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Abstract

The main purpose of this article is to investigate the qualitative effects of different factors on the response of forced vibration of shape memory alloy spring oscillator. Specially, the introduction of dry friction factor causes the governing equations to change from a smooth system to a (Filippov-type) non-smooth system in which the sliding phenomenon was observed. The mechanism, geometric structure, and analytic conditions of sliding bifurcations in a general n-dimensional piecewise smooth system were discussed in detail. The theoretical results obtained are verified by numerical analysis, and the feasibility of involved theories is estimated by calculation of sliding time.

Keywords

SMA spring oscillator bifurcation non-smooth sliding motions

Introduction

Shape memory alloys (SMA) have been manufactured into intelligent and parameter-controllable components for deformation and vibration control of engineering structures and mechanical equipment due to their unique mechanical properties, such as shape memory effect and superelasticity.¹ SMA helical spring² is the most widely used SMA component. It has been used in automotive thermostat, gas pipeline fire valve, robot claw and overcurrent protector, and other devices as a driver.³ At present, the researches on SMA spring mainly focus on its mechanical properties, deformation simulation, and design principle.⁴ There are also some reports on the dynamic characteristics of SMA spring oscillator,⁵ such as the response of the coupled SMA spring oscillator,⁶ the nonlinear thermo-mechanical response of the single SMA spring oscillator,^7,8 and the response of the non-smooth SMA spring oscillator.⁹ However, the theoretical research on different comprehensive factors (temperature, damping, external excitation, dry friction, etc.) of the system is still lacking. It is worth mentioning that except for dry friction, other factors considered in this article will not change the smoothness^10,11 of governing equations of the system. Therefore, new theoretical methods need to be found to study the effects of dry friction factor on the system.

In this article, the restoring force of SMA spring is described by the polynomial constitutive relation of SMA,^12,13 and the nonlinear dynamic model of forced vibration of SMA spring oscillator is established. The response of the system is analyzed by means of bifurcation diagrams, Lyapunov exponent spectrums, time history diagrams, phase portraits, and Poincaré cross-sectional drawings. The bifurcation and chaos of this system are mainly discussed. The non-smooth differential equations with dry friction are discussed separately due to its particularity.

Restoring force model of SMA spring

The constitutive model of SMA proposed by Falk¹⁴ is applied in this article. The constitutive equation of the model is

σ = a (T - T_{M}) ϵ - b ϵ^{3} + e ϵ^{5}

(1)

in which a, b, and e are material constants, $σ$ is stress, $ϵ$ is strain, T is temperature, and $T_{M}$ is critical temperature for martensitic transformation.

$T_{A}$ (austenite transformation critical temperature) is one of the important temperature parameters of SMA material. The relationship among $T_{M}$ , a, b, e, and $T_{A}$ can be written as

T_{A} = T_{M} + \frac{b^{2}}{4 ae}

(2)

Therefore, the restoring force model of SMA spring can be established as

F (x, T) = \bar{a} (T - T_{M}) x - \bar{b} x^{3} + \bar{e} x^{5}

(3)

in which $\bar{a} = aA / L$ , $\bar{b} = bA / L^{3}$ , and $\bar{e} = eA / L^{5}$ (A is the area of SMA spring wire, L is the original length of SMA spring, and x is the displacement).

Control equations for forced vibration of SMA spring oscillator

SMA spring oscillator system consists of a mass block with mass m, a linear damper with damping coefficient c, and the SMA spring. The mass block is subjected to harmonic excitation force $F (t) = \bar{F} \sin (Ω t)$ , as shown in Figure 1. The first part of this article focused on an ideal oscillator model without considering the dry friction factor.

Figure 1.

Forced vibration system of shape memory alloy (SMA) spring oscillator.

By Newton’s second law, the differential equation of vibration of the system can be expressed as

m \overset{\cdot\cdot}{x} + c \overset{\cdot}{x} + \bar{a} (T - T_{M}) x - \bar{b} x^{3} + \bar{e} x^{5} = \bar{F} \sin (Ω t)

(4)

The dimensionless transformation can be written as follows

\begin{matrix} X = \frac{x}{L}, τ = ω_{0} t, ω_{0}^{2} = \frac{aA T_{M}}{mL}, B = \frac{bA}{mL ω_{0}^{2}}, \\ E = \frac{eA}{mL ω_{0}^{2}}, ξ = \frac{c}{ω_{0}}, θ = \frac{T}{T_{M}}, H = \frac{\bar{F}}{mL ω_{0}^{2}}, \\ ω = \frac{Ω}{mL ω_{0}}, θ_{A} = \frac{T_{A}}{T_{M}} = 1 + \frac{B^{2}}{4 E} \end{matrix}

(5)

where $X, τ, ω, H, ξ$ , and $θ$ are dimensionless displacement, time, frequency of exciting force, amplitude of exciting force, damping coefficient, and temperature, respectively. The dimensionless equation of forced vibration of the system can be expressed as

\overset{\cdot\cdot}{X} + ξ \overset{\cdot}{X} + (θ - 1) X - B X^{3} + E X^{5} = H \sin (ω τ)

(6)

{\begin{matrix} \overset{\cdot}{X} = Y \\ \overset{\cdot}{Y} = H \sin (ω τ) - ξ Y - (θ - 1) X + B X^{3} - E X^{5} \end{matrix}

(7)

Formula (7) shows that the nonlinearity of the SMA material constitutive relation leads to the nonlinearity of the system state. The dynamic behaviors of the system are also affected by environmental temperature, damping, external exciting force, and other factors (such as dry friction), so the nonlinear dynamical response of the system is bound to be complex.

