# Case Sudy: Optimization of Parameters of Beam Excitation Waveform

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Case study background and problem formulations

Instructions for optimization with PSG Run-File, PSG MATLAB Toolbox, PSG MATLAB Subroutines and PSG R.

PROBLEM: problem_linear

minimizing linear
subject to
b1 ≤Ax ≤b2 (multiple linear constraints)
Box constraints (types of variables)
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Linear = Linear function
Box constraints = constraints on individual decision variables
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Dataset1 158 N/A 0.003127 33.45 # of Variables # of Scenarios Objective Value Solving Time, PC 3.14GHz (sec) Environments Run-File Problem Statement Data Solution Matlab Toolbox Data Matlab Subroutines Matlab Code Data R R Code Data

Instructions for importing problems from Run-File to PSG MATLAB.

Problem Datasets # of Variables # of Scenarios Objective Value Solving Time, PC 2.83GHz (sec)
Dataset2 Problem statement Data Solution 158 N/A 0.000218 6.56
Dataset3 Problem statement Data Solution 158 N/A 0.000015 9.41
Dataset4 Problem statement Data Solution 158 N/A 0.000006 13.11
Dataset5 Problem statement Data Solution 158 N/A 0.000003 12.93
Dataset6 Problem statement Data Solution 158 N/A 0.00001 8.37
Dataset7 Problem statement Data Solution 158 N/A 1.0269e-06 7.43
Dataset8 Problem statement Data Solution 158 N/A 0.000006 16.32
Dataset9 Problem statement Data Solution 158 N/A 0.000013 13.5
Dataset10 Problem statement Data Solution 158 N/A 1.3339e-05 17.35
Dataset11 Problem statement Data Solution 158 N/A 1.4316e-05 19.64
Dataset12 Problem statement Data Solution 158 N/A 1.6291e-05 28.29
Dataset13 Problem statement Data Solution 158 N/A 0.00002 2.83
Dataset14 Problem statement Data Solution 158 N/A 0.000026 18.04
Dataset15 Problem statement Data Solution 158 N/A 0.000037 8.11
Dataset16 Problem statement Data Solution 158 N/A 0.000053 10.31
Dataset17 Problem statement Data Solution 158 N/A 0.000079 14.77

CASE STUDY SUMMARY

This Case Study demonstrates optimization of parameters of hinged beam under the influence of a number of periodic concentrated forces for excitation and formation of wave motion. The use of mathematical methods that lead to optimization problems is the traditional problems in the design of mechanical devices with optimum characteristics specified criterial (Balandin (1995), Komkov (1972)). However, in some cases, an optimization approach leads to mixed multi-extremal problems of constrained minimization of functionals, which are difficult to solve (Banichuk (1990), Banichuk, and Klimov (1991)). In this case study we consider a model problem of the vibrations of hinged beam under the influence of a number of periodic concentrated forces. The problem is to choose the parameters of force influence (amount of forces, their amplitudes, phases, and points of their application) to install the vibrations of a beam well (according to some criterion) met the specified parameters waveform.
The Case Study solves the problem for a fixed frequency (k = 2.5) with a serial relaxing constraints imposed on the amount of forces (I = 2, 3, 5, 6, 7, 8, 9, 10). Besides we solve the problem to determine the optimal characteristics for 5 forces (I = 5) within a frequency range comprising the resonance (k = 2.1, 2.3, 2.5, 2.7, 2.9, 3.1, 3.3, 3.5, 3.7).

References
• Balandin D. (1995), “On the optimal vibration absorption in elastic objects”. Applied Mathematics and Mechanics, Vol. 59, # З (In Russian).
• Komkov V. (1972), Optimal control theory for the damping of vibrations of simple elastic systems. Springer Verlag. Berlin. Heidelberg, New York.
• Banichuk N.V. (1990), Introduction to Optimization of Structures, Springer-Verlag, New York.
• Banichuk N.V., and D.M.Klimov (1991), Dynamical Problems of Rigid-Elastic Systems and Structures, Springer-Verlag, , New York.