Arbitrary Forcing Functions Research Articles

Nonlinear Dynamics of Swinging Clapper Bells under Arbitrary or Resonant Forcing Functions

Study of swinging clapper bells involves aspects encompassing sound and acoustic engineering, mechanical engineering, and structural engineering. From the musical point of view, clapper bells are directly played idiophone instruments, where the playing device, the clapper, although directly excited, is not explicitly controlled by the bell ringer. The achievement of a clear and optimal sound mainly depends on the acoustic characteristics of the bell and on the regularity of the clapper strokes, which is not only governed by the ringing style and the relevant parameters of clapper and bell but also by the real time corrections to the excitation introduced by trained bell ringers. In fact, despite centuries of experience allowed to optimize the bell performances, standardizing proportions and mounting arrangements, effective sound control requires some fine tuning of the forcing function. Another crucial topic, especially in view of assessing existing structures, regards the evaluation of time histories of the actions transmitted by the bell to the pivots and the study of the interactions between the bell and the supporting structures, belfries, and bell-towers. “Ringability” of swinging bells and bell-structure interactions are usually tackled in the framework of rigid body dynamics, so arriving at an initial value problem, governed by a system of two second order nonlinear ordinary differential equations (ODEs), whose solutions are piecewise-defined functions. In the relevant literature, numerical solutions of the system are commonly sought using built-in algorithms provided in advanced software packages; since the use of such general algorithms is subject to some restrictions, especially regarding the forcing functions, validity of the results is often limited. The present study focuses on an innovative procedure to solve the equations of motion. The method, extremely fast and effective, is based on original numerical explicit-implicit predictor-corrector integration algorithms with constant time step, duly validated reproducing the outcomes of relevant reference case studies. Each time the clapper strikes the bell a new “piece” of the solution is initialized, so avoiding user interventions in the elaboration phase. Independently on the oscillation amplitude and on the duration of the considered time interval, the algorithms can successfully manage undamped oscillations; friction and viscosity damped oscillations; free oscillations in transient and stationary phases; and can be applied also to solve stiff equations. Furthermore, the capability of the proposed methods to deal with arbitrary forcing functions is particularly innovative. The outcomes of relevant case studies, regarding the oscillations of the old tenor bell of the Great St. Mary church in Cambridge, confirm the potentialities of the method, also highlighting some topical issues, involving, for example, the assessment of damping equivalence. Finally, a pioneering feature of the algorithms is their ability to handle and to define “resonant” forcing functions, continuously tuning the frequency of the excitation to the natural frequency of the oscillation, according to the oscillation amplitude.

Analysis of pressure diffusion-wave fields in matrix-fracture media using Green functions of frequency domain

A Green function-based analytic solution of pressure diffusive wave motion is introduced for investigating the transient pressure response in fluid saturated matrix-fracture media. One-dimensional solutions are presented for a time-harmonically forced problem with internal damping and are used to analyze the propagation and attenuation of pressure pulse in a semi-infinite spatial domain. The concise form of the solutions simplifies the calculation of pressure diffusion with arbitrary forcing functions at fixed boundaries. It indicates that the periodically forced function with internal damping has remarkable effects on pressure diffusion-wave motions. It is found that the characteristic delay-frequency separates the pressure diffusion-wave domain into matrix-dominated, transition, and fracture-dominated zones. The Green functions could physically predict any transient response of pressure fluctuations due to hydro-fracturing in geological reservoirs given proper physical parameters.

Approximate Analytical Solution to Diurnal Atmospheric Boundary-Layer Growth Under Well-Watered Conditions

Simplified numerical models of the atmospheric boundary layer (ABL) are useful both for understanding the underlying dynamics and potentially providing parsimonious modelling approaches for inclusion in larger models. Herein the governing equations of a simplified slab model of the uniformly mixed, purely convective, diurnal ABL are shown to allow immediate solutions for the potential temperature and specific humidity as functions of the ABL height and net radiation when expressed in integral form. By employing a linearized saturation vapour relation, the height of the mixed layer is shown to obey a non-linear ordinary differential equation with quadratic dependence on ABL height. A perturbation solution provides general analytical approximations, of which the leading term is shown to represent the contribution under equilibrium evaporation. These solutions allow the diurnal evolution of the height, potential temperature, and specific humidity (i.e., also vapour pressure deficit) of the mixed layer to be expressed analytically for arbitrary radiative forcing functions.

