Cool Concepts #4 - Math, Mechanics, Phantom, and Space.

Hello! Cool Concepts are back after a couple of months. In this edition, I am exploring a bunch of things that grabbed my attention recently — a little math, a lot on Space, and some daily problems. One of these topics came on a special request from my brother, and I hope he finds it informative. Feel free to skip and skim, and let me know any thoughts or additional information you may have.  

Compliant Mechanisms

A mechanism is a mechanical device that transfers or transforms motion, force, or energy. A pair of scissors, pliers, pumps, pistons, nutcrackers, lemon squeezers, etc., are examples of rigid body mechanisms, meaning they have multiple rigid parts attached by movable joints or levers. Mechanisms can be found in automobiles, robotics, hardware equipment, and anything that has moving parts. The main aim of these mechanisms is to generate more output force or energy than what was applied as input. Take a lemon squeezer, for example. The input force you give at one end is increased at the output, enough to squeeze a lemon. If you give the same force to a lemon directly in the same way, you won't be able to squeeze it.

Compliant mechanisms achieve the same kind of transformations using the flexibility of their parts rather than movable joints. Here's an example of a compliant plier mechanism.

They are a class of machines that transmits force and motion through elastic body deformation. These machines are constructed to use the flexibility of materials to our advantage. The above example is made from a single piece of material, but it generates the same output force as rigid type pliers with joints.

The main advantages of using compliant mechanisms are

1. Fewer parts — The number of parts required to accomplish a task is way less in compliant mechanisms than their rigid counterparts. The part count can be reduced by having flexible parts instead of screws, springs, hinges, etc.
2. Production processes — Production of compliant mechanisms is significantly easier. As they involve fewer and more flexible parts, they can be manufactured from single planes of materials. For example, the compliant pliers shown above were manufactured from a single polypropylene sheet. Compliant mechanisms can be manufactured using various methods, including machining, laser cutting, 3D printing, water-jet cutting, etc. It also reduces manufacturing costs, especially when they are made in bulk.
3. Performance — Compliant mechanisms don't have moveable joints, and even complex mechanisms used by NASA in space have fewer moveable pins, joints, and levers than their rigid counterparts. The mechanisms used in harsh environments such as space may adversely affect the joints and evaporate the lubricants used. The design of compliant mechanisms results in increased performance, reduced friction, and a reduced need for lubrication. Here are some more examples of compliant mechanisms.

Gallery | Compliant Mechanisms

Compliant Mechanisms

Voyager

The Voyagers

Voyager 1 and 2 are spacecraft built by NASA in the 1970s. They are two identical spacecraft made to be sent on different trajectories for the same purpose. Voyager 2 was launched first, and Voyager 1 was launched second, as Voyager 2 was launched on a longer trajectory and would reach Jupiter and Saturn after Voyager 1.

Voyagers 1 and 2 are identical spacecraft. Each is equipped with instruments to conduct ten different experiments. The instruments include television cameras, infrared and ultraviolet sensors, magnetometers, plasma detectors, and cosmic-ray and charged-particle sensors. In addition, the spacecraft radio is used to conduct experiments.

The Voyagers travel too far from the Sun to use solar panels; instead, they were equipped with power sources called radioisotope thermoelectric generators (RTGs). These devices, used on other deep space missions, convert the heat produced from the natural radioactive decay of plutonium into electricity to power the spacecraft instruments, computers, radio, and other systems.

Voyager 1's trajectory, designed to send the spacecraft closely past the giant moon Titan and behind Saturn's rings, bent the spacecraft's path inexorably northward out of the ecliptic plane — the plane in which most of the planets orbit the Sun. Voyager 2 was aimed to fly by Saturn at a point that would automatically send the spacecraft in the direction of Uranus.

Another interesting artifact of the Voyagers is a playable "time-capsule" record called the "Golden Record." The mission scientists decided to include identical records in both the Voyagers. If an extraterrestrial intelligent life form encounters the spacecraft, it intends to show them who we humans are, and it's a message to our extraterrestrial neighbors. The contents of the record were selected for NASA by a committee chaired by Carl Sagan of Cornell University et al. Dr. Sagan and his associates assembled 115 images and a variety of natural sounds, such as those made by surf, wind, thunder, birds, whales, and other animals. They added musical selections from different cultures and eras, spoken greetings from Earth-people in fifty-five languages, and printed messages from President Carter and U.N. Secretary General Waldheim.

The mission

The idea for the project started in the summer of 1965 when it was discovered that it is possible to send a spacecraft to the outer planets of the solar system, using the gravity of the planets themselves to swing the spacecraft from one planet to the next without using external propulsion systems. It was due to the rare alignment of Jupiter, Saturn, Uranus, and Neptune that happens once in 176 years. It was calculated that this would be possible if a spacecraft were launched in the late 1970s.

The Voyagers' original mission was to use this alignment and visit Jupiter and Saturn. As the mission progressed with the successful progression of all its milestones, it was discovered that two additional flybys of Uranus and Neptune were possible. So the mission was expanded to include Uranus and Neptune.

Timeline

Here is a timeline of significant events of the mission. Click on each milestone to view the details.

Voyager

NASA Jet Propulsion Laboratory California Institute of Technology

Logarithms

Almost all of us have learned logarithms in high school. In high school, we carried a booklet of Clarke's tables to look up the logarithms of numbers for math problems. Having seen so many calculations and graphs represented in linear scale versus logarithmic scale, I have always found it difficult to visualize logarithms in my mind. We have seen that logarithms and exponential functions are related, but I was never able to develop an intuition for it.

