matrix filled with inclusions having negative shear modulus: the cylindrical inclusions in antiplane strain conditions, as well as spherical and crack-like inclusions with negative shear modulus whose bulk modulus was the same as that of the matrix. It was shown that while the presence of inclusions with negative moduli might make the solution of the full elastic problem not unique, the effective moduli – the moduli that relate stress and strain averaged over the volume elements macroscopic with respect to the inclusions – are unique. It was also shown, using the differential self-consistent method (e.g. [37]) that the inclusions with negative shear modulus could both increase and decrease the effective moduli (as compared to those of the matrix) depending upon the value of the negative modulus. In both cases however the effective shear modulus remains positive as long as the concentration (volumetric fraction) of the inclusion is below a certain critical value. After reaching the critical value the effective shear modulus abruptly becomes negative and the composite loses stability. The critical concentration depends upon the value of negative stiffness. Furthermore, there exists a value of the negative stiffness when the critical concentration becomes zero meaning that such a composite is unstable, no matter how few inclusions it contains. Interestingly, for higher values of the negative shear modulus the critical concentration becomes non-zero again and grows as the value of the negative stiffness increases thus producing a new region of stability. The condition of the existence of this second region of stability – the high negative shear modulus – qualitatively corresponds to the stability condition obtained by Kochmann and Drugan [36].
6678
E. Pasternak et al. / Journal of Sound and Vibration 333 (2014) 6676–6687
A considerable body of research was devoted to studying the sonic or acoustic metamaterials. Some authors interpret the existence of the frequency gaps in these materials as wave reflection (the presence of negative refraction index) associated with negative effective dynamic stiffness or modulus (e.g., [38–45]) or, by association, with negative density or mass (e.g., [46–53]). In particular, the effective dynamic modulus or stiffness is defined in this interpretation as seff ¼ F(t)/x(t), where F(t) is the (harmonic) force applied and x(t) is the displacement (or stress and strain in the case of effective modulus). We note however that the negative effective stiffness defined in such a way can be found even in the ordinary linear oscillator (e.g., mass on a conventional spring). Indeed, consider a forced oscillator excited by a harmonic driving force: mx€ þ r x_ þ sx ¼ FðtÞ; where FðtÞ ¼ F 0 expðiωtÞ, m is the mass, r is resistance, s is the spring stiffness and F0 is the amplitude of the applied force. The steady-state solution for light damping reads (e.g. [54]): xðtÞ ¼ iF 0 ðωZ m Þ 1 expðωt þϕÞ, where Zm ¼ r þ iðωm s=ωÞ ¼ Z m expðiϕÞ. Then the frequency-dependent ‘effective’ stiffness defined as above as F(t)/x(t) can be expressed as seff ðωÞ ¼ FðtÞ=xðtÞ ¼ ðω2 m sÞ iωr ¼ m½ðω2 ω20 Þ þ iωr=m. Here ω0 ¼ ðs=mÞ1=2 is the eigenfrequency. Thus, when the frequency ω⪢ω0 the stiffness formally becomes negative, even if the spring stiffness, s, is positive. This negative stiffness is of course the effect of the phase change (e.g. [54]) – the fact, which is recognised in some works on metamaterials (e.g. [34]). In this paper we concentrate on investigating the stability of dynamic systems (that the trajectories are bounded) whose elements can exhibit true negative stiffness (modulus) i.e. negative stiffness in static loading. In order to analyse the stability we consider a very simple system – a chain of standard linear oscillators (equal masses, m, attached to springs) connected in series, Fig. 1. We assume that some springs can have negative stiffness. We presume that the system is linear that is the stiffnesses are independent of displacements. The nonlinear effects and snapback phenomena are therefore not considered. We investigate the stability of such a system first using the global (effective) stiffness (Section 2) and then conduct the complete stability analysis (Section 3). In the latter case we consider both conservative systems and systems with viscous damping. In the dynamic case we assume fixed boundaries at both ends of the chain. In Section 4 we generalise the stability analysis to a simple 2D case of square lattice with longitudinal springs. Section 5 discusses the obtained results and offers an outlook. 2. Statics of chains with positive and negative stiffness springs. Effective stiffness Consider a chain of n springs with stiffnesses k1,.., kn, Fig. 1. Since the force in each spring must be the same, f, the 1 displacement of (the last) mass n is un ¼ ∑ni¼ 1 f ki . Therefore, the overall (effective) stiffness, keff, reads n
1
1
keff ¼ ∑ ki
(1)
i¼1
By definition the effective stiffness is a stiffness of an equivalent continuous material (a 1D material in this case). Thermodynamically, such an equivalent material can only be stable if its stiffness is positive. Therefore, the stability condition reads keff 40. This condition would permit some springs (but not all of them) to have negative stiffness. This is however not a sufficient condition of stability as the chain of springs can still be unstable as demonstrated in Section 3. Consider now a particular case, when there are m negative stiffness springs with the same negative stiffness k- and n–m springs with the same positive stiffness k0. Then 1
1
1
keff ¼ mk þ ðn mÞk0
(2)
We will treat the positive stiffness springs as ‘matrix’ and negative stiffness springs as ‘inclusions’. We introduce the stiffness of the matrix (the stiffness in the absence of the inclusions), kmtx ¼k0/n. In order to make the analysis independent of the number of springs, we introduce the concentration of negative stiffness springs c ¼m/n; 0rc r1. Finally, we introduce the dimensionless value of negative stiffness κ ¼ k /k0. Then the effective stiffness can be expressed as keff ¼ kmtx ð1 c=ccr Þ 1
(3)
1
where ccr ¼κ(1 þκ) is the critical concentration starting from which the effective stiffness becomes negative and the chain loses stability. Fig. 2 shows the dependence of the effective stiffness upon the concentration of the negative stiffness phase (3). It is seen that (a) the negative stiffness springs do make the chain stiffer and (b) upon reaching the critical concentration, the effective stiffness abruptly drops to a negative value. This is similar to the stiffening effect of negative stiffness inclusions (e.g., [5, 10, 25]). The necessary condition of stability, co ccr can also be rewritten in terms of the dimensionless value of negative stiffness as κ 4κ cr ¼
c 1c
(4)
Fig. 1. A chain of oscillators consisting of n masses connected by springs with stiffnesses k1,.., kn. For the sake of simplicity all masses are assumed to be the same, equal to m. Under applied force f, the i-th mass assumes displacement ui.
E. Pasternak et al. / Journal of Sound and Vibration 333 (2014) 6676–6687
6679
Fig. 2. Dependence of the normalised effective stiffness of the chain upon the concentration of the negative stiffness phase.
Fig. 3. System of n 1 equal masses (m) connected to the stationary masses 0 and n (boundaries) by n springs.
Thus, in this example the stability requires the values of negative stiffness κ to exceed a certain critical value, κcr. 3. Stability of chain of oscillators with positive and negative stiffnesses 3.1. Conservative system We now investigate the stability of the chain of n 1 masses connected by the springs, Fig. 3. We write the general system and assume the boundary conditions according to which the end masses (mass 0 and mass n) are stationary, that is u0 ðtÞ ¼ u_ 0 ðtÞ ¼ un ðtÞ ¼ u_ n ðtÞ ¼ 0;
(5)
where the dot denotes the time derivative. We write Lagrange equations d ∂L ∂L ¼ f i; dt ∂q_ i ∂qi
i ¼ 0; ::; n
(6)
where fi are the forces acting on each mass and the Lagrangian is mu_ i 1 n 1 ∑ ½ki ðui ui 1 Þ2 þ ki þ 1 ðui þ 1 ui Þ2 : 2i¼1 i¼0 2 n
L ¼ T Π ¼ ∑
(7)
Using (7) we obtain the following system of differential equations for the chain: mu€ i þðki þki þ 1 Þui ki ui 1 ki þ 1 ui þ 1 ¼ f i ;
i ¼ 1; …n 1
(8)
The criterion of stability is the positive definiteness of the potential energy Π¼
i 1 n 1h ∑ ki ðui ui 1 Þ2 þ ki þ 1 ðui þ 1 ui Þ2 2i¼1
(9)
6680
E. Pasternak et al. / Journal of Sound and Vibration 333 (2014) 6676–6687
Positive definiteness of (9) is equivalent to the positive definiteness of the matrix of stiffnesses 0 1 k1 þk2 k2 B C k2 þk3 k3 B k2 C B C B C ⋱ ⋱ ⋱ B C An ¼ B C ki ki þki þ 1 ki þ 1 B C B C B C ⋱ ⋱ @ A kn 1 kn 1 þ kn
(10)
where the empty spaces refer to zero entries. The condition of the positive definiteness of An is that all main diagonal minors of An are positive: 2 3 a1;1 a1;2 ⋯ a1;i 6a 7 6 2;1 a2;2 ⋯ a2;i 7 7 40 (11) Δi ¼ det6 6 ⋮ ⋮ ⋱ ⋮ 7 4 5 ai;1 ai;2 ⋯ ai;i Substituting (10) into (11) leads to the following condition: n
n
∑ ∏ kj 40
(12)
i¼1j¼1 jai
This is obviously the necessary and sufficient condition of stability. These rules are cumulative, meaning that as additional oscillators are added to the system, all of the old rules continue to govern the system together with the newly introduced rules. These rules also allow renumbering, as can be shown by shifting rows and columns in A. For example, the first rule k1 þk2 4 0 becomes ki þki þ 1 40 8 i when there are more than two oscillators in the system. As an illustration, Fig. 4 shows a simplified system containing only two oscillators (two masses on 3 springs). Fig. 5 shows the stable regions for the combinations of stiffnesses. It is seen that the system can be dynamically stable even when the stiffness of one of the springs is negative. We now prove another necessary condition of stability: for any system of n oscillators in a 1-D chain to be stable, only one stiffness ki is allowed to be negative. Indeed, for a single mass, we have the rule, k1 þk2 40 therefore only one stiffness can be negative. Suppose for n 1 springs only one spring can have negative stiffness. From (12) we have kn 4
k1 k2 …kn 1 ; Δn 2
n 42
(13)
where, according to (11) Δn 2 4 0. Therefore, if there is already negative stiffness spring amongst n 1 springs then k1 k2 …kn 1 o 0 ) kn 4 0, that is the n-th spring must have positive stiffness. Alternatively, if all n 1 springs have positive stiffness, then the n-th spring is allowed to be negative. Obviously, the maximum value of the negative stiffness increases with the increase in the values of the positive stiffnesses as illustrated in Fig. 5. Therefore, if we have n springs and only one is allowed to have negative stiffness then the critical concentration of negative stiffness springs when the system can still be stable is ccr ¼
1 n
(14)
The condition of stability (12) is stronger than the one formulated in the previous section, keff 40, at least in that that only one spring is allowed to be negative. This shows that condition keff 40, which allows a number of springs to be negative, is only a necessary condition of stability. Further insight into the behaviour of the systems with negative stiffness springs can be obtained by considering the simplified two mass system, which is a coupled oscillator shown in Fig. 4. The governing equations read ( mu€ 1 þ ðk1 þ k2 Þu1 k2 u2 ¼ f 1 (15) mu€ 2 þ ðk2 þ k3 Þu2 k2 u1 ¼ f 2
Fig. 4. A coupled oscillator consisting of two masses connected by three springs.
