The primary purpose of this study is to provide methods that can be used to determine the most suitable drop height for shock testing military equipment, in an efficient and cost ineffective manner. Shock testing is widely employed to assess the performance of electronic systems, including military devices and civilian systems. Determining the height of the drop for a test item is an important step prior to performing the test. Dropping a test item from excessive height leads high Gpeak values to damage the test equipment. On the other hand, dropping an item from a low height may not generate the required Gpeak value and duration. Therefore, prior to performing shock tests, an optimal drop height must be established to ensure that the resulting Gpeak value and duration time match the required test values. The traditional trialanderror methods are timeconsuming and costineffective, often requiring many attempts. To improve the conventional approaches, this study proposes the application of regression analysis and backpropagation neural network for determining the most suitable drop height for freefall shock tests. A new method is suggested for determining the drop test height. The results of the model are verified, using the results of a series of experiments that are conducted to verify the accuracy of the suggested approaches. The results of the study indicate that both approaches are equally effective in providing an effective guideline for performing drop tests from heights that would result in the peak Gs and duration needed for testing electronic devices.
Shock testing has an increasingly important role in ensuring that military electronic devices meet the requirements for effective and reliable operation in harsh environments. Shock specifications of military electronic devices are generally expressed in terms of a simple acceleration pulse, such as a halfsine wave or a sawtooth wave in milliseconds, to simulate the shocks that the devices may experience in military environments. A shock test may include an assessment of the overall system integrity for safety purposes during handling, transportation, or use. The provision and application of shock simulation methods present many problems for laboratorybased teams [
A freefall shock test machine.
The waveform of a halfsine shock.
The purpose of the shock test is to obtain a waveform to match the solid line in Figure
Model of shock motion.
In Figure
The motion of the test item can be divided into two stages: before impact and after impact. According to Newton’s second law the equation for motion in the preimpact stage can be expressed as
In the postimpact stage, as the table impacts the programmer, the programmer will rebound with a reactive force
A detailed analysis of the reactive force is complex as the reactive force varies depending on the mechanical properties of the programmer material. In many cases, the reactive force is nonlinear result in computing the exact displacement,
Let
The area under the curve of acceleration is equal to the change in velocity between
The function of the programmer is to act like a spring. Let the elastic coefficient of the programmer be given by
Equation (
From (
The peak Gs is determined from
Regression analysis is used to fit the curve of the relationship between the input and output database. In this study, polynomial analysis and the ration linear polynomial regression analysis were performed on the data sets. The polynomial is denoted as follows:
The square error can be written as
To minimize the total square error
The optimal solutions of
It is clear that (
The backpropagation neural network (BPNN) was introduced by Rumelhart and McClelland in 1985. It is a multilayer and forwardfeedback perceptron with learning capability [
The partial derivative
The derivation was shown in (
The choice of an appropriate learning parameter
The momentum term can make the learning algorithm more stable and accelerate the convergence in flat regions of the error functions. However, determining the optimal value of the momentum parameter
This study applies BPNNs to determine the drop height and duration for the shock test with programmers.
The structure of the BPNN (Figure
The BPNN has an input layer expressing the input parameters in the network. The number of neurons depends on the problem’s complexity. The input layer is the first (bottom) layer in the structure.
Above the bottom layer is a middle hidden layer expressing the interaction between input parameters and processed neurons. The number of neurons cannot be precisely determined. The number of neurons is typically determined based on when the optimal result is obtained.
The third (top) layer of the BPNN is an output layer which denotes the network output. The number of neurons is also determined based on the problem’s complexity.
The BPNN includes a transfer function. The sigmoid function is chosen as the nonlinear transfer function; it is expressed as follows:
Structure of BPNN.
The following procedure is used to analyze the BPNN:
determine the number of neurons at each layer;
set the initial weights and bias values in the network randomly;
insert input and output vectors into the network for training the weights;
estimate the output values of the hidden and output layers;
calculate the difference in output values between the hidden layer and output layer;
establish the adjustment coefficients for weights and bias values;
update bias and weight values;
repeat steps
The proposed prediction scheme is composed of data collected under different conditions. The peak Gs and duration of the shock test are determined for different programmers. From previous (and new) experiments, the data sets are arranged so as to fit the optimal curve by regression analysis; the BPNN is also trained to develop a knowledge database until the learning structure is robust. The data sets for four cases of different programmers are gathered to train and recall the target using either a BPNN or regression analysis. Finally, the degree of accuracy of targets is compared for either BPNN or regression analysis in various situations.
Before collecting data sets, all programmers and freefall shock machines are calibrated for testing. Figure
Training data sets by freefall shock machine.
Set 



10  20  40  60  80  100  120  
Case 1 plastic programmer  
Gpk (g)  54.08  95.51  165.71  233.61  291.395  334.19  383.3 

0.1  0.1  0.1  0.1  0.1  0.1  0.1 


Case 2 three pieces of elastomer  
Gpk (g)  —  82.41  190.03  324.22  463.55  613.73  794.82 

—  0.3  0.3  0.3  0.3  0.3  0.3 


Case 3 two pieces of elastomer  
Gpk (g)  59.21  134.2  306.63  449.72  632.13  811.05  — 

