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Statics and dynamics of a nonlinear oscillator with quasi-zero stiffness behaviour for large deflections
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Statics and dynamics of a nonlinear oscillator with quasi-zero stiffness behaviour for large deflections Gianluca Gatti PII: DOI: Reference:
S1007-5704(19)30462-9 https://doi.org/10.1016/j.cnsns.2019.105143 CNSNS 105143
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Communications in Nonlinear Science and Numerical Simulation
Received date: Revised date: Accepted date:
4 October 2018 14 November 2019 1 December 2019
Please cite this article as: Gianluca Gatti , Statics and dynamics of a nonlinear oscillator with quasizero stiffness behaviour for large deflections, Communications in Nonlinear Science and Numerical Simulation (2019), doi: https://doi.org/10.1016/j.cnsns.2019.105143
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Highlights
An oscillator with quasi-zero stiffness behaviour at large deflections is presented It consists of four linear springs properly arranged to achieve desired nonlinearity A fundamental analytical insight is performed based on a simple model Easy-to-use relations are found to achieve a minimum resonance frequency
1
Statics and dynamics of a nonlinear oscillator with quasi-zero stiffness behaviour for large deflections
Gianluca Gatti* Department of Mechanical, Energy and Management Engineering University of Calabria V. P. Bucci 46C, 87036, Rende (CS), Italy [email protected]
N° of Figures: 9 N° of Tables: 1 N° of Videos: 1
Declarations of interest: none
This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors
*
Corresponding author
2
ABSTRACT One degree of freedom quasi-zero stiffness mechanical oscillators are characterized by having a spring with a very low stiffness at some point in the force-deflection curve, and their investigation has been mainly motivated for vibration isolation and control purposes. Unlike conventional quasi-zero stiffness mechanical oscillators, which exhibit a very low dynamic stiffness for small deflections, this paper presents a complementary case, not yet systematically addressed in the literature, where the low dynamic stiffness is achieved at large deflections. As a consequence, the equivalent spring exhibits a softening rather than a hardening behaviour, and a lower limit for the resonance frequency which bends to the lower frequencies may be derived. The oscillator is physically realized by using four springs properly arranged, rather than two, as classically reported in the literature. An approximate analytic insight is presented, as this allows the derivation of simple easy-to-use relationships to help the design strategy.
KEYWORDS: nonlinear oscillator; softening spring; quasi-zero stiffness; vibration isolator.
1
INTRODUCTION
Nonlinear springs have been widely investigated in the past decades, and introduced on purpose into dynamical systems to improve their performance in vibration control [1], respect to the corresponding linear counterparts. One of the main features that may be achieved using a spring with a nonlinear static force-deflection curve is the quasi-zero stiffness (QZS) behaviour [2], and this means that at some particular point on the spring force-deflection diagram, the stiffness may be designed to be very low. Such an effect has been widely adopted for vibration isolation purposes [3-6], as this allows to achieve high-static-low-dynamic stiffness oscillators, with a remarkable trade-off between high-static stiffness characteristics to limit static deflections, and 3
low-dynamic stiffness characteristics to improve isolation performance. In these cases, the nonlinear spring behaviour may be realized by arranging linear springs in a geometric configuration to achieve the desired effect. A classical arrangement consists of combining a pair of linear springs oriented perpendicularly to the direction of motion and providing negative stiffness, with one linear spring aligned to the direction of motion and providing positive stiffness [2-4]. The global nonlinear behaviour is of hardening type, avoiding negative stiffness around the equilibrium configuration, where the QZS effect is required. In fact, it is the parallel combination of positive stiffness elements with negative ones [7] which allows to achieve such a QZS behaviour. Negative stiffness elements may be also realized by using buckled slender beams [8] or magnets arranged in an attractive configuration [9]. Motivated by the interest of the scientific and engineering community towards QZS systems and the recent promising applications as foldable mechanisms [10], this paper investigates a nonlinear systems which exhibits a softening effect for small oscillations and tends to a QZS behaviour for large oscillations. Such a characteristics is realized by arranging four linear springs in a particular geometric configuration, avoiding instability due to the presence of negative stiffness components. The static analysis of the proposed oscillator is performed, and the effect of its parameters is investigated. An approximate and specific formulation for the spring force-deflection curve is then introduced and incorporated into the equation of motion, to allow the derivation of an approximate closed-form expression of the amplitude-frequency relation for subsequent dynamic analyses.
2
STATIC MODEL OF THE SYSTEM
4
The model of the system considered in this work is illustrated in Fig. 1. It consists of a single degree-of-freedom oscillator, where a mass m is suspended by four linear springs with stiffness k, arranged in an oblique and symmetric configuration, Fig. 1(a). The oscillation of the mass is assumed to be constrained to the horizontal motion only, so that when the mass moves, the springs rotate and change their length, Fig. 1(b).
[Figure 1 around here]
2.1
Static force-deflection curve
The relationships between the mass displacement x, and the applied static restoring force fs, is given by
f s 2k ll l0 cos lr l0 cos ,
(1)
where ll (lr) is the actual length of the springs on the left (right) side in Fig. 1(b), () is the actual angle between the line of action of the springs on the left (right) side with the horizontal axis, as illustrated in Fig. 1(b), and l0 is the natural length of the springs. In particular, l0, as shown in Fig. 1(a), is the geometric distance between the position of the mass at rest and each of the spring supports on ground, while is the spring natural length factor, defining whether the springs are assembled in tension < 1, compression, > 1, or unloaded = 1. Equation (1) may be conveniently expanded as a function of the mass displacement x and rewritten in non-dimensional form as
5
bˆ 2 1 fˆs 2 1 2 2 bˆ 1 xˆ
bˆ 2 1 1 xˆ 2 1 2 bˆ 2 1 xˆ
1 xˆ ,
(2)
where fˆs f s ak , bˆ b a and xˆ x a .
[Figure 2 around here]
Equation (2) is plotted as a function of the non-dimensional displacement xˆ , for different values of the geometric parameter bˆ and for fixed values of the factor , in Figs. 2(a-c). For the range of parameters used to plot Figs. 2(a-c), it can be seen that the force-deflection curves are characterized by inflexion points at around xˆ 1 , which can have a positive (dashed lines), negative (dotted lines) or quasi-zero (solid lines) stiffness characteristics, depending on the value of bˆ . In the approximate range 1 xˆ 1 the system thus behaves as an equivalent softening spring. Moreover, with reference to the case of QSZ behaviour (solid lines), it can be observed that the higher the value of the factor , the higher the value of the geometric parameter bˆ needed to achieve a QZS characteristic. This effect may be observed from Fig. 2(a) to (c). It is worth noting that bˆ represents the geometrical ratio between the distance b and a in Fig. 1(a), and thus if its value is less than 1, it means that the springs configuration at rest is such that they are more inclined towards the horizontal dashed line, which is sketched in Fig. 1(b). We will now focus our attention to the case where the force-deflection curve of the equivalent nonlinear spring has a QZS behaviour at around xˆ 1 .