E. Pasternak et al. / Journal of Sound and Vibration 333 (2014) 6676–6687
6681
Stable
Unstable
Fig. 5. The regions of stability and instability for system (8) with boundary conditions (5) for n¼ 3.
The homogeneous system has the following set of fundamental solutions: ! u1 ¼ e1 ½C 1þ expðiω1 tÞ þC 1 expð iω1 tÞ þ e2 ½C 2þ expðiω2 tÞ þ C 2 expð iω2 tÞ u2 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u u 2 tðk1 þ 2k2 þ k3 Þ þ ðk1 k3 Þ2 þ4k2 ; ω1 ¼ 2m vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u u 2 tðk1 þ 2k2 þ k3 Þ ðk1 k3 Þ2 þ 4k2 ω2 ¼ 2m
(16)
where e1 and e2 are eigenvectors of the matrix A2 of system (15) and C 17 ; C 27 are constants. The stability is ensured when k1 þk2 4 0;
k1 k2 þk1 k3 þk2 k3 40
(17)
Fig. 6 shows dependence of the eigenfrequencies, ω1, ω2 for different stiffnesses reaching negative values. We assume that after due normalisation, m ¼1 and k1 ¼1. It is seen that as the negative stiffness reaches its minimum value, frequency ω2 tends to zero. Another feature is that the negative stiffness does reduce the eigenfrequencies. It can be interpreted that the negative stiffness can only reduce the dynamic effective stiffness. This is opposite to the static effective stiffness, which as shown in the previous section, is increased by the presence of negative stiffness springs. This apparent contradiction will be resolved in Section 5. As soon as the negative stiffness exceeds its critical value, frequency ω2 becomes imaginary and exponentially divergent terms appear in the solution, which indicates the overall instability of the system. Fig. 7 shows the trajectories of both masses in the case of absence of the driving force (f1 ¼ f2 ¼0) and for k1 ¼ k3 ¼1 and different values of the central spring stiffness k2. The initial conditions are u1 ð0Þ ¼ u_ 1 ð0Þ ¼ u_ 2 ð0Þ ¼ 0; u2 ð0Þ ¼ 1. These trajectories are combinations of oscillations with two eigenfrequencies each depending differently on the stiffness of the central spring. In particular, according to Fig. 6a the first eigenfrequency reduces when k2 reduces from positive values to zero and becomes 1 when k2 becomes negative. Opposite to this the second eigenfrequency keeps equal to 1 when k2 is positive and reduces to zero when k2 is negative. Subsequently at all values of the central spring stiffness at least one eigenfrequency is always equal to one. This circumstance leads to complex dependence of the trajectories of both masses on the stiffness of the central spring shown in Fig. 7. 3.2. System with light linear damping When damping is introduced to the system, the Lagrange equations assume the dissipative term as follows (e.g., [55]) d ∂L ∂L ∂F ¼ f i ; i ¼ 1; ::; n (18) dt ∂q_ i ∂qi ∂q_ i
6682
E. Pasternak et al. / Journal of Sound and Vibration 333 (2014) 6676–6687
Fig. 6. Dependence of eigenfrequencies for a coupled oscillator, Fig. 4, upon different values of spring stiffnesses: (a) cases k1 ¼k2 ¼ 1 and k1 ¼ k3 ¼ 1, the minimum value of negative stiffness is 0.5; (b) cases k1 ¼ 1, k2 ¼ 3 and k1 ¼ 1, k3 ¼3, the minimum value of negative stiffness is 0.75.
Fig. 7. Trajectories of the coupled oscillator, Fig. 4, in the absence of the driving force for different values of the central spring stiffnesses: k1 ¼k3 ¼1, k2 ¼ 0.5, 0.1, 1: (a) the movement of ball 1; (b) the movement of ball 2. The initial conditions are u1 ð0Þ ¼ u_ 1 ð0Þ ¼ u_ 2 ð0Þ ¼ 0; u2 ð0Þ ¼ 1.
where 1 F ¼ ∑αik u_ i u_ k 2 i;k
i; k ¼ 1; ::; n
(19)
and αij form a symmetric matrix of damping coefficients, αij ¼αji. The stability of such a system is determined by matrix (10), which leaves the previous conclusion unchanged: the stable system can only have one negative stiffness spring. Further analysis will be performed assuming, for the sake of simplicity
E. Pasternak et al. / Journal of Sound and Vibration 333 (2014) 6676–6687
6683
that αik ¼ αδij , where δij is the Kronecker delta. The corresponding system of equations reads mu€ i þ αu_ i þ ðki þ ki þ 1 Þui ki ui 1 ki þ 1 ui þ 1 ¼ f i ;
i ¼ 1; …n 1
Consider, as an example a two mass oscillator from Fig. 4. ( mu€ 1 þ αu_ 1 þðk1 þk2 Þu1 k2 u2 ¼ f 1 mu€ 2 þ αu_ 2 þðk2 þk3 Þu2 k2 u1 ¼ f 2 where e1 and e2 are eigenvectors of the matrix A2 of system (21). The system has the following general solution: ! h α α i u1 t þ iω1α t þC 1 exp t iω1α t ¼ e1 C 1þ exp u2 2m 2m h α α i þ t þiω2α t þC 2 exp t iω2α t þ e2 C 2 exp 2m 2m qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 ðk þ 2k þ k Þ þ ðk k Þ þ 4k 1 2 3 1 3 α2 2 ; ω21;α ¼ 2m 4m2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ðk1 þ 2k2 þ k3 Þ ðk1 k3 Þ2 þ 4k2 α2 ω22;α ¼ 2m 4m2
(20)
(21)
(22)
here e1 and e2 are eigenvectors of the matrix A2 of system (15) and C 17 ; C 27 are constants. It can be seen that as long as conditions (17) satisfy, even if the eigenfrequencies become imaginary, the fundamental solutions still have negative coefficient at t, so the solutions are not increasing and the system is stable albeit with aperiodic modes. In other words, as long as the eigenfrequencies are real the eigenmodes are periodic and the damping coefficient is still α. When an eigenfrequency is imaginary, the corresponding eigenmode becomes aperiodic (the frequency is zero) and the damping coefficient changes. In order to investigate this further we introduce the effective dumping coefficients and the corresponding frequencies qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8 8 0; ω2j;α ðαÞ o0 < α 8 2m ω2j;α ðαÞ; ω2j;α ðαÞ o 0 < 7 n αj ¼ ; ωj ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffi ; j ¼ 1; 2 (23) : : ω2j;α ðαÞ; ω2j;α ðαÞ Z0 α; ω2j;α ðαÞ Z 0 Fig. 8 shows the dependence upon stiffnesses of the normalised frequency ωn2 =ω2 of the system with damping where ω2 is the eigenfrequency of the system without damping, and the relative effective damping coefficients α27 =α. Frequency ωn1 does not become imaginary for the values of stiffnesses keeping the system stable and the corresponding damping is unchanged. We again used the normalisation that leads to m ¼1 and k1 ¼1. It is seen that the negative stiffness element decreases the resonant frequencies but can both increase and decrease the damping. Also, according to (22) and (23) as ω2 -0; α2þ -2α; α2 -0. Thus in this simple system, depending upon the initial conditions, the damping coefficient can be either increased 2 times or made vanish by using a near critical value of negative stiffness. It is also seen that qualitatively the behaviour of the frequencies and the effective damping coefficients is similar for k1 ¼ k3 ¼ 1; k2 ¼ k and k1 ¼ k2 ¼ 1; k3 ¼ k. This result can easily be generalised to chains of n springs. Indeed, by transferring system (20) to principal directions (eigenvectors of matrix An), e1, e2,…, en, one can write each mode of oscillations as exp( iλjt), j¼1,…n, where: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi α 7 α2 4m2 ω2j (24) λj ¼ 2m here ωj is the eigenfrequency of eigenmode j without damping. Obviously, the boundary of stability of the system corresponds to vanishing of the determinant of matrix An, Eq. (10). This corresponds to vanishing of one of the eigenfrequencies; let it be ωj. Substitution of ωj ¼0 into (24) leads to aperiodic eigenmodes with damping coefficients either 2α or 0. 4. Directionally decoupled lattices The obtained results allow a simple generalisation to higher dimensions assuming that displacements in different directions are not coupled. Fig. 9 shows an example of a 2D square lattice. We assume that the displacements of mass (i,j), uij and vij do not interact such that dynamics of the lattice is reduced to independent oscillations in the horizontal and vertical directions. For n by n lattice there are n vertical and n horizontal independently oscillating chains. Each chain, in order to be stable, must have not more than one negative stiffness spring. From here the critical concentration of negative stiffness springs when the lattice can still be stable is ccr ð2Þ ¼
2 n
(25)
6684
E. Pasternak et al. / Journal of Sound and Vibration 333 (2014) 6676–6687
Fig. 8. Dependence of the normalised values of frequency, ωn2 =ω2 , and the effective damping coefficient α27 =α upon normalised stiffnesses of the springs: (a) k1 ¼ k3 ¼ 1; k2 ¼ k, (b) k1 ¼ k2 ¼ 1; k3 ¼ k.
A straightforward generalisation to p dimensions gives ccr ðpÞ ¼
p np 1
(26)
Obviously, the previous conclusions about changing the frequency and damping in some modes remain.
