The Nonlinear Behavior of Vibrational Conveyers with Single-Mass Crank-and-Rod Exciters

Abstract

The single-mass, crank-and-rod exciters vibrational conveyers have a trough supported on elastic stands which are rigidly fastened to the trough and a supporting frame. The trough is oscillated by a common crank drive. This vibration causes the load to move forward and upward. The moving loads jump periodically and move forward with relatively small vibration. The movement is strictly related to vibrational parameters. This is applicable in laboratory conditions in the industry which accommodate a few grams of loads, up to those that accommodate tons of loading capacity. In this study I explore the transitional behavior across resonance, during the starting of a single degree of freedom vibratory system excited by crank-and-rod. A loaded vibratory conveyor is more safe to start than an empty one. Vibrational conveyers with cubic nonlinear spring and ideal vibration exciter have been analyzed analytically for primary and secondary resonance by the Method of Multiple Scales, and numerically. The approximate analytical results obtained in this study have been compared with the numerical results and have been found to be well matched.

1. Introduction

In this work, the vibrating system consists of a cubic nonlinear spring and sinusoidal excitation (ideal source) is studied. It is an ideal system, if there is no coupling between motion of the rotor and vibrating system. In this case, the excitation is completely independent of the system response. Vibrating systems with ideal and nonideal excitations were investigated by number of authors. Kononenko [1] presented the first detailed study on the nonideal problem of passage through resonance. Some numerical studies on dynamic characteristics of a vertical pendulum whose base is actuated horizontally through a slider crank mechanism, where the crank is driven through a DC motor, were performed in [2, 3], and investigations on the properties of the transient response of this nonlinear and nonideal problem showed that near the fundamental resonance region, near a secondary resonance region, respectively. An overview of the main properties of nonideal vibrating systems was presented by Balthazar et al. [4, 5]. These authors analyzed that the physical model of the ideal vibrating system consists of linear spring and sinusoidal excitation (ideal source) [6], the physical model of the vibrating system consists of linear spring and nonideal source [7], and the physical model of the vibrating system consists of a cubic nonlinear spring and nonideal source [8].

In this study, vibrational conveyers are constituted by a trough and elastic stands of equal length connected to trough inclinedly. Forced vibration motion driven by the crank-and-rod mechanism is given to this system (Figure 1). Crank-and-rod mechanism with elastic connecting rod can also be used and an elastic component added to the operation mechanism slowly increases the amplitude of the system from a low value to the maximum operation amplitude; it functions as elastic component at the initial motion of the conveyer and as rigid component at the continuous operation state. During the vibration motion, the forward and upward motions of the load on the trough are provided. After each contact to the trough, the velocity of forward movement of the load increases. The velocity of forward movement of the load increases until it reaches the maximum velocity of trough. After reaching the maximum velocity, it continues its motion at this velocity [9, 10].

(a)

Figure 1 

(a) Single-mass, crank-and-rod-driven vibrational conveyer, (b) the coercive force applied on the trough.

The vibrational conveyers have been analyzed numerically and analytically by the Method of Multiple Scales, for the primary and subharmonic and superharmonic resonance. The stability is analyzed by using an approximate analytical solution. The frequency-response curves show the behavior of the oscillator for the variation of the control parameters. Numerical simulations are performed and the simulation results are visualized by means of the phase portrait, Poincaré map, and Phase portrait.

2. The Governing Equations of the Motion

The equations of motion for the vibrating model of system may be obtained by using Lagrange’s Equation, the conveyor empty: 𝑑𝑑𝑡𝜕𝑇𝜕𝑞̇_𝑖+𝜕𝐷𝜕𝑞̇_𝑖−𝜕𝑇𝜕𝑞_𝑖+𝜕𝑉𝜕𝑞_𝑖=𝑄_𝑖, (2.1) where 𝑀 is mass of the trough of the conveyor, 𝑇 is the kinetic energy 1𝑇=2𝑀𝑧̇², (2.2)𝑉, the potential energy, is 1𝑉=2𝑘₁𝑧²+14𝑘₂𝑧⁴, (2.3)𝐷, the Rayleigh Dissipation Function, is 1𝐷=2𝑐𝑧̇², (2.4) and 𝑞_𝑖 is the generalized coordinate. Applying Lagrange’s Equation for the coordinate 𝑞_𝑖=𝑧 gives the differential equations of motion.

