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NASA NTRS1993

Application of symbolic and algebraic manipulation software in solving applied mechanics problems

As its name implies, symbolic and algebraic manipulation is an operational tool which not only can retain symbols throughout computations but also can express results in terms of symbols. This report starts with a history of symbolic and algebraic manipulators and a review of the literatures. With the help of selected examples, the capabilities of symbolic and algebraic manipulators are demonstrated. These applications to problems of applied mechanics are then presented. They are the application of automatic formulation to applied mechanics problems, application to a materially nonlinear problem (rigid-plastic ring compression) by finite element method (FEM) and application to plate problems by FEM. The advantages and difficulties, contributions, education, and perspectives of symbolic and algebraic manipulation are discussed. It is well known that there exist some fundamental difficulties in symbolic and algebraic manipulation, such as internal swelling and mathematical limitation. A remedy for these difficulties is proposed, and the three applications mentioned are solved successfully. For example, the closed from solution of stiffness matrix of four-node isoparametrical quadrilateral element for 2-D elasticity problem was not available before. Due to the work presented, the automatic construction of it becomes feasible. In addition, a new advantage of the application of symbolic and algebraic manipulation found is believed to be crucial in improving the efficiency of program execution in the future. This will substantially shorten the response time of a system. It is very significant for certain systems, such as missile and high speed aircraft systems, in which time plays an important role.

Tsai, Wen-Lang↗

NASA NTRS1991

Soluble carbohydrate allocation to roots, photosynthetic rate of leaves, and nitrate assimilation as affected by nitrogen stress and irradiance

Upon resupply of exogenous nitrogen to nitrogen-stressed plants, uptake rate of nitrogen is enhanced relative to nonstressed plants. Absorption of nitrogen presumably is dependent on availability of carbohydrates in the roots. A buildup in soluble carbohydrates thus should occur in roots of nitrogen-stressed plants, and upon resupply of exogenous nitrogen the increased uptake rate should be accompanied by a rapid decline in carbohydrates to prestress levels. To evaluate this relationship, three sets of tobacco plants growing in a complete hydroponic solution containing 1.0 mM NO3- were either continued in the complete solution for 21 d, transferred to a minus-nitrogen solution for 21 d, or transferred to a minus-nitrogen solution for 8-9 d and then returned to the 1.0 mM NO3- solution. These nitrogen treatments were imposed upon plants growing at photosynthetic photon flux densities of 700 and 350 micromoles m-2 s-1. Soluble carbohydrate levels in roots increased during onset of nitrogen stress to levels that were fourfold greater than in roots of non-stressed plants. Following resupply of external nitrogen, a rapid resumption of nitrogen uptake was accompanied by a decline in soluble carbohydrates in roots to levels characteristic of nonstressed plants. This pattern of soluble carbohydrate levels in roots during onset of and recovery from nitrogen stress occurred at both irradiance levels. The response of net photosynthetic rate to nitrogen stress could be expressed as a nonlinear function of concentration of reduced nitrogen in leaves. The net photosynthetic rate at a given concentration of reduced nitrogen, however, averaged 10% less at the lower than at the higher irradiance. The decline in net photosynthetic rate per unit of reduced nitrogen in leaves at the lower irradiance was accompanied by an increase in the nitrate fraction of total nitrogen in leaves from 20% at the higher irradiance to 38% at the lower irradiance.

NASA Discipline Number 61-10↗

DOE OSTI2022

Nematic Colloidal Micro‐Robots as Physically Intelligent Systems

Abstract Physically intelligent micro‐robotic systems exploit information embedded in micro‐robots, their colloidal cargo, and their milieu to interact, assemble, and form functional structures. Nonlinear anisotropic fluids such as nematic liquid crystals (NLCs) provide untapped opportunities to embed interactions via their topological defects, complex elastic responses, and ability to dramatically restructure in dynamic settings. Here a four‐armed ferromagnetic micro‐robot is designed and fabricated to embed and dynamically reconfigure information in the nematic director field, generating a suite of physical interactions for cargo manipulation. The micro‐robot shape and surface chemistry are designed to generate a nemato‐elastic energy landscape in the domain that defines multiple modes of emergent, bottom‐up interactions with passive colloids. Micro‐robot rotation expands the ability to sculpt interactions; the energy landscape around a rotating micro‐robot is dynamically reconfigured by complex far‐from‐equilibrium dynamics of the micro‐robot's companion topological defect. These defect dynamics allow transient information to be programmed into the domain and exploited. Robust micro‐robotic manipulation strategies are demonstrated that exploit these diverse modes of nemato‐elastic interaction to achieve cargo docking, transport, release, and assembly of complex reconfigurable structures at multi‐stable sites. Such structures are of great interest to future developments of LC‐based advanced optical device and micro‐manufacturing in anisotropic environments.

