Reliability HotWire

Issue 114, August 2010

Using RENO to Enhance Traditional Engineering Analysis

Tables of material properties found in engineering reference books often provide only a point estimate of each property for each material. For example, the Young’s modulus for steel is often reported as 30x10⁶ psi. However, when measuring the Young’s modulus of steel in a laboratory, some test specimens will have a smaller value and some will have a larger value than the modulus reported in the table. Often, only the point estimate in the material properties table is used for computations, ignoring the variability of the property. However, if the variability in a material property is known, ReliaSoft’s RENO software offers an easy way to explore the variability in the results of engineering computations.

Elastic Coefficient Example

Consider a problem adapted from Shigley and Mischke (1989) [1]. The task is to find the parameters of the normal distribution for the elastic coefficient, C_p, of a gear set with the gear and the pinion both made from 4340 steel. The Young’s modulus follows a normal distribution with a mean of 29x10⁶ psi and a standard deviation of 0.73x10⁶ psi. The Poisson’s ratio follows a normal distribution with a mean of 0.287 and a standard deviation of 0.0072. According to Shigley, the elastic coefficient is defined as:

Where E_P is Young’s modulus and v_P is Poisson’s ratio for the pinion. E_G and v_G are the Young’s modulus and Poisson’s ratio for the gear.

We will solve the problem by calculating 100 values of C_p based on randomly generated values from the distributions for Young’s modulus and Poisson’s ratio, and then computing the mean and standard deviation of C_p.

Step 1: Define the Random Variables. In RENO, add a random variable called PistonPR, choose a normal distribution, and enter the mean (0.287) and standard deviation (0.0072) as shown in Figure 1. Repeat this step to create random variables representing the Young’s modulus of the piston, PistonYM, the Poisson’s ratio of the gear, GearPR, and the Young’s modulus for the gear, GearYM.

Figure 1: Defining the random variable for the Piston's Poisson ratio

Step 2: Define the Storage Variable. Add a storage variable called NextIterationNumber, set the start value to 1, and select the option to reset the variable after each simulation. This variable will be the counter that keeps track of how many times the elastic coefficient has been calculated during the simulation.

Step 3: Define the Constant. Add a constant called NumberOfPoints and set the value to 100. This constant will be the total number of times that the elastic coefficient is calculated.

Step 4: Create the Flowchart. Using the Flowchart toolbar, add the following items to the diagram:

• Flag Marker
• Standard Block
• Result Storage
• Counter Block
• Conditional Block
• Reset Block
• Go To Flag

Connect the items in the Flowchart as shown in Figure 2. Rename the items to match Figure 2 and make the following changes to the item properties:

• Calculate C_P: Type the following equation for C_p into the Evaluates to field "sqrt(1/pi()/((1-PistonPR^2)/PistonYM+(1-GearPR^2)/GearYM))".
• Store C_P: Set the block to store an Array of Results.
• Loop Control: Set the block to use a Simulation Counter, change the Properties Start Value to 1, and set the Storage Variable to NextIterationNumber.
• Iterate Again?: Set the Condition to be less than or equal to NumberOfPoints.
• Next Iteration: Select Start Loop as the Go to Flag Marker.

Figure 2: The completed flowchart

Step 5: Run the Simulation and Transfer the Data to a Spreadsheet. Set the Simulation Setting to use a seed value of 1 and click Simulate. When the simulation is finished, click Details and navigate to the Store C_P results storage block. The first 10 entries of the array are shown in Figure 3.

Figure 3: The first 10 calculated elastic coefficient values

Transfer the data to a spreadsheet by pressing F7 or clicking the Transfer to Spreadsheet icon. Enter Results as the name for the spreadsheet. Close the Simulation Results Explorer and the Simulation Console.

Note: You may also use Weibull++ to calculate the parameters of the distribution of C_p. To use this method, highlight the C_p values from the Simulation Results worksheet (i.e. cells B10 through B109 in the worksheet shown in Figure 3), then either press F9 or click the Send Selected Data to Weibull icon to calculate parameters.

Step 6: Calculate the Mean and Standard Deviation. Open the Results spreadsheet from the Project Explorer. In cell D1, type the equation "=average(B10:B109)" to find the mean of C_p; in cell D2 type the equation "=stdev(B10:B109)" to find the standard deviation of C_p. The mean of C_p is therefore 2242 with a standard deviation of 19.60.

References

[1] Shigley and Mischke, Mechanical Engineering Design, 5th ed. New York: McGraw-Hill, Inc., 1989.  

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