Numerical solution and analysis

Considering that the state equation of the system contains a quintic nonlinear term, the fourth-order Runge-Kutta method is used to solve the state equation of the system (7)

{\begin{matrix} \overset{\cdot}{y} = f (x, y) \\ y (x_{0}) = y_{0} \end{matrix}

(8)

Assuming that the solution of the typical initial value problem (8) is $y = y (x)$ , its Taylor expansion is

\begin{matrix} y (x_{n + 1}) = y (x_{n}) + hy' (x_{n}) + \frac{h^{2}}{2!} y ″ (x_{n}) \\ + \frac{h^{3}}{3!} y ″' (x_{n}) + \dots \end{matrix}

(9)

Runge-Kutta method is essentially a numerical method indirectly using Taylor series. For the difference quotient $\frac{y (x_{n + 1}) - y (x_{n})}{h}$ , according to differential mean value theorem, $\exists θ \in (0, 1)$ satisfying

\frac{y (x_{n + 1}) - y (x_{n})}{h} = y' (x + θ h)

(10)

Therefore, $y (x_{n + 1}) = y (x_{n}) + hf (x_{n} + θ h, y (x_{n} + θ h))$ . Defining $K^{*} = f (x_{n} + θ h, y (x_{n} + θ h))$ as the average gradient in $[x_{n}, x_{n + 1}]$ , the $K^{*}$ in the fourth-order Runge-Kutta method can be calculated from the following expressions

{\begin{matrix} y_{n + 1} & = y_{n} + \frac{h}{6} (K_{1} + 2 K_{2} + 2 K_{3} + K_{4}) \\ K_{1} & = f (x_{n}, y_{n}) \\ K_{2} & = f (x_{n} + \frac{h}{2}, y_{n} + \frac{h}{2} K_{1})) \\ K_{3} & = f (x_{n} + \frac{h}{2}, y_{n} + \frac{h}{2} K_{2}) \\ K_{4} & = f (x_{n} + h, y_{n} + h K_{3}) \end{matrix}

(11)

In each step of the fourth-order Runge-Kutta method, the function value f has to be calculated four times, so its truncation error is lower while calculation accuracy is higher.

Nonlinear dynamical response of the system is analyzed by means of bifurcation diagrams, Lyapunov exponential spectrums, phase portraits, time history diagrams, and Poincaré cross-sectional drawings. In numerical simulations, the values of fixed relevant parameters are as follows

B = 1.3 \times 10^{3}, E = 4.7 \times 10^{5}, θ_{A} = 1.9, ω = 1

(12)

Influence of temperature on system response

In this case, the system parameters are shown in Table 1. When $ξ = 0.1$ and $H = 0.06$ , the global bifurcation diagram and the corresponding Lyapunov exponential spectrum are obtained by changing the dimensionless temperature $θ$ . As can be seen from Figure 2, the system can be transformed between periodic and chaotic states by changing the temperature only.

Table 1.

Parameter table with varying $θ$ .

Coefficients	Values
B	$1.3 \times 10^{3}$
E	$4.7 \times 10^{5}$
$θ_{A}$	1.9
$ω$	1
$ξ$	0.1
H	0.06
$θ$	[0.0, 4.0]

Figure 2.

(a) The bifurcation diagram and (b) Lyapunov exponent spectrum with $ξ = 0.1, H = 0.06$ .

Longitudinal coordinate X denotes dimensionless displacement, while $λ_{\max}$ represents maximum Lyapunov exponent. For an n-dimensional dynamical system $x_{n + 1} = F (x_{n})$ , if the trajectories with two adjacent initial points in a dynamic system separated exponentially over time, this phenomenon can be described by the Lyapunov exponent $λ$ . Assume that the average exponent in the exponential separation caused by each iteration is $λ$ . The distance of two initial points with initial distance $ε$ after n-times iterations is $ε \cdot e^{n λ (x_{0})} = | F^{n} (x_{0} + ε) - F^{n} (x_{0}) |$ . Taking the limit $ε \to 0, n \to + \infty$ , Lyapunov exponent can be defined as

λ = lim_{n \to \infty} \frac{1}{n} \sum_{i = 1}^{n - 1} \ln {| \frac{d F (x)}{d x} |}_{x = x_{i}}

(13)

Lyapunov exponent can describe the motion characteristics of the system. It takes positive and negative values along a certain direction to indicate the average divergence or convergence rate of adjacent orbits along the direction of the system under the action of attractors for a long time. When Lyapunov exponent $λ < 0$ , the phase volume will shrink, the motion state of the system will tend to be stable, and the system is insensitive to the initial state. On the contrary, when $λ > 0$ , the two adjacent trajectories will gradually separate, making the system finally enter the chaotic state. If the system is in chaotic state, there must be a Lyapunov exponent greater than zero. Therefore, the sign of Lyapunov exponent is an important basis for judging whether a system is chaotic or not.

