AC 2010-958: AN IMPORTANT EXPERIMENT AND PROJECT IN THE FIRSTMEASUREMENT COURSE
Bijan Sepahpour, The College of New Jersey
Bijan Sepahpour is a Professional Engineer and a Professor of Mechanical Engineering at TheCollege of New Jersey (TCNJ). Currently, he is serving as the chairman of the department and isactively involved in the generation of design-oriented exercises and development of laboratoryapparatus and experiments in the areas of mechanics of materials and dynamics of machinery forundergraduate engineering programs. He has served as the Chair of the Division ofExperimentation and Laboratory Studies (DELOS) as well as the Mechanical EngineeringDivision of ASEE.
One of the important components of a first measurement course in an engineering curriculum should be the coverage of the fundamental concepts in probability, uncertainty, and statistical analysis. An experiment and Project are designed and offered to better instill the significance of the above concepts and tools in engineering measurements, data analysis, and decision making process. The experiment calls for the establishment of the “Statistical” Spring Rate value (K) in each of the several sets of springs. Groups of students equal to the number of sets are formed to accurately measure and obtain the necessary data for each of the samples. These groups are then divided into smaller teams which conduct a comprehensive statistical analysis for each set and compare its tendencies with the rest. The teams then must justify their final decisions based on the statistical conclusions that they have drawn. The process for the establishment of six distinct samples is described. Design of the associated apparatus and their costs are presented. The parameters influencing the choice of the sample are discussed. The sample size and how its optimal selection may enable the coordinators to create fifteen (15) different combinations of the sets are described. The required number of measurements and the process for the establishment of K-values are briefly discussed. A comprehensive assessment of how the experiment and the project have improved the learning curve of the students is presented. Sufficient details are provided for creation of a variation of this exercise utilizing electronic components. This alternative features reduced cost and time for assembly and machining. The handout of the project and the experiment as well as a sample of the data required for conducting the analysis is included in the Appendices.
I - Introduction Laboratory experimentation is a critical final link for a thorough understanding and appreciation of scientific and engineering theories and principles. Every possible effort should be made not to deprive the future engineers or educators from this vital component of their education1. It is therefore necessary to continue development of effective and efficient pedagogical methods and techniques for the engineering laboratory experience2. The ability to perform Statistical Analyses and developing a solid understanding of the parameters influencing the reliability requirements/considerations in the engineering decision making process may prove critical for a functional design team. An integrated experiment and project are designed to better instill the significance of the above parameters.
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The experiment and the project call for the establishment of the “Statistical” Spring Rate value (K) of each of the three (3) or four (4) samples of springs that (in a hypothetical scenario) are sent by different companies for winning a bid. Each sample is comprised of 25 springs. The springs are to be used in the design of 12 modules of the International Space Station (ISS). The modules are comprised of eight (8) identical subassemblies/panels. All 12 modules are to be transported to the station with only one trip of the Space Shuttle Atlantis. The cargo bay of the shuttle is not spacious enough to accommodate the transport of all 12 modules in their intended (fully expanded) mode. As a result, the eight (8) panels of each module need to be coupled with each other in a manner that transportation requirements may be met. The springs are to be used in the design of the mechanisms to satisfy the required modes/configurations of the modules during their: a) transport, b) expansion, and c) the final intended geometry. For full details of the scenario and the activity, refer to Appendix “A”. If there are to be four (4) groups of students [comprised of three (3) or four (4) members] for conducting the experiment, each group will collect data on only one of the sample sets. Each group will then share the results of their measurements with the other groups. In this process, each group has established an average “K” value for “each and all” of the 25 springs in one assigned set. Since five (5) measurements are required for “each” of the springs, each team then will have to conduct a total of 125 measurements. This will translate to a total of 500 measurements for the entire class. The completed and shared data for each of the four (4) sets will be used for the statistical analyses of all sets. Central tendencies of each set will be compared with the rest. Each group of four will then break down into two groups with only two members to address the requirements of the project component of the task. These teams of two students must each perform the required analysis and justify their final decisions and recommendations based on the (statistical) conclusions that they have drawn for the assessment of the quality of each set. II - Objectives of the Experiment and the Project The following major objectives were set at the inception of the project: 1. To develop an experiment and project for a complete review and a better understanding of the statistical parameters that may heavily influence the engineering/design decision making process. 2. To create an opportunity for collaborative research and design efforts between undergraduate engineering student(s) and faculty. 3. To design, produce, test, and optimize a cost-effective, reproducible apparatus with outstanding features. 4. To make all information necessary for fabrication of the apparatus and conducting the experiment and the project available to engineering programs nationwide. P
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It was therefore desirable to design an apparatus and experiment that would be feasible for replication in other educational institutions within a budget of $1,000 for materials and components. The package would include no less than four sets of distinct samples and require approximately 15 hours of machining and assembly time. III- Design of the Experiment and the Project
1. Pedagogy This project has been designed for sophomore level students. Pedagogical measures have been taken for its realistic effectiveness (nation-wide). Therefore, the framework of the project has been set at a level that sophomores may succeed in its implementation and developing deeper appreciation for some of the statistically based decision making processes. The students in the Mechanical Engineering, Civil Engineering, Bio-Medical Engineering, and Engineering Management at The College of New Jersey (TCNJ) are all required to take the first measurement course. In Chapter 4 of their “Theory and Design for Mechanical Measurements” text, Figliola and Beasley offer a concise coverage of the Probability and Statistics concepts and analytical tools that are most critical for engineering applications.3 This is followed by a detailed chapter in Uncertainty Analysis. All of Chapter 4 and the first four (4) articles of Chapter 5 (of the above text) are covered in the first measurement course at TCNJ. The premise for creation of the project is to further review and explore the potential applications of the following parameters and topics that may influence the engineering design and production decisions. In this process, the targeted audience may also develop additional appreciation for other considerations such as safety, reliability, expected life, quality control, production constraints, cost, etc.
