Mike-
I struggled in a big way with math in school, and the reason was that math teachers don't teach applied math, but just pure theory.
Pure math theory without any connection to the real world has no meaning to me.
I learned electricity using the water flowing in pipes analogy.
If you are very lucky, you can find that rare individual who is good at both math and explaining how to apply math to practical everyday problems.
When I was in school, we had to take an EIT (engineering in training, now called engineering fundamentals) test, and this test covered a wide variety of topics across many branches of engineering, such and electrical, mechanical, thermodynamics, statics, dynamics, you name it.
I still have my EIT study guide because it has a summary of the major formulas used for each of these fields.
If you look at the forumulas used in engineering, they are pretty much the same for mechanical, electrical, thermodynamics, aerodynamics, etc., but just written in different ways, such as V=I*R and PV=MRT.
The formulas for any branch of engineering all have to adhere to the same constraints regarding the laws of physics, and the law of physics says there must always be a conservation of energy. Electrical equations are often used to model mechanical systems, since it all boils down to the flow of energy, and the electrical formulas can easily describe the flow of energy.
Think in terms of a black box, which may be an engine or motor of some type.
You have to input a certain amount of energy into this black box, and no matter what, you cannot get more energy back out of the box than you put in, and will always get less energy back out, since some mechanical energy will always be converted to heat energy via friction, etc.
Understanding a machine is like the expression used for understanding people, "follow the money", but instead use "follow the energy" and you will understand the machine.
Generally there are a few basic formulas that define each branch of engineering, and you really just need to understand those formulas. The formulas are basically the same, but just written using different variables, so they seem different, but they are really generally about the same.
Mathematics is basically a language, and if you can read the language, then you can understand a machine and how it will operate.
E=mC squared is a simple little formula, but the meaning behind it is that if you split an atom, you get two smaller particles, and a huge amount of energy released.
Bottom line is that matter contains a huge amount of energy.
Geometery defines planes and lines which are drawn on these planes.
Algebra defines the fundamentals of addition, subtraction, multiplication and division.
Algebra is very basic, and came from some farmer wanting to sell 10 bushels of corn, or figure out how to grow 100 bushels instead of 10, or sell 1/2 of his 10 bushels, etc.
Statics applies geometry to vector equations. Vectors are variables with have both a magnitude (force) and a direction.
If you can add up all the vectors in a truss bridge, you can find out where the forces are concentrating, and design the bridge to function safely.
Static equations are applied to things that are stationary and not moving.
Dynamics used vectors that are moving, such as an ice skater who controls their spinning speed by moving their arms closer or further away from their body.
Once you get into dynamics, you get into calculus.
Many of the old steam engine books from the 1,800's are full of calculus, and it was well understood in the 1,800's and before.
Calculus is a lot simplier than most people think, and basically deals with rates of change (such as the rate of change of velocity), and integration which is just addition.
The reason you need calculus to design a steam engine is because you have a situation where the piston starts with zero velocity, then accelerates (acceleration is just the rate of change of velocity) to some maximum value when the crank is near 90 degrees, and then begins to decelerate to a zero velocity at the end of the stroke.
If you plot out the acceleration, you can also figure out the maximum forces that are generated on the parts, and design the part and fastener accordingly.
Changing the velocity or direction of a mass always requires energy.
You transfer the energy from the burning fuel to the water, converting it to steam, then let the steam expand in the cylinder, transferring some energy to the piston, then store some of the energy in the flywheel. Other energy is lost due to friction of the parts that make contact, and some energy is wasted just accelerating and decelerating parts. Any unbalanced forces have are transmitted into the frame and then the base of the engine, and the base then has to absorb this energy.
A percentage of the energy from the burning fuel is converted into useful work (generally a pretty small percentage).
Math is like machining, if you use it a lot, you generally get better at it, and understand it better.
Engineering school is like math boot camp, where you are forced to learn the language of math and how to apply it.
Difficult at first, just like machining, and easier over time. Unlike machining, I can use an eraser on my equations if they are not correct.
There are some good fundamentals books out there.
Hope this helps.
Pat J