5. Discussion and outlook The results of our analysis show that dynamic systems with negative stiffness elements can, under certain restrictions be stable. In particular if the system consists of independent one-dimensional chains, each chain can have one spring with negative stiffness given that the value of the stiffness does not exceed a certain critical value determined by the number and stiffnesses of the positive stiffness springs. With increase of the value of negative stiffness the eigenfrequencies decrease. Furthermore, as the negative stiffness approaches its critical value there exist eigenmodes with vanishing frequencies. The decrease of eigenfrequencies caused by the negative stiffness springs is in apparent contradiction with the result of Section 2: the effective stiffness in the stable region increases with the increase of the negative stiffness, Fig. 2. In order to understand the effect let us consider another static model, Fig. 10. In this model all springs are connected in parallel by rigid plates. In this case the effective stiffness is n
keff ¼ ∑ ki
(27)
i¼1
where ki are the stiffnesses of individual springs. It is clear that in this configuration even a single negative spring reduces the effective stiffness, the more the higher the value of negative stiffness. Therefore the behaviour of the above chains of oscillators, which are oscillators connected in series, resembles the behaviour of a system of springs connected in parallel. The nature of this resemblance becomes transparent when one recalls that the rigidity of the plates connecting the parallel springs, Fig. 10, makes the springs independent. Similarly the chain of oscillators being linear allows decomposition into independent oscillators whose oscillations correspond to the eigenmodes. It seems that the independence of springs or eigenmode oscillators is behind the similarity of the effect of negative stiffness in chains of oscillators connected in series and springs connected in parallel. The presence of the critical value of negative stiffness at which one of the eigenfrequencies becomes zero suggests a way of controlling the resonance frequency by changing the negative stiffness value. The most important thing is the degree of control achievable as the resonant frequency can, in principle, be brought to zero. It is interesting that such a tuning mechanism is likely to be realised in life nature; according to [26,27] the hair bundles in the ears have a negative stiffness region in the force–displacement curve. Furthermore, the positions of the region and, as followed from the plots presented in these papers the values of negative stiffness can be controlled by offsetting the hair bundle. The strong dependence of some resonant frequencies upon the value of negative stiffness opens up an avenue of designing highly tunable hybrid materials with negative stiffness elements. What is needed is a tuneable negative stiffness element. Tuneable negative stiffness elements based on specially arranged levers and springs are known (e.g. [56]), but the tuning requires changing the design, for instance the arms of the lever.
E. Pasternak et al. / Journal of Sound and Vibration 333 (2014) 6676–6687
6685
Fig. 9. An example of a square lattice (for the sake of illustration a 6 6 lattice is shown).
k1 k2
f
kn Fig. 10. System of springs of stiffnesses k1, k2,…,kn connected in parallel.
Here, we propose to use a negative stiffness element whose negative stiffness value depends upon the compression force applied to it and thus easily tuneable. For such an element we use the effect of apparent negative stiffness produced by rotating non-spherical disks suggested by Dyskin and Pasternak [25,57]. Such an element is shown in Fig. 11a. The moment equilibrium with respect to point O reads qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 1 ; δ¼ (28) T δ2 ð1 ξÞ2 ¼ Pð1 ξÞ; ξ ¼ d sin α sin α This relationship is shown in Fig. 11b. For small displacements, ξ⪡1, developing T as a function of ξ into Taylor series and keeping only linear term we obtain the linear stress–strain relationship with negative modulus between shear stress T and shear strain γ¼u/d ¼ξsinα T ¼ ku;
k¼
P δ3 2 d ðδ 1Þ3=2
(29)
Two things are apparent from this equation. Firstly, the relationship is linear (and reversible) and stiffness is negative, k o0. Secondly, the stiffness depends upon the compressive force P. Thus the compressive force can be used to tune the value of negative stiffness. Since the force can be applied using an actuator, the resonant frequency of the oscillator can be easily controlled and the control can even be exercised remotely. 6. Conclusion Negative stiffness, as well known, is prohibited by thermodynamics; subsequently materials or systems characterised by global negative stiffness cannot be stable. However negative stiffness is thermodynamically possible at the level of elements
6686
E. Pasternak et al. / Journal of Sound and Vibration 333 (2014) 6676–6687
Fig. 11. A tuneable negative stiffness element: (a) rotating block; (b) the force–displacement relationship is descending and reversible. Infinitesimally it is linear characterised by negative stiffness.
of a system or parts of a material as long as the positive stiffness system (material) containing the negative stiffness elements is stiff sufficient to stabilise them. This conclusion was illustrated in this paper by considering examples of chains of springs and masses with fixed ends. We showed that chains of springs and masses connected in series as well as systems made of independently oscillating chains containing springs with negative stiffness could still exhibit stable motion. The necessary criterion of stability stipulates that there should be no more than a single negative stiffness spring per chain. To make this condition also sufficient, the value of the stiffness should not exceed a certain critical value determined by the number and stiffnesses of the positive stiffness springs. As the negative stiffness approaches its critical value, one of the eigenfrequency vanishes, which means the appearance of very slow periodic motion. In systems with light viscous damping vanishing of an eigenfrequency does not yet lead to instability. Further increase in the value of negative stiffness leads to the appearance of aperiodic eigenmodes with light and high damping. At the critical negative stiffness the low dissipative mode becomes non-dissipative, while in the high dissipative mode the damping coefficient becomes as twice as high as the damping coefficient of the system. The presented analysis suggests that the value of negative stiffness can act as an efficient controlling parameter capable of making one of the eigenfrequencies vanish and producing aperiodic eigenmodes. We proposed a realisation of a tuneable negative stiffness based on a rotating block, which under the combined action of tangential and compressive force exhibits negative stiffness proportional to the compressive force magnitude. Thus the compressive force can be used as a controlling factor to control the value of negative stiffness and the chain resonances.
Acknowledgement The authors acknowledge the financial support through ARC Discovery Grant DP120102434. References [1] N.G.W. Cook, The failure of rock, International Journal of Rock Mechanics and Mining Science 2 (1965) 389–403. [2] M.D.G. Salamon, Stability, instability and design of pillar workings, International Journal of Rock Mechanics and Mining Science 7 (1970) 613–631. [3] Y. Estrin, A.V. Dyskin, E. Pasternak, S. Schaare, S. Stanchits, A.J. Kanel-Belov, Negative stiffness of a layer with topologically interlocked elements, Scripta Materialia 50 (2003) 291–294. [4] S. Schaare, A.V. Dyskin, Y. Estrin, S. Arndt, E. Pasternak, A. Kanel-Belov, Point loading of assemblies of interlocked cube-shaped elements, International Journal of Engineering Sciences 46 (2008) 1228–1238 http://dx.doi.org/10.1016/j.ijengsci.2008.06.012. [5] R.S. Lakes, Extreme damping in composite materials with a negative stiffness phase, Physical Review Letters 86 (2001) 2897–2900. [6] R.S. Lakes, Extreme damping in compliant composites with a negative-stiffness phase, Philosophical Magazine Letters 81 (2001) 95–100. [7] R.S. Lakes, T. Lee, A. Bersie, Y.C. Wang, Extreme damping in composite materials with negative stiffness inclusions, Nature 410 (2001) 565–567. [8] R.S. Lakes, W.J. Drugan, Dramatically stiffer elastic composite materials due to a negative stiffness phase? Journal of Mechanics and Physics of Solids 50 (2002) 979–1009. [9] Y-C. Wang, R.S. Lakes, Stable extremely-high-damping discrete viscoelastic systems due to negative stiffness elements, Applied Physics Letters 84 (2004) 4451–4453. [10] A.V. Dyskin, E. Pasternak, Elastic composite with negative stiffness inclusions in antiplane strain, International Journal of Engineering Science 58 (2012) 45–56. [11] J.M.T. Thompson, G.W. Hunt, A General Theory of Elastic Stability, John Wiley & Sons, London, 1973. [12] A.R. Champneys, G.W. Hunt, J.M.T. Thompson, Localisation and Solitary Waves in Solid Mechanics, World Scientific, Singapore, 1984, 1–28. [13] G.W. Hunt, H.-B. Mühlhaus, A.I.M. Whiting, Folding processes and solitary waves in structural geology, in: A.R. Champneys, G.W. Hunt, J.M. T. Thompson (Eds.), Localisation and Solitary Waves in Solid Mechanics, World Scientific, Singapore1984, pp. 332–348. [14] C.-M. Lee, A.H. Bogatchenkov, V.N. Goverdovskiy, A.I. Temnikov, Vibration control system with opened kinematic chain and elastic links of small stiffness., Proceedings KORUS 2003, The 7th Korea–Russia International Symposium on Science and Technology 1 (2003) (2003) 285–289.
