Nonlinear Response of Vibrational Conveyers with Nonideal Vibration Exciter: Superharmonic and Subharmonic Resonance

Abstract

Vibrational conveyers with a centrifugal vibration exciter transmit their load based on the jumping method. Common unbalanced-mass driver oscillates the trough. The motion is strictly related to the vibrational parameters. The transition over resonance of a vibratory system, excited by rotating unbalances, is important in terms of the maximum vibrational amplitude produced and the power demand on the drive for the crossover. The mechanical system is driven by the DC motor. In this study, the working ranges of oscillating shaking conveyers with nonideal vibration exciter have been analyzed analytically for superharmonic and subharmonic resonances by the method of multiple scales and numerically. The analytical results obtained in this study agree well with the numerical results.

1. Introduction

The load-carrying element of a horizontal shaking conveyer performs, as a rule, linear (or sometimes circular or elliptical) symmetrical harmonic oscillations—with a sinusoidal variation of exciting force. In vertical shaking conveyers, the load-carrying element performs double harmonic oscillations: linear along the vertical axis and rotational around that axis (i.e., longitudinal and torsional oscillations). Conveyer drives with centrifugal vibration exciters may have (1) a single unbalanced-mass, (2) two equal unbalancing masses, (3) a pendulum-type unbalanced-mass, (4) four unbalanced-masses in two shafts, (5) four rotating unbalanced-masses for three principal modes of oscillation, that is, linear, elliptical, and circular. To induce strictly oriented linear oscillations of the load-carrying element, the conveyer drive should be arranged so that the line of excitation force passes through the inertial centre of the entire oscillating system. Nonideal drives find application in suspended and supported vibrational conveyers and feeders [1].

By the characteristics and adjustment of the elastic support elements (oscillating system), we can distinguish between shaking conveyers with a resonant, subresonant, and superresonant system.

A practical difficulty with unbalanced-mass exciters, observed as early as 1904 by A Sommerfeld, is that local instabilities may occur in operating speed of such devices.

Rocard [2], Mazert [3], Panovko, and I. I. Gubanova [4] have studied the problem of the stability of the unbalanced-mass exciter.

The first detailed study on the nonideal vibrating systems is presented by Kononenko. He obtained satisfactory results by the comparison of the experimental analysis and the approximated method [5].

After this publication, the nonideal problem was presented by Evan-Ivanowski [6] or Nayfeh and Mook [7]. These authors showed that sometimes dynamical coupling between energy sources and structural response that had not to be ignored in reel engineering problems.

Theorical studies and computations of Ganapathy and Parameswaran have indicated the beneficial effect of the “material load” during the starting and transition phase of an unbalanced-mass-driven vibrating conveyor [8].

Bolla et al. analyzed through the multiple scales method a response of a simplified nonideal and nonlinear vibrating system [9].

Götzendorfer in [10] presented a macromechanical model for the transport of granular matter on linear and horizontal conveyors subject to linear, circular or elliptic oscillations and compared it to experimental results [10].

