Dynamic Modeling of a Hydraulic Excavator Stick by Introducing Multi-Case Synthesized Load Spectrum for Bench Fatigue Test

Abstract

A multi-case load spectrum compiling method is proposed in this study for dynamic modeling of a hydraulic excavator stick to simplify and accelerate the fatigue bench test. This new method includes a simplified criterion of small-load-omitting threshold based on the principle of invariable fatigue damage, an extreme value inference criterion based on the overflow characteristics of the hydraulic system, and a synthetic extrapolation method under various working conditions. Firstly, a one-dimensional spectrum of a medium-sized excavator stick was compiled. Then, the program load spectrum for the bench fatigue test was obtained by modifying the one-dimensional spectrum based on the damage consistency criterion and the damage equivalent principle. Lastly, the fatigue tests were conducted using the program load spectrum, as well as using the random spectrum. The comparison results demonstrate that the damage location and fatigue life distribution of the stick using these two spectra are generally consistent, with a relative error smaller than 8.8%; however, the proposed program load spectrum can accelerate the test process with less time consuming than that of the random spectrum. As a result, the multi-case load spectrum is feasible and reliable for dynamic modeling of the hydraulic excavator stick in practice.

1. Introduction

The fatigue life of the key structural components is a critical index that affects the safety and reliability of the hydraulic excavator. To evaluate the fatigue life, the bench fatigue test is widely adopted. One prerequisite of the bench fatigue test is that the load in the test must reflect the influence of the real load on the fatigue life of structural components. As the structural components of the excavator usually suffer random load in practical applications, the random load spectrum, which can consider the load sequences, is the optimal load choice. However, the random fatigue test has high requirements on the dynamic characteristics of the loading system. Moreover, the test process is complex and costly, and the test cycle is very long, so the random load spectrum is often converted into a program spectrum to perform the fatigue test.

Currently, there is a lack of a published standard to compile the fatigue test program spectrum for excavator hydraulic sticks, so the program spectrum can only be obtained from the measured data. In some studies, the measured load was used to compile the program spectrum for the bench fatigue test. For example, based on several actual engineering failure cases, Bošnjak, Srđan, and Arsić et al. [1,2,3,4,5] analyzed the causes of fatigue failure of key structural parts of bucket wheel excavators. However, in their research, the external load on the structure was not measured. Yin et al. [6] studied the external load test method for the electric drive BE-395B front shovel excavator, and obtained the hinge joint force, cable tension, and bucket tip force to make the program spectrum. Bae et al. [7] proposed a method to extract the fundamental fatigue load from the measured stress time histories collected by sensors arranged on the boom. The above studies did not carry out the fatigue life assessment of structural parts from the perspective of compiling the program spectrum to carry out the bench fatigue test. Taking the stress at the dangerous points of the stick and the boom as the intermediate quantities, Gao [8] and Shi [9] calculated the loading force of the bucket tip based on the stress equivalent principle. Then, the loading force was compiled into a program spectrum to conduct the bench fatigue test of an excavator with a fixed attitude. However, this method relies on the position selection of the dangerous points. Moreover, the method can only ensure that the stress states of the selected dangerous points are the same as the actual ones, while it cannot effectively evaluate the fatigue life of other dangerous points.