E. Pasternak et al. / Journal of Sound and Vibration 333 (2014) 6676–6687
6687
[15] W.J. Greenleaf, M.T. Woodside, E.A. Abbondanzieri, S.M. Block, Passive all-optical force clamp for high-resolution laser trapping, Physical Review Letters 95 (2005) 208102. [16] S-T. Park, T-T. Luu, Techniques for optimizing parameters of negative stiffness, Proceedings of the IMechE Vol. 221 Part C: J. Mechanical Engineering Science (2007) 505–511. [17] A. Carrella, M.J. Brennan, T.P. Waters, Demonstrator to show the effects of negative stiffness on the natural frequency of a simple oscillator, Proceedings of the IMechE Vol. 222 Part C: J. Mechanical Engineering Science (2008) 1189–1192. [18] C.-M. Lee, V.N. Goverdovskiy, Type synthesis of function-generating mechanisms for seat suspensions, International Journal of Automotive Technology 10 (2009) 37–48. [19] Z.P. Bažant, L. Cedolin, Stability of Structures: Elastic, Inelastic, Fracture, and Damage Theories, Oxford University Press, Oxford, 1991. [20] R.S. Lakes, P. Rosakis, A. Ruina, Microbuckling instability in elastomeric cellular solids, Journal of Material Science 28 (1993) 4667–4672. [21] H. Baughman, S. Stafström, C. Cui, S.O. Dantas, Materials with negative compressibilities in one or more dimensions, Science 279 (1998) 1522–1524. [22] V.R. Coluci, L.J. Hall, M.E. Kozlov, M. Zhang, S.O. Dantas, D.S. Galvão, R.H. Baughman, Modeling the auxetic transition for carbon nanotube sheets, Physical Review, B 78 (2008) 115408. [23] T. Jaglinski, D. Kochmann, D. Stone, R.S. Lakes, Composite materials with viscoelastic stiffness greater than diamond, Science 315 (2007) 620–622. [24] Z.G. Nicolaou, A.E. Motter, Mechanical metamaterials with negative compressibility transitions, Nature Materials (2012) 1–6, http://dx.doi.org/10.1038/ NMAT3331. [25] A.V. Dyskin, E. Pasternak, Mechanical effect of rotating non-spherical particles on failure in compression, Philosophical Magazine 92 (2012) 3451–3473. [26] P. Martin, A.D. Mehta, A.J. Hudspeth, Negative hair-bundle stiffness betrays a mechanism for mechanical amplification by the hair cell, Proceedings of the National Academy of Sciences of USA 97 (2000) 12026–12031. [27] L. Le Goff, D. Bozovic, A.J. Hudspeth, Adaptive shift in the domain of negative stiffness during spontaneous oscillation by hair bundles from the internal ear, PNAS 102 (2005) 16996–17001. (November 22, 2005). [28] G. Puglisi, L. Truskinovsky, Mechanics of a discrete chain with bi-stable elements, Journal of Mechanics and Physics of Solids 48 (2000) 1–27. [29] F. Cardin, M. Favretti, Dynamics of a chain of springs with non-convex potential energy, Mathematics and Mechanics of Solids 8 (2003) 651–668. [30] Y-C. Wang, Influences of negative stiffness on a two-dimensional hexagonal lattice cell, Philosophical Magazine 87 (2007) 3671–3688. [31] L. Dong, D.S. Stone, R.S. Lakes, Anelastic anomalies and negative Poisson's ratio in tetragonal BaTiO3 ceramics, Applied Physics Letters 96 (2010) 141904. [32] D.L. Platus, Negative-stiffness-mechanism vibration isolation systems, Proceedings of the SPIE Conference on Current Developments in Vibration Control 98 for Optomechanical Systems, 3786, SPIE, Denver, Colorado, 1999, 98–105 (July 1999). [33] Y-C. Wang, M. Ludwigson, R.S. Lakes, Deformation of extreme viscoelastic metals and composites, Materials Science and Engineering, A 370 (2004) 41–49. [34] Z Yang, H.M. Dai, N.H. Chan, G.C. Ma, P. Sheng, Acoustic metamaterial panels for sound attenuation in the 50–1000 Hz regime, Applied Physics Letters 96 (2010) 041906. [35] W.J. Drugan, Elastic composite materials having a negative-stiffness phase can be stable, Physical Review Letters 98 (2007) 055502. [36] D.M. Kochmann, W.J. Drugan, Dynamic stability analysis of an elastic composite material having a negative-stiffness phase, Journal of Mechanics and Physics of Solids 57 (2009) 1122–1138. [37] R.L. Salganik, Mechanics of bodies with many cracks, Mechanics of Solids 8 (1973) 135–143. [38] Z. Liu, X. Zhang, Y. Mao, Y.Y. Zhu, Locally resonant sonic materials, Science 289 (2000) 1734–1736. [39] N. Fang, D. Xi, J. Xu, M. Ambati, W. Srituravanich, C. Sun, X. Zhang, Ultrasonic metamaterials with negative modulus, Nature 5 (2006) 452–456. [40] S. Guenneau, A. Movchan, G. Pétursson, S.A. Ramakrishna, Acoustic metamaterials for sound focusing and confinement, New Journal of Physics 9 (2007) 399. [41] S.H. Lee, C.M. Park, Y.M. Seo, Acoustic metamaterial with negative modulus, Journal of Physics: Condensed Matter 21 (2009) 175704. [42] Y. Lai, Y. Wu, P. Sheng, Z.-Q. Zhang, Hybrid elastic solids, Nature Materials 10 (2011) 620–624, http://dx.doi.org/10.1038/NMAT3043. [43] Z. Liang, J. Li, Extreme acoustic metamaterial by coiling up space, Physical Review Letters 108 (2012) 114301. [44] A.-C. Hladky-Hennion, J.O. Vasseur, G. Haw, C. Croënne, L. Haumesser, A.N. Norris, Negative refraction of acoustic waves using a foam-like metallic structure, Applied Physics Letters 102 (2013) 144103. [45] K. Bi, G. Dong, X. Fu, J. Zhou, Ferrite based metamaterials with thermo-tunable negative refractive index, Applied Physics Letters 103 (2013) 131915. [46] Y. Ding, Z. Liu, C. Qiu, J. Shi, Metamaterial with simultaneously negative bulk modulus and mass density, Physical Review Letters 99 (2007) 093904. [47] Z. Yang, J. Mei, M. Yang, N. Chan, P. Sheng, Membrane-type acoustic metamaterial with negative dynamic mass, Physical Review Letters 101 (2008) 204301. [48] X. Zhou, G. Hu, Analytic model of elastic metamaterials with local resonances, Physical Review B 79 (2009) 195109. [49] E.P. Calius, X. Bremaud, B. Smith, A. Hall, Negative mass sound shielding structures: Early results, Physica Status Solidi B 246 (2009) 2089–2097. [50] X.N. Liu, G.K. Hu, G.L. Huang, C.T. Sun, An elastic metamaterial with simultaneously negative mass density and bulk modulus, Applied Physics Letters 98 (2011) 251907. [51] M.T. Islam, G. Newaz, Metamaterial with mass-stem array in acoustic cavity, Applied Physics Letters 100 (2012) 011904. [52] K.T. Tan, H.H. Huang, C.T. Sun, Optimizing the band gap of effective mass negativity in acoustic metamaterials, Applied Physics Letters 101 (2012) 241902. [53] X. Cai, Q. Guo, G. Hu, J. Yang, Particle focusing in a microchannel with acoustic metafluid, Applied Physics Letters 103 (2013) 031901. [54] H.J. Pain, The Physics of Vibrations and Waves, John Wiley & Sons Ltd; Chichester, England, 2005. [55] L.D. Landau, E.M. Lifshitz, Course of Theoretical Physics. Volume 1. Mechanics, 3rd ed. Butterworth-Heinemann; Oxford, England, 1976. [56] A. Carrella, M.J. Brennan, T.P. Waters, Demonstrator to show the effects of negative stiffness on the natural frequency of a simple oscillator, Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 222 (2008) 1189–1192. [57] A.V. Dyskin, E. Pasternak, Rock mass instability caused by incipient block rotation, in: Qihu Qian, Yingxin Zhou (Eds.), Harmonising Rock Engineering and the Environment, Proceedings of the 12th International Congress on Rock Mechanics, CRC Press, Balkema2012, p. 4.
Contents lists available at ScienceDirect
Journal of Sound and Vibration journal homepage: www.elsevier.com/locate/jsvi
Chains of oscillators with negative stiffness elements Elena Pasternak a, Arcady V. Dyskin b,n, Greg Sevel c,1 a
School of Mechanical and Chemical Engineering, The University of Western Australia, 35 Stirling Highway, Crawley WA 6009, Australia School of Civil, Environmental and Mining Engineering, The University of Western Australia, 35 Stirling Highway, Crawley WA 6009, Australia c 10 Nila Street, Wembley Downs 6019, WA, Australia b
a r t i c l e in f o
abstract
Article history: Received 10 November 2013 Received in revised form 26 June 2014 Accepted 29 June 2014 Handling Editor: L.N. Virgin Available online 6 August 2014
Negative stiffness is not allowed by thermodynamics and hence materials and systems whose global behaviour exhibits negative stiffness are unstable. However the stability is possible when these materials/systems are elements of a larger system sufficiently stiff to stabilise the negative stiffness elements. In order to investigate the effect of stabilisation we analyse oscillations in a chain of n linear oscillators (masses and springs connected in series) when some of the springs' stiffnesses can assume negative values. The ends of the chain are fixed. We formulated the necessary stability condition: only one spring in the chain can have negative stiffness. Furthermore, the value of negative stiffness cannot exceed a certain critical value that depends upon the (positive) stiffnesses of other springs. At the critical negative stiffness the system develops an eigenmode with vanishing frequency. In systems with viscous damping vanishing of an eigenfrequency does not yet lead to instability. Further increase in the value of negative stiffness leads to the appearance of aperiodic eigenmodes even with light damping. At the critical negative stiffness the low dissipative mode becomes non-dissipative, while for the high dissipative mode the damping coefficient becomes as twice as high as the damping coefficient of the system. A special element with controllable negative stiffness is suggested for designing hybrid materials whose stiffness and hence the dynamic behaviour is controlled by the magnitude of applied compressive force. & 2014 Elsevier Ltd. All rights reserved.
1. Introduction The force–displacement response of materials and structures, at least in their stable state is characterised by positive stiffness whereby an increase in displacement results in the corresponding increase in the reaction force. In this case the work done leads to increase in the system's potential energy. However, in some circumstances when the system achieves unstable state (e.g. post-buckling regime in columns or shells or post-peak softening stage of the loading of brittle materials and rocks) it exhibits negative stiffness (at least incrementally) such that the potential energy formally receives negative increments under loading. Of course, such a system is unstable: maintaining the stable state and further loading are only possible in the presence of a stabilising device, either a sufficiently stiff loading frame (e.g., [1–4]) or a matrix with conventional positive stiffness which the negative stiffness elements are immersed into [5,6]. This device (matrix) increases its own potential energy in the process and thus maintains the energy balance.
n
Corresponding author. E-mail addresses: [email protected] (E. Pasternak), [email protected] (A.V. Dyskin), [email protected] (G. Sevel). 1 Participated in this project while on employ at UWA.
http://dx.doi.org/10.1016/j.jsv.2014.06.045 0022-460X/& 2014 Elsevier Ltd. All rights reserved.
