2.3 KiB
Accelerance is a measure used in mechanical systems that relates acceleration to applied force.
Introduction
(Secondary articles descriptions#Machine Vibration)
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
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)