2.2
Quasi-zero stiffness behaviour 6
To consider this particular case, Eq. (2) is differentiated once and twice with respect to xˆ and set to zero. The equation resulting from the second derivative turns out to be independent on , and it is plotted on the xˆ bˆ plane as the thicker solid line in Fig. 3 – it has the shape of an overturned “”. The equation resulting from the first derivative is also plotted in Fig. 3 as thinner different lines for different values of . For each value of , the solutions to achieve a QZS behaviour are graphically found as the intersection points between the two corresponding curves (an animation is also reported in Video 1, for a better visualization).
[Figure 3 around here]
[Video-still 1 around here]
Excluding the QZS behaviour achieved at bˆ 0 , which is not of practical interest in this work, it may be noted that for less than about 1.25, there is only one condition for a QZS behaviour, which appears at values of displacement very close to 1. For values of greater that about 1.25 and less than approximately 1.36, another condition for a QZS behaviour exists, and this corresponds to a larger value of bˆ and a lower value of xˆ . For values of greater that about 1.36, there is no geometric configuration yielding to a QZS behaviour at large displacements. Some numerical solutions are reported in Table l, where the first two rows report the values of and bˆ , and the last two rows report the corresponding coordinates of the inflexion point in the force-deflection curve, for xˆ 0 . In the case when multiple solutions exist (as for greater than about 1.25), only the one which is closer to xˆ 1 is reported in the table.
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[Table 1 around here]
From Table 1, we can note that as the factor increases, from pre-tension to pre-compression, the geometric ratio between the dimension b and a in Fig. 1(a) also increases, and this indicates that the configuration of the spring supports changes from rectangular-like to square-like. Furthermore, we can note that as increases, the xˆP coordinated of the inflection point slightly decreases, but stays around 1, while fˆP largely decreases. For a better observation of such behaviour, the force-deflection curve and corresponding stiffness are plotted in Fig. 4(a) and (b) respectively, for some of the conditions reported in Table 1. It is worth reminding that the QZS behaviour considered in this work is different from that commonly considered in the literature [2,6], where it is located at the static equilibrium configuration, i.e. the equivalent spring stiffness is zero for very small displacement and then increases in a hardening way. In the case considered in this work, the equivalent spring stiffness is non-zero and positive at the static equilibrium configuration and then it decreases in a softening way until it reaches a quasi-zero condition. Beyond this point, the stiffness continues increasing in a hardening way, avoiding static instability.
[Figure 4 around here]
2.3
Approximate force-deflection curve by Taylor series expansion
To incorporate the static force-deflection curve characteristic into the dynamic equation of motion, and perform an approximate analytic insight, Eq. (2) should be conveniently approximated. 8
A first attempt is to consider a Taylor series expansion around the static equilibrium configuration, to yield
fˆs ,T kˆ1,T xˆ kˆ3,T xˆ 3 kˆ5,T xˆ 5 kˆ7,T xˆ 7
,
(3)
where
ˆ2 1 bˆ 2 1 ˆ ˆ2 4 b , kˆ1,T 4 , k 2 b 3,T 3 1 bˆ 2 1 bˆ 2
kˆ5,T
3 ˆ 2 8 12bˆ 2 bˆ 4 ˆ 1 ˆ 2 64 240bˆ 2 120bˆ 4 5bˆ6 b , k7,T b . 5 7 2 4 1 bˆ 2 1 bˆ 2
(4a-d)
The approximate force-deflection curve given in Eq. (3) may be investigated by considering its first derivative with respect to xˆ and setting it to zero, resulting into a cubic polynomial equation in terms of xˆ 2 . The existence of real positive solutions for xˆ 2 corresponds to the existence of extrema in the force-deflection curve, potentially related to a QZS behaviour. In particular, if the discriminant of the derivative of Eq. (3) is zero, at least two solutions coincide, and this happens on the thick solid curves in Fig. 5; if it is positive, which happens for values of the parameters between the two thick solid curves in Fig. 5, then there are three real solutions for xˆ 2 ; if it is negative (outside the two thick solid curves in Fig. 5), there is only one real solution.
[Figure 5 around here]
9
Furthermore, by examining the type of roots by Descartes rule of signs [11], it results that if the values of the parameters and bˆ are below (above) the horizontal thin dotted line in Fig. 5, then there is only one positive (negative) real root. This means that in the lighter-shaded region of Fig. 5, the approximate force-deflection curve of Eq. (3) exhibits only one maximum (for xˆ 0 ), resulting into an unstable system with negative stiffness for displacement values greater than the one corresponding to the maximum. In the white region of Fig. 5, the approximate force-deflection curve exhibits two extrema (for xˆ 0 ), resulting into an unstable behaviour for displacement values between the one corresponding to the maximum and the minimum, i.e. a bi-stable configuration. In the darker-shaded region of Fig. 5, the approximate force-deflection curve exhibits no extremum, which means that the stiffness is always strictly positive, and no QZS behaviour manifests. For the values of parameters corresponding to the upper branch of the solid curve in Fig. 5, the approximate force-deflection curve in Eq. (3) exhibits a QZS behaviour. Also, superimposed in Fig. 5, as a dashed line, is the curve representing the QZS condition for the exact force-deflection curve in Eq. (2), where markers denote some of the values reported in Table 1. From Fig. 5 it can be clearly noted that the QZS conditions for the exact and approximate forcedeflection curves, given in Eq. (2) and (3) respectively, do not correspond. Furthermore, for most of the values of bˆ , the approximate force-deflection curve predicts an unstable behaviour for large deflections, which do not manifest in the real system. This is better illustrated in Fig. 6(a-f), where exact (thick solid lines) and approximate (thin dash-dotted lines) force and stiffness curves are plotted for three different combinations of the parameters in Table 1. In particular, it may be noted in Fig. 6(a,d) and (b,e), that the approximate equivalent spring stiffness (thin dash-dotted lines) becomes negative for displacement values greater than that at which the maximum occurs,
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and this leads to static instability. In Fig. 6(c,f) it may be noted that the stiffness of the approximate model is always strictly positive and do not exhibit any QZS behaviour.
[Figure 6 around here]
2.4
Approximate force-deflection curve by fitting inflexion point
To overcome the limitations of the previous model, which is good for small displacement only, the exact force-deflection curve in Eq. (2) is approximated by a new polynomial equation with the following four boundary conditions: (i) the stiffness at zero deflection is equal to that from Taylor expansion; (ii) the stiffness at the inflection point is zero; (iii) the derivative of the stiffness at the inflection point is zero; (iv) the force value at the inflection point is equal to that of the exact expression in Eq. (2). To satisfy these conditions, a seventh-order odd polynomial equation is selected, and this motivated the former Taylor series expansion up to the same order for comparison. Equation (2) is thus approximated by
fˆs , F kˆ1, F xˆ kˆ3, F xˆ 3 kˆ5, F xˆ 5 kˆ7, F xˆ 7 ,
(5)
where
1 bˆ 2 1 ˆ 10 kˆ1, F kˆ1,T kˆ1 4 , k3, F kˆ5, F xˆP2 7kˆ7, F xˆP4 , 2 3 1 bˆ . ˆ 14kˆ xˆ 6 ˆ ˆ k f k 15 7, F P ˆ P kˆ5, F 1 , k7, F 1. 5 xˆP4 8 xˆP7 xˆP6
11
(6a-d)
Equation (5) and its derivative are also plotted in Fig. 6 as thicker dashed lines. It can be noted that, this latter approximation better fits the exact force-deflection curve of the mechanical oscillator illustrated in Fig. 1 up to the QZS condition, avoiding an erroneous prediction of instability for large deflections. It should be noted that no boundary condition has been set for xˆ 1 , so that there is no control on the stiffness behaviour over that region, except that it is
forced to be non-negative, as in the exact static behaviour of the mechanical oscillator.