E. Pasternak et al. / Journal of Sound and Vibration 333 (2014) 6676–6687
6677
There are many examples of systems that exhibit negative stiffness as a structural or effective property. In particular discrete systems of elastic springs, arches, link models as well as link and lever models with elastic springs are known to exhibit negative stiffness in the unloading part of the loading curve (e.g., [9,11–18]). These also include tubes and columns pre-buckled to an S-shaped configuration [6,19], single foam cells [20], ‘wine-rack’ [21] and zig-zag [22] structures, platelike interlocking structures of cubic elements [3,4] and materials with phase transitions (e.g., [7,23,24]). The effect of apparent negative stiffness can be produced by rotating non-spherical grains [25] (This mechanism of negative stiffness is explained below in Section 5.). In life nature, the negative stiffness regime was observed in the mechanical response of a hair cell of the bullfrog's sacculus, which is believed to be responsible for amplification of its sensitivity [26,27]. Negative stiffness elements also provide the simplest models of the effect of unstable regions created by nonlinear springs with non-convex energy [28,29]. The behaviour of systems made of conventional elements that under some conditions exhibit negative stiffness can be (and has been) analysed by the conventional means of structural mechanics. Thus, Wang [30] modelled the effect of a negative stiffness inclusion within a positive stiffness matrix by considering a double hexagonal structure in which external hexagon is made of standard elements, while the internal hexagon is made of negative stiffness elements. He found that depending on the value of the negative stiffness and the loading conditions, there exist zones of increased effective (positive) stiffness followed by abrupt transition to negative effective stiffness. Continuum analysis of the behaviour of systems with negative stiffness elements and especially of materials with negative stiffness inclusions is commonly performed employing standard methods of mechanics of solids with the signs of the stiffnesses (moduli) reversed. Thus, Lakes [5,6] and Lakes et al. [7] used the Voigt–Reuss and Hashin–Strikman bounds by replacing the positive moduli with the corresponding negative ones to infer that the presence of negative stiffness inclusions makes the material stiffer. Lakes and Drugan [8] came to the same conclusion by considering a matrix with negative stiffness spherical inclusions in dilute concentrations. Lakes et al. [7] report the results of experiments with composite materials consisted of inclusions of vanadium dioxide (ferroelastic material with transformation from monoclinic to tetragonal at 67 1C) in a pure (99.99 per cent) tin matrix. Despite the dilute concentration of inclusions (1 per cent volumetric fraction) the composite is observed to produce considerable effect on the mechanical damping (the loss tangent) and the stiffness, namely drop and then rise of the compliance at 65.5–66 1C as well as rise and then considerable drop in damping in the same temperature interval and then a smaller peak in damping at 67 1C. Anomalous fluctuations with temperature in the Young's modulus and loss tangent at very low frequencies (0.1 Hz) as well as negative Poisson's ratio in tetragonal BaTiO3 after ageing under bending were further reported in [31]; however no extremely high stiffness was found. Furthermore, in the applications of systems with negative stiffness elements to achieve the sound attenuation and isolation it is understood that the negative stiffness elements are responsible for reducing the overall stiffness of the system and hence reducing the resonant frequency below the excitation thus avoiding resonances (e.g. [18]). The use of negative stiffness elements in increasing damping and energy dissipation was discussed in [9,32–34]. In general, as the presence of the negative stiffness in a composite phase formally violates the condition of positive definiteness of elastic energy, it was necessary to analyse in which cases the phase with positive definite energy could stabilise the composite. Drugan [35] proved that a cylinder or sphere made from a material whose tensor of elastic moduli is strongly elliptic (a certain range of negative Young's and bulk moduli is permitted) can be stabilised by a thin coating made of a conventional material as long as it is sufficiently stiff. Kochmann and Drugan [36] extended the analysis of the dynamic stability to any thickness of the coating in the case of cylindrical inclusion. In particular, for the cylinder in an infinite matrix a simple criterion of stability was obtained: μII/μI 4 1 λI/μI, where λ and μ are the Lame constants and the superscripts I and II refer to the material of the cylinder and the matrix respectively. It is interesting, that in the case of negative shear modulus, μI ¼ mμII, this criterion reduces to m4 1þλI/μII, that is the stability requires high values of negative modulus. Dyskin and Pasternak [10,25] considered effective elastic moduli of a composite consisting of a conventional matrix filled with inclusions having negative shear modulus: the cylindrical inclusions in antiplane strain conditions, as well as spherical and crack-like inclusions with negative shear modulus whose bulk modulus was the same as that of the matrix. It was shown that while the presence of inclusions with negative moduli might make the solution of the full elastic problem not unique, the effective moduli – the moduli that relate stress and strain averaged over the volume elements macroscopic with respect to the inclusions – are unique. It was also shown, using the differential self-consistent method (e.g. [37]) that the inclusions with negative shear modulus could both increase and decrease the effective moduli (as compared to those of the matrix) depending upon the value of the negative modulus. In both cases however the effective shear modulus remains positive as long as the concentration (volumetric fraction) of the inclusion is below a certain critical value. After reaching the critical value the effective shear modulus abruptly becomes negative and the composite loses stability. The critical concentration depends upon the value of negative stiffness. Furthermore, there exists a value of the negative stiffness when the critical concentration becomes zero meaning that such a composite is unstable, no matter how few inclusions it contains. Interestingly, for higher values of the negative shear modulus the critical concentration becomes non-zero again and grows as the value of the negative stiffness increases thus producing a new region of stability. The condition of the existence of this second region of stability – the high negative shear modulus – qualitatively corresponds to the stability condition obtained by Kochmann and Drugan [36].
6678
E. Pasternak et al. / Journal of Sound and Vibration 333 (2014) 6676–6687
A considerable body of research was devoted to studying the sonic or acoustic metamaterials. Some authors interpret the existence of the frequency gaps in these materials as wave reflection (the presence of negative refraction index) associated with negative effective dynamic stiffness or modulus (e.g., [38–45]) or, by association, with negative density or mass (e.g., [46–53]). In particular, the effective dynamic modulus or stiffness is defined in this interpretation as seff ¼ F(t)/x(t), where F(t) is the (harmonic) force applied and x(t) is the displacement (or stress and strain in the case of effective modulus). We note however that the negative effective stiffness defined in such a way can be found even in the ordinary linear oscillator (e.g., mass on a conventional spring). Indeed, consider a forced oscillator excited by a harmonic driving force: mx€ þ r x_ þ sx ¼ FðtÞ; where FðtÞ ¼ F 0 expðiωtÞ, m is the mass, r is resistance, s is the spring stiffness and F0 is the amplitude of the applied force. The steady-state solution for light damping reads (e.g. [54]): xðtÞ ¼ iF 0 ðωZ m Þ 1 expðωt þϕÞ, where Zm ¼ r þ iðωm s=ωÞ ¼ Z m expðiϕÞ. Then the frequency-dependent ‘effective’ stiffness defined as above as F(t)/x(t) can be expressed as seff ðωÞ ¼ FðtÞ=xðtÞ ¼ ðω2 m sÞ iωr ¼ m½ðω2 ω20 Þ þ iωr=m. Here ω0 ¼ ðs=mÞ1=2 is the eigenfrequency. Thus, when the frequency ω⪢ω0 the stiffness formally becomes negative, even if the spring stiffness, s, is positive. This negative stiffness is of course the effect of the phase change (e.g. [54]) – the fact, which is recognised in some works on metamaterials (e.g. [34]). In this paper we concentrate on investigating the stability of dynamic systems (that the trajectories are bounded) whose elements can exhibit true negative stiffness (modulus) i.e. negative stiffness in static loading. In order to analyse the stability we consider a very simple system – a chain of standard linear oscillators (equal masses, m, attached to springs) connected in series, Fig. 1. We assume that some springs can have negative stiffness. We presume that the system is linear that is the stiffnesses are independent of displacements. The nonlinear effects and snapback phenomena are therefore not considered. We investigate the stability of such a system first using the global (effective) stiffness (Section 2) and then conduct the complete stability analysis (Section 3). In the latter case we consider both conservative systems and systems with viscous damping. In the dynamic case we assume fixed boundaries at both ends of the chain. In Section 4 we generalise the stability analysis to a simple 2D case of square lattice with longitudinal springs. Section 5 discusses the obtained results and offers an outlook. 2. Statics of chains with positive and negative stiffness springs. Effective stiffness Consider a chain of n springs with stiffnesses k1,.., kn, Fig. 1. Since the force in each spring must be the same, f, the 1 displacement of (the last) mass n is un ¼ ∑ni¼ 1 f ki . Therefore, the overall (effective) stiffness, keff, reads n
1
1
keff ¼ ∑ ki
(1)
i¼1
By definition the effective stiffness is a stiffness of an equivalent continuous material (a 1D material in this case). Thermodynamically, such an equivalent material can only be stable if its stiffness is positive. Therefore, the stability condition reads keff 40. This condition would permit some springs (but not all of them) to have negative stiffness. This is however not a sufficient condition of stability as the chain of springs can still be unstable as demonstrated in Section 3. Consider now a particular case, when there are m negative stiffness springs with the same negative stiffness k- and n–m springs with the same positive stiffness k0. Then 1
1
1
keff ¼ mk þ ðn mÞk0
(2)
We will treat the positive stiffness springs as ‘matrix’ and negative stiffness springs as ‘inclusions’. We introduce the stiffness of the matrix (the stiffness in the absence of the inclusions), kmtx ¼k0/n. In order to make the analysis independent of the number of springs, we introduce the concentration of negative stiffness springs c ¼m/n; 0rc r1. Finally, we introduce the dimensionless value of negative stiffness κ ¼ k /k0. Then the effective stiffness can be expressed as keff ¼ kmtx ð1 c=ccr Þ 1
(3)
1
where ccr ¼κ(1 þκ) is the critical concentration starting from which the effective stiffness becomes negative and the chain loses stability. Fig. 2 shows the dependence of the effective stiffness upon the concentration of the negative stiffness phase (3). It is seen that (a) the negative stiffness springs do make the chain stiffer and (b) upon reaching the critical concentration, the effective stiffness abruptly drops to a negative value. This is similar to the stiffening effect of negative stiffness inclusions (e.g., [5, 10, 25]). The necessary condition of stability, co ccr can also be rewritten in terms of the dimensionless value of negative stiffness as κ 4κ cr ¼
c 1c
(4)
Fig. 1. A chain of oscillators consisting of n masses connected by springs with stiffnesses k1,.., kn. For the sake of simplicity all masses are assumed to be the same, equal to m. Under applied force f, the i-th mass assumes displacement ui.
E. Pasternak et al. / Journal of Sound and Vibration 333 (2014) 6676–6687
6679
Fig. 2. Dependence of the normalised effective stiffness of the chain upon the concentration of the negative stiffness phase.
Fig. 3. System of n 1 equal masses (m) connected to the stationary masses 0 and n (boundaries) by n springs.