Here the measurement values are taken from the position where the components in the same directions of the vertical force applied by the leaf spring on itself and the weight force balances each other. If the mass of the leafs works as spring as well as the mass of the 𝐴𝐴₃ connecting rod, the changes in the angle 𝜓 and moment 𝑀₀ applied on the crank arm by the motor are neglected, and the coercive force in the vibration motion can be written as below (Figure 2(b)) [11, 12], then the coercive force becomes 𝐹(𝑡)𝜋=𝐹cos𝛾,𝛾=2−𝐹(𝑡)=𝑀𝜃+𝜓,cos𝛾=sin𝜃+𝜓,Θ=𝜃−𝜓,Θ=𝜔𝑡,_𝑂𝑟sin𝜔𝑡,𝐹=𝑚_𝑡𝑟𝜔²_𝑛=𝑚_𝑡𝑟𝑘₁𝑚_𝑡=𝑟𝑘₁, (2.5) where 𝑚_𝑡 is the total vibrating mass, 𝑟 is the length of the crank, then one can write the generalized force 𝑄₁𝑀𝑧̈+𝑐𝑧̇+𝑘₁𝑧+𝑘₂𝑧³(𝑡),̈𝑐=𝐹𝑧+𝑀̇𝑘𝑧+¹𝑀𝑘𝑧+²𝑀𝑧³=1𝑀𝐹(𝑡). (2.6) Rewriting (2.6) in terms of variables, it will obtained, where 𝑐 is damping coefficient, 𝑘₁ is the linear spring coefficient, and 𝑘₂ is the cubic spring coefficient, that 𝜔_𝑛 is the natural frequency: 𝜔²_𝑛=𝑘₁/𝑀, 𝜀=1/𝑀, 𝑧̈+𝜔²_𝑛1𝑧=𝑀𝐹(𝑡)−𝑐𝑧̇−𝑘₂𝑧³, (2.7) where 𝑏=𝑐/2𝑀, 𝛼=𝑘₂/𝑀, 𝑧̈+𝜔²_𝑛𝐹𝑧=𝑀sin𝜃+𝜓−2𝑏𝑧̇−𝛼𝑧³. (2.8) The conveyor was filled with material, during one of the motions of the trough, and the load may be in a ride, slide, or flight phase, depending on the actual accelerations, velocities, and displacements of the load and trough. These various phases of motion are analyzed.

(a)

Figure 2 

(a) Vibrating model of system, (b) material motion.