Impact-induced vibration in complex flexible multi-body systems: numerical and experimental investigation of press machines

This article is concerned with the numerical and experimental investigations of impact-induced vibrations in highly non-linear complex multibody system applications. In order to accurately model the impact response of such complex systems, a computational procedure based on the integration of finite element and multibody system algorithms is used. This procedure, which employs the finite element floating frame of reference formulation, allows for defining arbitrary joint constraint and forcing functions, for accurately capturing the coupling between the rigid body motion and the elastic deformation, and for modeling the impact forces between rigid and flexible bodies in the multibody system model. The system equations of motion are developed using the augmented Lagrangian formulation that allows for the use of sparse matrix techniques. The algorithm used in this study ensures that the kinematic constraint equations are satisfied at the position, velocity, and acceleration levels. The impact-induced vibration of a highly non-linear press machine is numerically and experimentally examined in this study. In order to accurately model the dynamic behavior of such a press machine, it is necessary to model the transfer feeder elastic deformations resulting from the impulsive impact forces. The deformations of the transfer press system as well as those of its frame are modeled in this investigation using the floating frame of reference formulation that allows for effectively reducing the number of degrees of freedom by filtering out very high-frequency modes of vibration using component mode synthesis techniques. The formulations of several relative angle and angular velocity kinematic constraints that are required to model this multibody system are developed in order to model transfer feeder complex rotational components. The numerical results obtained using the three-dimensional multibody system transfer press model developed in this study are validated using experimental results.

Fourier spectral embedded boundary solution of the Poisson’s and Laplace equations with Dirichlet boundary conditions

A Fourier spectral embedded boundary method, for solution of the Poisson’s equation with Dirichlet boundary conditions and arbitrary forcing functions (including zero forcing function), is presented in this paper. This iterative method begins by transformation of the Dirichlet boundary conditions from the physical boundaries to some corresponding regular grid points (which are called the numerical boundaries), using a second order interpolation method. Then the transformed boundary conditions and the forcing function are extended to a square, smoothly and periodically, via multiplying them by some suitable error functions. Instead of direct solution of the resulting extended Poisson’s problem, it is suggested to define and solve an equivalent transient diffusion problem on the regular domain, until achievement of the steady solution (which is considered as the solution of the original problem). Without need of any numerical time integration method, time advancement of the solution is obtained directly, from the exact solution of the transient problem in the Fourier space. Consequently, timestep sizes can be chosen without stability limitations, which it means higher rates of convergence in comparison with the classical relaxation methods. The method is presented in details for one- and two-dimensional problems, and a new emerged phenomenon (which is called the saturation state) is illustrated both in the physical and spectral spaces. The numerical experiments have been performed on the one- and two-dimensional irregular domains to show the accuracy of the method and its superiority (from the rate of convergence viewpoint) to the other classical relaxation methods. Capability of the method, in dealing with complex geometries, and in presence of discontinuity at the boundaries, has been shown via some numerical experiments on a four-leaf shape geometry.

Orthogonality condition for a multi-span beam, and its application to transient vibration of a two-span beam

This paper treats the orthogonality condition for a multi-span beam, and its application to forced (transient) vibration of a two-span beam. The beam is modeled as a Bernoulli–Euler beam. The boundary conditions for the particular case of two-span beam are clamped–pinned–pinned. An exact closed-form solution is obtained for this problem. Even though there has been an enormous amount of work on beam vibration, most of the studies are conducted on a single-span beam. There are some studies on the multi-span beam vibration. However, their treatment is rather specialized in terms of the applied loading and the initial conditions. None of the studies in the past treats an exact solution for a forced (transient) vibration of a general two-span beam with arbitrary initial conditions and arbitrary forcing functions. Therefore, the solution obtained in this paper is new. The key development in the solution is the orthogonality condition for a multi-span beam. The method of solution developed in this paper establishes a general methodology for the forced (transient) vibration of a multi-span beam. The closed-form solution obtained in this paper can be used as a benchmark solution for the transient vibration of a two-span beam.

