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Abstract

The controlled motion of a rigid body in the horizontal plane is investigated in this article. Three internal and acceleration-controlled masses are used to actuate the system. Dry friction acting between the system and the plane is isotropic. The dynamics of two basic motions of the system, that is, rectilinear and rotary motions, are first studied. Then by combining these two basic types of motions, planar locomotion of the system is constructed. Two typical planar trajectories of the system, that is, oblique lines and curve lines, are proposed and both approached with folding lines. The slope of the oblique lines and the curvature of the curves can be adjusted by varying the drive parameters, and the planar locomotion is thus controlled. To achieve a maximum average velocity, the drive parameters are optimized.

Keywords

Planar locomotion dry friction rectilinear motion rotary motion optimization

Introduction

With the rapid development of robot applications, micro-robots are attracting more and more attention of worldwide researchers. The reason lies in the huge potential of micro-robots in applications of pipeline inspection, earthquake rescue, medicine delivery, precision positioning, and so on. Conventional robots have outer components like legs, wheels and caterpillars, and complex transmission gears, which make it hard for them to be miniaturized. In contrast, vibration-driven systems are driven by the periodic motions of their internal masses. They are simple in design and easy to be manufactured in miniature size. In addition, their hermetic shapes make them adaptable to many extreme resistive media with high viscidity or corrosivity.

Chernous’ko, Bolotnik, Fang, and Xu have carried out intensive researches on the rectilinear motion of vibration-driven systems with one internal mass.^1–6 The vibration-driven system^1–4 is composed of a rigid body and one internal mass. The mass moves periodically inside the rigid body. Due to the anisotropic dry friction acting on the body by supporting plane, the system has a net displacement in each period. Two modes of the piecewise constant drives, that is, two-phase mode and three-phase mode, are applied to the internal mass. With the two-phase mode, Chernous’ko analyzed and optimized the rectilinear motion of the vibration-driven system subjected to dry friction, linear, and quadratic resistance. It is shown that for the isotropic quadratic resistance, the system performs progressive motion, while for the linear resistance it can only have net displacement in the medium with anisotropic resistance. Rectilinear motion of the vibration-driven system in three-phase mode is investigated in Chernous’ko¹ and Fang and Xu.⁵ In Fang and Xu,⁵ the method of averaging is utilized to analyze the steady state motion, and stick-slip effect is taken into account. The drive forms of vibration-driven systems still have many other types, for example, harmonic drive.^6–8 Fang and Xu⁶ investigated the rectilinear motion of the vibration-driven system in the presence of anisotropic dry friction. Their focus is on the stick-slip effect of the rectilinear motion, and the diagram of sliding bifurcation is analytically presented. Bolotnik and colleagues^7,8 greatly improved the rectilinear motion of the vibration-driven system by optimizing the harmonic motions of two internal masses. Moreover, robots like inverted pendulum⁹ and mini-robots¹⁰ verified the theory of the vibration-driven system and exhibited the potential of the system in miniaturization.

In order to realize planar locomotion, more internal drive components (translational or rotating masses) are added to vibration-driven systems. A vibration-driven system with four short legs is modeled by Volkova and Yatsun.¹¹ The system is actuated by the harmonic motions of two internal masses, which are restricted within two paralleled straight guides. An algorithm is proposed to approximate the “S” shape curve trajectory. In Vartholomeos and colleagues,^12–14 the planar trajectory of the vibration-driven system is controlled by three rotating masses with high accuracy. Additionally, the system is low in cost and energy supply. The layouts of the internal masses in the above system are symmetric. Besides, inphase and antiphase motions are performed by the internal masses to control the planar trajectories.

Based on the analysis and optimization of a box-like rigid body with one internal mass in Chernous’ko,¹ we introduce two more internal masses and extend to the study of the planar locomotion of the updated system. Two basic types of motions, that is, rectilinear motion and rotary motion, are proposed to construct the planar locomotion of the system. As a result, planar trajectories of the system are approximately provided.

Dynamics of the mechanical system

Description of the dynamic system

The vibration-driven system under consideration is a cuboid (called body M) with square top and bottom, which is placed in a horizontal plane (see Figure 1). Inside the body, three masses m ₁, m ₂, and m ₃ are mounted to the square bottom. On the bottom, coordinate frame $o_{1} - ξ η ζ$ is fixed at its center o ₁ with axes $ξ$ and $η$ parallel to the sides and $ζ$ perpendicular to the bottom. Rectangular frame o-xyz is inertial and its axis z is also perpendicular to the bottom. Figure 2 shows the internal layouts of the dynamic system. Masses m ₂ and m ₃ move periodically within the cavities or grooves, which are shown in shadow and symmetric about the symmetric axis l ₂ of body M. Mass m ₁ moves along the symmetric axis l ₁. The masses are in their initial positions, and the distances from the centers of masses m ₁ and m ₂ to l ₂ are equal and denoted by l. Due to the motions of these internal masses, body M may be actuated to move and rotate in the horizontal plane. To realize planar locomotion, first two types of basic motions of body M are studied.

Figure 1.

Scheme of the vibration-driven system.

Figure 2.

Layouts of the dynamic system.

Rectilinear and rotary motions

Rectilinear motions

Rectilinear motion of body M due to the motion of single mass m ₁

In this scenario, mass m ₁ moves periodically along the axis $ξ$ with masses m ₂ and m ₃ staying still in their initial positions (see Figure 3). Then, the governing equations of the system are set up as

(M + m_{2} + m_{3}) \overset{\cdot\cdot}{x} = F_{1} + R_{x}, m_{1} (\overset{\cdot\cdot}{x} + \overset{\cdot\cdot}{ξ}) = - F_{1}

(1)

Figure 3.

