J.K. Hammond

Parameter estimation for hysteretic systems

Abstract The differential system characterization of hysteretic system is well known. The problem of estimating the parameters of this system on the basis of input-output data, possibly noise corrupted, is considered. It turns out that the estimation problem is a non-linear optimization problem. The Gauss Newton method is used in setting up a two-stage iterative least squares algorithm. The usefulness of the algorithm is validated through its application to various simulated time histories from the hysteretic model.

On the response of single and multidegree of freedom systems to non-stationary random excitations

Abstract The early part of the paper indicates two current theoretical spectral methods of dealing with non-stationary processes. Then, by using Priestleys evolutionary spectral density, the non-stationary response of single and multimode systems to stationary excitations is derived. Finally, by constructing a non-stationary evolutionary spectral function, the corresponding non-stationary response of linear systems is investigated.

Evolutionary (frequency/time) spectral analysis of the response of vehicles moving on rough ground by using “covariance equivalent” modelling

Abstract In a number of physical problems such as the motion of vehicles travelling over rough ground or the noise emanating from a moving source, non-stationarity is induced by a non-linear time dilation (due to velocity variations or Doppler effects) of the source or excitation process so that the resulting process is “frequency modulated”. This is true even if the underlying process is homogeneous in another domain. Hitherto it has not been possible to apply the frequency/time analysis due to Priestley [1] to this class of problem but here, by introducing the concept termed “covariance equivalence” by the authors, this method can be seen to apply. An example of a vehicle moving with variable velocity on a rough surface is considered.

Approximate, time domain, non-stationary analysis of stochastically excited, non-linear systems with particular reference to the motion of vehicles on rough ground

Abstract An approximate state-space method for obtaining the time varying mean and covariance of non-linear systems excited by non-stationary random processes is presented. In particular the class of non-stationarity associated with the motion of a vehicle on rough ground (i.e., the process is “frequency modulated” as a result of the vehicles variable velocity) is of interest. The method is based on a technique of modelling the input process as a “shaping filter” in the spatial domain which may be linked to the vehicle dynamic equations through the velocity function. The non-linear problem is overcome by using the technique of statistical linearization. An example is briefly discussed.

Approximate eigenfunction analysis of first order non-linear systems with application to a cubic system

For stochastic systems excited by white Gaussian noise, the transition probability density is the solution to the Fokker—Planck—Kolmogorov (FPK) equation. If the system is non-linear the resulting FPK equation can be solved exactly only in a few special cases. Here an approximate expression is developed for the transition probability density function of a class of non-linear systems. By using this approximation the autocovariance and spectral density function for a certain first order system are obtained. The approximate results so obtained are compared with those obtained by direct digital simulation.

The use of signal envelopes to describe the bandlimited response of a class of systems and to define an alternative shock spectrum

A method for describing the behaviour of a class of linear, modally dense systems is given. The class of systems is such that each member of it responds to a particular input in a slightly different way. This means that while the gross features of the response may change little from member to member, the detailed response may be different. The class of systems is characterized in terms of the average bandlimited impulse response envelope. This is achieved by considering the natural frequencies, the amplitudes and relative phases of the modes in a particular frequency band to be random variables with known probability density functions. The statistics of the bandlimited impulse response envelope are discussed and an upper limit, in addition to the expression for the average bandlimited impulse response envelope, is derived. By generalizing this to different input signals an upper bound for the response of a system in a frequency band is generated from the input and impulse response envelopes in that band. Using this and the statistical description of the bandlimited impulse response envelope yields an upper bound for the response of a class of systems. The relationship between these envelope techniques and shock spectra is discussed and an alternative procedure for shock spectra generation is described that takes into account the characteristics of the system under test.