(i) Flight Phase (when 𝑦_𝑡^′′<−𝑔). A flight phase occurs if the normal force becomes zero. At the start of a flight the load and trough have the same 𝑦-coordinate and 𝑦^′-velocity. The load loses contact with the trough and is said to travel in a “flight phase.” The governing equations of motion then are as follows: for the trough: (2.8) (conveyor empty); for the material is moving under the action of gravity which, 𝑥_𝑚^′′=0,𝑦_𝑚^′′=−𝑔. (2.9) The end of flight phase can be derived from the condition of equal 𝑦 coordinates and terminates in an “impact” with the trough 𝑦_𝑚=𝑦_𝑡. (2.10) This oblique impact is considered to be instantaneous and “perfectly plastic” in the 𝑦 direction. Considering the principle of conservation of the total momentum and the Newton’s Law of Impact, 𝑥^′_𝑡=𝑦^′_𝑡cot𝛽. (2.11) “friction impact”, in the 𝑥 direction, “𝑡” and “𝑚” represent the parameters of the trough and the material particle, respectively. “𝑎” and “𝑏” represent the parameters “after” and “before” the impact, respectively (Figure 2(b)): 𝐼_𝑥=−𝜇𝐼_𝑦𝑦^′_𝑎=−𝑒𝑦^′_𝑏,𝑣_𝑎cos𝛼_𝑎=𝑒𝑣_𝑏cos𝛼_𝑏. (2.12) The change of the velocity of the load during an impact with the trough is related to the total linear impulse at time of flight phase, by the linear momentum equation 𝑚𝑉^→_𝑎−𝑚𝑉^→_𝑏𝐼=𝐼_𝑥=𝑚𝑣_𝑎sin𝛼_𝑎−𝑣_𝑏sin𝛼_𝑏,𝐼_𝑦=𝑚𝑣_𝑎cos𝛼_𝑎+𝑣_𝑏cos𝛼_𝑏𝑚𝑣_𝑎sin𝛼_{𝑎̇𝑥𝑎}−𝑣_𝑏sin𝛼_{𝑏̇𝑥𝑏}=−𝜇𝑚𝑣_𝑎cos𝛼_{𝑎̇𝑦𝑎}+𝑣_𝑏sin𝛼_{𝑏−̇𝑦𝑏}𝑥̇_𝑚𝑎−𝑥̇_𝑚𝑏=±𝜇𝑦̇_𝑚𝑎−𝑦̇_𝑚𝑏. (2.13) The coefficient of friction times the change in velocity of the load along the 𝑦 direction due to impact 𝑥̇_𝑚𝑎−𝑥̇_𝑚𝑏=sign𝜇𝑦̇_𝑚𝑎−𝑦̇_𝑚𝑏, (2.14) where the negative or positive signs for the frictional force 𝑓sign are +1.0 or −1.0 as (𝑥̇_𝑚𝑏−𝑥̇_𝑡𝑏)<0 or >0. If the flight ends with a tangential velocity of the load that is not equal to that of the trough, a positive or negative slide phase occurs, depending on the sign of this difference, until the velocities become equal.

(ii) Ride Phase. In this case the trough and the load are as one body. If the load’s velocity and trough’s velocity are equal: 𝑦̇_𝑚=𝑦̇_𝑡,𝑦_𝑚=𝑦_𝑡, (2.15) the conditions are such that 𝑥̈_𝑡≤𝜇𝑦̈_𝑡,𝑥̇+𝑔_𝑚=𝑥̇_𝑡. (2.16) For the trough, where 𝑚 is mass of the conveyed material on the trough of the conveyor, 𝑇 is the kinetic energy 1𝑇=2𝑀𝑧̇²+12𝑚𝑧̇², (2.17)𝑉, the potential energy, is 1𝑉=2𝑘₁𝑧²+14𝑘₂𝑧⁴+𝑚𝑔sin𝛽𝑧, (2.18)𝐷, the Rayleigh Dissipation Function, is 1𝐷=2𝑐𝑧̇²(𝑀+𝑚)𝑧̈+𝑐𝑧̇+𝑘₁𝑧+𝑘₂𝑧³(𝑡)̈𝑐=𝐹−𝑚𝑔sin𝛽𝑧+̇𝑘𝑀+𝑚𝑧+¹𝑘𝑀+𝑚𝑧+²𝑧𝑀+𝑚³=1𝐹(𝑡)−𝑀+𝑚𝑚𝑔𝑀+𝑚sin𝛽 (2.19)(2.20) rewriting (2.20) in terms of variables, 𝑚 is mass of the conveyed material on the trough of the conveyor, 𝑔 is acceleration due to gravity, 𝜔²_𝑛=𝑘₁/𝑚+𝑀, 2𝑏=𝑐, 𝜀=𝑚𝑔/𝑀+𝑚, 𝛼=𝑘₂, 𝐺=𝑚𝑔/𝑀+𝑚, 𝑧̈+𝜔²_𝑛𝑧=𝜀𝐹sin𝜃+𝜓−2𝑏𝑧̇−𝛼𝑧³−𝐺sin𝛽. (2.21) And, for the material, 𝑥_𝑚=𝑥_𝑡,𝑥̇_𝑚=𝑥̇_𝑡. (2.22)

(iii) Slide Phase. Sliding may start when the accelerations of the load and the trough are equal. Vertical component of trough’s acceleration is greater than acceleration of gravity 𝑦̈_𝑡>−𝑔,|𝑥̈|>𝜇𝑔+𝑦̈_𝑡or𝑥̇_𝑚≠𝑥̇_𝑡. (2.23) If 𝑚𝑥̈ is greater than the maximum value of the friction force, the load starts to slide on the trough.

