Motion of an Unbalanced Impact Body Colliding with a Moving Belt

Abstract

In the field of mechanical engineering, conveyors and moving belts are frequently used machine parts. In many working regimes, they are subjected to sudden loading, which can be a source of irregular motion in the impacting bodies and undesirable behavior in the working machine. This paper deals with a mechanical model where colisions between an impact body and a moving belt take place. The impact body is constrained by a flexible rope, the upper end of which is excited by a slider in the vertical direction. The behavior of the system was investigated in terms of its dependence on the amplitude and frequency of excitation given by the movement of the slider, and the eccentricity of the center of gravity of the impact body. Outputs of the computations indicate that different combinations of the analyzed parameters lead to high complexity of the system’s movement. The bifurcation analysis shows multiple periodic areas changed by chaotic regions. The research carried out provides more details about the behavior and properties of strongly nonlinear mechanical systems resulting from impacts and dry friction. The obtained information will enable designers to propose parameters for industrial machines that make it possible to avoid their working at undesirable operating levels.

1. Introduction

Moving belts are machine parts that are frequently utilized in the area of mechanical engineering. They are applied for the transport of bodies or bulk materials and their processing. In many cases, the belts of conveyors operating in underground or surface mines, and crushing or milling plants, are subjected to impact loading caused by falling stones, blocks of rock, or crushing hammers. The impact forces, together with dry friction in the contact area, have a significant influence on the behavior of these devices, which is directly connected to the safety of the working process, adequate maintenance, and service life of the machinery. Because of this, conveyor belt machines have become a subject of analysis at many workplaces. Further technological motivations for the proposed research and applications can be found, e.g., in [1]; see also references therein.

Moon and Wickert [2] investigated the vibration of a power transmission belt system excited by pulleys exhibiting slight eccentricity through experiments and analytical methods. The experimentally observed jump and hysteresis phenomena in the near-resonant response were studied with a model that took into account nonlinear stretching of the belt. Andreus and Casini [3] studied the movement of a body excited by a moving belt and colliding with a rigid and flexible obstacle. The varying nature of discontinuities arising in the combined problem of friction and impact was recognized and discussed. The authors presented closed-form solutions for both transient and steady-state responses, assuming Coulomb friction law and rigid stop-limiting motion. Chatterjee [4] presented a new approach to controlling friction-induced self-excited vibration (see also [5]). The investigated system consisted of a body flexibly connected to the frame and sliding on a moving belt. The control action was achieved by modulating the normal load at the frictional interface based on the state of the oscillatory system. These research results had a major impact on the theory of non-smooth dynamical systems that was developed in recent decades; see, e.g., [6,7]. This paper deals with searching for the control parameters.

This paper focuses on the impacts of an unbalanced impact body coupled with an elastic rope to a moving belt. The rope is set into motion at the upper end in the vertical direction. The impact body is unbalanced, which leads to eccentric collisions. Computational simulations were used to perform the research. The research outputs indicate that the character (regular or chaotic) of the system vibration induced by the impacts, dry friction, and motion of the belt depends on the frequency of the excitation, its amplitude, and the eccentricity of the impact body. This paper is a natural research continuation of the work with a much simpler model reported in [8], where the impact body was assumed to be balanced and results, contrary to this paper, were obtained for the parameters of two systems. In [9], a balanced case model is described with the assumption of a special case of an impact body that was taken as a cylinder.

The paper starts with the introduction, followed by the analyzed system (Section 2), and setting up of the mathematical model (Section 3). To achieve the aim of the paper, typical motion regimes are constructed (Section 4). Next, dynamics quantifiers and qualifiers are applied to the ranges of three parameters (Section 5) and, finally, bifurcations are observed (Section 6). Lastly, concluding remarks are made (Section 7).