Thus, in this example the stability requires the values of negative stiffness κ to exceed a certain critical value, κcr. 3. Stability of chain of oscillators with positive and negative stiffnesses 3.1. Conservative system We now investigate the stability of the chain of n 1 masses connected by the springs, Fig. 3. We write the general system and assume the boundary conditions according to which the end masses (mass 0 and mass n) are stationary, that is u0 ðtÞ ¼ u_ 0 ðtÞ ¼ un ðtÞ ¼ u_ n ðtÞ ¼ 0;
(5)
where the dot denotes the time derivative. We write Lagrange equations d ∂L ∂L ¼ f i; dt ∂q_ i ∂qi
i ¼ 0; ::; n
(6)
where fi are the forces acting on each mass and the Lagrangian is mu_ i 1 n 1 ∑ ½ki ðui ui 1 Þ2 þ ki þ 1 ðui þ 1 ui Þ2 : 2i¼1 i¼0 2 n
L ¼ T Π ¼ ∑
(7)
Using (7) we obtain the following system of differential equations for the chain: mu€ i þðki þki þ 1 Þui ki ui 1 ki þ 1 ui þ 1 ¼ f i ;
i ¼ 1; …n 1
(8)
The criterion of stability is the positive definiteness of the potential energy Π¼
i 1 n 1h ∑ ki ðui ui 1 Þ2 þ ki þ 1 ðui þ 1 ui Þ2 2i¼1
(9)
6680
E. Pasternak et al. / Journal of Sound and Vibration 333 (2014) 6676–6687
Positive definiteness of (9) is equivalent to the positive definiteness of the matrix of stiffnesses 0 1 k1 þk2 k2 B C k2 þk3 k3 B k2 C B C B C ⋱ ⋱ ⋱ B C An ¼ B C ki ki þki þ 1 ki þ 1 B C B C B C ⋱ ⋱ @ A kn 1 kn 1 þ kn
(10)
where the empty spaces refer to zero entries. The condition of the positive definiteness of An is that all main diagonal minors of An are positive: 2 3 a1;1 a1;2 ⋯ a1;i 6a 7 6 2;1 a2;2 ⋯ a2;i 7 7 40 (11) Δi ¼ det6 6 ⋮ ⋮ ⋱ ⋮ 7 4 5 ai;1 ai;2 ⋯ ai;i Substituting (10) into (11) leads to the following condition: n
n
∑ ∏ kj 40
(12)
i¼1j¼1 jai
This is obviously the necessary and sufficient condition of stability. These rules are cumulative, meaning that as additional oscillators are added to the system, all of the old rules continue to govern the system together with the newly introduced rules. These rules also allow renumbering, as can be shown by shifting rows and columns in A. For example, the first rule k1 þk2 4 0 becomes ki þki þ 1 40 8 i when there are more than two oscillators in the system. As an illustration, Fig. 4 shows a simplified system containing only two oscillators (two masses on 3 springs). Fig. 5 shows the stable regions for the combinations of stiffnesses. It is seen that the system can be dynamically stable even when the stiffness of one of the springs is negative. We now prove another necessary condition of stability: for any system of n oscillators in a 1-D chain to be stable, only one stiffness ki is allowed to be negative. Indeed, for a single mass, we have the rule, k1 þk2 40 therefore only one stiffness can be negative. Suppose for n 1 springs only one spring can have negative stiffness. From (12) we have kn 4
k1 k2 …kn 1 ; Δn 2
n 42
(13)
where, according to (11) Δn 2 4 0. Therefore, if there is already negative stiffness spring amongst n 1 springs then k1 k2 …kn 1 o 0 ) kn 4 0, that is the n-th spring must have positive stiffness. Alternatively, if all n 1 springs have positive stiffness, then the n-th spring is allowed to be negative. Obviously, the maximum value of the negative stiffness increases with the increase in the values of the positive stiffnesses as illustrated in Fig. 5. Therefore, if we have n springs and only one is allowed to have negative stiffness then the critical concentration of negative stiffness springs when the system can still be stable is ccr ¼
1 n
(14)
The condition of stability (12) is stronger than the one formulated in the previous section, keff 40, at least in that that only one spring is allowed to be negative. This shows that condition keff 40, which allows a number of springs to be negative, is only a necessary condition of stability. Further insight into the behaviour of the systems with negative stiffness springs can be obtained by considering the simplified two mass system, which is a coupled oscillator shown in Fig. 4. The governing equations read ( mu€ 1 þ ðk1 þ k2 Þu1 k2 u2 ¼ f 1 (15) mu€ 2 þ ðk2 þ k3 Þu2 k2 u1 ¼ f 2
Fig. 4. A coupled oscillator consisting of two masses connected by three springs.
E. Pasternak et al. / Journal of Sound and Vibration 333 (2014) 6676–6687
6681
Stable
Unstable
Fig. 5. The regions of stability and instability for system (8) with boundary conditions (5) for n¼ 3.
The homogeneous system has the following set of fundamental solutions: ! u1 ¼ e1 ½C 1þ expðiω1 tÞ þC 1 expð iω1 tÞ þ e2 ½C 2þ expðiω2 tÞ þ C 2 expð iω2 tÞ u2 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u u 2 tðk1 þ 2k2 þ k3 Þ þ ðk1 k3 Þ2 þ4k2 ; ω1 ¼ 2m vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u u 2 tðk1 þ 2k2 þ k3 Þ ðk1 k3 Þ2 þ 4k2 ω2 ¼ 2m
(16)
where e1 and e2 are eigenvectors of the matrix A2 of system (15) and C 17 ; C 27 are constants. The stability is ensured when k1 þk2 4 0;
k1 k2 þk1 k3 þk2 k3 40
(17)
Fig. 6 shows dependence of the eigenfrequencies, ω1, ω2 for different stiffnesses reaching negative values. We assume that after due normalisation, m ¼1 and k1 ¼1. It is seen that as the negative stiffness reaches its minimum value, frequency ω2 tends to zero. Another feature is that the negative stiffness does reduce the eigenfrequencies. It can be interpreted that the negative stiffness can only reduce the dynamic effective stiffness. This is opposite to the static effective stiffness, which as shown in the previous section, is increased by the presence of negative stiffness springs. This apparent contradiction will be resolved in Section 5. As soon as the negative stiffness exceeds its critical value, frequency ω2 becomes imaginary and exponentially divergent terms appear in the solution, which indicates the overall instability of the system. Fig. 7 shows the trajectories of both masses in the case of absence of the driving force (f1 ¼ f2 ¼0) and for k1 ¼ k3 ¼1 and different values of the central spring stiffness k2. The initial conditions are u1 ð0Þ ¼ u_ 1 ð0Þ ¼ u_ 2 ð0Þ ¼ 0; u2 ð0Þ ¼ 1. These trajectories are combinations of oscillations with two eigenfrequencies each depending differently on the stiffness of the central spring. In particular, according to Fig. 6a the first eigenfrequency reduces when k2 reduces from positive values to zero and becomes 1 when k2 becomes negative. Opposite to this the second eigenfrequency keeps equal to 1 when k2 is positive and reduces to zero when k2 is negative. Subsequently at all values of the central spring stiffness at least one eigenfrequency is always equal to one. This circumstance leads to complex dependence of the trajectories of both masses on the stiffness of the central spring shown in Fig. 7. 3.2. System with light linear damping When damping is introduced to the system, the Lagrange equations assume the dissipative term as follows (e.g., [55]) d ∂L ∂L ∂F ¼ f i ; i ¼ 1; ::; n (18) dt ∂q_ i ∂qi ∂q_ i
6682
E. Pasternak et al. / Journal of Sound and Vibration 333 (2014) 6676–6687
Fig. 6. Dependence of eigenfrequencies for a coupled oscillator, Fig. 4, upon different values of spring stiffnesses: (a) cases k1 ¼k2 ¼ 1 and k1 ¼ k3 ¼ 1, the minimum value of negative stiffness is 0.5; (b) cases k1 ¼ 1, k2 ¼ 3 and k1 ¼ 1, k3 ¼3, the minimum value of negative stiffness is 0.75.
Fig. 7. Trajectories of the coupled oscillator, Fig. 4, in the absence of the driving force for different values of the central spring stiffnesses: k1 ¼k3 ¼1, k2 ¼ 0.5, 0.1, 1: (a) the movement of ball 1; (b) the movement of ball 2. The initial conditions are u1 ð0Þ ¼ u_ 1 ð0Þ ¼ u_ 2 ð0Þ ¼ 0; u2 ð0Þ ¼ 1.
