Engineering Documents Should Not Be Like Snowflakes

Posted on 20-January-2016 by mathscinotes

Quote of the Day

No management system can substitute for hard work. A manager who does not work hard or devote extra effort cannot expect his people to do so. He must set the example. The manager may not be the smartest or the most knowledgeable person, but if he dedicates himself to the job and devotes the required effort, his people will follow his lead.

— Hyman Rickover

Figure 1: Many People Believe that Every Snowflake is Unique.

Figure 1: Many People Believe that Every
Snowflake is Unique (Source).

I had dinner the other night with a networking engineer who works as a contractor. He is known as a person who writes well, and he is finding that many of the companies that he works for are asking him to determine how their networks operate and to write down what he discovers.

He related a story from a recent assignment where he was hired by a small, high-tech firm to reconfigure the routers and switches on their network. When that task was done, they asked him to investigate other aspects of their network and to write down what he finds. He asked his manager if the company had a documentation template that they wanted him to use. The manager responded as follows:

Documentation template? You need to understand that our documents are like snowflakes – every document is unique.

I cannot imagine a manager responding like this – think of all the problems this lack of standardization creates:

  • Engineers waste time on unproductive style decisions
  • Every engineer now need to know how to create tables of contents, appendixes, etc.
  • When the original author is not available to maintain the document, someone else now has to figure out each documents style system rather than working with one company  standard.
  • The documents will have a wide variation in look, feel, and completeness.

As I have stated before, I am quite emphatic on documentation standards and templates. I make sure that every engineer knows the various document standards (e.g. short form, long form, mathematics, requirements) and how to use the templates.

Unfortunately, the tools used by most organizations do not promote easy standardization. When I worked as a proposal manager for ATK, I would frequently work with partners who had different documentation tools that ATK, so I have used every documentation tool known to man short of stone tablets. I only have worked with a few that made documentation standardization easy.

Secure Computing had the best documentation system I have ever used. It was based on LaTex and enforcing standards was easy because they used a standard template file. Their requirements were stored in DOORS, and they did a very good job of linking the DOORS database to their documents.

The worst tool I have used for engineering documents is Microsoft Word. Its style system is okay if you have disciplined authors working on small, one-person documents, but it is simply too difficult to enforce standards when being used on documents with multiple authors (e.g. proposals). I cannot tell you how many hours I have spent cleaning up formatting issues in large Word documents.

Posted in Technical Writing | Comments Off on Engineering Documents Should Not Be Like Snowflakes

A Quick Look at the Walmart Closings

Posted on 18-January-2016 by mathscinotes

Quote of the Day

The secret of joy in work is contained in one word -- excellence. To know how to do something well is to enjoy it.

- Pearl Buck. I also enjoy when I have mastered a skill.

Figure 1: List of Walmart Facilities Being Closed.

Figure 1: List of Walmart Facilities Being Closed.

I live in Minnesota where Target is a major employer. Target recently greatly overexpanded into Canada and had to close its stores there. There were numerous business issues associated with their Canadian operations, such as poor locations and inadequate logistics. Target chose to give up and simply close their Canadian operations.

I saw in the news that Walmart is in the process of closing 269 of its stores, including 154 in the US. I became curious where the US store closures were occurring, so I downloaded the closings file and generated a pivot table of the data (Figure 1). It looks like most of the store closures involve:

  • Walmart Express stores
  • Stores that are geographically close to other Walmart stores.
  • Over half the stores are in six states: TX, NC, AR, CA, AL, KS.

As with Target, it looks like a classic case of overexpansion.

My father-in-law used to work for the Small Business Administration (SBA) and helped start hundreds of small businesses. He told me that the most dangerous time for a small business is its first attempt at expansion. He often discussed how a viable trucking firm with one truck often would fail when it expanded to two trucks. The business owners often underestimated the challenges with:

  • finding the time needed to find work to fully utilize trucks instead of one
  • hiring good staff
  • having to work through other people rather than doing everything yourself

Look, I get it. I understand. Starting your own business can be hard, and it is even harder to keep it afloat for it to be the successful company you know it can be. In today's business climate, one major difficulty faced by budding entrepreneurs is to find people who have the necessary skill sets for the job and also know how to effectively contribute to the company, thus making it very important to safeguard both the company and its valued assets by utilizing options like keyman insurance for top employees; especially in the initial stages of a business, risk management and mitigation should be a priority if the ship must remain steady during stormy waters.

