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Inline 4 Engines: Balance and Vibrations

Inline 4 Engines: Balance and Vibrations

April 1, 2026

TL;DR

Getting right the slider-crank mechanism (2D) via MBSD to later form a 3D ICE I4 Vibration Simulation

Video | Docs ↗ Racing | Docs ↗

Intro

This channel has awsome videos explaining engine balance

But i wanted to make my own.

Enough Theory for Balance

In other words: whats the minimum physics context for an LLM to do this right?

+++ some references below

Complete mathematical equations for the slider-crank mechanism used in engine balance simulation.

  1. Coordinate System & Geometry

1.1 Coordinate System Definition

2D Cartesian system:

  • Origin: At crank pivot (crankshaft axis)
  • X-axis: Horizontal, pointing toward the slider (piston) at $\theta = 0°$
  • Y-axis: Vertical, perpendicular to X (right-hand rule)
  • Rotation: Crank rotates counterclockwise (positive $\theta$ direction)

Units (all equations use SI base + mm for length):

  • Length: mm
  • Mass: kg
  • Force: N (= kg⋅mm/s²)
  • Torque: N⋅mm
  • Angular velocity: rad/s
  • Time: s

1.2 Mechanism Definition

Parameters:

  • $r$ = crank radius (L_crank) [mm]
  • $L$ = connecting rod length (L_rod) [mm]
  • $\theta$ = crank angle from X-axis [rad]
  • $\varphi$ = connecting rod angle from X-axis [rad]
  • $x$ = slider position along X-axis [mm]

Constraint equations:

$$r \cos(\theta) + L \cos(\varphi) = x \quad \text{…(1.1)}$$

$$r \sin(\theta) + L \sin(\varphi) = 0 \quad \text{…(1.2)}$$

  1. Kinematics

2.1 Slider Position

From constraint (1.2): $\sin(\varphi) = -\frac{r \sin(\theta)}{L}$

Therefore: $\cos(\varphi) = \sqrt{1 - \sin^2(\varphi)} = \frac{\sqrt{L^2 - r^2 \sin^2(\theta)}}{L}$

Slider position:

$$x(\theta) = r \cos(\theta) + \sqrt{L^2 - r^2 \sin^2(\theta)}$$

Standard form with $\lambda = r/L$:

$$x(\theta) = r\left[\cos(\theta) + \sqrt{1 - \lambda^2 \sin^2(\theta)}\right]$$

Physical range:

  • Minimum: $x_{\min} = L - r$ (when $\theta = \pi$)
  • Maximum: $x_{\max} = L + r$ (when $\theta = 0$)
  • Stroke: $S = x_{\max} - x_{\min} = 2r$

2.2 Slider Velocity

Using the chain rule: $v_x = \frac{dx}{dt} = \frac{dx}{d\theta} \cdot \omega$

Slider velocity:

$$v_x(\theta) = \omega r \sin(\theta) \frac{\sqrt{L^2 - r^2 \sin^2(\theta)} + r \cos(\theta)}{\sqrt{L^2 - r^2 \sin^2(\theta)}}$$

Compact form:

$$v_x = \frac{r \omega \sin(\theta)}{[1 - (r/L)^2 \sin^2(\theta)]^{1/2}} \times \left[1 + \frac{(r/L) \cos(\theta)}{\sqrt{1 - (r/L)^2 \sin^2(\theta)}}\right]$$

2.3 Slider Acceleration

Acceleration formula:

$$a_x(\theta) = r \omega^2 \left[\cos(\theta) + \frac{(r/L) \cos(2\theta)}{\sqrt{1 - (r/L)^2 \sin^2(\theta)}}\right]$$

Alternative expanded form:

$$a_x = r \omega^2 \cos(\theta) + \frac{r^2 \omega^2 \cos(2\theta)}{\sqrt{L^2 - r^2 \sin^2(\theta)}}$$

Physical interpretation:

  • First term: primary acceleration (2× fundamental frequency)
  • Second term: secondary acceleration component (4× fundamental frequency)

2.4 Connecting Rod Angle

From constraint (1.2):

$$\varphi(\theta) = -\arcsin\left(\lambda \sin(\theta)\right) \quad \text{where} \quad \lambda = \frac{r}{L}$$