Yao, Tianyi↗

NASA NTRS1983

NASA/RAE collaboration on nonlinear control using the F-8C digital fly-by-wire aircraft

A cooperative advanced digital research experiment (CADRE) was established by the National Aeronautics and Space Administration (NASA) and the Royal Aircraft Establishment (RAE), in which nonlinear control algorithms developed by the RAE were tested on the F-8C digital fly-by-wire (DFBW) aircraft based at the Dryden Flight Research Facility. In the initial phase of the collaboration, some variable-gain algorithms, referred to collectively as variable integral control to optimize response (VICTOR) algorithms, were fight tested. With VICTOR, various measures available within the control system are used to vary gains and time-constants within the closed loop and thereby enhance the control capability of the system, while reducing the adverse effects of sensor noise on the control surfaces. A review of design procedures for VICTOR and results of preliminary flight tests are presented. the F-8C aircraft is operated in the remotely augmented vehicle (RAV) mode, with the control laws implemented as FORTRAN programs on a ground-based computer. Pilot commands and sensor information are telemetered to the ground, where the data are processed to form surface commands which are then telemetered to the ground, where the data are processed to form surface commands which are then telemetered back to the aircraft. The RAV mode represents a single-string (simplex) system and is therefore vulnerable to a hardover since comparison monitoring is not possible. Hence, extensive error checking is conducted on both the ground and airborne computers to prevent the development of potentially hazardous situations. Experience with the RAV monitoring and validation procedures is described.

Butler, G. F.↗

NASA NTRS1983

Development of monofilar rotor hub vibration absorber

A design and ground test program was conducted to study the performance of the monofilar absorber for vibration reduction on a four-bladed helicopter. A monofilar is a centrifugal tuned two degree-of-freedom rotor hub absorber that provides force attenuation at two frequencies using the same dynamic mass. Linear and non-linear analyses of the coupled monofilar/airframe system were developed to study tuning and attenuation characteristics. Based on the analysis, a design was fabricated and impact bench tests verified the calculated non-rotating natural frequencies and mode shapes. Performance characteristics were measured using a rotating absorber test facility. These tests showed significant attenuation of fixed-system 4P hub motions due to 3P inplane rotating-system hub forces. In addition, detuning effects of the 3P monofilar modal response were small due to the nonlinearities and tuning pin slippage. However, attenuation of 4P hub motions due to 5P inplane hub forces was poor. The performance of the 5P monofilar modal response was degraded by torsional motion of the dynamic mass relative to the support arm which resulted in binding of the dynamic components. Analytical design studies were performed to evaluate this torsional motion problem. An alternative design is proposed which may alleviate the torsional motion of the dynamic mass.

Duh, J.↗

DOE OSTI2024

MHD-induced beta limits in the Large Helical Device

Using the extended-magnetohydrodynamics code, M3D-C1, we perform a systematic numerical study of the effect of externally applied heating on the achievable plasma beta in a ten field-period heliotron. Heat sources of varying intensity are applied to a vacuum magnetic field that is representative of the standard configuration of the Large Helical Device, with R 0 = 3.66 m, where R 0 is the radial position of the magnetic axis in vacuum. As the system is driven to a state that is unstable to low-n magnetohydrodynamic (MHD) modes, nonlinear mode interactions lead to the formation of chaotic magnetic fields. With sufficiently strong heating, a collapse of the electron temperature profile is observed. This demonstrates the necessity of simulating the self-consistent evolution of plasma profiles, without imposing assumptions on the structure of the magnetic field, to accurately determine transport properties in stellarator plasmas. It also highlights the value of these advanced simulation capabilities for accelerating the development of high-performance stellarator operating scenarios.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

NASA NTRS1996

Momentum Transfer Studies and Studies of Linear and Nonlinear Optical Properties of Metal Colloids and Semiconductor Quantum Dots