But to be able to succeed, business owners must be able to meet the requirements that are listed above. Time management, teamwork, and more importantly, hiring good staff are all factors that must be considered if you want to be successful. Trust me, I know how hard it can be, especially if you haven't taken the time to enlist the help of somewhere like these Montreal executive search firms to assist you with the hiring process. They are experts at their chosen craft, and if you want to make sure that you can overcome any challenges that your business throws your way, you must have good staff who can lend you a helping hand along the way. Of course, it's not always that easy.

Sometimes you might need to think out of the box to get the desired results. As a small company, you may not have the manpower or the resources to look at every facet of running a business. And despite the availability of ERP software packages and solution providers, business heads tend to want to do things themselves because many feel that they are in the best position to streamline business processes. While that may be true, it can also take a tonne of time; time that could be utilized in areas of core proficiency. ERP consulting isn't about the technology - it's about aligning people and processes. Taking help from the right professionals can get you a solution roadmap for your business, one that has an evolving growth phase, and give a more real-time understanding of business activities.

Having said that, remember that many fail at the first time of trying. Others keep trying until they have tried everything in the book to make sure their business has the potential to go all the way. Sometimes it just depends on the owner. If you have an owner who knows that cheap merchant services aren't always cheap but decides to hire one anyway because they know how much they can positively impact a business, you could be well on your way to success. You need to take these risks sometimes, especially if you are a small business that is trying to find its feet.

Unfortunately for some business owners, and the owners of the truck company, things just don't work out. And it looks to be the case with Target and Walmart too.
It looks like they are showing that expanding a large business can also be a challenge.

Posted in Management | Comments Off on A Quick Look at the Walmart Closings

Book Review: Against The Tide

Posted on 17-January-2016 by mathscinotes

Quote of the Day

People may not find themselves prepared for the situation in which they find themselves in life.The good leader recognizes this and reassigns them to a place or position where they can be successful. If there in no appropriate role in which they can flourish, the best course of action for all is to sever the relationship. The only clearly wrong decision is to demand that an unprepared person face a challenging climb alone.

— Admiral Dave Oliver. Man do I buy into this quote. I cannot tell you how many times I have seen people thrown into hopeless situations where they could not possibly succeed.

Figure 1: Admiral Hyman Rickover (Source).

Figure 1: Admiral Hyman Rickover (Source).

I have been working my way through a number of management books lately. The best of the lot, Against the Tide, was written by Admiral Dave Oliver about the management principles of Hyman G. Rickover (Figure 1). Years ago, I worked for a retired sub captain named Ernie Fischer. He had a number of interesting stories about serving on a nuclear submarine and about Hyman G. Rickover, the man responsible for creating the modern nuclear navy.

Because of Ernie's stories, I decided to read a biography on Rickover written by the well-known naval author Norman Polmar. The biography was comprehensive, but did not go into details on how Rickover managed people – Dave Oliver's Against The Tide fills this gap.

The book is short (178 pages including index), well-written, and provides some insight into how Rickover managed the introduction of a very complex technology into a very conservative organization (i.e. US Navy). Rickover was a difficult man in many ways, but it would be difficult to argue with his success.

I could summarize the book here, but Dave Oliver does an excellent job in the following Youtube video.

Posted in Management, Military History | Comments Off on Book Review: Against The Tide

Worked Calculation Example of Lithium Battery Capacity Versus Load

Posted on 16-January-2016 by mathscinotes

Quote of the Day

Successful ... politicians are insecure and intimidated men. They advance politically only as they placate, appease, bribe, seduce, bamboozle or otherwise manage to manipulate the demanding and threatening elements in their constituencies.

— Walter Lippmann

Introduction

Figure 1: 18650 Lithium -Ion Cylindrical Battery (Source).

Figure 1: 18650 Lithium -Ion
Cylindrical Battery (right)
versus a standard AA battery
(left).  (Source)

I frequently am asked to comment on data that other engineers send me. This morning I received some test data obtained from an engineer measuring the backup time of an Uninterruptible Power Source (UPS) containing multiple lithium-ion (Li-Ion) batteries. The engineer was disappointed with the backup time provided by this UPS and wanted to know if his test results were reasonable considering the battery capacity of the UPS. While there were numerous circuit parameters measured during this testing, the critical information was the battery voltage versus time.

The engineer's routine calculations showed that the UPS should provide 4 hours of backup time, but he measured 2.7 hours. While batteries are listed with a nominal charge  capacity, their actual capacity varies strongly with the load presented to the battery. His testing was performed at room temperature and with a load of 0.8 A.

