Vibration of Mass-Spring-Damper System
Two types of vibration:
- Discrete system – mass and spring elements are separate and concentrated – represented by lumped-parameter model
- Continuous system – mass and spring elements are continuously spread over space
Nomenclature
| \(f_n\) | natural frequency [Hz] | the number of complete cycles per unit time |
| \(\tau =\dfrac{1}{f_n} = \dfrac{2\pi}{\omega_{n}}\) | period [s] | the time required for one complete motion cycle |
| \(\omega_{n} = \sqrt{\dfrac{k}{m}}\) | natural circular frequency [rad/s] (= undamped natural frequency), resonant frequency | the frequency at which resonance occurs if the driving frequency matches it |
| \(\zeta = \dfrac{c}{2m\omega_{n}}\) | damping ratio [-] | a measure of the severity of the damping |
| \(\omega\) | driving frequency [rad/s] | the frequency of the applied harmonic force |
| \(\omega_{d} = \omega_{n}\sqrt{1-\zeta^2} \) | damped natural frequency [rad/s] | the angular displacement per unit time |
Damped forced vibration
Consider a mass-spring-damper system subject to a sinusoidal input force \(F = F_o \sin{\omega t}\).
(1) Complementary solution \(x_c\)
The complementary solution \(x_c\) is also termed the transient solution since it is exponentially decaying.
The complementary solution is the solution of the homogeneous ODE
\[\ddot{x} + 2\zeta \omega_{n} \dot{x} + \omega_{n}^2 x = 0 \tag{2′} \]
To solve the ODE, try \(x = e^{\lambda t}\), \(\dot{x} = \lambda e^{\lambda t} \) and \(\ddot{x} = \lambda^2 e^{\lambda t}\) then \[ \lambda = \omega_n (-\zeta \pm \sqrt{\zeta^2 – 1} ) \]
| (i) for overdamping \( \zeta>1\) | \( \lambda_{1}\),\(\lambda_{2}\) are both distinct real negative values | \[x_c = A_{1}e^{\lambda_{1}t} + A_{2}e^{\lambda_{2}t}\] |
| (ii) for critical damping\( \zeta=1 \) | \(\lambda_{1} = \lambda_{2} = -\omega_n \) | \[x_c = (A_{1} + A_{2}t) e^{-\omega_{n}t} \] |
| (iii) for underdamping \( 0 < \zeta < 1 \) | \( \lambda = \omega_n (-\zeta \pm i\sqrt{1-\zeta^2}) \) | \[\begin{align} x_c &= ( A_{1}e^{i \omega_{d}t} + A_{2}e^{-i \omega_{d}t} ) e^{-\zeta \omega_n t} \\ &= (A_3 \cos{(\omega_{d}t)} + A_4 \sin{(\omega_{d}t)}) e^{-\zeta \omega_n t} \\ &= C e^{-\zeta \omega_n t} \sin{ (\omega_{d}t + \psi)} \end{align} \] where \(\omega_{d} = \omega_{n} \sqrt{1-\zeta^2} \) |
N.B. \(\lambda\) is the root \(s\) of the characteristic equations for the mass-spring-damper system.
(2) Particular solution \(x_p\)
The particular solution \(x_p\) is also termed the steady-state solution.
The particular solution is the solution to the complete ODE \[\ddot{x} + 2\zeta \omega_{n} \dot{x} + \omega_{n}^2 x = \dfrac{F_o \sin{\omega t}}{m} \tag{2} \]
To solve the ODE, try \(x_p = X_{1} \cos{\omega t} + X_{2} \sin{\omega t} \) or \(x_p = X \sin{(\omega t – \phi)} \)
i.e. \(x_p\) lags \(F\) by \(\psi\) angle.
then Equation (2) gives
\[
\left\{
\begin{array}{l} \begin{align}
X &= \dfrac{ F_{o} / k}{\sqrt{ \{1- (\dfrac{\omega}{\omega_n})^2\}^2 + (2\zeta \dfrac{\omega}{\omega_n})^2 }} \\
\phi &= \tan^{-1} \{ \dfrac{2\zeta(\dfrac{\omega}{\omega_n})}{1 – (\dfrac{\omega}{\omega_n})^2} \}
\end{align} \end{array}
\right.
\]
The complete solution \(x\)
For an underdamping system \( 0 < \zeta < 1 \), \[ \begin{array}{l} \begin{align} x &= x_c + x_p \\ &= C e^{-\zeta \omega_n t} \sin{ (\omega_{d}t + \psi)} + X \sin{(\omega t - \phi)} \end{align} \end{array} \]| \(C\), \(\psi\) | – | determinable from the initial conditions |
| \(\zeta\), \(\omega_n\), \(\omega_d\), \(X\), \(\omega\), \(\phi\) | – | properties of the system |
N.B. Remarks on the particular solution \(x_p\)
Amplitude ratio \(M\) (= magnification factor)
The amplitude of the particular solution \(X\) normalised with the static deflection \(\dfrac{F_o}{k}\) caused by the external force \[ M = \dfrac{X}{(\dfrac{F_o}{k})} = \dfrac{1}{\sqrt{ \{1 – (\dfrac{\omega}{\omega_n})^2 \}^2 + (2\zeta\dfrac{\omega}{\omega_n})^2 }} \]
To avoid resonance, either increase the damping \(\zeta\) or alter the driving frequency \(\omega\).
Phase lag \(\phi\) of the particular solution \(x_p\)
Reference
[1] Meriam and Kraige, Engineering Mechanics: Dynamics.