Multiple limit cycles of a dual-cylinder dual-pad brake system caused by dry friction

2.1. System mechanical model and mathematical equations

The structure of the dual-cylinder dual-pad brake is schematically depicted in Fig. 1. The disc brake is fixed on the rim and rotates accordingly. The disc brake is regarded as a plate revolving around point $o$. The two brake pads are fixed on the steering knuckle through brake calipers. The brake pads are elastically connected with each other. The forward direction coordinate $xoy$ is shown in Fig. 1(a), and the side direction coordinate $yoz$ is shown in Fig. 1(b). $v_0$ is the tangential speed of contact point between disc and pad.

Fig. 1Dual-cylinder dual-pad brake

a)

b)

Fig. 2Mechanical model of dual-cylinder dual-pad brake system

Only the tangential motion of discs and pads is studied in this paper. The brake system in Figs. 1(a) and (b) is simplified into a 2DOF mechanical model illustrated in Fig. 2. The disc brake is regarded as a belt moving with a constant speed $v_0$, and the brake pads are represented by two blocks with $m_1$ and $m_2$ mass. Under the effect of constantly actuating speed $v_0$, the two blocks reciprocate from left to right on the belt. The two blocks are connected with the rigid wall by linear spring with varying stiffness $k_1$, $k_2$ and viscous damper $c_1$, $c_2$. The pressing forces between the two mass blocks and the belt are $F_{N1}$ and $F_{N2}$. The two blocks are connected with each other by linear spring $k_0$. The entire system is simplified as a double mass-belt motion.

The differential equations of $x$ direction motion of the system based on Newton’s Second Law are expressed as follows:

where $F_i$ is the dry friction between mass blocks and the belt; $F_i=u{F}_{Ni}$ and $i=\mathrm{}$1, 2 represent the mass block $m_1$ and $m_2$; $u$ is the friction coefficient; $F_N$ is the pressing force between mass blocks and the belt; ${\left(\right)}^{\mathrm{\text{'}}}=d\left(\right)/dt$, ${\left(\right)}^{\text{'}\text{'}}=d^2\left(\right)/d{t}^2$; $t$ is the time.

The stiffness in the aforementioned studies is assumed to be linear; however, the stiffness shows nonlinear effect on system dynamics. This paper focuses on the vibration of the brake pads in the $x$ direction, with the nonlinear stiffness $k$ changing with varying displacement on the $x$ direction. The model [12, 13] of variable stiffness $k$ is expressed as follows:

where $k_{i1}$ is the linear stiffness coefficient, $k_{i2}$ is the square stiffness coefficient, and $k_{i3}$ is the cube stiffness coefficient.

2.2. Selection of brake dry friction model

Dry friction damping force is caused by the absence of lubrication on the interface of two mutually contacting objects, which impedes their relative motion. A sign function is usually used to describe this type of dry friction. The value of dry friction is proportional to the normal pressure between the two contacting objects with its direction opposite to the direction of motion. However, this function describes the physical property of kinetic friction without considering the hysteretic nature of dry friction when the relative velocity is zero. Friction hysteresis is a significant nonlinear behavior mainly displayed as the multi-value and unsmooth nature of friction. This property can generate bifurcation, chaos, and other types of complicated dynamic behavior in the system. Three types of typical hysteresis friction models are currently applied in the brake system.

2.2.2. Mcmillan hysteresis model

In 1997, Mcmillan [16] proposed a new hysteresis dry friction model (Fig. 4). The model divides dry friction into two parts: state dry friction $F_1$ and kinetic dry friction $F_2$. Both are related with relative speed and acceleration on the interface. The mathematical model is expressed as follows:

4

$F_1=-{\mu }_k{F}_N\left(\frac{2}{\pi }\right)\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{t}\mathrm{a}\mathrm{n}\left(\frac{v{\mu }_k}{\left|x^{\mathrm{\text{'}}\mathrm{\text{'}}}\right|\tau }\right),$

