Spacecraft detumbling through energy dissipation

The attitude motion of a tumbling, rigid, axisymmetric spacecraft is considered. A methodology for detumbling the spacecraft through energy dissipation is presented. The differential equations governing this motion are stiff, and therefore an approximate solution, based on the variation of constants method, is developed and utilized in the analysis of the detumbling strategy. Stability of the detumbling process is also addressed

ADAMS model validation for an all terrain vehicle using test track data

MD ADAMS R is widely used for vehicle suspension modeling. In this paper we present modeling, simulation, and test track evaluation of an all terrain recreational vehicle. Our intention is to study the degree to which simplified ADAMS modeling actually matches human-driven vehicle response. For suspension model validation, a vehicle is generally tested on a four-post test rig and base-excitation is applied at four ground-wheel contacts. However, actual driving experience does not match idealized testing conditions. In this work the vehicle is manually driven on a variety of tracks at different speeds, and the vertical accelerations at four axle locations and four body points are measured. The same are then compared in detail against predictions from ADAMS simulation with vertical base excitation. The contribution of this paper is in identifying those aspects of the simulation results that match experiments well, and identifying possible sources for the observed mismatch, especially under more severe test conditions

Semi-implicit Integration and Data-Driven Model Order Reduction in Structural Dynamics with Hysteresis

Structural damping is known to be approximately rate-independent in many cases. Popular models for rate-independent dissipation are hysteresis models; and a highly popular hysteresis model is the Bouc-Wen model. If such hysteretic dissipation is incorporated in a refined finite element model, then the mathematical model includes the usual structural dynamics equations along with nonlinear nonsmooth ordinary differential equations for a large number of internal hysteretic states at Gauss points, to be used within the virtual work calculation for dissipation. For such systems, numerical integration becomes difficult due to both the distributed non-analytic nonlinearity of hysteresis as well as the very high natural frequencies in the finite element model. Here we offer two contributions. First, we present a simple semi-implicit integration approach where the structural part is handled implicitly based on the work of Pich\'e, and where the hysteretic part is handled explicitly. A cantilever beam example is solved in detail using high mesh refinement. Convergence is good for lower damping and a smoother hysteresis loop. For a less smooth hysteresis loop and/or higher damping, convergence is observed to be roughly linear on average. Encouragingly, the time step needed for stability is much larger than the time period of the highest natural frequency of the structural model. Subsequently, data from several simulations conducted using the above semi-implicit method are used to construct reduced order models of the system, where the structural dynamics is projected onto a small number of modes and the number of hysteretic states is reduced significantly as well. Convergence studies of error against the number of retained hysteretic states show very good results

Solution of planar elastic stress problems using stress basis functions

The use of global displacement basis functions to solve boundary-value problems in linear elasticity is well established. No prior work uses a global stress tensor basis for such solutions. We present two such methods for solving stress problems in linear elasticity. In both methods, we split the sought stress $\sigma$ into two parts, where neither part is required to satisfy strain compatibility. The first part, $\sigma_p$, is any stress in equilibrium with the loading. The second part, $\sigma_h$, is a self-equilibrated stress field on the unloaded body. In both methods, $\sigma_h$ is expanded using tensor-valued global stress basis functions developed elsewhere. In the first method, the coefficients in the expansion are found by minimizing the strain energy based on the well-known complementary energy principle. For the second method, which is restricted to planar homogeneous isotropic bodies, we show that we merely need to minimize the squared $L^2$ norm of the trace of stress. For demonstration, we solve eight stress problems involving sharp corners, multiple-connectedness, non-zero net force and/or moment on an internal hole, body force, discontinuous surface traction, material inhomogeneity, and anisotropy. The first method presents a new application of a known principle. The second method presents a hitherto unreported principle, to the best of our knowledge

New approximations, and policy implications, from a delayed dynamic model of a fast pandemic

