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

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.


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 .

The Governing Equations of the Motion
The equations of motion for the modified rocard system may be obtained by using Lagrange's equation where T , the kinetic energy, is 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 where the constants k 1 and k 2 are the linear and nonlinear elastic coefficients, respectively, D, the Rayleigh dissipation function, is 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 k 1 y k 2 y 3 me θ 2 sin θ −θ cos θ ,

2.5
Equation 2.5 can be rewritten as follows: 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 θ that is the active torque generated by the electric circuit of the DC motor, shown in Figure 1.

Analytical Solution
Ideal system: if there is no coupling between motion of the rotor and vibrating system anḋ θ constant θ ωt,θ 0 , 2.6 becomes On the right side of the equation, a function of time is present. Nonideal system:θ where E θ M m θ − H θ is the difference between the torque generated by the motor and the resistance torque. Function E θ u 1 − u 2θ is approximated by a straight line, where u 1 is a control parameter that can be changed according to the voltage, and u 2 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: where the fast scale T 0 t and slow scale T 1 εt. The time derivatives transform according to where D n ∂/∂T n , n 0, 1, . . . ; then, Mathematical Problems in Engineering 3.5 Substituting 3.5 into 3.2 , we will obtain and equating coefficients of a like powers ε, we obtain i for ε 0 ii for ε 1

3.9
The solution of 3.7 can be written as Mathematical Problems in Engineering

Subharmonic
Resonances Ω ≈ 3ω 0 (Table 1) Near resonance: The solution of 3.13 can be written as

3.15
Taking where a and β are real, and substituting it into 3.11 , we will obtain

3.17
Secular terms will be eliminated from the particular solution of 3.8 if we choose A to be a solution of −2εσ E θ .

3.18
In addition to the terms proportional to e ±iω 0 T 0 or proportional to e ±i Ω−2ω 0 T 0 there is another term that produces a secular term in 3.9 . We express Ω − 2ω 0 T 0 in terms of ω 0 T 0 according to 3.19

3.20
In addition to the terms proportional to e ±iω 0 T 0 in 3.9 , there is another term that produces a secular term in 3.9 . This is −αΛ 3 e ±3iΩT 0 . T 0 eliminate the secular terms, we express 3 ΩT 0 in terms of ω 0 T 0 according to

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 : 3.22

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 T 1 −T 1 σ T 1 , γ sup β T 1 −T 1 σ T 1 . 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:

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 ≈ 3ω 0 and superharmonic resonance 3 Ω ≈ ω 0 . 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.

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.