2. The Analyzed System

The investigated system is presented in Figure 1 and it is composed of: a slider (1), impact body (2), flexible rope (3), conveyor belt (4), and rigid platform (5), see [8,9]. The slider performs a reciprocating sliding motion in the vertical direction. The impact body is of a complex shape consisting of a central cylinder and two end parts. The rope connects the slider with the center of the impact body. The belt moves in a horizontal direction at a constant velocity. The impact body performs a general planar motion. It can slide in the vertical and horizontal directions, and it can rotate about its axis. The rope transmits only the tensile force. All bodies of the system (except the rope) can be considered absolutely rigid. The flexibility of the rope makes it possible for the impact body to collide with the moving belt. Collisions can occur between the belt and the cylindrical part of the impact body. The material of the colliding bodies, in a small area near to the impact point and in a normal direction to the contact surfaces, is considered to be linearly elastic, exhibiting some material damping. As the impact body is unbalanced, the impacts are eccentric; the center of gravity of the impact body is marked T in Figure 1. The friction in the contact area is considered to be of a Coulomb type and the connection of the rope with the impact body and the slider is considered neutral (without resistances against the relative motion). The stiffness of the rope is linear, as is the damping of the impact body caused by the environment. From the point of view of time history, vertical motion of the slider is harmonic with a short transient period at the beginning.

3. Mathematical Model of the System

The Lagrange equations of the second kind have been used to set up the motion equations:

$$\begin{array}{ccc}\hfill m\ddot{x}-m{e}_T\ddot{\phi }sin(\phi +{\psi }_T)-m{e}_T{\dot{\phi }}^{2}cos(\phi +{\psi }_T)& =& Q_x,\hfill \\ \hfill m\ddot{y}+m{e}_T\ddot{\phi }cos(\phi +{\psi }_T)-m{e}_T{\dot{\phi }}^{2}sin(\phi +{\psi }_T)& =& Q_y,\hfill \\ \hfill -m{e}_T\ddot{x}sin(\phi +{\psi }_T)+m{e}_T\ddot{y}cos(\phi +{\psi }_T)+(J_T+m{e}_T^2)\ddot{\phi }& =& Q_{\phi }\hfill \end{array}$$

(1)

where

$$\begin{array}{ccc}{Q}_x\hfill & =\hfill & F_x+F_t-b\dot{x},\hfill \end{array}$$

(2)

$${\displaystyle \begin{array}{ccc}{Q}_y\hfill & =\hfill & F_y+F_c-mg-b\dot{y},\hfill \end{array}}$$

(3)

$$\begin{array}{ccc}{Q}_{\phi }\hfill & =\hfill & F_{t}R-mg{e}_{T}cos(\phi +{\psi }_T)-b_{\phi }\dot{\phi }\hfill \end{array}$$

(4)

and x and y denote the position of the impact body center, and $\phi$ its angular rotation, see [8,9]. $F_x$, $F_y$, $F_t$, and $F_c$ are x and y components of the forces, by which the suspension acts on the impact body; the tangential component of the contact force (friction force), and the normal component of the contact force acting on the impact body, respectively. ($\dot{}$) and ($\ddot{}$) stand for the first and second derivative with respect to time. The notations of the other quantities and their descriptions are shown in

Table 1. Parameters of the system (1).

Quantity	Value	Description
m	10 kg	mass of the impact body
$J_T$	0.2 kg m${}^2$	moment of inertia of the impact body
		referred to its center of gravity
${\psi }_T$	0.4 rad	angular phase shift of the center of gravity
$L_0$	0.35 m	length of the unloaded suspension of the impact body
$h_z$	0.89 m	the mean hight of the excitation body above the belt
R	0.04 m	radius of the impact body
$k_c$	1 $\times {10}^7$ Nm${}^{-1}$	contact stiffness (impact body–belt)
$b_c$	100 Nsm${}^{-1}$	the coefficient of contact damping (impact body–belt)
$k_l$	200 N m${}^{-1}$	stiffness of the impact body suspension
$b_l$	5 Nsm${}^{-1}$	the damping coefficient of the impact body suspension
g	9.80665 m s${}^{-2}$	the gravity acceleration
$\alpha$	1 s${}^{-1}$	the run up coefficient
$v_p$	0.5 ms${}^{-1}$	velocity of the belt
b	0.01 Ns m${}^{-1}$	the outer damping coefficient of the impact body
		(sliding motion)
$b_{\phi }$	0.001 Nms rad${}^{-1}$	the outer damping coefficient of the impact body
		(rotational motion)
f	0.2	coefficient of friction
a	100 s m${}^{-1}$	mathematical constant defining the shape
		of the friction characteristic

.

