Rigid Bodies Mayhem (ENGMEC 1-3)

TARGET AUDIENCE: Students, Self (Especially if you are taking these courses, here’s my take and cheatsheets on how I passed them! Satin satin lang to :» )

Note: Taglish

OVERVIEW: One of the few dreaded things that I took in college is this trilogy of courses that summarize behavior of rigid bodies in boundary conditions. I never particularly liked it because it was for civil and mech eng majors heavy and I was more inclined in the electronics side of my specialization. NGL, I had my scholarship thats why I needded to pass this out of necessity kaya there are few instances that made me reflect that I almost failed this courses in my college stay. Deliks ako kaya di to isang post for a sure-fire way to ace these courses.

Nonetheless, the learnings in these courses evolved into a deep appreciation on the known physics to define the behavior of rigid bodies. This progressed into learnings for machine elements and finite element analysis that we did for computational works.

My preferred resources!

There are many resources out there. Some siyempre, will be the course modules provided by the instructor. I have diligently skimmed them and will provide both lesson slides as well as my handwritten notes (wow) in succeeding parts below. What I want to share here is more on what free online resources helped me survive these courses. No gatekeeping, here are the resources that I used:

• YT channel of Jeff Hanson: This guy covered the entire topic under rigid and deformable bodies so to be fair, its practically a free online course in itself!! I binged watched his content malala and not only was the topic and teachings godsent since some topics we’re not covered by instructor, there were moments that made me laugh since he’s a funny and very grounded person. Dimo na kailangan makinig sa prof kung makikinig ka sa kanya!
• Book Solutions on LIT Solutions: Often, the assignment is a modified version seen in the reference book. With this website, you just have to simply check what part of the book the question is and it will provide the solution! No Chegg, no other solution scheme with a paywall; Hands-down the only resource you need for getting by-the-book solution. Of course, take this with a grain of salt since like any other resource, it’s sometimes erroneous kaya studying the concept and principles is literally a given. No slouch or AI slop can solve these solutions anyway kaya need i-verify ung solutions, no shortcuts.
• Getting Books and Resources on Anna Archive:: Need ng reference or copy of the book since ayaw magbigay ng kopya ung instructor? Holy grail for keeping knowledge is through this website! Minsan binibigay nga ung copy of the solution manual here so keep your eyes peeled and prompt properly!

These three is what you need lang talaga to pass these courses. As in, you dont need anything else but to study and review the work. I typically do that via writing physical notes since you have something tangible to look and read at rather than on a screen. Plus side rin is that, you get to focus on your notes only and not get distracted easily what other apps kapag u take notes digitally. Old teknik but it works for me. Tulungan lang here! lmk if I missed something like other free resources that will help others. Comment in giscus below!

Profs to Pick:

Engr. Lucero (CIV); Engr. Ventanilla (CIV); Doc Nic Roxas (MEM)

ENGMEC1:

ENGMEC1 OVERVIEW:

Statics of Rigid bodies cover the study of forces and their effects on rigid bodies at rest. Ideally, deformations under load is negligible so distances remain constance. Name of the game: 1.) Determine the resultant forces and moments.; 2.) Establish conditions of equilibrium; 3.) Apply equations to analyze structures/machines.

I remember this since tinuro to ni sir Lucero. Astig na instructor that was very patient to students. For me, i just hate truss analysis lalo na kung usapang bridge kasi diko ma-visualize lalo na kung weird shape para pahirapan ung tanong. I have no TRUSS in may solutions talaga. May different supports that have different forces like roller or fixed. They have different counter-forces acting on that point so thats the confusing part. If nagets mo yan, pasado ka na rito.

ENGMEC1 CHEATSHEET:

Characteristics:

• Magnitude: ( P )
• Direction: ( \theta )
• Point of Application: ( A )

Rectangular Components:

\[F_x = F \cos \theta, \quad F_y = F \sin \theta\] \[F = \sqrt{F_x^2 + F_y^2}, \quad \theta = \tan^{-1}\left(\frac{F_y}{F_x}\right)\]

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Vector Addition

• Parallelogram Law
• Triangle Law

Law of Cosines: \(R^2 = P^2 + Q^2 - 2PQ \cos B\)

Law of Sines: \(\frac{A}{\sin A} = \frac{B}{\sin B} = \frac{C}{\sin C}\)

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Moment of a Force

\[M = F \cdot d\]

• Sign convention: Clockwise (+), Counterclockwise (−)
• Varignon’s Theorem: Moment of a force = sum of moments of its components

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Couple

\[M = F \cdot d\]

• Equal, opposite, parallel forces not collinear
• Moment is the same about all centers

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Force-Couple System

• Moving a force introduces a couple moment.