5

$F_2=\left\{\begin{array}{ll}-\frac{\left|x^{\mathrm{\text{'}}\mathrm{\text{'}}}\right|\tau }{v-x^{\mathrm{\text{'}}\mathrm{\text{'}}}\frac{\tau }{{\mu }_k}}{F}_N,& \mathrm{s}\mathrm{g}\mathrm{n}\left(x^{\text{'}\text{'}}\right)v>0\mathrm{}\mathrm{}\cup \mathrm{}\mathrm{s}\mathrm{g}\mathrm{n}\left(x^{\mathrm{\text{'}}\mathrm{\text{'}}}\right)>2\left|x^{\mathrm{\text{'}}\mathrm{\text{'}}}\right|\frac{\tau }{{\mu }_k},\\ \mathrm{s}\mathrm{g}\mathrm{n}\left(x\mathrm{\text{'}}\mathrm{\text{'}}\right){\mu }_s\mathrm{s}\mathrm{i}\mathrm{n}(\mathrm{\Omega }v+\mathrm{\Phi })F_N,& 0<\mathrm{s}\mathrm{g}\mathrm{n}\left(x^{\mathrm{\text{'}}\mathrm{\text{'}}}\right)<2\left|x^{\mathrm{\text{'}}\mathrm{\text{'}}}\right|\frac{\tau }{\mu },\end{array}\right.$

where:

6

$\left\{\begin{array}{l}\mathrm{\Phi }=\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{s}\mathrm{i}\mathrm{n}\left(\frac{{\mu }_k}{{\mu }_s}\right),\\ \mathrm{\Omega }=\frac{{\mu }_k\left(\pi -\mathrm{\Phi }\right)}{x^{\mathrm{\text{'}}\mathrm{\text{'}}}\tau },\end{array}\right.$

where ${\mu }_k$ is the kinetic friction coefficient, ${\mu }_s$ is the static friction coefficient, $v$ is the relative velocity, and $\tau$ is the delay time.

Fig. 3Bilinear hysteresis model

a) Linear part

b) Hysteretic part

c) Linear along with hysteresis

Fig. 4Mcmillan hysteresis model

Fig. 5Wojewoda hysteresis model

The Mcmillan hysteresis model can accurately express the effect of hysteresis. However, considerable physical quantities are present in this model, and some even have unclear physical meanings. Parameters are hard to be selected. Meanwhile, the simulation results are remarkable when the value of acceleration $x^{\text{'}\text{'}}$ is fixed, but unsatisfactory when $x\text{'}\text{'}$ varies. The acceleration of two contacting surfaces always varies at any time, which limits the calculation of this hysteresis friction model.

2.2.3. Wojewoda hysteresis model

Wojewoda [17, 18] proposed a new friction model to address the deficiency of the previous models (Fig. 5). The mathematical model is expressed as follows:

7

$u=\left\{\begin{array}{ll}{u}_{st}\mathrm{s}\mathrm{g}\mathrm{n}\left(v\right),& u_{st}<u_{d+}\mathrm{}\cup \mathrm{}\mathrm{s}\mathrm{g}\mathrm{n}\left(vv\mathrm{\text{'}}\right)>0,\\ u_{d+}\mathrm{s}\mathrm{g}\mathrm{n}\left(v\right),& u_{st}>u_{d+}\mathrm{}\cup \mathrm{}\mathrm{s}\mathrm{g}\mathrm{n}\left(vv\mathrm{\text{'}}\right)>0,\\ u_{d_{}}\mathrm{s}\mathrm{g}\mathrm{n}\left(v\right),& \mathrm{s}\mathrm{g}\mathrm{n}\left(vv\mathrm{\text{'}}\right)<0,\end{array}\right.$

8

$F=u{F}_N.$

Among them:

$\left\{\begin{array}{l}{u}_{st}=\frac{1}{2}\frac{k}{F_N}\frac{v^2}{\left|v\mathrm{\text{'}}\right|}-u_o,\\ u_o=2u_c-u_s,\\ u_{d+}=u_c\left(1+\frac{u_s\left(v\right)-u_c}{u_c}g\left(v,v^{\mathrm{\text{'}}}\right)\right),\\ u_s\left(v\right)=u_s+\mathrm{\Delta }{u}_s\frac{1}{1+\left|\frac{v}{v_A}\right|},\\ g\left(v,v^{\mathrm{\text{'}}}\right)=\frac{1}{1+{\left(\frac{v-\tau v^{\mathrm{\text{'}}}}{v_A}\right)}^2},\\ u_{d-}=u_c\left(1-\frac{u_s-u_c}{u_c}g\left(v,v^{\mathrm{\text{'}}}\right)\right),\end{array}\right.$