Phase 1 of this work involved design work on a momentum transfer device. The progress on design and testing will be presented. Phase 2 involved the systematic study of the MPD thruster for dual uses. Though it was designed as a thruster for space vehicles, the characteristics of the plasma make it an excellent candidate for industrial applications. This project sought to characterize the system for use in materials processing and characterization. The surface modification on ZnCdTe, CdTe, and ZnTe will be presented. Phase 3 involved metal colloids and semiconductor quantum dots. One aspect of this project involves a collaborative effort with the Solid State Division of ORNL. The thrust behind this research is to develop ion implantation for synthesizing novel materials (quantum dots wires and wells, and metal colloids) for applications in all optical switching devices, up conversion, and the synthesis of novel refractory materials. The ions of interest are Au, Ag, Cd, Se, In, P, Sb, Ga, and As. The specific materials of interest are: CdSe, CdTe, InAs, GaAs, InP, GaP, InSb, GaSb, and InGaAs. A second aspect of this research program involves using porous glass (25-200 A) for fabricating materials of finite size. The results of some of this work will also be reported.

Collins, W. E.↗

DOE OSTI2025

Basic Research Needs for Inverse Methods for Complex Systems under Uncertainty

Inverse problems, which aim to infer unknown properties of a system using experimental and observational data, are central to addressing many of the U.S. Department of Energy’s (DOE) most critical scientific and engineering challenges. Accurate, computationally efficient, and data-efficient solutions to inverse problems are essential for advancing DOE mission-critical science drivers, including analyzing data from large-scale experimental facilities, optimizing fusion reactor performance, accelerating materials discovery, enhancing geophysical imaging, improving wildfire predictions, and enabling autonomous systems and digital twins. However, these problems are becoming increasingly complex, often involving nonlinear, highdimensional, and interconnected systems and models that span multiple physics and scales, while relying on data with varying quantity, quality, and information content. Compounding these challenges is the uncertainty inherent in DOE-relevant systems, where errors in inputs, noise in data, incompleteness of data, and discrepancies between models and reality constrain the accuracy and precision of solutions. At the same time, the convergence of recent scientific computing trends—scientific machine learning, artificial intelligence, and computing advances such as exascale computing—is creating unprecedented opportunities for tackling these challenges. The cross-cutting nature of inverse problems, combined with their growing complexity and rapidly evolving data and algorithmic demands, strongly motivates the formulation of a prioritized research agenda to maximize their capabilities and impact. In response to this need, DOE’s Advanced Scientific Computing Research (ASCR) program in the Office of Science convened the Workshop on Basic Research Needs for Inverse Problems for Complex Systems Under Uncertainty in June 2025. This workshop brought together experts across disciplines to identify grand challenges and major opportunities in the field. Through collaborative discussions, the workshop defined transformative research directions aimed at addressing the mathematical, statistical, and computational challenges posed by inverse problems under uncertainty. As a result of these efforts, four priority research directions (PRDs) were identified to guide future research and development in this area. These PRDs, summarized below, represent a roadmap for advancing the foundational science and mathematics of inverse problems, enabling robust, scalable, and uncertainty-aware solutions that are critical for DOE applications.

97 MATHEMATICS AND COMPUTING↗

NASA NTRS1989

Assessment of SPAR elements and formulation of some basic 2-D and 3-D elements for use with testbed generic element processor

The initial Computational Structural Mechanics (CSM) Testbed was based on Level 13 of the SPAR finite element computer program. Until recently, the element library of the Testbed has been limited to those elements in Level 13 of SPAR. The development of a generic element processor has enabled element researchers to develop, implement and assess element formulations with relative ease. An assessment of new elements as well as the existing SPAR Level 13 elements has revealed some definite shortcomings with the SPAR Level 13 2-D and 3-D elements. The SPAR S81 solid element does not pass the patch test problem proposed by MacNeal-Harder. These deficiencies are identified here. The 2-D elements, however, seem to perform well taking into account the limitations imposed by the theory used to formulate them, (i.e., thin plates only). Common deficiencies of the 2-D and 3-D elements in SPAR have to do with their adaptability to the nonlinear analysis utilities developed by Lockheed Palo Alto Research Lab. Also, the EFIL format of the SPAR element data does not conform to the standard format of the Testbed.