In this post, I will show that his test results are reasonable when when the effects of the load current are taken into account. I will perform the calculations three ways:

  • Method 1: Simple capacity/load modeling (ignoring current load on capacity).
  • Method 2: Using a graphical curve of capacity versus load.
  • Method 3: Using numerical methods for interpolating digitized capacity graphs.

Background

Definitions

C-Rate
C-rate is the charge/discharge current normalized to the battery capacity. A charge/discharge rate of one C for one hour draws a charge equal to the battery capacity. For example, the 1C discharge rate for a 2.2 A-hour battery is 2.2 A.
Cutoff Voltage (VCutoff)
The battery voltage at which the UPS terminates the discharge of the battery.
Capacity
The available charge in the battery, which is a function of the load current. A battery's charge capacity is generally specified at some low current draw. For example, the capacity of lead-acid batteries is specified at a 20 hour (C/20 or 0.05C) rate.

UPS Characteristics

Here are the key facts about this UPS:

  • It contains 4 series-connected, Li-Ion, 18650 cylindrical batteries (Figure 1).
    • These batteries have a nominal cell voltage of 3.7 V.
    • I assume that each 18650 battery is rated to have a nominal 2800 mA-hour charge capacity. The actual rating is 2700 mA-hour (minimum) and 2900 mA-hour (typical). I will average the two values for this analysis.
  • The UPS load is a device that requires an input voltage between 10 V and 16 V.
    • The voltage range requirement is met by connecting the batteries in series.
    • The engineer modeled the load using 800 mA of constant current draw.
  • Most UPS hardware stops discharging the battery at  VCutoff, which for Li-Ion batteries is typically near 3.3. V. In this case, the UPS cutoff was ~2.9 V.

Analysis

Measured Battery Voltage Versus Time for Constant Load Current

Figure 2 shows the data that was emailed to me. I have literally seen hundreds of these test plots. This UPS I include a calculation on Figure 3 that shows that the charged cell voltage for the batteries in this 4 cell-string pack is 3.92 V. This is below the manufacturers 4.2 V charge voltage in their specification. The cutoff voltage is 2.87 V.

Figure 3: Raw Battery Data That Was Emailed To Me.

Figure 2: Raw Battery Data That Was Emailed To Me.

This data was measured by instrumenting the battery series inside the UPS. We will need the initial cell voltage and the final cell voltage in order to estimate the charge drawn from the battery.

Method 1 will not use this data, but Method 2 and 3 will.

Method 1: Simple Calculation Ignoring Load Current Impact on Capacity

Figure 3 shows how to estimate the backup time provided by this UPS assuming nominal battery characteristics.

Figure 2: Nominal Calculation Example.

Figure 3: Nominal Calculation Example.

The key problem with this analysis is that it assumes that the battery capacity does not depend on the load. I model the effect of load current on battery capacity any time the load currents exceed 0.1C.

Method 2: Graphical Analysis

Figure 4 shows how to compute the expected backup time using the battery's capacity versus load chart. All the calculations are shown on the graph and I obtain a backup time estimate of 2.8 hours.

The calculations shown on the graph can be described as follows:

  • Us the initial cell voltage to determine how much charge is lost because the UPS did not fully charge the battery.
  • Use the final cell voltage to determine how much charge is available from a fully charged battery.
  • Determine the difference between the final and initial charge, which reflects the charge available for backup energy.
  • Divide the available charge (in mA-hours)  by the load current, which give the backup time.
Figure M: Computing Run Time Using Capacity Chart.

Figure 4: Computing Run Time Using Capacity Chart.

Method 3 : Numerical Analysis

Manufacturer's Capacity Rating Versus Load

Figure 5 shows the typical discharge specification for  Li-Ion battery from Panasonic with a nominal rating of 2900 mA-hour (2700 mA-hour minimum).  As you can see in Figure 4, the typical capacity is measured when the battery has a minimal load (0.2C). One unusual aspect of this chart is that you also get full capacity with a high load current (2C). I only rarely see this characteristic on a capacity chart.

Figure M: Discharge Capacity Versus Load Current.

Figure 5: Discharge Capacity Versus Load Current.

I digitized this data using Dagra and pasted it into Mathcad. Figure 6 shows the  digitized the data and a routine to interpolate the data. I will assume that batteries actual capacity is 2800 mA-hour, the mean of the minimum and maximum.

Figure 5: Digitized and Interpolated Charge Data.

Figure 6: Digitized and Interpolated Charge Data.

