Abstract
This paper is concerned with the vibration-driven system which can move due to the periodic motion of the internal mass and the dry friction; the system can be modeled as Filippov system and has the property of stick-slip motion. Different periodic solutions of stick-slip motion can be analyzed through sliding bifurcation, two-parameter numerical continuation for sliding bifurcation is carried out to get the different bifurcation curves, and the bifurcation curves divide the parameters plane into different regions which stand for different stick-slip motion of the periodic solution. Furthermore, continuations with additional condition are carried out for the directional control of the vibration-driven system in one period; the curves divide the parameter plane into different progressions.
1. Introduction
Recently, mobile mechanisms that can move due to the vibration of the internal mass have been widely researched, and these mechanisms have many advantages over conventional mobile systems (driven by legs, wheels, wings, etc.), for example, easy fabrication, hermetic structure, and locomotion in the narrow environment. Thus they have extensive application in pipeline inspection, life detection in disaster, and medical endoscopy.
Chernousko [1] first proposed the horizontal motion of the system driven by the movable internal mass; the friction which acted on the body is anisotropic, which means the coefficient of friction in forward and backward direction is different. The two periodic control modes, velocity-controlled mode and acceleration-controlled mode, are constructed for the relative motion of the internal mass, and optimal parameters of periodic control were decided to realize the maximum mean velocity of the body. Fang et al. [2] used the method of averaging to obtain an approximate expression of the average steady-state velocity when the stick-slip phenomenon was not considered, optimal parameters of the internal controlled mass were determined to maximize the average velocity, and some control strategies were given to control the motion of system under the stick-slip effect. Liu et al. [3] studied the vibroimpact capsule system which has a main body interacting with an internal harmonically driven mass, when the internal mass contact with the plate impact occurs, and the parameters for the maximum mean velocity can be determined through nonlinear dynamics analysis, the energy consumption was also considered, and the parameters for the maximum mean velocity and the minimum energy consumption were not the same. Fang et al. [4, 5] and Zimmermann et al. [6–8] studied the two and more modules vibration-driven systems; the approximate expression of steady-state motion was obtained when the friction is small and the optimal parameters were got to achieve the maximum mean velocity. Bolotnik et al. [9] modeled the system driven by the movable internal mass which can move in the horizontal direction and the vertical direction (change the normal force for anisotropic friction in the different direction). Then the approximate expression of average steady-state velocity was obtained through the method of averaging; optimal parameters (the amplitude and the phase shift of the horizontal and vertical vibration excitation forces) were determined to realize the maximum average velocity and to control the direction of motion. In the paper, we study the model.
The dry friction plays an important role in vibration-driven system motion. The systems with dry friction belong to Filippov piecewise-smooth dynamical systems [10, 11]. The Filippov systems may exhibit different types of limit cycles caused by the interaction of a trajectory with the boundary of the sliding regions; the features of Filippov system are called the sliding bifurcation. Kowalczyk et al. [12] investigated a dry friction oscillator through numerical continuation of sliding bifurcation and revealed the codimension two sliding bifurcation points. Marcel Guardia et al. [13] analytically considered sliding bifurcations of periodic orbits in the dry friction oscillator, and the results agreed with the numerical calculation [12]. Fang et al. [14] studied the vibration-driven system through sliding bifurcation, and a two-parameter bifurcation problem was theoretically analyzed. For the numerical continuation of piecewise-smooth system, the software SlideCont [15] and TC-HAT [16] based on ATUO have been developed. Joseph Páez Chávez used the software TC-HAT to study the bifurcation of some mechanical models, the nonsmooth Jeffcott rotor [17], the impact oscillator [18], the piecewise-linear capsule system [19], etc. The continuation toolbox COCO was developed for continuation and bifurcation analysis of smooth and nonsmooth dynamical systems [20, 21]; the soft impact oscillator [22] and the impulsively coupled oscillators [23] were analyzed through the COCO. In this paper, the COCO will be employed to numerically study the sliding bifurcation of vibration-driven system; the sliding bifurcation will help us to understand the stick-slip property of periodic solution and give some instructive ideas to design and control the system.
The paper is organized as follows. In Section 2, the model of the vibration-driven system is described. The mathematical model of vibration-driven system is studied in detail in order to perform the numerical analysis by the mean of COCO in Section 3. Two-parameter sliding bifurcations are analyzed and the directional control of the vibration-driven system is tackled by the numerical continuation in Section 4, and some conclusions are given in Section 5.
2. Modeling of Vibration-Driven System
The vibration-driven system is considered as depicted in Figure 1. The vibration-driven system is composed of a rigid body and an internal mass; the rigid body realizes the translational motion along a straight line in the resistive environment. The internal mass can move relative to the rigid body in horizontal and vertical direction. The internal mass is considered a point mass. Dry friction acts between the rigid body and the ground.