When I looked at a video about the Riemann Hypothesis, I unexpectedly stumbled upon an explanation of logarithms. It was so simple that I loathed myself for not understanding it correctly. So, here is an attempt to articulate the logic of logarithms in a way I don't forget.

Definition and examples

A logarithm (l) is the number of times a number (x) must be multiplied together to make another number (y). This sounds very familiar. In simpler terms, a logarithm reverses an exponent, like how division reverses multiplication and how subtraction reverses addition. The notation for this is l = log_x(y), read as "log y to the base x."

For example, we know that 2³ = 8, meaning if we multiply x = 2 with itself thrice, we get y = 8. Now, the logarithm of 8 is the number of times 2 should be multiplied to arrive at it, which is thrice (l = 3). It is expressed as log₂8 = 3. The base is the initial number we multiply to get to the final number.

Similarly, to find log₅625, all we have to ask is, how many times should I multiply 5 to get 625. The answer is 4 (5 x 5 x 5 x 5 = 625). So log₅625 = 4.

Of course, there are more concepts like common logarithms (base = 10), natural logarithms (base = e), change of base rule, etc., but the above method is to grasp the basics.

Timeshare

Timeshare is an interesting concept I heard during a team lunch with my colleagues. A timeshare is a shared ownership model on a property, based on specific periods of time. Timeshares are typically used for vacation homes, condominiums, campsites, and resorts.

Timeshare is a type of fractional ownership where the buyers purchase the right to occupy a unit of real estate over specific periods in a year. For example, buying one week of a timeshare means the buyer owns 1/52 of the property, and a month of a timeshare means the buyer owns 1/12 of the property. Timeshare is most popular in the vacation business, and there are three types of a timeshare.

1. Fixed Week — The buyers purchase a fixed week (or weeks) of property ownership every year and own the property for that week (or weeks). But the disadvantage with this plan is that it is not practical to plan a vacation or usage of the property on the exact week every year.
2. Floating week — The buyers purchase the duration (week or weeks), but the period is not fixed. This gives some flexibility, but it often results in conflicts between the owners trying to occupy the property at the same time of the year (summer vacation homes, for instance).
3. Points — In this type, points are used to represent ownership based on factors such as location, property size, time availability, etc. This type is mainly used by different resort owners on a commercial scale.

Timeshare is not considered a good investment in real estate, and people buy it more for ownership and usage rather than as an investment. However, the concept of shared ownership based on time was interesting and something I had never heard of.

Phantom Traffic Jams

Traffic Jams are an inevitable part of our daily lives. We have all been in a slow-moving traffic jam, which seems to have happened without any apparent reason, such as an accident, a road closure, a stop light, a change in the speed limit, or something else. It is more frustrating knowing that there is nothing evidently causing them. These are known as Phantom Traffic Jams.

A Phantom Traffic Jam occurs due to small changes in the driving habits of individuals on the road. It mainly occurs on highways with heavy traffic when the number of vehicles on the road exceeds a critical density causing dynamic instability. For example, when a driver breaks or slows down without any significant reason on a highway with heavy traffic, it causes a traffic wave that propagates through the rest of the traffic and can continue for miles.

So, the primary cause of this type of phantom traffic jam is the variance in speed of the vehicles on the road with a particular speed limit. Another cause is the improper and unequal spacing of vehicles on the road (typically less than 35 m apart). Check out this simulation of a phantom traffic jam from MIT on a circular route with no blockage. It has been proven with real-life experiments too.

How to avoid it?

Avoiding phantom traffic jams is a collective effort between drivers on the road. Drivers cause a phantom traffic jam when they don't have a good sense of what is happening on the road. Here are some tips to avoid them.

1. We should try to break gradually and be alert on the road. Breaking gradually and in advance can significantly reduce phantom traffic jams.
2. We should try not to accelerate when the traffic is moving smoothly. One driver accelerating can alert other drivers to brake preemptively, causing a phantom traffic jam to the vehicles behind us. Accelerating unnecessarily can also cause us to brake again further down the road, thus starting another phantom traffic jam.
3. We should avoid going too slow or too fast (cutting through the traffic) on roads with considerable amounts of traffic.
4. Using navigation apps helps us foresee upcoming traffic to prepare for unforeseen braking or plan for a detour.
5. Avoiding phantom traffic is one of the potential advantages of autonomous vehicles. They are mostly equipped with live traffic data and are programmed to maintain minimum distances from the surrounding vehicles and match the average speed of the overall flow. This leads to controlled braking, smooth lane changes, and other optimal maneuvers. In a recent experiment, one autonomous vehicle with 20 human drivers could dampen the traffic wave and reduce phantom traffic.

So, those were the concepts I learned recently. This post marks my 26th week of writing, and I hope it was informative and helpful in some ways. What are some cool things you learned recently? I'd love to know. As always, thanks for reading and your support and feedback over the last 6 months. Peace out!

References

1. https://www.compliantmechanisms.byu.edu/about-compliant-mechanisms
2. Why Machines that Bend Are Better — Veritasium.
3. https://voyager.jpl.nasa.gov/mission/
4. https://voyager.jpl.nasa.gov/mission/timeline/#event-a-once-in-a-lifetime-alignment
5. A Brief History of Voyager 1 and 2.
6. The Riemann Hypothesis, Explained — Quanta Magazine.
7. https://math.stackexchange.com/questions/837180/how-to-see-the-logarithm-as-the-inverse-function-of-the-exponential
8. https://www.investopedia.com/terms/t/timeshare.asp
9. What is phantom traffic and why is it ruining your life? — TedEd.