2. The Governing Equations of the Motion

The equations of motion for the modified rocard system may be obtained by using Lagrange’s equation 𝑑𝑑𝑡𝜕𝑇𝜕𝑞̇_𝑖+𝜕𝐷𝜕𝑞̇_𝑖−𝜕𝑇𝜕𝑞_𝑖+𝜕𝑉𝜕𝑞_𝑖=0, (2.1) where 𝑇, the kinetic energy, is 1𝑇=2𝑀𝑦̇²+12𝑚𝑦̇+𝜃̇𝑒cos𝜃²+𝜃̇𝑒sin𝜃²+12𝐼_mot𝜃̇², (2.2) where 𝑒 is the eccentricity of the mass 𝑚, 𝑚 is the unbalanced-mass, 𝑀 is the mass of the trough and the conveyed material on the trough of the conveyor, 𝜃 is the angle of rotation of the shafts carrying unbalanced-masses, 𝐼_mot is the moment of inertia of the rotating parts in the motor, 𝑉, the potential energy, is 1𝑉=2𝑘₁𝑦²+14𝑘₂𝑦⁴+𝑚𝑔𝑒(1+sin𝜃), (2.3) where the constants 𝑘₁ and 𝑘₂ are the linear and nonlinear elastic coefficients, respectively, 𝐷, the Rayleigh dissipation function, is 1𝐷=2𝑐𝑦̇²+12𝐾𝜔_𝑠−𝜃̇, (2.4) and 𝑞_𝑖 is the generalized coordinate. Applying Lagrange’s equation for the two coordinates 𝑞_𝑖=𝑦 and 𝑞_𝑖=𝜃 gives the differential equations of motion 𝑀𝑦̈+𝑚𝑦̈+𝑐𝑦̇+𝑘₁𝑦+𝑘₂𝑦³=𝑚𝑒𝜃̇²̈,𝐼sin𝜃−𝜃cos𝜃_mot+𝑚𝑒²_𝐼sys̈̈=1𝜃+𝑚𝑒𝑦cos𝜃+𝑔cos𝜃2𝜕𝐾𝜕𝜃̇𝜃̇−𝜔_𝑠+𝐾. (2.5) Equation (2.5) can be rewritten as follows: 𝑦̈+𝜔²₀𝑦=𝜀−2𝜇𝑦̇−𝛼𝑦³−𝑒𝜃̈cos𝜃+𝜅𝜃̇²,̈̈𝜃̇sin𝜃𝜃=𝜀−𝐼cos𝜃𝑦+𝑔+𝐸, (2.6)(2.7) where ()𝑒𝐼=𝑚+𝑀𝐼_sys,𝜅=𝑚𝑒𝑐𝑚+𝑀,2𝜇=𝑚𝑘,𝛼=²𝑚,𝑘₁𝑚+𝑀=𝜔²₀,𝑚̇𝜃=()𝐿𝜃̇𝑚+𝑀=𝜀,𝐸𝑚+𝑀𝑚𝐼_sys,𝐿𝜃̇=12𝜕𝐾𝜕𝜃̇𝜃̇−𝜔_𝑠+𝐾, (2.8) where 𝑐 is the damping coefficient of the vibrating conveyor, 𝑔 is acceleration due to gravity, 𝐼_sys=(𝐼_𝑚+𝐼_𝑀+𝐼_mot) is the total moment of inertia of all the rotating parts in the system, and 𝜔_𝑠 is the synchronous angular speed of the induction motor [9]. 𝐾 is the instantaneous drive torque available at the shafts. Note that 𝐸 contains 𝐿(𝜃̇) that is the active torque generated by the electric circuit of the DC motor, shown in Figure 1.

Figure 1 

Vibrating model of the system.

3. Analytical Solution

Ideal system: if there is no coupling between motion of the rotor and vibrating system and 𝜃̇=constant (𝜃=𝜔𝑡, 𝜃̈=0), (2.6) becomes 𝑦̈+𝜔²₀𝑦=𝜀−2𝜇𝑦̇−𝛼𝑦³+𝜅𝜔²sin𝜔𝑡. (3.1) On the right side of the equation, a function of time is present.

Nonideal system: ̈̈𝜃̇,̈𝜃=𝜀−𝐼cos𝜃𝑦+𝑔+𝐸𝑦+𝜔²_𝑛𝑦=𝜀−2𝜇𝑦̇−𝛼𝑦³−𝑒𝜃̈cos𝜃+𝜅𝜃̇²sin𝜃, (3.2) where 𝐸(𝜃̇)=𝑀_𝑚(𝜃̇)−𝐻(𝜃̇) is the difference between the torque generated by the motor and the resistance torque. Function 𝐸(𝜃̇)=𝑢₁−𝑢₂𝜃̇ is approximated by a straight line, where 𝑢₁ is a control parameter that can be changed according to the voltage, and 𝑢₂ is a constant parameter, characteristic for the model of the motor.