Mathcad Model of Battery Capacity Data

Given the discharge data shown in Figure 7, I can determine the effective capacity of the UPS battery at a 0.8 A load. Figure 6 shows the my interpolated results for the manufacturer's data and my interpolation for a 0.8 A load. The graph shows that the effective battery capacity is ~2779 mA-hour with an initial charge voltage of 4.2 V. I also show that 554 mA of charge is missing if the battery is initially charged to only 3.92 V, which is this case. In the next section, I will show how to algebraically obtain these results.

Figure 6: Plot of Battery Capacity at 0.8 A Load (0.29C).

Figure 7: Plot of Battery Capacity at 0.8 A Load (0.29C).

Estimated Discharge Time Assuming that Capacity is Load-Dependent

In Figure 8, I use 2-dimensional interpolation to compute the battery capacity assuming VCutoff = 3.3 V and ILoad = 0.8 A.

Figure 7: Calculation of Discharge Time (2.5 hours) Assuming a Load-Dependent Charge.

Figure 8: Calculation of Discharge Time (2.8 hours) Assuming a Load-Dependent Charge.

My calculated discharge time of 2.8 hours roughly agrees with the measured discharge time of 2.7 hours.

Conclusion

My estimate for the battery operating time is 2.8 hours with an 800 mA load. We actually measured 2.7 hours, so the estimate is ~4%. This is error is within reason for batteries – they are subject to individual variation.

Posted in Batteries | 21 Comments

Old West Cavalry Distributions of Age and Countries of Origin

Posted on 14-January-2016 by mathscinotes

Quote of the Day

You will never reach your destination if you stop and throw stones at every dog that barks.

— Winston Churchill

Introduction

Figure 1: US Cavalry Field Uniforms in 1876 (Source).

Figure 1: US Cavalry Field Uniforms in 1876 (Source).

When I was a boy, my father often told me stories of his grandfather, Louis Bauer, who was a member of the US Cavalry on the American Frontier. In fact, my father left my brother Tim the watch, spurs, and shaving cup that Louis used when he served in the cavalry. I used to wear the spurs for fun as a kid.

Louis Bauer was a recent immigrant from Germany who volunteered for the US Army and served in the 7th Cavalry nine years after the Battle of the Little Bighorn. I always thought it was interesting that a young immigrant would join the military shortly after arriving in the US.

Last weekend, I was watching a show on CSPAN that mentioned that many members of the US Cavalry were German-born – he also commented that the same was true during the Civil War. I decided I would look up some information on the 7th Cavalry and generate a breakdown of the troopers by country of origin and age. The easiest information to find was about the 7th Cavalry under Custer.

Background

Everything in this post is taken from this paper containing the muster roll of the 7th Cavalry at the time of the Battle of the Little Bighorn in June 1876 (Source).  I imported this data into Excel and generated a pivot table. The muster rolls list an AP reporter as a staff member of the 7th Cavalry – I left his data in my results. Today, we would call him an embedded reporter. I also left in a number of troopers who were not present on the day of the battle. In Appendix A, I show a list given at the Little Bighorn Battlefield National Monument for the troops at the battle.

Analysis

Countries of Origin

Figure 2 show the countries of birth for the 7th Cavalry troopers. Over 40% of the troopers were foreign-born – they were from seventeen different countries with over half having been born in  Ireland and Germany.

Figure 2: Countries of Origin for 7th Calvary.

Figure 2: Countries of Origin for 7th Calvary.

I should point out that some troopers were from countries that no longer exist (e.g. Prussia). I replaced these country names with their modern equivalent (e.g. Prussia as Germany).

Age Distribution

Figures 3(a) and 3(b) show the age distribution of the 7th Cavalry troopers. They actually are a bit older than I would have imagined based on the average ages of the GIs during 20th century conflicts.  For example, the average age of 7th Cavalry troopers versus military forces in WW2 and Vietnam is given below.

  • 7th Cavalry: 28.7 years
  • WW2: 26 years
  • Vietnam: 22 year
Figure 3(a): Pivot Table of 7th Cavalry Ages. Figure 3(b): Pivot Chart of 7th Cavalry Ages.

I should mention that the total trooper count differs between Figures 2 and 3. This is because some troopers did not have their date of birth listed.

Conclusion

It appears that over 40% of the 7th Cavalry troopers were foreign-born. I cannot imagine arriving into a new country and ending up in the middle of conflict for which you have little knowledge.