Matrix Formulation for Minimum Response of Undamped Structure

A useful method for the determination of frequencies that correspond to points of local minima in the frequency response of finite element structure is presented. The method is based on the application of linear matrix algebra methods in conjunction with the fundamental definitions for existence of local minima. The procedure was initially developed for determination of antiresonant frequencies, and is now extended for application to minimum response. The derivations of the procedures for antiresonance and minimum response are both included in this paper. A significant advantage of this method is that it allows for the formulation of sensitivity relationships similar to what is commonly used for resonant frequency. Thus, there is potential for expanded use of minimum response information for structural optimization, damage detection, and other related tasks. In addition, generality in the formulation is retained such that the response may be defined as any linear combination of the coordinate responses, and any arbitrary forcing function may be considered. A numerical example, with a simple one-dimensional spring-mass system, is presented to provide for illustration and validation of the approach.

Two-point boundary value problems, Green's functions, and product integration

When the Green's function for a two-point boundary value problem can be found, the solution for any forcing term reduces to a quadrature. Here we investigate using this as the basis for numerical schemes for boundary value problems and parabolic initial-boundary value problems. Application of Gaussian quadrature is disappointing, only converging slowly due to the discontinuous derivative of the Green's functions; however, rapid convergence is recovered by the use of product integration; that is, the prior accurate evaluation of Green's integral for a set of basis functions and the subsequent evaluation for arbitrary forcing functions by a matrix-vector multiplication. The method requires more operations than finite differencing for the same number of nodes, but converges far more rapidly. For small numbers of nodes, the performance is similar to that for orthogonal collocation; however, at high resolution, collocation accuracy deteriorates due to ill-conditioning whereas the present method is stable.

Seismic analysis of structures with a fractional derivative model of viscoelastic dampers

Viscoelastic dampers are now among some of the preferred energy dissipation devices used for passive seismic response control. To evaluate the performance of structures installed with viscoelastic dampers, different analytical models have been used to characterize their dynamic force deformation characteristics. The fractional derivative models have received favorable attention as they can capture the frequency dependence of the material stiffness and damping properties observed in the tests very well. However, accurate analytical procedures are needed to calculate the response of structures with such damper models. This paper presents a modal analysis approach, similar to that used for the analysis of linear systems, for solving the equations of motion with fractional derivative terms for arbitrary forcing functions such as those caused by earthquake induced ground motions. The uncoupled modal equations still have fractional derivatives, but can be solved by numerical or analytical procedures. Both numerical and analytical procedures are formulated. These procedures are then used to calculate the dynamic response of a multi-degree of freedom shear beam structure excited by ground motions. Numerical results demonstrating the response reducing effect of viscoelastic dampers are also presented.

Accurate and stable electromagnetic transient simulation using root-matching techniques

The method of substituting the trapezoidal integrator, developed by Dommel, is generally applied in electromagnetic transient simulations. The trapezoidal rule is based on a truncated Taylor series and therefore contains truncation errors. These truncation errors cause numerical oscillations when the time step is large relative to some of the time constants in the network. Various techniques have been developed to reduce the numerical oscillations, each with strengths and drawbacks. In this paper an exponential form of the difference equation is developed, using root-matching techniques, that eliminates the truncation error and provides a highly efficient and accurate time domain regardless of the time step used. This exponential form of the difference equation generates a solution at each time point that is exact for the step response and a good approximation for an arbitrary forcing function. The exponential form is compatible with Dommel's method.

Teaching Vibration Analysis Using Symbolic Computation Software

This paper presents a general approach to modelling and learning vibration analysis for simple beams using symbolic computation software. The emphasis here is on the fact that a complete solution of the beam vibrations problem, for different boundary conditions and arbitrary forcing functions, is made very simple by using symbolic computation software. The heart of the procedure is to convert the beam vibration problem into a system of simultaneous algebraic linear equations and then use symbolic computation software to solve it. The analysis here also shows how to obtain models suitable to design controllers for flexible systems.