It can be seen from Figure 2 that when $θ$ changes between $[0.05, 0.84]$ , the corresponding region on the bifurcation diagram is a set of dense points. Meanwhile, the maximum Lyapunov exponent of the corresponding region on the Lyapunov exponent spectrum is positive, indicating that the corresponding system motions in this region are chaotic.

When $θ$ varies in other intervals, the bifurcation diagram turns to be a finite number of points, and the maximum Lyapunov exponent is negative, implying that the corresponding motions of the system in these regions are periodic.

At bifurcation points $θ = 0.05$ and $θ = 0.84$ , the bifurcation diagram and the Lyapunov exponential spectrum have mutations. The results show that with the increase of dimensionless temperature $θ$ , the motion of the system changes from periodic motion to chaotic motion then to periodic motion.

The phase portraits and Poincaré cross-sectional drawings of the system at two temperatures are presented in Figures 3 and 4 where ordinate Y represents dimensionless velocity. In the first case with $θ = 0.70$ , the phase trajectory is an irregular curve, and strange attractors appear on the Poincaré section. Besides, the maximum Lyapunov exponent is $0.2103$ , indicating that the system is in chaotic motion.

Figure 3.

(a) The phase portrait and (b) Poincaré cross-sectional drawing with $θ = 0.70$ .

Figure 4.

(a) The phase portrait and (b) Poincaré cross-sectional drawing with $θ = 1.76$ .

When $θ = 1.76$ , the phase trajectory is a closed curve. There is only one fixed point on the Poincaré section with $λ_{\max} = - 0.0505$ , implying that the motion of the system is periodic.

Influence of damping coefficient on system response

In this case, the coefficients are shown in Table 2. When the amplitude of dimensionless excitation force is fixed at $H = 0.06$ and the dimensionless temperatures $θ$ are fixed at 0.7, 1.7, and 2.7, the global bifurcation diagrams and corresponding Lyapunov exponential spectrums of the system with the varying dimensionless damping coefficient $ξ$ are shown in Figures 5 –7. It can be clearly observed that when the damping coefficient increases gradually, the system undergoes chaotic, quasi-periodic motions and finally becomes periodic.

Table 2.

Parameter table with varying $ξ$ .

Coefficients	Values
B	$1.3 \times 10^{3}$
E	$4.7 \times 10^{5}$
$θ_{A}$	1.9
$ω$	1
H	0.06
$θ$	0.7/1.7/2.7
$ξ$	[0.0, 0.4]

Figure 5.

(a) The bifurcation diagram and (b) Lyapunov exponent spectrum with $θ = 0.7$ .

Figure 6.

(a) The bifurcation diagram and (b) Lyapunov exponent spectrum with $θ = 1.7$ .

Figure 7.

(a) The bifurcation diagram and (b) Lyapunov exponent spectrum with $θ = 2.7$ .

It can be seen from the Figure 5 that the system is in chaotic motion at low temperature $(θ < 1)$ with a wide range of damping coefficient $(0 \leq ξ \leq 0.21)$ . When the value of dimensionless damping coefficient $ξ$ is greater than the value of bifurcation point $0.21$ , the system enters periodic and quasi-periodic motion.

Under the intermediate temperature state $(1 < θ < θ_{A})$ and high temperature state $(θ > θ_{A})$ , the range of $ξ$ with the system in chaotic motion state is very narrow $(0 \leq ξ \leq 0.005)$ .

When $θ = 1.7$ , the system goes through quasi-periodic motion and enters periodic motion with the increase of dimensionless damping coefficient $ξ$ .

When $θ = 2.7$ , the system enters the periodic motion directly. The periodic-4 motion occurs when $ξ = 0.02$ , while the periodic-5 can be obtained when $ξ = 0.05$ . Except for the two cases, the other periodic motions are all in the form of periodic-1.

Taking the medium temperature state as an example to verify the validity of bifurcation analysis results, the evolution of system trajectory is observed in detail. When $H = 0.06$ and $θ = 1.7$ , the corresponding time history diagrams, phase portraits, and Poincaré cross-sectional drawings of $ξ$ at $0.0001, 0.0007, 0.0008$ , and $0.01$ are shown in Figures 8 –11, respectively.

Figure 8.

(a) The time history diagram, (b) phase portrait, and (c) Poincaré cross-sectional drawing with $ξ = 0.0001$ .

Figure 9.

(a) The time history diagram, (b) phase portrait, and (c) Poincaré cross-sectional drawing with $ξ = 0.0007$ .

Figure 10.

(a) The time history diagram, (b) phase portrait, and (c) Poincaré cross-sectional drawing with $ξ = 0.0008$ .