0.45  0.45  0.45  0.45  0.45  0.45  — 


Case 4 one piece of elastomer  
Gpk (g)  82.82  199.28  443.36  647.99  862.61  —  — 

0.9  0.9  0.9  0.9  0.9  —  — 
Test data sets by freefall shock machine.
Set 
 

90  60  60  40  
Gpk (g)  314.04  326.54  436.34  445.68 

0.1  0.3  0.45  0.9 

2.98  2.92  1.66  1.30 
The setup of the freefall shock machine.
The elastomer of felt on the machine.
The plastic elastomer on the machine.
Representative shock test results.
Tables
MSE of heights by BPNN and regression analysis.
Item  Case  

Case 1 (cm)  Case 2 (cm)  Case 3 (cm)  Case 4 (cm)  Average MSE (cm)  
Training patterns  
BPNN  2.026 × 10^{−18}  5.845 × 10^{−21}  8.067 × 10^{−21}  1.338 × 10^{−21}  5.103 × 10^{−19} 
Regression  0.0887  4.14 × 10^{−27}  8.84 × 10^{−26}  1.73 × 10^{−27}  0.02218 
Test patterns  
BPNN  0.0462  0.1406  3.227  0.013  0.8567 
Regression  0.178  0.112  3.411  0.046  0.9368 
MSE of time duration by BPNN and regression analysis.
Item  Case  

Case 1 (ms)  Case 2 (ms)  Case 3 (ms)  Case 4 (ms)  Average MSE (ms)  
Training patterns  
BPNN  5.210 × 10^{−25}  3.324 × 10^{−27}  5.941 × 10^{−24}  7.60 × 10^{−28}  1.617 × 10^{−24} 
Regression  2.051 × 10^{−5}  0.0248  2.77 × 10^{−28}  0.0014  0.00656 
Test patterns  
BPNN  7.67 × 10^{−4}  2.551 × 10^{−6}  9.20 × 10^{−4}  5.87 × 10^{−5}  4.36 × 10^{−4} 
Regression  5.39 × 10^{−4}  0.0905  0.002  0.003  0.024 
The 24 data sets were selected for drop heights of 10, 20, 40, 60, 80, 100, and 120 cm (Table
Fitting curves of
Fitting curves of
Fitting curves of
Fitting curve of
In the same arrangement used for the training network in Table
Maximum errors (ME) of height by BPNN and regression analysis.
Item  Case  

Case 1 (cm)  Case 2 (cm)  Case 3 (cm)  Case 4 (cm)  Average ME (cm)  
Training patterns  
BPNN  2.404 × 10^{−9}  1.532 × 10^{−10}  1.608 × 10^{−10}  6.280 × 10^{−11}  6.952 × 10^{−10} 
Regression  0.498  1.421 × 10^{−13}  5.4 × 10^{−13}  8.530 × 10^{−14}  0.1245 
Test patterns  
BPNN  0.215  0.375  1.796  0.1147  0.6252 
Regression  0.178  0.112  3.411  0.046  0.937 
Maximum errors (ME) of time duration by BPNN and regression analysis.
Item  Case  

Case 1 (ms)  Case 2 (ms)  Case 3 (ms)  Case 4 (ms)  Average ME (ms)  
Training patterns  
BPNN  1.172 × 10^{−12}  9.415 × 10^{−14}  1.377 × 10^{−13}  5.596 × 10^{−14}  3.65 × 10^{−13} 
Regression  0.0076  0.3071  3.575 × 10^{−14}  0.0583  0.0932 
Test patterns  
BPNN  0.0277  0.0016  0.0303  0.0077  0.0168 
Regression  0.0232  0.301  0.0428  0.0502  0.1043 
Training of epochs by ANN.
The fitting curves obtained by regression analysis do not completely match the fitting curves obtained by the BPNN. The curves of the BPNN are spiral on regressive curves (Figures
Performance comparison: regression analysis versus BPNN.
Condition  Manner  

Regression analysis  BPNN  
Ability to handle more than 2 variables  NO  YES 
Degree of accuracy  GOOD  EXCELLENT 
Ability to develop approximate equations  YES  NO 
Need to collect cases  YES  YES 
Time efficiency  FAST  FAST 
Need for recalling  YES  YES 
Ability to solve complicated nonlinear problems  NO  YES 
Regression analysis and a BPNN are applied to analyze the nonlinear relationships and estimate the optimal height and duration for freefall shock tests in this study. The conventional “trialanderror” approach for determining drop height is timeconsuming as it requires repeated trials. The goal of this study is to improve the conventional method (in terms of time and cost) of conducting freefall shock tests. The BPNN accurately estimates the values of height and duration during shock testing without limiting the number of variables. The results of this study indicate that both approaches are equally effective in providing an effective guideline for performing drop tests.
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors would like to thank the Environmental Engineering and Testing Section, System Development Center of National ChungShan Institute of Science & Technology, for the data sets that were used in this research. This work was supported in part by the National Science Council in Taiwan, under the Project title: CaltechTaiwan collaboration on energy researchuncertainty mitigation for renewable energy integration, Project no. NSC 1013113P008001.