Two issues exist for applying the measured load to compile the program spectrum used for the bench fatigue test of the key structural parts of the excavator: (1) It is difficult to obtain the external load of the excavator working device through directly test, because the working environment of the excavator is harsh, the load conditions under different working media are quite different, and the magnitude and direction of the external load of the working device are constantly changing during operations. It is necessary to propose a special load identification method. (2) Directly compiling the measured load into the bench fatigue test program spectrum is also hard due to the contradiction between the fixed attitude of the whole machine fatigue test and the time-varying attitude of the actual operation. Hence, it is necessary to propose an equivalent method to convert the measured complex load to the unidirectional load required for the bench fatigue test to ensure that the bench fatigue test can reflect the actual stress state of the structure. Considering that the external load is difficult to test, a cross-section stress testing method based on strain testing was proposed [10,11], and a three-dimensional axis pin force sensor was developed [12]. These two methods can measure all the loads of the excavator working device including the side load and eccentric load. In view of the contradiction between the fixed attitude of the whole machine fatigue test and the time-varying attitude of the actual operation, a method of respectively arranging the load spectrum of the stick in their local coordinate systems and performing bench fatigue tests was put forward [13]. At the same time, based on the damage consistency criterion of key fatigue points, the multi-directional load of the measured hinge point was equivalent to the unidirectional load required for the bench fatigue test. This method overcomes the problem that the relationship between the forces cannot be reproduced when the load components of each hinge point of the stick are compiled into a load spectrum for the fatigue test, whilst ensuring that the fatigue test of the bench can reflect the actual stress state as much as possible.

Based on our previous work, this study further investigates the compilation of the load spectrum. A program spectrum compilation method for bench fatigue test of hydraulic excavators is proposed in this study, and the following problems will be solved: (1) How to consider the diversity of the actual working medium of the excavator in the compiled load spectrum; (2) How to determine the threshold value of cyclic elimination of small-value loads in spectral compilation since external loads vary greatly in different operating media; (3) What is the influence of the dynamic characteristics of the excavator hydraulic system and the overload protection device on the extrapolation of extreme values during spectrum compilation; (4) How to standardize the compiled program spectrum to improve its applicability since there are many excavator manufacturers of the same tonnage. All these problems have not been resolved yet. To bridge this research gap, this study proposed a multi-case load spectrum compiling method to solve these problems.

The remainder of this work is organized as follows. Section 2 introduces the proposed method. Analysis and results are presented in Section 3. Section 4 concludes the main findings.

2. Materials and Methods

The compilation process of the bench fatigue test of the excavator stick is summarized in Figure 1. This paper mainly focuses on the following key steps: the threshold value of small load omitting; the extreme value inference criterion of the mean and amplitude; the synthesis and extrapolation of various typical operating conditions; and the correction, acceleration, and normalization of the program load spectrum.

2.1. Determining Threshold of Small Amplitude Cyclic Load

Many load cycles will be obtained after peak-valley extraction of random load-time history, among which the load cycles with smaller amplitudes account for a large proportion, which will cause a large workload to the rain flow counting. However, these load cycles do not necessarily cause fatigue damage to the structure, or do not affect the fatigue life of the structure, and can be omitted in advance.

The criteria for determining the omitting threshold value of small amplitude cyclic load (hereinafter referred to as ‘small loads’) have not been finalized or widely used because the relevant parameters in the calculation model are difficult to obtain [14,15]. The common omitting criteria are divided into three categories: ① Omit according to the percentage of the maximum load amplitude or maximum cyclic load range [16,17]; ② Omit according to the percentage of the material fatigue limit, such as the load less than 50~70% of the fatigue limit is usually removed [18,19]; ③ If the load belongs to the normal distribution, the load less than 1.75σ will be omitted (σ is the standard deviation of the cyclic load) [20]. However, these three empirical methods cannot reflect the contribution of the small load to fatigue damage, which may cause a large deviation in the structural life evaluation.

Considering the engineering practicability, this paper proposes a simple and convenient method to determine the omitting threshold value of small load from the perspective of fatigue damage. In this method, the omitting threshold value is determined as the critical value on the premise of keeping the fatigue damage at the dangerous point unchanged before and after the small load is omitted from the full load-time history. In other words, the omitting threshold is determined according to whether fatigue damage is caused to the structure. The “fatigue damage” here is calculated according to Miner’s rule and the S-N curve of the structural details at specific dangerous points. It is worth noting that this threshold is not the fatigue limit in smooth materials.