Appendix A: Little Bighorn Battlefield Nat. Monument Trip

I was able to visit the site of the Little Bighorn battlefield on October 16, 2016. While touring the site, I found the following chart that summarized birth places of cavalry troops present at the battle (Figure 4).

Figure 4: Birthplaces of 7th Cavalry Troopers.

Figure 4: Birthplaces of 7th Cavalry Troopers.

Because my photo is not as clear as I would like, I captured the data on the chart in Table 1. Note that two troopers are listed as born "at sea." The muster rolls show only one trooper as born at sea, with the additional comment that he was from Ireland. In the data above, I listed that trooper as "from Ireland."

Table 1: 7th Cavalrymen Countries of Origin.
Country Numbers Percentages
Foreign Total
 368 43.8%
Greece 1 0.3%
Russia 1 0.3%
Spain 1 0.3%
Hungary 2 0.5%
Poland 2 0.5%
Sweden 2 0.5%
At Sea 2 0.5%
Norway 3 0.8%
Denmark 6 1.6%
Italy 7 1.9%
France 8 2.2%
Switzerland 12 3.3%
Canada 15 4.1%
United Kingdom 53 14.4%
Germany 125 34.0%
Ireland 128 34.8%
United States Total 472 56.2%
New York 101 21.4%
Pennsylvania 82 17.4%
Ohio 60 12.7%
Massachusetts 45 9.5%
Indiana 22 4.7%
Kentucky 19 4.0%
Illinois 18 3.8%
Maryland 16 3.4%
Maine 12 2.5%
Missouri 12 2.5%
New Jersey 11 2.3%
Connecticut 8 1.7%
New Hampshire 7 1.5%
Rhode Island 7 1.5%
Michigan 7 1.5%
Wisconsin 7 1.5%
Vermont 5 1.1%
Virginia 5 1.1%
West Virginia 4 0.8%
North Carolina 4 0.8%
Iowa 3 0.6%
District of Columbia 3 1.6%
Louisiana 2 0.4%
Georgia 2 0.4%
South Carolina 2 0.4%
Tennessee 2 0.4%
Washington 1 0.2%
California 1 0.2%
New Mexico 1 0.2%
Delaware 1 0.2%
Texas 1 0.2%
Kansas 1 0.2%
Grand Total 840 100.0%
Posted in History Through Spreadsheets, Military History | Comments Off on Old West Cavalry Distributions of Age and Countries of Origin

Derivation of Pejsa Point-Blank Range Formula

Posted on 11-January-2016 by mathscinotes

Quote of the Day

Almost everything you do will seem insignificant, but it is important that you do it.

— Mahatma Gandhi. I think about this statement frequently. I often encounter situations where I wonder if I can make a difference, but I still keep trying.

Introduction

Figure 1: Illustration of Point-Blank Range.

Figure 1: Illustration of Point-Blank Range.

A reader asked me if I could work through Pejsa's formulas related to calculating Point-Blank Range (PBR). During my earlier review of Pejsa's work, I chose not to cover this material just because it was not some of my favorite material – taste is definitely a part of mathematics. However, it is not difficult material to work through, and I like to answer questions when I can.

Background

Definitions

Figure 1 illustrates the key variables involved in computing the zero characteristics for a rifle setup for a specific point-blank range.

Line of Sight (LOS)
The line passing through the viewing axis of the rifle sight.
Near Zero (ZN)
The point nearest the shooter at which the bullet intersects the LOS.
Far Zero (ZR)
The point farthest from the shooter at the which the bullet intersects the LOS.
Midpoint (ZR)
The range along the LOS at the bullet attains it maximum height (Hm).
Point-Blank Range (ZPBR )
The maximum distance where the trajectory of the bullet does not exceed a predetermined amount of rise or drop, also the maximum distance a specific round can be fired and hit a given target without any compensation for bullet drop (Source).

Shortcomings of Pejsa's Approach

The primary advantage of Pejsa's approach is that it is algebraic – no iteration is required. However, the model does have shortcomings:

  • The formulas are focused on projectiles that are similar in shape to the US military's M2 30 caliber bullet.

    The M2 bullet shape has a drag coefficient as shown in Figure M by the line labeled GP or Pejsa. The Pejsa formulas scale the G1 ballistic coefficients so that it will properly fit the GP shape. Always use the G1 ballistic coefficient for modeling using Pejsa formulas.

  • The formulas are not directly usable with other coefficients, e.g. G7.

    Fortunately, G1 coefficients are available for nearly every bullet. However, G1 is considered very "old-school" compared to more modern shapes. To compare the drag coefficients of the various shapes, see Appendix A.