Extension of the impact body suspension $\Delta l$ and its rate can be expressed as follows:

$$\Delta l=\sqrt{x^2+{(y-y_z-h_z)}^2}-L_0,$$

$$\Delta \dot{l}=1/2{\left(x^2+{(y-y_z-h_z)}^2\right)}^{-1/2}\left(2x\dot{x}+2(y-y_z-h_z)(\dot{y}-{\dot{y}}_z)\right).$$

Stiffness and material damping of the rope are considered to be linear. In addition, the rope transmits only the tensile force. This is expressed by

$$\begin{array}{ccc}\hfill F_l=k_l\Delta l+b_l\Delta \dot{l}& \mathrm{for}& \Delta l>0\mathrm{and}{k}_l\Delta l+b_l\Delta \dot{l}>0,\hfill \\ \hfill F_l=0& \mathrm{for}& \Delta l\le 0\mathrm{or}{k}_l\Delta l+b_l\Delta \dot{l}\le 0.\hfill \end{array}$$

Then, for the horizontal axial force components acting on the impact body, it holds:

$$\begin{array}{ccc}\hfill F_x& =& -F_{l}cos\left(\beta \right),\hfill \\ \hfill F_y& =& -F_{l}sin\left(\beta \right)\hfill \end{array}$$

(5)

where $\beta$ follows from the solution of the following equations; see Figure 2:

$$cos\left(\beta \right)=\frac{x}{\sqrt{x^2+{(y-y_z-h_z)}^2}},$$

$$sin\left(\beta \right)=\frac{y-y_z-h_z}{\sqrt{x^2+{(y-y_z-h_z)}^2}}.$$

The contact stiffness and damping are considered to be linear and the contact force can be only compressive. Then, for each magnitude, it holds that

$$\begin{array}{ccc}{F}_c=k_c(R-y)-b_c\dot{y}\hfill & \mathrm{for}& y<R\mathrm{and}{k}_c(R-y)-b_c\dot{y}>0,\hfill \end{array}$$

(6)

$$\begin{array}{ccc}{F}_c=0\hfill & \mathrm{for}& y\ge R\mathrm{or}{k}_c(R-y)-b_c\dot{y}\le 0.\hfill \end{array}$$

(7)

For the friction force acting on the impact body, it holds that

$$F_t=-\frac{2F_{c}f}{\pi }\mathrm{atan}\left(a(\dot{x}+R\dot{\phi }-v_p)\right).$$

(8)

The vertical position of the driving body is given by

$$y_z\left(t\right)=z_a(1-{\mathrm{e}}^{-\alpha t})sin\left(\omega t\right)$$

(9)

where $z_a$ is the amplitude of the driving body kinetic excitation, $\alpha$ is the run up coefficient, and $\omega$ stands for the excitation frequency. The time history of the position of the sliding body is $y_z$, which excites the system as shown in Figure 3.

At the beginning, the system takes the rest position defined by the following initial conditions:

$$\begin{array}{ccc}x\left(0\right)=0,\hfill & \hfill & \dot{x}\left(0\right)=0,\hfill \\ y\left(0\right)=h_z-L_0-mg/k_l,\hfill & \hfill & \dot{y}\left(0\right)=0,\hfill \\ \phi \left(0\right)=-\pi /2-{\psi }_T,\hfill & \hfill & \dot{\phi }\left(0\right)=0.\hfill \end{array}$$

4. Periodic, Quasi-Periodic, Chaotic Motion

The examinated system (1) is described by 20 parameters, 17 of which (given in Table 1) are fixed for all further simulations. The last three, the excitation frequency $\omega$, amplitude $z_a$, and eccentricity $e_T$, will be variable, illustrating the richness of dynamics that the model accounts for.

Firstly, three study cases are given: periodic, quasi-periodic, and chaotic. In all cases, there are given time histories, phase portraits with the Fourier spectra, and corresponding forces. The contact force $F_c$ given by ((6) and (7)), the friction force $F_t$ derived in (8), and the elastic force in the rope

$$F_r=\sqrt{F_x^2+F_y^2}$$

(10)

underline the movement character of entire bodies.