For large and complex welded structures such as excavator sticks, the sensitivity of fatigue damage at each dangerous point of the structure to the load is not consistent when external loads are applied. Therefore, the omitting threshold value of each dangerous point should be determined respectively, and the minimum value of them should be taken as the final threshold value. The calculation process is shown in Figure 2.

Table 1. The small-load-omitting threshold value of the unidirectional equivalent stick load.

Threshold Value Calculated by the Proposed Method (kN)		19.185
10~15% of the maximum cyclic load range (kN)	Category I: Loose soil	15.962~23.943
	Category II: Original soil	20.764~31.145
	Category III: Clay with small stones	18.762~28.143
	Category IV: Heavy clay containing large stones	21.083~31.624

compares the omitting threshold value calculated by the proposed method with the values of 10~15% of the maximum cyclic load range under four typical digging conditions of a medium-sized excavator’s stick. It can be seen that the threshold value determined by this method, 19.185, is close to but not completely within the range of 10~15% of the maximum cyclic load range, indicating that it is reasonable but not completely reliable to use 10~15% of the maximum cyclic load range to determine the threshold value. In addition, the omitting threshold value can remain a constant under the different digging conditions.

2.2. Load Extremum Extrapolation Criterion

In engineering, the load with an occurrence probability of 1 × 10⁻⁶ is generally regarded as the extreme load. Several commonly used distributions, such as normal distribution, log-normal distribution, two-parameter Weibull distribution, three-parameter Weibull distribution, Gamma distribution, and extremal distribution are respectively used to conduct the Kolmogorov–Smirnov test (KS distribution test) on the load average and load amplitude of the medium-sized excavator’s stick. Taking the Category IV digging object as an example, the distribution test results are shown in the

Table 2. The different distributions’ D value of the load amplitude of the stick using the KS distribution test method.

Critical D Value	Normal Distribution	Log-Normal Distribution	Two-Parameter Weibull Distribution	Three-Parameter Weibull Distribution	Gamma Distribution	Extremal Distribution
0.0355	0.1510	0.1726	0.1500	0.0303	0.0843	0.0380

.

It can be seen that the load amplitude value distribution is more in line with the three-parameter Weibull distribution (D value 0.0303 < critical D value 0.0355). So, the extreme load Y_{i_max} of amplitude value under the i-th digging object is:

$$Y_{i\_\max }=c_i+a_i\sqrt[b_i]{-\ln P}$$

(1)

where a_i, b_i, and c_i are the scale, shape, and position parameter of the three-parameter Weibull distribution of the equivalent load amplitude under the i-th digging object; P = 1 × 10⁻⁶ is the probability of amplitude extremum.

The mean value obtained from the rain flow counting of the equivalent load-time history was fitted with log-normal distribution, so the extreme load X_{i_max} of the mean value under the i-th digging object is:

$$X_{i\_\max }=e^{-U_P{\sigma }_i+{\mu }_i}$$

(2)

where μ_i and σ_i are the mean and standard deviation of the log-normal distribution of the equivalent load mean value under the i-th digging object; U_P can be obtained from the standard normal distribution.

Therefore, the extreme amplitude value Y_max after the synthesis of the four digging objects is:

$$Y_{\max }=\max \left(Y_{i\_\max }\right)$$

(3)

The extreme mean value X_max after the synthesis of the four digging objects is:

$$X_{\max }=\max \left(X_{i\_\max }\right)$$

(4)

With the above equations, the theoretical value of the extreme load of mean and amplitude are derived.

However, the hydraulic system of the excavator usually has an overload protection system. Therefore, limited by the set pressure of the main safety valve and the actuator relief valve in the hydraulic system, the theoretical extreme value may not be reached. If the main safety valve or the actuator relief valve overflows during the load spectrum test, it indicates that the load spectrum test has recorded the extreme load that may be generated in practice, and there is no need to infer the extreme value according to the Conover theory. Therefore, the cylinder pressure curves measured during the digging of the primary soil were analyzed, as shown in Figure 3.

