MSR/ressources/Accelerance.md

Accelerance is a measure used in mechanical systems that relates acceleration to applied force.

# Introduction
([[Secondary articles descriptions#Machine Vibration|source]])

In a dynamic world, $F = mr\omega^2$. The mass and stiffness are not constant anymore and we cannot continue to use Newtonian physics ($F=ma$) and Hooke's law ($F=kx$).

To preserve the linearity of Newton's $2^{nd}$ law a dynamic mass is defined : 
$m(\omega)$.
The reciprocal of dynamic mass is accelerance, and is also a function of frequency : Accelerance $= \frac{1}{m(\omega)} = \frac{a(\omega)}{F(\omega)}$

**Symmetry is bad practice because it support resonant modes**

Force is a wave that travels at the speed of sound.

# Formalisation
([web source](https://eng.libretexts.org/Bookshelves/Electrical_Engineering/Signal_Processing_and_Modeling/Introduction_to_Linear_Time-Invariant_Dynamic_Systems_for_Students_of_Engineering_(Hallauer)/10%3A_Second_Order_Systems/10.05%3A_Common_Frequency-Response_Functions))

We define the dimensionless excitation frequency ratio, the excitation frequency relative to the system undamped natural frequency :
$$\beta \equiv \frac{\omega}{\omega_{n}}\tag{1}$$
And we define :
$$\omega_{n}=\sqrt{\frac{k}{m}}, \quad \zeta \equiv \frac{c}{2 m \omega_{n}}=\frac{c}{2 \sqrt{m k}} \equiv \frac{c}{c_{c}}, \quad u(t) \equiv \frac{1}{k} f_{x}(t)$$

For an $m-c-k$ system, from Laplace transformation of the ODE (*ordinary differential equation*) $m\ddot{x}+c\dot{x}+kx = f_x(t)$, and with use of notation defined in Equations $(1)$ and $(2)$, the equation for complex mechanical admittance is :

$$\left\{\frac{L[x(t)]}{L\left[f_{x}(t)\right]}\right\}_{s=j \omega}=\frac{1}{\left(k-\omega^{2} m\right)+j \omega c}=\frac{1}{k}\left[\frac{1}{\left(1-\beta^{2}\right)+j 2 \zeta \beta}\right]\tag{3}$$

(The inverse being *dynamic stiffness*)

For _accelerance_ (also known as _inertance_), the subject variable is an acceleration, and the reference variable is an action. Since $L[\ddot{x}(t)]=s^{2} \times L[x(t)]$, the accelerance of an $m-c-k$ system, from Equation $(3)$, is
$$\left\{\frac{L[\ddot{x}(t)]}{L\left[f_{x}(t)\right]}\right\}_{s=j\omega}=\frac{(j\omega)^{2}}{\left(k-\omega^{2}m\right)+j\omega c}=\frac{1}{m}\left[\frac{-\beta^{2}}{\left(1-\beta^{2}\right)+j2\zeta\beta}\right]\tag{4}$$
(The inverse of accelerance is called _apparent mass_)