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Resultants

Coplanar, Parallel Force System:
\(R = \Sigma F, \quad R \cdot d = \Sigma M_o\)

Non-Concurrent, Non-Parallel:
\(R_x = \Sigma F_x, \quad R_y = \Sigma F_y\)

\[R = \sqrt{R_x^2 + R_y^2}, \quad \theta = \tan^{-1}\left(\frac{R_y}{R_x}\right)\] \[R \cdot d = \Sigma M_o\]

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Equilibrium

Conditions: \(\Sigma F = 0, \quad \Sigma M = 0\)

Coplanar Force Systems:

• Concurrent:
  \(\Sigma F_x = 0, \quad \Sigma F_y = 0\)

• Parallel:
  \(\Sigma F = 0, \quad \Sigma M = 0\)

• General:
  \(\Sigma F_x = 0, \quad \Sigma F_y = 0, \quad \Sigma M = 0\)

---

Analysis of Structures

Truss:

• Members = two-force members (tension/compression)
• Methods:
  • Method of Joints
  • Method of Sections

Frame:

• Members subjected to bending and axial loads
• Method: Method of Members

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Friction

Laws of Dry Friction:
\(F = \mu N\)

\[F_s \leq \mu_s N, \quad F_k = \mu_k N\]

Angle of Friction:
\(\tan \phi = \mu\)

Belt Friction:
\(\frac{T_2}{T_1} = e^{\mu \beta}\)

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Centroids

\[\bar{x} = \frac{\int x \, dA}{\int dA} = \frac{\Sigma (x_i A_i)}{\Sigma A_i}\] \[\bar{y} = \frac{\int y \, dA}{\int dA} = \frac{\Sigma (y_i A_i)}{\Sigma A_i}\]

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Area Moment of Inertia

\[I = \int \rho^2 \, dA\] \[I_x = \int y^2 \, dA, \quad I_y = \int x^2 \, dA\]

Polar Moment: \(J = I_x + I_y\)

Radius of Gyration: \(r = \sqrt{\frac{I}{A}}\)

Parallel Axis Theorem: \(I = I_o + A d^2\)

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Equilibrium in 3D

Components: \(F_x, \; F_y, \; F_z\)

Direction Cosines: \(\cos \theta_x = \frac{F_x}{F}, \quad \cos \theta_y = \frac{F_y}{F}, \quad \cos \theta_z = \frac{F_z}{F}\)

Equilibrium Equations: \(\Sigma F_x = 0, \quad \Sigma F_y = 0, \quad \Sigma F_z = 0\)

\[\Sigma M_x = 0, \quad \Sigma M_y = 0, \quad \Sigma M_z = 0\]

Below is a consolidated lesson slide and personal notes I have for this course.

ENGMEC2:

ENGMEC2 OVERVIEW:

This course now deals with the movement of the rigid body in respect to space and time. Of course, added complexity yan if we consider also forces that interact with the body aside from gravity which can be friction or collisions (elastic or inelastic momentum). I think for my instructor, Engr. Ventanilla(?), very cool lang sa pagexplan and light in regards to the requirements asking us. Aside from the normal assessment load, she ensured that we learned the topic after each F2F session. So that stuck with me what type of instructor she was.

Overall, there is no easy way to say this but kabisaduhin lang talaga ung equations T_T. We have to prepare on how to apply it for the questions that will appear in the assessments/exams. Mas ok kung nakikita mo ung application kasi yan talaga end goal. For me, I keep thinking of my robot instead of the given (e.g. train) crashing. (baduy na kung baduy!) Visualize lang talaga malala and how we can apply it in the future :”> Better than memorizing, unedrstanding when to apply equations/formulas is equally important. Malilito ka kung anong kinematic equations ung gagamitin or if wave mechanics ung tanong. You have to know the questions too. Equivalent in importance.