Figure 11.

(a) The time history diagram, (b) phase portrait, and (c) Poincaré cross-sectional drawing with $ξ = 0.01$ .

As shown in Figure 8, when $ξ = 0.0001$ , the time history diagram is a random aperiodic curve. The phase portrait is an irregular trace, and the Poincaré cross-sectional drawing consists of a set of scattered points with layers, so the system can be judged to be chaotic.

As shown in Figure 9, when $ξ = 0.0007$ , the time history diagram has definite periodicity. The phase trajectory is a closed curve, and there are only a finite number of fixed points on the Poincaré section, indicating that the system is in periodic motion.

As shown in Figure 10, when $ξ = 0.0008$ , the time history diagram is a non periodic oscillation though it seems to be periodic. The phase portrait is not a closed curve, but the trajectory concentrated in a closed zone, and the Poincaré cross-sectional drawing approximates a closed curve, implying that the system is in quasi-periodic motion. Finally, the system enters the periodic state at $ξ = 0.01$ (Figure 11). The whole evolution process of phase trajectory is consistent with the corresponding process on its bifurcation diagram Figure 6(a).

Influence of external excitation amplitude on system response

In this case, parameters are set as shown in Table 3. Figures 12 –14 are the global bifurcation diagrams and Lyapunov exponent spectrums of the system varying with the amplitude H of the dimensionless exciting force. It can be seen that the coincidence degree between bifurcation diagram and exponential spectrum in the relevant interval is high.

Table 3.

Parameter table with varying H.

Coefficients	Values
B	$1.3 \times 10^{3}$
E	$4.7 \times 10^{5}$
$θ_{A}$	1.9
$ω$	1
$ξ$	0.1
$θ$	0.7/1.7/2.7
H	[0.0, 0.2]

Figure 12.

(a) The bifurcation diagram and (b) Lyapunov exponent spectrum with $θ = 0.7$ .

Figure 13.

(a) The bifurcation diagram and (b) Lyapunov exponent spectrum with $θ = 1.7$ .

Figure 14.

(a) The bifurcation diagram and (b) Lyapunov exponent spectrum with $θ = 2.7$ .

It can be seen from the graphs that the motion of the system changes alternately between periodic motion and chaotic motion with the increase of H at any kind of temperature states. At different temperatures, corresponding periodic and chaotic motion intervals are different.

It is noteworthy that the displacement of the periodic motion jumps with the change of H, which is mainly caused by the selection of the initial value instead of bifurcation. Different initial point selection leads to the convergence of phase trajectories to different equilibrium states.

When $θ = 2.7$ , the system has a period doubling bifurcation leading to chaotic motion with H varying between $[0.1, 0.12]$ . To illustrate the bifurcation process, the time history diagrams, phase portraits, and Poincaré cross-sectional drawings of the system with $ξ = 0.1, θ = 2.7$ , and different H are given in Figures 15 –18.

Figure 15.

(a) The time history diagram, (b) phase portrait, and (c) Poincaré cross-sectional drawing with $H = 0.1$ .

Figure 16.

(a) The time history diagram, (b) phase portrait, and (c) Poincaré cross-sectional drawing with $H = 0.108$ .

Figure 17.

(a) The time history diagram, (b) phase portrait, and (c) Poincaré cross-sectional drawing with $H = 0.11$ .

Figure 18.

(a) The time history diagram, (b) phase portrait, and (c) Poincaré cross-sectional drawing with $H = 0.12$ .

It can be seen from Figure 15 that when $H = 0.1$ , there is only one fixed point on Poincaré cross-section, so the system is in periodic-1 motion.

When $H = 0.108$ , there are two fixed points on Poincaré cross-section, which indicates that the system is in periodic-2 motion.

For the same reason, Figure 16 shows that system is in periodic-4 motion.

When $H = 0.12$ , strange attractor occurs on Poincaré cross-section, resulting in the chaotic motion of the system.

The motion characteristics of the system with different values of H can also be verified by the time history diagrams and phase portraits. Due to the article length limitation, it is omitted here.

Influence of dry friction factor on system response

To explore the mechanisms of non-smooth SMA oscillator,¹⁵ a conveyer belt with a constant velocity $v_{c}$ was employed to provide dry friction (see in Figure 1). The control equations for the non-smooth SMA spring oscillator can be written as

m \overset{\cdot\cdot}{x} + c \overset{\cdot}{x} + \bar{a} (T - T_{M}) x - \bar{b} x^{3} + \bar{e} x^{5} = \bar{F} \sin (Ω t) + f (v_{rel})

(14)

where $v_{rel} = \overset{\cdot}{x} - v_{c}$ and

f (v_{rel}) = {\begin{matrix} - f_{0}, & v_{rel} > 0 \\ [- f_{0}, + f_{0}], & v_{rel} = 0 \\ + f_{0}, & v_{rel} < 0 \end{matrix}

(15)

