Non-linear response of a gyrostabilized platform to transient input

In this paper the response of a gyrostabilized platform subjected to a transient torque has been analyzed by deliberately introducing non-linearity into the command of the servomotor. The resulting third-order non-linear differential equation has been solved by using a transformation technique involving the displacement variable. The condition under which platform oscillations may grow with time or die with time are important from the point of view of platform stabilization. The effect of deliberate addition of non-linearity with a view to achieving the ideal response—that is, to bring the platform back to its equilibrium position with as few oscillations as possible—has been investigated. The conditions under which instability may set in on account of the small transient input and small non-linearity has also been discussed. The analysis is illustrated by means of a numerical example. The results of analysis are compared with numerical solutions obtained on a digital computer.

Non-linear response of soil N2O emissions to nitrogen fertiliser in a cotton-fallow rotation in sub-tropical Australia

Nitrogen fertiliser is a major source of atmospheric N2O and over recent years there is growing evidence for a non-linear, exponential relationship between N fertiliser application rate and N2O emissions. However, there is still high uncertainty around the relationship of N fertiliser rate and N2O emissions for many cropping systems. We conducted year-round measurements of N2O emission and lint yield in four N rate treatments (0, 90, 180 and 270 kg N ha-1) in a cotton-fallow rotation on a black vertosol in Australia. We observed a nonlinear exponential response of N2O emissions to increasing N fertiliser rates with cumulative annual N2O emissions of 0.55 kg N ha-1, 0.67kg N ha-1, 1.07 kg N ha-1 and 1.89 kg N ha-1 for the four respective N fertiliser rates while no N response to yield occurred above 180N. The N fertiliser induced annual N2O EF factors increased from 0.13% to 0.29% and 0.50% for the 90N, 180N and 270N treatments respectively, significantly lower than the IPCC Tier 1 default value (1.0 %). This non-linear response suggests that an exponential N2O emissions model may be more appropriate for use in estimating emission of N2O from soils cultivated to cotton in Australia. It also demonstrates that improved agricultural N management practices can be adopted in cotton to substantially reduce N2O emissions without affecting yield potential.

The non-linear response of a gyrostabilized platform: Effects of correction torque time delay and of random inputs

First, the non-linear response of a gyrostabilized platform to a small constant input torque is analyzed in respect to the effect of the time delay (inherent or deliberately introduced) in the correction torque supplied by the servomotor, which itself may be non-linear to a certain extent. The equation of motion of the platform system is a third order nonlinear non-homogeneous differential equation. An approximate analytical method of solution of this equation is utilized. The value of the delay at which the platform response becomes unstable has been calculated by using this approximate analytical method. The procedure is illustrated by means of a numerical example. Second, the non-linear response of the platform to a random input has been obtained. The effects of several types of non-linearity on reducing the level of the mean square response have been investigated, by applying the technique of equivalent linearization and solving the resulting integral equations by using laguerre or Gaussian integration techniques. The mean square responses to white noise and band limited white noise, for various values of the non-linear parameter and for different types of non-linearity function, have been obtained. For positive values of the non-linear parameter the levels of the non-linear mean square responses to both white noise and band-limited white noise are low as compared to the linear mean square response. For negative values of the non-linear parameter the level of the non-linear mean square response at first increases slowly with increasing values of the non-linear parameter and then suddenly jumps to a high level, at a certain value of the non-linearity parameter.

The non-linear response of a gyrostabilized platform: Effects of correction torque time delay and of random inputs

First, the non-linear response of a gyrostabilized platform to a small constant input torque is analyzed in respect to the effect of the time delay (inherent or deliberately introduced) in the correction torque supplied by the servomotor, which itself may be non-linear to a certain extent. The equation of motion of the platform system is a third order nonlinear non-homogeneous differential equation. An approximate analytical method of solution of this equation is utilized. The value of the delay at which the platform response becomes unstable has been calculated by using this approximate analytical method. The procedure is illustrated by means of a numerical example. Second, the non-linear response of the platform to a random input has been obtained. The effects of several types of non-linearity on reducing the level of the mean square response have been investigated, by applying the technique of equivalent linearization and solving the resulting integral equations by using laguerre or Gaussian integration techniques. The mean square responses to white noise and band limited white noise, for various values of the non-linear parameter and for different types of non-linearity function, have been obtained. For positive values of the non-linear parameter the levels of the non-linear mean square responses to both white noise and band-limited white noise are low as compared to the linear mean square response. For negative values of the non-linear parameter the level of the non-linear mean square response at first increases slowly with increasing values of the non-linear parameter and then suddenly jumps to a high level, at a certain value of the non-linearity parameter.

