Restoring Force Function Research Articles

Improved harmonic balance method for analyzing asymmetric restoring force functions in nonlinear vibration of mechanical systems

The non-linear differential equation governing some mechanical systems, such as the non-linear vibrations of FG beams resting on a nonlinear foundation, behaves similarly to the oscillations of a non-linear oscillator with an asymmetric restoring force function that includes both odd and non-odd non-linear terms. The objective of this research is to obtain higher-order approximate analytical solutions for such problems by introducing an enhanced harmonic balance method. By building upon the original problem, two new symmetric systems with odd non-linear terms are introduced, and their higher-order approximate analytical solutions are derived using a novel approach. In proposed method, the restoring force function is represented by its Fourier series expansion. Unlike previous papers, linearizing the equation or taking the first derivative of the Fourier series in the subsequent iteration is unnecessary. However, to enhance the solution’s accuracy, the remaining error from each iteration is utilized in the next one. Finally, by combining the results from the two introduced systems, the analytical period and corresponding periodic solution of the original problem can be obtained. This method is applied to the governing differential equation of FG beams resting on non-linear foundations, a physical non-natural oscillator, and conservative Toda oscillator. The key advantages of this approach are its simplicity and its ability to provide highly precise solutions for both small and large amplitudes in a single iteration.

Generic numerical and analytical methods for solving nonlinear oscillators

Nonlinear oscillations are a challenging problem due to the high complexity of the underlying differential equations. This paper focuses on the nonlinear oscillator with cubic and harmonic restoring force as it represents a very general case. In particular, many well-known oscillators (e.g., the Duffing equation, the capillary oscillator, the cubic-quintic Duffing oscillator, the simple pendulum, etc) are subcases of it. Accurate solutions in terms of Fourier series functions for different oscillation amplitudes were derived using an energy-based numerical method and fitting processes. In addition, an analytical approach was developed by finding the derivative of the restoring force function at the point x=c0A, where A is the amplitude of the oscillation, and c0 is a constant. Using this approach, the error was negligible (∼0.02–0.22%) even for A → ∞. The methods and solutions for the nonlinear oscillator with cubic and harmonic restoring force can be equally applied to all its subcases.

Analysis and design of power and bandwidth of nonlinear vibration energy harvesters with hardening restoring forces

This paper presents a theoretical foundation for designing the power and bandwidth performance of an electromagnetic nonlinear vibration energy harvester with a hardening resonator. To this end, the steady-state solution derived via first-order approximation is investigated to establish a graphical approach, which gives a clear perspective on how the resonance peak point on the frequency–displacement curve is determined in terms of the excitation amplitude, nonlinear restoring force function, and other design parameters. Then, a ν-power bandwidth, a generalized version of the half-power bandwidth, is newly introduced and its approximate analytical formulation is derived. Based on the findings of the parameter study on the ν-power bandwidth, a design scheme is proposed, which begins with specifying the location of the resonance peak point that guarantees the existence of the high-energy branch under possible variations of the excitation, followed by determining the linear natural frequency that makes the ν-power bandwidth as large as reasonably possible. Design examples, assuming a specific class of hardening nonlinearity of the restoring force, are shown to demonstrate how the proposed design scheme yields vibration energy harvesters with maximized power bandwidth performance while incorporating given design requirements.

Guided identification of nonlinear reduced-order models via the incorporation of von Kármán beam theory