1. Finite VS Infinite Statistics
7. Number of Measurements Required
2. Mean, Median, Mode, and True Mean
8. Probability Density Function
3. Variance and Standard Deviation
9. Central Tendencies
4. Histograms and Frequency Distribution
10. Probability Linked to Reliability
5. Normal Distribution, Chi-Squared, etc.
11. Measurements and Sources of Error
6. Confidence and Precision Intervals 12. [Introduction to] Uncertainty
The author invited two rising junior engineering students for collaboration. The details of their involvement and contributions are provided in Appendix “C”.
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2. Choosing of the Sample The choice of the Sample is quite important and it may be considerably influenced by the following factors;
1. Desired Sample Size [as this may be the single most important parameter in most Statistical Analyses] ,
2. The Number of Students and Intended Groups in each section, 3. The Number of Sections in a given semester/academic year, 4. Time Limitations for Conducting the Measurements and Recording the Data, 5. Desire to Repeat the Experiment (and Project) in Future with Sufficient Modifications to
avoid Repetition of Previously Used Data, 6. Cost [depending on the quality and the sample size(s)] , 7. Availability, Durability, and Aesthetics, 8. The Intended/Desired Spectral Density of the Sets/Data, 9. Degree of Difficulty in creating the Required Number of Different Sets [with different
ranges and frequencies] , 10. Probability of obtaining the intended Spectral Density and the Ranges for the target Sets, 11. Degree of Difficulty, Feasibility, and Safety in Performing the Required Measurements, 12. The Number of Necessary Measurements for obtaining a relatively meaningful set(s) of
statistics for the targeted parameter(s)] , 13. The Cost and the Complexity of the Design of the Associated Apparatus, 14. The Tools required for performing the necessary measurements, 15. Creation of (Numbered/Colored) Housing Units/Cradles to Prevent the Potential Mixing
of Samples in one set with others, and 16. Ease of Maintenance and Storage for Continuous Reuse [in other sections and future] .
In the previous iterations of this exercise, the following choices for the samples have been made. As shown in Table (1), for some, the groups of students had to physically take measurements of the provided samples and for others; the data for the different sets was provided. Table 1. The Choices and the Types of Samples in the Previous Iterations
# Sample Type Specifications Size of the Samples Actual / Physical
Simulated / Pre-measured Data
1
3 or 4 different (Colored) Sets of Washers
1.5 in O.D. 0.25 in I.D. t = 0.05
50 (in each of the 3/4)
√√√
2
3 or 4 different (Colored) Sets of Washers
2.5 in O.D. 0.375 in I.D. t = 0.05
30 (in each of the 3/4)
√√√
3
Ball Bearings 4 Sets
1.00 in Dia. 75 (in each of the 4)
√√√
4
Roller Bearings 4 Sets
0.50 in Dia. 2.50 in Length
50 (in each of the 4)
√√√
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For the proposed case, the goal is to establish the Index (K) of the spring. This index is reflective of its load carrying capacity/sensitivity. Specifically, it is an indicator for the magnitude of the load required for the extension or contraction of the spring for One Unit of Length. At first a small set of Stainless Steel Tension Springs with low Index values were attempted. Since the loading for Tension Springs must be done in Tension mode, there is the possibility of damaging the spring inadvertently-especially by students in their first measurement course. To reduce this possibility, another set with a higher K-value was attempted. The concern in this case may be the need for a larger set of loads which in turn may pose issues related to injury. More importantly, the major difficulty surfacing with the choice of the Extension Springs is the complexity involved in conducting the required measurements for the establishment of the K-value of each spring. It is important to note that for each spring, there should be about 4 to 6 measurements to get a reliable result. Although, in the current laboratories of the mechanical engineering program at TCNJ, there is access to an apparatus capable of facilitating such required measurements (with sufficient accuracy), the use of this device may be an overdesign and may pose complexities in its replication at other institutions. The alternative then was to consider Compression Springs that could be fully compressed without being damaged. Generally, these types of springs enjoy a better Central Tendency than the Tension Springs and the K-values are significantly closer to the mean of the sample.
3. Specifications of the Sample The specifications of the springs should meet the requirements listed in section III-2. To consider and address these parameters, a certified, 4-in long, stainless steel compression spring, with an index that could have been established with the use of a set of laboratory loads [within the range of four (4) to five (5) lbs.] was chosen. See Figure (1).
Figure 1. The Chosen (Compression Spring) Sample 4. Size of the Sample In the author’s institution, three (3) or four (4) sections of the First Measurement Course are offered every Spring, and there are approximately 16 students in each section. Usually, four groups [comprised of three (3) or four (4) members] are formed. Therefore, the Number of the Sets was set to be four (4). The choice of the “sample size of 25” was influenced by the desire for creation of six (6) distinct sets.