For the trough, the kinetic energy is 1𝑇=2𝑀𝑍̇²+12𝑚𝑍̇sin𝛽², (2.24) the potential energy is 1𝑉=2𝑘₁𝑍²+14𝑘₂𝑍⁴+𝑚𝑔sin𝛽𝑍, (2.25) and the Rayleigh Dissipation Function is 1𝐷=2𝑐𝑍̇², (2.26) If Newton’s second law of motion is written along the 𝑌-axis (vertical) direction for the load on trough, 𝑓=𝜇𝑁. (2.27) Here 𝑓 is the friction force affecting on the load and 𝑎_𝑦 is vertical component of trough’s acceleration: 𝐹_𝑌=𝑚𝑎_𝑌=𝑚𝑌̈_𝑡̈̈.̈,𝑄,−𝑚𝑔+𝑁=𝑚𝑍sin𝛽,𝑁=𝑚−𝑔+𝑍sin𝛽𝑓=∓𝜇𝑚−𝑔+𝑍sin𝛽_𝑍=∓𝜇𝑚𝑍̈sin𝛽−𝑔cos𝛽. (2.28) If it is taken 𝑧 the displacement of the trough from the equilibrium position 0 (positive upward). 𝑀𝑔sin𝛽=(𝑘₁𝛿_𝑠𝑡+𝑘₂𝛿³_𝑠𝑡), where 𝛿_𝑠𝑡 represents the elongation of the spring at the equilibrium position: 𝑀+𝑚sin²𝛽±𝜇sin𝛽cos𝛽𝑍̈+𝑐𝑍̇+𝑘₁𝑍+𝑘₂𝑍³(𝑡)̈=𝐹−𝑚𝑔sin𝛽±𝜇cos𝛽𝑍+𝜔²_𝑛1𝑍=𝑀+𝑚sin²𝐹(𝑡)̇𝛽±𝜇sin𝛽cos𝛽−𝑚𝑔sin𝛽±𝜇cos𝛽−𝑐𝑍−𝑘₂𝑍³ (2.29)(2.30) Rewriting (2.29), where 𝜇 is the coefficient of friction between the materials and the trough, 𝜔²_𝑛=𝑘₁/(𝑀+𝑚(sin²𝛽±𝜇sin𝛽cos𝛽)), 𝜀=1/(𝑀+𝑚(sin²𝛽±𝜇sin𝛽cos𝛽)), 2𝑏=𝑐, 𝛼=𝑘₂, 𝐺 = 𝑚𝑔/(𝑀+𝑚(sin²𝛽±𝜇sin𝛽cos𝛽)), and 𝑍̈+𝜔²_𝑛𝑍=𝜀𝐹sin𝜃+𝜓−2𝑏𝑍̇−𝛼𝑍³−𝐺sin𝛽±𝜇cos𝛽. (2.31)