We will obtain an approximate analytical solution to (3.2) by using the multiple scales method: 𝜃(𝑡,𝜀)≈𝜃₀𝑇₀,𝑇₁+𝜀𝜃₁𝑇₀,𝑇₁+𝜀²𝜃₂𝑇₀,𝑇₁,𝑦(𝑡,𝜀)≈𝑦₀𝑇₀,𝑇₁+𝜀𝑦₁𝑇₀,𝑇₁+𝜀²𝑦₂𝑇₀,𝑇₁,𝑇_𝑛=𝜀^𝑛𝑡,𝑛=0,1,…,𝑇₀=𝑡,𝑇₁=𝜀𝑡 (3.3) where the fast scale 𝑇₀=𝑡 and slow scale 𝑇₁=𝜀𝑡. The time derivatives transform according to 𝑑=𝑑𝑡𝑑𝑇₀𝜕𝑑𝑡𝜕𝑇₀+𝑑𝑇₁𝜕𝑑𝑡𝜕𝑇₁+⋯=𝐷₀+𝜀𝐷₁𝑑+⋯₂𝑑𝑡²=𝐷²₀+2𝜀𝐷₀𝐷₁+⋯, (3.4) where 𝐷_𝑛=𝜕/𝜕𝑇_𝑛, (𝑛=0,1,…); then, 𝜃̇=𝑑𝜃𝑑𝑡≈𝐷₀𝜃₀+𝜀𝐷₁𝜃₀+𝐷₀𝜃₁̈𝑑+⋯,𝜃=²𝜃𝑑𝑡²≈𝐷²₀𝜃₀+𝜀2𝐷₀𝐷₁𝜃₀+𝐷²₀𝜃₁̇+⋯,𝑦=𝑑𝑦𝑑𝑡≈𝐷₀𝑦₀+𝜀𝐷₁𝑦₀+𝐷₀𝑦₁̈𝑑+⋯,𝑦=²𝑦𝑑𝑡²≈𝐷²₀𝑦₀+𝜀2𝐷₀𝐷₁𝑦₀+𝐷²₀𝑦₁+⋯. (3.5) Substituting (3.5) into (3.2), we will obtain 𝐷²₀𝜃₀=𝜀−2𝐷₀𝐷₁𝜃₀−𝐷²₀𝜃₁−𝐼cos𝜃𝐷²₀𝑦₀𝜃̇,𝐷+𝑔+𝐸₂₀𝑦₀+𝜀2𝐷₀𝐷₁𝑦₀+𝐷²₀𝑦₁+𝜔²_𝑛𝑦₀+𝜀𝑦₁=𝜀−2𝜇𝐷₀𝑦₀−𝛼𝑦₀³−𝑒𝐷²₀𝜃₀cos𝜃+𝜅𝐷₀𝜃₀+𝜀𝐷₁𝜃₀+𝐷₀𝜃₁²sin𝜃, (3.6) and equating coefficients of a like powers 𝜀, we obtain(i)for 𝜀⁰𝐷²₀𝜃₀𝐷=0,₂₀𝑦₀+𝜔²₀𝑦₀=𝜅𝐷₀𝜃₀²sin𝜃, (3.7)(ii)for 𝜀¹𝐷²₀𝜃₁=−2𝐷₀𝐷₁𝜃₀−𝐼cos𝜃𝐷²₀𝑦₀+𝑔+𝐸𝜃̇, (3.8)𝐷²₀𝑦₁+𝜔²₀𝑦₁=−2𝐷₀𝐷₁𝑦₀−2𝜇𝐷₀𝑦₀−𝛼𝑦₀³−𝑒𝐷²₀𝜃₀cos𝜃+𝜅2𝐷₀𝜃₀𝐷₁𝜃₀+𝐷₀𝜃₁sin𝜃. (3.9) The solution of (3.7) can be written as 𝜃₀=𝐵𝑇₀+𝜎𝑇₁𝑦₀,=Λsin𝑇₀Ω+𝐴𝑇₁𝑒^{𝑖𝜔0𝑇0}+𝐴𝑇₁𝑒^{−𝑖𝜔0𝑇0}, (3.10)(3.11) where cos𝜃₀+𝜀𝜃₁=cos𝜃₀+𝑂(𝜀),sin𝜃₀+𝜀𝜃₁=sin𝜃₀+𝑂(𝜀),Λ=−𝜅Ω²Ω²−𝜔²₀. (3.12)

3.1. Subharmonic Resonances Ω≈3𝜔₀ (Table 1)

Table 1 

Vibrational conveyer parameters in SI units.