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In order to obtain reliable estimates, four (4) to six (6) measurements are to be taken for each of the springs. It was decided to run five (5) measurements for each spring. With a sample size of 25, this arrangement would require 125 measurements for each set translating to a total of 500 measurements for all four (4) sets. This required time for conducting such high number of measurements may be problematic. As a result, it was decided to have “each” one of the four (4) teams to perform the necessary measurements and organize the data on “only one” of the four (4) sample sets. Each group would then share the results of their measurements with the other groups. Through this process, all four (4) groups would have complete information on the average “K” value for “each” of the 25 springs in “each” of the four (4) sets. The compiled and shared data for the four (4) sets would be used for the statistical analysis of “each of the sets” and its “comparison” with the statistics and the central tendencies of the other sets. The next step was the ordering of 100 certified springs from a reliable source and hoping that four (4) distinct sets [with different ranges and statistics] may be created from this pool. If successful, then the same four (4) sets would be available for all of the three or four sections. Each set would have its own color-code to avoid possible mix-ups. It is advantageous to have different sets of data for each section (as well as each semester/year). It was realized that with six sets of 25 springs (instead of four), the experiment (and the project) may be conducted with 15 different combinations. So, it was decided to increase the number of sets (of 25 springs). The fact that the cost decreases as the number of ordered springs increase, was an added encouraging factor in this decision. The cost of 200 springs would be no more than 60% (more than the 100). Table (D1), in Appendix “D” reflects on the possible combinations of four (4) sets of springs based on the availability of six (6) distinct sets. Table (D2), provides data for possible combinations of three (3) sets of springs
5. Establishment of the Desired Range and Frequency of the Data Sets The next step in the process is the creation of six distinct sets of samples. To accomplish this task, the combined effect of the ranges and the frequencies of each set must be unique. There are many possibilities. However, the spectral density of the samples might not deliver all of the initially planned combinations (and histograms). Although Tables (2) and (3) show a promising set of such unique features (and their corresponding Histograms), at the time, there was no guarantee to physically compile such distinct sets from the 200 available springs. Fortunately, (as will be shown in section V-1,) the recommended springs provide normal distribution. Additionally, the fact that only 150 out of the 200 springs would be utilized (for creation of the six (6) sets of 25 springs,) makes the choices available for the desired ranges and frequencies significantly broader. It should be clear that while another group may be able to replicate the sets chosen by this group, there is a large array of other possibilities to create/ choose from. This group has been quite impressed by the quality of the recommended springs. However, due to different settings at different times of manufacturing, it is probable that the statistics of a newly acquired set would be (slightly) different than this and that of any other set.
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Table 2. The Predetermined Distributions of the Six (6) Sets [of the 25 Springs]
Distribution for Set # 1 Distribution for Set # 2
Range Frequency Range Frequency
-3j to -2.5j 1 -3j to -2.5j 0
-2.5j to -2j 1 -2.5j to -2j 1
-2j to -1.5j 2 -2j to -1.5j 1
-1.5j to -1j 2 -1.5j to -1j 2
-1j to -0.5j 3 -1j to -0.5j 3
-0.5j to 0 5 -0.5j to 0 5
0 to 0.5j 4 0 to 0.5j 6
0.5j to 1j 3 0.5j to 1j 4
1j to 1.5j 2 1j to 1.5j 2
1.5j to 2j 1 1.5j to 2j 1
2j to 2.5j 1 2j to 2.5j 0
2.5j to 3j 0 2.5j to 3j 0
Distribution for Set # 3 Distribution for Set # 4
Range Frequency Range Frequency
-3j to -2.5j 0 -3j to -2.5j 0
-2.5j to -2j 0 -2.5j to -2j 0
-2j to -1.5j 0 -2j to -1.5j 1
-1.5j to -1j 1 -1.5j to -1j 3
-1j to -0.5j 3 -1j to -0.5j 5
-0.5j to 0 4 -0.5j to 0 7
0 to 0.5j 6 0 to 0.5j 4
0.5j to 1j 5 0.5j to 1j 2
1j to 1.5j 3 1j to 1.5j 2
1.5j to 2j 2 1.5j to 2j 1
2j to 2.5j 1 2j to 2.5j 0
2.5j to 3j 0 2.5j to 3j 0
Distribution for Set # 5 Distribution for Set # 6
Range Frequency Range Frequency
-3j to -2.5j 0 -3j to -2.5j 0
-2.5j to -2j 0 -2.5j to -2j 0
-2j to -1.5j 0 -2j to -1.5j 0
-1.5j to -1j 0 -1.5j to -1j 3
-1j to -0.5j 2 -1j to -0.5j 6
-0.5j to 0 6 -0.5j to 0 9
0 to 0.5j 8 0 to 0.5j 5
0.5j to 1j 6 0.5j to 1j 2
1j to 1.5j 3 1j to 1.5j 0
1.5j to 2j 0 1.5j to 2j 0
2j to 2.5j 0 2j to 2.5j 0
2.5j to 3j 0 2.5j to 3j 0
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IV- Design of the Apparatus
An apparatus and a procedure for measuring the K-values of the springs must be in place before an attempt for testing to see if the desired (on paper) sets may be physically generated.
The elementary equation of F= K. may be used to obtain the index of a spring.4 By knowing
the exact value of the mass/load and the induced contraction, K=F/ should provide a good estimate of the Index of the Spring (K). However, one such application and estimate may not be sufficient. Application of 4 to 6 incrementally increasing loads may suffice. A table for recording of the corresponding contractions should be constructed. With the plot of data points for each spring, it would be possible to obtain the slope of the generated line which is reflective of the K-value of that particular spring (only) in that set. This design of the apparatus is premised on meeting/delivery of the following characteristics:
1. Simple to Operate, 2. Cost-Effective, 3. Safe, 4. Reliable for Delivery of the Intended Accuracy, 5. Simple to Construct and Replicate at Other Institutions, 6. Light but Sturdy, 7. Independent, and Portable, 8. Durable, 9. Environmentally Sound, and 10. Aesthetically Pleasing.
As shown in Figures (2) and (3), the apparatus has four major components; a top and a bottom plate, a post with two bushings (at top and bottom portions), and a (surface) hardened (and polished) rod. The vertical post is coupled with the base plate. The top plate not only holds the rod, it will also support the loads concentrically (using the rod’s geometry). The measurement of the contraction of the springs is accomplished by inserting the polished rod through the spring and passing its free end through the cylindrical opening of the vertical post. The top plate is rotated so that (a part of) it would be over the tip of the 2”- Travel Dial Indicator. The top plate would come into contact with the indicator before the (first) load is exerted to the spring. A recording of this “gap” has to be made and then used to correct the recorded values of the contractions. It should be clear that the first load is actually the mass/weight of the upper assembly of the apparatus. Figure (4) shows the sequential loading of a Spring. Table (4) provides comprehensive listing of the components of the apparatus and the sources for obtaining them. It also reflects on the associated costs for the 200 samples, the required materials and the components. Table (5) takes the additional factors of machining and assembly requirements into account. It also lists the time required for running the necessary measurements for creation of the six (6) [distinct] sets with 25 springs in each.
