Bilinear Transform of the Ideal Mass

Starting with the driving point impedance

$$R(s) \isdef \frac{F(s)}{V(s)} = ms$$

the bilinear transform gives the digital impedance

$$\frac{F_d(z)}{V_d(z)} \isdef R_d(z) = R\left(c\frac{1-z^{-1}}{1+z^{-1}}\right) = mc\frac{1-z^{-1}}{1+z^{-1}}$$

Multiplying out

$$F_d(z) + z^{-1}F_d(z) = mc V_d(z) - mc z^{-1}V_d(z)$$

and taking the inverse $z$ transform gives

$$f_n + f_{n-1} = mc \left(v_n - v_{n-1}\right)$$

or

$$\zbox{v_n = v_{n-1} + \frac{1}{mc}\left(f_n + f_{n-1}\right)}$$

(The $f_{n-1}$ term is new relative to the FDA.)

Can check: Equivalent to trapezoid rule for numerical integration