  • The formulas could be modified to work with other bullet shapes, e.g. G7.

    I will not be the person to take that task on. See this web site for some work in this area.

Objectives

I will derive Equations 1 -4 from Pejsa's Modern Practical Ballistics.

Eq. 1 \displaystyle M=\frac{1}{{\frac{2}{{F0}}+\frac{G}{{{{V}_{0}}\cdot \sqrt{{{{H}_{m}}+S}}}}}}
Eq. 2 \displaystyle {{Z}_{N}}=\frac{{1-SH}}{{\frac{1}{{F0}}-\frac{1}{M}}}
Eq. 3 \displaystyle {{Z}_{F}}=\frac{{1+SH}}{{\frac{1}{{F0}}+\frac{1}{M}}}
Eq. 4 \displaystyle {{Z}_{{PBR}}}=\frac{{1+SQ}}{{\frac{1}{{F0}}+\frac{{SQ}}{M}}}

where

  • S is the sight height above the LOS.
  • Hm is the maximum height achieved by the bullet above the LOS.
  • \displaystyle SH=\sqrt{{1+\frac{S}{{{{H}_{m}}}}}}
  • \displaystyle SQ=\frac{{SH}}{{\sqrt{2}}}
  • M is the point along the trajectory at which the bullet has the high positive deviation from the LOS. M is shorthand for "Midpoint," which is a bit of a misnomer – it is not the geometric midpoint of the trajectory.
  • G =41.67, a constant that appears throughout Pejsa's work. It is a scaled version of the gravitational constant, g. I derive its value here.

I will compare the results produced by these four formulas against a web-based calculator in my analysis.

Analysis

Midpoint Formula

I covered the derivation of the midpoint formula in this post, and I will not discuss it further here.

Near Zero Formula

Figure 2 shows the derivation of the near-zero formula.

Figure M: Derivation of Near Zero Formula.

Figure 2: Derivation of Near Zero Formula.

Far Zero Formula

Figure 3 shows the derivation of the far-zero formula.

Figure M: Far Zero Formula Derivation.

Figure 3: Far Zero Formula Derivation.

Point-Blank Range Formula

Figure 4 shows the derivation of the point-blank range formula.

Figure M: Derivation of Point-Blank Range Formula.

Figure 4: Derivation of Point-Blank Range Formula.

Excel Implementation

I always appreciate when an author includes an implementation of their results so that I can experiment with them. I have created an Excel version of Equations 1 – 4, which you can download here.

In the worksheet, I include a comparison between the results from the Pejsa formulas and my reference web-based calculator (Figure 5).

Figure M: Comparison Between Pejsa and Web Site Values.

Figure 5: Comparison Between Pejsa and Web Site Values.

Conclusion

This post shows that the Pejsa formula provides simple, algebraic results that compare well with results from a often-used web site calculator. I provided a tool (i.e. Excel workbook) that will allow others to experiment with Pejsa formulas.

Appendix A: Comparison of Drag Coefficients.

Figure 6 the drag coefficients for various projectile shapes. The A shape is for an air gun projectile.

Figure M: Drag Coefficients For Different Ballistic Shapes.

Figure 6: Drag Coefficients For Different Ballistic Shapes (Source).

Posted in Ballistics | 2 Comments

Work Allocations Relative to the Ansoff Matrix

Posted on 7-January-2016 by mathscinotes

Quote of the Day

I was almost a sorry witness of his doings, knowing that just a little theory and calculation would save him 90 percent of the labor.

— Tesla about Edison

Introduction

FIgure 1: The Ansoff Matrix Provides a Framework For Evaluating Product Development Approaches.

Figure 1: The Ansoff Matrix Provides a
Framework For Evaluating Product
Development Approaches (Source).

I am often asked to evaluate the focus of my team's efforts relative to company's priorities. There are different ways of expressing these priorities – one common approach is to look at how your group resources are allocated relative to the Ansoff matrix (Figure 1).

Technology firms are usually focused on developing products that expand their firm's Total Addressable Market (TAM), and it is important to ensure that your labor allocations reflect that priority. On the Ansoff matrix, diversification reflects new products for new markets.

Analysis

Figure 2 shows how I approached the problem in Excel. For each employee, I load the employees workload allocation per the four Ansoff categories. I then compute a weighted-average allocation for my total group.

Figure 2: Excel Table Showing My Workload Allocations Per the Ansoff Matrix.

Figure 2: Excel Table Showing My Workload Allocations Per the Ansoff Matrix.