ENGMEC2 CHEATSHEET:

Sadly, I dont have lecture notes since si Engr. Ven dapat need mo pumasok para makuha mo ung turo/modules. Below though is my consolidated notes as well as my answers to some of the activities if that helps 🥀🥀

Kinematics of Rotation

• Angular displacement, velocity, acceleration: \(\theta(t), \quad \omega = \frac{d\theta}{dt}, \quad \alpha = \frac{d\omega}{dt}\)

• Rotational kinematic equations (constant acceleration): \(\omega = \omega_0 + \alpha t\) \(\theta = \theta_0 + \omega_0 t + \tfrac{1}{2} \alpha t^2\) \(\omega^2 = \omega_0^2 + 2 \alpha (\theta - \theta_0)\)

• Relation between linear and angular motion: \(v = r \omega, \quad a_t = r \alpha, \quad a_c = \frac{v^2}{r} = r \omega^2\)

Dynamics (Newton–Euler Equations)

• Newton’s 2nd Law (linear): \(\vec{F} = m \vec{a}\)

• Rotational form (torque): \(\vec{\tau} = I \vec{\alpha}\)

• General rigid body (translation + rotation): \(\Sigma \vec{F} = m \vec{a}_G, \quad \Sigma \vec{M}_G = I_G \vec{\alpha}\)

Momentum

• Linear momentum: \(\vec{p} = m \vec{v}\)

• Angular momentum (about a point O): \(\vec{H}_O = \vec{r} \times m \vec{v}\)

• Angular momentum for rigid body about center of mass: \(\vec{H}_G = I_G \vec{\omega}\)

• Impulse-momentum principle: \(\Sigma \vec{F} \, \Delta t = \Delta \vec{p}\) \(\Sigma \vec{\tau} \, \Delta t = \Delta \vec{H}\)

Energy Equations

• Kinetic energy of rigid body: \(T = \tfrac{1}{2} m v_G^2 + \tfrac{1}{2} I_G \omega^2\)

• Work-energy principle: \(W_{1 \to 2} = \Delta T\)

• Power of a torque: \(P = \tau \omega\)

• Potential energy (gravity): \(V = m g h\)

Moment of Inertia

• Definition: \(I = \int r^2 \, dm\)

• Parallel axis theorem: \(I = I_G + m d^2\)

• Perpendicular axis theorem (planar body): \(I_z = I_x + I_y\)

• Radius of gyration: \(k = \sqrt{\tfrac{I}{m}}\)

Equilibrium & Elasticity

• Equilibrium conditions: \(\Sigma F_x = 0, \quad \Sigma F_y = 0, \quad \Sigma F_z = 0\) \(\Sigma M_x = 0, \quad \Sigma M_y = 0, \quad \Sigma M_z = 0\)

• Stress: \(\sigma = \frac{F}{A}\)

• Strain: \(\epsilon = \frac{\Delta L}{L}\)

• Hooke’s Law (elasticity): \(\sigma = E \epsilon\)

• Shear stress/strain: \(\tau = G \gamma\)

• Bulk modulus: \(K = \frac{p}{\Delta V / V}\)

Oscillations

• Simple harmonic motion (SHM): \(x(t) = A \cos(\omega t + \phi)\)

• Equation of motion: \(m \ddot{x} + kx = 0\)

• Natural frequency: \(\omega = \sqrt{\tfrac{k}{m}}\)

• Physical pendulum: \(\omega = \sqrt{\tfrac{m g d}{I}}\)

• Torsional pendulum: \(\omega = \sqrt{\tfrac{k_\theta}{I}}\)

Below is my handwritten notes and solutions that I have thankfully digitized :)

ENGMEC3:

ENGMEC3 OVERVIEW:

We are going to the good parts which we now considers deformable bodies. This considers deflection and perceived effect of the loading and stress being subject in area of interest. I remember Doc Nicanor teaching this class. All F2F classes where the lecture slides was not shared so you really have to go to his F2F class and listen. This was the time where I really watched Jeff Hanson YT since I needed to really engrained the problems by just taking many (x3) practice problems which worked!!

I don’t have those practice problems but you can follow the playlist he made for deformable bodies. I watched that playlist twice haha no bullshit, it was a routine, 1hr before sleeping answer one mechanics problem. Punishment kung mali was to try and re-answer it before checking the explanation i believe haha

ENGMEC3 CHEATSHEET:

I’ll just provide my notes here since its the definitive cheatsheet that I used for my exam and I believe at the time this was also I put in my index card which was my weapon along with the sci-cal and compass to create the mohr cicle!

Reflections:

Looking back at this experience, it just reaffirms to me that effort really pays off. Bobo talaga ako sa subject na to HAHA. Kaya when courses like these come in the syllabus, kahit boring and tedious, sooner or later, engineering students will come to realize that need mo matuto to since this will become your foundation for more advanced projects. Bread and butter nga. Choosing to learn this means only that I choose this lifestyle and career; way of life talaga.

Kaya as reflection, it just a statement that kahit parang out of line to sa pagiging mechatronics ko, need to malaman since in the future we will be dealing with projects that requires us to do these type of calculations!

Is this a decent take? Lmk in the comments below~~