Application of the Harmonic Balance Method to Ground Moling Machines Operating in Periodic Regimes

Abstract The paper describes current research into mathematical modelling of a novel vibro-impact ground moling system. Experimental and theoretical studies suggest periodic responses are required to achieve the optimal penetrating conditions for the ground moling process, as this results in reduced soil penetration resistance. Therefore, there is a practical need for a robust and efficient methodology to calculate periodic responses for a wide range of operational parameters. Due to the structural complexity of a real vibro-impact moling system, the dynamic response of an idealised impact oscillator has been investigated in the first instance. This paper presents a detailed study of periodic responses of the impact oscillator under harmonic forcing using alternating frequency-time harmonic balance method. Recommendations of how to effectively adapt the alternating frequency-time harmonic balance method for a stiff impacting system are given. The periodic motion is represented algebraically by a truncated Fourier series and the systematic methodology employed allows for convergence. The idea central to this procedure is that the linear oscillator is explicitly solvable analytically, and this allows for the initial set of Fourier coefficients. The clearance value is then adjusted so that contact with the secondary stiffness is slight and the nonlinearity is weak. The solution to this subsequent system is obtainable as the initial guess is close to the required solution.

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RATIONAL ENERGY BALANCE METHOD TO NONLINEAR OSCILLATORS WITH CUBIC TERM

Asian-European Journal of Mathematics ◽

10.1142/s1793557113500198 ◽

2013 ◽

Vol 06 (02) ◽

pp. 1350019 ◽

Cited By ~ 15

Author(s):

M. Daeichin ◽

M. A. Ahmadpoor ◽

H. Askari ◽

A. Yildirim

Keyword(s):

Energy Balance ◽

Harmonic Balance ◽

Harmonic Balance Method ◽

Nonlinear Problems ◽

Nonlinear Oscillators ◽

Initial Guess ◽

Second Order ◽

Balance Method ◽

Strong Nonlinearity ◽

Novel Approach

In this paper, a novel approach is proposed for solving the nonlinear problems based on the collocation and energy balance methods (EBMs). Rational approximation is employed as an initial guess and then it is combined with EBM and collocation method for solving nonlinear oscillators with cubic term. Obtained frequency amplitude relationship is compared with exact numerical solution and subsequently, a very excellent accuracy will be revealed. According to the numerical comparisons, this method provides high accuracy with 0.03% relative error for Duffing equation with strong nonlinearity in the second-order of approximation. Furthermore, achieved results are compared with other types of modified EBMs and the second-order of harmonic balance method. It is demonstrated that the new proposed method has the highest accuracy in comparison with different approaches such as modified EBMs and the second-order of harmonic balance method.

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Imbalance Response of Nonlinear Rotor-SFD Dynamic Systems With Structure Modelled As FRFs Using Harmonic Balance Method

Volume 2: Structures and Dynamics; Renewable Energy (Solar, Wind); Inlets and Exhausts; Emerging Technologies (Hybrid Electric Propulsion, UAV, ...); GT Operation and Maintenance; Materials and Manufacturing (Including Coatings, Composites, CMCs, Additive Manufacturing); Analytics and Digital Solutions for Gas Turbines/Rotating Machinery ◽

10.1115/gtindia2017-4749 ◽

2017 ◽

Author(s):

Manoj Settipalli ◽

Rahul Chandran ◽

Venkatarao Ganji ◽

Theodore Brockett

Keyword(s):

Dynamic Systems ◽

Harmonic Balance ◽

Degrees Of Freedom ◽

Fourier Coefficients ◽

Harmonic Balance Method ◽

Rotor System ◽

Balance Method ◽

Squeeze Film ◽

Static Structure ◽

The Impact

Squeeze-film-dampers (SFDs) used to couple rotor dynamic systems to linear static structures, such as those in aircraft engines and turbochargers, are often approximated as linear connections in dynamic simulations. Linearized stiffness and damping coefficients of the SFDs can be reasonably estimated for circular centered orbits. Selection of linearized properties for the SFD is challenged under more general whirling conditions, such as those occurring in non-centered dampers with steady gravity loading. In this paper, an efficient method for coupling the rotor system to a static structure modeled as frequency-response-functions (FRFs) through nonlinear SFDs is illustrated. The harmonic balance method (HBM) with arc length continuation technique is employed in the frequency domain to obtain the system periodic response. Degrees-of-freedom participating in the non-linear SFD model, when separated from the remaining linear degrees-of-freedom, are expanded in terms of Fourier coefficients. The algorithm allows the Fourier coefficients approximating the nonlinearity to be iteratively determined at each frequency of interest. The approach has a tremendous time advantage over a traditional nonlinear transient analysis. The method can be used to efficiently predict vibration response on the engine static structure to typical imbalance on the rotors to assess the risk of meeting the low vibration requirements typical of new designs. The prediction includes the primary driving frequencies and their harmonics in the vibration estimate. A flexible rotor system connected to structure through an SFD is used to demonstrate the approach and discuss the impact of results.

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