Undamped dynamic vibration absorber

(a) Principle

A small auxiliary spring-mass system ($m_2$, $k_2$) is attached to the main system ($m_1$, $k_1$), which is excited by $F_0\sin\omega t$. When $\sqrt{k_2/m_2} = \omega$, the absorber's force on the main mass cancels the exciting force and the main mass stays still. The absorber mass vibrates instead.

  F0 sin(wt)
     |
   [ m1 ]--k2--[ m2 ]
     |
     k1
   ///

Equations of motion:

$$m_1\ddot{x}_1 + (k_1+k_2)x_1 - k_2x_2 = F_0\sin\omega t, \qquad m_2\ddot{x}_2 + k_2(x_2 - x_1) = 0$$

Put $x = X\sin\omega t$:

$$\begin{bmatrix} k_1 + k_2 - m_1\omega^2 & -k_2 \\ -k_2 & k_2 - m_2\omega^2 \end{bmatrix}\begin{Bmatrix} X_1 \\ X_2 \end{Bmatrix} = \begin{Bmatrix} F_0 \\ 0 \end{Bmatrix}$$ $$X_1 = \frac{(k_2 - m_2\omega^2)F_0}{(k_1+k_2-m_1\omega^2)(k_2 - m_2\omega^2) - k_2^2}, \qquad X_2 = \frac{k_2F_0}{(\ldots)}$$

$X_1 = 0$ when $k_2 - m_2\omega^2 = 0$, i.e. $\omega = \sqrt{k_2/m_2}$. Then $X_2 = -\dfrac{F_0}{k_2}$, so the absorber spring force $k_2X_2 = -F_0$ exactly cancels the exciting force. The absorber works only at the one tuned frequency; away from it, the combined system has two resonances.

(b) Design

Operating frequency: $\Omega = \dfrac{2\pi \times 1500}{60} = 157.08$ rad/s. The main system is at resonance, so $k_1 = m_1\Omega^2 = 100 \times 157.08^2 = 2467401$ N/m.

Tuning: $k_2 = m_2\Omega^2 = 10 \times 157.08^2 = 246740\ \text{N/m} = 246.7\ \text{kN/m}$.

New natural frequencies: the mass ratio is $\mu = m_2/m_1 = 0.1$. The absorber and the main system have the same natural frequency $\Omega$. With $r = \omega/\Omega$, the frequency equation is

$$r^4 - (2+\mu)r^2 + 1 = 0 \;\Rightarrow\; r^2 = \frac{2.1 \pm \sqrt{2.1^2 - 4}}{2} = 0.7298,\ 1.3702$$ $$\omega_{n1} = \Omega\sqrt{0.7298} = 134.19\ \text{rad/s}\ (1281\ \text{rpm}), \qquad \omega_{n2} = \Omega\sqrt{1.3702} = 183.87\ \text{rad/s}\ (1756\ \text{rpm})$$

The two new resonant speeds lie about 15% below and 17% above the operating speed, so the machine runs between them, safely away from both.

Answer: Absorber stiffness $= 246.7$ kN/m; new natural speeds $\approx 1281$ rpm and $1756$ rpm.