Quaternions and the rotation of a rigid body

The orientation of an arbitrary rigid body is specified in terms of a quaternion based upon a set of four Euler parameters. A corresponding set of four generalized angular momentum variables is derived (another quaternion) and then used to replace the usual three-component angular velocity vector to specify the rate by which the orientation of the body with respect to an inertial frame changes. The use of these two quaternions, coordinates and conjugate moments, naturally leads to a formulation of rigid-body rotational dynamics in terms of a system of eight coupled first-order differential equations involving the four Euler parameters and the four conjugate momenta. The equations are formally simple, easy to handle and free of singularities. Furthermore, integration is fast, since only arithmetic operations are involved.     

Assessing the influence of a rapid water drawdown on the seismic response characteristics of a reservoir rock slope using time–frequency analysis

Acta Geotechnica - To investigate the influence of a rapid water drawdown (RWD) on the seismic response characteristics of reservoir rock slopes, numerical dynamic analyses and shaking table tests...     

Numerical modeling approach for design of water-retaining dams in underground hard rock mines—a case example

During transition from open pit extractions to underground mining of an orebody, often both the open pit and the underground workings operate simultaneously, before the former is closed. To avoid the risk of inundation, the underground workings connecting or driven closer to the open pit are isolated using bulkheads. In this paper, the authors reviewed some of the theoretical equations and norms followed worldwide for determining the safe dimensions of a bulkhead to withstand water pressure. It is found that the theoretical equations are insufficient to represent the actual mode of failure and the ultimate pressure–bearing capacity of a bulkhead, as they were developed based on only one mode of failure of the dam construction material. For better representation of the bulkhead failure and its strength determination, it is found prudent to conduct strain-softening numerical modeling simulating a real mining scenario. Mode of dam failure and effect of parameters such as dam thickness and roadway dimensions on the ultimate pressure–bearing capacity of an arched bulkhead are studied. Numerical modeling studies show that the failure initiates with tensile cracking of the dam surface, but the bulkhead ultimately fails in a combination of tension and shear yielding. On comparison, it is found that the tensile failure theories underestimate the pressure-bearing capacity of a dam, while the shear strength– and crushing strength–based equations overestimate the same. Further, an application of the numerical modeling technique for design of water-retaining dams at an underground mine for its safe isolation from the open pit is presented.     

Observations and Impacts from the 2010 Chilean and 2011 Japanese Tsunamis in California (USA)

The coast of California was significantly impacted by two recent teletsunami events, one originating off the coast of Chile on February 27, 2010 and the other off Japan on March 11, 2011. These tsunamis caused extensive inundation and damage along the coast of their respective source regions. For the 2010 tsunami, the NOAA West Coast/Alaska Tsunami Warning Center issued a state-wide Tsunami Advisory based on forecasted tsunami amplitudes ranging from 0.18 to 1.43 m with the highest amplitudes predicted for central and southern California. For the 2011 tsunami, a Tsunami Warning was issued north of Point Conception and a Tsunami Advisory south of that location, with forecasted amplitudes ranging from 0.3 to 2.5 m, the highest expected for Crescent City. Because both teletsunamis arrived during low tide, the potential for significant inundation of dry land was greatly reduced during both events. However, both events created rapid water-level fluctuations and strong currents within harbors and along beaches, causing extensive damage in a number of harbors and challenging emergency managers in coastal jurisdictions. Field personnel were deployed prior to each tsunami to observe and measure physical effects at the coast. Post-event survey teams and questionnaires were used to gather information from both a physical effects and emergency response perspective. During the 2010 tsunami, a maximum tsunami amplitude of 1.2 m was observed at Pismo Beach, and over $3-million worth of damage to boats and docks occurred in nearly a dozen harbors, most significantly in Santa Cruz, Ventura, Mission Bay, and northern Shelter Island in San Diego Bay. During the 2011 tsunami, the maximum amplitude was measured at 2.47 m in Crescent City Harbor with over $50-million in damage to two dozen harbors. Those most significantly affected were Crescent City, Noyo River, Santa Cruz, Moss Landing, and southern Shelter Island. During both events, people on docks and near the ocean became at risk to injury with one fatality occurring during the 2011 tsunami at the mouth of the Klamath River. Evaluations of maximum forecasted tsunami amplitudes indicate that the average percent error was 38 and 28 % for the 2010 and 2011 events, respectively. Due to these recent events, the California tsunami program is developing products that will help: (1) the maritime community better understand tsunami hazards within their harbors, as well as if and where boats should go offshore to be safe, and (2) emergency managers develop evacuation plans for relatively small “Warning” level events where extensive evacuation is not required. Because tsunami-induced currents were responsible for most of the damage in these two events, modeled current velocity estimates should be incorporated into future forecast products from the warning centers.     

Space- and Time-Dependent Probabilities for Earthquake Fault Systems from Numerical Simulations: Feasibility Study and First Results

In weather forecasting, current and past observational data are routinely assimilated into numerical simulations to produce ensemble forecasts of future events in a process termed “model steering”. Here we describe a similar approach that is motivated by analyses of previous forecasts of the Working Group on California Earthquake Probabilities (WGCEP). Our approach is adapted to the problem of earthquake forecasting using topologically realistic numerical simulations for the strike-slip fault system in California. By systematically comparing simulation data to observed paleoseismic data, a series of spatial probability density functions (PDFs) can be computed that describe the probable locations of future large earthquakes. We develop this approach and show examples of PDFs associated with magnitude M > 6.5 and M > 7.0 earthquakes in California.