Electromagnetic transient simulation of power systems using root-matching techniques

For electromagnetic transient simulations the method of substituting the trapezoidal integrator, developed by Dommel, is generally applied. The trapezoidal rule is based on a truncated Taylor series and therefore contains truncation errors. These truncation errors cause numerical oscillations when the time step is large relative to some of the time constants in the network. This leaves the user unsure whether observed oscillations are numerical or true artefacts of the system simulated. Various techniques have been developed to reduce the numerical oscillations, each with strengths and drawbacks. An exponential form of the difference equation is developed, using root-matching techniques and demonstrated on a simplified power system. This technique eliminates the truncation error and provides highly efficient and accurate time domain regardless of the time step used. This exponential form of difference equation generates a solution at each time point that is exact for the step response and a good approximation for an arbitrary forcing function.

Experimental Analysis of Impact-Damped Flexible Beams

Statistical analysis of the phenomenon of impact damping is performed on 60 steady-state vibration tests of a flexible beam to relate the system's modal damping to system parameters. Compared with other measures of effectiveness of the impact damper, modal damping ratios are useful in predicting the response of continuous systems under arbitrary forcing functions. Contour plots show that the modal damping ratio is nonlinearly dependent on the damper's mass and gap. However, these are not the only parameters that control the effectiveness of the impact damper. Multiple nonlinear regression analysis of the data is used to test the dependence of the system's modal damping on a number of parameters. It is hypothesized that the system's parameters are the mass and gap of the impact damper, the excitation frequency, the peak of the frequency response function, and the modal amplitude at the location of the impact damper. Multiple nonlinear regression analysis shows the validity of the hypothesis and the ability of the derived formula to predict the modal damping ratio of the beam.

NUMERICAL SOLUTIONS OF FORCED VIBRATION AND WHIRLING OF A NON-LINEAR STRING USING THE THEORY OF A COSSERAT POINT

O'Reilly and Holmes [1] performed experiments and analysis of non-linear motion of a string subjected to vertical oscillation of one end. Their analysis suggested that differences between theoretical and experimental results were mainly due to uncertainty in the assumed form for the forcing function that was used to satisfy the forced boundary condition. Here, a numerical solution of the non-linear string problem is considered which incorporates the forced boundary condition directly and thus avoids the need to assume an arbitrary forcing function. Specifically, the numerical formulation of string problems based on the theory of a Cosserat point is used. It is shown that the director inertia coefficient in the Cosserat theory can be chosen so that the theory predicts the first modal frequency of small deformation free vibration exactly for each level of discretizational. Also, examples of forced in-plane motion and free non-linear whirling motion are considered to validate the numerical model. The resulting simulations indicate that the forcing amplitude for the onset of persistent whirling and aperiodic response is about five times smaller than that observed in the experiments even when the uncertainty in the forcing function is removed from the analysis. Additional attempts to bridge the gap between theory and experiment suggest that there is still some mechanism in the experiment that is not properly modelled.

Evaluation of Time Integration Methods. 3rd Report. Error of Forced Vibration Solution to the Input of Arbitray Forcing Functions.

The authors have been conducting research on the evaluation of errors caused by time integration methods. They proposed, in the previous paper, a method of evaluating errors in the solution of time integration methods when the harmonic input is applied to the first-or second-order system. The method is extended, in this paper, to the case where an arbitrary forcing function is applied to the system. Based on this method, a conventional method to evaluate the error in the numerical solution of forced vibration is also proposed in this paper.

An orthogonal modes approach to fluid-loaded elastic structures with arbitrary forcing functions

A numerical scheme is presented wherein the structural equations for an elastic structure are coupled to the acoustic radiation equations and recast in canonical form. The acoustic impedance matrix, which may be found via boundary element methods or by the superposition method, is expanded in a power series on angular frequency prior to coupling it with the structural matrices. The resulting system of equations is then decomposed to find an orthogonal basis set which uncouples the original equations. Once the basis set has been determined, the structural response and acoustic radiation spectra may be easily reconstructed for any arbitrary forcing function on the structure. Results will be given for two water-loaded examples, a finite plate in an infinite rigid baffle and a spherical shell.