Rectilinear motion of body M with single mass m ₁ moving.

where M, m ₁, m ₂, and m ₃ are the mass of body M, masses m ₁, m ₂, and m ₃, respectively; F ₁ is the force acted on body M by mass m ₁; and R_x is the dry friction acted by the supporting plane, which is defined as

R_{x} = {\begin{matrix} - (M + m_{1} + m_{2} + m_{3}) gf for \overset{\cdot}{x} > 0 \\ (M + m_{1} + m_{2} + m_{3}) gf for \overset{\cdot}{x} < 0 \end{matrix}

(2)

and for $\overset{\cdot}{x} = 0$

| R_{x} | \leq (M + m_{1} + m_{2} + m_{3}) gf

(3)

where f is the dry friction coefficient and g is the gravitational acceleration. Eliminating F ₁ in equation (1) yields

\overset{\cdot\cdot}{x} = - μ_{1} \overset{\cdot\cdot}{ξ} + a_{1}

(4)

where

μ_{1} = \frac{m_{1}}{(M + m_{1} + m_{2} + m_{3})}, a_{1} = \frac{R_{x}}{(M + m_{1} + m_{2} + m_{3})}

(5)

2. Rectilinear motion of body M due to the inphase motions of masses m ₂ and m ₃

Now masses m ₂ and m ₃ perform inphase motions parallel to the axis $η$ , that is, masses m ₂ and m ₃ are always symmetric about the axis $η$ with mass m ₁ fixed in its initial position (see Figure 4). F ₂ and F ₃ are the forces in the plane xoy acted on body M by masses m ₂ and m ₃, respectively. Simplifying the forces F ₂ and F ₃ to the center o ₁ of body M, one can obtain the principal vector F _y and the principle moment M _z, given by

F_{y} {= F}_{2} {+ F}_{3}, M_{z} = r_{2} \times F_{2} + r_{3} \times F_{3}

(6)

Figure 4.

Rectilinear motion of body M with the inphase motions of masses m ₂ and m ₃.

Due to the symmetry of masses m ₁ and m ₂, one has

| r_{2} | = | r_{3} |, | F_{2} | = | m_{2} \overset{\cdot\cdot}{η} |, | F_{3} | = | m_{3} \overset{\cdot\cdot}{η} |

(7)

Assuming

m_{2} = m_{3} = m

(8)

we obtain

| F_{y} | = | F_{2} {+ F}_{3} | = 2 | F_{2} | = 2 | m_{2} \overset{\cdot\cdot}{η} | = 2 | m \overset{\cdot\cdot}{η} |, M_{z} = 0

(9)

The directions of position vectors $r_{2}$ and $r_{3}$ and forces $F_{2}$ , $F_{3}$ , and $F_{y}$ are shown in Figure 4.

Thus, the governing equations of body M in this scenario are given by

\begin{matrix} (M + m_{1}) \overset{\cdot\cdot}{y} & = F_{2} + F_{3} + R_{y}, m_{2} (\overset{\cdot\cdot}{y} + \overset{\cdot\cdot}{η}) \\ = - F_{2}, m_{3} (\overset{\cdot\cdot}{y} + \overset{\cdot\cdot}{η}) = - F_{3} \end{matrix}

(10)

where dry friction R_y is defined as

R_{y} = {\begin{matrix} - (M + m_{1} + m_{2} + m_{3}) gf for \overset{\cdot}{y} > 0 \\ (M + m_{1} + m_{2} + m_{3}) gf for \overset{\cdot}{y} < 0 \end{matrix}

(11)

and for $\overset{\cdot}{y} = 0$

| R_{y} | \leq (M + m_{1} + m_{2} + m_{3}) gf

(12)

Eliminating F ₂ and F ₃ in equation (10) yields

\overset{\cdot\cdot}{y} = - μ_{2} \overset{\cdot\cdot}{η} + a_{2}

(13)

where

\begin{matrix} μ_{2} = \frac{2 m_{2}}{(M + m_{1} + m_{2} + m_{3})}, \\ a_{2} = \frac{R_{y}}{(M + m_{1} + m_{2} + m_{3})} \end{matrix}

(14)

Letting

m_{1} = m_{2} + m_{3} = 2 m_{2} = 2 m

(15)

we obtain

μ_{2} = μ_{1} = \frac{2 m}{(M + m_{1} + m_{2} + m_{3})} = μ

(16)

Comparing equations (4) and (13), we represent them with

{\overset{\cdot\cdot}{D}}_{i} = - μ_{i} {\overset{\cdot\cdot}{d}}_{i} + a_{i} (i = 1, 2)

(17)

where

D_{1} = x, D_{2} = y, d_{1} = ξ, d_{2} = η

(18)

Up to now, the first basic type of body M’s motion is presented. From equation (17), one may find that rectilinear motions in the x- and y-directions have no differences although they are realized through different internal motions. In the next section, we will see how the rotary motion is constructed.

Rotary motion

To achieve a rotary motion of body M, masses m ₂ and m ₃ are expected to perform antiphase motions parallel to the axis $η$ , which means masses m ₂ and m ₃ are always antisymmetric to each other about the axis $η$ , while mass m ₁ stays still in the process. In Figure 5, $θ$ is the rotation angle of body M and its positive direction is counterclockwise. ${\bar{F}}_{2 ξ}$ , ${\bar{F}}_{2 η}$ , ${\bar{F}}_{3 ξ}$ , and ${\bar{F}}_{3 η}$ are the forces in the plane xoy acted on body M by masses m ₂ and m ₃ and ${\bar{r}}_{i}$ position vectors from the center o ₁ to the center of mass m_i (i = 2, 3). Simplifying the forces to the center o ₁ of body M, one can obtain the principal vector ${\bar{F}}_{xy}$ and the principle moment ${\bar{M}}_{z}$ , which are given as

\begin{matrix} {\bar{F}}_{xy} {= \bar{F}}_{2 ξ} + {\bar{F}}_{2 η} {+ \bar{F}}_{3 ξ} + {\bar{F}}_{3 η}, \\ {\bar{M}}_{z} = {\bar{r}}_{2} \times ({\bar{F}}_{2 ξ} + {\bar{F}}_{2 η}) + {\bar{r}}_{3} \times ({\bar{F}}_{3 ξ} + {\bar{F}}_{3 η}) \end{matrix}

(19)

Figure 5.

