Interpolation of Mode Shapes: A Matrix Scheme Using Two-Way Spline Curves

Done, G.T.S.

doi:10.1017/s0001925900003577

Cited by 21 publications

(5 citation statements)

References 10 publications

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“…For example, during the flight of missiles and other aircraft, the airflow and aerodynamic were complexes around and inside the missile body, such as flow fluctuations during water treatment, hydropower, and pipeline transportation [7]. There are many problems in mechanics, some of which have not been well solved in scientific research, and some or due to various reasons, there is no way to conduct ground experimental research [8,9]. Such problems, more accurate simulations are possible, however, through statistical processes and calculations that can then be learned and finally well established.…”

Section: Interpolation Methods For Visualmentioning

confidence: 99%

[Retracted] Interpolation Method for Visual Simulation of Engine Exhaust Flame

Cheng¹,

Liu

et al. 2021

Journal of Sensors

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Propulsive force and exhaust fluid temperature are important indicators in the performance of an engine. An investigation of the effects of propellant composition, plane flight conditions, and engine operating environment on rocket thrust and the range of smoke plume temperature can provide references in the design of engine mechanics at the optimization of propellant composition, in monitoring of target identification and in the evolving of stealth of stealth technology. In order to understand the characteristics of the engine tail flame, a visual simulation of the engine tail flame was carried out by combining the engine operating conditions with the tail flame conditions. Based on the advantages of the bicubic spline interpolation algorithm and the Kriging interpolation algorithm, this paper proposes a hybrid interpolation algorithm, which performs color mapping and three-dimensional space separation in the engine plume data set and model, and visualizes the engine and engine plume. The simulation realizes real-time monitoring of the functions of various engine components through characteristic colors. The research results show that the hybrid interpolation method can effectively visualize the engine exhaust flame. The simulated plume has a relatively obvious temperature peak at 0.7 m, and the temperature of the plume flow field is significantly higher than that of the frozen plume flow field by about 200 ~1000 K. This shows that the algorithm in this paper helps to visualize the expression of engine tail flame information.

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Section: Interpolation Methods For Visualmentioning

confidence: 99%

[Retracted] Interpolation Method for Visual Simulation of Engine Exhaust Flame

Cheng¹,

Liu

et al. 2021

Journal of Sensors

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“…2 By splining in two directions, interpolations for points on a surface can be performed. 3 A generalization of the splining concept is the surface spline in which a plate is passed through the given points. 4 In general, splining interpolation is based on the assumption of a constant El beam or a constant thickness plate.…”

Section: Numerical Resultsmentioning

confidence: 99%

“…It is very well possible, however, to use a nonuniform beam or plate for the splining, or even the actual structure, as was hinted at in Ref. 3.…”

Section: Numerical Resultsmentioning

confidence: 99%

Elimination of degrees of freedom by structural splining.

Hassig¹

1973

AIAA Journal

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5 10 15 REDESIGN CYCLES Fig. 1 Design cycles for 5-element panel; (n = 5, Y\ = 0.8, A u = 1 0 + 2.0,work the derivatives dkjdpi were evaluated numerically by incrementing successive values of p t . However, other techniques 7 for evaluating derivatives will be used in future studies in order to reduce solution times.Since the constraint equation is not linear in the design parameters, the direction of VA cr , the gradient of /l cr , is a function of the p .'s. In carrying out computations in the weight-minimization mode using Eq. (14) it was found that the value of k cr for the new design was not exactly A 0 . Hence, the following procedure was employed: the weight-minimization mode was employed until A cr (p) fell below a specified value. At this point a "lambdamodification mode" was entered in which the weight was held constant at the last value attained by weight minimization, while the value of X cr was increased along the direction of steepest ascent subject to the weight constraint. The following equation governs the lambda-modification mode 5 :where c 2 is a positive constant. Numerical ResultsA computer program was written in which the element stiffness, aerodynamic, and mass matrices were generated according to Eq. (9) for a specified structural mass fraction, 77, and a specified set of design parameters, p.. These were assembled to form an eigenvalue problem of the form given by Eq. (10). The value of a was systematically varied until the critical value for the given design was obtained. Equations (14) and (15) were employed to determine design changes for weight-minimization steps and for lambda-modification steps, respectively.The results shown in Figs. 1 and 2 represent improved designs and not optimal designs and represent only preliminary studies of the use of the weight-minimization, lambda-modification procedure. Figure 1 is for a panel with structural mass fraction r\ -0.8 and having five finite elements, i.e., n = 5. The value of A 0 , as determined for n = 5, was A 0 =. 342.901.In Fig. 1 weight-minimization proceeded until A cr fell below a limit set at A L = ^0 -2.0. Lambda-modification was allowed to raise A cr above A 0 with an upper limit of A u = A 0 + 2.0. It is seen that a weight reduction of 3.6% of the total weight was possible with A cr = 343.190. The best design gives a total weight reduction of 4.0% with A cr = 341.315. Reference 5 contains similar figures for a 5-element panel with A u = A 0 and for a 9-element panel. The 9-element problem was terminated after four weightminimization steps with no lambda-modification. Figure 2 shows the shape of the best designs for the 5-element and 9-element panels.Although it might be possible to obtain better designs than the best design in Fig. 1, it is significant to note that the percent weight saved is considerably more than was obtained by Turner, 3 and the shape of the best designs shown in Fig. 2 indicate that Turner's Use of n = 3 did not allow the design to assume a form resembling an optimum design.Weisshaar 8 recently published resu...

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“…Since the slopes can be computed from these spline functions, the slope vector a a at the aerodynamic control points can be written as <x a =D as R s (3) where D as are the slopes derived from the vectors of F as .…”

Section: Transformation Matricesmentioning

confidence: 99%