where $v=x\mathrm{\text{'}}-v_0$ represents the relative velocity, $v_0$ is the initial velocity, $u_c$ is the kinetic friction coefficient, $u_s$ is the static friction coefficient, $\mathrm{\Delta }{u}_s$ is the conditioning parameter of maximum static friction, $v_A$ is the conditioning parameter of mean velocity, $\tau$ is the delay time, and $k$ represents stiffness.

This model introduces transient relative motion acceleration and considers varying static friction nature, maximum static friction nature, Stribeck effect, pre-slip effect, and friction hysteresis to accurately reflect the features of dry friction. Moreover, the formulation of this model is simple and easy to understand and calculate. Hence, this model is selected for further computation and analysis in this paper.

On the basis of Eq. (3) and Table 1, the properties of hysteretic friction are presented in Fig. 6.

Fig. 6Characteristic diagram of hysteretic friction

Table 1Parameters for calculation of dry friction properties

Nomenclature	Physical quantities/Unit	Value
Static friction coefficient	$u_s$	0.6
Kinetic friction coefficient	$u_c$	0.4
Delay time	$\tau /s$	0.002
Conditioning parameter of mean velocity	$v_A$	0.2
Conditioning parameter of maximum static friction	$\mathrm{\Delta }{u}_s$	0.03

3.1. Numerical calculation of multiple limit cycle properties of dual-cylinder dual-pad brake system

With aid from the cooperating brake manufacturer, the dual-cylinder dual-pad brake system is proposed. Subsequently, the dynamic model of the brake system is developed. Wojewoda hysteresis model is selected to search the phenomenon of multiple limit cycles caused by dry friction. The Runge-Kutta method is employed to calculate the multiple limit cycle properties of the proposed brake. The data needed in calculation are listed in Tables 1 and 2.

Brake pressure during braking with varying time is calculated using the selected parameters [19]. The results are illustrated in Fig. 7.

Fig. 7 shows that the brake pressure $F_N$ is stabilized in around 6000 N. To calculate system dynamics during braking, $F_N=\mathrm{}$6000 N is selected. Considering the symmetry of the two brake pads, only the vibration of the left brake (brake pad 1) is studied. The diagram of displacement of brake pad 1 under different initial excitation and the bifurcation diagram of initial brake speed are shown in Fig. 8.

Fig. 8 shows the bifurcation relationship between the displacement of brake pad 1 and speed under different initial excitation. Brake pad 1 tends to the equilibrium point when initial brake speed $v_0>\mathrm{}$10 m/s. With the gradual decrease of initial brake speed, Hopf bifurcation, which generates limit cycles, appears in the system, with the corresponding $v_0=\mathrm{}$10 m/s initial brake speed of Hopf. When the initial brake speed 1.8 m/s$<v_0<\mathrm{}$10 m/s, weak initial excitation induces small stable limit cycles, and the amplitude of vibration of the brake system is relatively small, and vice versa. When the initial brake speed decreases further until $v_0<\mathrm{}$1.8 m/s, the system generates relatively large limit cycles regardless of weak or strong initial excitation, which means that the amplitudes of vibration of the brake system are relatively large.

Fig. 7Diagram of brake pressure varying with time

Fig. 8Bifurcation diagram of m1x1-v0 with FN= 6000 N brake pressure

Different initial speeds are determined as $v_0=\mathrm{}$12 m/s, 10 m/s, and 1.8 m/s, and the dynamics of brake pad 1 under different initial excitations are calculated (Fig. 9).

Fig. 9Dynamics of m1 given v0= 12 m/s: a) x1-x1' phase diagram of brake pad 1 under weak initial excitation; b) x1-x1' phase diagram of brake pad 1 under strong initial excitation

a)

b)

Fig. 9 describes the motion dynamics of brake pad 1 under different initial excitations and 12 m/s initial braking speed. The phase diagram of brake pad 1 gradually tends to an equilibrium point without stick-slip motion regardless of strong or weak initial excitation. The final equilibrium point is kept unchanged under different initial excitations.

