Green’s functions for a loaded rolling tyre

AbstractA new formulation to determine the unit impulse response (Green’s) functions of a loaded rotating tyre in the vehicle-fixed (Eulerian) reference frame for tyre/road noise predictions is presented. The proposed formulation makes use of the set of eigenfrequencies and eigenmodes for the statically loaded tyre obtained from a finite element (FE) model of the tyre. A closed-form expression for the Green’s functions of a rotating tyre in the Eulerian reference system as a function of the eigenfrequencies and eigenmodes of the statically loaded tyre is found. Non-linear effects during loading are accounted for in the FE model, while the frequency shift due to the rotational velocity is included in the calculation of the Green’s functions. In the literature on tyre/road noise these functions are generally used to determine the tyre response during tyre/road contact calculations. The presented formulation opens the possibility to solve the contact problem directly in the Eulerian reference frame and to include local tyre softening due to non-linear effects while keeping the computational advantage of describing the tyre dynamics as a set of impulse response functions. The advantage of obtaining the Green’s functions in the Eulerian reference system is that only the Green’s functions corresponding to the potential contact zone need to be determined, which significantly reduces the computational cost of solving the tyre/road contact and since the mesh is fixed in space, a finer mesh can be used for the potential contact zone, improving the accuracy of the contact force calculations. Although these effects might be less pronounced if a more accurate tyre model is used, it is found that using the Green’s functions of the loaded tyre in a contact force calculation leads to smaller forces than in the unloaded case, lower frequencies are present in the response and they decrease faster as the rotational velocity increases

Determining thermo-acoustic stability of a system whose boundary conditions are represented by strictly positive real transfer functions

The ultimate goal of the present research is to establish a methodology using which one can characterize the thermo-acoustic quality (figure of merit) of a given burner with flame. For this purpose, the probability of a certain burner/flame to be in either a stable or unstable regime when it is embedded in a randomly selected acoustic environment (similar to a combustor appliance) should be evaluated. An approach presented in this contribution consists of performing multiple calculations for the (in)-stability of a system composed of a known burner with flame and acoustically passive arbitrary upstream and downstream reflection coefficients. In this paper, a low order analytical network model of the acoustic system is used. Properties of strictly positive real functions are used to model the random frequency dependence of passive reflections. The implementation and testing of this particular method to generate random, frequency dependent acoustic embedding for the burner is the core subject of the present contribution. Within this method, initially, the roots of a Hurwitz polynomial are randomly selected and this polynomial is taken as the denominator of the impedance function, subsequently, the corresponding numerator polynomial coefficients are computed to obtain an impedance function that is strictly positive real. Then, this function is transformed to represent a reflection coefficient function in complex variable 𝑠𝑠 and is used as an embedding to evaluate the given burner’s flame stability by calculating the system’s complex eigen frequencies for various upstream and downstream reflection coefficients

Intrinsic thermo-acoustic instability criteria based on frequency response of flame transfer function

A study of Intrinsic Thermo-Acoustic (ITA) instability behavior of flames anchored to a burner deck is performed by introducing a mapping between the Flame Transfer Function, FTF(s), defined in the complex (Laplace) domain and the experimentally measured Flame Frequency Response, FFR(iω). The conventional approach requires a system identification procedure to obtain the FTF(s) from the measured FFR(iω). Next, root-finding techniques are applied to define the complex eigenfrequencies. The common practice is to fit the FTF(s) by a rational function that may lead to artifacts like spurious poles and zeros. The purpose of the present work is to establish instability criteria which are directly applicable in the frequency domain. The particular case is considered where the acoustic boundary conditions at both sides of the flame are anechoic. Therefore, the pure ITA mode is treated. First, the causality of the measured FFR(iω) is checked. Then, the criteria of the ITA mode instability applicable to the FFR(iω) phase and magnitude, are derived. Causality properties are used to find the unstable frequency, growth rate, and even the maximum possible value of the linear growth rate. In addition, a procedure is explained to reconstruct the flame transfer function in the complex plane s from the measured flame frequency response which could be an alternative method to study the FTF behavior in the complex domain instead of its estimation with a rational function