where 1 F ¼ ∑αik u_ i u_ k 2 i;k
i; k ¼ 1; ::; n
(19)
and αij form a symmetric matrix of damping coefficients, αij ¼αji. The stability of such a system is determined by matrix (10), which leaves the previous conclusion unchanged: the stable system can only have one negative stiffness spring. Further analysis will be performed assuming, for the sake of simplicity
E. Pasternak et al. / Journal of Sound and Vibration 333 (2014) 6676–6687
6683
that αik ¼ αδij , where δij is the Kronecker delta. The corresponding system of equations reads mu€ i þ αu_ i þ ðki þ ki þ 1 Þui ki ui 1 ki þ 1 ui þ 1 ¼ f i ;
i ¼ 1; …n 1
Consider, as an example a two mass oscillator from Fig. 4. ( mu€ 1 þ αu_ 1 þðk1 þk2 Þu1 k2 u2 ¼ f 1 mu€ 2 þ αu_ 2 þðk2 þk3 Þu2 k2 u1 ¼ f 2 where e1 and e2 are eigenvectors of the matrix A2 of system (21). The system has the following general solution: ! h α α i u1 t þ iω1α t þC 1 exp t iω1α t ¼ e1 C 1þ exp u2 2m 2m h α α i þ t þiω2α t þC 2 exp t iω2α t þ e2 C 2 exp 2m 2m qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 ðk þ 2k þ k Þ þ ðk k Þ þ 4k 1 2 3 1 3 α2 2 ; ω21;α ¼ 2m 4m2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ðk1 þ 2k2 þ k3 Þ ðk1 k3 Þ2 þ 4k2 α2 ω22;α ¼ 2m 4m2
(20)
(21)
(22)
here e1 and e2 are eigenvectors of the matrix A2 of system (15) and C 17 ; C 27 are constants. It can be seen that as long as conditions (17) satisfy, even if the eigenfrequencies become imaginary, the fundamental solutions still have negative coefficient at t, so the solutions are not increasing and the system is stable albeit with aperiodic modes. In other words, as long as the eigenfrequencies are real the eigenmodes are periodic and the damping coefficient is still α. When an eigenfrequency is imaginary, the corresponding eigenmode becomes aperiodic (the frequency is zero) and the damping coefficient changes. In order to investigate this further we introduce the effective dumping coefficients and the corresponding frequencies qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8 8 0; ω2j;α ðαÞ o0 < α 8 2m ω2j;α ðαÞ; ω2j;α ðαÞ o 0 < 7 n αj ¼ ; ωj ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffi ; j ¼ 1; 2 (23) : : ω2j;α ðαÞ; ω2j;α ðαÞ Z0 α; ω2j;α ðαÞ Z 0 Fig. 8 shows the dependence upon stiffnesses of the normalised frequency ωn2 =ω2 of the system with damping where ω2 is the eigenfrequency of the system without damping, and the relative effective damping coefficients α27 =α. Frequency ωn1 does not become imaginary for the values of stiffnesses keeping the system stable and the corresponding damping is unchanged. We again used the normalisation that leads to m ¼1 and k1 ¼1. It is seen that the negative stiffness element decreases the resonant frequencies but can both increase and decrease the damping. Also, according to (22) and (23) as ω2 -0; α2þ -2α; α2 -0. Thus in this simple system, depending upon the initial conditions, the damping coefficient can be either increased 2 times or made vanish by using a near critical value of negative stiffness. It is also seen that qualitatively the behaviour of the frequencies and the effective damping coefficients is similar for k1 ¼ k3 ¼ 1; k2 ¼ k and k1 ¼ k2 ¼ 1; k3 ¼ k. This result can easily be generalised to chains of n springs. Indeed, by transferring system (20) to principal directions (eigenvectors of matrix An), e1, e2,…, en, one can write each mode of oscillations as exp( iλjt), j¼1,…n, where: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi α 7 α2 4m2 ω2j (24) λj ¼ 2m here ωj is the eigenfrequency of eigenmode j without damping. Obviously, the boundary of stability of the system corresponds to vanishing of the determinant of matrix An, Eq. (10). This corresponds to vanishing of one of the eigenfrequencies; let it be ωj. Substitution of ωj ¼0 into (24) leads to aperiodic eigenmodes with damping coefficients either 2α or 0. 4. Directionally decoupled lattices The obtained results allow a simple generalisation to higher dimensions assuming that displacements in different directions are not coupled. Fig. 9 shows an example of a 2D square lattice. We assume that the displacements of mass (i,j), uij and vij do not interact such that dynamics of the lattice is reduced to independent oscillations in the horizontal and vertical directions. For n by n lattice there are n vertical and n horizontal independently oscillating chains. Each chain, in order to be stable, must have not more than one negative stiffness spring. From here the critical concentration of negative stiffness springs when the lattice can still be stable is ccr ð2Þ ¼
2 n
(25)
6684
E. Pasternak et al. / Journal of Sound and Vibration 333 (2014) 6676–6687
Fig. 8. Dependence of the normalised values of frequency, ωn2 =ω2 , and the effective damping coefficient α27 =α upon normalised stiffnesses of the springs: (a) k1 ¼ k3 ¼ 1; k2 ¼ k, (b) k1 ¼ k2 ¼ 1; k3 ¼ k.
A straightforward generalisation to p dimensions gives ccr ðpÞ ¼
p np 1
(26)
Obviously, the previous conclusions about changing the frequency and damping in some modes remain.
5. Discussion and outlook The results of our analysis show that dynamic systems with negative stiffness elements can, under certain restrictions be stable. In particular if the system consists of independent one-dimensional chains, each chain can have one spring with negative stiffness given that the value of the stiffness does not exceed a certain critical value determined by the number and stiffnesses of the positive stiffness springs. With increase of the value of negative stiffness the eigenfrequencies decrease. Furthermore, as the negative stiffness approaches its critical value there exist eigenmodes with vanishing frequencies. The decrease of eigenfrequencies caused by the negative stiffness springs is in apparent contradiction with the result of Section 2: the effective stiffness in the stable region increases with the increase of the negative stiffness, Fig. 2. In order to understand the effect let us consider another static model, Fig. 10. In this model all springs are connected in parallel by rigid plates. In this case the effective stiffness is n
keff ¼ ∑ ki
(27)
i¼1
where ki are the stiffnesses of individual springs. It is clear that in this configuration even a single negative spring reduces the effective stiffness, the more the higher the value of negative stiffness. Therefore the behaviour of the above chains of oscillators, which are oscillators connected in series, resembles the behaviour of a system of springs connected in parallel. The nature of this resemblance becomes transparent when one recalls that the rigidity of the plates connecting the parallel springs, Fig. 10, makes the springs independent. Similarly the chain of oscillators being linear allows decomposition into independent oscillators whose oscillations correspond to the eigenmodes. It seems that the independence of springs or eigenmode oscillators is behind the similarity of the effect of negative stiffness in chains of oscillators connected in series and springs connected in parallel. The presence of the critical value of negative stiffness at which one of the eigenfrequencies becomes zero suggests a way of controlling the resonance frequency by changing the negative stiffness value. The most important thing is the degree of control achievable as the resonant frequency can, in principle, be brought to zero. It is interesting that such a tuning mechanism is likely to be realised in life nature; according to [26,27] the hair bundles in the ears have a negative stiffness region in the force–displacement curve. Furthermore, the positions of the region and, as followed from the plots presented in these papers the values of negative stiffness can be controlled by offsetting the hair bundle. The strong dependence of some resonant frequencies upon the value of negative stiffness opens up an avenue of designing highly tunable hybrid materials with negative stiffness elements. What is needed is a tuneable negative stiffness element. Tuneable negative stiffness elements based on specially arranged levers and springs are known (e.g. [56]), but the tuning requires changing the design, for instance the arms of the lever.
E. Pasternak et al. / Journal of Sound and Vibration 333 (2014) 6676–6687
6685
Fig. 9. An example of a square lattice (for the sake of illustration a 6 6 lattice is shown).
k1 k2
f
kn Fig. 10. System of springs of stiffnesses k1, k2,…,kn connected in parallel.
Here, we propose to use a negative stiffness element whose negative stiffness value depends upon the compression force applied to it and thus easily tuneable. For such an element we use the effect of apparent negative stiffness produced by rotating non-spherical disks suggested by Dyskin and Pasternak [25,57]. Such an element is shown in Fig. 11a. The moment equilibrium with respect to point O reads qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 1 ; δ¼ (28) T δ2 ð1 ξÞ2 ¼ Pð1 ξÞ; ξ ¼ d sin α sin α This relationship is shown in Fig. 11b. For small displacements, ξ⪡1, developing T as a function of ξ into Taylor series and keeping only linear term we obtain the linear stress–strain relationship with negative modulus between shear stress T and shear strain γ¼u/d ¼ξsinα T ¼ ku;
k¼
P δ3 2 d ðδ 1Þ3=2
(29)
Two things are apparent from this equation. Firstly, the relationship is linear (and reversible) and stiffness is negative, k o0. Secondly, the stiffness depends upon the compressive force P. Thus the compressive force can be used to tune the value of negative stiffness. Since the force can be applied using an actuator, the resonant frequency of the oscillator can be easily controlled and the control can even be exercised remotely. 6. Conclusion Negative stiffness, as well known, is prohibited by thermodynamics; subsequently materials or systems characterised by global negative stiffness cannot be stable. However negative stiffness is thermodynamically possible at the level of elements
6686
E. Pasternak et al. / Journal of Sound and Vibration 333 (2014) 6676–6687
Fig. 11. A tuneable negative stiffness element: (a) rotating block; (b) the force–displacement relationship is descending and reversible. Infinitesimally it is linear characterised by negative stiffness.
of a system or parts of a material as long as the positive stiffness system (material) containing the negative stiffness elements is stiff sufficient to stabilise them. This conclusion was illustrated in this paper by considering examples of chains of springs and masses with fixed ends. We showed that chains of springs and masses connected in series as well as systems made of independently oscillating chains containing springs with negative stiffness could still exhibit stable motion. The necessary criterion of stability stipulates that there should be no more than a single negative stiffness spring per chain. To make this condition also sufficient, the value of the stiffness should not exceed a certain critical value determined by the number and stiffnesses of the positive stiffness springs. As the negative stiffness approaches its critical value, one of the eigenfrequency vanishes, which means the appearance of very slow periodic motion. In systems with light viscous damping vanishing of an eigenfrequency does not yet lead to instability. Further increase in the value of negative stiffness leads to the appearance of aperiodic eigenmodes with light and high damping. At the critical negative stiffness the low dissipative mode becomes non-dissipative, while in the high dissipative mode the damping coefficient becomes as twice as high as the damping coefficient of the system. The presented analysis suggests that the value of negative stiffness can act as an efficient controlling parameter capable of making one of the eigenfrequencies vanish and producing aperiodic eigenmodes. We proposed a realisation of a tuneable negative stiffness based on a rotating block, which under the combined action of tangential and compressive force exhibits negative stiffness proportional to the compressive force magnitude. Thus the compressive force can be used as a controlling factor to control the value of negative stiffness and the chain resonances.