Rather than go through the equations, it is easier just to include the spreadsheet.

Conclusion

While this is as simple problem to solve, it is a common type of management math problem.

Posted in Management | 2 Comments

A Mathcad Utility Function for Air Pressure and Density

Posted on 6-January-2016 by mathscinotes

Quote of the Day

Fantasy is hardly an escape from reality. It’s a way of understanding it.

— Lloyd Alexander

Introduction

Figure 1: Example of Moist Air Density Versus Pressure (Source).

Figure 1: Example of Moist Air Density
Versus Pressure and Temperature.
(Source)

I have some work to do that requires a model for air density as a function of altitude, temperature, and pressure. As usual, I will turn to the Wikipedia to provide me a working model. I will then check the results of my model against results published by other sources.

Figure 1 shows some typical air density data for different pressure and temperature values for air with a dew point of 10 °C. I am going to write some Mathcad programs that compute the density of air as a function of combinations of temperature, altitude, pressure, and relative humidity. I always write Mathcad versions of the software routines that I am creating so that I have results that I can use to verify my code.

I should note that humidity is a small effect, but it is one that I wanted to include in my modeling efforts.

Background

Definitions

Standard Atmosphere
The U.S. Standard Atmosphere is an atmospheric model of how the pressure, temperature, density, and viscosity of the Earth's atmosphere change over a wide range of altitudes or elevations (Source).
Relative Humidity
Relative humidity (abbreviated RH) is the ratio of the partial pressure of water vapor to the equilibrium vapor pressure of water at the same temperature. Relative humidity depends on temperature and the pressure of the system of interest (Source).
Dew Point
The dew point is the temperature at which the water vapor in a sample of air at constant barometric pressure condenses into liquid water at the same rate at which it evaporates (Source).
Lapse Rate
The lapse rate is defined as the rate at which atmospheric temperature decreases with an increase in altitude (Source).

Objectives

In a sense, this is a simple post because I am just implementing empirical air pressure and density formulas from the Wikipedia. However, it is useful to see how to implement empirical relationships in Mathcad.

There are two key formulas: pressure versus altitude, sea level temperature, sea level pressure (Equation 1); and  density versus altitude, sea level pressure and sea level density  (Equation 2).

Eq. 1 \displaystyle p\left( {h,{{T}_{0}},{{p}_{0}}} \right)={{p}_{0}}\cdot {{\left( {1-\frac{L}{{{{T}_{0}}}}\cdot h} \right)}^{{\frac{{g\cdot M{{W}_{{air}}}}}{{{{R}_{d}}\cdot L}}}}}
Eq. 2 \displaystyle \rho \left( {h,{{p}_{0}},{{\rho }_{0}}} \right)={{\rho }_{0}}\cdot \frac{{p\left( {h,{{T}_{0}},{{p}_{0}}} \right)}}{{{{p}_{0}}}}\cdot \frac{{{{T}_{0}}}}{{{{T}_{0}}-L\cdot h}}

where

  • p is the estimated air pressure as measured at the altitude h.
  • Rd is the universal gas constant.
  • ρ0 is the air density at the reference level.
  • MWair is the average molecular weight of an air molecule.
  • g is the acceleration due to gravity at the Earth's surface.
  • p0 is the air pressure as measured at our reference level (e.g. sea level).
  • T0 is the air temperature as measured at our reference level (e.g. sea level).
  • h is the change in altitude as measured from our reference level.
  • L is the lapse rate, which is the assumed constant rate of temperature reduction with increase in altitude.

Analysis

Dry Air Characteristics

Figure 2 shows how I implemented the dry air pressure and density formulas from the Wikipedia. Nothing special here – just grind it out.

Figure 2: Derivation of Dry Air Pressure and Density Formulas.

Figure 2: Derivation of Dry Air Pressure and Density Formulas.

Lapse Rate Definition Common Pressure Approximation Plot of Standard Atmosphere Characteristics

Humid Air Density Corrections

Figure 3 shows how I implemented the moist air model for density given in the Wikipedia. I parameterized the formula two ways:

  • pressure, temperature, relative humidity

    There is some debate about whether to use dew point or relative humidity. As far as I am concerned, they are equivalent measures because one can readily convert from one to the other using the formulas shown in Appendix A.

  • height, sea level temperature, relative humidity

    This model assume the characteristics of the 1976 US Standard Atmosphere, and specifically the 6.5 K/km lapse rate.

I also show how the model generates results similar to those from various web sources.