We study an SEIQR (Susceptible-Exposed-Infectious-Quarantined-Recovered) model for an infectious disease, with time delays for latency and an asymptomatic phase. For fast pandemics where nobody has prior immunity and everyone has immunity after recovery, the SEIQR model decouples into two nonlinear delay differential equations (DDEs) with five parameters. One parameter is set to unity by scaling time. The subcase of perfect quarantining and zero self-recovery before quarantine, with two free parameters, is examined first. The method of multiple scales yields a hyperbolic tangent solution; and a long-wave approximation yields a first order ordinary differential equation (ODE). With imperfect quarantining and nonzero self-recovery, the long-wave approximation is a second order ODE. These three approximations each capture the full outbreak, from infinitesimal initiation to final saturation. Low-dimensional dynamics in the DDEs is demonstrated using a six state non-delayed reduced order model obtained by Galerkin projection. Numerical solutions from the reduced order model match the DDE over a range of parameter choices and initial conditions. Finally, stability analysis and numerics show how correctly executed time-varying social distancing, within the present model, can cut the number of affected people by almost half. Alternatively, faster detection followed by near-certain quarantining can potentially be even more effective

Complete dimensional collapse in the continuum limit of a delayed SEIQR network model with separable distributed infectivity

We take up a recently proposed compartmental SEIQR model with delays, ignore loss of immunity in the context of a fast pandemic, extend the model to a network structured on infectivity, and consider the continuum limit of the same with a simple separable interaction model for the infectivities $\beta$. Numerical simulations show that the evolving dynamics of the network is effectively captured by a single scalar function of time, regardless of the distribution of $\beta$ in the population. The continuum limit of the network model allows a simple derivation of the simpler model, which is a single scalar delay differential equation (DDE), wherein the variation in $\beta$ appears through an integral closely related to the moment generating function of $u=\sqrt{\beta}$. If the first few moments of $u$ exist, the governing DDE can be expanded in a series that shows a direct correspondence with the original compartmental DDE with a single $\beta$. Even otherwise, the new scalar DDE can be solved using either numerical integration over $u$ at each time step, or with the analytical integral if available in some useful form. Our work provides a new academic example of complete dimensional collapse, ties up an underlying continuum model for a pandemic with a simpler-seeming compartmental model, and will hopefully lead to new analysis of continuum models for epidemics

Stable compensators in parallel to stabilize arbitrary proper rational SISO plants

We consider stabilization of linear time-invariant (LTI) and single input single output (SISO) plants in the frequency domain from a fresh perspective. Compensators that are themselves stable are sometimes preferred because they make starting the system easier. Such starting remains easy if there is a stable compensator in parallel with the plant rather than in a feedback loop. In such an arrangement, we explain why it is possible to stabilize all plants whose transfer functions are proper rational functions of the Laplace variable $s$. In our proposed architecture we have (i) an optional compensator $C_s(s)$ in series with the plant $P(s)$, (ii) a necessary compensator $C_p(s)$ in parallel with $C_s(s)P(s)$, along with (iii) a feedback gain $K$ for the combined new plant $C_s(s)P(s)+C_p(s)$. We show that stabilization with stable $C_s(s)$ and $C_p(s)$ is always possible. In our proposed solution the closed-loop plant is biproper and has all its zeros in the left half plane, so there is a $K_0$ such that the plant is stable for $K>K_0$. We are not aware of prior use of parallel compensators with such a goal. Our proposed architecture works even for plants that are impossible to stabilize with stable compensators in the usual single-loop feedback architecture. Several examples are provided

Microeconomics of the ideal gas like market models

We develop a framework based on microeconomic theory from which the ideal gas like market models can be addressed. A kinetic exchange model based on that framework is proposed and its distributional features have been studied by considering its moments. Next, we derive the moments of the CC model (Eur. Phys. J. B 17 (2000) 167) as well. Some precise solutions are obtained which conform with the solutions obtained earlier. Finally, an output market is introduced with global price determination in the model with some necessary modifications.Comment: 15pp. Revised & a reference added. An appeal in Appendix-annex (section 8; not for publication) also added. Physica A (accepted for publication