3
DYNAMIC MODEL OF THE SYSTEM
3.1
Amplitude-frequency equation
The equation of motion of the system depicted in Fig. 1(a) is given by
mx cx f s x F cos t ,
(7)
where the static spring restoring force f s x is given by Eq. (1), the overdots denote differentiation respect to time t, a harmonic force with amplitude F and angular frequency is assumed to excite the oscillator mass m, and a viscous damping c is introduced as a dissipative term. By using one of the approximate expressions for the spring restoring force given in Eq. (3) or (5), Eq. (7) may be conveniently written in non-dimensional form as
xˆ 2 xˆ xˆ xˆ 3 xˆ 5 xˆ 7 cos ,
12
(8)
where 1t and 1 are the non-dimensional time and frequency respectively,
1 kkˆ1 m , c 2m1 is the damping ratio, F akkˆ1 is the non-dimensional force amplitude, primes denote differentiation respect to , a phase is introduced for convenience, and kˆ3 kˆ1 , kˆ5 kˆ1 , kˆ7 kˆ1 , where kˆ3 , kˆ5 , kˆ7 may be set equal to Eqs. 4(b-d) or 6(b-d), depending on the approximation adopted. To solve Eq. (8) in closed-form in terms of the amplitude-frequency equation, it is further assumed that the system response is predominately harmonic at the excitation frequency, i.e.
xˆ Xˆ cos , and this is substituted into Eq. (8), where a first-order harmonic balance approximation is applied to yield
4 Xˆ 2 2 4 2 Xˆ 2 2 Xˆ G Xˆ G Xˆ
2
2 0 ,
(9)
3 5 35 where G Xˆ Xˆ Xˆ 3 Xˆ 5 Xˆ 7 . 4 8 64
It can be noted that the approximate expression in Eq. (9) is quadratic in 2 , it may thus be solved in closed-form and plotted into an amplitude-frequency diagram, as detailed in Section 3.3 down below.
3.2
Stability of the harmonic solution
The stability of the approximate steady-state solutions of Eq. (9) is calculated by applying Floquet theory as described in [12]. To this aim, a disturbance u is added to the approximate harmonic solution and substituted into Eq. (8) to yield the following linearized equation 13
u 2 u u 3 xˆ 2u 5 xˆ 4u 7 xˆ 6u 0 .
(10)
Equation (10) admits a solution in the form of u e [13], where is a periodic function of period 2 , which can be expanded in Fourier series to the first term as
C cos S sin , and substituted back into Eq. (10). By then equating the harmonic coefficients on both sides of the resulting equation, a system of two linear homogeneous algebraic equations in the unknowns C and S is obtained, which admits a nontrivial solution if and only if the determinant of the coefficient matrix is equal to zero. This leads to the following fourth-order polynomial equation in terms of
3 ˆ 2 5 ˆ 4 35 ˆ 6 2 2 2 1 X X X 4 8 64 , 9 ˆ 2 25 ˆ 4 245 ˆ 6 2 2 2 2 X 4 0 2 1 X X 4 8 64
(11)
which can be solved numerically for specific values of the system parameters. The harmonic response derived in Section 3.1 is then asymptotically stable if and only if all the solutions for have a negative real part, while it is unstable if at least one of them has a positive real part. In the frequency response curves (FRCs) reported in this paper, approximate stable solutions are represented by solid lines, while approximate unstable solutions are represented by dashed lines.
3.3
Frequency response curves 14
The FRCs are plotted in Figs. 7(a-f) for three different combinations of the system parameters and bˆ , as reported in Table 1. Figures 7(a-c) show the FRCs obtained from Eq. (9) when the coefficients in Eqs. 4(a-d) are used, while Figs. 7(d-f) show the FRCs obtained when the coefficients in Eqs. 6(a-d) are adopted. Also, superimposed in Figs. 7(a-f) as markers, are the numerical solutions for validation. In particular, the numerical solutions were obtained by numerically integrating the equation of motion, where the exact expression for the static spring restoring force from Eq. (2) is used. Fourier coefficients were then extracted from the calculated time histories at steady-state. Only those coefficients corresponding to the excitation frequency are then plotted as black markers in the FRCs shown in the paper. Nevertheless, it was verified that, for the parameters used in this work, the amplitudes of the higher and lower order harmonics never exceeded 5% of the amplitude at the fundamental one.
[Figure 7 around here]
The classical softening effect which causes the resonance peak to bend towards the lower frequencies can be seen, whereas for a hardening system, it bends to the higher frequencies [13]. It can be noted that the approximation proposed in Section 2.4 better predicts the numerical solution around the resonance, and this is more evident at lower frequencies (and higher amplitudes) where the Taylor series approximation presented in Section 2.3 incorrectly predicts an instability due to negative stiffness. In particular, the FRCs in Fig. 7(a) and (b), which are based on the Taylor series expansion, show an instability (dashed line) of the harmonic solution in the upper branch of the FRC, which does not manifest in the numerical solution (black markers), while the FRCs in Fig. 7(d) and (e), based on the approximation proposed in Section
15
2.4, better match the numerical solution. The FRC in Fig. 7(c) shows a less evident softening effect at resonance than that shown in Fig. 7(f), and this is due to the presence of the QZS behaviour, which is not captured by the Taylor series expansion – rather, it is well detected by the approximation derived in Section 2.4.
3.4
Resonance frequency and corresponding displacement amplitude
The approximate closed-form expression of the amplitude-frequency equation reported in Eq. (9) can be further exploited to get an insight into the relation between the resonance peak and its corresponding frequency. In fact, it is noted that the resonance frequency occurs when the peak displacement is a maximum, and this happens when the two solutions to Eq. (9) in terms of 2 are equal. This condition is determined by the following equation
4 2 Xˆ 2 2 Xˆ G Xˆ 4 Xˆ 2 G Xˆ 2
2
2 0 ,
(12)
which can be solved in terms of the damping ratio to give
2 r
G Xˆ G Xˆ
2
2 Xˆ
2 .
(13a,b)
The approximate expression for the resonance frequency is then obtained by substituting Eq. (13a,b) into Eq. (9) and rearranging to give
16
2 r
G Xˆ
Xˆ
2
2 ,
(14a,b)
where it can be noted that a real solution only occurs for a damping ratio given by Eq. (13b). Equation (14b) is thus plotted in Fig. 8(a), for the combination of parameters corresponding to
in Table 1, and for different values of the excitation . For the range of parameters used to plot Fig. 8(a), it can be noted that there is a minimum value for the resonance frequency, and this is substantially independent from . Furthermore, the corresponding displacement amplitude at this lowest resonance frequency is also independent from [Figure 8 around here] An approximate value for such a lowest resonance frequency and its corresponding displacement amplitude can be obtained by differentiating the square root of Eq. (14b) respect to Xˆ for = 0, equating to zero and solving numerically to give Xˆ r 1.3 and r 0.767 . It is worth noting that for = 0, Eq. (14b) gives the expression of the backbone curve of the undamped, unforced response of the nonlinear oscillator. It is now interesting to set Xˆ Xˆ r into Eq. (13b) and plot it as a function of the excitation in Fig. 8(b). It may then be seen how the relation between the excitation amplitude and the damping ratio to achieve the lowest resonance frequency is almost linear for the range of values used to plot Fig. 8(b), being r 0.5 .