Near resonance: 𝐷₀𝜃₀=Ω,Ω=3𝜔₀+𝜀𝜎. (3.13)(3.14) The solution of (3.13) can be written as 𝜃₀=Ω𝑇₀=3𝜔₀𝑇₀+𝜎𝑇₁. (3.15) Taking 𝐴𝑇₁,𝑇₂=𝑎𝑇₁2𝑒^{𝑖𝛽[𝑇1]}, (3.16) where 𝑎 and 𝛽 are real, and substituting it into (3.11), we will obtain 𝑦₀=12𝑒^{−𝑖𝑇0𝜔0−𝑖𝛽(𝑇1)}𝑎𝑇₁+12𝑒^{𝑖𝑇0𝜔0+𝑖𝛽(𝑇1)}𝑎𝑇₁+Λsin𝑇₀Ω,𝜃₁=18Ω³−Ω𝜔²₀²𝑒^{−𝑖(𝑇0𝜔0+𝛽(𝑇1))}×𝐼−4Ω²𝜔²₀𝑎𝑇₁×1+𝑒^{2𝑖(𝑇0𝜔0+𝛽(𝑇1))}Ω²+𝜔²₀cos𝑇₀Ω−2𝑖−1+𝑒^{2𝑖(𝑇0𝜔0+𝛽(𝑇1))}Ω𝜔₀sin𝑇₀Ω−2𝑒^{𝑖(𝑇0𝜔0+𝛽(𝑇1))}Ω²−𝜔²₀cos𝑇₀Ω−4𝑔+−𝜅Ω²+Λ𝜔²₀sin𝑇₀Ω,𝑦₁=𝑒^{−𝑖(3𝑇0Ω+4𝑇0𝜔0+3𝛽(𝑇1))}×−8𝑒^{3𝑖𝛽(𝑇1)}ΩΩ³+3Ω²𝜔₀−Ω𝜔²₀−3𝜔³₀×𝑖𝑒^{𝑖𝑇0(4Ω+3𝜔0)}𝜔²₀3𝛼Λ³+8𝑖Λ𝜇Ω+𝐼𝜅²Ω²−𝐼𝜅Λ𝜔²₀36Ω⁴−13Ω²𝜔²₀+𝜔⁴₀+𝑖𝑒^{𝑖𝑇0(2Ω+3𝜔0)}𝜔²₀−3𝛼Λ³+8𝑖Λ𝜇Ω−𝐼𝜅²Ω²+𝐼𝜅Λ𝜔²₀36Ω⁴−13Ω²𝜔²₀+𝜔⁴₀+4𝑒^{𝑖𝑇0(Ω+3𝜔0)}𝑔𝐼𝜅𝜔²₀9Ω⁴−10Ω²𝜔²₀+𝜔⁴₀+4𝑒^{𝑖𝑇0(5Ω+3𝜔0)}𝑔𝐼𝜅𝜔²₀×9Ω⁴−10Ω²𝜔²₀+𝜔⁴₀+8𝑒^{3𝑖𝑇0(Ω+𝜔0)}𝑔𝐼𝜅36Ω⁶−49Ω⁴𝜔²₀+14Ω²𝜔⁴₀−𝜔⁶₀−𝑖𝑒^{3𝑖𝑇0(2Ω+𝜔0)}𝜔²₀4Ω⁴−5Ω²𝜔²₀+𝜔⁴₀