Aminpour, Mohammad A.↗

DOE OSTI2022

Space Charge Driven Emittance Growth and the Effect of Octupoles in IOTA

The Integrable Optics Test Accelerator (IOTA) at Fermilab is a small machine dedicated to a broad frontier accelerator physics program. An important aspect of this program is to investigate the potential benefits of the resonance free tune spread achievable with integrable optics to store and accelerate high intensity proton beams for which space charge is significant. In this context, a good understanding of proton beam emittance growth and particle loss mechanisms is essential. Assuming nominal design parameters, simulations show that for a bunched beam, the bulk of emittance growth takes place immediately following injection, typically within tens of turns. We attempt to account for this growth using a simplified RMS mismatch theory; some of its limitations and possible improvements are briefly discussed. We then compare theoretical predictions to simulations performed using the PIC code pyORBIT. Further exploring ways to mitigate emittance growth and reduce particle loss, we compare two beam matching strategies: (1) matching at the injection point (2) matching at the center of the nonlinear (octupole) insertion region where $\beta_x = \beta_y$. To observe how nonlinearity affects emittance growth and whether it dominates growth due to mismatch, we track two different distributions. Finally, we explore the potential of using octupoles in a quasi-integrable configuration to mitigate growth using a variety of initial distributions both at reduced and full intensities.

43 PARTICLE ACCELERATORS↗

NASA NTRS1991

Linear and nonlinear dynamic analysis by boundary element method

An advanced implementation of the direct boundary element method (BEM) applicable to free-vibration, periodic (steady-state) vibration and linear and nonlinear transient dynamic problems involving two and three-dimensional isotropic solids of arbitrary shape is presented. Interior, exterior, and half-space problems can all be solved by the present formulation. For the free-vibration analysis, a new real variable BEM formulation is presented which solves the free-vibration problem in the form of algebraic equations (formed from the static kernels) and needs only surface discretization. In the area of time-domain transient analysis, the BEM is well suited because it gives an implicit formulation. Although the integral formulations are elegant, because of the complexity of the formulation it has never been implemented in exact form. In the present work, linear and nonlinear time domain transient analysis for three-dimensional solids has been implemented in a general and complete manner. The formulation and implementation of the nonlinear, transient, dynamic analysis presented here is the first ever in the field of boundary element analysis. Almost all the existing formulation of BEM in dynamics use the constant variation of the variables in space and time which is very unrealistic for engineering problems and, in some cases, it leads to unacceptably inaccurate results. In the present work, linear and quadratic isoparametric boundary elements are used for discretization of geometry and functional variations in space. In addition, higher order variations in time are used. These methods of analysis are applicable to piecewise-homogeneous materials, such that not only problems of the layered media and the soil-structure interaction can be analyzed but also a large problem can be solved by the usual sub-structuring technique. The analyses have been incorporated in a versatile, general-purpose computer program. Some numerical problems are solved and, through comparisons with available analytical and numerical results, the stability and high accuracy of these dynamic analysis techniques are established.

Ahmad, Shahid↗

NASA NTRS1998

Fuzzy Adaptive Control for Intelligent Autonomous Space Exploration Problems

The principal objective of the research reported here is the re-design, analysis and optimization of our newly developed neural network fuzzy adaptive controller model for complex processes capable of learning fuzzy control rules using process data and improving its control through on-line adaption. The learned improvement is according to a performance objective function that provides evaluative feedback; this performance objective is broadly defined to meet long-range goals over time. Although fuzzy control had proven effective for complex, nonlinear, imprecisely-defined processes for which standard models and controls are either inefficient, impractical or cannot be derived, the state of the art prior to our work showed that procedures for deriving fuzzy control, however, were mostly ad hoc heuristics. The learning ability of neural networks was exploited to systematically derive fuzzy control and permit on-line adaption and in the process optimize control. The operation of neural networks integrates very naturally with fuzzy logic. The neural networks which were designed and tested using simulation software and simulated data, followed by realistic industrial data were reconfigured for application on several platforms as well as for the employment of improved algorithms. The statistical procedures of the learning process were investigated and evaluated with standard statistical procedures (such as ANOVA, graphical analysis of residuals, etc.). The computational advantage of dynamic programming-like methods of optimal control was used to permit on-line fuzzy adaptive control. Tests for the consistency, completeness and interaction of the control rules were applied. Comparisons to other methods and controllers were made so as to identify the major advantages of the resulting controller model. Several specific modifications and extensions were made to the original controller. Additional modifications and explorations have been proposed for further study. Some of these are in progress in our laboratory while others await additional support. All of these enhancements will improve the attractiveness of the controller as an effective tool for the on line control of an array of complex process environments.

