Technical Brief

Analytical Approximations for Stick–Slip Amplitudes and Frequency of Duffing Oscillator

[+] Author and Article Information
Devarajan K

Department of Mechanical Engineering,
Amrita School of Engineering,
Amrita Vishwa Vidyapeetham,
Amrita University,
Coimbatore, Tamil Nadu 641 116, India
e-mail: k_devarajan@cb.amrita.edu

Bipin Balaram

Department of Mechanical Engineering,
Amrita School of Engineering,
Amrita Vishwa Vidyapeetham,
Amrita University,
Coimbatore, Tamil Nadu 641 116, India
e-mail: b_bipin@cb.amrita.edu

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received December 7, 2015; final manuscript received August 11, 2016; published online January 20, 2017. Assoc. Editor: D. Dane Quinn.

J. Comput. Nonlinear Dynam 12(4), 044501 (Jan 20, 2017) (8 pages) Paper No: CND-15-1412; doi: 10.1115/1.4034734 History: Received December 07, 2015; Revised August 11, 2016

Linear spring mass systems placed on a moving belt have been subjected to numerous investigations. Dynamical characteristics like amplitude and frequency of oscillations and bifurcations have been well studied along with different control mechanisms for this model. But the corresponding nonlinear system has not received comparable attention. This paper presents an analytical investigation of the behavior of a Duffing oscillator placed on a belt moving with constant velocity and excited by dry friction. A negative gradient friction model is considered to account for the initial decrease and the subsequent increase in the frictional forces with increasing relative velocity. Approximate analytical expressions are obtained for the amplitudes and base frequencies of friction-induced stick–slip and pure-slip phases of oscillations. For the pure-slip phase, an expression for the equilibrium point is obtained, and averaging procedure is used to arrive at approximate analytical expressions of the periodic amplitude of oscillations around this fixed-point. For stick–slip oscillations, analytical expressions for amplitude are arrived at by using perturbation analysis for the finite time interval of the stick phase, which is linked to the subsequent slip phase through conditions of continuity and periodicity. These analytical results are validated by numerical studies and are shown to be in good agreement with them. It is shown that the pure-slip oscillation phase and the critical velocity of the belt remain unaffected by the nonlinear term. It is also shown that the amplitude of the stick–slip phase varies inversely with nonlinearity. The effect of different system parameters on the vibration amplitude is also studied.

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Grahic Jump Location
Fig. 1

(a) The classical mass-on-moving-belt system: A mass at position X on a belt that moves at constant speed Vb. (b) Friction coefficient as a function of the relative velocity as given by Eq. (4) with μs = 0.4, μm = 0.25, and vm = 0.5.

Grahic Jump Location
Fig. 2

One period of stick–slip displacements u(t) (top row), velocities u˙(t) (middle), and phase plane orbits u˙(u) (bottom row) for three values of belt velocity vb (a)–(c). (––) Analytical prediction (Eqs. (34) and (41)); (–––) numerical simulation of Eq.(1) with Eqs. (4) and (9). The simulations were performed for the following parameter values: β = 0.05, μs = 0.4, μm = 0.25, vm = 0.5, L = 0.1, α = 0.3, K = 1, and (a) vb = 0.05, (b) vb = 0.25, (c) vb = 0.3944.

Grahic Jump Location
Fig. 3

(a) Displacement amplitude of stick–slip and pure-slip vibration amplitudes as a function of belt velocity vb. (b) Influence of nonlinear term in stick–slip and pure-slip amplitudes. The same parameters were used here as used in Fig. 2.

Grahic Jump Location
Fig. 5

(a) Analytical prediction of stick–slip and pure-slip vibration amplitudes for different friction gradients (μs−μm)/vm=0.075,0.099,0.15,0.3,0.6. (b) Influence of nondimensional parameter γ in stick–slip and pure-slip vibration amplitudes for (μs−μm)/vm=0.3. The same parameters were used here as used in Fig. 2.

Grahic Jump Location
Fig. 4

(a) Fundamental natural frequency of periodic motion as a function of belt velocity vb. (b) Influence of nonlinear term in fundamental frequency of stable periodic motion. The same parameters were used here as used in Fig. 2.



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