Static and Modal Analysis and Optimization of Cross-Sectional Area of the Manipulator Arm

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Figure 2. Arm of manipulator

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2. The Calculation of Loading for Manipulator Arm Based on the Static Loading

Calculation of arm LA 425 as a part of manipulator K1 - K5 was made on the basis of static loading 20 000 [N for the highest extension of the arm where the loading is the highest. Using the volume elements, the finite – element model (Figure 3) was made for manipulator arm LA 425 (Figure 2).

Figure 3. Finite element model of the manipulator arm LA425

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Arm is made of material 11 523.1.

Stability equations are solved in this way ^[1]:

(1)

Static analysis of the manipulator arm includes solution of the stability equations (1). System of equation is solved by Gaussian method of the elimination. This method includes positive definite symetric system of equation. The algorithm performs minimal operation number. There are not performed L^TDL operations with zero elements.

After the calculation of the nodal point displacements, calculation of the loading of elements is made. The relation between deformation vector and displacement are expressed by equation:

(2)

where [B] is transformation of deformation – matrix of displacement. Loading vector represents a conjunction of the transformation deformation - matrix loading [D ] with the deformation vector

(3)

Usage of (2) in (3), the relation between loading vectorand vector of nodal point displacements can be obtained

(4)

Distribution of the stresses in arm of manipulator is given in MPa and it can be seen in the Figure 4 and detail of the distribution of stresses is shown in the Figure 5.

Figure 4. Distribution of stresses in [MPa]

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Figure 5. Detail of distribution of stresses in [MPa]

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3. The Calculation of the Eigenfrequencies and Eigenshapes for Manipulator arm

Mathematical model can be expressed in form (5)

(5)

where M is the mass matrix, C is the damping matrix, K is the stiffness matrix, is the time varying loading vector and,, are the displacement, velocity and acceleration vectors, respectively.

In relation to the calculation of the arm LA 425 of manipulator K1 - K5, the static loading was 4 000 kg. The computational model (Figure 2) representing the manipulator arm LA 425 (Figure 3) was created by help of the eightnodal volume elements.

Modal and frequency analysis were done according to known equation in the form:

(6)

where is the i-th eigenfrequency, is the eigenvector representing the nodal shape of the i-th eigenfrequency. We calculated eigenfrequencies and from the eigenfrequencies and on the basis of the eigenfrequencies, the frequency characteristics with the modal damping  = 0,001 was calculated The motion equation for the i-th eigenshape is in accordance with:

(7)

This relationship is generalized problem of the eigenvalues where solution is made by the semi-automatic-space iterative method. This method is based on the idea of the inverse iteration conversion with several vectors at the same time.

In the Figure 6, we can see the first three shapes of the eigenfrequencies and in Table 1, there are the first 10 eigenfrequencies calculated by the help of the underspace method.

Table 1. The first ten eigenfrequencies

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Tables index

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Figure 6. The first three shapes of the eigenfrequencies

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4. Optimization of Cross-sectional Area of the Manipulator Arm

In relation to the material saving, long service life, reliable and safe operation of mechanical equipment, nowadays strict requirements call for new approaches in putting various technical ideas into practice. It can be fulfilled only with the usage of suitable computing systems for fast and efficient study of the parameters which influence the static or dynamic characteristics of a machine in a greater or a lesser extent.

4.1. General Formulation of the Optimization Problem

In the first step, it is necessary to create correct physical machine model (static or dynamic). Second step is focused on the investigation and model description by the proper mathematic means – creation of the computational model (equations system, couplings conditions, etc.) The third and often most difficult procedure of optimum construction designing is connected with mathematical formulation of the optimization problem. It can be formulated as looking for extreme conditions (it is minimum value of extreme conditions) of the objective function:

(8)

Furthermore, it is based on additional (limiting) conditions which are given by irregularities which determinate admissible area:

(9)

where is vector of designing (design) variables. Equations (8) and (9) contain the general non-linear functions of designing parameters. They belong to problems of mathematic programming which is often based on non-linearity. For research of global minimum, KuhnTucker obligatory conditions were used:

(10)

