Reliability-based weight reduction optimization of forearm of bucket-wheel stacker reclaimer considering multiple uncertainties

Abstract

This paper focuses on the weight reduction optimization of a forearm of a bucket-wheel stacker reclaimer considering uncertainties of structural parameters, material properties, loads, and surrogate model. However, the optimization problem is a high-dimensional problem, with dozens of independent variables, which has negative effects on the optimization efficiency. Considering that millions of iterations are required for the reliability-based optimization, the finite element model can cause overwhelming computational cost. In addition, due to its complex structure and working conditions, multiple uncertainties exist in practical applications and affect the reliability of a design, especially the uncertainty of the surrogate model. To address these challenges, the sensitivity analysis is performed to improve the optimization efficiency by selecting main factors. The Kriging model with high accuracy is constructed to reduce the computational cost. In order to improve the optimization efficiency further, the deterministic optimization is performed firstly, and the optimal design is used as the initial point of the reliability-based optimization algorithm. For estimating the reliability, the multiple uncertainty models are constructed. Finally, according to the design requirements and taking the multiple uncertainties into account, the reliability-based optimization is proposed and carried out. The result proves that the weight is reduced greatly and the reliability is kept at a high level.

Appendices

Appendix 1: Morris method

Since the number of model evaluations required for a sensitivity analysis is positively correlated with the number of variables, and the Morris method requires fewer evaluations than other methods (Saltelli et al. 2004), the Morris method is used for the sensitivity analysis in this paper. The Morris method calculates elementary effects (EE) for each variable based on several groups of OAT (one-factor-a-time) experiments (Richter et al. 2010; Touhami et al. 2013), and analyzes the overall influences and nonlinearities of the variables on the output by calculating the mean of the absolute values μ_i and standard deviation σ_i of their EEs. Suppose that r groups of OAT experiments are designed. The Morris method is expressed as

$$ {d}_i(x)=\frac{y\left({x}_1,\cdots, {x}_{i-1},{x}_i+\varDelta, {x}_{i+1},{x}_n\right)-y\left(\mathbf{x}\right)}{\varDelta } $$

(31)

$$ {\mu}_i=\frac{1}{r}\sum \limits_{j=1}^r\left|{d}_i^{(j)}\left(\mathbf{x}\right)\right| $$

(32)

$$ {\sigma}_i=\sqrt{\frac{1}{r}\sum \limits_{j=1}^r{\left({d}_i^{(j)}(x)-\frac{1}{r}\sum \limits_{k=1}^r{d}_i^{(k)}\left(\mathbf{x}\right)\right)}^2} $$

(33)

where x = [x₁,x₂,...,x_n] is the normalized vector of an input, each component of which is set to [0, 1], y(x) is the output, d_i(x) is the EE of the ith variable, Δ is the change of the ith variable, μ_i and σ_i represent the overall influence and the nonlinearity of the ith variable on the output, respectively, and they are two indicators for evaluating the sensitivity in the Morris method. A large value of μ_i indicates that the overall influence of the ith variable is large and a large value of σ_i indicates that the ith variable is significantly nonlinear or has a significant correlation with others. Therefore, a variable with a large value of μ_i or σ_i should be regarded as a main factor.

The results of the sensitivity analysis performed on the weight, the maximum stress, the maximum displacement, the first-order natural frequency under no-load situation, and the second-order natural frequency under full-load situation are shown in Fig. 13, where the center points represent μ_is, the one-side lengths of the error bars represent σ_is, and the ranges of μ_i ± σ_i for each variable are shown. Herein, if μ_i + σ_i ≥ 500 kg for the weight, μ_i + σ_i ≥ 8 MPa for the maximum stress, μ_i + σ_i ≥ 0.005 m for the maximum displacement, and μ_i + σ_i ≥ 0.01 Hz for the natural frequencies, the ith variable is considered as a variable with great influence on the outputs. It is found that all the variables have great influences on the weight while only some of them on the other outputs. Since the goal is to minimize the weight which is positively correlated with all the variables, main factors can be selected based on the latter four outputs and the remaining variables are set to their lower bounds. Finally, the selected main factors are h, l_d, l_u, t₁₃, t₁₄, t₁₅, t₁₆, t₂₃, t₂₄, t₂₅, w₁₃, w₁₄, w₁₅, w₁₆, w₂₃, w₂₄, w₂₅, w₂₆, w₃₃, w₃₅, w.