Figure M: Model Including Humidity Correction.

Figure 3: Model Including Humidity Correction.

Calculation Example

In Figure 4, I use the formulas of Figures 2 and 3 to recreate Figure 1.

Figure M: My Recreating Figure 1 Using the Formulas of Figure M.

Figure 4: My Recreating Figure 1 Using the Formulas of Figure 3.

Density of Air Water Vapor Pressure Reference Data Reference Data Reference Data

Conclusion

All this work was performed using information from the Wikipedia. What an amazing resource for the working engineer! I was quickly able to develop automated routines for computing air pressure and density as functions of pressure, temperature, altitude, and relative humidity. I will now code these same formulas into a Java routine for use in some design tools I am building. I will use the Mathcad routines written above to generate test cases to validate my Java implementation.

Appendix A: Converting Dew Point to Relative Humidity

Figure 5 shows my Mathcad implementation of formulas to convert between Relative Humidity (RH) and Dew Point Temperature (TD). I grabbed these formulas from this site.

Figure M: Dewpoint and Relative Humidity Conversion Formula (Source).

Figure 5: Dew Point and Relative Humidity Conversion Formula (Source).

Appendix B: Exponential Form of the Pressure Formula.

I occasionally see Equation 1 written in the form of an exponential. Figure 6 demonstrates how an exponential function can be used to approximate Equation 1.

Figure 6: Exponential Approximation to Equation 1.

Figure 6: Exponential Approximation to Equation 1.

 

Posted in General Science | Comments Off on A Mathcad Utility Function for Air Pressure and Density

A Most Unlucky Man

Posted on 3-January-2016 by mathscinotes

Quote of the Day

Almost everything you do will seem insignificant, but it is important that you do it.

— Mahatma Gandhi

Figure 1: Grave of Joe Kieyoomia. I Cannot Find His Photograph.

Figure 1: Grave of Joe Kieyoomia. I Cannot Find
His Photograph (Source).

I watched an interesting video about the survivors of Hiroshima on CSPAN this weekend. While the main topic of the video was interesting, I was surprised to learn that there were a number of Americans POWs that were present at Hiroshima and died. They also mentioned an American POW that was near Nagasaki when it was bombed named Joe Kieyoomi. Joe, a Navajo soldier, has a war story that is almost unbelievable.

  • Survivor of the Baatan Death March.
  • He was mistaken by the Japanese as an American of Japanese descent. This subjected Joe to additional torture because he was viewed as a traitor to the Japanese people.
  • Since Joe spoke Navajo, the Japanese used him to try to break the code developed by the Navajo code talkers, which he did not know. When he failed to break the code, the Japanese subjected him to more torture.
  • He ended his imprisonment in Japan at a POW site 60 miles south of Nagasaki. He was at that camp when the atomic bomb was dropped.

His WW2 service literally spanned the entire American-portion of the war from beginning to end. You can read the details yourself on Google books.
Posted in Personal | 1 Comment

Continue To Be Vigilant For Lead Exposure In Children

Posted on 2-January-2016 by mathscinotes

Quote of the Day

If you've been playing poker for half-an-hour and you don't know who the patsy is – your're the patsy.

— Warren Buffet

Figure 1: I See Writing Like This On Sidewalks Near My Home All The Time.

Figure 1: I See Writing Like This On Sidewalks
Near My Home All The Time (Source).

Last week, I had a conversation with a coworker about a mystery that he  recently solved that I feel is worth sharing here. He has a small daughter who was found to have high lead levels in her blood. As any good father would do, he tried to find the source of lead in his home. His home is new, so there is no lead paint in it. They tested everything that their daughter drank or ate, and they found no lead in anything – yet her blood's high lead level persisted.

As he watched his daughter play, he noticed that much of her play time involved using sidewalk chalk to make drawings on their sidewalks. He decided to have the chalk tested for lead, and the test results were positive. The paint used to color the chalk contained lead. After replacing the chalk, his daughter's lead levels reduced. The chalk was manufactured in China. I should point out that face paint from China was recently shown to have lead in it.

While this story has a happy ending, we need to maintain our vigilance about lead. While lead is not used for paint in the US, I have encountered products manufactured in other countries that still use lead-based paint. After having a few surprises myself, I now always check.

Postscript

I should also mention that a number of years ago a coworker encountered lead in some painted wood blocks that were given to his children as gifts. I believe these blocks were part of this recall. This lead issue has been going on for years.

Posted in Personal | Comments Off on Continue To Be Vigilant For Lead Exposure In Children