2.5 Rod Angular Velocity

$$\omega_{\text{rod}}(\theta) = -\frac{\omega r \cos(\theta)}{\sqrt{L^2 - r^2 \sin^2(\theta)}} = -\frac{\omega r \cos(\theta)}{L \cos(\varphi)}$$

2.6 Rod Angular Acceleration

$$\alpha_{\text{rod}} = \frac{d\omega_{\text{rod}}}{dt} = \frac{d^2\varphi}{dt^2}$$

  1. Dynamics & Forces

3.1 Assumptions

  1. Rigid bodies: No deformation; all masses are point masses or have defined inertias
  2. No friction: Ideal pins/bearings
  3. Constant angular velocity: Crankshaft rotates at constant $\omega$
  4. Massless constraints: Slider pin and crank pin have negligible mass
  5. 2D planar motion: All motion in X–Y plane; no Z-axis components
  6. No combustion forces: Initial phase uses external motor to drive crank

3.2 Component Masses & Inertias

Crank:

  • Mass: $m_{\text{crank}}$ [kg]
  • Moment of inertia about pivot: $I_{\text{crank}}$ [kg⋅mm²]
  • Center of mass location: $R_{\text{cg,crank}}$ [mm]

Connecting Rod:

  • Mass: $m_{\text{rod}}$ [kg]
  • Length: $L$ [mm]
  • Moment of inertia about center of mass: $I_{\text{rod,cm}}$ [kg⋅mm²]

Slider (piston):

  • Mass: $m_{\text{slider}}$ [kg]

3.3 Inertial Forces on Slider

$$F_{\text{inertia}} = m_{\text{slider}} \cdot a_x(\theta) = m_{\text{slider}} \cdot r \omega^2 \left[\cos(\theta) + \frac{(r/L) \cos(2\theta)}{\sqrt{1 - (r/L)^2 \sin^2(\theta)}}\right]$$

3.4 Force in Connecting Rod

Rod internal force (for massless rod):

$$F_{\text{rod,axial}} = \frac{m_{\text{slider}} \cdot a_x}{\cos(\varphi)}$$

3.5 Crank Pin Force

Components:

$$F_{\text{rod,x}} = F_{\text{rod}} \cos(\varphi)$$

$$F_{\text{rod,y}} = F_{\text{rod}} \sin(\varphi)$$

Magnitude:

$$|F_{\text{rod}}| = \sqrt{F_{\text{rod,x}}^2 + F_{\text{rod,y}}^2}$$

3.6 Torque on Crankshaft

Torque from slider inertial force:

$$\tau_{\text{slider}} = F_{\text{rod}} \cdot r \sin(\theta - \varphi) = \frac{m_{\text{slider}} a_x \cdot r \sin(\theta - \varphi)}{\cos(\varphi)}$$

Simplified (massless rod, constant $\omega$):

$$\tau(\theta) = m_{\text{slider}} r \omega^2 \left[\cos(\theta) + \frac{(r/L) \cos(2\theta)}{\sqrt{1 - (r/L)^2 \sin^2(\theta)}}\right] \sin(\theta - \varphi(\theta))$$

3.7 Ground Reaction Force on Slider

Force balance on slider:

$$m_{\text{slider}} a_x = F_{\text{rod}} \cos(\varphi) - F_{\text{ground,x}}$$

Components:

$$F_{\text{ground,x}} = F_{\text{rod}} \cos(\varphi) - m_{\text{slider}} a_x$$

$$F_{\text{ground,y}} = -F_{\text{rod}} \sin(\varphi)$$

Magnitude:

$$|F_{\text{ground}}| = \sqrt{F_{\text{ground,x}}^2 + F_{\text{ground,y}}^2}$$

  1. Reaction Forces at Bearings

4.1 Crank Bearing Force

The crank bearing must support:

  1. The reaction force from the rod
  2. The centrifugal force of the crank mass
  3. Weight of crank (if vertical support needed)

Acceleration of crank center-of-mass:

Position of crank CG: $x_{\text{cg}} = R_{\text{cg}} \cos(\theta)$, $y_{\text{cg}} = R_{\text{cg}} \sin(\theta)$

Acceleration: $$a_{\text{cg,x}} = -R_{\text{cg}} \omega^2 \cos(\theta)$$ $$a_{\text{cg,y}} = -R_{\text{cg}} \omega^2 \sin(\theta)$$