Fig. 13

Sensitivity analysis results. (a) Sensitivity analysis on weight. (b) Sensitivity analysis on the maximum stress. (c) Sensitivity analysis on the maximum displacement. (d) Sensitivity analysis on the first-order natural frequency. (e) Sensitivity analysis on second-order natural frequency

Appendix 2: Load calculation

Loads on the forearm can be divided into three parts: wind load, loads on the bucket wheel and loads on the conveyer.

• (1) Wind load

According to the design rules of cranes, the maximum wind pressure during the work is set to 250 Pa. In order to allow the bucket-wheel stacker reclaimer to work normally at the maximum wind pressure, the wind pressure p is 250 Pa. The wind load is calculated as

$$ {F}_{\mathrm{w}}= CpA{\sin}^2\theta $$

(34)

where C is the wind load coefficient, set as 1.7, A is the area of the windward surface, and θ is the angle between the wind direction and the normal direction of the surface. In order to maximize the effect of the wind load, θ is set to 90°.

For a double-side structure, the wind load on the leeward side decreases due to the windshield of the windward side. It is calculated as

$$ {F}_{\mathrm{w}}^{\prime }=\eta {F}_{\mathrm{w}} $$

(35)

where η is the windshield coefficient, determined by the interval ratio and the full ratio of the windward side.

As the wind load is applied as a line pressure, and according to (36), where l is the length of the beam and f is the line pressure, the value of the line pressure is set as 1.7pw₃ on the windward side and 1.7ηpw₃ on the leeward side.

$$ {F}_{\mathrm{w}}=1.7 pl{w}_3= fl $$

(36)

$$ {F}_{\mathrm{w}}^{\prime }=1.7\eta pl{w}_3={f}^{\prime }l $$

(37)

• (2) Loads on the bucket wheel

Loads on the bucket wheel are mainly composed of its own weight m₀, the weight of material in the bucket wheel m₁ and the cutting force F, where m₀ is known in advance.

The weight of the material m₁ is calculated at the situation of 1/4 full load of all the buckets (Li 2013), which is written as

$$ {m}_1=\frac{1}{4} V\rho z $$

(38)

where V is the volume of a bucket, ρ is the density of the material, and z is the number of the buckets.

The cutting force is calculated according to the power of the drive motor, which is written as (Li 2013)

$$ P+{P}_{\mathrm{h}}=\eta {P}_{\mathrm{a}} $$

(39)

$$ P=\frac{F{v}_{\mathrm{c}}}{1000} $$

(40)

$$ {P}_{\mathrm{h}}={Q}_{\mathrm{L}}\rho gh $$

(41)

where P is the power of the cutting force, P_h is the power consumed for lifting the material, η is the transmission efficiency, P_a is the rated power of the drive motor, F is circumferential cutting force, v_c is the circumferential speed on the tip of the bucket wheel, Q_L is the theoretical productivity, and h is the lifting height of the material, approximately equal to the radius of the bucket wheel.

There is also a pressure along with the radius of the bucket wheel from the material on the bucket. According to the empirical formula (Li 2013; Yang 2013), it is written as

$$ {F}^{\prime }=0.3F $$

(42)

• (3) Loads on the conveyer

Loads on the conveyer are composed of the weight of the conveyer system m₂ known in advance, and the weight of material m₃ calculated as (Li 2013)

$$ {m}_3=\frac{fQL}{3600{v}_{\mathrm{b}}} $$

(43)

where f is the dynamic load factor, Q is the rated production capacity (the unit is kg/h), L is the length of the forearm, and v_b is the speed of the conveyer.

Appendix 3: Replication of results

In order to facilitate the replication of results in this paper, the MATLAB code files for calculating the POFs of a design are provided as supplementary material. Table 11 shows brief descriptions of all the files. The reliability-based weight optimization can be conducted conveniently with the help of the files.

Table 11 MATLAB code files in the supplementary material

There are 4 files in total, where the “ModelInfo.mat” contains all the messages used to construct the Kriging model. “POF_calculation.m” is the main program which is used to calculate the POFs of all the constraints under a design. “rlh.m” is used to generate a random Latin hypercubic sampling. “ReturnX.m” is used to convert each design variable value to a number between [0, 1].