A reduction in the number of Degrees Of Freedom (DOFs) of a nonlinear structural model can significantly decrease the computational time of dynamic analyses, especially in problems where long analyses are required. A Reduced-Order Model (ROM) of a structural system is commonly generated using some combination of dimensionality reduction, such as substitution of a modal basis or static condensation of membrane DOFs. Perhaps the most common and widespread framework for developing low-order models of continuous systems is the Ritz method in conjunction with a smooth modal basis. The Ritz method requires access to the closed form governing differential equations, and it is difficult to implement for problems with complex boundary conditions. In recent decades, an alternative framework that instead constructs a low-order model from an existing Finite Element Model (FEM) has been developed. Such models are typically known as NonLinear Reduced-Order Models (NLROMs). Such NLROMs can be used to handle complex boundary conditions so long as the FEM can do the same, and they allow the analyst to start from an existing finite element package as opposed to directly working with the governing differential equations. The form of the NLROM restoring force function is assumed by the analyst prior to identifying the model, and the parameters of the function are identified to achieve a high-quality fit to the FEM load–displacement data. This research aims to examine the connection between classically derived models using the Ritz method and NLROMs in order to predict the coefficients of NLROMs and to develop simplifying assumptions about their structure. Specifically, the Condensed Von Kármán (CVK) beam equation is used as a basis for the study of nonlinear beam structures. The Ritz method is applied to the CVK beam equation, and the matrix form of the equilibrium equation is determined. The equivalence between the CVK coefficients and the NLROM coefficients is shown, and a simplified, tangent stiffness-based identification procedure is proposed to determine the coefficients of the CVK-based model. An arc-length continuation algorithm is used to predict the static load–displacement response of the models studied and to assess the accuracy of the ROMs. In the case of initially straight, post-buckled beams, the proposed identification procedure is shown to generate a model that generally outperforms a corresponding NLROM identified using a more traditional displacement-based procedure.

Nonlinear dynamic model for linear guideway with crowning subjected to external load in five directions

The motion accuracy of linear guideway is adversely affected by ball passage vibration (BPV) when rolling balls within carriage move in a loop. Reduced vibration amplitude can be realized by a carriage with crowning design by now. However, few dynamic models are availability for the BPV analysis when linear guideway is subjected to actual external load in five directions. In this paper, a nonlinear dynamic model is proposed to investigate the periodic micro fluctuation of linear guideway with crowning subjected to load in five directions. The idea of coordinate transformation is adopted for the easier derivations of time-dependent and piecewise restoring force function. The different contact states of each ball are described in detail due to load distribution. The numerical simulation is employed to deal with equation of motion and discuss the effects of ball grouping, external load, preload and carriage speed. Experiment is also conducted for the verification of proposed model. The analytical model developed here is promising to clarify the mechanism of BPV from a dynamic perspective and serve the walking parallelism accuracy monitoring of linear guideway.

Nonlinear dynamic analysis of four-station rotary tool holder subjected to harmonic excitation

Four-station rotary tool holder on a lathe is fabricated to achieve transposition operation of turning tools and carry the violent cutting force prior to feed system. Its dynamic characteristic has a potential influence on the structural dynamics and cutting stability of a machine tool. This paper develops a mechanical model for the dynamic characteristic analysis and vibration response prediction. Using the fractal contact theory, the analytical model highlights the rough joint interfacial contact of main components including tool holder body, trapezoidal meshing gear plate and screw shaft. The elastic supports of thrust bearing unit and hollow shaft are also considered. The multi-degree-of-freedom system involving the coupled displacement and piecewise restoring force function is established using lumped mass method. The modal and harmonic excitation tests are performed to investigate the natural property and verify the proposed model. Furthermore, numerical simulation evaluates the effect of external excitation, locking force, fractal parameter, bearing preload and shaft stiffness on dynamic characteristic and stability. The proposed model is promising for the structural optimal design of four-station rotary tool holder.

A comprehensive nonlinear dynamic model for ball screw feed system with rolling joint characteristics

Modern tendency of machine tools design requires more accurate model to predict the system dynamics, in order to anticipate its interaction with machining process. In this paper, a comprehensive dynamic model of ball screw feed system (BSFS) considering nonlinear kinematic joints is introduced to investigate the varying dynamic characteristics when worktable is subjected to combined load from six directions. The load–deformation relationship of each kinematic joint is dealt with a set of translational and angular spring elements. The nonlinear restoring force function of each joint involving coupling displacement is calculated. Based on the lumped mass method, the analytical 18-DOF dynamic equation is formulated by the analysis of the interaction force between joints. Model verification tests are conducted. The worktable response exhibits the abundant and fascinating nonlinear phenomena arising in nonlinear joint and coupling effect. The nonlinear behavior behaves significant difference owing to the variations of excitation, platform position, screw length, preload and damping of joints. Thus, the model is promising for comprehension of machine dynamic behavior and for development of sophisticated control strategy.