Bearing reaction force:

$$\vec{F}{\text{bearing}} = \vec{F}{\text{rod}} + m_{\text{crank}} \vec{a}_{\text{cg}}$$

$$F_{\text{bearing,x}} = F_{\text{rod,x}} - m_{\text{crank}} R_{\text{cg}} \omega^2 \cos(\theta)$$

$$F_{\text{bearing,y}} = F_{\text{rod,y}} - m_{\text{crank}} R_{\text{cg}} \omega^2 \sin(\theta)$$

4.2 Multi-Cylinder: Total Bearing Force

For an engine with multiple cylinders at angles $\theta_1, \theta_2, \ldots, \theta_N$:

$$F_{\text{total,x}}(\theta) = \sum_i \left[F_{\text{rod,i}} \cos(\varphi_i) - m_{\text{crank}} R_{\text{cg}} \omega^2 \cos(\theta_i)\right]$$

$$F_{\text{total,y}}(\theta) = \sum_i \left[F_{\text{rod,i}} \sin(\varphi_i) - m_{\text{crank}} R_{\text{cg}} \omega^2 \sin(\theta_i)\right]$$

Magnitude:

$$|F_{\text{bearing,total}}| = \sqrt{F_{\text{total,x}}^2 + F_{\text{total,y}}^2}$$

  1. Vibration Analysis

5.1 Harmonic Decomposition (FFT)

The bearing reaction force can be decomposed into harmonics:

$$F_{\text{bearing}}(t) = F_0 + \sum_{n=1}^{\infty} \left[F_n \cos(n \omega t + \phi_n)\right]$$

Harmonic amplitude extraction:

$$F_n = \frac{1}{\pi} \int_0^{2\pi} F_{\text{bearing}}(\theta) \cos(n \theta) , d\theta$$

  • Primary imbalance (1st order): $n = 1$, frequency = $1 \times \text{RPM}$
  • Secondary imbalance (2nd order): $n = 2$, frequency = $2 \times \text{RPM}$
  • Tertiary imbalance (3rd order): $n = 3$, frequency = $3 \times \text{RPM}$

5.2 Frequency Response

For each harmonic, the vibration amplitude depends on resonance:

$$\text{Vibration amplitude} \propto \frac{F_n}{|k - n^2 \omega^2 m + j n \omega c|}$$

where:

  • $k$ = stiffness [N/mm]
  • $m$ = mass [kg]
  • $c$ = damping [N·s/mm]
  • $j$ = imaginary unit
  1. Special Cases & Simplifications

6.1 Infinite Rod Approximation ($L \to \infty$)

When rod length is much larger than crank ($L » r$), the rod angle $\varphi \approx 0$:

$$x(\theta) \approx r \cos(\theta) + L$$

$$v_x(\theta) \approx -r \omega \sin(\theta)$$

$$a_x(\theta) \approx -r \omega^2 \cos(\theta) \quad \text{(simple harmonic)}$$

6.2 Four-Stroke Cycle with Gas Pressure

If we include combustion/pressure forces:

$$F_{\text{gas}}(\theta) = p(\theta) \cdot A_{\text{piston}} \quad [\text{N}]$$

where $p(\theta)$ is the gas pressure curve and $A_{\text{piston}}$ is the piston area.

Total force on slider:

$$F_{\text{total}} = F_{\text{inertia}} + F_{\text{gas}} - F_{\text{ground}}$$

Wait…arent you gonna extend the MBSD 2D simulator already to 3D for this?

Nop, im not.

This is going to be equation based.

No point reference coordinates this time :)

Engine Balance

Balance…what is it?

Vibrations?

FFT?

Forces?

I just wanted to create my engine balance simulations as proof:

#git clone https://github.com/JAlcocerT/mbsd
cd ./mbsd

You might do this without a mbsd framework like the one ive built.

MBSD | Repo 3Design | Repo

Just…make the magic of the model and equation happen.

You got x4 slider-cranks connected to a common axis

Yes, this one is based on remotion :)

So you can model it as one axis that receives variable forces (the reaction ones that makes the crank stay in position) over time as it rotates

cd ./mbsd/2D-Dynamics
make run-slider-crank-no-gravity

Or you can start from the beginning.

Remotion x Video | Repo

Inline 4

After all of this modelling: how about visualizing?