It can be seen from Figure 3 that the pressure in the rodless chamber of the stick cylinder exceeded the preset pressure 316 bar, which is the main safety valve of the system and was obtained through the maximum digging force test. The overflow phenomenon also occurred: the measured peak pressure exceeded the set pressure of the system, then gradually attenuated and fluctuated, and finally stabilized at the rated pressure. This phenomenon is caused by the dynamic characteristics of the relief valve when the main safety valve suddenly changes from the closed state to the rated pressure state. This phenomenon occurred many times in the whole load spectrum test process, which indicates that the extreme load had been captured in the load spectrum test.

The extreme amplitude values of the test prototype excavator’s stick determined from three different perspectives, namely, the Conover theoretical extreme value, the extreme value measured in the static maximum digging force test, and the extreme values captured from the actual working conditions are compared in

Table 3. Extrapolation of extreme amplitude load of the test prototype of a medium-sized excavator.

① Conover Theoretical Extreme Value	② Extreme Value Measured in Static Maximum Digging Force Test	③ Extreme Value Measured in Actual Digging Process
212.2 kN	68.6 kN	105.4 kN

. It is seen that the extremum extrapolated according to the Conover theory was much larger than the extremum value measured in the actual digging process with overflow phenomenon, while the extremum value measured in the static maximum digging force test was much smaller than the extreme value measured in the digging process. Such a large difference existing between the extremum values obtained by different extrapolation criteria would inevitably cause great differences in the results of the load spectrum compilation. Therefore, during the compilation of the program load spectrum, the extreme load should be determined according to the actual digging process, instead of extrapolating according to the Conover theory.

2.3. Extrapolation and Synthetic Method of Load Spectrum

In previous work [21], the common working conditions of hydraulic excavators were investigated, and four different digging objects and their proportions were determined. In order to reflect the actual load situation as much as possible, the excavator load spectrum should be measured while digging these four objects, respectively; hence, the data synthesis under different digging objects should be considered when compiling the program load spectrum. For this reason, a new method is proposed to simultaneously complete the data synthesis and frequency extrapolation of multiple digging objects in one step.

① According to the distribution parameters of the mean and amplitude in various digging objects, the corresponding joint probability density function f(x,y) and joint probability distribution function F(x,y) of the mean and amplitude could be obtained:

$$f(x,y)=\frac{1}{x\sigma \sqrt{2\pi }}\exp \left(\frac{-{\left(\ln x-\mu \right)}^2}{2{\sigma }^2}\right)\left(\frac{b}{a}\right){\left(\frac{y-c}{a}\right)}^{b-1}\exp \left(-{\left(\frac{(y-c)}{a}\right)}^b\right)$$

(5)

$$F(x,y)={\displaystyle {\int }_{-\infty }^y{\displaystyle {\int }_0^{x}f(x,y)dxdy}}$$

(6)

② Calculate the total frequency, N_i, of the mean and amplitude of the i-th digging object within the full sample length T:

$$N_i=n_i{k}_{1i}T/t$$

(7)

where T is the full sample length (number of digging cycles/bucket); k_1i is the proportion of the i-th digging object obtained from the investigation; t is the actual collection bucket number of the i-th digging object; n_i is the total number of cycles obtained from equivalent load rain flow counting of the i-th digging object; i = 1, 2, 3,…, q, and q is the total number of digging object types.

③ Calculate the extrapolation frequency, N_i′, of the i-th digging object:

$$N_i’=k_{2i}\times {10}^6$$

(8)

where k_2i is the frequency extrapolation coefficient of the i-th digging object; k_2i = N_i/(∑N_i). i = 1, 2, 3,…, q, q is the total number of digging object types. It can be seen that the extrapolation frequency N_i′ of the i-th digging object is independent of the full sample length T.