Fig. 10 illustrates the motion dynamics of brake pad 1 under different initial excitations with 10 m/s initial braking speed. When the initial excitation is weak, the phase path of brake pad 1 finally tends to a limit cycle, wherein brake pad 1 obeys a single periodic motion with 0.042 mm amplitude and 223 Hz frequency [Fig. 10(a)]. When the initial excitation is strong, the phase path of brake pad 1 finally tends to a limit cycle, wherein brake pad 1 obeys a single periodic motion with 13.3 mm amplitude and 223 Hz frequency [Fig. 10(b)]. When the initial braking speed is 10 m/s, a small stable limit cycle and a large limit cycle under different initial excitations exist. An unstable limit cycle with 9.6 mm amplitude exists between these two cycles.

Fig. 10Dynamics of m1 given v0= 10 m/s: a) x1-x1' phase diagram, Poincare sections, and FFT spectrogram of brake pad 1 under weak initial excitation; b) x1-x1' phase diagram, Poincare sections, and FFT spectrogram of m1 under strong initial excitation; c) multiple limit cycles of the system

a)

b)

c)

Fig. 11 represents the motion dynamics of brake pad 1 under different initial excitations with 1.8 m/s initial braking speed. When the initial excitation is weak, the phase path of brake pad 1 finally tends to a limit cycle, wherein brake pad 1 obeys a single periodic motion with 6.5 mm amplitude and 223 Hz frequency [Fig. 11(a)]. When the initial excitation is strong, the phase path of brake pad 1 finally tends to a limit cycle, wherein brake pad 1 obeys a single periodic motion with 15.5 mm amplitude and 223 Hz frequency [Fig. 11(b)]. When the initial braking speed is 1.8 m/s, a small stable limit cycle and a large limit cycle under different initial excitations exist. An unstable limit cycle with 1.65 mm amplitude exists between these two cycles.

On the basis of Figs. 8 to 12, the peculiarities of multiple limit cycles of the brake system under different initial speeds are summarized in Table 3.

Based on Figs. 8-11 and Table 3, the conclusions are drawn as follows:

1) The brake frequency of dual-cylinder dual-pad brake is $f=$220 Hz, which implies oscillation of the braking process. Oscillation occurs when dry friction generates stick-slip motion between brake pads and disc brakes. Oscillation is the phenomenon wherein stable limit cycles appear on the phase diagram.

Fig. 11Dynamics of m1 given v0= 1.8 m/s: a) x1-x1' phase diagram, Poincare sections, and FFT spectrogram of m1 under weak initial excitation; b) x1-x1' phase diagram, Poincare sections, and FFT spectrogram of m1 under strong initial excitation; c) multiple limit cycles of the system

a)

b)

c)

2) Regarding the dual-cylinder dual-pad brake system, the number of its limit cycles changes with varying initial braking speed. A small and a large stable limit cycle exist in the system when the initial braking speed $v_0$ is in the range of 1.8 mm to 10 mm. Moreover, an unstable limit cycle exists between the two stable limit cycles. However, when the initial braking speed $v_0<\mathrm{}$1.8 m/s, only the small stable limit cycle exists. The large and the unstable limit cycles disappear. Although the amplitude of the small limit cycle is small, its frequency of vibration is high enough to cause damage to the entire system.

3) The amplitudes of the limit cycles of the dual-cylinder dual-pad brake system also change with varying initial braking speed. With increasing initial braking speed, the amplitude of the large stable limit cycle gradually increases; the amplitude of the small stable limit cycle gradually decreases; and the amplitude of the unstable limit cycle gradually decreases. However, when the initial braking speed is less than a certain value, both the small stable and unstable limit cycles disappear, leaving one large unstable limit cycle.

3.2. Effects of brake pressure on peculiarities of multiple limit cycles

To analyze the effects of brake pressure on the peculiarities of multiple limit cycles, $F_N=\mathrm{}$6000 N, $F_N=\mathrm{}$5000 N, $F_N=\mathrm{}$2000 N are selected for the investigation according to Fig. 7. The obtained system $x_1$-$v_0$ bifurcation diagrams are shown in Figs. 12(a) to 12(c).