Two Cartesian reference frames are introduced: the inertial reference frame and the coordinate system attached to the rigid body. The - and -axes are horizontal; the and -axes are vertical. We denote as the coordinate of the point in the inertial reference frame , denotes the displacement of the rigid body, and denote the coordinates of the internal body in the reference frame , and are the masses of the rigid body and the internal body, respectively, and is the gravitational acceleration.
The dynamics equation of the system along axis can be governed by Newton’s second laws as follows:where is the frictional force. The force is described by Coulomb law: where is the resultant force on the body except for the dry friction in the horizontal direction, is the normal force exerted on the system by the ground, and is the coefficient of dry friction. The asymmetrical friction in forward and backward motions arises from the time-varying normal force . The force and force can be expressed: The rigid body keeps the contact with the ground; therefore, the force must be satisfied the inequality . So we have
Next, we assume the control acceleration is harmonic function with the same frequency but shifted in the phase; particularly,
Here and are the driving-amplitudes, respectively, is the driving-frequency, and is the phase difference between the forces.
Substituting (6) into (1)-(5),where , , and .
To reduce the number of parameters of the system, the nondimensional variables and and the parameters and are introduced:Substituting these variables above into (7)-(9) (omit the asterisks),The expressions and stand for the normalized dry friction and the normal force, respectively. The parameter represents the ratio of the possible maximum value of the dry friction force to the amplitude . We assume the value of the parameter is in the interval . Due to periodicity, the phase difference ranges from 0 to .
3. Modeling of the Vibration-Driven System as Filippov System
The mathematical model of vibration-driven system can be defined as a piecewise-smooth system of the Filippov type. We can transform (11) into vector fields, event functions, and reset functions through the approach of multisegment periodic orbits. Let and represent the state variables of the system and the parameters, respectively.
The multisegment periodic orbits of the vibration-driven system consist of two or more segments, which can be modeled as follows.
Stick: This segment occurs when ; the motion during this segment is governed by the equation
This segment terminates when the resultant force on the body except for the dry friction equals the threshold of the dry friction force. The event functions are detected as follows: (transition to forward slip) (transition to backward slip)
The next segment initial point is defined by the reset function .
Forward slip: this segment occurs when the force is larger than the maximum value of the dry friction; that is, ; the motion during this segment is governed by the equation
This segment ends when the velocity becomes zero; that is, . The next segment is connected by the reset function .
Backward slip: this segment occurs when ; the motion of the system during this segments is described by the equation
This segment ends when the velocity becomes zero; the event function is defined: . The next segment is connected by the reset function .
Stick : this segment is introduced to keep the variable within the interval , when , the motion of the system is governed by (14), the segment ends when , and the reset function is
Forward slip : this segment is introduced to keep the variable within the interval , when , the motion of the system is governed by (15), the segment ends when , and the reset function is .
Backward slip : this segment is introduced to keep the variable within the interval, when , the motion of the system is governed by (16), the segment ends when , and the reset function is .
A periodic solution of Filippov system can be described as a sequence of triplet ; the segment of system is governed by the vector field , terminates at the event function , and connects the next segment by the reset function . Any periodic trajectory of the system is described by solution signature ; is the length of signature. Therefore the periodic solution of the vibration-driven can be described by combinations of the seven triplets corresponding to above statement:
In the Filippov system, there are four possible sliding bifurcations in the limit cycle because of the interaction of a trajectory with the boundary of a sliding region, including crossing-sliding bifurcation, gazing-sliding bifurcation, switching-sliding bifurcation, and adding-sliding bifurcation. Nondegeneracy conditions for the four sliding bifurcation are given [10, 11]. The sliding bifurcation does not change the number and stability of the system’s solutions, but it will induce the different interaction between the limit cycle and the sliding regions.
4. Numerical Bifurcation Analysis
In this section, we will perform the sliding bifurcation and directional control to analyze the dynamics response of the vibration-driven system.
4.1. Sliding Bifurcation Analysis
When , the periodic trajectory of the motion is shown by Figure 2. The cycle signature is . We start the numerical continuation of the periodic solution by the method of path-following using the parameter value as an initial value. The additional boundary may be applied to the start point of the third segment for the crossing-sliding bifurcation continuation. We will use the COCO to carry out the numerical continuation concerning parameters and ; the curve which is the result of this numerical continuation is shown in Figure 3.