By the dimensionless transformation $(5)$ and

\begin{matrix} V_{c} = \frac{v_{c}}{L ω_{0}}, G (v_{rel}) = \frac{f (v_{rel})}{mL ω_{0}^{2}} \\ G_{0} = \frac{f_{0}}{mL ω_{0}^{2}}, x_{1} = X, x_{2} = \overset{\cdot}{X} \end{matrix}

(16)

the control equations can be described as

{\begin{matrix} {\overset{\cdot}{x}}_{1} = x_{2} \\ {\overset{\cdot}{x}}_{2} = H \sin (ω τ) - ξ x_{2} - (θ - 1) x_{1} + {Bx}_{1}^{3} - {Ex}_{1}^{5} + G (x_{2} - V_{c}) \end{matrix}

(17)

Through coordinate transformation $y_{1} = x_{1}, y_{2} = x_{2} - V_{c}$ , the governing equations can be written as

{\begin{matrix} {\overset{\cdot}{y}}_{1} = y_{2} + V_{c} \\ {\overset{\cdot}{y}}_{2} = H \sin (ω τ) - ξ (y_{2} + V_{c}) - (θ - 1) y_{1} + {By}_{1}^{3} - {Ey}_{1}^{5} + G (y_{2}) \end{matrix}

(18)

It is a typical Filippov system (piecewise smooth system).^16,17 The parameters are set as shown in Table 4. The phase portraits are shown in Figure 19 where $y_{1}$ represents dimensionless displacement and $y_{2}$ is the dimensionless velocity with a set of parameters fixed as $H = 3.0, ξ = 0.02, θ = 2.0, V_{c} = 0.2, B = 1.3 \times 10^{3}, E = 4.7 \times 10^{5}, G_{0} = 1.65$ . The sliding phenomenon¹⁸ of trajectory can be clearly seen when the trajectory reached the non-smooth boundary $y_{2} = 0$ in Figure 19(a) where $A_{1}$ denotes the starting point, $A_{2}$ represents the ending point of sliding process while $B_{1}$ is the traversing point.

Table 4.

Parameter table with varying H.

Coefficients	Values
B	$1.3 \times 10^{3}$
E	$4.7 \times 10^{5}$
H	3.0
$ξ$	0.02
$θ$	2.0
$G_{0}$	1.65
$V_{c}$	0.2
$ω$	2.5/4.5

Figure 19.

Phase portrait with (a) $ω = 2.5$ and (b) $ω = 4.5$ .

Actually, in the engineering practice, the sliding (sticking) phenomenon is not always expected. For instance, the occurrence of sliding phenomenon of the lathes for producing molds with SMA spring means the production of defective products.¹⁹ Therefore, finding out a way to eliminate the sliding phenomenon is of great concern. As shown in Figure 19(b), controlling the frequency of external excitation is a relatively simple and effective solution because the change of external excitation invalidates the necessary condition of sliding bifurcation which will be discussed in detail in the next subsection.

Because of discontinuity of the vector field resulting from $G (y_{2})$ , traditional methods, though can be employed to cope with the dynamics of the system in different smooth regions, cannot be directly used to explore the behaviors of the trajectory contacting the non-smooth boundary.

Mechanism and geometric structure of sliding bifurcation

Consider the general n-dimensional piecewise smooth system

\overset{\cdot}{Y} = {\begin{matrix} F_{1} (Y) & , & h (Y) > 0 \\ F_{2} (Y) & , & h (Y) < 0 \end{matrix}

(19)

The definition of the non-smooth boundary can be expressed as $Σ = {Y \in R^{n + 1} | h (Y) = 0}$ . The vector field of sliding regions can be described by the convex closure of $F_{1} (Y)$ and $F_{2} (Y)$ as

F_{s} (Y) = \frac{F_{1} (Y) + F_{2} (Y)}{2} + \frac{F_{1} (Y) - F_{2} (Y)}{2} α (Y)

(20)

in which $α (Y)$ satisfies the condition $- 1 \leq α (Y) \leq 1$ and $h_{Y} F_{S} (Y) = 0$ where $h_{Y} = (\frac{\partial h}{\partial y_{1}}, \frac{\partial h}{\partial y_{2}})^{T} = \nabla h$ (normal vector $h_{Y}$ is perpendicular to the non-smooth boundary, while $F_{S} (Y)$ is tangent to the non-smooth boundary). Therefore, $α (Y)$ can be written as

α (Y) = - \frac{h_{Y} (F_{1} + F_{2})}{h_{Y} (F_{2} - F_{1})}

(21)

The specific expression of the sliding area and its boundary is

\hat{Σ} = {Y \in Σ | - 1 \leq α (Y) \leq 1}

(22)

\partial {\hat{Σ}}^{\pm} = {Y \in Σ | α (Y) = \pm 1}

(23)

$\partial {\hat{Σ}}^{-}$ is the tangent point of $F_{1} (Y)$ and the non-smooth boundary $Σ$ , indicating that $h_{Y} F_{1} (Y) = 0$ . Similarly, $\partial {\hat{Σ}}^{+}$ is the tangent point of $F_{2} (Y)$ and the non-smooth boundary $Σ$ , satisfying the condition $h_{Y} F_{2} (Y) = 0$ .