Remotion x Video | Repo Engine Balance | Repo

For this post, i want to test how good the current LLMs are without much context:

#git init && git add . && git commit -m "Initial commit: Starting engine balance repo" && gh repo create engine-balance --private --source=. --remote=origin --push
git clone https://github.com/JAlcocerT/engine-balance
#claude

Will it be able to generate a proper physics engine

with an output data model that can be re-used for remotion animations?

oh and also…

being able to say in the future which of the engine configuration is best for the balance

with…proper why’s?

i want to create a way to simulate in python a slider-crank, being able to animate it and to have in any given point the position, speed, aceleration and forces (including the reaction forces between the bars). 

The later goal will be to take few of this slider cranks and make simulation of engine balance at the axis for few configurations, like I4, I6, V8 etc

can we create a brd.md to define all of this?
thats sounds good, with the export data we would be able to animate it later on with other frameworks like remotionJS, blender etc?
  • BRD.md: “What & Why” (requirements, architecture, scope)
  • equations.md: “The Physics” (math model, coordinate system, all equations)
  • dev-phases.md: “The Code” (concrete tasks, references, examples, testing)
before starting, can you define a Out of Scope section? Also, for starters we can make the tests with matplotlib, I will let you know later own which ultimamte presentation layer I want, as the output will go mostlikely to  youtube

Here is where you noticed those P1 estimations of 2-3 weeks still.

When its…20-30 minutes already:

then create a dev-phases.md where you can reference from brd and equations.md what we are doing, the uv python environment is ready for you
make explore #from the p3.1

With this we can understand the expected engine balance in a table:

alt text

And with this one, the force locus of the I4:

. .venv/bin/activate && pytest tests/test_balance.py::TestForceAnimation -v

alt text

It’s a 3D curve showing how the magnitude and direction of the total bearing force evolves from 0° to 360° of crank rotation.

I also made couple more tinkering with the p3.3

make decompose              # Output: output/i4_harmonics/, output/i6_harmonics/
make rpm-sweep              # Output: output/i4_rpm/, output/i6_rpm/

What it is: 3D trajectory of bearing force over one crank cycle

The 3D space: X (Fx) × Y (Fy) × Z (crank angle), color-coded by progression

Engine comparisons:

  • I4: Large tilted bowtie (secondary-heavy, needs balancer shafts)
  • I6: Thin vertical line near origin (nearly perfect balance)
  • I3/I5: Symmetric stars (excellent primary balance)
  • V8: Moderate bowtie (good compromise)
  • V12: Smallest locus (pinnacle of smoothness)

How to interpret: Shape = balance quality, size = imbalance magnitude, pattern = harmonic content

Why it matters: Makes abstract force numbers visceral — seeing I4’s force vector wildly swing vs I6’s barely moving is instant physics understanding

Generation methods: make explore, make animate-forces, or Python script

Physical explanation: Why I4 needs balancer shafts, why I6 is perfect despite having 6 cylinders

About Engine Balance

The best video I found about engine balance:

Then provide a summary and pros and cons of the balance of each engine type discussed

The video titled “ENGINE BALANCE: Inline 6 vs. V6 vs. VR6 vs. Flat / Boxer 6” compares the balance, packaging, strengths, and weaknesses of four common six-cylinder engine layouts: the inline-six, V6, VR6, and flat-six/boxer-six.

Inline 6

The inline-six is widely regarded as the smoothest and simplest of the group. Its layout gives it excellent inherent balance and an even firing interval, which helps it run very smoothly without needing major balancing aids. It is also mechanically straightforward, typically using only one cylinder head and a simpler overall design than split-bank engines. Its main drawback is packaging: because all six cylinders are arranged in a straight line, the engine is long and can be difficult to fit in tighter engine bays, especially in transverse layouts.

Pros: Simple design, excellent inherent balance, smooth operation, even firing interval. Cons: Long overall length makes packaging more difficult.

V6

The V6 was largely developed to solve the packaging problems of the inline-six. By splitting the cylinders into two banks, it becomes much shorter and easier to fit into a wider range of vehicles, including transverse-engine layouts. The tradeoff is that it does not usually have the same natural balance as an inline-six, so smoothness often depends more heavily on bank angle, crankshaft design, and sometimes balancing aids. It is also mechanically more complex, since it typically uses two cylinder heads and more duplicated components.