𝛼Λ³+𝐼𝜅𝜅Ω²−Λ𝜔²₀−𝑖𝑒^{3𝑖𝑇0𝜔0}𝜔²₀4Ω⁴−5Ω²𝜔²₀+𝜔⁴₀−𝛼Λ³+𝐼𝜅−𝜅Ω²+Λ𝜔²₀+2𝑒^{2𝑖𝛽(𝑇1)}𝜔²₀36Ω⁵+108Ω⁴𝜔²₀−39Ω²𝜔³₀+Ω𝜔⁴₀+3𝜔⁵₀×𝑒^{𝑖𝑇0(5Ω+2𝜔0)}Ω+𝜔₀²2𝐼𝜅Ω𝜔²₀+3𝛼Λ²−Ω+𝜔₀+𝑒^{𝑖(𝑇0(Ω+4𝜔0)+2𝛽(𝑇1))}Ω+𝜔₀²2𝐼𝜅Ω𝜔²₀+3𝛼Λ²−Ω+𝜔₀+𝑒^{𝑖(𝑇0(Ω+2𝜔0)}Ω−𝜔₀²2𝐼𝜅Ω𝜔²₀−3𝛼Λ²Ω+𝜔₀+𝑒^{𝑖(5𝑇0Ω+4𝑇0𝜔0+2𝛽(𝑇1))}×Ω−𝜔₀²2𝐼𝜅Ω𝜔²₀−3𝛼Λ²Ω+𝜔₀𝑎𝑇₁+24𝑖𝑒^{𝑖(𝑇0(2Ω+𝜔0)+𝛽(𝑇1))}𝛼×ΛΩ𝜔²₀−−1−2𝑒^{2𝑖(𝑇0𝜔0+𝛽(𝑇1))}+2𝑒^{2𝑖(𝑇0(Ω+𝜔0)+𝛽(𝑇1))}+𝑒^{2𝑖(𝑇0(Ω+2𝜔0)+2𝛽(𝑇1))}Ω+−1+6𝑒^{2𝑖(𝑇0𝜔0+𝛽(𝑇1))}−6𝑒^{2𝑖(𝑇0(Ω+𝜔0)+𝛽(𝑇1))}+𝑒^{2𝑖(𝑇0(Ω+2𝜔0)+2𝛽(𝑇1))}𝜔₀×36Ω⁶−49Ω⁴𝜔²₀+14Ω²𝜔⁴₀−𝜔⁶₀𝑎𝑇₁²+𝑒^3𝑖𝑇0Ω1+𝑒^{6𝑖(𝑇0𝜔0+𝛽(𝑇1))}×Ω𝛼Ω₂−𝜔²₀²36Ω⁵+108Ω⁴𝜔₀−13Ω³𝜔²₀−39Ω²𝜔²₀+Ω𝜔⁴₀+3𝜔⁵₀𝑎𝑇₁³×cos𝑇₀𝜔₀+𝑖sin𝑇₀𝜔₀)/64Ω6Ω²−5Ω𝜔₀+𝜔²₀×−Ω²𝜔₀+𝜔³₀²×6Ω³+23Ω²𝜔₀+16Ω𝜔²₀+3𝜔³₀. (3.17) Secular terms will be eliminated from the particular solution of (3.8) if we choose 𝐴 to be a solution of −2𝜀𝜎^′+𝐸𝜃̇. (3.18)