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The FRCs for two different values of excitation and for a damping ratio given from Fig. 8(b) are plotted in Fig. 9(a) and (b) as thicker lines. It can be noted that the resonance frequency in both cases is at about 0.77 and the amplitude at resonance is about 1.3. In both figures, thinner lines are used to plot the FRCs for slightly different values of damping, to show that the damping value given from Fig. 8(b) is the one to achieve the lowest value for the resonance frequency.
[Figure 9 around here]
It can be noted that the shape of the frequency response curve of the system shown in Fig. 9, recalls that of an asymmetric Duffing oscillator [14], which has been adopted to investigate the characteristics of a mistuned isolator with QZS behaviour [15-16]. However, this latter model is conceptually different from that described in this paper - in particular, the asymmetric Duffing oscillator has only one configuration with QZS and it has a fundamentally hardening behaviour for large deflections, while the system considered here, has two symmetric configurations with QZS conditions and it has a fundamentally softening behaviour in the region of interest. Such softening behaviour, arising from the QZS conditions for large deflections, has also been exploited to increase the elastic potential energy in a spring-like system [17].
4
CONCLUSIONS
This paper has investigated the static and approximate dynamic characteristics of a nonlinear oscillator with a softening characteristic and a QZS behaviour for large deflections. The specific nonlinearity is realized by assembling four linear springs in a geometric arrangement to achieve the desired characteristic. The proposed configuration allows to exploit the benefit of a softening system up to its fundamental limit. The approximate dynamic model developed avoids an 18
erroneous prediction of instability due to the presence of negative stiffness components and the use of a different approximation. The static force-deflection curve is approximated by a polynomial equation with specific coefficients, and it is incorporated into the dynamic equation of motion to derive an approximate closed-form expression of the amplitude-frequency equation. The effect of the system parameters is investigated, and it is found that a simple relation among them exists such that a minimum value for the resonance frequency is achieved. It is believed that the proposed system configuration, and the fundamental insight developed in this paper, will stimulate further theoretical and experimental advances towards a better understanding and exploitation of nonlinear phenomena, and have potential benefits in the design of engineering systems and structures with improved and targeted dynamic performance.
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Figure Captions List Fig. 1
Nonlinear oscillator under study: (a) undeformed configuration at rest, (b) deformed configuration
Fig. 2
Non-dimensional force-deflection curve for different values of the system parameters: (a) = 0.8 and bˆ = 0.2 (dotted line), bˆ = 0.4 (solid line), bˆ = 0.6 (dashed line); (b) = 1.0 and bˆ = 0.3 (dotted line), bˆ = 0.5 (solid line), bˆ = 0.7 (dashed line); (c) = 1.2 and bˆ = 0.5 (dotted line), bˆ = 0.7 (solid line), bˆ = 0.9 (dashed line)
Fig. 3
Graphical solution of xˆ and bˆ to achieve a QZS behaviour. Thicker solid curve represents the second derivative of Eq. (2) set to zero. Other thinner lines represent the first derivative of Eq. (2) set to zero for = 1 (dotted line), = 1.2 (dashed line), = 1.3 (solid line) and = 1.4 (dashed-dot line).
Fig. 4
(a) Non-dimensional force-deflection curve and (b) stiffness curve for = 0.7 and bˆ = 0.3763 (dotted line), = 1.0 and bˆ = 0.5955 (solid line), = 1.3 and bˆ = 1.0117 (dashed line)
Fig. 5
Region of parameters identifying the stiffness behaviour of the approximate force-defection curve in Eq. (3), in terms of existence of extrema. In the lighter shaded region, the approximate stiffness presents one maximum; in the white region, it presents two extrema (one maximum and one minimum); in the darker shaded region, it does not 22
present any extremum. The dashed line connecting markers is plotted from Table 1. Fig. 6
Non-dimensional force-deflection curve (a-c) and stiffness curve (d-f) for: (a,d) = 0.8 and bˆ = 0.4417; (b,e) = 1.1 and bˆ = 0.6926; (c,f) = 1.2 and bˆ = 0.8173. Solid lines are from Eq. (2), thin dash-dotted lines are from Eq. (3), and dashed lines are from Eq. (5)
Fig. 7
Approximate analytic FRCs using Eq. (9) with parameters defined in Eqs. 4(a-d) for (a-c), and with parameters defined in Eqs. 6(a-d) for (d-f), for = 0.02, = 0.04 and: (a,d) = 0.8 and bˆ = 0.4417; (b,e) = 1.0 and bˆ = 0.5955; (c,f) = 1.2 and bˆ = 0.8173. Approximate analytic stable (solid lines) and unstable solutions (dashed lines), numerical solutions (black markers).
Fig. 8
(a) Resonance frequency as a function of displacement amplitude, for = 0.02 (dashed line), = 0.04 (dash-dot line), = 0.06 (dotted line), = 0.1 (solid line). (b) Damping ratio to achieve the lowest resonance frequency as a function of excitation.
Fig. 9
FRCs for = 1.0 and bˆ = 0.5955, for (a) = 0.1 and (b) = 0.02, with damping given from Fig. 8(b) to achieve the lowest resonance frequency. Approximate analytic stable (solid lines) and unstable solutions (dashed lines), numerical solutions (black markers). Thinner lines denote the FRC for slightly different values of damping as labelled.
23
Video Captions List Video 1
Relations among the systems parameters to achieve a QZS behaviour
fig .1
24
fig 2
25
fig. 3
26
fig. 4
27
fig. 5
28
fig. 6
29
fig. 7
30
fig. 8
31
fig. 9
32
fig. 10
33
Author Contribution Statement There is only one author for this paper, who contributes to all the aspects described therein.