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Figure 2. Components of the Apparatus Figure 3. Assembled Apparatus
Laboratory apparatus is generally expensive due to low production levels, specialized features and significantly higher Design Costs built into the final cost. However, if blueprints of the designs of a (desired) apparatus are available, and on site machining capabilities exist, a major cut may be expected in the final cost.5 The blueprints of the apparatus may be obtained from the author.
Figure 4. Sequential Loading of one of the Springs
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Table 4. Parts List and Cost for the Samples and the Components of the Associated Apparatus
Part Source Quantity Price ($) Sub Total ($)
4” Long Springs with a 5.63 “ ID and
a “K“ Value of: 3 lb/in ≤ K ≤ 4 lb/in
Lee Spring 200
(For 6 Sets) NA 295
Dial Indicators with 2” Range
With No Stand (needed)
Machine Shop
Discounts 4 35 140
1/4“ x 4” Flat Aluminum Stock McMaster-Carr 1 35 35
1/8“ x 3” Flat Aluminum Stock McMaster-Carr 1 20 20
3/8“ Dia. x 12“ Long
Precision Steel Rods
McMaster-Carr 4 18 72
1/8“ thick x 24“x 36“
Poly-Carbonate Peg Board
McMaster-Carr 1 15 15
3/8“ ID x 1“ Long
Precision Brass Bushing
McMaster-Carr 4 x 2 1.50 12
1/4-20 x 2“ Long Coupling Nuts McMaster-Carr 5x6 + 8x4 1.20 75
Hardware McMaster-Carr - 45 45
Flat Weights/
Large Slotted Mass Set
PASCO
Scientific
2 Sets
(for 4 Groups) 129 258
Shipping and Handling
60
Total Cost: $995
Table 5. Breakdown of the cost and the Required Hours
1 Overall Cost of the Materials for the 200 Samples and the Components ≤ $ 1000
2
Required
Machining and
Assembly Time
I - Average Machining About 10 Hrs.
II - Above Average Machining Zero
III- Assembly of the Cradles and the Apparatus 4-5 hours
3
Testing of the Springs and the Creation of the Six (6) Sets of 25 Springs
8-10 Hrs.
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V- Classification of the Springs into Physical Sets
1. The Required Measurements for Rating of the Springs from (- 3 ) to (+ 3 ) About (200 x 5 =) 1000 measurements has to be conducted to fully classify all of the 200 springs with the hope that the six intended (distinct) sets may be assembled. For each of the five/six measurements, a graph has to be generated. As discussed earlier, the slope of the graph would provide a good estimate of the K-value of the spring. Table (6) shows a sample of the data collected on the deformations and the corresponding K-Values of five (5) of the springs during the classification Process. Table (7) shows the corresponding graphs (and slopes) for the first four springs listed in Table (6). The complete version of the measurements shown here along with the associated graphs and the corresponding calculated K-values and the Histogram are included in Appendices “D, E, F, and G”. This package is representative of part of the analysis that the small teams have to perform during the data collection and organization phase. The required time in the task of performing reliable measurements opened the eyes of the team to the fact that conducting measurements on all four (4) sets (by a single team) was not realistic. This led to the decision of “each team being assigned to one and only one set”. The laboratory handout would then be developed with the intention of trying to keep the time required for completion of the measurement component between 50 to 60 minutes. To review the results obtained by each of the groups, software would need to be utilized for recording, organizing and (partially) analyzing the measured data. Table 6. A Sample of the Data Collected on the Deformations and the Corresponding K-Values of five Springs during the classification Process.
K-Values K = ѐ (lb/in) KA (lb/in) KB (lb/in) KC (lb/in) KD (lb/in) KE (lb/in)
Spring 1
0.225 0.495 0.769 1.023 1.171 1.279
3.853 3.786 3.753 3.804 3.775 3.819
Spring 2
0.236 0.510 0.772 1.035 1.174 1.294
3.674 3.675 3.738 3.760 3.765 3.775
Spring 3
0.215 0.474 0.743 1.008 1.154 1.264
4.033 3.954 3.884 3.861 3.830 3.865
Spring 4
0.218 0.497 0.764 1.039 1.174 1.281
3.977 3.771 3.777 3.746 3.765 3.813
Spring 5
0.211 0.482 0.749 1.020 1.159 1.279
4.109 3.888 3.853 3.816 3.814 3.819
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Table 7. Graphs and Slopes Corresponding to the Data for the First Four Springs in Table (6)
- 1.5 - 1 - 0.5 + 0.5 + 1 + 1.5
Figure 5. Classification of the Rating of the Springs and Clear Indication of a Normal Distribution
Spring 2
y = 3.8061x - 0.0478
0.000
1.000
2.000
3.000
4.000
5.000
6.000
0.000 0.500 1.000 1.500
Spring 1
y = 3.7987x - 0.0043
0.000
1.000
2.000
3.000
4.000
5.000
6.000
0.000 0.500 1.000 1.500
Spring 3
y = 3.8035x + 0.0577
0.000
1.000
2.000
3.000
4.000
5.000
6.000
0.000 0.500 1.000 1.500
Spring 4
y = 3.7579x + 0.0226
0.000
1.000
2.000
3.000
4.000
5.000
6.000
0.000 0.500 1.000 1.500
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Every spring that its K-value got established, would be housed on a specific “Row and Column” of a matrix/cradle for identification purposes. With this information and system at hand, the
entire stock of the 200 springs were classified from (- 3 ) to (+ 3 ). Figure (5) displays the classification of the springs and reflects the presence of Normal Distribution. 2. Housing for Each of the Six (6) Sets It is obvious that the six (6) distinct sets have to be stored in a manner that their samples could not be mixed with those of the others. [Extreme measures should be taken to prevent such possibility.] Secondly, it is preferred to have an arrangement that makes the springs easily accessible for measurement [and back to the quarter(s)]. Therefore, for each set, a corresponding cradle was designed and assembled. The machining requirements for these (six) units did not exceed two hours. However, the assembly time (for the six) was about 3 hours (for one person). The total cost of the components and the materials for the six cradles was about $90. These units are efficient, durable, and aesthetically pleasing (as all of the chosen materials and the hardware are non-corrosive). Figures (6) and (7) show the blank and loaded modes of these units. They also served as the matrix for the identification purposes during the classification process.