Mathematic programming belongs to direct methods of optimization problems solution. In consequence of general non-linear character of objective function and limitation functions, equations (8), (9), (10), there will be non-linearity. In majority cases, this state can not be solved analytically. Then it is necessary to apply iterative methods, which allow computing vectors chain (r = 1, 2.....) which converge to optimum. Direct equation application (10) in optimization process impacts on the problems with the limitations in non-equations form and therefore, there are other methods of non-linear programming used. The most common application in optimization of problems relates to methods of “zero” place value (comparator) and methods of “first” place value (gradient) but the second one mentioned methods are more effective. Their general disadvantage is that there is the problem to perform the analysis of sensitivity. Fundamentally, it deals with various modified iterative methods of type. General idea of these iterative approaches can be expressed by equation:

(11)

Just like in the equation (10), the operator ∇ expresses gradient of objective function in the k-th iterative step for direction of progress optimum finding. It has been shown in practice that direct optimizing methods involve rather demanding computations which can be quite complicated from the technical aspect. However, there are also indirect methods that do not operate directly with the target function and they are used only for evaluation of the limitations (usually stresses). One of such methods is the “Fully stress design”, and we will utilize its basic principles and procedures hereinafter. The method is based on the idea of designing the structural variables in such a manner that the stresses in the entire structure will be at the permissible limit.

4.2. The method of “Fully Stress Design”

This method is used to find an optimum solution utilizing the overall loading capacity of the material at the given loading. The limiting conditions for the stresses may be written as follows:

(12)

where isthe stress for the given point and is the maximum permitted stress. As an example, an optimum design of beam structures will be used with respect to its sectional characteristics.

4.3. Optimization of Sectional Parameters by Beam Elements

We consider duo node beam finite element with constant cross-section. In relation to this element, there are slide forces, axial forces, torque constant and elastic moments changing in a linear way between both ending nodal points. Maximum stress will be for one or other end of the beam, therefore points of calculated stresses will be there. Except this fact, we have define also position of points, for which we compute stresses in cross-sectional area. Final stresses will be calculated according to equations:

axial compound of normal stress:

(13)

slide stress from slide forces:

(14)

slide stress from torque:

(15)

where F_x, F_y, F_z are compounds of internal force in given cross-section, M_x, M_y, M_z are internal moments, A is cross-sectional surface, J_y and J_z are axial quadratic moments of persistence, J_k is cross-sectional torque moment, S_y and S_z are static moments of cross-sectional part dedicated by point area where we calculate the stress, c_y and c_z are point distances in section plane, in which we calculate stresses from cross-sectional centre of gravity, e_y and e_z are coordinates of eccentricity (distance between centre of gravity and slide centre), r_T is effective radius of torsion. Equivalent stress is calculated according to HMH hypothesis (von Misses):

(16)

Computational points for beam cross-section have to be defined for the sites where the extreme values can be expected. Furthermore, we have to define stiffness matrix of three-dimensional beam finite element and calculation of internal force parameters, from which the stresses in endpoint nodes will be assigned according to equations (13), (14), (15). If axis y, z will be central, then S_y= 0, S_z= 0, D_yz= 0 and we obtain known, in literature very often published, stiffness matrix of beam element. There are many approaches in optimization of beam cross-section. One of the approaches can be optimization of total height, width or both cross-sectional variables. The optimization modulus in Cosmos program can be used for this work.

4.4. Optimization of Cross-sectional Area of the Manipulator Arm

Optimization of the manipulator arm was done in program COSMOS 2.85 and the optimum designing of the model (Figure 2) led to optimization of the whole area of beam arm.

The cross-sectional area of arm for individual beam elements was changed with element width of arm and element height was constant. Stress limitation value was specified at σ_idov = 150 [MPa]. The arm was loaded by own weight and maximal (standard) force 20 000 N at one and second side of arm. Arm weight before optimization was 48.9 kg.. Stress distribution in manipulator arm before optimization is in the Figure 4. Stress distribution in manipulator arm after optimization is at Figure 7.

Figure 7. Stress distribution after optimization in [MPa]

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Optimization results for individual iterative steps are in the Figure 8.

Figure 8. Convergence graph – objective function

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