Rotary motion of body M with the antiphase motions of masses m ₂ and m ₃.

Due to the antisymmetry of masses m ₂ and m ₃, one has

{\bar{r}}_{2} = - {\bar{r}}_{3}, {\bar{F}}_{2 ξ} = - {\bar{F}}_{3 ξ}, {\bar{F}}_{2 η} = - {\bar{F}}_{3 η}

(20)

Consequently

{\bar{F}}_{xy} {= \bar{F}}_{2 ξ} + {\bar{F}}_{2 η} {+ \bar{F}}_{3 ξ} + {\bar{F}}_{3 η} = 0

(21)

from which one may know that in this scenario, body M simply rotates about its center o ₁ without translation.

To set up the governing equation of the rotation of body M, first, we need to acquire the total kinetic energy of the system. In Figure 5, the relative position vectors are

{\bar{r}}_{2} = {(l, η)}^{T}, {\bar{r}}_{3} = {(- l, - η)}^{T}

(22)

Then, the absolute position vectors of masses m ₂ and m ₃, are, respectively, given by

\begin{matrix} x_{2} = T_{r} {\bar{r}}_{2} = (\begin{matrix} l \cos θ - η \sin θ \\ l \sin θ + η \cos θ \end{matrix}), \\ x_{3} = T_{r} {\bar{r}}_{3} = (\begin{matrix} - l \cos θ + η \sin θ \\ - l \sin θ - η \cos θ \end{matrix}) \end{matrix}

(23)

where the transformation matrix

T_{r} = (\begin{matrix} \cos θ & - \sin θ \\ \sin θ & \cos θ \end{matrix})

(24)

The total kinetic energy T of the system can be given as

\begin{matrix} T & = \frac{1}{2} m_{2} {\overset{\cdot}{x}}_{2}^{2} + \frac{1}{2} m_{3} {\overset{\cdot}{x}}_{3}^{2} + \frac{1}{2} J_{m} {\overset{\cdot}{θ}}^{2} \\ = m [{\overset{\cdot}{η}}^{2} + 2 l \overset{\cdot}{η} \overset{\cdot}{θ} + (l^{2} + η^{2}) {\overset{\cdot}{θ}}^{2}] + \frac{1}{2} J_{m} {\overset{\cdot}{θ}}^{2} \end{matrix}

(25)

where the dots denote the derivative with respect to time t, and J_m is the sum of rotational inertias of all components of the system with respect to their own centers.

Utilizing the Lagrange equation of the second kind

\frac{d}{dt} (\frac{\partial T}{\partial \overset{\cdot}{θ}}) - \frac{\partial T}{\partial θ} = Q_{θ}

(26)

in which $Q_{θ}$ is the generalized force corresponding to the coordinate $θ$ ; we obtain the governing equation of the rotary motion as

(J_{m} + 2 {ml}^{2} + 2 m η^{2}) \overset{\cdot\cdot}{θ} + 2 ml \overset{\cdot\cdot}{η} + 4 m η \overset{\cdot}{η} \overset{\cdot}{θ} = M_{ν}

(27)

Here, some remarks on the rotary resistance $M_{ν}$ are given. The distribution of dry friction in the contact area is complicated, which is affected by many factors like normal pressure, material of contact surfaces, and surface roughness.¹⁵ In the case described in Figure 5, masses m ₂ and m ₃ are always antisymmetric about the center o ₁, and as a result, the center of mass of the system as a whole is at the center o ₁ all the time. Thus, based on this fact and to simplify our discussion, it is assumed that the normal pressure between body M and the horizontal plane is uniformly distributed over the contact area; without loss of generality, body M is assumed to rotate counterclockwise (see Figure 6). Then the elementary force $R_{0}$ at the point $(ξ, η, 0)$ is expressed as

R_{0} = κ \sin γ e_{1} - κ \cos γ e_{2} + 0 e_{3}

(28)

Figure 6.

Analysis of the friction acting on body M when it rotates.

where e ₁, e ₂, and e ₃ are the unit vectors of axes $ξ$ , $η$ , and $ζ$ , respectively, and

\begin{matrix} κ = \frac{(M + m_{1} + m_{2} + m_{3}) g}{4 s^{2}}, \\ \sin γ = \frac{η}{\sqrt{ξ^{2} + η^{2}}}, \cos γ = \frac{ξ}{\sqrt{ξ^{2} + η^{2}}} \end{matrix}

(29)

where s is half of the side length of the square bottom and $κ$ is the magnitude of the uniformly distributed normal pressure. $R_{0}$ is perpendicular to the position vector

r = (ξ e_{1} + η e_{2} + 0 e_{3})

(30)

of the point $(ξ, η, 0)$ . Then, the net force $R$ and torque $M_{ν}$ due to friction can be calculated as

\begin{matrix} R & = \int_{- s}^{s} \int_{- s}^{s} R_{0} d ξ d η \\ = \int_{- s}^{s} \int_{- s}^{s} κ (\sin γ e_{1} - \cos γ e_{2} + 0 e_{3}) d ξ d η \\ = κ \int_{- s}^{s} \int_{- s}^{s} (\frac{η}{\sqrt{ξ^{2} + η^{2}}} e_{1} - \frac{ξ}{\sqrt{ξ^{2} + η^{2}}} e_{2} + 0 e_{3}) d ξ d η = 0 \end{matrix}