Fig. 12x1-v0 bifurcation diagram of m1 under different brake pressure

a)$F_N=$6000 N

b)$F_N=$5000 N

c)$F_N=$2000 N

Fig. 12 presents the bifurcation relationship between the displacements and speeds of brake pad 1 under different initial excitations. This bifurcation diagram is obtained by using Poincare sections method. $x_1^{\mathrm{\text{'}}}=0$ is used to extract the phase diagram obtaining two corresponding values of $x_1$ and is used to select the larger value to draw the bifurcation diagram. According to Figs. 10 and 11, the large stable limit cycle is symmetric around the vertical axis; hence, the displacement of $x_1$ is extracted and regarded as the approximation of the amplitude. The small limit cycle is not symmetric around the vertical axis but to a point which means the extracted value of $x_1$ is not the true value of the amplitude. However, the variation trend of amplitudes can be reflected on the bifurcation diagram.

Fig. 12 describes the peculiarities of the multiple limit cycles of brake pad 1 under $F_N=\mathrm{}$6000 N brake pressure. Hopf bifurcation of braking oscillation in the brake system appears when the initial breaking speed is 10 m/s. Multiple limit cycles appear when the initial braking speed ranges from 1.8 m/s to 10 m/s. The small stable and unstable limit cycles disappear when the initial breaking speed is 1.7 m/s. The maximum amplitude of the large stable limit cycle is 15.7 mm. Moreover, the small stable limit cycle gradually increases with decreasing speed. Hence, the corresponding maximal amplitude of small stable limit cycle appears when the initial braking speed is 1.8 m/s. The unstable limit cycle gradually decreases with decreasing speed. Therefore, the corresponding maximal amplitude of unstable limit appears when the initial breaking speed is 10 m/s.

Fig. 12(b) describes the peculiarities of multiple limit cycles of brake pad 1 under $F_N=\mathrm{}$5000 N brake pressure. Hopf bifurcation of braking oscillation in the brake system appears when the braking speed is 8.4 m/s. Multiple limit cycles appear when the initial braking speed ranges from 1.6 m/s to 8.4 m/s. The small stable and unstable limit cycles disappear when the initial braking speed is 1.5 m/s. The maximal amplitude of the large stable limit cycle is 13 mm. Furthermore, the corresponding maximal amplitude of the small stable limit cycle appears when the initial braking speed is 1.6 m/s. The maximal amplitude of the unstable limit cycle appears when the initial braking speed is 8.4 m/s.

Fig. 12(c) illustrates the peculiarities of multiple limit cycles of brake pad 1 under $F_N=\mathrm{}$2000 N brake pressure. Hopf bifurcation of braking oscillation appears in the brake system when the initial braking speed is 3.3 m/s. Multiple limit cycles appear when the initial braking speed ranges from 1.2 m/s to 3.3 m/s. The small stable and unstable limit cycles disappear when the initial braking speed is 1.1 m/s. The maximal amplitude of the large stable limit cycle is 5.1 mm. Moreover, the corresponding maximal amplitude of the small stable limit cycle appears when the initial braking speed is 1.2 m/s. The maximal amplitude of the unstable limit cycle appears when the initial braking speed is 3.3 m/s.

To study the influence of brake pressure on the maximal amplitude of the small stable and unstable limit cycles, according to the aforementioned analysis, (a) $F_N=\mathrm{}$6000 N, $v_0=\mathrm{}$10 m/s; (b) $F_N=\mathrm{}$6000 N, $v_0=\mathrm{}$1.8 m/s; (c) $F_N=\mathrm{}$5000 N, $v_0=\mathrm{}$8.4 m/s; (d) $F_N=\mathrm{}$5000 N, $v_0=\mathrm{}$1.5 m/s; (e) $F_N=\mathrm{}$2000 N, $v_0=\mathrm{}$3.3 m/s; and (f) $F_N=\mathrm{}$2000 N, $v_0=\mathrm{}$1.2 m/s are selected to obtain the diagrams of motion peculiarities of brake pad 1 under different initial excitations.