(a) (b) Figure 6. (a) Unloaded View of Cradle “A”, (b) Cradle “A” Loaded with the Springs of “Set A”
Figure 7. Cradles “A”, “B”, and “C” Housing Sets “A”, “B”, and “C” Respectively
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VI- Assessment Completion of a Rating and Assessment form is an integral part of the requirements for this exercise. This form is included in Appendix “B”. The main objective of the survey is the continuous fine-tuning of this activity for further improving the learning curve in the future iterations. The survey was conducted for three (3) section of the course in the Spring of 2008 and for four (4) sections in the Spring of 2009. The total number of the surveyed students was 108. Tables (8) through (12) provide detailed summaries of the results for five of the (more measurable) questions on the project’s assessment form. About 80% of the 108 surveyed students believe that they would be able to complete the same task between 50 to 70 percent of the time it took them in the first trial. Nearly all would incorporate an activity of this nature should they get the opportunity to teach a first course in measurement. The assessment results clearly reflect on the fact that there is (nearly perfect) consensus that the project is a balanced activity that is highly valued by the members of the seven (7) surveyed sections. Table (13) reflects on the level of performance and the scores achieved for the experiment and the project. The data is highly indicative of the effectiveness of the proposed activity for improving the learning curve in this vital area for the engineers to be. Table 8. Summary of the Results for the First Measurable Question on the Project Assessment Form
Question # 1: How would you rate the time required for completion of this Project?
xxxxxxxxxx Rating
Semester Section # N Too
Short
Short About
Right
Long Too
Long
Spring
2008
1 17 - - 12 5 -
2 16 - - 10 5 1
3 18 - - 12 6 -
Spring
2009
1 14 - - 10 4 -
2 16 - - 10 5 1
3 12 - - 8 4 -
4 15 - - 10 5 -
Total
7
Sections
108
-
-
72
34
2
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Table 9. Summary of the Results for the Second Measurable Question on the Project Assessment Form
Question # 2: If you had to do this experiment/activity again, how long would it take the second time? Use the Percentages listed below.
xxxxxxxxxx Rating
Semester Section # N 30- 40 %
40- 50 %
50- 60 %
60- 70 %
70- 80 %
80- 90 %
Almost The Same
Can Not Predict
Spring
2008
1 17 - 1 9 5 2 - - -
2 16 - 1 4 6 1 1 1 2
3 18 1 - 8 6 2 1 - -
Spring
2009
1 14 - 1 7 4 - 1 - 1
2 16 - 2 8 5 1 - - -
3 12 - - 7 3 1 - - 1
4 15 - 2 6 3 2 1 0 1
Total
7
Sections
108
1
7
49
32
9
4
1
5
Table 10. Summary of the Results for the Third Measurable Question on the Project Assessment Form
Question # 3: Would the experience gained in this activity help you optimize your approach the next time you have to deal with a similar task?
xxxxxxxxxx Rating
Semester Section # N Highly
Unlikely
Unlikely Probably Very
Likely
Definitely
Spring
2008
1 17 - - - 7 10
2 16 - - 1 5 10
3 18 - - - 6 12
Spring
2009
1 14 - - - 3 11
2 16 - - - 6 10
3 12 - - - 4 8
4 15 - - - 6 9
Total
7 Sections
108
-
-
1
37
70
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Table 11. Summary of the Results for the Fourth Measurable Question on the Project Assessment Form
Question # 4: How would you rate the overall Value of this Experiment and Project?
xxxxxxxxxx Rating
Semester Section # N Very Low Low Medium High Very High
Spring
2008
1 17 - - - 7 10
2 16 - - 1 3 12
3 18 - - - 5 13
Spring
2009
1 14 - - - 4 10
2 16 - - - 5 11
3 12 - - - 3 9
4 15 - - - 5 10
Total
7 Sections
108
-
-
1
32
75
Table 12. Summary of the Results for the Fifth Measurable Question on the Project Assessment Form
Question # 5: If you get to teach a similar course, would you incorporate such an activity in your course? If yes, what changes would you recommend or introduce?
xxxxxxxxxx Rating
Semester Section # N Highly
Unlikely
Unlikely Probably Very
Likely
Definitely
Spring
2008
1 17 - - - 6 11
2 16 1 5 10
3 18 - - - 7 11
Spring
2009
1 14 - - - 6 8
2 16 - - - 7 9
3 12 - - - 2 10
4 15 - - - 5 10
Total
7 Sections
108
-
-
1
38
69
Page 15.153.18
Table 13. Summary of the performance Record of the Two-Member Teams.
Semester Section # N Number of
Teams
Range
(%)
Average
(%)
Spring
2008
1 17 8* 85- 100 91
2 16 8 80 - 98 89
3 18 9 83 - 99 90
Spring
2009
1 14 7 90- 100 87
2 16 8 82 – 96 90
3 12 6 83 – 99 88
4 15 7* 78 - 98 90
Total
7 Sections
108
53
80 - 100
89.4
* With One Three-Member Team
VII- An Alternative/Variation of the Experiment and Project The use of electrical resistors for the creation of different sets of samples may prove quite advantageous. The cost is truly insignificant and the volt-meters for measuring the resistance of the samples may already be in place.