In addition to the terms proportional to 𝑒^{(±𝑖𝜔0𝑇0)} or proportional to 𝑒^{[±𝑖(Ω−2𝜔0)𝑇0]} there is another term that produces a secular term in (3.9). We express (Ω−2𝜔₀)𝑇₀ in terms of 𝜔₀𝑇₀ according to Ω−2𝜔₀𝑇₀=𝜔₀𝑇₀+𝜀𝜎𝑇₀=𝜔₀𝑇₀+𝜎𝑇₁. (3.19)

3.2. Superharmonic Resonances Ω≈(1/3)𝜔₀ (Table 2)

Table 2 

Vibrational conveyer parameters in SI units.

We consider3Ω=𝜔₀+𝜀𝜎. (3.20)

In addition to the terms proportional to 𝑒^{(±𝑖𝜔0𝑇0)} in (3.9), there is another term that produces a secular term in (3.9). This is −𝛼Λ³𝑒^{(±3𝑖Ω𝑇0)}. 𝑇₀ eliminate the secular terms, we express 3Ω𝑇₀ in terms of 𝜔₀𝑇₀ according to 3Ω𝑇₀=𝜔₀+𝜀𝜎𝑇₀=𝜔₀𝑇₀+𝜎𝜀𝑇₀=𝜔₀𝑇₀+𝜎𝑇₁. (3.21) Eliminating secular terms in equations, taking (3.16), and separating real and imaginary parts, the system of equations solved it is as follows [11]: 𝑦₀=12𝑒^{−𝑖𝑇0𝜔0−𝑖𝛽(𝑇1)}𝑎𝑇₁+12𝑒^{𝑖𝑇0𝜔0+𝑖𝛽(𝑇1)}𝑎𝑇₁+Λsin𝑇₀Ω,𝜃₁=18(Ω³−Ω𝜔²₀)²𝑒^{−𝑖(𝑇0𝜔0+𝛽(𝑇1))}×𝐼−4Ω²𝜔²₀𝑎𝑇₁×1+𝑒^{2𝑖(𝑇0𝜔0+𝛽(𝑇1))}Ω²+𝜔²₀cos𝑇₀Ω−2𝑖−1+𝑒^{2𝑖(𝑇0𝜔0+𝛽(𝑇1))}Ω𝜔₀sin𝑇₀Ω−2𝑒^{𝑖(𝑇0𝜔0+𝛽(𝑇1))}Ω²−𝜔²₀cos𝑇₀Ω−4𝑔+−𝜅Ω²+Λ𝜔²₀sin𝑇₀Ω,𝑦₁=𝑒^{−𝑖(2𝑇0Ω+5𝑇0𝜔0+3𝛽(𝑇1))}×−8𝑒^{3𝑖𝛽(𝑇1)}ΩΩ⁴−10Ω²𝜔²₀+9𝜔⁴₀×4𝑒^{𝑖𝑇0(4Ω+5𝜔0)}𝑔𝐼𝜅𝜔²₀Ω²−𝜔²₀−4𝑒^{5𝑖𝑇0𝜔0}𝑔𝐼𝜅𝜔²₀−Ω²+𝜔²₀+𝑖𝑒^{𝑖𝑇0(3Ω+5𝜔0)}𝜔²₀4Ω²−𝜔²₀3𝛼Λ³+8𝑖Λ𝜇Ω+𝐼𝜅²Ω²−𝐼𝜅Λ𝜔²₀+𝑒^{𝑖𝑇0(Ω+5𝜔0)}𝜔²₀−4Ω²+𝜔²₀3𝑖𝛼Λ³+8Λ𝜇Ω+𝑖𝐼𝜅²Ω²−𝑖𝐼𝜅Λ𝜔²₀+8𝑒^{𝑖𝑇0(2Ω+5𝜔0)}𝑔𝐼𝜅4Ω⁴−5Ω⁴𝜔²₀+𝜔⁴₀+2𝑒^{2𝑖𝛽(𝑇1)}𝜔²₀4Ω⁴−37Ω²𝜔²₀+9𝜔₀×𝑒^{4𝑖𝑇0(Ω+𝜔0)}Ω+𝜔₀²2𝐼𝜅Ω𝜔²₀+3𝛼Λ²−Ω+𝜔₀+𝑒^{2𝑖(3𝑇0𝜔0+𝛽(𝑇1))}Ω+𝜔₀²×2𝐼𝜅Ω𝜔²₀+3𝛼Λ²−Ω+𝜔₀+𝑒^{4𝑖𝑇0𝜔0}Ω−𝜔₀²2𝐼𝜅Ω𝜔²₀−3𝛼Λ²Ω+𝜔₀+𝑒^{2𝑖(2𝑇0Ω+3𝑇0𝜔0+𝛽(𝑇1))}Ω−𝜔₀²2𝐼𝜅Ω𝜔²₀−3𝛼Λ²Ω+𝜔₀𝑎𝑇₁−24𝑖𝑒^{𝑖𝛽(𝑇1)}𝛼ΛΩ𝜔²₀4Ω⁴−5Ω²𝜔²₀+𝜔⁴₀×2𝑒^{𝑖(3𝑇0Ω+5𝑇0𝜔0+2𝛽(𝑇1))}Ω²−9𝜔²₀−2𝑒^{𝑖(𝑇0(Ω+5𝜔0)+2𝛽(𝑇1))}Ω²−9𝜔²₀−𝑒^{𝑖𝑇0(Ω+3𝜔0)}Ω²−4Ω𝜔₀+3𝜔²₀+𝑒^{𝑖(3𝑇0Ω+7𝑇0𝜔0+4𝛽(𝑇1))}Ω²−4Ω𝜔₀+3𝜔²₀+𝑒^{3𝑖𝑇0(Ω+𝜔0)}Ω²+4Ω𝜔₀+3𝜔²₀−𝑒^{𝑖(𝑇0(Ω+7𝜔0)+4𝛽(𝑇1))}×Ω²+4Ω𝜔₀+3𝜔²₀𝑎𝑇₁²+𝑒^{2𝑖𝑇0(Ω+𝜔0)}+𝑒^{2𝑖(𝑇0(Ω+4𝜔0)+3𝛽(𝑇1))}𝛼ΩΩ²−𝜔²₀²×4Ω⁴−37Ω²𝜔²₀+9𝜔⁴₀𝑎𝑇₁³/64−Ω²𝜔₀+𝜔³₀²4Ω⁵−37Ω³𝜔²₀+9Ω𝜔⁴₀. (3.22)