Esogbue, Augustine O.↗

DOE OSTI2025

Leveraging FPGA Advantages for Quicker Data Processing for LBNF

The Long Baseline Neutrino Facility (LBNF) will deliver a 2.4 MW muon neutrino beam from Fermilab to the Deep Underground Neutrino Experiment (DUNE), requiring unprecedented precision in beamline alignment to achieve DUNE's neutrino oscillation measurement goals. Vertical misalignments of beamline components as small as 0.5 mm can contribute 6-7\% uncertainty in predicted neutrino flux, necessitating sub-0.1 mm alignment monitoring capabilities. The Horn Location Sensor (HLS) system employs frequency sweep interferometry (FSI) in a distributed hydrostatic leveling network to achieve the required precision under harsh radiation conditions up to 5000 kRad/year. Traditional FSI implementations suffer from laser sweep nonlinearities that degrade resolution and require computationally intensive post-processing corrections using gas reference cells. This work presents a real-time FPGA-based implementation of the HLS data acquisition and processing system using a sweep tracker interferometer for dynamic sweep linearization. The system utilizes a PYNQ-Z2 FPGA with programmable logic implementing parallel 16k-point FFT processing across four channels, synchronized by the sweep tracker signal to eliminate post-processing requirements. Spectral performance testing demonstrates significant improvements in peak sharpness compared to traditional fixed-frequency digitization. The FPGA implementation enables real-time displacement monitoring with processing speeds orders of magnitude faster than software-based approaches, essential for the operational requirements of LBNF's eventual distributed sensor network. This advancement in real-time FSI processing directly supports DUNE's precision neutrino physics program by providing the rapid feedback necessary for maintaining stringent beamline alignment tolerances during high-power beam operations.

Rossel, Jacob↗

DOE OSTI2025

Real-Time FPGA Implementation For Frequency Sweep Interferometry In The LBNF Complex

Rossel, A. Jacob [Fermilab; Unlisted]↗

NASA NTRS2019

Modal Analysis of the Orion Capsule Two Parachute System

As discussed in Ref [1], it is apparent from flight tests that the system made up of two main parachutes and a capsule can undergo several distinct dynamical behaviors. The most significant and problematic of these is the pendulum mode in which the system develops a pronounced swinging motion with an amplitude of up to 24 deg. Large excursions away from vertical by the capsule could cause it to strike the ground at a large horizontal or vertical speed and jeopardize the safety of the astronauts during a crewed mission. In reference [1], Ali et al. summarized a series of efforts taken by the Capsule Parachute Assembly System (CPAS) Program to understand and mitigate the pendulum issue. The period of oscillation and location of the system's pivot point are determined from post-flight analysis. Other noticeable but benign modes include: 1) flyout (scissors) mode, where the parachutes move back and forth symmetrically with respect to the vertical axis similar to the motion of a pair of scissors; 2) maypole mode, where the two parachutes circle around the vertical axis at a nearly constant radius and period; and 3) breathing mode, in which deformation of the non-rigid canopies affects the axial acceleration of the system in an oscillatory manner. Because these modes are relatively harm- less, little effort has been devoted to analyzing them in comparison with the pendulum motion. Motions of the actual system made up of two parachutes and a capsule are extremely complicated due to nonlinearities and flexibility effects. Often it is difficult to obtain insight into the fundamental dynamics of the system by examining results from a multi-body simulation based on nonlinear equations of motion (EOMs). As a part of this study, the dynamics of each mode observed during flight is derived from first principles on an individual basis by making numerous simplifications along the way. The intent is to gain a better understanding into the behavior of the complex multi-body system by studying the reduced set of differential equations associated with each mode. This approach is analogous to the traditional modal analysis technique used to study airplane flight dynamics, in which the full nonlinear behavior of the airframe is decomposed into the phugoid and short period modes for the longitudinal dynamics and the spiral, roll-subsidence, and dutch-roll modes for the lateral dynamics. It is important to note that the study does not address the mechanisms that cause the system to transition from one mode to another, nor does it discuss motions during which two or more modes occur simultaneously.