(31)

and

\begin{matrix} M_{ν} & = \int_{- s}^{s} \int_{- s}^{s} r \times R_{0} d ξ d η \\ = \int_{- s}^{s} \int_{- s}^{s} (ξ e_{1} + η e_{2} + 0 e_{3}) \times κ (\sin γ e_{1} - \cos γ e_{2} + 0 e_{3}) d ξ d η \\ = - κ \int_{- s}^{s} \int_{- s}^{s} (ξ \cos γ + η \sin γ) d ξ d η e_{3} \\ = - κ \int_{- s}^{s} \int_{- s}^{s} \sqrt{ξ^{2} + η^{2}} d ξ d η e_{3} \\ = - \frac{1}{3} [\sqrt{2} + \ln (1 + \sqrt{2})] (M + m_{1} + m_{2} + m_{3}) gs e_{3} \end{matrix}

(32)

Similarly, if body M rotates clockwise, we obtain that

M_{ν} = \frac{1}{3} [\sqrt{2} + \ln (1 + \sqrt{2})] (M + m_{1} + m_{2} + m_{3}) gs e_{3}

(33)

Denoting

ν = [\sqrt{2} + \ln (1 + \sqrt{2})] s / 3

(34)

we summarize that

M_{ν} = {\begin{matrix} - (M + m_{1} + m_{2} + m_{3}) g ν for \overset{\cdot}{θ} > 0 \\ (M + m_{1} + m_{2} + m_{3}) g ν for \overset{\cdot}{θ} < 0 \end{matrix}

(35)

For $\overset{\cdot}{θ} = 0$

| M_{ν} | \leq (M + m_{1} + m_{2} + m_{3}) g ν

(36)

From equations (35) and (36), one may find that $M_{ν}$ in equation (27) varies with the sign of body M’s angular velocity $\overset{\cdot}{θ}$ .

To obtain the angular velocity $\overset{\cdot}{θ}$ , we integrate equation (27) from both sides with respect to time t, which yields

\begin{matrix} (J_{m} + 2 {ml}^{2} + 2 m η^{2}) \overset{\cdot}{θ} = - 2 ml \overset{\cdot}{η} + M_{ν} t \\ + (J_{m} + 2 {ml}^{2} + 2 m η^{2} (t_{0})) \overset{\cdot}{θ} (t_{0}) + 2 ml \overset{\cdot}{η} (t_{0}) - M_{ν} t_{0} \end{matrix}

(37)

where t ₀ denotes the initial moment. It is easy to see from equation (37) that

\begin{matrix} \overset{\cdot}{θ} (t) = \\ \frac{- 2 ml \overset{\cdot}{η} + M_{ν} t + (J_{m} + 2 {ml}^{2} + 2 m η^{2} (t_{0})) \overset{\cdot}{θ} (t_{0}) + 2 ml \overset{\cdot}{η} (t_{0}) - M_{ν} t_{0}}{J_{m} + 2 {ml}^{2} + 2 m η^{2}} \end{matrix}

(38)

Up to now, governing equations of the two basic types of body M’s motion are established. It is also known to us that vibration-driven systems move due to the vibrations of some components of their own. However, variables $\overset{\cdot}{ξ}, \overset{\cdot\cdot}{ξ}, \overset{\cdot}{η}, and \overset{\cdot\cdot}{η}$ in equations (17) and (27) concerning internal drives of the system are still unknown. Thus, before explaining the construction of the planar locomotion of body M, we first introduce the drive modes to which three internal masses are subject.

Internal acceleration-controlled motion

In order to actuate body M, internal masses m ₁, m ₂, and m ₃ are assumed to move periodically within it. Relative displacements of them are constrained in a prescribed length L given by

0 \leq d_{i} (t) \leq L (i = 1, 2)

(39)

Moreover, relative accelerations, velocities, and displacements of the internal masses are periodic functions with the same period T_i . At the initial moment of each period, the internal masses are supposed to be at rest in their initial positions, and they reach the maximum relative displacements L at $t = β_{i}$ . Thus, we have the following conditions

d_{i} (0) = 0, d_{i} (T_{i}) = 0, d_{i} (β_{i}) = L, β_{i} \in [0, T_{i}]

(40)

and

{\overset{\cdot}{d}}_{i} (0) = 0, {\overset{\cdot}{d}}_{i} (T_{i}) = 0 (i = 1, 2)

(41)

Here, we introduce the drive mode of the internal masses (see Figure 7). The drive mode is divided into three segments and the length of each segment is denoted as $τ_{i 1}$ , $τ_{i 2}$ , and $τ_{i 3}$ (i = 1, 2) and relative accelerations $w_{ij}$ (i = 1,2; j = 1, 2, 3) in each segment are constant and are expressed as (see Figure 7(a))

{\begin{matrix} {\overset{\cdot\cdot}{d}}_{i} (t) = w_{i 1} for t \in (0, τ_{i 1}) \\ {\overset{\cdot\cdot}{d}}_{i} (t) = - w_{i 2} for t \in (τ_{i 1}, τ_{i 1} + τ_{i 2}) \\ {\overset{\cdot\cdot}{d}}_{i} (t) = w_{i 3} for t \in (τ_{i 1} + τ_{i 2}, T_{i}) \end{matrix} (i = 1, 2)

(42)

Figure 7.