④ Synthesize the four digging objects to obtain the two-dimensional mean-amplitude program load spectrum. On the basis of the extreme mean value and extreme amplitude value determined above, the amplitude of various digging objects was divided into 8 levels according to the 1, 0.95, 0.85, 0.725, 0.575, 0.425, 0.275, and 0.125 times of the extreme amplitude value, and the mean value was divided into 8 levels with equal intervals. The frequency n_iuv on each mean-amplitude level of the i-th digging object is calculated by:

$$n_{iuv}=N_i’{\displaystyle {\int }_{R_u}^{R_{u+1}}{\displaystyle {\int }_{M_v}^{M_{v+1}}\frac{1}{x\sigma \sqrt{2\pi }}\exp \left(\frac{-{\left(\ln x-\mu \right)}^2}{2{\sigma }^2}\right)\left(\frac{b}{a}\right){\left(\frac{y-c}{a}\right)}^{b-1}}}\exp \left(-{\left(\frac{(y-c)}{a}\right)}^b\right)dxdy$$

(9)

where M_v and M_v+1 (v = 1, 2, 3,…, 8) are the lower and upper integral limits of the mean value of the v-th level under the i-th digging object; R_u and R_u+1 (u = 1, 2, 3,…, 8) are the lower and upper integral limits of the amplitude value of the u-th level under the i-th digging object.

Then, the frequency n_uv corresponding to the v-th mean value and the u-th amplitude of the synthesized two-dimensional program load spectrum is:

$$n_{uv}={\displaystyle \sum _{i=1}^q{n}_{iuv}}$$

(10)

Finally, a two-dimensional program load spectrum compiled according to the above steps of the stick of a medium-sized excavator can be obtained, and it is presented in

Table 4. The mean-amplitude program load spectrum of a medium-sized excavator stick.

Level	Amplitude (kN)	Mean (kN)
		−48.17	−23.94	0.29	24.52	48.75	72.98	97.20	121.43
1	13.18	7829	75,035	182,958	201,256	137,814	70,008	29,144	15,612
2	28.99	1838	17,819	44,317	49,747	34,609	17,769	7449	4015
3	44.80	671	6507	16,186	18,178	12,656	6503	2729	1474
4	60.61	247	2394	5950	6679	4650	2391	1004	543
5	76.42	87	841	2086	2340	1628	838	352	191
6	89.60	31	298	738	827	576	296	125	68
7	100.14	11	106	262	293	204	105	44	24
8	105.41	17	167	412	460	320	165	69	38

.

2.4. One-Dimensional Program Load Spectrum

In order to facilitate the test loading, the fluctuation central method was used to transform the two-dimensional program spectrum into one-dimensional to keep the amplitudes of each level in the two-dimensional program spectrum unchanged. Then, the eight-level mean values corresponding to the amplitude of each level were weighted and averaged. The mean value M_u corresponding to the u-th level amplitude is calculated as follows:

$$M_u={\displaystyle \sum _{v=1}^8{M}_v{n}_{uv}}/{\displaystyle \sum _{v=1}^8{n}_{uv}}$$

(11)

The frequency n_u corresponding to the u-th level amplitude is:

$$n_u={\displaystyle \sum _{v=1}^8{n}_{uv}}$$

(12)

Considering that the actual load is random, that is, the load does not appear in a specific order, in order to reduce the difference between the program spectrum loading and the actual load, the test should be repeated several times. The size of the subroutine block should be determined according to the preliminary life estimation results so that the whole test includes at least 10 subroutine blocks.

The fatigue life represented by the spectrum in Table 3 is denoted as l_P. Given that the full life of the structure is N and the total number of subroutine blocks is C, then the life represented by one subroutine block is: l = N/C; When the spectrum in Table 3 was extended to a subroutine block, the frequency corresponding to the u-th level amplitude is modified to n_u′:

$$n_u’=\beta n_u$$

(13)

where β = l/l_p is the expansion factor.