For if $e_T=$ 0.015 m, $\omega =$ 4 rad s${}^{-1}$, and $z_a=$ 0.009 m, the movement is periodic, as shown in Figure 4, Figure 5 and Figure 6. In this case, the time histories (Figure 4) generate a T1 periodic loop in the phase portrait (Figure 5—left), with detectable contact between the impact body and belt. The corresponding FFT (Figure 5—right) shows single frequencies that correspond to periodicity. The forces (Figure 6) are given for completeness.

Next, let $e_T=$ 0.02 m, $\omega =$ 2.8 rad s${}^{-1}$, and $z_a=$ 0.0056 m; then, the movement is quasi-periodic, as shown in Figure 7, Figure 8 and Figure 9. The time histories (Figure 7) used for the construction of the phase portrait (Figure 8—left) behave in a quasi-periodic mode; contacts between the impact body and belt are detectable. The corresponding FFT (Figure 8—right) confirms the quasi-periodic trajectory, since, on every main frequency, only few-sided frequencies appear. The forces (Figure 9) are given for completeness.

Finally, if parameters are set as $e_T=$ 0.015 m, $\omega =$ 2.2 rad s${}^{-1}$, and $z_a=$ 0.0028 m, then the movement is chaotic, as is visible in Figure 10, Figure 11 and Figure 12. In this case, the time histories (Figure 10) together with the phase portrait (Figure 11—left) detect contacts between the impact body and the belt. The corresponding FFT (Figure 11—right) validates chaos, since, on every main frequency, there are superposed wide bands of both-sided frequencies. The forces (Figure 12) are given for completeness.

Hence, it can be concluded that:

Property 1.

There are system (1) parameters $e_T$, ω, and $z_a$ such that its movement character is:

• periodic (for, e.g., $e_T=$ 0.015 m, $\omega =$ 4 rad s${}^{-1}$, and $z_a=$ 0.009 m),
• quasi-periodic (for, e.g., $e_T=$ 0.02 m, $\omega =$ 2.8 rad s${}^{-1}$, and $z_a=$ 0.0056 m), and
• chaotic (for, e.g., $e_T=$ 0.015 m, $\omega =$ 2.2 rad s${}^{-1}$, and $z_a=$ 0.0028 m).

The proof of Property 2 is given by computer graphics analysis studying the phase diagrams together with FFT (Figure 5, Figure 8, and Figure 11—left) underlined by their time histories in Figure 4, Figure 7, and Figure 10. This dynamic behavior coincides with K and $E_{Samp}$ shown in Figure 13, Figure 14 and Figure 15 parts (b,c). It is also worth remarking that the same movement’s patterns are visible on related forces in Figure 6, Figure 9, and Figure 12.

Consequently, it can be concluded that:

Property 2.

Let $e_T\in \{0.01,0.015,0.02\}$. If the system’s (1) trajectory is chaotic, then the impact body (body 2, Figure 1) comes into contact with the belt (body 4, Figure 1).

5. Three Parameters’ Dynamics Depiction

In the previous section, for characterization of the movement, the combination of phase portraits with FFT was used. This method is applicable only in single cases. Hence, for further investigation, a combination of newly established tests is used. These are the 0–1 test for chaos (K for brevity) and sample entropy (denoted $E_{Samp}$).

The dynamics quantifier K was introduced by [10] (see also [11]) and gives binary outputs. If K is close to zero (denoted by blue in our results), regular movements appear, and when K approaches one, there is chaos (marked in red). It is possible to apply this test to the full variety of the time series corresponding to the real data [12] and time series derived by continuous [13,14] or discrete models [15].

As a dynamical quantifier, $E_{Samp}$ is utilized. This tool was introduced by [16] and used in wide range of research areas [17,18,19]. As an output, $E_{Samp}$ gives a value that measures the complexity of the system; as $E_{Samp}$ increases, the complexity increases.

