Root Sum Square Research Articles

Sheet Metal Joint Configurations and Their Variation Characteristics

In sheet metal assembly, joints are designed to facilitate welding the parts. The three basic joints used in sheet metal assemblies are lap (slip) joints, butt joints, and butt-lap (corner) joints. Each joint configuration has its own variation characteristics. However, the currently available variation analysis methods, such as worst case analysis, root sum squares, etc., are not applicable to deformable sheet metal because they are based on rigid bodies. This paper analyzes the variation characteristics of simple assemblies constructed from the three basic joints, using Mechanistic Variation Simulation. Mechanistic Variation Simulation combines engineering structural models with statistical analysis in predicting deformable sheet metal assembly variation. Furthermore, the variation characteristics of the boxes constructed from the three basic joints are also evaluated. The developed models and analysis provide an improved understanding of sheet metal product design and process design.

A hydrodynamic ocean tide model improved by assimilating a satellite altimeter‐derived data set

An upgraded version of the tidal solutions (FES94.1) is presented, obtained by assimilating an altimeter‐derived data set in the finite element hydrodynamic model, following the representer approach. The assimilated data are drawn from the CSR2.0 Texas solutions sampled on a 5° × 5° grid. The assimilation is applied over the Atlantic, Indian, and Pacific Oceans. The standard release of the new FES95.2 solutions is a 0.5° × 0.5° gridded version of the full finite element solutions. The associated tidal prediction model includes 26 constituents. The eight major constituents are drawn directly from the hydrodynamic model: K1, O1, Q1, M2, S2, N2, K2, and 2N2, corrected by assimilation except K2 and 2N2. The other 18 constituents are derived by admittance. Among them are μ2, ν2, L2, T2, M1, P1, J1, and OO1. The quality of these solutions is evaluated by reference to a standard sea truth data set of 95 stations. This quality is significantly improved after the assimilation process is applied: the root‐sum‐square (RSS) of the differences between solutions and observations, for the eight major constituents, is reduced from 3.8 cm for FES94.1 to 2.8 cm for FES95.2, i.e., a gain of 1 cm. The performances of the prediction model are evaluated by comparing tidal predictions with observations at 59 sites distributed over the world ocean and by looking at the level of variance of the sea surface variability observed by the T/P altimeter at its cross‐over track points after tidal correction. These evaluations lead to the same conclusion: this new prediction model performs much better than the one based on FES94.1, because of correction of the major constituents by T/P data assimilation and because of the increase in the number of constituents from 13 to 26. The tidal predictions are at the level of accuracy of those produced by the best recent T/P empirical models.

Study of an assembly tolerance allocation model based on Monte Carlo simulation

The traditional design methods of assembly tolerance allocation are usually based on engineers' experience, or the worst on worst tolerance analysis (WOW) method, or the root sum square tolerance analysis (RSS) method. However, the above-mentioned methods, whilst used frequently in the analysis of a single-dimensional chain, are not suitable for the analysis of geometrical tolerance and multi-dimensional chains. Also, the relationship between tolerance and manufacturing cost is not considered and a suitable tolerance allocation based on minimum manufacturing cost can not be obtained. Some research works have applied linear or non-linear programming methods to optimize the tolerance allocation of each part in an assembly. However, the convergence of the solution is not ensured. The purpose of this study is to provide an integrated approach, including tolerance design, manufacturing cost analysis and multiple chains consideration, using the Monte Carlo method to optimize the tolerance allocation with minimum cost. The Monte Carlo method, a statistical simulation method, was used to simulate the dimension variance of each part and each dimensional chain. The contribution percentage of each part on each dimensional chain was calculated. Tolerance cost was chosen as an object function and the tolerance allocation model as a constraint condition, the optimum tolerances of each part being obtained by the iteration method. Results of computer simulation for several examples were compared with published data for demonstrating the feasibility of the proposed method. It can be concluded that the tolerance-allocation model combined with a tolerance-cost relationship can provide a very practical and useful approach for design engineers.