A time-dependent dynamic model for ball passage vibration analysis of recirculation ball screw mechanism

Ball passage vibration arising during recirculation of rolling elements is a limitation of high-precision performance associated with ball screw. A relatively deep understanding of this phenomenon when rolling balls leave load area and enter circulation channel has been achieved by now. However, few mechanical models are established for calculation of rolling ball contact deformation and force inside nut while moving by taking into account transient phase at import and export of circulation channel. In this work, we present a general time-dependent dynamic model for description of ball motion and axial periodic micro fluctuation within ball screw subjected to external load. The numerical simulation is employed to deal with equation of motion including segmented restoring force function. Effects of applied load, preload and screw rotational speed on dynamic response are discussed theoretically. Reduced vibration amplitude may be realized by small ball diameter and phase difference between ball groupings. Proposed model is experimentally illustrated.

Friction-Induced Vibration Suppression via the Tuned Mass Damper: Optimal Tuning Strategy

Friction-induced vibrations are a significant problem in various engineering applications, while dynamic vibration absorbers are an economical and effective tool for suppressing various kinds of vibrations. In this study, the archetypal mass-on-moving-belt model with an attached dynamic vibration absorber was considered. By adopting an analytical procedure, the optimal tuning of the absorber’s parameters was defined. Furthermore, the bifurcations occurring at the loss of stability were analytically investigated; this analysis illustrated that a properly chosen nonlinearity in the absorber’s stiffness permits controlling the supercritical or subcritical character of the bifurcation. However, a numerical analysis of the system’s dynamics, despite confirming the analytical results, also illustrated that the system’s global behavior is only slightly affected by the bifurcation character. Indeed, a dynamic vibration absorber possessing a perfectly linear restoring force function seems to provide the optimal performance; namely, it minimizes the velocity range for which stick–slip oscillations exists.

Dynamic characteristics and reliability analysis of ball screw feed system on a lathe

For ball screw feed system on a lathe, the randomness of geometric parameters deriving from the inevitable manufacturing and assembly errors significantly affected the system dynamic characteristics and turning accuracy. In this study, a novel lumped dynamic model accounting for the analytical piecewise restoring force function involving the overall axial deformation is proposed. The equivalent mechanical model of bearing joints including the change of nominal contact angle is made a reasonable simplification to facilitate acquisition of the system analytical equation of motion. Additionally, the approximate analytical solution of the proposed equation is obtained by the modified Lindstedt–Poincaré method (MLP). The advanced first order second moment method (AFOSM) is employed to evaluate the reliability of dynamic response error. Furthermore, the numerical simulation aims to demonstrate the capabilities of the MLP and AFOSM method in the practical applications. Eventually, the effects of external excitation, sliding platform position, and nominal contact angles on the dynamic response are discussed and the reliability and sensitivity analysis in accordance with actual turning accuracy level is presented.

Extended IHB method for dynamic analysis of structures with geometrical and material nonlinearities

The incremental harmonic balance (IHB) method is popularly used in the nonlinear analysis of dynamic systems. It is, however, only applicable to polynomial type geometrical nonlinearity which greatly limits its application to more general nonlinear systems. The IHB method is extended (EIHB) in this report to overcome this limitation with a simple procedure for evaluating the stiffness matrix. The EIHB is versatile in dealing with any type of continuous nonlinear functions including the piecewise linear function and hysteretic restoring force function. Several nonlinear Single-Degree-of-Freedom (SDoF) systems are adopted for illustration of the proposed method. One more simple 7-DoFs suspension bridge deck and one more complicated 48-Dofs frame structure with buckling restrained braces are analyzed. New observations are noted from the analysis which would otherwise be difficult to obtain via existing methods. The proposed method is concluded to be simpler and convenient for nonlinear dynamic analysis of complicated structural system than existing methods.