3. Approximate Analytical Solution

The method of multiple scales is used to obtain approximate analytical solution of (2.8), (2.21), and (2.31) [13–16]. We seek a second-order expansion in the form 𝑧(𝑡,𝜀)≈𝑧₀𝑇₀,𝑇₁+𝜀𝑧₁𝑇₀,𝑇₁+…, (3.1) where the fast scale 𝑇₀=𝑡 and the slow scale 𝑇₁=𝜀𝑡. The time derivatives transform according to 𝑑𝑑𝑡=𝐷₀+𝜀𝐷₁𝑑+…,²𝑑𝑡²=𝐷²₀+2𝜀𝐷₀𝐷₁+…, (3.2) where 𝐷_𝑛=𝜕/𝜕𝑇_𝑛. Then, 𝑧^′≈𝐷₀𝑧₀+𝜀𝐷₁𝑧₀+𝐷₀𝑧₁,𝑧^′′≈𝐷²₀𝑧₀+𝜀2𝐷₀𝐷₁𝑧₀+𝐷²₀𝑧₁. (3.3) Substituting (3.1)–(3.3) into (2.8) and equating coefficients of like powers 𝜀, where Ω is excitation frequency, we obtain 𝐷²₀𝑧₀+𝜔²_𝑛𝑧₀𝐷=0,₂₀𝑧₁+𝜔²_𝑛𝑧₁=−2𝐷₀𝐷₁𝑧₀−2𝑏𝐷₀𝑧₀−𝛼𝑧³₀+𝐹sinΨ+𝑇₀Ω. (3.4) Substituting (3.1)–(3.3) into (2.21) and equating coefficients of like powers 𝜀, we obtain 𝐷²₀𝑧₀+𝜔²_𝑛𝑧₀𝛽,𝐷=𝐺sin₂₀𝑧₁+𝜔²_𝑛𝑧₁=−2𝐷₀𝐷₁𝑧₀−2𝑏𝐷₀𝑧₀−𝛼𝑧³₀+𝐹sinΨ+𝑇₀Ω. (3.5) Substituting (3.1)–(3.3) into (2.31) and equating coefficients of like powers 𝜀, we obtain: 𝐷²₀𝑧₀+𝑧₀𝛽𝛽,𝐷=𝐺𝜇cos−𝐺sin₂₀𝑧₁+𝜔²_𝑛𝑧₁=−2𝐷₀𝐷₁𝑧₀−2𝑏𝐷₀𝑧₀−𝛼𝑧³₀+𝐹sinΨ+𝑇₀Ω. (3.6)

This system analyzed the stability  (𝑎,𝛾) in the equilibrium point, using (3.9), (3.12), (3.14), (3.18), (3.20), (3.22), (3.25), (3.27), and (3.29), where 𝐽 is Jacobian matrix of (3.10), and stability of the approximate solutions depends on the value of the eigenvalues of the Jacobian matrix 𝐽. The solutions are unstable if the real part of the eigenvalues is positives [13, 14]. (Figure 3) shows the frequency-response curves for primary, subharmonic, and superharmonic resonance of the vibratory conveyor. These curves show that the nonlinearity bends the frequency-response curves. The bending of the frequency-response curves leads to multivalued amplitudes and hence to jump phenomenon. In contrast linear systems, the mass-nonlinear spring system exhibits no resonance (Figure 3).

Figure 3 

Primary resonance, subharmonic resonance and superharmonic resonance, frequency-response curve with stability, ~ stable, ⋯ unstable: (a1) conveyer empty, (𝜎=7.1179–8.72643, 𝑎=0.35701–0.57001), (b1) slide mode, (𝜎=9.41227–16.9651, 𝑎=0.39201–0.79101); (c1) ride mode, (𝜎=9.19323–15.9651, 𝑎=0.35401-0.69801); (a2) conveyer empty, (𝜎=0.853726–2.51325, 𝑎=0.1186–0.2001), (b2) slide mode, (𝜎=1.04791–1.68339, 𝑎=0.11631–0.15001), and (c2) ride mode, (𝜎=0.885482–1.96345, 𝑎=0.09681–0.15001). and (a3) conveyer empty, (𝜎=0.534199–0.667969, 𝑎=0.0305–0.0625), (b3) slide mode, (𝜎=0.554249–0.703279, 𝑎=0.0305–0.0636), and (c3) ride mode, (𝜎=0.880504–1.21958, 𝑎=0.0302–0.0765).

3.1. Primary Resonances: Ω≈𝜔₀ (Table 1)

Table 1 

Primary resonance: vibratory conveyer parameters in SI units.

Eliminating secular terms in equations, for analyzing the case of primary resonance, we chose the detuning parameter 𝜎, where 𝑎 and 𝛽 are real, Ω=𝜔₀+𝜀𝜎 (3.7) taking 𝐴𝑇₁,𝑇₂=𝑎𝑇₁2𝑒^{𝑖𝛽[𝑇1]} (3.8) and, separating real and imaginary parts, the system of equations is solved.