Variation Simulation for Deformable Sheet Metal Assemblies Using Finite Element Methods

Traditional variation analysis methods, such as Root Sum Square method and Monte Carlo simulation, are not applicable to sheet metal assemblies because of possible part deformation during the assembly process. This paper proposes the use of finite element methods (FEM) in developing mechanistic variation simulation models for deformable sheet metal parts with complex two or three dimensional free form surfaces. Mechanistic variation simulation provides improved analysis by combining engineering structure models and statistical analysis in predicting the assembly variation. Direct Monte Carlo simulation in FEM is very time consuming, because hundreds or thousands of FEM runs are required to obtain a realistic assembly distribution. An alternative method, based on the Method of Influence Coefficients, is developed to improve the computational efficiency, producing improvements by several orders of magnitude. Simulations from both methods yield almost identical results. An example illustrates the developed methods used for evaluating sheet metal assembly variation. The new approaches provide an improved understanding of sheet metal assembly processes.

Two Visually Meaningful Correlation Measures for Comparing Calculated and Measured Response Histories

Two visually meaningful correlation measures are proposedfor comparing calculated and measured response histories. One is an error index which is a simplification of RSS (root-sum-square) error factor, and the other is an inequality index that is a simplification of Theil's inequality coefficient. The first compares the difference be­ tween the calculated and the measured histories to the measured history. The second compares the difference between the two histories to the sum of the two. The proposed correlation measures are compared to other existing measures, namely, Geers' Error Factors, RSS Error Factor, and Theil's Inequality Coefficient for ease of interpret a­ tion and visualization. © 1994 John Wiley & Sons, Inc.

Two Visually Meaningful Correlation Measures for Comparing Calculated and Measured Response Histories

Two visually meaningful correlation measures are proposed for comparing calculated and measured response histories. One is an error index which is a simplification of RSS (root-sum-square) error factor, and the other is an inequality index that is a simplification of Theil’s inequality coefficient. The first compares the difference between the calculated and the measured histories to the measured history. The second compares the difference between the two histories to the sum of the two. The proposed correlation measures are compared to other existing measures, namely, Geers’ Error Factors, RSS Error Factor, and Theil's Inequality Coefficient for ease of interpretation and visualization.

Root Sum Squares Tolerance Analysis with Nonlinear Problems

Elementary tolerance analysis with nonlinear assembly equations or functions can lead to significant errors when either Worst Limits or Root Sum Squares are applied. A previous paper [1] presented the errors and a corrective design method for Worst Limit elementary tolerance analysis. In this paper, Root Sum Squares tolerance analysis is applied to a nonlinear assembly equation, which gives significant errors in the assembly acceptance fraction. Advanced statistical analysis can be used in an iterative design procedure to specify the independent variables which meet the specified assembly in spite of the nonlinearities.

Spacecraft attitude determination using a second-order nonlinear filter

The stringent attitude determination accuracy and faster slew maneuver requirements demanded by present-day spacecraft control systems motivate the development of recursive nonlinear filters for attitude estimation. This paper presents the second-order filter development for the estimation of attitude quaternion using three-axis gyro and star tracker measurement data. Performance comparisons have been made by computer simulation of system models and filter mechanization. It is shown that the second-order filter consistently performs better than the extended Kalman filter when the performance index of the root sum square estimation error of the quaternion vector is compared. The second-order filter identifies the gyro drift rates faster than the extended Kalman filter. The uniqueness of this algorithm is the online generation of the time-varying process and measurement noise covariance matrices, derived as a function or the process and measurement nonlinearity, respectively.

Aperture efficiency of Itapetinga 45-ft antenna at λ = 3.3 mm

The aperture efficiency at <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\lambda = 3.3</tex> mm of the 45-ft Itapetinga reflector was found to be of about 25 percent. This radiometric result suggests an equivalent surface accuracy of less than 0.25 mm, root sum square (rss) assuming a Gaussian distribution of errors, which is much smaller than the actual rss estimated for the dish (i.e., 0.37 mm, rss) and, represents a 5 dB improvement in gain. This result confirms that non-Gaussian errors across a smooth surface have negligible effects on the aperture efficiency. It is demonstrated that there is little effect on gain produced by error correlation regions that are small compared to the antenna diameter and that antenna efficiency may be predicted from mechanical measurements when the nature of such errors across the surface is know. It is possible to predict that relatively simple mechanical readjustments of the reflector and subreflector may raise its efficiency to 30 percent.