A Modified Newton–Harmonic Balance Approach to Strongly Odd Nonlinear Oscillators

Since combinations of the Newton’s method and the harmonic balance (HB) method require, at each iteration step, calculating the first or the first- and second-order derivatives of the restoring force function, and expanding the function, its first- and second-order derivatives into Fourier series, the procedural costs are high and sometimes difficult to achieve algebraically. It is thus preferable to avoid expensive re-linearization or computation of the second-order derivative. A new approach is proposed to construct accurate analytical approximation solutions to strongly nonlinear conservative oscillators with odd nonlinearities. The approach is based on a combination of a modified Newton method and the HB method. For the modified Newton method, two simplified Newton steps are taken between each Newton step where only one linearization of the restoring force function is required. The resulting equations are solved by applying the HB method appropriately. Using only one modified Newton iteration step may achieve highly accurate analytical approximation solutions to the strongly nonlinear oscillators. Three examples with physical implications are used to illustrate the proposed method. Through the modified Newton iteration step, the multiple cumbersome linearizations of the restoring force function are replaced by only one linearization, and the corresponding governing equations can be properly solved by the HB method. The current work is expected to extend to the study of other nonlinear oscillations.

An Iterative Approach to Obtaining the Nonlinear Frequency of a Conservative Oscillator with Strong Nonlinearities

An iterative procedure is suggested for obtaining the higher-order approximate solutions of a conservative system comprising an oscillator with cubic and quintic restoring force function. The proposed method is similar to the traditional harmonic balance methods but, unlike them, the errors obtained from the previous step are considered at the current step to increase the accuracy of the solution. A comparison of results with those obtained by exact solution and other approximate analytical techniques confirms the accuracy of the method. It is shown that the achieved approximate solutions are valid for both small and large amplitudes of oscillation and can meet the exact solutions with a high level of accuracy in the lower-order of approximations. Furthermore, using the obtained analytical solutions, the effect of cubic and quintic terms on the frequency is discussed.

Nonlinear Piecewise Restoring Force in Hydrokinetic Power Conversion Using Flow-Induced Vibrations of Two Tandem Cylinders

Flow-induced vibrations (FIVs) of two tandem, rigid, circular cylinders with piecewise continuous restoring force are investigated for Reynolds number 24,000 ≤ Re ≤ 120,000 with damping, and restoring force function as parameters. Selective roughness is applied to enhance FIV and increase the hydrokinetic energy captured by the vortex-induced vibration for aquatic clean energy (VIVACE) converter. Experimental results for amplitude response, frequency response, interactions between cylinders, energy harvesting, and efficiency are presented and discussed. All experiments were conducted in the low-turbulence free-surface water (LTFSW) Channel of the MRELab of the University of Michigan. The main conclusions are as follows: (1) the nonlinear-spring converter can harness energy from flows as slow as 0.33 m/s with no upper limit; (2) the nonlinear-spring converter has better performance at initial galloping than its linear-spring counterpart; (3) the FIV response is predominantly periodic for all nonlinear spring functions used; (4) the influence from the upstream cylinder is becoming more dominant as damping increases; (5) optimal power harnessing is achieved by changing the linear viscous damping and tandem spacing L/D; (6) close spacing ratio L/D = 1.57 has a positive impact on the harnessed power in VIV to galloping transition; and (7) the interactions between two cylinders have a positive impact on the upstream cylinder regardless of the spacing and harness damping.

Asymptotic analysis and accurate approximate solutions for strongly nonlinear conservative symmetric oscillators

• This paper focuses on a strongly nonlinear conservative symmetric oscillator. • A new method for constructing analytical approximate solutions to the oscillator is presented. • Exploring the full efficiency and capability of a second-order Taylor expansion and the harmonic balance method. • With one single iteration, explicit yet very brief and accurate analytical approximate solutions have been derived. A new accurate iterative and asymptotic method is introduced to construct analytical approximate solutions to strongly nonlinear conservative symmetric oscillators. The method is based on applying a second-order expansion with the harmonic balance method and it excludes the requirement of solving a set of coupled nonlinear algebraic equations. Newton's iteration or the linearized model may be readily deduced by considering only the first-order terms in the model. According to this iterative approach, only Fourier series expansions of restoring force function, its first- and second-order derivatives for each iteration are required. It is concluded here that by only one single iteration, very brief and yet accurate analytical approximate solutions can be attained. Three physical examples are solved and accurate solutions are presented to illustrate the physics of the system and the effectiveness of the proposed asymptotic method.

