J F Dunne

A spectral method for estimation of non-linear system parameters from measured response

The problem of estimating the parameters in a randomly excited single degree of freedom non-linear system, from measurements of the response alone, is addressed. It is shown that, in the case where the bandwidth of the excitation is significantly greater than that of the response, it is possible to estimate the overall damping, as measured by the equivalent linear damping factor, and the linear and non-linear stiffness parameters, using a spectral technique. To separate the contributions from linear and non-linear damping terms a further estimation stage is required, based on an analysis of the probability distribution of the energy envelope. To validate the proposed estimation method it is applied to some simulated data, corresponding to both white noise and correlated noise inputs.

Stochastic estimation methods for non-linear ship roll motion

Abstract A method is developed for estimating the parameters in a single-degree-of-freedom non-linear equation of motion, suitable for representing the rolling motion of a ship in an irregular sea. It is assumed that the excitation is unmeasurable but can be modelled as a stochastic process. The estimation technique is thus based on the processing of a record of the roll motion alone. For the purpose of validating the method, it is applied to simulated data corresponding to both white noise and correlated noise inputs. The simulation results enable the accuracy of the estimated result, with respect both to bias and variability, to be assessed in a range of typical cases.

NARX Neural Network Modelling of Hydraulic Suspension Dampers for Steady-state and Variable Temperature Operation

Summary Two NARX-type neural networks are developed for modelling nonlinear dynamic characteristics of passive twin-tube hydraulic dampers used in vehicle suspension systems. Quasi-isothermal and variable temperature NARX models are rigorously tested and compared with a state-of-the-art physical model proposed by Duym and Reybrouck (1998) and Duym (2000). Measured damper data, generated under isothermal and temperature varying conditions, is used for NARX training, physical model calibration, and predictive comparisons. Test kinematics include high amplitude sinusoidal displacements up to 14 Hz, and realistic random road profiles. The NARX models are trained via ‘teacher forcing’ and the feedforward backpropagation algorithm using both ‘Early Stopping’ and Bayesian Regularisation. Stable network design is also examined using the minimum posterior prediction error as the criterion for selecting a good network from a small number of tests. Calibration of the physical model proves highly complicated owing to considerable nonlinearity-in-the-parameters, requiring use of Sequential Quadratic Programming with an implicitly nonlinear constraint. The paper shows that NARX neural network modelling is vastly superior in terms of calibration efficiency, and prediction times, whilst offering roughly similar, if not better, model accuracy.

Estimation of ship roll parameters in random waves

The problem of estimating the parameters in an equation of roll motion from roll measurements only, taken in an irregular sea, is discussed. A single degree of freedom equation of motion is assumed, with a wide-band stochastic input and with a linear-in-the-parameters representation of both the damping and restoration terms. A method based on the Markov property of the energy envelope process, associated with the roll motion, is developed which enables all the relevant parameters to be estimated. The method is validated by applying it to some simulated data, for which the true parameters are known.

A model for simulating the instantaneous crank kinematics and total mechanical losses in a multicylinder in-line engine

Abstract A two-degree-of-freedom dynamic model is constructed to simulate the instantaneous crank kinematics and total mechanical losses arising in a multicylinder gasoline engine coupled to a dynamometer. The simulation model is driven using specified cylinder gas pressures, and loaded by nominal brake torque and total friction losses. Existing semi-empirical torque loss models (based on calibrated single-cylinder diesel engine data) are used to account for the instantaneous friction losses in the piston-ring assembly, in bearings, and in auxiliaries. The model is specialized to the simulation of crank kinematics and matched brake torque for a three-cylinder in-line direct injection spark ignition (DISI) engine, without a gearbox. This allows the total friction loss to be separated from the brake torque for an engine not fitted with the very large number of sensors otherwise needed to calibrate analytical friction models. An equivalent simulation model is also constructed using GT-Crank, which excludes explicit reference to friction. In using both models to simulate steady state operation at a specified mean engine speed, the output torque is matched by iteration. The GT-Crank model necessarily compensates for internal losses by exaggerating the total output torque. Both simulation models are compared with measured crank kinematics and brake torque obtained from a dynamometer-loaded I3 DISI engine. The paper shows that by comparing the matched output torque from simulation with the measured output torque from the engine, the proposed model gives a very good high-speed prediction of the total mechanical losses. At low speed, the instantaneous model is still not accurate. It is also shown, however, that apart from the no-load condition, use of an average torque to compensate for friction (as in GT-Crank) is wholly acceptable for simulating instantaneous crank kinematics. This is the first reported instance of a simulation model (which includes the particular form of semi-empirical friction loading) being comprehensively compared and verified using multicylinder DISI engine data.