The author and the collaborating students recommend the use of 1000 = 1K resistors for this experiment. Our team has tried several other resistors with different rates and has come to the conclusion that this class has a better chance for the creation and classification of the required (distinct) sets. The compactness, low cost, and ease of the storage are all considerable pluses in this selection. Figures (8) and (9) display the sets and the packs of fifty (50) 1K resistors.
Figure 8. A Pack of 50 Resistors for Set “A” Figure 9. Six Packs of (50) Resistors for Six Sets
Page 15.153.19
Since the measurement component of this package is not as challenging, our team recommends bringing a new dimension to the project. This may be accomplished by obtaining a set of four (4) good quality Volt-Meters [as shown in Figures (10) and (11)] and examining their specifications for potential application(s) in Uncertainty and Error Analysis. Part of the Specifications of the Meter is displayed in Figure (12). A review of the contents of Table (14) reveals the interesting and attractive aspects of this alternative package.
Figure 10. A High Quality Volt-Meter Figure 11. Set of Four Volt Meters
Figure 12. Specification of the Meters for use in Uncertainty and Error Analysis Table 14. Breakdown of the cost and the Required Hours for the Alternative Experiment
1 Overall Cost of the Materials (resistors) and the Four (4) Voltmeters ≤ $ 500
2 Required Machining and Assembly Time Zero
3 Testing of the Resistors and the Creation of the Six (6) Sets 3-4 Hours
Page 15.153.20
VIII – Summary and Conclusions The importance of the fundamental concepts in Probability, Uncertainty, and Statistical Analysis in the engineering decision making process has been further explored through a proposed experiment and project in a first measurement course. The scenario for the challenge raises the interest of the students. All of the necessary tools for addressing the requirements of the project are available through the texts used for the measurement courses in an engineering curriculum. The experiment calls for the comparison of the quality of different sets of springs by running a comprehensive “Statistical analysis”. Upon completion of the analysis, logical deductions have to be presented for identifying the set with the superior samples. The process for the design of the experiment and the associated apparatus for accurately conducting the necessary measurements have been presented. Parameters influencing the choice of the samples and the demanding steps necessary for the classification of these certified springs into several distinct and unique sets have been explored. The sample size and how its optimal selection may enable the coordinators to create fifteen (15) different combinations of the sets for use in the other sections or in the future years have been discussed. Detailed cost analysis has been presented for the components of the apparatus and the chosen samples. Comprehensive details for the classification and creation of the sets of data in other institutions have been provided. Upon request, blueprints of the apparatus would be available for its replication. The data collected from seven (7) sections of the first measurement course at TCNJ have been organized to assess how the experiment and the project have: a) enhanced the appreciation of the subject, b) raised the interest, and c) improved the learning curve of the students. Sufficient details have been provided for creation of a variation of this exercise utilizing electronic components leading to reduced cost and time. The handout of the project and the experiment as well as a sample of the data required for conducting the analysis is included in the Appendices for reference and potential modifications. It is hoped that the engineering educators find this exercise worthy of being added to the archives of the experiments/projects in their undergraduate programs and share their experience(s) with the author. Acknowledgments The author expresses special thanks to Alexander Michalchuk (department senior technician and machinist) for his continuous support and dedication to the project. He also thanks the collaborating students (David McNally and Daniel Lee) for their contributions. They certainly made a difference. References
1. Sepahpour, B., “Design of an Affordable Model Laboratory for Mechanical and Civil Engineering Programs”, Proceedings of ASEE 2003 National Conference, Nashville, TN, June 2003.
2. Sepahpour, B., Clark, E. and Limberis, L. “Modular Lumped Mass Experiment”, Proceedings of ASEE 2004 National Conference, Salt Lake City, Utah, June 2004.
3. Figliola, R. S. and Beasley D. E. Theory and Design for Mechanical Measurements. John Wiley & Sons, 2006.
4. Shigley, Joseph E. Mechanical Engineering Design, Third Edition, McGraw Hill, 1980. 5. Sepahpour, B., “Involving Undergraduate Students in Design of Experiments”, Proceedings of ASEE
2002 National Conference, Montreal, Canada, June 2002. 6. Zydney, et al. 2002. “Impact of Undergraduate Research Experience in Engineering”, Journal of
Engineering Education. 91(2): 151-157. 7. Navidi, William Statistics for Engineers and Scientists. McGraw-Hill, 2006.
Page 15.153.21
Page 15.153.22
The Engineers at the company have finalized the Design and are prepared to move into the production stage.
The Coupling and Expansion Mechanism for each pair of the panels requires the housing of 85 Compression Springs
that with a specified Index Value (K) will deliver the force required to achieve the unfolded/final configuration (for
each pair). Therefore, there will be total of [12 Modules x (7 Hinges/Module) x (85 Springs/Hinge) =] 7140 springs
required for the assembly and the operation.
The springs need to be customized to the specifications provided (below) by the designers:
Material High Strength
Stainless Steel
Index Value
K = 3.75 lb/in [± 0.06 lb/in]
Length L = 4.00 in [± 0.1 in]
Outside Diameter OD = 0.5625 in [± 0.05 in]
In the Folded / Transport Mode, the springs
need to be compressed to: 3.00 in [± 0.1 in]
Four (4) different manufacturers of springs have sent their samples to the company. At 10 a.m., the Chief Engineer
at the plant hands in the samples to the two (Juniors to be) Engineering interns (from TCNJ) to run a complete
Statistical Analysis on each of the samples and make a decision on which sample should be chosen.
They are informed that a 15 minute meeting/conference will be held at 2:30 p.m. (ON THE SAME DAY) for the
justification of the decision.
Measurements and Data Collection 1. Form four groups with four (4) or three (3) members. Samples of 25 springs are available in each of the
provided Sets of "A", "B", "C", or “F”
2. Each group will be assigned one of the sets (above). Using your specific assigned set:
a) Examine the Material of the Springs. How can you (quickly and without the use of any
chemicals) find out if the material is actually Stainless Steel?
b) Measure the OD and the Length of each of the springs in your designated set.