DIS Nothing to declare
34
Table 1. Conditions for the presence of an inflection point in the force-deflection curve, and its coordinates as function of the spring natural length factor
Pre-tension
Unloaded
Pre-compression
0.7
0.8
0.9
1
1.1
1.2
1.3
bˆ
0.3763
0.4417
0.5137
0.5955
0.6926
0.8173
1.0117
Inflexion
xˆ P
0.9996
0.9991
0.9981
0.9962
0.9924
0.9841
0.9601
Point
fˆP
2.5299
2.2920
2.0400
1.7690
1.4712
1.1308
0.6996
35
Statics and dynamics of a nonlinear oscillator with quasi-zero stiffness behaviour for large deflections Gianluca Gatti PII: DOI: Reference:
S1007-5704(19)30462-9 https://doi.org/10.1016/j.cnsns.2019.105143 CNSNS 105143
To appear in:
Communications in Nonlinear Science and Numerical Simulation
Received date: Revised date: Accepted date:
4 October 2018 14 November 2019 1 December 2019
Please cite this article as: Gianluca Gatti , Statics and dynamics of a nonlinear oscillator with quasizero stiffness behaviour for large deflections, Communications in Nonlinear Science and Numerical Simulation (2019), doi: https://doi.org/10.1016/j.cnsns.2019.105143
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Highlights
An oscillator with quasi-zero stiffness behaviour at large deflections is presented It consists of four linear springs properly arranged to achieve desired nonlinearity A fundamental analytical insight is performed based on a simple model Easy-to-use relations are found to achieve a minimum resonance frequency
1
Statics and dynamics of a nonlinear oscillator with quasi-zero stiffness behaviour for large deflections
Gianluca Gatti* Department of Mechanical, Energy and Management Engineering University of Calabria V. P. Bucci 46C, 87036, Rende (CS), Italy [email protected]
N° of Figures: 9 N° of Tables: 1 N° of Videos: 1
Declarations of interest: none
This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors
*
Corresponding author
2
ABSTRACT One degree of freedom quasi-zero stiffness mechanical oscillators are characterized by having a spring with a very low stiffness at some point in the force-deflection curve, and their investigation has been mainly motivated for vibration isolation and control purposes. Unlike conventional quasi-zero stiffness mechanical oscillators, which exhibit a very low dynamic stiffness for small deflections, this paper presents a complementary case, not yet systematically addressed in the literature, where the low dynamic stiffness is achieved at large deflections. As a consequence, the equivalent spring exhibits a softening rather than a hardening behaviour, and a lower limit for the resonance frequency which bends to the lower frequencies may be derived. The oscillator is physically realized by using four springs properly arranged, rather than two, as classically reported in the literature. An approximate analytic insight is presented, as this allows the derivation of simple easy-to-use relationships to help the design strategy.
KEYWORDS: nonlinear oscillator; softening spring; quasi-zero stiffness; vibration isolator.
1
INTRODUCTION
Nonlinear springs have been widely investigated in the past decades, and introduced on purpose into dynamical systems to improve their performance in vibration control [1], respect to the corresponding linear counterparts. One of the main features that may be achieved using a spring with a nonlinear static force-deflection curve is the quasi-zero stiffness (QZS) behaviour [2], and this means that at some particular point on the spring force-deflection diagram, the stiffness may be designed to be very low. Such an effect has been widely adopted for vibration isolation purposes [3-6], as this allows to achieve high-static-low-dynamic stiffness oscillators, with a remarkable trade-off between high-static stiffness characteristics to limit static deflections, and 3
low-dynamic stiffness characteristics to improve isolation performance. In these cases, the nonlinear spring behaviour may be realized by arranging linear springs in a geometric configuration to achieve the desired effect. A classical arrangement consists of combining a pair of linear springs oriented perpendicularly to the direction of motion and providing negative stiffness, with one linear spring aligned to the direction of motion and providing positive stiffness [2-4]. The global nonlinear behaviour is of hardening type, avoiding negative stiffness around the equilibrium configuration, where the QZS effect is required. In fact, it is the parallel combination of positive stiffness elements with negative ones [7] which allows to achieve such a QZS behaviour. Negative stiffness elements may be also realized by using buckled slender beams [8] or magnets arranged in an attractive configuration [9]. Motivated by the interest of the scientific and engineering community towards QZS systems and the recent promising applications as foldable mechanisms [10], this paper investigates a nonlinear systems which exhibits a softening effect for small oscillations and tends to a QZS behaviour for large oscillations. Such a characteristics is realized by arranging four linear springs in a particular geometric configuration, avoiding instability due to the presence of negative stiffness components. The static analysis of the proposed oscillator is performed, and the effect of its parameters is investigated. An approximate and specific formulation for the spring force-deflection curve is then introduced and incorporated into the equation of motion, to allow the derivation of an approximate closed-form expression of the amplitude-frequency relation for subsequent dynamic analyses.
2
STATIC MODEL OF THE SYSTEM
4
The model of the system considered in this work is illustrated in Fig. 1. It consists of a single degree-of-freedom oscillator, where a mass m is suspended by four linear springs with stiffness k, arranged in an oblique and symmetric configuration, Fig. 1(a). The oscillation of the mass is assumed to be constrained to the horizontal motion only, so that when the mass moves, the springs rotate and change their length, Fig. 1(b).
[Figure 1 around here]
2.1
Static force-deflection curve
The relationships between the mass displacement x, and the applied static restoring force fs, is given by
f s 2k ll l0 cos lr l0 cos ,
(1)
where ll (lr) is the actual length of the springs on the left (right) side in Fig. 1(b), () is the actual angle between the line of action of the springs on the left (right) side with the horizontal axis, as illustrated in Fig. 1(b), and l0 is the natural length of the springs. In particular, l0, as shown in Fig. 1(a), is the geometric distance between the position of the mass at rest and each of the spring supports on ground, while is the spring natural length factor, defining whether the springs are assembled in tension < 1, compression, > 1, or unloaded = 1. Equation (1) may be conveniently expanded as a function of the mass displacement x and rewritten in non-dimensional form as
5
bˆ 2 1 fˆs 2 1 2 2 bˆ 1 xˆ
bˆ 2 1 1 xˆ 2 1 2 bˆ 2 1 xˆ
1 xˆ ,
(2)
where fˆs f s ak , bˆ b a and xˆ x a .
[Figure 2 around here]
Equation (2) is plotted as a function of the non-dimensional displacement xˆ , for different values of the geometric parameter bˆ and for fixed values of the factor , in Figs. 2(a-c). For the range of parameters used to plot Figs. 2(a-c), it can be seen that the force-deflection curves are characterized by inflexion points at around xˆ 1 , which can have a positive (dashed lines), negative (dotted lines) or quasi-zero (solid lines) stiffness characteristics, depending on the value of bˆ . In the approximate range 1 xˆ 1 the system thus behaves as an equivalent softening spring. Moreover, with reference to the case of QSZ behaviour (solid lines), it can be observed that the higher the value of the factor , the higher the value of the geometric parameter bˆ needed to achieve a QZS characteristic. This effect may be observed from Fig. 2(a) to (c). It is worth noting that bˆ represents the geometrical ratio between the distance b and a in Fig. 1(a), and thus if its value is less than 1, it means that the springs configuration at rest is such that they are more inclined towards the horizontal dashed line, which is sketched in Fig. 1(b). We will now focus our attention to the case where the force-deflection curve of the equivalent nonlinear spring has a QZS behaviour at around xˆ 1 .
2.2
Quasi-zero stiffness behaviour 6
To consider this particular case, Eq. (2) is differentiated once and twice with respect to xˆ and set to zero. The equation resulting from the second derivative turns out to be independent on , and it is plotted on the xˆ bˆ plane as the thicker solid line in Fig. 3 – it has the shape of an overturned “”. The equation resulting from the first derivative is also plotted in Fig. 3 as thinner different lines for different values of . For each value of , the solutions to achieve a QZS behaviour are graphically found as the intersection points between the two corresponding curves (an animation is also reported in Video 1, for a better visualization).
[Figure 3 around here]
[Video-still 1 around here]
Excluding the QZS behaviour achieved at bˆ 0 , which is not of practical interest in this work, it may be noted that for less than about 1.25, there is only one condition for a QZS behaviour, which appears at values of displacement very close to 1. For values of greater that about 1.25 and less than approximately 1.36, another condition for a QZS behaviour exists, and this corresponds to a larger value of bˆ and a lower value of xˆ . For values of greater that about 1.36, there is no geometric configuration yielding to a QZS behaviour at large displacements. Some numerical solutions are reported in Table l, where the first two rows report the values of and bˆ , and the last two rows report the corresponding coordinates of the inflexion point in the force-deflection curve, for xˆ 0 . In the case when multiple solutions exist (as for greater than about 1.25), only the one which is closer to xˆ 1 is reported in the table.