Fig. 13Motion peculiarities of m1 under different initial speeds given FN= 6000 N

a)$v_0=\mathrm{}$10 m/s

b)$v_0=\mathrm{}$1.8 m/s

On the basis of the aforementioned analysis and Fig. 12, the amplitude of unstable limit cycle reaches the maximum 9.6 mm when $F_N=\mathrm{}$6000 N and $v_0=\mathrm{}$10 m/s. In addition, the amplitude of small stable limit cycle reaches its maximum 0.65 mm when $v_0=\mathrm{}$1.8 m/s.

The amplitude of unstable limit cycle reaches the maximum 8.2 mm when $F_N=\mathrm{}$5000 N and $v_0=\mathrm{}$8.4 m/s, and the amplitude of small stable limit cycle reaches its maximum 0.5 mm when $v_0=\mathrm{}$1.6 m/s (Fig. 14).

The amplitude of unstable limit cycle reaches the maximum 3 mm when $F_N=\mathrm{}$2000 N and $v_0=\mathrm{}$3.3 m/s, and the amplitude of small stable limit cycle reaches its maximum 0.2 mm when $v_0=\mathrm{}$1.2 m/s (Fig. 15).

On the basis of the aforementioned analysis, we obtain the effects of different pressure on the peculiarities of multiple limit cycles (see Table 4).

Fig. 14Motion peculiarities of m1 under different initial speeds given FN= 5000 N

a)$v_0=\mathrm{}$8.4 m/s

b)$v_0=\mathrm{}$1.6 m/s

Fig. 15Motion peculiarities of m1 under different initial speeds given FN= 2000 N

a)$v_0=\mathrm{}$3.3 m/s

b)$v_0=\mathrm{}$1.2 m/s

Based on Figs. 12 to 15 and Table 4, the conclusions are drawn as follows:

1) With increasing brake pressure, the maximal amplitudes of large and small stable limit cycles and unstable limit cycles correspondingly increase.

2) With increasing brake pressure, the speed of Hopf bifurcation point increases. Moreover, the corresponding initial speed when the small stable limit cycle and the unstable limit cycle disappear accordingly increases. The speed range where multiple limit cycles exist also increases.

3.3. Effects of combination of kinetic and static friction coefficients on peculiarities of multiple limit cycles.

To investigate the effects of friction coefficients on the peculiarities of multiple limit cycles, combinations of different kinetic and static friction coefficients are selected and considered as the research object to obtain different combination of $x_1$-$v_0$ bifurcation diagrams (see Figs. 16(a) to 16(c)).

Fig. 16 describes the peculiarities of multiple limit cycles of brake pad 1 under combinations of different kinetic and static friction coefficients. Figs. 16(a) and 16(b) describe the peculiarities of multiple limit cycles under combinations of $u_c=$ 0.4, $u_s=$0.6 and $u_c=$ 0.4, $u_s=\mathrm{}$0.8, respectively. The bifurcation characteristics of the two combinations are identical. The initial braking speed when the brake system yields Hopf bifurcation of brake oscillation is 10 m/s. The range of initial braking speed when multiple limit cycles appear is 1.8 m/s to 10 m/s. The initial braking speed when the small stable limit cycle and the unstable limit cycle disappear is 1.7 m/s. The maximal amplitude of the large stable limit cycle is 15.7 mm. The corresponding maximal amplitude of the small stable limit cycle appears when the initial braking speed is 1.8 m/s. The maximal amplitude of the unstable limit cycle appears when the initial braking speed is 10 m/s.

Fig. 16(c) shows the peculiarities of multiple limit cycles for the combination of $u_c=\mathrm{}$0.3, $u_s=\mathrm{}$0.6. The initial braking speed when the brake system yields Hopf bifurcation of brake oscillation is 7.6 m/s. The range of the initial braking speed when multiple limit cycles appear is 1.6 m/s to 7.6 m/s. The initial braking speed when small stable limit cycle and the unstable limit cycle disappear is 1.5 m/s. The maximal amplitude of the large stable limit cycle is 11.5 mm. The corresponding maximal amplitude of the small stable limit cycle appears when the initial braking speed is 1.6 m/s. The maximal amplitude of the unstable limit cycle appears when the initial braking speed is 7.6 m/s.

