Part 4 - Models for mechanical systems

Introduction

In this part we’ll see how we can model mechanical systems with controls.

Models for mechanical systems

In this series we’ll mainly deal with springs and dampers. Mainly translation motion.

For springs, we’ll of course use Hooke’s law: $$ F_s = ky $$

Where $k$ is the spring constant in $\left[\dfrac{N}{m}\right]$.

For dampers, we’ll use: $$ F_y = b \dot{y} $$

Where $b$ is the damping constant in $\left[\dfrac{Ns}{m}\right]$.

Example mass-spring-damper model

Imagine we have a mass attached to both a spring and damper.

We have a downward force called $F_d$. Assume the system is rest at the start, using Newton’s second law: $$ F = m\ddot{y} $$

$$ F_d - ky - b\dot{y} = m\ddot{y} $$

$$ F_d = m\ddot{y} + ky + b\dot{y} $$

Now we have a second-order differential equation that describes the system. Lets Laplace transform it and find its transfer function from $F_d \to y$.

Meaning $G_{F_dy}(s) = \dfrac{Y(s)}{F_d(s)}$

$$ F_d(s) = ms^2 Y(s) + kY(s) + bs Y(s) $$

$$ F_d(s) = Y(s) (ms^2 + bs + k) $$

$$ \boxed{G_{F_{dy}}(s) = \dfrac{Y(s)}{F_d(s)} = \dfrac{1}{ms^2 + bs + k}} $$

Example tank system

Given a tank system, we can write using Bernoulli’s equation as: $$ P_1 + \rho g h_1 + \dfrac{\rho v_1^2}{2} = P_2 + \rho g h_2 + \dfrac{\rho v_2^2}{2} $$

Let $q_out$ and $q_in$ be volume flow. Let $h_2 = 0$, $P_1 = P_2 = P$ and that $v_1 \approx 0$.

This yields: $$ \rho g h_1 = \dfrac{\rho v_2^2}{2} $$

$$ v_2 = \sqrt{2gh_1} $$

We define $q_out$ with $q_{out}= a \cdot v$, where $a$ is the area. $$ q_{out} = a \sqrt{2gh_1} = a \sqrt{2g} \cdot \sqrt{h_1} \ | \ \text{non-linear} $$

Let $V = A \cdot h_1$. This means that $\dot{V}$ is the rate of change of the volume per second. $\dot{V} = q_{in} - q_{out}$.

Which means: $$ A\dot{h} = q_{in} - q_{out} $$

$$ \dot{h} = \dfrac{1}{A}(q_{in} - q_{out}) $$

$$ \dot{h} = \dfrac{1}{A}(q_{in} - a \sqrt{2g} \cdot \sqrt{h_1}) $$

In the next part, we’ll see how we can linearize this function to obtain the so-called space-state representation of this system.