7
[Table 1 around here]
From Table 1, we can note that as the factor increases, from pre-tension to pre-compression, the geometric ratio between the dimension b and a in Fig. 1(a) also increases, and this indicates that the configuration of the spring supports changes from rectangular-like to square-like. Furthermore, we can note that as increases, the xˆP coordinated of the inflection point slightly decreases, but stays around 1, while fˆP largely decreases. For a better observation of such behaviour, the force-deflection curve and corresponding stiffness are plotted in Fig. 4(a) and (b) respectively, for some of the conditions reported in Table 1. It is worth reminding that the QZS behaviour considered in this work is different from that commonly considered in the literature [2,6], where it is located at the static equilibrium configuration, i.e. the equivalent spring stiffness is zero for very small displacement and then increases in a hardening way. In the case considered in this work, the equivalent spring stiffness is non-zero and positive at the static equilibrium configuration and then it decreases in a softening way until it reaches a quasi-zero condition. Beyond this point, the stiffness continues increasing in a hardening way, avoiding static instability.
[Figure 4 around here]
2.3
Approximate force-deflection curve by Taylor series expansion
To incorporate the static force-deflection curve characteristic into the dynamic equation of motion, and perform an approximate analytic insight, Eq. (2) should be conveniently approximated. 8
A first attempt is to consider a Taylor series expansion around the static equilibrium configuration, to yield
fˆs ,T kˆ1,T xˆ kˆ3,T xˆ 3 kˆ5,T xˆ 5 kˆ7,T xˆ 7
,
(3)
where
ˆ2 1 bˆ 2 1 ˆ ˆ2 4 b , kˆ1,T 4 , k 2 b 3,T 3 1 bˆ 2 1 bˆ 2
kˆ5,T
3 ˆ 2 8 12bˆ 2 bˆ 4 ˆ 1 ˆ 2 64 240bˆ 2 120bˆ 4 5bˆ6 b , k7,T b . 5 7 2 4 1 bˆ 2 1 bˆ 2
(4a-d)
The approximate force-deflection curve given in Eq. (3) may be investigated by considering its first derivative with respect to xˆ and setting it to zero, resulting into a cubic polynomial equation in terms of xˆ 2 . The existence of real positive solutions for xˆ 2 corresponds to the existence of extrema in the force-deflection curve, potentially related to a QZS behaviour. In particular, if the discriminant of the derivative of Eq. (3) is zero, at least two solutions coincide, and this happens on the thick solid curves in Fig. 5; if it is positive, which happens for values of the parameters between the two thick solid curves in Fig. 5, then there are three real solutions for xˆ 2 ; if it is negative (outside the two thick solid curves in Fig. 5), there is only one real solution.
[Figure 5 around here]
9
Furthermore, by examining the type of roots by Descartes rule of signs [11], it results that if the values of the parameters and bˆ are below (above) the horizontal thin dotted line in Fig. 5, then there is only one positive (negative) real root. This means that in the lighter-shaded region of Fig. 5, the approximate force-deflection curve of Eq. (3) exhibits only one maximum (for xˆ 0 ), resulting into an unstable system with negative stiffness for displacement values greater than the one corresponding to the maximum. In the white region of Fig. 5, the approximate force-deflection curve exhibits two extrema (for xˆ 0 ), resulting into an unstable behaviour for displacement values between the one corresponding to the maximum and the minimum, i.e. a bi-stable configuration. In the darker-shaded region of Fig. 5, the approximate force-deflection curve exhibits no extremum, which means that the stiffness is always strictly positive, and no QZS behaviour manifests. For the values of parameters corresponding to the upper branch of the solid curve in Fig. 5, the approximate force-deflection curve in Eq. (3) exhibits a QZS behaviour. Also, superimposed in Fig. 5, as a dashed line, is the curve representing the QZS condition for the exact force-deflection curve in Eq. (2), where markers denote some of the values reported in Table 1. From Fig. 5 it can be clearly noted that the QZS conditions for the exact and approximate forcedeflection curves, given in Eq. (2) and (3) respectively, do not correspond. Furthermore, for most of the values of bˆ , the approximate force-deflection curve predicts an unstable behaviour for large deflections, which do not manifest in the real system. This is better illustrated in Fig. 6(a-f), where exact (thick solid lines) and approximate (thin dash-dotted lines) force and stiffness curves are plotted for three different combinations of the parameters in Table 1. In particular, it may be noted in Fig. 6(a,d) and (b,e), that the approximate equivalent spring stiffness (thin dash-dotted lines) becomes negative for displacement values greater than that at which the maximum occurs,
10
and this leads to static instability. In Fig. 6(c,f) it may be noted that the stiffness of the approximate model is always strictly positive and do not exhibit any QZS behaviour.
[Figure 6 around here]
2.4
Approximate force-deflection curve by fitting inflexion point
To overcome the limitations of the previous model, which is good for small displacement only, the exact force-deflection curve in Eq. (2) is approximated by a new polynomial equation with the following four boundary conditions: (i) the stiffness at zero deflection is equal to that from Taylor expansion; (ii) the stiffness at the inflection point is zero; (iii) the derivative of the stiffness at the inflection point is zero; (iv) the force value at the inflection point is equal to that of the exact expression in Eq. (2). To satisfy these conditions, a seventh-order odd polynomial equation is selected, and this motivated the former Taylor series expansion up to the same order for comparison. Equation (2) is thus approximated by
fˆs , F kˆ1, F xˆ kˆ3, F xˆ 3 kˆ5, F xˆ 5 kˆ7, F xˆ 7 ,
(5)
where
1 bˆ 2 1 ˆ 10 kˆ1, F kˆ1,T kˆ1 4 , k3, F kˆ5, F xˆP2 7kˆ7, F xˆP4 , 2 3 1 bˆ . ˆ 14kˆ xˆ 6 ˆ ˆ k f k 15 7, F P ˆ P kˆ5, F 1 , k7, F 1. 5 xˆP4 8 xˆP7 xˆP6
11
(6a-d)
Equation (5) and its derivative are also plotted in Fig. 6 as thicker dashed lines. It can be noted that, this latter approximation better fits the exact force-deflection curve of the mechanical oscillator illustrated in Fig. 1 up to the QZS condition, avoiding an erroneous prediction of instability for large deflections. It should be noted that no boundary condition has been set for xˆ 1 , so that there is no control on the stiffness behaviour over that region, except that it is
forced to be non-negative, as in the exact static behaviour of the mechanical oscillator.