According to industry experience, the full life target of excavator working device structure is generally about 8000~10,000 hours. In this paper, let a subroutine block represents 1000 h, and then the spectrum in Table 3 is further expanded in frequency according to Equation (13) to obtain the one-dimensional program load spectrum representing 1000 h.

Taking the test prototype of a medium-sized excavator as an example, the spectrum after frequency expansion is shown in

Table 5. One-dimensional program load spectrum with variable mean for a medium-size excavator stick (1000 working hours).

Level	Amplitude (kN)	Mean (kN)	Frequency
1	13.18	26.91	1,120,745
2	28.99	27.67	276,525
3	44.80	27.70	101,077
4	60.61	27.70	37,155
5	76.42	27.69	13,024
6	89.60	27.70	4608
7	100.14	27.64	1634
8	105.41	27.63	2566

.

2.5. Compilation of Program Load Spectrum

The program spectrum compiling process consists of many steps, and errors will inevitably occur in each step, eventually resulting in the error between the damage reproduced in the bench fatigue test and the damage in the actual service state. Therefore, the one-dimensional program load spectrum in Table 4 should be corrected before it is used in the bench fatigue test to reduce the error between the bench fatigue test and the actual service state.

2.5.1. Fatigue Damage Consistency Correction of the Program Load Spectrum for Bench Fatigue Test

It is a very complicated problem to study the error of each step in detail. In this paper, the damage consistency criterion and the damage of the key fatigue measuring points on the structure were used to modify the load spectrum. The damage of the key fatigue measuring points on the structure caused by the program load spectrum should be close to the damage caused by the actual digging process on the premise of ensuring the damage of the key fatigue measuring points on the structure is not greater than or equal to the actual damage of each measuring point calculated by the measured stress spectrum. The specific steps are as follows:

Firstly, the stress time history of each fatigue key measuring point on the structure was compiled into a stress spectrum. Then, according to the S-N curve of welded joint and the Miner cumulative damage rule, the actual damage D_rj of each measuring point is calculated by:

$$D_{rj}={\displaystyle \sum _{i=1}^{n_s}\frac{n_{ij}{({\sigma }_{ij})}^{m_j}}{C_j}}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}j=1,2,\dots ,h$$

(14)

where h is the number of key fatigue measuring points; m_j and C_j are the S-N curve constants of the j-th key fatigue measuring point, and n_s is the number of the stress spectrum levels; σ_ij and n_ij represent the i-th level amplitude value and corresponding frequency of the stress spectrum of the j-th key fatigue measuring point, respectively.

Secondly, the damage D_pj of the key fatigue measuring points using the one-dimensional program load spectrum listed in Table 4 is calculated by

$$D_{pj}={\displaystyle \sum _{k=1}^{n_p}\frac{n_k{(P_k{k}_j)}^{m_j}}{C_j}}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}j=1,2,\dots ,h$$

(15)

where n_p is the number of the one-dimensional program load spectrum levels, here, n_p = 8; P_k and n_k denote the k-th level amplitude value and its corresponding frequency of the one-dimensional program load spectrum, respectively. k_j is the load-stress proportional coefficient between P_k and the stress of the j-th key fatigue measuring point, which is only related to the location of the key fatigue measuring point on the structure.

Then, the correction factor γ is used to correct the one-dimensional program load spectrum, and the corrected amplitude of each level is P_k′ = γP_k. The damage D_pj′ of the key fatigue measuring point caused by the modified one-dimensional program spectrum is:

$$D_{pj}’={\displaystyle \sum _{k=1}^{n_p}\frac{n_k{(\gamma P_k{k}_j)}^{m_j}}{C_j}}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}j=1,2,\dots ,h$$

(16)

The damage consistency correction optimization model is established as follows:

Objective function: $\min \left\{{\displaystyle \sum _{j=1}^h{\left(D_{rj}-D_{pj}’\right)}^2}\right\}$

Constraint condition: $D_{rj}\le D_{pj}’$

Finally, Equations (14) and (16) were substituted into the optimization model to acquire the correction coefficient γ, and the program spectrum P_k′ = γP_k after damage consistency correction that would be used as the bench fatigue test program spectrum is obtained. In the example of this article, the correction coefficient of the stick program load spectrum solved by the optimization model is 1.104.