Determination of the quantized hall resistance value by using a calculable capacitor at ETL

We report the International System (SI) value of the quantized Hall resistance (R <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</inf> ) determined by using a calculable capacitor at the electrotechnical laboratory (ETL). As the result of our measurements at ETL, a most reliable value of h / e <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> has been estimated as 25 812.8036 Ω <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">SI</inf> with a systematic uncertainty of 0.25 ppm root sum square (rss) and a random error of 0.20 ppm one standard deviation (1σ).

A New Tolerance Analysis Method for Designers and Manufacturers

Even when all manufactured parts for an assembly are produced within limits, these parts still may not assemble properly if the assembly tolerance analysis was inadequately performed. Naturally occurring shifts in a process can produce biased distributions which can result in increased assembly problems and a greater number of rejects than anticipated. The most common methods of analysis of assembly tolerance buildup are worst case and root sum squares. The limitations of each of these methods are discussed and a simple new method is proposed which accounts for expected bias. This new method includes both worst case and root sum squares as extreme cases.

Uncertainty Analysis of Heat Exchanger Thermal-Hydraulic Designs

A simple method of assessing the uncertainties of thermal hydraulic parameters in the design of heat exchangers is described. Assuming a Gaussian form of data distribution for “uncertain” parameters leads to a root-sum-square (RSS) method of establishing the probability, or confidence level, that the heat exchanger design will meet Us thermal hydraulic performance requirements. An example is provided to illustrate the method. Various other design considerations to provide proper margins for the final sizing of the heat exchanger are also discussed.

Control techniques to improve Space Shuttle solid rocket booster separation

The present Space Shuttle's control system does not prevent the Orbiter's main engines from being in gimbal positions that are adverse to solid rocket booster separation. By eliminating the attitude error and attitude rate feedback just prior to solid rocket booster separation, the detrimental effects of the Orbiter's main engines can be reduced. In addition, if angular acceleration feedback is applied, the gimbal torques produced by the Orbiter's engines can reduce the detrimental effects of the aerodynamic torques. This paper develops these control techniques and compares the separation capability of the developed control systems. Currently with the worst case initial conditions and each Shuttle system dispersion aligned in the worst direction (which is more conservative than will be experienced in flight), the solid rocket booster has an interference with the Shuttle's external tank of 30 in. Elimination of the attitude error and attitude rate feedback reduces that interference to 19 in. Substitution of angular acceleration feedback reduces the interference to 6 in. The two latter interferences can be eliminated by atess conservative analysis techniques, that is, by using a root sum square of the system dispersions.

Tidal and geodetic observations for the SEASAT altimeter calibration experiment

A calibration experiment for the SEASAT radar altimeter was conducted in the Bermuda calibration area during September and October of 1978 by a group of Federal Government agencies, universities, and research organizations. As part of the SEASAT calibration activities, a tide gage was installed by the National Ocean Survey at an open coastal location on Bermuda to provide a determination of the instantaneous sea surface height during the SEASAT overflights of the island. The tide gage was geodetically tied to the laser tracking station on Bermuda, so that SEASAT's position relative to the sea surface could be determined independently and compared with the value provided by the altimeter measurements. The root sum square error in the determination of the vertical position of the laser, relative to the sea surface, has been estimated to be 4.0 cm exclusive of possible errors arising from the present lack of precise information on the elevation of the geoid at Bermuda.