(i) Flight phase: 𝑎^′𝛾=−𝑎𝑏−𝐹cos2𝜔_𝑛𝛾^′=−3𝑎³𝛼+8𝑎𝜎𝜔_𝑛+4𝐹sin𝛾8𝑎𝜔_𝑛,𝛾=𝑇₁𝜎+𝜓−𝛽𝑇₁,𝑓₁=𝑎^′,𝑓₂=𝛾^′,𝑓=𝜕_𝑎𝑓₁,𝜕_𝛾𝑓₁,𝜕_𝑎𝑓₂,𝜕_𝛾𝑓₂𝐽=𝑏−𝑏−𝑎𝑏𝑎−3𝑎𝛼4𝜔_𝑛−𝜎+3𝑎²𝛼8𝜔_𝑛. (3.9)(3.10)(3.11)

(ii) Ride phase: 𝑎^′𝛾=−𝑎𝑏−𝐹cos2𝜔_𝑛𝛾^′=−6𝑎𝐺²𝛼−3𝑎³𝛼𝜔⁴_𝑛+8𝑎𝜎𝜔⁵_𝑛+6𝑎𝐺²𝛼cos2𝛽+4𝐹𝜔⁴_𝑛sin𝛾8𝑎𝜔⁵_𝑛,𝛾=𝑇₁𝜎+𝜓−𝛽𝑇₁𝑏𝐽=−𝑏−𝑎𝑏𝑎−3𝑎𝛼4𝜔_𝑛6𝐺²𝛼+𝜔⁴_𝑛3𝑎²𝛼−8𝜔_𝑛−6𝐺²𝛼cos2𝛽8𝜔⁵_𝑛. (3.12)(3.13)

(iii) Slide phase: 𝑎^′𝛾=−𝑎𝑏−𝐹cos2𝜔_𝑛𝛾^′=18𝑎𝜔⁵_𝑛𝑎−6𝐺²𝛼1+𝜇²+𝜔⁴_𝑛−3𝑎²𝛼+8𝜎𝜔_𝑛+6𝐺²𝛼−−1+𝜇²cos2𝛽+2𝜇sin2𝛽+4𝐹𝜔⁴_𝑛sin𝛾,𝛾=𝑇₁𝜎+𝜓−𝛽𝑇₁𝐽=𝑏−𝑏−𝑎𝑏𝑎−3𝑎𝛼4𝜔_𝑛𝒜, (3.14) where 𝒜 denotes (1/8𝜔⁵_𝑛)(6𝐺²𝛼(1+𝜇²)+𝜔⁴_𝑛(3𝑎²𝛼−8𝜎𝜔_𝑛)+6𝐺²𝛼((−1+𝜇²)cos(2𝛽)−2𝜇sin(2𝛽))).

3.2. Subharmonic Resonances: Ω≈3𝜔₀ (Table 2)

Table 2 

Subharmonic resonance: vibratory conveyer parameters in SI units.

Eliminate secular terms in equations, for analyzing the case of subharmonic resonance, where cos𝜃₀+𝜀𝜃₁=cos𝜃₀+𝑂(𝜀),sin𝜃₀+𝜀𝜃₁=sin𝜃₀+𝑂(𝜀),Λ=−𝜅Ω²Ω²−𝜔²₀ (3.15) near resonance 𝐷₀𝜃₀=Ω,Ω=3𝜔₀+𝜀𝜎. (3.16) The solution of (3.17) can be written as 𝜃₀=Ω𝑇₀=3𝜔₀𝑇₀+𝜎𝑇₁ (3.17) and, separating real and imaginary parts, the system of equations is solved.

(i) Flight phase: 𝑎^′𝑎=−8𝑏𝜔_𝑛+3𝑎𝛼Λcos𝛾8𝜔_𝑛𝛾^′=−9𝛼𝑎²+2Λ²+8𝜎𝜔_𝑛+9𝑎𝛼Λsin𝛾8𝜔_𝑛,𝛾=𝑇₁𝜎+𝜓−3𝛽𝑇₁,Λ=𝐾Ω²Ω²−𝜔²_𝑛𝐽=−𝑏+2𝜎3−3𝛼8𝑎²+2Λ²4𝜔_𝑛−−𝑎𝑏3𝑏𝑎−9𝑎𝛼4𝜔_𝑛𝑎−𝜎+9𝛼²+2Λ²8𝜔_𝑛. (3.18)(3.19)