3
DYNAMIC MODEL OF THE SYSTEM
3.1
Amplitude-frequency equation
The equation of motion of the system depicted in Fig. 1(a) is given by
mx cx f s x F cos t ,
(7)
where the static spring restoring force f s x is given by Eq. (1), the overdots denote differentiation respect to time t, a harmonic force with amplitude F and angular frequency is assumed to excite the oscillator mass m, and a viscous damping c is introduced as a dissipative term. By using one of the approximate expressions for the spring restoring force given in Eq. (3) or (5), Eq. (7) may be conveniently written in non-dimensional form as
xˆ 2 xˆ xˆ xˆ 3 xˆ 5 xˆ 7 cos ,
12
(8)
where 1t and 1 are the non-dimensional time and frequency respectively,
1 kkˆ1 m , c 2m1 is the damping ratio, F akkˆ1 is the non-dimensional force amplitude, primes denote differentiation respect to , a phase is introduced for convenience, and kˆ3 kˆ1 , kˆ5 kˆ1 , kˆ7 kˆ1 , where kˆ3 , kˆ5 , kˆ7 may be set equal to Eqs. 4(b-d) or 6(b-d), depending on the approximation adopted. To solve Eq. (8) in closed-form in terms of the amplitude-frequency equation, it is further assumed that the system response is predominately harmonic at the excitation frequency, i.e.
xˆ Xˆ cos , and this is substituted into Eq. (8), where a first-order harmonic balance approximation is applied to yield
4 Xˆ 2 2 4 2 Xˆ 2 2 Xˆ G Xˆ G Xˆ
2
2 0 ,
(9)
3 5 35 where G Xˆ Xˆ Xˆ 3 Xˆ 5 Xˆ 7 . 4 8 64
It can be noted that the approximate expression in Eq. (9) is quadratic in 2 , it may thus be solved in closed-form and plotted into an amplitude-frequency diagram, as detailed in Section 3.3 down below.
3.2
Stability of the harmonic solution
The stability of the approximate steady-state solutions of Eq. (9) is calculated by applying Floquet theory as described in [12]. To this aim, a disturbance u is added to the approximate harmonic solution and substituted into Eq. (8) to yield the following linearized equation 13
u 2 u u 3 xˆ 2u 5 xˆ 4u 7 xˆ 6u 0 .
(10)
Equation (10) admits a solution in the form of u e [13], where is a periodic function of period 2 , which can be expanded in Fourier series to the first term as
C cos S sin , and substituted back into Eq. (10). By then equating the harmonic coefficients on both sides of the resulting equation, a system of two linear homogeneous algebraic equations in the unknowns C and S is obtained, which admits a nontrivial solution if and only if the determinant of the coefficient matrix is equal to zero. This leads to the following fourth-order polynomial equation in terms of
3 ˆ 2 5 ˆ 4 35 ˆ 6 2 2 2 1 X X X 4 8 64 , 9 ˆ 2 25 ˆ 4 245 ˆ 6 2 2 2 2 X 4 0 2 1 X X 4 8 64
(11)
which can be solved numerically for specific values of the system parameters. The harmonic response derived in Section 3.1 is then asymptotically stable if and only if all the solutions for have a negative real part, while it is unstable if at least one of them has a positive real part. In the frequency response curves (FRCs) reported in this paper, approximate stable solutions are represented by solid lines, while approximate unstable solutions are represented by dashed lines.
3.3
Frequency response curves 14
The FRCs are plotted in Figs. 7(a-f) for three different combinations of the system parameters and bˆ , as reported in Table 1. Figures 7(a-c) show the FRCs obtained from Eq. (9) when the coefficients in Eqs. 4(a-d) are used, while Figs. 7(d-f) show the FRCs obtained when the coefficients in Eqs. 6(a-d) are adopted. Also, superimposed in Figs. 7(a-f) as markers, are the numerical solutions for validation. In particular, the numerical solutions were obtained by numerically integrating the equation of motion, where the exact expression for the static spring restoring force from Eq. (2) is used. Fourier coefficients were then extracted from the calculated time histories at steady-state. Only those coefficients corresponding to the excitation frequency are then plotted as black markers in the FRCs shown in the paper. Nevertheless, it was verified that, for the parameters used in this work, the amplitudes of the higher and lower order harmonics never exceeded 5% of the amplitude at the fundamental one.
[Figure 7 around here]
The classical softening effect which causes the resonance peak to bend towards the lower frequencies can be seen, whereas for a hardening system, it bends to the higher frequencies [13]. It can be noted that the approximation proposed in Section 2.4 better predicts the numerical solution around the resonance, and this is more evident at lower frequencies (and higher amplitudes) where the Taylor series approximation presented in Section 2.3 incorrectly predicts an instability due to negative stiffness. In particular, the FRCs in Fig. 7(a) and (b), which are based on the Taylor series expansion, show an instability (dashed line) of the harmonic solution in the upper branch of the FRC, which does not manifest in the numerical solution (black markers), while the FRCs in Fig. 7(d) and (e), based on the approximation proposed in Section
15
2.4, better match the numerical solution. The FRC in Fig. 7(c) shows a less evident softening effect at resonance than that shown in Fig. 7(f), and this is due to the presence of the QZS behaviour, which is not captured by the Taylor series expansion – rather, it is well detected by the approximation derived in Section 2.4.
3.4
Resonance frequency and corresponding displacement amplitude
The approximate closed-form expression of the amplitude-frequency equation reported in Eq. (9) can be further exploited to get an insight into the relation between the resonance peak and its corresponding frequency. In fact, it is noted that the resonance frequency occurs when the peak displacement is a maximum, and this happens when the two solutions to Eq. (9) in terms of 2 are equal. This condition is determined by the following equation
4 2 Xˆ 2 2 Xˆ G Xˆ 4 Xˆ 2 G Xˆ 2
2
2 0 ,
(12)
which can be solved in terms of the damping ratio to give
2 r
G Xˆ G Xˆ
2
2 Xˆ
2 .
(13a,b)
The approximate expression for the resonance frequency is then obtained by substituting Eq. (13a,b) into Eq. (9) and rearranging to give
16
2 r
G Xˆ
Xˆ
2
2 ,
(14a,b)
where it can be noted that a real solution only occurs for a damping ratio given by Eq. (13b). Equation (14b) is thus plotted in Fig. 8(a), for the combination of parameters corresponding to
in Table 1, and for different values of the excitation . For the range of parameters used to plot Fig. 8(a), it can be noted that there is a minimum value for the resonance frequency, and this is substantially independent from . Furthermore, the corresponding displacement amplitude at this lowest resonance frequency is also independent from [Figure 8 around here] An approximate value for such a lowest resonance frequency and its corresponding displacement amplitude can be obtained by differentiating the square root of Eq. (14b) respect to Xˆ for = 0, equating to zero and solving numerically to give Xˆ r 1.3 and r 0.767 . It is worth noting that for = 0, Eq. (14b) gives the expression of the backbone curve of the undamped, unforced response of the nonlinear oscillator. It is now interesting to set Xˆ Xˆ r into Eq. (13b) and plot it as a function of the excitation in Fig. 8(b). It may then be seen how the relation between the excitation amplitude and the damping ratio to achieve the lowest resonance frequency is almost linear for the range of values used to plot Fig. 8(b), being r 0.5 .
17
The FRCs for two different values of excitation and for a damping ratio given from Fig. 8(b) are plotted in Fig. 9(a) and (b) as thicker lines. It can be noted that the resonance frequency in both cases is at about 0.77 and the amplitude at resonance is about 1.3. In both figures, thinner lines are used to plot the FRCs for slightly different values of damping, to show that the damping value given from Fig. 8(b) is the one to achieve the lowest value for the resonance frequency.