The evaluation of the movement character was performed in the free software environment R [20] using the packages Chaos01 [21] and TSEntropies [22]. This dynamics mining process was applied on simulation outputs that were obtained by the Runge–Kutta method of the fourth order using MATLAB [23] (solver ode45). Each simulation was performed on ${10}^5$ s with recorded ${10}^7$ values. To eliminate the distortions of the system, only the last 20% of simulated data were examined. The imput vector $\varphi$ of the K and $E_{Samp}$ tests is assembled as the Euclidean norm of state variables x and y:

$$\varphi \left(j\right)=\sqrt{x{\left(j\right)}^2+y{\left(j\right)}^2}.$$

As mentioned above, there were three driving parameters detected, for which the following simulations were performed:

$$e_T\times z_a\times \omega =\{0.01,0.015,0.02\}\times [0.001,0.01]\times [1,7].$$

Individual cases of $e_T\in \{0.01,0.015,0.02\}$ are shown in Figure 13, Figure 14 and Figure 15. There are given dynamics characteristics K and $E_{Samp}$ together with the output of the contact test of the impact body with the belt. It is worthy to note, that the results of K and $E_{Samp}$ coincide and the complexity (or number of chaotic cases) increases while $e_T$ increases (see Figure 13, Figure 14 and Figure 15 parts (b,c)). The area of $z_a\times \omega$ parameters for which contacts appears is slightly variable in respect to $e_T$. Moreover, it is observable that small contacts regions are surrounded by non-contact cases and vice versa (compare Figure 13 and Figure 15 parts (a); in these figures, contact cases are marked in black and non-contact by red).

The proof of Property 2 is given by computer graphics analysis studying the 0–1 test for chaos K and sample entropy $E_{Samp}$ comparing to their corresponding contact tests, which is given in Figure 13, Figure 14 and Figure 15. Note that the inverse implication of Property 2 is not valid. As a counter-example, the system’s (1) parameters can be as those related in Figure 4, Figure 5 and Figure 6.

6. Bifurcation Analysis

To show the genesis of the high complexity movement from the low one, and vice versa, the bifurcation analysis on a restriction of $z_a\times \omega$ is given for $z_a=0.008$ m supported by K and $E_{Samp}$ characteristics. For this purpose, bifurcation diagrams depending on $\omega \in [2,7]$ rad s${}^{-1}$, and its magnification for $\omega \in [2,3]$ rad s${}^{-1}$, are constructed. The bifurcation diagrams (Figure 16 and Figure 17) were constructed using the projection technique of local maxima (dark gray) together with local minima (light gray); the maximum (green line) and minimum (red line) values are also given for completeness. Figure 16 corresponds to the x variable and Figure 17 to y, while the parts indexed with “2” (right column figures) show a magnification of relevant “1” indexes (left column figures). These bifurcation diagrams show an increase in complexity when $e_T$ increases. It is also observable that there are numerous periodic windows, with a period doubling effect, surrounded by chaotic cases.

7. Conclusions

In this paper, a newly constructed mechanical system consisting of an unbalanced impact body constrained by a flexible rope and a moving belt is derived. This model was researched under massive numerical simulation, and its impressive dynamics were revealed.

To accomplish this, a model of three degrees of freedom was built and then simulated in MATLAB [23] using the Runge–Kutta method as an in-function ode45 adaptive solver, and a final evaluation was performed in R [20] applying the Chaos01 [21] and TSEntropies [22] packages.

The primary results show that all possible types of movement character can be stimulated (Property 1); period, quasi-period, and chaos (Figure 5, Figure 8, and Figure 11). For the completeness of the study, all forces are also given: $F_c$, $F_t$, and $F_r$ (Figure 6, Figure 9, and Figure 12).

Next, the dynamics qualifier K and quantifier $E_{Samp}$ were applied to a range of three parameters: $e_T\times z_a\times \omega$; Figure 13, Figure 14 and Figure 15. The necessary condition for chaotic movement (contact) was observed (Property 2). It is also noteworthy that the statistical success of the 0–1 test for chaos was over 96% (see

Table 2. Statistical success of the 0–1 test for chaos.

${\mathit{e}}_{\mathit{T}}$ [m]	# Reg. Cases	# Chaotic Cases	# Decidable Cases	Test Success
0.01	5293	162	5455	98.27%
0.015	4808	542	5350	96.38%
0.02	4740	593	5333	96.07%

). Here, successful outputs of K are marked (in blue for regularity and red for chaos) if their value is in the interval $(0,0.1)$ for regular cases and $(0.9,1)$ for chaotic ones.

Finally, bifurcation analysis was performed, showing multiple periodic windows followed by chaotic bands (Figure 16 and Figure 17).