Velocity transfer function of two-mass system with analogies

Displacements $x_1$ ($m_1$) and $x_2$ ($m_2$, force $f$). Output: velocity of $m_2$, $v_2 = \dot x_2$, so $V_2(s) = sX_2(s)$.

Equations of motion

• $m_1$: damper $B_1$ to the left wall, spring $k_1$ to $m_2$.
• $m_2$: spring $k_1$ to $m_1$, $k_2$ to the left wall, $B_2 \parallel k_3$ to the right wall.

$$\begin{aligned} m_1\ddot x_1 + B_1\dot x_1 + k_1(x_1 - x_2) &= 0 \\ m_2\ddot x_2 + B_2\dot x_2 + (k_2 + k_3)x_2 + k_1(x_2 - x_1) &= f(t) \end{aligned}$$

Laplace form:

$$\begin{aligned} (m_1s^2 + B_1s + k_1)X_1 - k_1X_2 &= 0 \\ -k_1X_1 + (m_2s^2 + B_2s + k_1 + k_2 + k_3)X_2 &= F \end{aligned}$$

Transfer function

$$\frac{X_2}{F} = \frac{m_1s^2 + B_1s + k_1}{\Delta}, \qquad \Delta = (m_1s^2 + B_1s + k_1)(m_2s^2 + B_2s + k_1 + k_2 + k_3) - k_1^2$$

Velocity output:

$$\frac{V_2(s)}{F(s)} = \frac{s(m_1s^2 + B_1s + k_1)}{\Delta}$$

with

$$\begin{aligned} \Delta =\; & m_1m_2s^4 + (B_1m_2 + B_2m_1)s^3 + \left[B_1B_2 + m_1(k_1 + k_2 + k_3) + m_2k_1\right]s^2 \\ & + \left[B_1(k_1 + k_2 + k_3) + B_2k_1\right]s + k_1(k_2 + k_3) \end{aligned}$$

F-V analogy (mesh, velocity $\to$ current)

   L1=m1  R1=B1             L2=m2   R2=B2   C=1/(k2+k3)
 +-LLLL---/\/\--+----------LLLL----/\/\------||-----+
 |              |                                   |
 |           C1=1/k1                               (~) e
 |     i1       |              i2                   |
 +--------------+-----------------------------------+

($k_2$ and $k_3$ both act on $x_2$ alone, so they combine into one capacitor $1/(k_2+k_3)$, i.e. $C_2 = 1/k_2$ and $C_3 = 1/k_3$ in series.) The required output $V_2$ corresponds to mesh current $i_2$.

F-I analogy (node, velocity $\to$ voltage)

 v1      L1    v2
 +----+--LLLL--+----+----+----+----+
 |    |        |    |    |    |    |
 C1   R1       C2   R2   L2   L3  (i)
 |    |        |    |    |    |    |
 +----+--------+----+----+----+----+ gnd

($C_1 = m_1$, $R_1 = 1/B_1$, $L_1 = 1/k_1$, $C_2 = m_2$, $R_2 = 1/B_2$, $L_2 = 1/k_2$, $L_3 = 1/k_3$.) The output velocity $v_2$ is the voltage of node 2.