Similarly, the numerical continuation is performed in parameter space with different cycle signatures and different additional boundary condition for different segment boundary point and the results are depicted in Figure 3. The signature of cycle trajectory in is , when the additional conditions and are applied to the point of the segment according to nondegeneracy conditions of adding-sliding bifurcation; hence we can get the adding-sliding bifurcation curve . The adding-sliding bifurcation branch is got similarly. The curves , , , and represent crossing-sliding bifurcation branches. The sliding bifurcation curves divide the two parameters plane into eight regions and there are six different stick-slip periodic solutions in the parameter plane. The periodic solution of system in with signature is depicted in Figure 4(a), which means the velocity of the system is always greater than or equal to 0. The signature of the periodic trajectory in is as depicted in Figure 4(b), which means the velocity of the system is always lower than or equal to 0. The velocity in the two regions does not change its sign; it is important for the practical application to do some work, such as medical robot for intestinal therapy. Furthermore, when the velocity of system changes its sign, the efficiency will decrease because of more energy dissipated by opposite slip. The limit cycle of system in can be described by the signature depicted in Figure 4(c). The signature of periodic solution in is showed in Figure 4(d); the signature of the periodic solution in is depicted in Figure 4(e). The cyclic signature in Figure 4(f) is .
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The result of two-parameter continuation for the sliding bifurcation with respect to the parameters and by fixing different value ((a) ; (b) ; (c) ) is shown in Figure 5. The regions and are shrinking and the regions , , , and are expanding as the increases. Therefore it is easy to realize directional motion in or through changing the parameters when the parameter increases.
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We carry out the numerical continuation with respect to the parameters in for the different and the results are depicted in Figure 6 ((a) ; (b) ; (c) ). The periodic solutions in region in Figures 3 and 6 have the same stick-slip motion; the curve represents the switching-sliding bifurcation branch. As shown from Figure 6, there are different regions of stick-slip motion of system in the parameter plane when is different; the motion in which is always equal to or lower than 0 can be realized by changing the parameter and when , but the motion in could not happen no matter the value of and . The velocity of system which is always equal to or larger than zero can be controlled through changing the parameter when is or . Therefore the value of is important for realizing directional motion.
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The numerical continuation for the parameters , in the by fixing the value of the parameter is carried out and the results are depicted in Figure 7. From Figure 7, there are no regions and in the parameter plane when , but the regions and appear in the parameter plane as increases to 0.6; the regions and expand when .
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4.2. Directional Control
Based on the above analysis, we can see that the direction of the system progression in one period can be forward (the region in Figure 3) or backward (the region in Figure 3) owing to different parameters.
The average velocity of the system in one period isThe regions of different directional progression in one period can be determined by implementing the parameters continuation of the periodic solution with additional condition .
When , there are six different stick-slip period solutions in the parameter plane in Figure 3; two parameters can be continued numerically based on different period solutions with the additional condition . The result is presented in Figure 8. The curves divide the parameters into three regions: forward drift (), backward drift (), and zero drift (, on the curves). They are shown in Figures 9(a), 9(b), and 9(c), respectively. According to Figure 8, it can be seen that, for any value of the parameter , the direction of system progression can be controlled by changing the phase , the direction of progression is forward when is in , and the direction of progression is backward when is in . When , the direction of progression is forward when is in ; and the direction of progression is backward when is in and .
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When , there is no solution with additional condition in the parameter plane, which indicates that the direction of the progression does not change. The direction of the progression is forward because the average velocity is larger than zero in the region seen from Figure 6(b).
When , the parameter plane for directional continuation is presented in Figure 10, some conclusions are draw from Figure 10 similarly: for any parameter , the direction of system progression can be controlled by changing the phase ; the direction of progression is forward when is in ; and the direction of progression is backward when is in and . When , the direction of progression is forward when is in ; and the direction of progression is backward when is in and .
5. Conclusions
This paper studies the dynamical response of the vibration-driven system which is composed of a body with movable internal mass. The asymmetry of friction in forward and backward direction is essential to the motion of system, which arises from the normal force change due to the vertical motion of the internal mass. The vibration-driven system involving dry friction belongs to the Filippov system, the cycle trajectory of the system can be divided into smooth segments, the event functions defined the terminal point of the segments, and the reset functions connected the segments. We take advantage of the software COCO to carry out the bifurcation analysis.
Two-parameter sliding bifurcations are carried out by performing the numerical continuation. Different period solutions of stick-slip motion are obtained though the sliding bifurcation curves. For directional control of the vibration-driven system, the drift of the vibration-driven system in one period may change sign, the continuation with additional condition is carried out in the parameters plane, and the curves are obtained. The curves divide the parameters plane into three modes of drift (backward, forward, and zero). So the direction of the vibration-driven system progression can be controlled by changing the parameters.
The particular contribution of this research is the numerical continuation of the parameters for the vibration-driven system and detailed classifications of parameter space where different system dynamic behaviors can be obtained. The bifurcation analysis improves our understanding of the dynamical behaviors of the vibration-driven system and is of benefit to devise control strategies for the vibration-driven system.
Data Availability
No data were used to support this study.
Disclosure
The research did not receive specific funding.
Conflicts of Interest
The authors declare that they have no conflicts of interest.