(ii) Ride phase: 𝑎^′𝑎=−8𝑏𝜔_𝑛+3𝑎𝛼Λcos𝛾8𝜔_𝑛𝛾^′=−18𝐺²𝛼−9𝛼𝑎²+2Λ²𝜔⁴_𝑛+8𝜎𝜔⁵_𝑛+18𝐺²𝛼cos2𝛽+9𝑎𝛼Λ𝜔⁴_𝑛sin𝛾8𝜔⁵_𝑛,𝛾=𝑇₁𝜎+𝜓−3𝛽𝑇₁,Λ=𝐾Ω²Ω²−𝜔²_𝑛−𝐽=ℬ−𝑎𝑏3𝑏𝑎−9𝑎𝛼4𝜔_𝑛𝒞, (3.20)(3.21) where ℬ denotes  −(1/12𝜔⁵_𝑛)(18𝐺²𝛼+𝜔⁴_𝑛(9𝑎²𝛼+18𝛼Λ²+12𝑏𝜔_𝑛−8𝜎𝜔_𝑛)−18𝐺²𝛼cos(2𝛽)) and 𝒞 denotes (1/8𝜔⁵_𝑛)(18𝐺²𝛼+𝜔⁴_𝑛(9𝛼(𝑎²+2Λ²)−8𝜎𝜔_𝑛)−18𝐺²𝛼cos(2𝛽)).

(iii) Slide phase: 𝑎^′=−0.05𝑎−0.429898𝑎²𝛾𝛾cos_′=1.37347×10⁻⁶−155304.+9274.12−45000.0010125+𝑎²𝛾+78.507𝜎+939005.𝑎sin,𝛾=𝑇₁𝜎+𝜓−3𝛽𝑇₁,𝒟−𝐽=−0.05𝑎0.15𝑎−114.64𝑎ℰ, (3.22)(3.23) where 𝒟 denotes  −9.15646×10⁻⁷(155304.+4.217335×10⁷(0.0010125+𝑎²)+364042.(0.15−2𝜎)) and ℰ denotes 1.37347×10⁻⁶(155304.+9274.12(4500(0.0010125+𝑎²)−78.507𝜎)).

3.3. Superharmonic Resonances: Ω≈1/3𝜔₀ (Table 3)

Table 3 

Superharmonic resonance: vibratory conveyer parameters in SI units.

Eliminating secular terms in equations, for analyzing the case of superharmonic resonance, where 3Ω=𝜔₀+𝜀𝜎3Ω𝑇₀=𝜔₀+𝜀𝜎𝑇₀=𝜔₀𝑇₀+𝜎𝜀𝑇₀=𝜔₀𝑇₀+𝜎𝑇₁ (3.24) and, separating real and imaginary parts, the system of equations is solved.

(i) Flight phase: 𝑎^′=−𝑎𝑏+𝛼Λ³cos𝛾8𝜔_𝑛𝛾^′=−3𝑎³𝛼+6𝑎𝛼Λ²−8𝑎𝜎𝜔_𝑛+𝛼Λ³sin𝛾8𝑎𝜔_𝑛,𝛾=𝑇₁𝜎+3𝜓−𝛽𝑇₁𝑏𝐽=−𝑏−𝑎𝑏𝑎−3𝑎𝛼4𝜔_𝑛𝑎−𝜎+3𝛼²+2Λ²8𝜔_𝑛. (3.25)(3.26)

(ii) Ride phase: 𝑎^′=−𝑎𝑏+𝛼Λ³cos𝛾8𝜔_𝑛𝛾^′=𝑎−6𝐺²𝛼−3𝛼𝑎²+2Λ²𝜔⁴_𝑛+8𝜎𝜔⁵_𝑛+6𝑎𝐺²𝛼cos2𝛽−𝛼Λ³𝜔⁴_𝑛sin𝛾8𝑎𝜔⁵_𝑛,𝛾=𝑇₁𝜎+3𝜓−𝛽𝑇₁𝑏𝐽=−𝑏−𝑎𝑏𝑎−3𝑎𝛼4𝜔_𝑛6𝐺²𝛼+𝜔⁴_𝑛3𝑎²𝛼+6𝛼Λ²−8𝜎𝜔_𝑛−6𝐺²𝛼cos2𝛽8𝜔⁵_𝑛. (3.27)(3.28)