Multicylinder engine pressure reconstruction using NARX neural networks and crank kinematics

Abstract A NARX neural network is adapted for cylinder pressure trace reconstruction on a multicylinder engine. Following a systematic study to establish the required NARX input information (using measured pressure traces and simulated crank kinematics), two fully recurrent training algorithms are developed and applied to real engine data. These include a back-propagation-through-time algorithm (BPTT) and an extended Kalman filter (EKF). For multi-cylinder engines, two cases are examined, both assuming crank kinematics is obtained from a single shaft-encoder fitted at the forward end of the crankshaft. In one case, a NARX model is constructed to provide an inverse relationship between the kinematics at the encoder location and the pressure trace in an arbitrary cylinder. In the second case, by transforming the kinematics (to emulate a local encoder), a different NARX model is constructed to relate the kinematics at the crank location of a particular cylinder to the corresponding pressure trace. The accuracy and efficiency of both NARX models is examined for application to a three-cylinder in-line DISI engine (in which pressure traces are measured on all cylinders). The paper shows that the computational requirements of training are substantial and, although the efficiency of the EKF algorithm is better than the BPTT, the fitting accuracies are similarly good. For generalization, however (to unseen data), neither method is yet sufficiently accurate (even for steady state engine operation) unless substantially more training data are used to achieve the target accuracy of ± 4 per cent. The overall conclusion of the paper is that the NARX model has the correct architecture for multicylinder pressure reconstruction.

Efficient Extreme Value Prediction for Nonlinear Beam Vibrations Using Measured Random Response Histories

Predicted extreme exceedance probabilities associated withexperimental measurements of highly non-linear clamped-clamped beamvibrations driven by band-limited white-noise, are compared using twodifferent approaches for application to short data sets. The firstapproach uses response history measurements to calibrate a discretedynamic model using a Markov moment method appropriately matched toextreme value prediction via finite element solution of theFokker–Planck (FPK) equation. The dynamic model is obtained via theWoinowsky–Krieger equation with added empirical damping. Stationary FPKsolutions are used to obtain mean crossing rates, and for the purpose ofextreme value prediction, crossings are assumed to be independent. Thesecond approach uses a Weissman type I asymptotic estimator, justifiedby use of the Hasofer–Wang hypothesis test. Both methods are comparedwith exceedance probabilities obtained using data from ‘long’ experiments in which dependence between extreme values is excluded. Thepaper shows that by exploiting the Weissman estimator in a ‘forward’predictive mode, very accurate exceedance probabilities can be obtainedfrom relatively small amounts of measured data. The calibrated modelbased predictions are consistently in error as a result of non-linearcoupling effects not included in the model – this coupling isimplicitly accounted for in the Weissman predictions.

Parametric estimation for randomly excited non- linear systems

The problem of estimating unknown parameters in a non-linear randomly excited dynamic system, when the excitation is unmeasurable, is considered. It is shown that, if the excitation is modelled stochastically as a Gaussian process, with a prescribed spectral form, it is possible to estimate the parameters from response data alone using either moment equations or a spectral input-output relationship. When applied to simulated data for a particular non-linear oscillator, as an example, it is found that the use of moment equations leads to a very good estimation of the stiffness parameters but is incapable of yielding estimates of the absolute level of damping. However the latter can be found accurately by applying a spectral relationship. Improvements in the accuracy of estimation for the damping parameters, and the input intensity, are achieved by using a theoretical expression for the distribution of the energy envelope of the response in combination with statistical linearisation.

A note on the accuracy of stochastic averaging applied to non-linear oscillator response prediction

Abstract Stationary response statistics derived from a non-linear single-degree-of-freedom oscillator driven by broadband noise are obtained by using the stochastic averaging method. A fully numerical implementation of the method is used to approximate large-amplitude mean threshold crossing intervals, and comparisons with time domain simulations are presented. By uniformly increasing the level of non-linear damping both the accuracy and the limitations of the method are demonstrated.

A review of evaporative cooling system concepts for engine thermal management in motor vehicles

The evaporative cooling system concepts proposed over the past century for engine thermal management in automotive applications are examined and critically reviewed. The purposes of this review are to establish the evident system shortcomings and to identify the remaining research questions that need to be addressed to enable this important technology to be adopted by vehicle manufacturers. Initially, the benefits of the evaporative cooling systems are restated in terms of the improved engine efficiency, the reduced carbon dioxide emissions and the improved fuel economy. This is followed by a historical coverage of the proposed concepts dating back to 1918. Possible evaporative cooling concepts are then classified into four distinct classes and critically reviewed. This culminates in an assessment of the available evidence to establish the reasons why no system has yet been approved for serial production commercially. Then, by systematic examination of the critical areas in evaporative cooling systems for application to automotive engine cooling, the remaining research challenges are identified.