Acceleration-controlled mode of the internal masses: (a) relative acceleration of the internal masses during one period, (b) relative velocity of the internal masses during one period, and (c) relative displacement of the internal masses during one period.

With some calculations, we represent $τ_{i 1}$ , $τ_{i 2}$ , and $τ_{i 3}$ with $w_{i 1}$ , $- w_{i 2}$ , and $w_{i 3}$ , namely

\begin{matrix} τ_{i 1} = {[\frac{2 w_{i 2} L}{w_{i 1} (w_{i 1} + w_{i 2})}]}^{\frac{1}{2}}, \\ τ_{i 2} = {(\frac{2 L}{w_{i 2}})}^{\frac{1}{2}} [{(\frac{w_{i 1}}{w_{i 1} + w_{i 2}})}^{\frac{1}{2}} + {(\frac{w_{i 3}}{w_{i 2} + w_{i 3}})}^{\frac{1}{2}}] \\ τ_{i 3} = {[\frac{2 w_{i 2} L}{w_{i 3} (w_{i 2} + w_{i 3})}]}^{\frac{1}{2}}, \\ T_{i} = τ_{i 1} + τ_{i 2} + τ_{i 3} = {(\frac{2 L}{w_{i 2}})}^{\frac{1}{2}} [{(\frac{w_{i 1} + w_{i 2}}{w_{i 1}})}^{\frac{1}{2}} \\ + {(\frac{w_{i 2} + w_{i 3}}{w_{i 3}})}^{\frac{1}{2}}] (i = 1, 2) \end{matrix}

(43)

In practical application, there should be constraints on the relative accelerations of internal masses, thus

0 < w_{ij} \leq W (i = 1, 2; j = 1, 2, 3)

(44)

Furthermore, to actuate the system, the maximum drive force and moment should exceed the resistances; thus, it can be derived from equations (17) and (27) that

W > max {\frac{a_{i}}{μ_{i}}, \frac{2 g ν}{μ l}} (i = 1, 2)

(45)

By integrating equation (42), we obtain the relative velocity of internal masses (see Figure 7(b))

{\begin{matrix} {\overset{\cdot}{d}}_{i} = w_{i 1} t for t \in [0, τ_{i 1}] \\ {\overset{\cdot}{d}}_{i} = w_{i 1} τ_{i 1} - w_{i 2} (t - τ_{i 1}) for t \in [τ_{i 1}, τ_{i 1} + τ_{i 2}] \\ {\overset{\cdot}{d}}_{i} = w_{i 1} τ_{i 1} - w_{i 2} τ_{i 2} + w_{i 3} (t - τ_{i 1} - τ_{i 2}) for t \in [τ_{i 1} + τ_{i 2}, T_{i}] \end{matrix} (i = 1, 2) (46)

(46)

and the relative displacement (see Figure 7(c))

{\begin{matrix} d_{i} = \frac{w_{i 1} t^{2}}{2} & for & t \in [0, τ_{i 1}] \\ d_{i} = \frac{w_{i 1} τ_{i 1}^{2}}{2} + w_{i 1} τ_{i 1} (t - τ_{i 1}) - \frac{w_{i 2} {(t - τ_{i 1})}^{2}}{2} & for & t \in [τ_{i 1}, τ_{i 1} + τ_{i 2}] \\ d_{i} = \frac{w_{i 1} τ_{i 1}^{2}}{2} + w_{i 1} τ_{i 1} τ_{i 2} - \frac{w_{i 2} τ_{i 2}^{2}}{2} \\ + (w_{i 1} τ_{i 1} - w_{i 2} τ_{i 2}) (t - τ_{i 1} - τ_{i 2}) + \frac{w_{i 3} {(t - τ_{i 1} - τ_{i 2})}^{2}}{2} & for & t \in [τ_{i 1} + τ_{i 2}, T_{i}] \end{matrix} (i = 1, 2) (47)

(47)

For the excitation shown in equation (42), the rectilinear or rotary motion of body M may have two modes, that is, modes a and b (see Figure 8). In mode a, the velocity of body M is negative at first, then turns positive, and finally decreases to 0 until the end of this period. In mode b, the velocity of body M has only positive value. After reaching a peak, it vanishes and keeps at 0 until the end of the period. For a more natural, simple, and convenient practical realization, only mode b is considered in this article.

Figure 8.

Two motion modes of body M under the acceleration-controlled mode: (a) mode a and (b) mode b.

For the rectilinear motion in the x- or y-direction, substituting equation (42) into equation (17) and integrating equation (17) for mode b yield

{\begin{matrix} {\overset{\cdot}{D}}_{i} (t) = 0 & for & t \in [0, τ_{i 1}] \\ {\overset{\cdot}{D}}_{i} (t) = (μ_{i} w_{i 2} - a_{i}) (t - τ_{i 1}) & for & t \in [τ_{i 1}, τ_{i 1} + τ_{i 2}] \\ {\overset{\cdot}{D}}_{i} (t) = (μ_{i} w_{i 2} - a_{i}) τ_{i 2} - (μ_{i} w_{i 3} + a_{i}) (t - τ_{i 1} - τ_{i 2}) & for & t \in [τ_{i 1} + τ_{i 2}, t_{i 1}] \\ {\overset{\cdot}{D}}_{i} (t) = 0 & for & t \in [t_{i 1}, T_{i}] \end{matrix} (i = 1, 2) (48)

(48)

where

t_{i 1} = τ_{i 1} + τ_{i 2} + \frac{μ_{i} w_{i 2} - a_{i}}{μ_{i} w_{i 3} + a_{i}} τ_{i 2} (i = 1, 2)

(49)

To ensure mode b, it needs to be satisfied that

μ_{i} w_{i 1} \leq a_{i}, μ_{i} w_{i 2} > a_{i}, μ_{i} w_{i 3} \leq a_{i} (i = 1, 2)