Taking the derivative of equation (21) with respect to Y, one may find that

\begin{matrix} α_{Y} = - \frac{{(h_{Y} (F_{1} + F_{2}))}_{Y} (h_{Y} (F_{2} - F_{1})) - (h_{Y} (F_{1} + F_{2})) {(h_{Y} (F_{2} - F))}_{Y}}{{(h_{Y} (F_{2} - F_{1}))}^{2}} \\ = - \frac{(h_{Y} F_{1 Y} + h_{YY} F_{1} + h_{Y} F_{2 Y} + h_{YY} F_{2}) (h_{Y} F_{2} - h_{Y} F_{1})}{{(h_{Y} F_{2} - h_{Y} F_{1})}^{2}} \\ + \frac{(h_{Y} F_{2 Y} + h_{YY} F_{2} - h_{Y} F_{1 Y} - h_{YY} F_{1}) (h_{Y} F_{1} + h_{Y} F_{2})}{{(h_{Y} F_{2} - h_{Y} F_{1})}^{2}} \\ = \frac{- (h_{Y} F_{1 Y} + h_{Y} F_{2 Y}) (h_{Y} F_{2} - h_{Y} F_{1}) + (h_{Y} F_{2 Y} - h_{Y} F_{1 Y}) (h_{Y} F_{1} + h_{Y} F_{2})}{{(h_{Y} F_{2} - h_{Y} F_{1})}^{2}} \end{matrix}

(24)

For $Y \in \partial {\hat{Σ}}^{-}$ , $F_{S} (Y) = F_{1} (Y)$ , so $h_{Y} F_{1} = 0$ . According to equation (24), the normal vector of $\partial {\hat{Σ}}^{-}$ can be written as

α_{Y} = - \frac{2}{h_{Y} F_{2}} h_{Y} F_{1 Y}

(25)

There are four possible kinds of sliding bifurcation according to the way that the periodic solution of the system intersects with the boundary of the sliding region $\partial {\hat{Σ}}^{\pm}$ ²⁰ as shown in Figure 20 where $D_{+}$ and $D_{-}$ represent the two sub-phase spaces separated by the non-smooth boundary.

Figure 20.

(a) The crossing sliding bifurcation, (b) The grazing sliding bifurcation, (c) The switching sliding bifurcation, and (d) The adding sliding bifurcation.

Figure 20(a) is used to describe the crossing sliding bifurcation. With the parameter varying, the trajectory traversed the boundary of the sliding region at the bifurcation point and left the sliding area (see trajectory b in Figure 20(a)). With further changes of parameters, the trajectory entered the sliding region, resulting in the appearance of sliding motion (see trajectory c in Figure 20(a)).

Figure 20(b) is the sketch map of grazing sliding bifurcation.²¹ With the parameter varying, the trajectory is tangent across the boundary of the sliding region at the bifurcation point. (see trajectory b in Figure 20(b)). With the further changes of parameters, the trajectory entered the sliding region, resulting in the appearance of sliding motion (see trajectory c in Figure 20(b)).

Figure 20(c) shows the switching sliding bifurcation which is similar to the first kind of sliding bifurcation. The difference between them is that the trajectory remained in the sliding region after crossing the boundary of it (see trajectory b in Figure 20(c)). With further changes of parameters, a segment of the trajectory remained in the sliding region (see trajectory c in Figure 20(c)).

Figure 20(d) describes the adding bifurcation. The main difference between this bifurcation and the above three bifurcations is that the bifurcation trajectory of the system is completely located in the sliding region and tangent to the boundary of the sliding region at the bifurcation point (see trajectory b in Figure 20(d)). With further change of parameters, a segment of the trajectory was still in the sliding region (see trajectory c in Figure 20(d)).

Analytic conditions for sliding bifurcation

The four sliding bifurcations described in “Mechanism and geometric structure of sliding bifurcation” can be accurately described by analytical conditions which are given below.

1. The crossing sliding bifurcation and the grazing sliding bifurcation.

First, because the bifurcation point $Y^{*}$ is on the non-smooth boundary, the following condition must be satisfied

h (Y^{*}) = 0

(26)

Besides, the bifurcation point $Y^{*} \in \partial {\hat{Σ}}^{-}$ , so it also has to satisfy the condition

\frac{d}{dt} h (ϕ_{1} (Y^{*}, t)) |_{t = 0} = h_{Y} F_{1} (Y^{*}) = 0

(27)

where $ϕ$ is the smooth manifold corresponding to the trajectory. In addition, the trajectory left the sliding region after crossing the boundary of it. Therefore

\frac{d}{d t} α (ϕ_{1} (Y^{*}, t)) |_{t = 0} = α_{Y} F_{1} (Y^{*}) < 0

(28)

By the law of derivation of the dot product and the combination of formulas (25), (27), and (28)

\frac{d^{2}}{d t^{2}} h (ϕ_{1} (Y^{*}, t)) |_{t = 0} = h_{Y} F_{1 Y} F_{1} (Y^{*}) > 0