3. Using the mass/load set and the apparatus provided (for each group), you need to establish the K-value for
each of the 25 springs in your assigned set by:
a) Recalling that the K-value of a spring is reflective of its load carrying capacity/sensitivity.
Specifically, it is an indicator for the magnitude of the load required for the extension or
contraction of the spring for One Unit of Length. For example, a spring with an Index value of
K = 50 lb/in requires a load of 50 lb to extend/contract by one inch.
Page 15.153.23
Therefore, the elementary equation of F= K. may be used to obtain the index of a spring.
So, knowing the exact value of the mass/load and the induced contraction, K=F/ should
provide a good estimate of the Index of the Spring (K). However, one such application may
not be sufficient. Including the Load Support Plate, each group is supplied with five (5)
different loads that should be applied incrementally (in an increasing mode) to the spring.
b) A table for recording of the corresponding contractions should be constructed. With the plot of
five data points for each spring, you should be able to obtain the slope of the generated line
which is reflective of the K-value of that particular spring in the set. Although not required, but
you should check the potential Hysteresis of the measurement system by incrementally
decreased loading (at least for a few springs).
c) It is important to note that the Dial Indicator’s Starting/initial setting value must be accounted for in your measurements.
d) You need to first obtain the exact values of each of the masses/loads used in your incremental
measurements.
4. After obtaining the K-values of all of the springs in your set, you need to list them on a central data bank for
sharing the statistics of “your set” with all members of your class. The compiled data for ALL of the sets will
be available for comparison of the quality of the springs in each of the sets. This may be used to draw
conclusions on the best choice of the springs and which company has won the bid.
Analysis (and the Project) In groups of two (or a maximum of three), perform the following analyses and TABULATE all important and
interesting results in a highly organized manner. Refer to Appendix B of your Text for a Guide to Technical Writing.
1. Compute the Histogram and Frequency Distribution of the K-value (only) of the Data of EACH set.
[Hint: Refer to the Approach and Display of example(s) of the text and generate comparable charts and tables, etc.]
You may use existing software to accomplish part/all of the above task(s).
2. Using INFINITE Statistics Criteria, calculate and tabulate the following for EACH of the sets of data;
(1) Mean, X '
(2) Variance, ı 2
(3) Standard Deviation, ı
3. Assuming that all sets of data follow Normal Distribution, show (for EACH set) that the Probability that a
measurement will yield a value within X ' ± 1 ı is about 0.68 or 68%.
4. If measurements are taken to obtain the K-value of a Single spring, what is the probability that the measured value
will be between:
X ' – 2ı < X < X ' +2ı ?
Page 15.153.24
5. Using FINITE Statistics Criteria, calculate and tabulate the following for EACH of the sets of data;
(1) Mean, x
(2) Variance, Sx2
(3) Standard Deviation, Sx
(4) The interval of values in which 95% of the measurements of x should lie.
(5) If a 26th. Data (for K-value) were to be generated (for Each set), in what Range of values should this
measurement lie for a 95% probability?
(6) What is the Estimated TRUE mean value for each set?
6. Using the Sample Variance obtained for Each set of data, and Chi-Squared Distribution, obtain the true variance
expected at 95% confidence (i.e., obtain the Precision Interval for the Variance.)
7. Test the hypothesis that the measured data of Each set is described by a Normal Distribution and provide
statistical justification for the test results of Each set.
8. Using RATIOS, calculate (for Each set) the number of measurements required to suppress the precision interval of
the mean values to within 1 UNIT if the variance is 20 UNITS.
9. Using RATIOS, calculate (for Each set) the total number of measurements required for a 95% precision interval of
30 UNITS if S # 1 (or S # 2, S # 3, S # 4, etc.) = 100 UNITS.
10. Comment on ALL the potential sources of error during the collection of data leading to the final values of the
spring Indexes.
11. Considering the equation used for the calculation of the “K”, examine the effect of Relative Uncertainty of each
measurement on the resulting values of the final “K”. If the Uncertainties are set at 2%, what is the (Percentage of )
Uncertainty in the “K”s?
12. Superimpose the Distribution of the obtained data (for “K”) of ALL sets to COMPARE :
a) The Frequency Distributions,
b) The Central Tendencies, and the
c) Probability Density Functions.
Decision / Conclusion Using the information obtained in steps 1 through 12, make a comprehensive (but highly condensed) Comparative
Analysis of the sets of samples and Decide (by Statistical Justification) the springs of which set are superior and
should be used in the Module.
Page 15.153.25
Appendix B: Rating and Assessment Form of the Activity
RATING AND ASSESSMENT
1. How many members formed your group? [ ]
2. Indicate the number, duration, and place of EACH of your meetings. (Use the following TABLE for tabulation)
Meeting #
DATE
DAY
TIME
PLACE
Total Time Expended:
3. How would you rate the time required for completion of this Project?
[Use a Mark in the blank box of your choice.]
Too
Short
Short About
Right
Long Too
Long
4. If you had to do this experiment/activity again, how long would it take the second time? Use the Percentages listed below.
(30-40)%
(40-50)%
(50-60)%
(60-70)%
(70-80)%
(80-90)% Almost
The Same Can’t Predict
Page 15.153.26
5. Would the experience gained in this activity help you optimize your approach the next time you have to deal with a similar task?