Fig. 16x1-v0 bifurcation diagrams of m1 displacements under combinations of different kinetic and static friction coefficients

a)$u_c=\mathrm{}$0.4, $u_s=\mathrm{}$0.6

b)$u_c=\mathrm{}$0.4, $u_s=\mathrm{}$0.8

c)$u_c=\mathrm{}$0.3, $u_s=\mathrm{}$0.6

Fig. 17Motion peculiarities of m1 under different initial speeds given uc= 0.4, us= 0.6

a)$v_0=\mathrm{}$10 m/s

b)$v_0=\mathrm{}$1.8 m/s

Fig. 18Motion peculiarities of m1 under different initial speeds given uc= 0.4, us= 0.8

a)$v_0=\mathrm{}$10 m/s

b)$v_0=\mathrm{}$1.8 m/s

Fig. 19Motion peculiarities of m1 under different initial speeds given uc= 0.3, us= 0.6

a)$v_0=\mathrm{}$7.6 m/s

b)$v_0=\mathrm{}$1.6 m/s

The corresponding initial braking speeds of the maximal amplitudes of small stable limit cycles and unstable limit cycles under different combinations are selected as follows to study the peculiarities of multiple limit cycles.

According to the aforementioned analysis and Fig. 17, if $u_c=$ 0.4, $u_s=\mathrm{}$0.6: 1) When $v_0=\mathrm{}$10 m/s, the amplitude of unstable limit cycle reaches the maximum value of 9.6 mm; and 2) when $v_0=\mathrm{}$1.8 m/s, the amplitude of small stable limit cycle reaches the maximum value of 0.65 mm.

If $u_c=$ 0.4, $u_s=$ 0.8: 1) When $v_0=\mathrm{}$10 m/s, the amplitude of unstable limit cycle reaches the maximum value of 9.6 mm; and 2) when $v_0=\mathrm{}$1.8 m/s, the amplitude of small stable limit cycle reaches the maximum value of 0.65 mm (Fig. 18).

If $u_c=$ 0.3, $u_s=\mathrm{}$0.45: 1) When $v_0=\mathrm{}$7.6 m/s, the amplitude of unstable limit cycle reaches the maximum value of 7.5 mm; and 2) when $v_0=\mathrm{}$1.6 m/s, the amplitude of small stable limit cycle reaches the maximum value of 0.45 mm (Fig. 19).

The effects of combinations of different static and kinetic friction coefficients on the peculiarities of multiple limit cycles are summarized in Table 5.

Based on Figs. 16-19 and Table 5, the conclusions are drawn as follows:

1) Through comparison of the data between $u_c=$ 0.4, $u_s=\mathrm{}$0.6 and $u_c=$ 0.4, $u_s=\mathrm{}$0.8, the amplitudes of limit cycles and the speed range where multiple limit cycles exist are identical under the same kinetic friction coefficient and different static friction coefficients. This phenomenon verifies that static friction coefficient has few effects on the peculiarities of multiple limit cycles.

2) Through comparison of the data between $u_c=$ 0.4, $u_s=$ 0.6 and $u_c=$ 0.3, $u_s=\mathrm{}$0.6, under the same static friction coefficient and different kinetic friction coefficients, the larger the kinetic friction coefficient is, the larger the amplitudes of limit cycles and speed range where multiple limit cycles appear are. This phenomenon shows the significant influence of kinetic friction coefficients on the peculiarities of limit cycles.

3.4. Effects of mass ratio on peculiarities of multiple limit cycles

Assuming the mass ratio of $m_1$ to $m_2$ to be $\delta$, given $v_0=\mathrm{}$6 m/s, the obtained $x_1$-$\delta$ bifurcation diagram is shown in Fig. 20.

Fig. 20 illustrates the peculiarities of multiple limit cycles of brake pad 1 with different mass ratio. The amplitude of the large stable limit cycle progressively increases with increasing mass ratio [Fig. 20(a)]. The amplitude of small stable limit cycle initially increases and then decreases with increasing mass ratio and the amplitude reaches maximum value when $\delta =\text{1}$ [Fig. 20(b)].

Fig. 20x1-δ bifurcation diagram of the displacements of m1 given v0= 6 m/s

a) Global diagram

b) Diagram of partially enlarged details