Pros: More compact, easier to package, well suited to many engine bays. Cons: Generally less inherently balanced than an inline-six, more mechanically complex.

VR6

The VR6 is a narrow-angle six-cylinder design that blends characteristics of the inline-six and V6. Its main advantage is compactness: it is narrower than a conventional V6 and can be packaged very efficiently, while also avoiding some of the bulk of a traditional split-bank design. However, it should not be thought of as having the same inherent balance or simplicity as an inline-six. It is best understood as a clever packaging compromise that offers a unique middle ground between size, smoothness, and complexity.

Pros: Very compact, efficient packaging, simpler overall package than some conventional V6 designs. Cons: Still more of a compromise than an inline-six, not as naturally balanced, can involve uneven port geometry and added engineering complexity.

Flat / Boxer 6

The flat-six, especially in true boxer form, is also an extremely smooth and well-balanced engine design. It benefits from a low center of gravity and a short crankshaft, which can help both handling and responsiveness. However, its width makes packaging more difficult, and this layout can be more complex and expensive to manufacture depending on the application. While it offers major dynamic advantages, it is not always easy to fit into a vehicle platform.

Pros: Excellent balance, low center of gravity, short crankshaft, strong performance character. Cons: Wide layout makes packaging difficult, often more expensive and complex to manufacture.

Each six-cylinder layout reflects a different engineering priority. The inline-six emphasizes smoothness and simplicity, the V6 prioritizes compact packaging, the VR6 is a specialized compromise designed to save space, and the flat-six/boxer-six delivers excellent balance and a low center of gravity at the cost of width and complexity. Rather than there being one universally best design, each configuration makes sense in different applications depending on what engineers value most.

Primary Balance

This is the obvious one.

The pistons are moving in between and stop at the top/bottom.

Secondary Balance

Blame it to the boogie Archimedes.

Some time ago I recorded this unexpected video in a restaurant:

Who said Remotion?

And the thing goes from this RemotionJS experience

if someone would like to take the script you have prepared to make a V8 engine video using the python scripts that are having perfectly defined the slider-cranks and the angles and so on, what should it know?

create a hand over for the next agent to take advantage of all our work at a summary-for-remotion.md
#git clone /engine-balance
#claude #/skills 
if someone would like to take the script you have prepared to make a V8 engine video using the python scripts that are having perfectly defined the slider-cranks and the angles and so on, what should it know?

create a hand over for the next agent to take advantage of all our work at a summary-for-remotion.md
 lets write these choices at a summary-remotion-choices.md

  i want a 3D realistic engine model were we can see the pistons moving according to the data
  generated by python. for now, lets create a single static video (i guess we can add
  diferent remotion scripts later on). The data generation part is done in python, read the
  summary-for-remotion and see the makefile to know how. Id like a modern polished UI.
  1920x1080 and 30fps is fine. No forces with augmented reality for now.
#npx create-video@latest
#npx skills add remotion-dev/skills
#npx skills list

cd ./engine-balance/remotion-v8-video
npm install
npm run dev

When you are happy enough, render that composition:

npx remotion render V8CrankMechanism v8-test.mp4

waste of tokens .mp4 :)

Or in other words: nah, AI is a joke

Conclusions

Someone told me that she sees the difference when I talk with passion

Versus when I talk for the money

If you want to convince her that there is no such gap:

Consulting ServicesConsulting Services
Consulting - Tier of Service

FAQ

Cars for Track Days?

Some say that the BMW z4 e86 is underated.

You have ~2007 units for ~10k$ as shown here

Racing | Docs ↗

Im well aware that some people do cool stuff on the tracks with their bmw (BMW e86).

Fast Fourier Transform?

The good old FFT…

References

Engine Balance Theory:

  • Norton, R.L. “Design of Machinery” 5th ed., Chapter 14 (Engine Balance)
  • Heywood, J.B. “Internal Combustion Engine Fundamentals”, Chapter 3 (Kinematics & Dynamics)
  • SAE J670e “Engine Sound Quality”

FFT & Harmonic Analysis:

This Project:

  • BRD.md § 4.3 — Balance Analysis Requirements
  • equations.md § 3-4 — Kinematics & Dynamics Equations
  • dev-phases.md § 2.2 — Phase 2.2 Specification
From Idea to ResolutionOOP 101 for Engineers - Pendulum x 3Bodies