---
tags: Mathematical modeling and optimization
title: 2.T. Generic multi-body and FVM analysis
---

# Tutorials

## 1. Quater car suspension system

_Consider a simplified representation of a car's suspension system, focusing on one corner of the vehicle in one dimension $z$. The quarter car suspension system consists of a car body (mass $M$) and a suspension system (tire mass $m$), where the car body is equipped with a spring damper system._

<img src="https://live.staticflickr.com/65535/53539872136_bb9e5ee7be_z.jpg" width=80%>

_Supposed the car load is 350 kg and the tire weights 50 kg. Height of tire and car body are 20 cm and 60 cm respectively. Giving the spring and damper coefficient as 2.5 N/mm and 1.0 N s/mm, and the representative spring coefficient of tire as 14 N/mm._

_The suspension system is passing through a bumping ground level with 5 cm amplitude of 0.8 Hz frequency._ __*How is the vibration level for the passenage who is sitting in the car?*__

### Components of the Free Body Diagram:

__Forces Analysis__:
- Car Body ($M_s$):
- Movement upwards (including the gravity acting downward) ($M_s\cdot \vec{a_s}$).
- Suspension force from the spring ($F_s$) and damper ($F_d$) acting downward.
- Suspension Mass ($M_u$):
- Movement upwards (including the gravity acting downward) ($M_u\cdot \vec{a_u}$).
- Reaction of the spring ($F_s$) and damper ($F_d$) suspension force acting upward.
- Suspension force ($F_t$) from the tire acting downward.

<img src="https://live.staticflickr.com/65535/53541984816_17be9f3c38_z.jpg" width=100%>

__Equations for the Quarter Car Suspension System__:

Now, let's briefly discuss the equations of motion for both the car body and the suspension mass (tire). Supposed the upwards direction as positive.

- Equation for Car Body ($m_s$):
- Applying Newton's second law:

$\qquad \displaystyle m_s \ a_s = - F_s - F_d$

Where
- $a_s$ is the acceleration of the car body.
- $F_s$ is the force acting from the spring, acting as spring coefficient by displacement ($\kappa_s\ \Delta y_s$).
- $F_d$ is the force acting from the damper, modeled as damping coefficient by velocity ($d_s\ \Delta u_s$)

- Equation for Suspension Mass ($m_u$):
- Applying Newton's second law:

$\qquad \displaystyle m_u\ a_u = F_s + F_d - F_t$

Where
- $a_s$ is the acceleration of the car body.
- $F_t$ is the force acting from the tire, modeled as spring function $k_t\ y_u$

Considering the relative displacement of car body as ($y_s - y_u$), the relative velocity ($u_s -u_u$) as ($y_s^\prime -y_u^\prime$) and the relative displacement of tire as ($y_u-y_g$), the equation system is formulated to:
$$
\begin{align}
& m_s \ y_s^{\prime\prime} + \kappa_s \ (y_s -y_u) + d_s\ (y_s^\prime - y_u^\prime) = 0\\
\\
&m_u\ y_u^{\prime\prime} - \kappa_s \ (y_s -y_u) - d_s\ (y_s^\prime - y_u^\prime) + \kappa_t \ (y_u - y_g) = 0
\end{align}
$$

[<ins> __Sample solution__](https://cocalc.com/share/public_paths/452dfdb26ee660b3bf0d3b0a63e4df12f4dacbd4)

<img src="https://live.staticflickr.com/65535/53546016431_240bc827b8_z.jpg">

---

## 2. Metallic fin for passive cooling

_Thinking on designing a passive cooling system for a high-power electronics device (e.g., a CPU, power converter, or LED array). The heat is conducted through a 1D metallic fin or heat sink attached to the hot component._

<img src="https://www.boydcorp.com/wp-content/uploads/Thermal/Air-Cooling/Skived-Fin-4-566x300-1.jpg" width="65%"> <img src="https://live.staticflickr.com/65535/54616351908_08e72c46c7_b.jpg" width="30%">

_Dimension of single fin are 10cm height with cross sectional perimeter of 4cm and area of 0.1 cm². The fin is made of copper-aluminum compund, for which the thermal conductivity is ~400/237 W/(m K) and the convection coefficient is set for 100 (W/m²·K)._

_Supposed there is no laternal heat transfer on the fin. The temperature of the bottom side is 80°C and the temperature along the fin height can achieve the steady state solution_. _**If the length of cooper-aluminum (cooper on bottom, aluminum on top) 5cm -5cm, how is the temperature distribution along the fin height?**_

### Applying Finite Volume Method (FVM)

Since the thermal conductivity is non-homogeneous, FVM is the easiest way to model the physics. Following the [__Diffusion Scenario__](https://hackmd.io/LDFLDDmZREKzGpMbmj3B9g?both#Scenario---1D-Diffusion), governing equation in matrix form yields
$$(\frac{kA_c}{\Delta x})T_W + (-2 \frac{kA_c}{\Delta x} - h P \Delta x)T_P + (\frac{kA_c}{\Delta x})T_E = −hP\Delta x T_\infty$$

To avoid inconsistent flux, the thermal condutivity for material interface can be expressed using harmonic mean, for which
$$k_{\text{h}} = \frac{2 k_1k_2}{k_1+k_2}$$.

Referring to the index of faces and neighboring cells $W$ and $E$.
<img src="https://live.staticflickr.com/65535/54616354194_a269dbfbd5_b.jpg">

Equation for the matrix system is extended by the face information to
$$(\frac{k[i]A_c}{\Delta x})T[i-1] + (- \frac{(k[i]+k[i+1])A_c}{\Delta x} - h P \Delta x)T[i] + (\frac{k[i+1]A_c}{\Delta x})T[i+1] = −hP\Delta x T_\infty$$

Implementation of boundary conditions can follow the same procedure as in [the tutorial](https://hackmd.io/LDFLDDmZREKzGpMbmj3B9g?both#Scenario---1D-Diffusion).

[__Sample solution__](https://cocalc.com/share/public_paths/64c2aa87227880d2a7bee47c0d7de7e82a2a0ba0)

<img src="https://live.staticflickr.com/65535/54615297067_c983b23be6.jpg">