Linear Hyperbolic Model for Alluvial Channels

The exact linear solutions for the sediment transport and bed form evolution for one‐dimensional sediment‐water two‐phase motion have been obtained using the St. Venant shallow‐water equations with the assumption of quasisteady flow. These solutions are applicable to alluvial channels of infinite length, initially at equilibrium followed by an arbitrary forcing function of either sediment transport or bed elevation imposed as an upstream boundary condition. The solutions have been used to predict aggradation in a channel due to constant overloading. Comparison of the results with available experimental data and with the solution obtained from a parabolic model is satisfactory. The present theory is significant conceptually since it provides valuable insight into the physical phenomenon as well as into the mathematical behavior of the solutions.

Quick algorithms for computing either displacement, velocity or acceleration of an oscillator

AbstractAccurate and efficient algorithms are derived for computing any desired response quantity of a single‐degree‐of‐freedom linear oscillator subjected to an arbitrary forcing function. These algorithms are based on linear interpolation of the excitation between discrete points sampled uniformly in time. In the special case of the undamped oscillator, the displacement algorithm is equivalent to a previously published algorithm for this situation which was derived using a finite element approach. However, contrary to the claim made by the authors of this latter algorithm, it does not give an exact solution of the equation of undamped motion for an arbitrary forcing function.

Resonance sound transmission into arbitrarily loaded submerged cylindrical shells

We develop the fundamental exact analytical and computational model required to study the transmission of incident plane sound waves into submerged elastic cylindrical shells subjected to arbitrary forcing functions on their outer surface. We use the superposition principle in this linear problem to separate the contributions to the internally transmitted field caused by the incident wave from that of the surface excitation. This basic model uses the exact (2-D) formulation of elastodynamics to describe the shell motions, and that of general linear acoustics to describe the wave motion in the inner and outer fluids. We display the isobaric contours of the internally transmitted pressure fields, exhibiting their caustics and their progressive development as the frequency is increased within the band 0≤k3a≤10. The contour plots are generated for all possible combinations of forcing functions and insonification conditions with a view toward computing an advantage ratio PR which measures the gains that would result from sensing the internally transmitted field, rather than the field external to the shell at its surface. This advantage ratio is shown always to be greater than unity, and in many cases, much greater than unity. The effects of two basically different types of loads are analyzed via internal isobaric plots, and also by means of spectral plots computed at selected fixed points inside the shell. We observe an encouraging overall permanence of the caustic locations, at fixed frequencies, as the types of surface loads are varied. The present investigation of sound transmission into arbitrarily surface-loaded shells clearly demonstrates the filtering behavior of the shell in the frequency domain, and its focusing action in space. Hence, for moderate surface loads, the air-filled submerged shell analyzed here acts as a very effective sound concentrator, particularly near the resonances of its monopole mode.

A comparison of lumped and distributed solutions to nonlinear, transient, magnetic field problems

The transient magnetization of highly nonlinear materials is investigated using finite-element techniques and a multimode equivalent circuit approach. The use of constant eddy inductances in the equivalent circuit is shown to be correct only for linear materials or for small field perturbations in nonlinear materials. During large perturbations, the nonuniform permeability of nonlinear materials causes standard equivalent circuit techniques to be grossly inaccurate. Using numerical results, a field-dependent nonlinear eddy inductance has been defined that substantially increases the accuracy of nonlinear lumped analyses and is applicable to arbitrary forcing functions. This nonlinear inductance is a function of the differential permeability of the material surface and of a bulk differential permeability calculated from the average flux density in the material. To verify the accuracy of this approach, the transient total flux predicted by an equivalent circuit analysis incorporating nonlinear inductances is compared to the total flux from a transient finite-element analysis. Comparisons are given for the total flux versus time following the step application and the step removal of a surface field to an infinite flat plate and a long solid cylinder. The response of a toroidal coil with a saturable core to an instantaneous voltage change is also investigated using lumped and distributed techniques.