Free vibration analysis of a structural system with a pair of irrational nonlinearities

An alternative method of deriving accurate and simple analytical approximate solutions to a structural dynamical system governed by a pair of strong irrational restoring forces is presented. This system can be used to represent mathematical models in various engineering problems. Prior to solving the problem, a rational approximation of the nonlinear restoring force function is applied to achieve a convergent truncation. Analytical solutions are then obtained using the combination of the harmonic balance method and Newton's method. This approach shows that lower-order analytical procedures can yield highly accurate and exact solutions that are difficult to obtain with an analytical expression.

Experimental Identification of the Nonlinear Parameters of an Industrial Translational Guide for Machine Performance Evaluation

Prediction of machine dynamics at the design stage is a challenge due to lack of adequate methods for identifying and handling the nonlinearities in the machine joints, which appear as the nonlinear restoring force function of relative displacement and relative velocity across the joint. This paper discusses identification of such a nonlinear restoring force function for an industrial translational guide for use with the Nonlinear Receptance Coupling Approach (NLRCA) to evaluate machine dynamic characteristics. Translational guides are among the most commonly used joints in machine tools. Both parametric and nonparametric techniques have been employed to identify the nonlinearities. A novel parametric model based on Hertzian contact mechanics has been derived for the translational guide. A nonparametric method based on two-dimensional Chebyshev polynomials is also used. The models derived from the two techniques, i.e., parametric and nonparametric, are fitted to the experimental data derived from static and dynamic tests to get the restoring force as a function of relative displacement and relative velocity across the joint. The nonlinear representation obtained from both techniques is later converted into the describing function representation which is needed for evaluation of machine dynamic characteristics using the NLRCA. The describing function representations obtained from the two approaches are compared. The design of experiments for evaluating the nonlinearities in such industrial machine tool joints is a challenge, requiring careful alignment and calibration, because they are typically very stiff. This constrains the dynamic experiments to be carried out at high frequencies (e.g. 2000—7000 Hz) where the experimental readings are very sensitive to errors in geometry and calibration.

A convenient technique for evaluating angular frequency in some nonlinear oscillations

In this paper, a convenient technique for evaluating angular frequency in some nonlinear oscillations is proposed. It is well known that once the restoring force function is given beforehand, the period of motion can be determined by an integral. The angular frequency has a relation with the period of motion as well as the integral. One makes a little modification for the integrand in the integral and a change of variable. It is found that if the three divisions are chosen on the integration interval and the trapezoid quadrature rule is used, a higher accurate result for the angular frequency can be achieved. For the restoring force being an odd function, three numerical examples are presented. The eardrum-type oscillation is studied as well. Higher accurate results for the angular frequency are obtained in all those examples.

Analysis of a kind of Duffing oscillator system used to detect weak signals

The stability of the periodic solution of the Duffing oscillator system in the periodic phase state is proved by using the Yoshizaw theorem, which establishes a theoretical basis for using this kind of chaotic oscillator system to detect weak signals. The restoring force term of the system affects the weak-signal detection ability of the system directly, the quantitative relationship between the coefficients of the linear and nonlinear items of the restoring force of the Duffing oscillator system and the SNR in the detection of weak signals is obtained through a large number of simulation experiments, then a new restoring force function with better detection results is established.

STIFFNESS NON-LINEARITY CLASSIFICATION THROUGH STRUCTURED RESPONSE COMPONENT ANALYSIS USING VOLTERRA SERIES

Most non-linear analysis problems, consider only the Duffing oscillator as a representative case. In engineering analysis, it is however, also important to recognise the type of non-linearity actually influencing the system. A procedure, involving structured higher-order FRF analysis based on Volterra theory is suggested in the present work, to distinguish a polynomial form of non-linearity from other possible forms. Volterra theory provides concepts of linear, bilinear, trilinear, etc. kernels, which upon convolution with the excitation force and subsequent summation can be employed to represent the response of a non-linear system. The kernels of the system are understood as multidimensional unit impulse response functions. The Volterra series response representation is employed in this work to facilitate its processing in a structured manner, to extract characteristic features, which can help in placing the system non-linearity in an appropriate class. The Volterra series platform is also employed to make a distinction between symmetric and asymmetric forms of the restoring force function. A multi-tone excitation procedure is further suggested, through which higher-order kernels of the system can be constructed for identification of the structure of the polynomial representing the restoring force. The procedures are illustrated through numerical simulation.