(50)

Under this constraint, we can calculate the rectilinear displacement of body M as

\begin{matrix} D_{i} (T_{i}) & = \frac{{\overset{\cdot}{D}}_{i} (τ_{i 1} + τ_{i 2}) (t_{1} - τ_{i 1})}{2} \\ = \frac{μ_{i} L (w_{i 2} + w_{i 3}) (μ_{i} w_{i 2} - a_{i})}{w_{i 2} (μ_{i} w_{i 3} + a_{i})} \\ (\sqrt{\frac{w_{i 1}}{w_{i 1} + w_{i 2}}} + \sqrt{\frac{w_{i 3}}{w_{i 2} + w_{i 3}}}) (i = 1, 2) \end{matrix}

(51)

and the average rectilinear velocity of body M as

\begin{matrix} V_{i} = \frac{D_{i} (T_{i})}{T_{i}} \\ = μ_{i} {(\frac{L}{2})}^{\frac{1}{2}} \frac{(w_{i 2} + w_{i 3}) (μ_{i} w_{i 2} - a_{i}) (\sqrt{\frac{w_{i 1} + w_{i 2}}{w_{i 1}}} + \sqrt{\frac{w_{i 2} + w_{i 3}}{w_{i 3}}})}{w_{i 2}^{\frac{1}{2}} (μ_{i} w_{i 3} + a_{i}) \frac{w_{i 1} + w_{i 2}}{w_{i 1}} \frac{w_{i 2} + w_{i 3}}{w_{i 3}}} \\ (i = 1, 2) \end{matrix}

(52)

For the rotary motion in the $θ$ direction, it follows from equation (38) that

{\begin{matrix} \overset{\cdot}{θ} (t) = 0 & for & t \in [0, τ_{i 1}] \\ \overset{\cdot}{θ} (t) = \frac{(μ {lw}_{i 2} - g ν) (t - τ_{i 1})}{J + μ l^{2} + μ η^{2}} & for & t \in [τ_{i 1}, τ_{i 1} + τ_{i 2}] \\ \overset{\cdot}{θ} (t) = \frac{(μ {lw}_{i 2} - g ν) τ_{i 2} - (μ {lw}_{i 3} + g ν) (t - τ_{i 1} - τ_{i 2})}{J + μ l^{2} + μ η^{2}} & for & t \in [τ_{i 1} + τ_{i 2}, t_{i 1}] \\ \overset{\cdot}{θ} (t) = 0 & for & t \in [t_{i 1}, T_{i}] \end{matrix} (i = 1, 2) (53)

(53)

where $J = J_{m} / (M + m_{1} + m_{2} + m_{3})$ . It is easy to see that $J + μ l^{2} + μ η^{2} > 0$ . Thus, the sign of $\overset{\cdot}{θ} (t)$ is determined by its numerator, this is why both rectilinear and rotary motions share modes a and b. Furthermore, by comparing the numerator of $\overset{\cdot}{θ} (t)$ and ${\overset{\cdot}{D}}_{i} (t)$ , one may find that they have the same form. Similarly, to ensure mode b, we have

μ {lw}_{i 1} \leq g ν, μ {lw}_{i 2} > g ν, μ {lw}_{i 3} \leq g ν

(54)

Correspondingly, we calculate the rotary displacement of body M in one period as

\begin{matrix} θ (T_{i}) & = \frac{\overset{\cdot}{θ} (τ_{i 1} + τ_{i 2}) (t_{1} - τ_{i 1})}{2} \\ = \frac{L}{l^{2}} \frac{(μ {lw}_{i 2} - g ν) (w_{i 2} + w_{i 3}) {[\sqrt{\frac{w_{i 1}}{w_{i 1} + w_{i 2}}} + \sqrt{\frac{w_{i 3}}{w_{i 2} + w_{i 3}}}]}^{2}}{(μ {lw}_{i 3} + g ν) w_{i 2} [J / μ l^{2} + 1 + {(\frac{w_{i 2}}{w_{i 2} + w_{i 3}})}^{2}]} \end{matrix}

(55)

and the average rotary velocity of body M in one period as

\begin{matrix} ω (T_{i}) = \frac{θ (T_{i})}{T_{i}} = \frac{1}{l^{2}} {(\frac{w_{i 2} L}{2})}^{\frac{1}{2}} \\ \frac{(μ {lw}_{i 2} - g ν) (w_{i 2} + w_{i 3}) (\sqrt{\frac{w_{i 1} + w_{i 2}}{w_{i 1}}} + \sqrt{\frac{w_{i 2} + w_{i 3}}{w_{i 3}}})}{(μ {lw}_{i 3} + g ν) w_{i 2} [J / μ l^{2} + 1 + {(\frac{w_{i 2}}{w_{i 2} + w_{i 3}})}^{2}] (\sqrt{\frac{w_{i 1} + w_{i 2}}{w_{i 1}}} + \sqrt{\frac{w_{i 2} + w_{i 3}}{w_{i 3}}})} \end{matrix}

(56)

For further optimization, we introduce the dimensionless variables as

p_{ij} = \frac{μ_{i} w_{ij}}{a_{i}}, q_{ij} = \frac{μ {lw}_{ij}}{g ν} (i = 1, 2; j = 1, 2, 3)

(57)