4. Stability Analysis

We analyzed in this system the stability (𝑎,𝛾) in the equilibrium point, using (4.1), where 𝐹 is the Jacobian matrix of (4.2), and 𝛾_sub=3𝛽[𝑇₁]−𝑇₁𝜎[𝑇₁], 𝛾_sup=[𝛽[𝑇₁]−𝑇₁𝜎[𝑇₁]]. Stability of the approximate solutions depends on the value of the eigenvalues of the Jacobian matrix 𝐹 [12]. The solutions are unstable if the real part of the eigenvalues is positives Figures 2(a1) and 2(a2). Figure 2 shows the frequency-response curves for the subharmonic and superharmonic resonance of the unbalanced vibratory conveyor: 𝑎^′_sub3=−𝜇𝑎−8𝜔₀𝛼Λ𝑎²𝛾,𝛾cos_′sub3=−2𝐼𝜅Ω²𝜔²₀+3𝛼Λ²−Ω²+𝜔²₀4Ω²−𝜔²₀𝜔₀+98𝜔₀𝛼𝑎²−98𝜔₀𝛾𝑎𝛼Λ𝑎sin−𝜎,_′sub3=−𝜇𝑎−8𝜔₀𝛼Λ³+𝐼𝜅𝜅Ω²−Λ𝜔²₀𝛾,𝛾cos_′sub=−2𝐼𝜅Ω²𝜔²₀+3𝛼Λ²−Ω²+𝜔²₀4Ω²−𝜔²₀𝜔₀+3𝛼𝑎²8𝜔₀+𝛼Λ³+𝐼𝜅𝜅Ω²−Λ𝜔²₀sin𝛾8𝜔₀𝑎𝜕−𝜎,𝐹=_𝑎𝑓₁,𝜕_𝛾𝑓₁,𝜕_𝑎𝑓₂,𝜕_𝛾𝑓₂,𝑓₁=𝑎^′,𝑓₂=𝛾^′. (4.1)(4.2)

Figure 2 

Subharmonic resonance and superharmonic resonance. (a) Frequency-response curves with stability, −−− stable, ⋯ unstable {(a1)∶𝑎=2.78–10, 𝜎=0.43258–2.62092 and (a2): 𝑎=2.73–7.21, 𝜎=0.6782–0.986448}, (b) effect of detuning parameter, and (c) effect of damping parameter.

5. Numerical Results

The numerical calculations of the vibrating system are performed with the help of the software Mathematica [13, 14]. We analyze the subharmonic resonance Ω≈3𝜔₀ and superharmonic resonance 3Ω≈𝜔₀. Figure 3 shows the power spectrum, phase portrait and the Poincaré map for superharmonic resonance and Figure 3 shows the power spectrum for superharmonic resonance.

(a)

(b)

(c)

(d)

Figure 3 

Subharmonic resonance: (a) power spectrum and superharmonic resonance, (b) power spectrum, (c) phase portrait, and (d) Poincaré sections.

6. Conclusions

The vibrating system is analyzed, analytically, and numerically for superharmonic and subharmonic resonance by the method of multiple scales. Very often in the motion of the system near resonance the jump phenomenon occurs. The frequency-response curves of the subharmonic resonance consist of two branches; the left one is stable and the right one is unstable (saddle node bifurcation). The frequency-response curves of the superharmonic resonance consist of three branches; the left one is stable, the middle one is unstable, and the right one is stable (pitchfork bifurcation). The stable motions of the oscillator are shown with one peaks in the power spectrum for superharmonic resonance and with two peaks in the power spectrum for subharmonic resonance. Both analytical and numerical results that we have obtained are in good agreement. The system studied here exhibits chaotic behaviour in case of strong nonlinearity. This will be reported in the forthcoming paper. Furthermore, control methods in the passage through resonance await for future publication.

Acknowledgment

The authors greatly appreciate the comments of an anonymous referee.