(Use the Rating and the Space provided below)
6. How would you rate the overall value of this Experiment and Project?
7. If you get to teach a similar course, would you incorporate such an
activity in your course? If yes, what changes would you recommend
or introduce? (Use the Rating and the Space provided below)
Highly
Unlikely
Unlikely
Probably
Very Likely
Definitely
Recommended Changes:
Lowest Highest
1 2 3 4 5
Highly Unlikely Definitely
1 2 3 4 5
Page 15.153.27
Appendix C:
Student Collaboration Two rising junior engineering students were invited to collaborate with the author and assist him in achieving the goals of the project. The parameters in successful implementation of the processes involved were comprehensively discussed, outlined, and a preliminary Gantt chart was generated. Through four weekly scheduled meetings, alternative designs and approaches for each of the components and processes were evaluated, ranked, and chosen. It took two weeks to fabricate, modify, and test the reliability of the apparatus and its feasibility for replication in other institutions. Another two weeks were necessary to successfully classify the samples into categories that would produce six (6) distinct sets with the predetermined ranges and frequencies. This was the first Summer Internship experience for the two rising Juniors involved. They did participate in the design of other apparatus. It was truly a pleasure to observe the growth of these young engineers-to-be during this period. By the end of the Summer, it was clear that: a) they have developed a sense of ownership of the laboratory, b) are proud of their creations, c) being quite protective of the laboratory. A survey of alumni from the College of Engineering at the University of Delaware reveals that "Alumni with research experience were more likely to pursue graduate degrees, and they reported greater enhancement of important cognitive and personal skills. In addition, respondents who had been involved in research were much more likely to have reported that they had a faculty member play an important role in their career choice." 6 Both of the students involved in the design and development of this (and other) projects and experiment are currently pursuing graduate studies.
Page 15.153.28
Appendix D: Table D1. Possible Combinations of Four Sets of Springs Based on the Availability of Six Sets
Ci
Combination
#
Trial #
YEAR/
Section
SET: A
SET: B
SET: C
SET: D
SET: E
SET: F
C1 1 2008
C2 1 2009
C3 1 2010
C4
C5
C6
C7
C8
C9
C10
C11
C12
C13
C14
C15
Possible Number of Combinations for Four (4) Sets out of Six (6) Sets 7:
6C4 =
Page 15.153.29
Appendix D (Cont.):
Table D2. Possible Combinations of Three Sets of Springs Based on the Availability of Six Sets
YEAR
Trial #
SET: A
SET: B
SET: C
SET: D
SET: E
SET: F
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
Possible Number of Combinations for Three (3) Sets out of Six (6):
6C3 =
Page 15.153.30
Appendix E: Sample of the Data Collected for Obtaining the K-values for Each Springs in a Set
Contractions ѐS ѐA ѐB ѐC ѐD ѐE
K-values K = F/ѐ KA KB KC KD KE
Spring 1 0.225 0.495 0.769 1.023 1.171 1.279
3.853 3.786 3.753 3.804 3.775 3.819
Spring 2 0.236 0.510 0.772 1.035 1.174 1.294
3.674 3.675 3.738 3.760 3.765 3.775
Spring 3 0.215 0.474 0.743 1.008 1.154 1.264
4.033 3.954 3.884 3.861 3.830 3.865
Spring 4 0.218 0.497 0.764 1.039 1.174 1.281
3.977 3.771 3.777 3.746 3.765 3.813
Spring 5 0.211 0.482 0.749 1.020 1.159 1.279
4.109 3.888 3.853 3.816 3.814 3.819
Spring 6 0.219 0.474 0.756 0.992 1.134 1.264
3.959 3.954 3.817 3.923 3.898 3.865
Spring 7 0.214 0.483 0.758 1.011 1.171 1.286
4.051 3.880 3.807 3.850 3.775 3.799
Spring 8 0.213 0.479 0.753 1.015 1.145 1.268
4.070 3.912 3.833 3.834 3.860 3.853
Spring 9 0.191 0.449 0.731 0.993 1.144 1.267
4.539 4.174 3.948 3.919 3.864 3.856
Spring 10 0.230 0.500 0.765 1.022 1.168 1.290
3.770 3.748 3.773 3.808 3.784 3.787
Spring 11 0.215 0.495 0.759 1.015 1.163 1.308
4.033 3.786 3.802 3.834 3.801 3.735
Spring 12 0.213 0.485 0.762 1.016 1.161 1.275
4.070 3.864 3.787 3.831 3.807 3.831
Spring 13 0.219 0.493 0.771 1.036 1.188 1.286
3.959 3.801 3.743 3.757 3.721 3.799
Spring 14 0.221 0.493 0.759 1.022 1.155 1.276
3.923 3.801 3.802 3.808 3.827 3.828
Spring 15 0.195 0.450 0.722 0.968 1.118 1.254
4.446 4.164 3.997 4.021 3.953 3.896
Spring 16 0.227 0.504 0.779 1.045 1.170 1.284
3.819 3.718 3.705 3.724 3.778 3.805
Spring 17 0.215 0.486 0.753 1.020 1.155 1.290
4.033 3.856 3.833 3.816 3.827 3.787
Spring 18 0.220 0.478 0.748 1.012 1.150 1.294
3.941 3.921 3.858 3.846 3.843 3.775
Spring 19 0.233 0.515 0.777 1.038 1.191 1.319
3.721 3.639 3.714 3.750 3.711 3.704
Spring 20 0.221 0.483 0.749 1.005 1.169 1.290
3.923 3.880 3.853 3.873 3.781 3.787
Spring 21 0.223 0.487 0.772 1.034 1.159 1.294
3.888 3.848 3.738 3.764 3.814 3.775
Spring 22 0.230 0.497 0.776 1.030 1.180 1.316
3.770 3.771 3.719 3.779 3.746 3.712
Spring 23 0.219 0.477 0.758 1.015 1.148 1.307
3.959 3.929 3.807 3.834 3.850 3.738
Spring 24 0.202 0.479 0.746 1.010 1.167 1.296
4.292 3.912 3.869 3.853 3.787 3.769
Spring 25 0.212 0.488 0.742 1.022 1.145 1.278
4.090 3.840 3.889 3.808 3.860 3.822
Page 15.153.31
Appendix F: Graphs and Slopes Corresponding to Each of the 25 Springs in the Set of App. D