With them, we change equations (50), (51), (52), (54), (55), and (56) into dimensionless forms, given by

p_{i 1} \leq 1, p_{i 2} > 1, p_{i 3} \leq 1

(58)

\begin{matrix} D_{i} (T_{i}) = μ_{i} L \frac{(p_{i 2} + p_{i 3}) (p_{i 2} - 1)}{p_{i 2} (p_{i 3} + 1)} \\ {[{(\frac{p_{i 1}}{p_{i 1} + p_{i 2}})}^{\frac{1}{2}} + {(\frac{p_{i 3}}{p_{i 2} + p_{i 3}})}^{\frac{1}{2}}]}^{2} \end{matrix}

(59)

\begin{matrix} V_{i} (T_{i}) = {(\frac{μ aL}{2})}^{\frac{1}{2}} \frac{(p_{i 2} + p_{i 3}) (p_{i 2} - 1)}{\sqrt{p_{i 2}} (p_{i 3} + 1)} \\ (\frac{1}{\sqrt{1 + \frac{p_{i 2}}{p_{i 1}}} (1 + \frac{p_{i 2}}{p_{i 3}})} + \frac{1}{(1 + \frac{p_{i 2}}{p_{i 1}}) \sqrt{1 + \frac{p_{i 2}}{p_{i 3}}}}) \end{matrix}

(60)

q_{i 1} \leq 1, q_{i 2} > 1, q_{i 3} \leq 1

(61)

θ (T_{i}) = \frac{L}{l^{2}} \frac{(q_{i 2} - 1) (q_{i 2} + q_{i 3}) {[\sqrt{\frac{q_{i 1}}{q_{i 1} + q_{i 2}}} + \sqrt{\frac{q_{i 3}}{q_{i 2} + q_{i 3}}}]}^{2}}{q_{i 2} (q_{i 3} + 1) [J / μ l^{2} + 1 + {(\frac{q_{i 2}}{q_{i 2} + q_{i 3}})}^{2}]}

(62)

\begin{matrix} ω (T_{i}) = \frac{1}{l^{2}} {(\frac{g ν L}{2 μ l})}^{\frac{1}{2}} \\ \frac{(q_{i 2} - 1) (q_{i 2} + q_{i 3}) {(\sqrt{\frac{q_{i 1} + q_{i 2}}{q_{i 1}}} + \sqrt{\frac{q_{i 2} + q_{i 3}}{q_{i 3}}})}^{2}}{\sqrt{q_{i 2}} (q_{i 3} + 1) [J / μ l^{2} + 1 + {(\frac{q_{i 2}}{q_{i 2} + q_{i 3}})}^{2}] (\sqrt{\frac{q_{i 1} + q_{i 2}}{q_{i 1}}} + \sqrt{\frac{q_{i 2} + q_{i 3}}{q_{i 3}}})} \end{matrix}

(63)

Based on these nondimensionalized equations, the planar locomotion on body M is constructed in the next section.

Planar locomotion and optimized drive mode

If the trajectory of a system is planar, then we consider its motion to be planar motion. In this section, we classify the planar trajectories into two kinds of lines, that is, oblique lines and curve lines. The trajectory of the planar locomotion constructed is composed of these two kinds of lines. However, it is hard or even impossible for a vibration-driven system to move exactly along an oblique line or a curve line, so we use folding lines to approach the two kinds of lines along which body M is expected to move. In the process, the motion of body M in one period is divided into two stages. In each stage, one type of the two basic motions of body M, that is, rectilinear motion and rotary motion, is chosen. Thus, with different combinations of the basic motions, the trajectory of body M may be different folding lines. In this way, planar trajectories of body M in oblique lines or curve lines are approximated. To this end, we start from the oblique lines.

Oblique lines

Construction of the oblique lines

To approach oblique lines, we design the relative accelerations of internal masses as shown in Figure 9. In the first stage, mass m ₁ moves alone, while in the second stage masses m ₂ and m ₃ perform inphase motions spontaneously. The motions in the first and second stages are governed by equations (4) and (13), respectively. The displacements of body M in one period $T_{k}$ under a periodic excitation are shown in Figure 10. Body M moves from point A to B in the first stage and from B to C in the second stage. Thus, the final displacement of body M is from A to C. That is why we can approach oblique lines with folding lines in this way.

Figure 9.

Acceleration-controlled mode for oblique lines.

Figure 10.

Trajectory of body M in one period designed to approach oblique lines.

Slope of oblique lines

From the former discussion, one knows that the route AB plus BC is equivalent to the route AC, which is an oblique line. The slope k of AC is given as

k = \tan α = \frac{y (T_{2})}{x (T_{1})}

(64)

It is to see that by changing the sign of the drive modes or reversing the order of the two stages, one can control body M to move along oblique lines with different slopes. Then, it is sufficient to study the oblique lines with the slope

k \in [0, 1]

(65)

Substituting equation (64) into equation (65) yields

x (T_{1}) \geq y (T_{2})

(66)

Thus, to achieve a maximum average velocity, it is assumed that body M moves with the optimized drive parameters $w_{1 j}$ (j = 1, 2, 3) in the first stage, while the drive parameters $w_{2 j}$ (j = 1, 2, 3) in the second stage are half-optimized, which is partly determined by the slope k. To optimize the oblique trajectory, we calculate the partial derivatives of $V_{i} (T_{i})$ with respect to p_ij (j = 1, 2, 3) and we have

\frac{\partial V_{i} (T_{i})}{\partial p_{ij}} > 0 (j = 1, 2, 3)

(67)

Considering equation (58), we choose that

p_{11} = 1, p_{12} = P, p_{13} = 1

(68)

Substituting equation (68) into equation (59) yields

x (T_{1}) = 2 μ L (1 - \frac{1}{P})

(69)

and it follows from equation (64) that

y (T_{2}) = kx (T_{1}) = 2 k μ L (1 - \frac{1}{P})

(70)

To optimize the motion in the second stage, we choose

p_{21} = 1, p_{23} = 1

(71)

Substituting equations (70) and (71) into equation (59) yields

2 k μ L (1 - \frac{1}{P}) = μ L \frac{(p_{22} + 1) (p_{22} - 1)}{2 p_{22}} {[2 {(\frac{1}{1 + p_{22}})}^{\frac{1}{2}}]}^{2}

(72)

from which it can be solved that

p_{22} = \frac{1}{1 - k (1 - \frac{1}{P})}

(73)

Letting k = 1 in equation (73), we can see that $p_{22} = P$ , then rectilinear motion in the second stage is also optimized. For $k \in [0, 1)$ , $p_{22} < P$ , that is, the motion is not optimized. Thus, the rectilinear motion in the second stage is optimized only when k = 1, that is why we call it “half-optimized.”