Pei, J.↗

DOE OSTI2020

Comparison between PARFUME and Bison Using the AGR-2 Irradiation Experiment

This report documents comparisons between Fuel Model (PARFUME) model predictions versus Bison for selected compacts from the second irradiation test of the Advanced Gas Reactor (AGR) program that occurred from June 2010 to October 2013 in the Advanced Test Reactor (ATR) at Idaho National Laboratory (INL). PARFUME is a fuel performance analysis and modeling code, used for evaluating gas-reactor tristructural isotropic (TRISO) coated particle fuel for prismatic, pebble bed, plate, and cylindrical type fuel geometries. PARFUME is an integrated mechanistic computer code that evaluates the thermal, mechanical, and physico-chemical behavior of TRISO coated-fuel particles and the probability for fuel failure given the particle-to-particle statistical variations in physical dimensions and material properties that arise during the fuel fabrication process. Bison is a nuclear fuel performance application built using the Multiphysics Object-Oriented Simulation Environment (MOOSE) finite element library (Permann 2020). Bison is capable of modeling multiple fuel forms in a wide variety of dimensions and geometries. It solves coupled nonlinear partial differential equations, including heat conduction, mechanics, fission product species transport etc., in a fully implicit manner. Comparisons between PARFUME and Bison were performed using four compacts from the AGR-2 experiment. Selected outputs were chosen based on their impact on the probability of the SiC layer failing. In general there was good agreement between PARFUME and Bison with the exception of predicting the gap formed between the buffer and IPyC. Further comparisons between PARFUME and Bison are planned to further develop Bison’s capabilities.

11 NUCLEAR FUEL CYCLE AND FUEL MATERIALS↗

NASA NTRS1996

[Kelvin-Helmholtz instability in protostellar jets]

NASA grant NAG 5 2866, funded by the Astrophysics Theory Program, enabled the study the Kelvin-Helmholtz instability in protostellar jets. In collaboration with co-investigator Philip Hardee, the PI derived the analytic dispersion relation for the instability in including a cooling term in the energy equation which was modeled as one of two different power laws. Numerical solutions to this dispersion relation over a wide range of perturbation frequencies, and for a variety of parameter values characterizing the jet (such as Mach number, and density ratio) were found It was found that the growth rates and wavelengths associated with unstable roots of the dispersion relation in cooling jets are significantly different than those associated with adiabatic jets, which have been studied previously. In collaboration with graduate student Jianjun Xu (funded as a research associate under this grant), hydrodynamical simulations were used to follow the growth of the instability into the nonlinear regime. It was found that asymmetric surface waves lead to large amplitude, sinusoidal distortions of the jet, and ultimately to disruption Asymmetric body waves, on the other hand, result in the formation of shocks in the jet beam in the nonlinear regime. In cooling jets, these shocks lead to the formation of dense knots and filaments of gas within the jet. For sufficiently high perturbation frequencies, however, the jet cannot respond and it remains symmetric. Applying these results to observed systems, such as the Herbig-Haro jets HH34, HH111 and HH47 which have been observed with the Hubble Space Telescope, we predicted that some of the asymmetric structures observed in these systems could be attributed to the K-H modes, but that perturbations on timescales associated with the inner disk (about 1 year) would be too rapid to cause disruption. Moreover, it was discovered that weak shock 'spurs' in the ambient gas produced by ripples in the jet surface due to nonlinear, modes of surface and/or body waves could accelerate the ambient gas to low velocity. This latter effect represents a new mechanism by which supersonic jets can accelerate low velocity outflows.

Stone, James↗

NASA NTRS1974

Testing techniques for determining static mechanical properties of Pneumatic tires

Fore-aft, lateral, and vertical spring rates of model and full-scale pneumatic tires were evaluated by testing techniques generally employed by industry and various testing groups. The purpose of this experimental program was to investigate what effects the different testing techniques have on the measured values of these important static tire mechanical properties. The testing techniques included both incremental and continuous loadings applied at various rates over half, full, and repeated cycles. Of the three properties evaluated, the fore-aft stiffness was demonstrated to be the most affected by the different testing techniques used to obtain it. Appreciable differences in the fore-aft spring rates occurred using both the increment- and continuous-loading techniques; however, the most significant effect was attributed to variations in the size of the fore-aft force loop. The dependence of lateral stiffness values on testing techniques followed the same trends as that for fore-aft stiffness, except to a lesser degree. Vertical stiffness values were found to be nearly independent of testing procedures if the nonlinear portion of the vertical force-deflection curves is avoided.

Dodge, R. N.↗