Acknowledgement The authors acknowledge the financial support through ARC Discovery Grant DP120102434. References [1] N.G.W. Cook, The failure of rock, International Journal of Rock Mechanics and Mining Science 2 (1965) 389–403. [2] M.D.G. Salamon, Stability, instability and design of pillar workings, International Journal of Rock Mechanics and Mining Science 7 (1970) 613–631. [3] Y. Estrin, A.V. Dyskin, E. Pasternak, S. Schaare, S. Stanchits, A.J. Kanel-Belov, Negative stiffness of a layer with topologically interlocked elements, Scripta Materialia 50 (2003) 291–294. [4] S. Schaare, A.V. Dyskin, Y. Estrin, S. Arndt, E. Pasternak, A. Kanel-Belov, Point loading of assemblies of interlocked cube-shaped elements, International Journal of Engineering Sciences 46 (2008) 1228–1238 http://dx.doi.org/10.1016/j.ijengsci.2008.06.012. [5] R.S. Lakes, Extreme damping in composite materials with a negative stiffness phase, Physical Review Letters 86 (2001) 2897–2900. [6] R.S. Lakes, Extreme damping in compliant composites with a negative-stiffness phase, Philosophical Magazine Letters 81 (2001) 95–100. [7] R.S. Lakes, T. Lee, A. Bersie, Y.C. Wang, Extreme damping in composite materials with negative stiffness inclusions, Nature 410 (2001) 565–567. [8] R.S. Lakes, W.J. Drugan, Dramatically stiffer elastic composite materials due to a negative stiffness phase? Journal of Mechanics and Physics of Solids 50 (2002) 979–1009. [9] Y-C. Wang, R.S. Lakes, Stable extremely-high-damping discrete viscoelastic systems due to negative stiffness elements, Applied Physics Letters 84 (2004) 4451–4453. [10] A.V. Dyskin, E. Pasternak, Elastic composite with negative stiffness inclusions in antiplane strain, International Journal of Engineering Science 58 (2012) 45–56. [11] J.M.T. Thompson, G.W. Hunt, A General Theory of Elastic Stability, John Wiley & Sons, London, 1973. [12] A.R. Champneys, G.W. Hunt, J.M.T. Thompson, Localisation and Solitary Waves in Solid Mechanics, World Scientific, Singapore, 1984, 1–28. [13] G.W. Hunt, H.-B. Mühlhaus, A.I.M. Whiting, Folding processes and solitary waves in structural geology, in: A.R. Champneys, G.W. Hunt, J.M. T. Thompson (Eds.), Localisation and Solitary Waves in Solid Mechanics, World Scientific, Singapore1984, pp. 332–348. [14] C.-M. Lee, A.H. Bogatchenkov, V.N. Goverdovskiy, A.I. Temnikov, Vibration control system with opened kinematic chain and elastic links of small stiffness., Proceedings KORUS 2003, The 7th Korea–Russia International Symposium on Science and Technology 1 (2003) (2003) 285–289.
E. Pasternak et al. / Journal of Sound and Vibration 333 (2014) 6676–6687
6687
[15] W.J. Greenleaf, M.T. Woodside, E.A. Abbondanzieri, S.M. Block, Passive all-optical force clamp for high-resolution laser trapping, Physical Review Letters 95 (2005) 208102. [16] S-T. Park, T-T. Luu, Techniques for optimizing parameters of negative stiffness, Proceedings of the IMechE Vol. 221 Part C: J. Mechanical Engineering Science (2007) 505–511. [17] A. Carrella, M.J. Brennan, T.P. Waters, Demonstrator to show the effects of negative stiffness on the natural frequency of a simple oscillator, Proceedings of the IMechE Vol. 222 Part C: J. Mechanical Engineering Science (2008) 1189–1192. [18] C.-M. Lee, V.N. Goverdovskiy, Type synthesis of function-generating mechanisms for seat suspensions, International Journal of Automotive Technology 10 (2009) 37–48. [19] Z.P. Bažant, L. Cedolin, Stability of Structures: Elastic, Inelastic, Fracture, and Damage Theories, Oxford University Press, Oxford, 1991. [20] R.S. Lakes, P. Rosakis, A. Ruina, Microbuckling instability in elastomeric cellular solids, Journal of Material Science 28 (1993) 4667–4672. [21] H. Baughman, S. Stafström, C. Cui, S.O. Dantas, Materials with negative compressibilities in one or more dimensions, Science 279 (1998) 1522–1524. [22] V.R. Coluci, L.J. Hall, M.E. Kozlov, M. Zhang, S.O. Dantas, D.S. Galvão, R.H. Baughman, Modeling the auxetic transition for carbon nanotube sheets, Physical Review, B 78 (2008) 115408. [23] T. Jaglinski, D. Kochmann, D. Stone, R.S. Lakes, Composite materials with viscoelastic stiffness greater than diamond, Science 315 (2007) 620–622. [24] Z.G. Nicolaou, A.E. Motter, Mechanical metamaterials with negative compressibility transitions, Nature Materials (2012) 1–6, http://dx.doi.org/10.1038/ NMAT3331. [25] A.V. Dyskin, E. Pasternak, Mechanical effect of rotating non-spherical particles on failure in compression, Philosophical Magazine 92 (2012) 3451–3473. [26] P. Martin, A.D. Mehta, A.J. Hudspeth, Negative hair-bundle stiffness betrays a mechanism for mechanical amplification by the hair cell, Proceedings of the National Academy of Sciences of USA 97 (2000) 12026–12031. [27] L. Le Goff, D. Bozovic, A.J. Hudspeth, Adaptive shift in the domain of negative stiffness during spontaneous oscillation by hair bundles from the internal ear, PNAS 102 (2005) 16996–17001. (November 22, 2005). [28] G. Puglisi, L. Truskinovsky, Mechanics of a discrete chain with bi-stable elements, Journal of Mechanics and Physics of Solids 48 (2000) 1–27. [29] F. Cardin, M. Favretti, Dynamics of a chain of springs with non-convex potential energy, Mathematics and Mechanics of Solids 8 (2003) 651–668. [30] Y-C. Wang, Influences of negative stiffness on a two-dimensional hexagonal lattice cell, Philosophical Magazine 87 (2007) 3671–3688. [31] L. Dong, D.S. Stone, R.S. Lakes, Anelastic anomalies and negative Poisson's ratio in tetragonal BaTiO3 ceramics, Applied Physics Letters 96 (2010) 141904. [32] D.L. Platus, Negative-stiffness-mechanism vibration isolation systems, Proceedings of the SPIE Conference on Current Developments in Vibration Control 98 for Optomechanical Systems, 3786, SPIE, Denver, Colorado, 1999, 98–105 (July 1999). [33] Y-C. Wang, M. Ludwigson, R.S. Lakes, Deformation of extreme viscoelastic metals and composites, Materials Science and Engineering, A 370 (2004) 41–49. [34] Z Yang, H.M. Dai, N.H. Chan, G.C. Ma, P. Sheng, Acoustic metamaterial panels for sound attenuation in the 50–1000 Hz regime, Applied Physics Letters 96 (2010) 041906. [35] W.J. Drugan, Elastic composite materials having a negative-stiffness phase can be stable, Physical Review Letters 98 (2007) 055502. [36] D.M. Kochmann, W.J. Drugan, Dynamic stability analysis of an elastic composite material having a negative-stiffness phase, Journal of Mechanics and Physics of Solids 57 (2009) 1122–1138. [37] R.L. Salganik, Mechanics of bodies with many cracks, Mechanics of Solids 8 (1973) 135–143. [38] Z. Liu, X. Zhang, Y. Mao, Y.Y. Zhu, Locally resonant sonic materials, Science 289 (2000) 1734–1736. [39] N. Fang, D. Xi, J. Xu, M. Ambati, W. Srituravanich, C. Sun, X. Zhang, Ultrasonic metamaterials with negative modulus, Nature 5 (2006) 452–456. [40] S. Guenneau, A. Movchan, G. Pétursson, S.A. Ramakrishna, Acoustic metamaterials for sound focusing and confinement, New Journal of Physics 9 (2007) 399. [41] S.H. Lee, C.M. Park, Y.M. Seo, Acoustic metamaterial with negative modulus, Journal of Physics: Condensed Matter 21 (2009) 175704. [42] Y. Lai, Y. Wu, P. Sheng, Z.-Q. Zhang, Hybrid elastic solids, Nature Materials 10 (2011) 620–624, http://dx.doi.org/10.1038/NMAT3043. [43] Z. Liang, J. Li, Extreme acoustic metamaterial by coiling up space, Physical Review Letters 108 (2012) 114301. [44] A.-C. Hladky-Hennion, J.O. Vasseur, G. Haw, C. Croënne, L. Haumesser, A.N. Norris, Negative refraction of acoustic waves using a foam-like metallic structure, Applied Physics Letters 102 (2013) 144103. [45] K. Bi, G. Dong, X. Fu, J. Zhou, Ferrite based metamaterials with thermo-tunable negative refractive index, Applied Physics Letters 103 (2013) 131915. [46] Y. Ding, Z. Liu, C. Qiu, J. Shi, Metamaterial with simultaneously negative bulk modulus and mass density, Physical Review Letters 99 (2007) 093904. [47] Z. Yang, J. Mei, M. Yang, N. Chan, P. Sheng, Membrane-type acoustic metamaterial with negative dynamic mass, Physical Review Letters 101 (2008) 204301. [48] X. Zhou, G. Hu, Analytic model of elastic metamaterials with local resonances, Physical Review B 79 (2009) 195109. [49] E.P. Calius, X. Bremaud, B. Smith, A. Hall, Negative mass sound shielding structures: Early results, Physica Status Solidi B 246 (2009) 2089–2097. [50] X.N. Liu, G.K. Hu, G.L. Huang, C.T. Sun, An elastic metamaterial with simultaneously negative mass density and bulk modulus, Applied Physics Letters 98 (2011) 251907. [51] M.T. Islam, G. Newaz, Metamaterial with mass-stem array in acoustic cavity, Applied Physics Letters 100 (2012) 011904. [52] K.T. Tan, H.H. Huang, C.T. Sun, Optimizing the band gap of effective mass negativity in acoustic metamaterials, Applied Physics Letters 101 (2012) 241902. [53] X. Cai, Q. Guo, G. Hu, J. Yang, Particle focusing in a microchannel with acoustic metafluid, Applied Physics Letters 103 (2013) 031901. [54] H.J. Pain, The Physics of Vibrations and Waves, John Wiley & Sons Ltd; Chichester, England, 2005. [55] L.D. Landau, E.M. Lifshitz, Course of Theoretical Physics. Volume 1. Mechanics, 3rd ed. Butterworth-Heinemann; Oxford, England, 1976. [56] A. Carrella, M.J. Brennan, T.P. Waters, Demonstrator to show the effects of negative stiffness on the natural frequency of a simple oscillator, Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 222 (2008) 1189–1192. [57] A.V. Dyskin, E. Pasternak, Rock mass instability caused by incipient block rotation, in: Qihu Qian, Yingxin Zhou (Eds.), Harmonising Rock Engineering and the Environment, Proceedings of the 12th International Congress on Rock Mechanics, CRC Press, Balkema2012, p. 4.