Curve lines with rectilinear and rotary motions

Construction of the curve lines

As for the curve lines, we design the relative accelerations of internal masses as shown in Figure 11. In the first stage, mass m ₁ moves alone, while in the second stage masses m ₂ and m ₃ perform antiphase motions spontaneously. Equations (4) and (27) govern the motions in the first and second stages, respectively. The displacements of body M in one period $T_{ρ}$ under the excitation are shown in Figure 12. In the first stage, body M moves from point E to F and has an angular displacement in the second stage. In the following period, it moves from point F to G and then rotates again. Repeat the process, then body M will move roughly along the circle at point D with radius R.

Figure 11.

Acceleration-controlled mode for curve lines.

Figure 12.

Trajectory of body M in one period designed to approach curve lines.

Curvature of the curve lines

From the discussion above, we know that points E, F, and G are on the circumcircle with radius R. After some geometric processing, the curvature of the curve line is represented as

ρ = \frac{1}{R} = \frac{2 \sin \frac{θ (T_{2})}{2}}{x (T_{1})}, θ \in [0, \frac{π}{2}]

(74)

To achieve a maximum velocity, it is also assumed that the rectilinear motion in the first stage is optimized. The drive parameters are chosen as displayed in equation (68). For the second stage, we calculate the partial derivatives of $ω (T_{2})$ with respect to q ₂₁ and q ₂₃ and find

\frac{\partial ω (T_{2})}{\partial p_{ij}} > 0 (j = 1, 3)

(75)

Considering equation (61), we choose

q_{21} = 1, q_{23} = 1

(76)

Substituting equations (62), (69), and (76) into equation (74) yields

ρ = \frac{\sin [\frac{L}{l^{2}} \frac{q_{22} - 1}{q_{22} [J / μ l^{2} + 1 + {(\frac{q_{22}}{q_{22} + 1})}^{2}]}]}{μ L (1 - \frac{1}{P})}

(77)

from which one may obtain that

[A (B + 1)] q_{22}^{3} + (2 AB - 1) q_{22}^{2} + (AB + 1) q_{22} + 1 = 0

(78)

where

\begin{matrix} A = & \frac{l^{2}}{L} \arcsin [ρ μ L (1 - \frac{1}{P})], B = J / (μ l^{2}) + 1, \\ \arcsin [μ ρ L (1 - \frac{1}{P})] \in [0, \frac{π}{2}] \end{matrix}

(79)

By solving equation (78), one may obtain dimensionless drive parameter $q_{22} \in (1, W)$ .

Up to now, the drive parameters of the second pattern of motion are determined, that is, according to equations (68), (76), and (78), body M can move roughly along curves with given curvature $ρ$ .

Conclusions

A vibration-driven system with three internal and acceleration-controlled masses is modeled to perform the planar locomotion. With the inphase and antiphase motions of the internal masses, the system can move in a straight line or rotate about its center, respectively. The two basic types of motions are utilized to construct the planar motion of the vibration-driven system.

Two typical trajectories of the dynamic system, that is, oblique lines and curve lines, are considered in the article. For the oblique lines, we construct them by alternating rectilinear motions of body M in the x- and y-directions. For the curve lines, body M is controlled to have a linear displacement followed with an angular displacement in each period. As a result, both the planar lines are approached by folding lines. Assuming that the displacements of body M in the two basic motions are not very large or even tiny, the actual trajectory of the dynamic system is close enough to the oblique and curve lines.

Two characteristic parameters of body M’s trajectories, that is, the slope of the oblique lines and the curvature of the curve lines, are defined and expressed in the drive parameters w_ij (i = 1, 2; j = 1, 2, 3). The planar trajectory changes with the drive parameters varying. Thus, one can construct planar trajectories of body M with different slopes or curvatures.

To achieve a maximum velocity of the system, drive parameters w_ij (i = 1, 2; j = 1, 2, 3) are both optimized for rectilinear and rotary motion. We find that the velocities of rectilinear and rotary motions both grow monotonically with the increase in the drive parameters w_i ₁ and w_i ₃. Thus, in the optimized motions, w_i ₁ and w_i ₃ take their maximum values while w_i ₂ is utilized to control the geometric shape of the folding lines.

Footnotes

Academic Editor: Fen Wu

Declaration of conflicting interests

The authors declare that there is no conflict of interest.

Funding

This research was supported by National Natural Science Foundation of China under Grant No. 11272236.

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Locomotion analysis of a vibration-driven system with three acceleration-controlled internal masses

Abstract

Keywords

Introduction

Dynamics of the mechanical system

Description of the dynamic system

Rectilinear and rotary motions

Rectilinear motions

Rotary motion

Internal acceleration-controlled motion

Planar locomotion and optimized drive mode

Oblique lines

Construction of the oblique lines

Slope of oblique lines

Curve lines with rectilinear and rotary motions

Construction of the curve lines

Curvature of the curve lines

Conclusions

Footnotes

Declaration of conflicting interests

Funding

References