2.5.2. Acceleration and Normalization of Program Load Spectrum for Bench Fatigue Test

In order to accelerate the test, the above program load spectrum needs to be concentrated in time main. The small load less than the fatigue cut-off line will be omitted, but the omission should have no influence on the damage of each fatigue key point on the structure.

In addition, to facilitate the test loading, the eighth level load with larger load values but fewer loading cycles was converted to the seventh level according to the principle of equal damage, and the new frequency n₇’ was calculated according to Equation (17); meanwhile, the second level load with smaller load values but more loading cycles was converted to the third level, and the new frequency n₃’ was calculated according to Equation (18).

$$n_7’=n_7+\frac{n_8{\sigma }_8^{m_8}{C}_7}{{\sigma }_7^{m_7}{C}_8}$$

(17)

where σ₇ and n₇ represent the seventh load level and corresponding frequency in the above eight-level program load spectrum; m₇ and C₇ represent the constants of the S-N curve corresponding to the seventh load level; σ₈ and n₈ represent the eighth load level and corresponding frequency in the above eight-level program load spectrum; m₈ and C₈ represent the constants of the S-N curve corresponding to the eighth load level.

$$n_3’=n_3+\frac{n_2{\sigma }_2^{m_2}{C}_3}{{\sigma }_3^{m_3}{C}_2}$$

(18)

It is known that the tonnage of excavators varies significantly, even for those belonging to the same tonnage category. For instance, the tonnage of the medium-sized excavator ranges from 12.5 to 25.5 tons, while the large-sized excavator ranges from 35.8 to 49.2 tons. Because different tonnage excavators suffer different external loads, the program load spectrum should be different for different tonnage excavators. However, making program load spectra according to each tonnage level is unrealistic, as this process is time consuming and expensive. Hence, the program load spectrum is normalized to make it applicable for various excavators with different tonnages. Considering the maximum bucket cylinder digging force is a characteristic parameter that indicates the working capacity of the excavator, the maximum bucket cylinder digging force of the medium-sized excavator test prototype, 138 kN, is taken as unit 1, and the relative values of the upper and lower load limits in the program load spectrum are given in

Table 6. Relative values of programmed load spectrum of different tonnage excavators.

Level	Upper Limit of the Amplitude (kN)	Lower Limit of the Amplitude (kN)	Frequency
1	0.56	−0.16	157,683
2	0.69	−0.28	37,155
3	0.81	−0.41	13,025
4	0.92	−0.52	4608
5	1.00	−0.60	4628

. It is worth noting that this normalization is an approximation for engineering applications. The premise of this method is that the external load encountered in the actual work is basically proportional to the “maximum bucket cylinder digging force”.

For an excavator with a different tonnage, its program spectrum can be obtained by multiplying the relative values in Table 5 by the maximum bucket cylinder digging force.

During the bench fatigue test, the subroutine of low-high-low sequence was repeated to load the tested object. For example, Figure 4 shows the program load spectrum loading sequence of the bench fatigue test of the medium-sized excavator’s stick.

3. Bench Fatigue Test

The fatigue life virtual prediction technology was used to verify the validity of the bench fatigue test program load spectrum compiled in this paper. The bench fatigue test program load spectrum and the random equivalent load-time history before compiled, which is hereinafter referred to as a random spectrum, were respectively used in the fatigue life tests, and the results using these two spectra are compared to evaluate the proposed program load spectrum compiling method.

3.1. Structural Stress Analysis under Unit Load

The random load at the hinge points of the excavator are low-frequency loads and does not include the natural frequency of the structure. Therefore, the stress results of static finite element analysis were used as the input of the fatigue analysis.