(iii) Slide phase: 𝑎^′=−𝑎𝑏−𝛼Λ³cos𝛾8𝜔_𝑛𝛾^′=18𝑎𝜔⁵_𝑛𝑎−6𝐺²𝛼1+𝜇²+𝜔⁴_𝑛−3𝛼𝑎²+2Λ²+8𝜎𝜔_𝑛+6𝐺²𝛼−−1+𝜇²cos2𝛽+2𝜇sin2𝛽+𝛼Λ³𝜔⁴_𝑛sin𝛾𝛾=𝑇₁𝜎+𝜓−𝛽𝑇₁𝐽=𝑏−𝑏−𝑎𝑏𝑎−3𝑎𝛼4𝜔_𝑛ℱ, (3.29)(3.30) where ℱ denotes (1/8𝜔⁵_𝑛)(6𝐺²𝛼(1+𝜇²)+𝜔⁴_𝑛(3𝑎²𝛼+6𝛼Λ²−8𝜎𝜔_𝑛)+6𝐺²𝛼((−1+𝜇²)cos(2𝛽)−2𝜇sin(2𝛽))).

4. Numerical Results

The numerical calculations of the vibrating system are performed with the help of the software Mathematica [17, 18]. Figures 4, 5, and 6 show the displacement-time response, the power spectrum, phase portrait, and Poincaré map for the primary, subharmonic, and superharmonic resonance. By Poincaré map I concludes that the motion of the oscillator is periodic with period-1.

Figure 4 

Primary resonance: (a) displacement-time response, (b3) Poincaré map (before stabilization), fixed points (stable focus), (b1), (b2) Poincaré map (after stabilization), (c1) power spectrum, (c2) phase portrait (after stabilization), (c3) phase portrait (before stabilization).

Figure 5 

Subharmonic resonance: (a) displacement-time response, (b3) Poincaré map (before stabilization), fixed points (stable focus), (b1), (b2) Poincaré map (after stabilization), (c1) power spectrum, (c2) phase portrait (after stabilization), (c3) phase portrait (before stabilization).

Figure 6 

Superharmonic resonance: (a) displacement-time response, (b3) Poincaré map (before stabilization), fixed points (stable focus), (b1), (b2) Poincaré map (after stabilization), (c1) power spectrum, (c2) phase portrait (after stabilization), (c3) phase portrait (before stabilization).

In Figures 4, 5, and 6, the curves obtained numerically and analytically by solving (2.8), (2.21), and (2.31) are plotted. The main characteristic values used in this study are given in Tables 1, 2, and 3.

5. Conclusions

In this study, the transition over resonance of a nonlinear vibratory system, excited by crank-and-rod, is important in terms of the maximum vibrational amplitude produced on the drive for the crossover. The maximum amplitude of vibration is then of interest in determining the structural safety of the vibrating members. The shaded region in the amplitude-frequency plot is unstable; the extend of unstableness depends on a number of factors such as the amount of damping present, nonlinearity of spring, and the rate of chance of the exciting frequency (Figure 3). Results of the numerical simulations, obtained from the analytical equations, showed that the important dynamic characteristics of the system, such as damping, nonlinearity and the amplitude excitations effects, still presented a periodic behavior for these situations. The bending of the response is due to the nonlinearity, and is responsible for a jump phenomenon. In the motion of the system near resonance the jump phenomenon occurs. A periodic solution in the case of the angular velocity taken above the resonance (after stabilization) is illustrated in Figure 4.

Comparing the results obtained by applying the approximate analytic method with those obtained numerically it is concluded that the difference is negligible, proving the correctness of the analytic procedure used.

In the future, it is possible to investigate this system with the nonideal source, and nonideal vibrating systems are those for which the power supply is limited. Nonideal problems are more realistic, and the model should take into account also the influence of the dynamics of the oscillating mechanical elements on electrical properties of the DC motor [5].

Acknowledgment

The author greatly appreciates the comments of an anonymous referee.