(29)

2. The switching sliding bifurcation.

It satisfies two basic conditions of equations (26), (27), and

\frac{d}{d t} α (ϕ_{1} (Y^{*}, t)) |_{t = 0} = α_{Y} F_{1} (Y^{*}) > 0

(30)

\frac{d^{2}}{d t^{2}} h (ϕ_{1} (Y^{*}, t)) |_{t = 0} = h_{Y} F_{1 Y} F_{1} (Y^{*}) < 0

(31)

3. The adding bifurcation.

It satisfies two basic conditions of equations (26) and (27). In this case, however, the bifurcation trajectory is completely located in the sliding region and is tangent to the boundary of it at the bifurcation point. Therefore

\frac{d}{d t} α (ϕ_{1} (Y^{*}, t)) |_{t = 0} = α_{Y} F_{1} (Y^{*}) = 0

(32)

Besides, the bifurcation trajectory in this case has a minimum (The first-order derivative is equal to 0 while the second derivative is greater than 0.) at $\partial {\hat{Σ}}^{-}$ . Therefore, the non-degenerate condition can be written as

\frac{d^{2}}{d t^{2}} α (ϕ_{1} (Y^{*}, t)) |_{t = 0} = α_{Y} F_{1 Y} F_{1} (Y^{*}) > 0

(33)

By the law of derivation of the dot product and the combination of formulas (25), (27), and (33)

\frac{d^{3}}{d t^{3}} h (ϕ_{1} (Y^{*}, t)) |_{t = 0} = h_{Y} {(F_{1 Y})}^{2} F_{1} (Y^{*}) < 0

(34)

With all the results obtained, it is easy to calculate the sliding region of the SMA spring oscillator as follows

F_{1} = [\begin{matrix} y_{2} + V_{c} \\ H \sin (ω τ) - ξ (y_{2} + V_{c}) - (θ - 1) y_{1} + {By}_{1}^{3} - {Ey}_{1}^{5} - G_{0} \end{matrix}]

(35)

F_{2} = [\begin{matrix} y_{2} + V_{c} \\ H \sin (ω τ) - ξ (y_{2} + V_{c}) - (θ - 1) y_{1} + {By}_{1}^{3} - {Ey}_{1}^{5} + G_{0} \end{matrix}]

(36)

h (Y) = y_{2}

(37)

Σ = {Y \in R^{3} | h (Y) = y_{2} = 0}

(38)

where $Y = (y_{1}, y_{2})^{T}$ . Normal vector of the non-smooth boundary $h_{Y} = (0, 1)$ . The vector field in the sliding region can be obtained by equation (20)

F_{s} = [\begin{matrix} y_{2} + V_{c} \\ H \sin (ω τ) - ξ (y_{2} + V_{c}) - (θ - 1) y_{1} + {By}_{1}^{3} - {Ey}_{1}^{5} + α G_{0} \end{matrix}]

(39)

where $α (Y)$ can be obtained by equation (21)

\begin{matrix} α (Y) = & - \frac{H \sin (ω τ)}{G_{0}} + \frac{ξ (y_{2} + V_{c})}{G_{0}} + \frac{(θ - 1) y_{1}}{G_{0}} \\ - \frac{({By}_{1}^{3})}{G_{0}} + \frac{({Ey}_{1}^{5})}{G_{0}} \end{matrix}

(40)

So the sliding region can be described as

\hat{Σ} = {Y \in Σ | - 1 \leq - \frac{H \sin (ω τ)}{G_{0}} + \frac{ξ (y_{2} + V_{c})}{G_{0}} + \frac{(θ - 1) y_{1}}{G_{0}} - \frac{({By}_{1}^{3})}{G_{0}} + \frac{({Ey}_{1}^{5})}{G_{0}} \leq 1}

(41)

and the boundary of it can be written as

\partial {\hat{Σ}}^{+} = {Y \in Σ | - \frac{H \sin (ω τ)}{G_{0}} + \frac{ξ (y_{2} + V_{c})}{G_{0}} + \frac{(θ - 1) y_{1}}{G_{0}} - \frac{({By}_{1}^{3})}{G_{0}} + \frac{({Ey}_{1}^{5})}{G_{0}} = 1}

(42)

\partial {\hat{Σ}}^{-} = {Y \in Σ | - \frac{H \sin (ω τ)}{G_{0}} + \frac{ξ (y_{2} + V_{c})}{G_{0}} + \frac{(θ - 1) y_{1}}{G_{0}} - \frac{({By}_{1}^{3})}{G_{0}} + \frac{({Ey}_{1}^{5})}{G_{0}} = - 1}

(43)

Numerical verification and discussion

In order to better explore and understand the mechanism of sliding bifurcation, the sliding process in Figure 19(a) is numerically analyzed as follows, and the feasibility of the above theory is verified by calculating the sliding time.