Index of refraction turbulence effects on thermal blooming in laboratory experiments

A laboratory experiment has been carried out to determine the effect of index of refraction turbulence on thermal blooming for cw CO(2) laser radiation. The turbulence is generated by the forced mixing of two gases of different indices of refraction; helium and nitrogen. The thermal blooming is obtained by adding small amounts of propylene to the mixture. The construction of the test cell and methods of measuring the turbulence are discussed in detail. The experimental results are compared with the root-sum-squared (RSS) theory, i.e., that the blooming and turbulence effects may be considered separately and that the total beam area is given by A = (AO) + A(B) + A(T), where A(B) and A(T) are the increase in the time-averaged area of the beam due to blooming and turbulence. Agreement is excellent for the particular case considered.

Archaeological evidence for tectonic activity in the region of the Haifa-Qishon Graben, Israel

Abstract Thirty-five archaeological sites between Lebanon and Gaza on the Mediterranean shore of Israel were surveyed for evidence of relative changes of sea level; and published and unpublished data were compiled on a further 23 sites. Remains of occupation levels were classified into 13 archaeological periods from Neolithic to Crusader, giving a good dating scale over the last 9000 years. The Bronze Age shoreline (4000 years B.P.) is shown to be very close to the present shore along most of the coast, with vertical changes of less than 1.0 m at most sites. For later periods the accuracy of measurement of relative changes of level was improved to a root sum square error of 28 cm. The mean and modal sea levels derived from all sites of the same archaeological period were plotted against time to indicate the most probable “eustatic” sea-level curve, though it is accepted that hydro-isostatic adjustment and broad-scale slow earth movements cannot be separated statistically from eustatic changes on a 200-km stretch of coast. The relative sea level is shown to have been between −70 cm and present sea level for the last 4000 years, with no evidence for oscillations of amplitude more than ±30 cm at periodicities longer than 400 years, and no high sea level still stands. Relative vertical displacement between sites of the same archaeological age is attributed to tectonism. The sites on the alluvium of the Haifa—Qish on graben show no signs of vertical displacement, but there is maximum activity to north and south. Acco, on the north margin, shows submergence of nearly 2.0 m since A.D. 1300, with probable uplift preceding that date. Mount Carmel, south of the graben, shows signs of only very slight relative change, possibly attributable to eustatic change, in the immediate vicinity of the graben. But the southern flank of the mountain, 20–80 km south of the southern fault of the Graben, shows active vertical movements on the whole coast. Dor and Caesarea indicate multiple vertical movements ranging from 1.0 to 5.0 m in the last 2000 years, and a slump or fault has intersected the harbour area of Caesarea about 150 m offshore. These observations are related briefly to recent geodetic levelling of the coast and graben area; observations on the Dead Sea Rift, and hypotheses of fault control of the linearity of the Israel Mediterranean coastline.

Response spectrum techniques for asymmetric buildings

AbstractA scheme is proposed to calculate the effect of torsion on each lateral load resisting element of asymmetrical buildings in the context of the response spectrum technique. The scheme consists of: (i) Obtain the modal shear and torque on the building by the response spectrum technique, (ii) Compute the total modal shear forces on each frame by resolving the modal shear and torque on the building according to principles of structural mechanics. The shears on each frame due to the lateral load effect and torsional effect are combined algebraically, (iii) Obtain the total shear force on each frame by combining the total modal shears on that frame in a root sum square manner.Since the proper phase relationship between the lateral load effect and torsional effect is accounted for on a modal basis, it is believed that the proposed scheme provides a more realistic load estimate on the frames than the conventional approach. An example of a simplified mono‐symmetrical frame structure is chosen to illustrate the accuracy of the proposed scheme, using dynamic time‐history analysis as a standard for comparison.

Performance predictions for high altitude self-contained satellite navigation systems

This paper summarizes a study of self-contained navigation system accuracies for high altitude orbital missions projected through 1985. It is found that root-sum-square (RSS) satellite position errors as low as a few hundred feet are obtainable with range measuring systems. Totally autonomous, nonradiating systems are operationally more attractive, and some promising candidates in this category give RSS position errors from about one-half to a few miles, depending on the orbit. The extended Kalman filter and extensive Monte Carlo simulations are used as the basic analysis tools. Favorable comparisons between covariance analysis and Monte Carlo results are reported. Time-average statistics are also found to provide useful figures of merit.