Non linear response of planar asymmetric systems

The seismic behaviour of one-storey asymmetric structures has been studied since 1970s by a number of researches studies which identified the coupled nature of the translational-to-torsional response of those class of systems leading to severe displacement magnifications at the perimeter frames and therefore to significant increase of local peak seismic demand to the structural elements with respect to those of equivalent not-eccentric systems (Kan and Chopra 1987). These studies identified the fundamental parameters (such as the fundamental period TL normalized eccentricity e and the torsional-to-lateral frequency ratio Ωϑ) governing the torsional behavior of in-plan asymmetric structures and trends of behavior. It has been clearly recognized that asymmetric structures characterized by Ωϑ >1, referred to as torsionally-stiff systems, behave quite different form structures with Ωϑ <1, referred to as torsionally-flexible systems. Previous research works by some of the authors proposed a simple closed-form estimation of the maximum torsional response of one-storey elastic systems (Trombetti et al. 2005 and Palermo et al. 2010) leading to the so called “Alpha-method” for the evaluation of the displacement magnification factors at the corner sides. The present paper provides an upgrade of the “Alpha Method” removing the assumption of linear elastic response of the system. The main objective is to evaluate how the excursion of the structural elements in the inelastic field (due to the reaching of yield strength) affects the displacement demand of one-storey in-plan asymmetric structures. The system proposed by Chopra and Goel in 2007, which is claimed to be able to capture the main features of the non-linear response of in-plan asymmetric system, is used to perform a large parametric analysis varying all the fundamental parameters of the system, including the inelastic demand by varying the force reduction factor from 2 to 5. Magnification factors for different force reduction factor are proposed and comparisons with the results obtained from linear analysis are provided.

Non-linear resonance in strings under narrow band random excitation, part I: Planar response and stability

Non-linear planar response of a string to planar narrow band random excitation is investigated in this paper. A response equation for the mean square deflection σ2 is obtained under a single mode approximation by using the equivalent linearization technique. It is shown that the response is triple valued, as in the case of harmonic excitation, if the centre frequency of excitation Ω lies in a certain specified range. The triple valued response occurs only if the excitation bandwidth β is smaller than a critical value βcrit which is a monotonically increasing function of the intensity of excitation. An approximate method of investigating the almost sure asymptotic stability of the solution is presented and regions of instability in the Ω-σ2 plane have been charted. It is shown that planar response can become unstable either due to an unbounded growth of the in-plane component of motion or due to a spontaneous appearance of an out-of-plane component.

Transient response of coupled non-linear non-conservative systems

This paper deals with an approximate method of analysis of non-linear, non-conservative systems of two degrees of freedom. The approximate equations for amplitude and phase are obtained by a generalized averaging technique based on the ultraspherical polynomial approximation. The method is illustrated by an example of a spring-mass-damper system.

The pulse response of non-linear third order systems

The response of a third order non-linear system subjected to a pulse excitation is analysed. A transformation of the displacement variable is effected. The transformation function chosen is the solution of the linear problem subjected to the same pulse. With this transformation the equation of motion is brought into a form in which the method of variation of parameters is applicable for the solution of the problem. The method is applied to a single axis gyrostabilized platform subjected to an exponentially decaying pulse. The analytical results are compared with digital and analog computer solutions.

Approximate methods for step function response of undamped non-linear spring mass systems with arbitrary hardening type spring characteristic

The possibility of applying two approximate methods for determining the salient features of response of undamped non-linear spring mass systems subjected to a step input, is examined. The results obtained on the basis of these approximate methods are compared with the exact results that are available for some particular types of spring characteristics. The extension of the approximate methods for non-linear systems with general polynomial restoring force characteristics is indicated.

Response spectrum of non-linear spring mass system subjected to transient disturbances

The transient response spectrum of a cubic spring mass system subjected to a step function input is obtained. An approximate method is adopted where non-linear restoring force characteristic is replaced by two linear segments, so that the mean square error between them is a minimum. The effect of viscous damping on the peak response is also discussed for various values of the damping constant and the non-linearity restoring force parameter.

Step function response of non-linear spring mass systems in the presence of Coulomb damping

The transient response of non-linear spring mass systems with Coulomb damping, when subjected to a step function is investigated. For a restricted class of non-linear spring characteristics, exact expressions are developed for (i) the first peak of the response curves, and (ii) the time taken to reach it. A simple, yet accurate linearization procedure is developed for obtaining the approximate time required to reach the first peak, when the spring characteristic is a general function of the displacement. The results are presented graphically in non-dimensional form.