The finite element model of the stick was established in the finite element analysis software, ANSYS R15.0. The two kinds of load spectra to be applied were unidirectional load. Therefore, in the model, the 5 degrees of freedom of the hinge point of the stick and the boom and the 5 degrees of freedom of the fixed end of the stick cylinder rod were constrained, respectively. Then, the unit vertical force was applied at the hinge point of the stick and the bucket, as shown in Figure 5.

3.2. Fatigue Analysis and Discussion

The results of ANSYS were imported into the fatigue analysis software, nCode DesignLife, to predict the fatigue life. A fatigue analysis process that can withstand the above two load spectra is built. The random spectrum applied in the fatigue simulation process only represented the working time of 5.011 hours, while the representative working time of the applied bench fatigue test program load spectrum was 1000 hours. Therefore, the life of the structural details was the life value of the simulation results multiplied by the representative working time of the corresponding spectrum.

Figure 6 compares the overall life distribution cloud diagram of the stick structure under the action of the two load spectra. It can be seen that the life distribution of the stick structure was basically the same, and there were four areas with short life: ① The front end of the bucket cylinder support ear plate located on the upper wing plate of the stick; ② Near the reaming hole on the ear plate of the stick cylinder support; ③ Part of the weld near the reaming hole of the stick and the boom; ④ Intersection of butt weld on the web and the lower flange of the stick. The comparisons of these four areas with local enlarged figures are shown in Figure 7, Figure 8, Figure 9 and Figure 10, respectively.

The fatigue life conversion values of the above four local areas of the stick structure under the two load spectra are presented in

Table 7. Comparison of the fatigue life of several local details of the stick under two load spectra.

Local Details		Random Spectrum (kN)	Program Load Spectrum (kN)	Relative Deviation
① Front end of the bucket cylinder support ear plate located on the upper wing plate of the stick	left	4603.69	4364.22	−5.2%
	right	4745.47	4492.22	−5.3%
② Near the reaming hole on the ear plate of the stick cylinder support		3472.9	3298.83	-5.0%
③ Part of the weld near the reaming hole of the stick and the boom		6603.18	6304.32	-4.5%
④ Intersection of butt weld on the web and the lower flange of the stick		9924.8	10,800.54	8.8%

. It can be seen that the life distribution of the stick structure and the life values were relatively consistent, and the maximum relative deviation was merely about 8.8%, indicating that the programming method of the load spectrum was reasonable. Assuming that the maximum loadable frequency of the test bench is 1Hz, the fatigue lifetime test using the random spectrum takes 1000 h to represent 1000 h of working time. In contrast, the fatigue lifetime test using the proposed program load spectrum only consumes 60.31 h, which largely shortens the time expense. Hence, the program load spectrum outperforms the random spectrum in terms of time efficiency.

4. Conclusions

This paper focuses on solving several key issues in the compilation of the bench fatigue test program load spectrum of the hydraulic excavator stick:

(1)

A simple and convenient calculation method to determine the omitting threshold value of small amplitude cyclic load was proposed. The threshold value was determined as the critical value to keep the fatigue damage at the dangerous point unchanged before and after the small load was omitted from the full load-time history.

(2)

The load extremum extrapolation criterion, the synthetic method of various digging objects, and the correction and acceleration method based on the damage consistency criterion and the damage equivalent principle were proposed. Taking a medium-sized excavator’s stick as an example, its programmed load spectrum for fatigue bench test was obtained and further normalized to adapt to other excavators with different tonnages.

(3)

The validity of the program load spectrum is verified by using the finite element method and fatigue analysis theory. The comparison showed that the stick structure had the same damage position and life distribution under the action of the bench fatigue test program load spectrum and the random spectrum.

The program load spectrum compilation method also had a certain reference value for compiling the load spectrum of other multi-condition hydraulic construction machinery.