Substantiating the excitation conditions of a two-module vibration-driven locomotion system with two unbalanced rotors

2.1. Dynamic diagram and mathematical model of the vibration-driven locomotion system

Let us consider the wheeled two-module vibration-driven locomotion system, whose dynamic diagram is shown in Fig. 1. The system consists of two movable platforms, which can roll along a smooth horizontal surface. The platforms are connected by a spring-damper element characterized by stiffness $k$ and damping coefficient $c$. To actuate the system, two inertial vibration exciters (unbalanced rotors) are mounted on each platform. The rotary motions of the rotors are described by the corresponding laws ${\phi }_1\left(t\right)$ and ${\phi }_2\left(t\right)$, while the positions of the movable platforms relative to the inertial reference system $xOy$ are defined by the generalized coordinates $x_1$ and $x_2$. The masses of the front and rear platforms are denoted as $m_1$, $m_2$, respectively, and the masses of the corresponding unbalanced rotors are $m_3$, $m_4$. In addition, the rotors are characterized by the eccentricities $r_1=A_1{B}_1$, $r_2=A_2{B}_2$.

Fig. 1Dynamic diagram of the wheeled two-module vibration-driven locomotion system

While performing further investigations, let us assume zero-friction conditions when the wheels are rolling along a smooth horizontal surface. Let us study the case of the steady-state operational conditions of the inertial exciters when the rotors’ angular speeds are constant and equal to ${\omega }_1$ and ${\omega }_2$. Denoting the initial positions (phases of oscillations) of the front and rear unbalanced rotors as ${\phi }_{10}$ and ${\phi }_{20}\text{,}$ their motion laws can be described as follows: ${\phi }_1\left(t\right)={\phi }_{10}+{\omega }_{1}t$, ${\phi }_2\left(t\right)={\phi }_{20}+{\omega }_{2}t$. The mathematical model describing the locomotion conditions of the considered wheeled two-module vibration-driven system with two unbalanced rotors can be derived using the Euler-Lagrange equations and presented in the following form:

where the dots above $x_1\left(t\right)$ and $x_2\left(t\right)$ denote the first- and second-order derivatives of these displacements with respect to time, i.e., the corresponding velocities and accelerations.

Taking into account the fact that the considered locomotion system is semidefinite, its first natural frequency equals zero, which shows that the platforms move (oscillate) as a single body. The second natural frequency of the two-mass vibratory system can be determined as follows:

Eq. (2) allows for determining the spring stiffness of the considered vibration-driven locomotion system providing its energy-efficient near-resonance operational conditions:

where $\xi$ is the correction coefficient providing the near-resonance operation.

Therefore, when the system’s inertial parameters ($m_1$, $m_2$, $m_3$, $m_4$) are known, and the forced frequency ${\omega }_f$ is prescribed, the necessary natural frequency can be determined as follows: ${\omega }_n={\omega }_f/\xi$, where $\xi$ can take different values depending on a particular design and operational peculiarities of the system $(\xi =$ 0.93…0.99 [16]). Then, Eq. (3) can be used for calculating the necessary spring stiffness providing the system’s near-resonance locomotion.

Fig. 2Simulation model of the two-module vibration-driven system developed in MapleSim software

2.2. Simulation model of the two-module vibration-driven system

The simplified simulation model of the two-module vibration-driven system is developed in the MapleSim software (Fig. 2). Two prismatic sliders P₁ and P₂ with the connected masses RB₁ and RB₂ are used for simulating the movable platforms. The rods RBF₁, RBF₂, RBF₃, RBF₄ with the revolute joints $R_1$, $R_2$ and joined masses RB₃, RB₄ simulate the unbalanced rotors.

To actuate the unbalanced rotors, the corresponding constant-speed motion drivers (motors) CS₁ and CS₂ are used. The sliders P₁ and P₂ are connected by slider P₃ equipped with the spring-damper elements SD₁ and S₁. The fixed frames FF₁ and FF₂ are used for prescribing the initial positions and motion directions of the sliders P₁ and P₂. To define the kinematic characteristics of the platforms (masses RB₁, RB₂), the corresponding sensors Probe1 and Probe2 are applied.

3.1. Numerical modeling of the system locomotion in the Mathematica software

Let us analyze the system’s locomotion conditions in the Mathematica software by numerical solving of the system of differential Eq. (1) using the Runge-Kutta methods. The following inertial and excitation parameters are considered: $m_1=m_2=$ 0.25 kg, $m_3=m_4=$ 0.025 kg, ${\omega }_1={\omega }_2=$ 157 s^-1, $c=$2 (N∙s)/m, ${\phi }_{10}=$0. Using the Eq. (3), the spring stiffness can be calculated: $k\approx$3.7∙10³ N/m. To perform further numerical modeling, the following values of the phase shift angle ${\phi }_{20}$ of the rear unbalanced rotor are adopted: 0, 45° ($\pi$/4), 90° ($\pi$/2), 135° (3$\pi$/4), 180° ($\pi$), 225° (5$\pi$/4), 270° (3$\pi$/2), 315° (7$\pi$/4). The corresponding modeling results are shown in Fig. 3 using the black, blue, green, purple, red, gray, brown, and orange curves, respectively. Figs. 3(a-c) show time dependencies of the front platform’s basic kinematic characteristics (displacement, velocity, acceleration) at different phase shift angles ${\phi }_{20}$. In Fig. 3(d), the dependence of the front platform’s average speed on ${\phi }_{20}$ is presented. Fig. 3(e) describes the relative positions (displacements) of the front and rear platforms in time. The initial distance between the platforms is set equal to 0.06 m.

The distance traveled by the front platform during the time period of 1 s, as well as the motion direction, significantly depends on the phase shift angle ${\phi }_{20}$ (see Fig. 3). The angles 45°, 90°, and 135° provide the backward motion, while at 225°, 270°, and 315° the platform moves forward. The largest average speeds of about 0.23 m/s are observed at ${\phi }_{20}=$90° and ${\phi }_{20}=$ 270°, while the lowest ones are equal to zero at ${\phi }_{20}=$ 0° and ${\phi }_{20}=$ 180°. The maximal instantaneous values of the platform’s speed and acceleration reach 3.7 m/s and 550 m/s², respectively, at ${\phi }_{20}=$ 135°. The maximal relative distance between the front and rear platforms exceeds 0.1 m, while the minimal one is about 0.015 m at ${\phi }_{20}=$ 180°. The obtained results will be used while designing the experimental prototype of the wheeled vibration-driven robot.