[Figure 9 around here]
It can be noted that the shape of the frequency response curve of the system shown in Fig. 9, recalls that of an asymmetric Duffing oscillator [14], which has been adopted to investigate the characteristics of a mistuned isolator with QZS behaviour [15-16]. However, this latter model is conceptually different from that described in this paper - in particular, the asymmetric Duffing oscillator has only one configuration with QZS and it has a fundamentally hardening behaviour for large deflections, while the system considered here, has two symmetric configurations with QZS conditions and it has a fundamentally softening behaviour in the region of interest. Such softening behaviour, arising from the QZS conditions for large deflections, has also been exploited to increase the elastic potential energy in a spring-like system [17].
4
CONCLUSIONS
This paper has investigated the static and approximate dynamic characteristics of a nonlinear oscillator with a softening characteristic and a QZS behaviour for large deflections. The specific nonlinearity is realized by assembling four linear springs in a geometric arrangement to achieve the desired characteristic. The proposed configuration allows to exploit the benefit of a softening system up to its fundamental limit. The approximate dynamic model developed avoids an 18
erroneous prediction of instability due to the presence of negative stiffness components and the use of a different approximation. The static force-deflection curve is approximated by a polynomial equation with specific coefficients, and it is incorporated into the dynamic equation of motion to derive an approximate closed-form expression of the amplitude-frequency equation. The effect of the system parameters is investigated, and it is found that a simple relation among them exists such that a minimum value for the resonance frequency is achieved. It is believed that the proposed system configuration, and the fundamental insight developed in this paper, will stimulate further theoretical and experimental advances towards a better understanding and exploitation of nonlinear phenomena, and have potential benefits in the design of engineering systems and structures with improved and targeted dynamic performance.
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[11] Korn GA, Korn TM. Mathematical Handbook for Scientists and Engineers. New York: McGraw-Hill; 1961. [12] Malatkar P, Nayfeh AH. Steady-state dynamics of a linear structure weakly coupled to an essentially nonlinear oscillator. Nonlinear Dyn 2007;47:167-79. https://doi.org/10.1007/s11071006-9066-4
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[13] Ramlan R, Brennan MJ, Kovacic I, Mace BR, Burrow SG. Exploiting knowledge of jumpup and jump-down frequencies to determine the parameters of a Duffing oscillator. Commun Nonlinear Sci 2016;37:282-91. http://dx.doi.org/10.1016/j.cnsns.2016.01.017 [14] Kovacic I, Brennan MJ , Lineton B. On the resonance response of an asymmetric Duffing oscillator. Int J Nonlin Mech 2008;43:858-67. https://doi.org/10.1016/j.ijnonlinmec.2008.05.008 [15] Huang X, Liu X, Sun J, Zhang Z, Hua H. Effect of the system imperfections on the dynamic response of a high-static-low-dynamic stiffness vibration isolator. Nonlinear Dyn 2014;76:1157-67. https://doi.org/10.1007/s11071-013-1199-7 [16] Abolfathi A, Brennan MJ, Waters TP, Tang B. On the Effects of Mistuning a Force-Excited System Containing a Quasi-Zero-Stiffness Vibration Isolator. J Vib Acoust 2015;137:044502. https://doi.org/10.1115/1.4029689
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Figure Captions List Fig. 1
Nonlinear oscillator under study: (a) undeformed configuration at rest, (b) deformed configuration
Fig. 2
Non-dimensional force-deflection curve for different values of the system parameters: (a) = 0.8 and bˆ = 0.2 (dotted line), bˆ = 0.4 (solid line), bˆ = 0.6 (dashed line); (b) = 1.0 and bˆ = 0.3 (dotted line), bˆ = 0.5 (solid line), bˆ = 0.7 (dashed line); (c) = 1.2 and bˆ = 0.5 (dotted line), bˆ = 0.7 (solid line), bˆ = 0.9 (dashed line)
Fig. 3
Graphical solution of xˆ and bˆ to achieve a QZS behaviour. Thicker solid curve represents the second derivative of Eq. (2) set to zero. Other thinner lines represent the first derivative of Eq. (2) set to zero for = 1 (dotted line), = 1.2 (dashed line), = 1.3 (solid line) and = 1.4 (dashed-dot line).
Fig. 4
(a) Non-dimensional force-deflection curve and (b) stiffness curve for = 0.7 and bˆ = 0.3763 (dotted line), = 1.0 and bˆ = 0.5955 (solid line), = 1.3 and bˆ = 1.0117 (dashed line)
Fig. 5
Region of parameters identifying the stiffness behaviour of the approximate force-defection curve in Eq. (3), in terms of existence of extrema. In the lighter shaded region, the approximate stiffness presents one maximum; in the white region, it presents two extrema (one maximum and one minimum); in the darker shaded region, it does not 22
present any extremum. The dashed line connecting markers is plotted from Table 1. Fig. 6
Non-dimensional force-deflection curve (a-c) and stiffness curve (d-f) for: (a,d) = 0.8 and bˆ = 0.4417; (b,e) = 1.1 and bˆ = 0.6926; (c,f) = 1.2 and bˆ = 0.8173. Solid lines are from Eq. (2), thin dash-dotted lines are from Eq. (3), and dashed lines are from Eq. (5)
Fig. 7
Approximate analytic FRCs using Eq. (9) with parameters defined in Eqs. 4(a-d) for (a-c), and with parameters defined in Eqs. 6(a-d) for (d-f), for = 0.02, = 0.04 and: (a,d) = 0.8 and bˆ = 0.4417; (b,e) = 1.0 and bˆ = 0.5955; (c,f) = 1.2 and bˆ = 0.8173. Approximate analytic stable (solid lines) and unstable solutions (dashed lines), numerical solutions (black markers).
Fig. 8
(a) Resonance frequency as a function of displacement amplitude, for = 0.02 (dashed line), = 0.04 (dash-dot line), = 0.06 (dotted line), = 0.1 (solid line). (b) Damping ratio to achieve the lowest resonance frequency as a function of excitation.
Fig. 9
FRCs for = 1.0 and bˆ = 0.5955, for (a) = 0.1 and (b) = 0.02, with damping given from Fig. 8(b) to achieve the lowest resonance frequency. Approximate analytic stable (solid lines) and unstable solutions (dashed lines), numerical solutions (black markers). Thinner lines denote the FRC for slightly different values of damping as labelled.
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Video Captions List Video 1
Relations among the systems parameters to achieve a QZS behaviour
fig .1
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fig 2
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fig. 3
26
fig. 4
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fig. 5
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fig. 6
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fig. 7
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fig. 8
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fig. 9
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fig. 10
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Author Contribution Statement There is only one author for this paper, who contributes to all the aspects described therein.
DIS Nothing to declare
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Table 1. Conditions for the presence of an inflection point in the force-deflection curve, and its coordinates as function of the spring natural length factor
Pre-tension
Unloaded
Pre-compression
0.7
0.8
0.9
1
1.1
1.2
1.3
bˆ
0.3763
0.4417
0.5137
0.5955
0.6926
0.8173
1.0117
Inflexion
xˆ P
0.9996
0.9991
0.9981
0.9962
0.9924
0.9841
0.9601
Point
fˆP
2.5299
2.2920
2.0400
1.7690
1.4712
1.1308
0.6996
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