It is noteworthy that at the starting point $A_{1}$ , the signs of vector field ( ${\overset{\cdot}{y}}_{1}$ ) on both sides of the non-smooth boundary $y_{2} = 0$ are opposite and the difference is exactly equal to $2 G_{0} = 2 \times 1.65 = 3.3$ . Based on the analytic expression of the sliding region of the upper section, it can be found that the vector field of the sliding region has a special structure as shown in Figure 21(a).

Figure 21.

(a) The structure of the vector field in sliding region and (b) The structure of the vector field in traversing region.

Sliding phenomena occur when there are opposite vector fields ( ${\overset{\cdot}{y}}_{1}$ ) on both sides of the non-smooth boundary $y_{2} = 0$ . When the trajectory is forced by the opposite vector field near the boundary, it can only oscillate along the boundary. When the limit of step size of simulation approaches zero, it behaves as an ideal sliding. On the contrary, the direction of vector fields ( ${\overset{\cdot}{y}}_{1}$ ) at traversing point $B_{1}$ on both sides of the interface remains the same indicating that when the same sign vector field exists on both sides of the non-smooth boundary, the trajectory traverses the interface directly, as shown in Figure 21(b). Furthermore, the characteristic of ending point $A_{2}$ is that its vector field ( ${\overset{\cdot}{y}}_{1}$ ) on one side of the non-smooth boundary decreases gradually to zero and eventually reverses; thus, the phase trajectory finally departs from the interface. These results are consistent with the conclusion of analytical expression.

In addition, the sliding time can be estimated according to the variation of excitation $(H \sin (ω τ))$ in Table 5. For $A_{1}$ , $H \sin (ω τ_{1}) = - 1.5965$ . For $A_{2}$ , $H \sin (ω τ_{2}) = 1.6628$ . Therefore, the sliding time can be approached as $| τ_{1} - τ_{2} | = 0.4595$ which coincides with the numerical results $Δ τ = 801.972 - 801.503 = 0.4699$ in the time history of dimensionless velocity $y_{2}$ with $ω = 2.5$ shown in Figure 22.

Table 5.

Simulation data near points of special behaviors.

Motion state	$y_{1}$	$y_{2}$	${\overset{\cdot}{y}}_{1}$	$H \sin (ω τ)$
$A_{1}$ (sliding-starting point)	0.0717	$\approx 0, > 0$	−2.7658	−1.5965
	0.0715	$\approx 0, < 0$	0.5320	−1.5902
$A_{2}$ (sliding-ending point)	0.0202	$\approx 0, > 0$	$\approx 0, < 0$	1.6628
	0.0204	$\approx 0, > 0$	$\approx 0, > 0$	1.6690
$B_{1}$ (traversing point)	0.1051	$\approx 0, > 0$	−3.3209	2.9597
	0.1053	$\approx 0, < 0$	−0.0689	2.9610

Figure 22.

Time history of $y_{2}$ with $ω = 2.5$ .

Conclusion

With the amplitude of exciting force and damping fixed at small values, periodic or chaotic motion can be observed with the temperature varying. When the amplitude of exciting force is fixed at a small value, with the increase of damping, chaotic, quasi-periodic, and periodic motion appear alternatively in the system. In the low temperature state $(θ < 1)$ , the damping coefficient range of the system in the chaotic state is wide, while in the middle temperature state $(1 < θ < θ_{A})$ and the high temperature state $(θ > θ_{A})$ , the damping coefficient range of the system in the chaotic state is very narrow. If the temperature and damping remain unchanged, the system leads to chaos by means of period doubling bifurcation with the variation of exciting force.

The introduction of dry friction factor changes the smoothness of the original system $(7)$ . The special sliding motion resulting from sliding bifurcations can be observed in the Filippov-type system $(18)$ . With different analytic conditions, sliding bifurcations can be divided into four categories according to the corresponding geometric characteristics. The specific expression of the sliding area and its boundary along with the special structures of vector field are presented. It is found that vector fields on both sides of non-smooth boundary are opposite in the sliding region. The feasibility of involved methods is estimated by the calculation of sliding time. The engineering value of the theoretical results obtained is that by means of controlling the frequency of external excitation rather than changing the property of the material itself, the sliding bifurcation can be eliminated.

In addition, this article only discusses the simple case in which dry friction is considered as a single variable temporarily. The coupling between dry friction and other factors will be discussed separately in other literatures.

Footnotes

Handling Editor: Yunn-Lin Hwang

Data Availability

All the data used to support the findings of this study are available from the corresponding author upon request.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research was supported by National Natural Science Foundation of China (Grant no. 11632008).

ORCID iD

Rui Qu

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Forced vibration of shape memory alloy spring oscillator and the mechanism of sliding bifurcation with dry friction

Abstract

Keywords

Introduction

Restoring force model of SMA spring

Control equations for forced vibration of SMA spring oscillator

Numerical solution and analysis

Influence of temperature on system response

Influence of damping coefficient on system response

Influence of external excitation amplitude on system response

Influence of dry friction factor on system response

Mechanism and geometric structure of sliding bifurcation

Analytic conditions for sliding bifurcation

Numerical verification and discussion

Conclusion

Footnotes

Data Availability

Declaration of conflicting interests

Funding

ORCID iD

References