13.5: Summary
In this chapter we have analysed vibrations, a periodic type of motion that is important in many fields of science and engineering. Vibrations originate when a position dependent restoring force, like that generated by a spring, acts on a mass. When there are no external forces one obtains free vibrations, while if there are periodic external forces acting, we have forced vibrations. If a velocity dependent force from a damper acts on the mass, the vibrations are damped. There are two main challenges in analysing vibrations: 1. Deriving the correct EoM using the FBD and Newton’s or Euler’s second law. 2. Solving the EoM, which is mainly the mathematical challenge of solving a second order linear ODE.
- For systems with a restoring force, like mass-spring systems, the EoM as determined using Newton’s second law has a special form that causes the motion to exhibit vibrations.
- First determine the static equilibrium position and then determine vibrational motion \(y(t)\) with respect to this position.
- Free vibrations
If there are no time dependent external driving forces, we deal with free vibrations and the following EoM:
\[m_{e} \ddot{y}_{e}+c_{e} \dot{y}_{e}+k_{e} y_{e}=0 \tag{13.87} \label{13.87}\]
- If \(c_{e}=0\), then we are dealing with free undamped vibration. In that case the solution of the EoM is \(y_{e}(t)=\Re A e^{i \omega_{n} t+\varphi_{0}}=A \cos \left(\omega_{n} t+\varphi_{0}\right)\), with natural resonance frequency \(\omega_{n}=\sqrt{k_{e} / m_{e}}\).
- If \(c_{e} \neq 0\) we are dealing with free damped vibration.
Determine the discriminant \(\Delta=c_{e}^{2}-4 m_{e} k_{e}\).
- If \(\Delta>0\) we are dealing with overdamped free vibration with \(y_{A}=A_{+} e^{\lambda_{+} t}+A_{-} e^{\lambda_{-} t}\) and \(\lambda_{ \pm}=\frac{-c_{e} \pm \sqrt{\Delta}}{2 m_{e}}\).
- If \(\Delta<0\) we are dealing with underdamped free vibration with \(y_{A}=A e^{-\frac{c_{e}}{2 m_{e}} t} \cos \left(\omega_{d} t+\varphi_{0}\right)\) and \(\omega_{d}=\sqrt{\omega_{n}^{2}-c_{e}^{2} /\left(4 m_{e}^{2}\right)}\).
- If \(\Delta=0\) we are dealing with critically damped free vibration with \(y_{A}=A_{0} e^{-\frac{c_{e}}{2 m_{e}} t}+A_{t} t e^{-\frac{c_{e}}{2 m_{e}} t}\).
- The system will show forced vibration if the EoM can be written in the form:
\[m_{e} \ddot{y}_{e}+c_{e} \dot{y}_{e}+k_{e} y_{e}=F_{0, e} \cos (\omega t) \tag{13.88} \label{13.88}\]
- If \(c_{e}=0\), then we are dealing with forced undamped vibration. In that case \(\varphi_{0}=0\) such that the steady-state solution of the EoM is \(y_{e}(t)=A \cos (\omega t)\), with \(A=F_{0, e} /\left(k_{e}-m_{e} \omega^{2}\right)\).
- If \(c_{e} \neq 0\) we are dealing with forced damped vibration.
The steady-state solution of the EoM is:
\(y_{e, f}(t)=A \cos \left(\omega t+\varphi_{0}\right)\).
The amplitude of the steady-state solution is:
\(A=F_{0, e} / \sqrt{m_{e}^{2}\left(\omega_{n}^{2}-\omega^{2}\right)^{2}+\omega^{2} c_{e}^{2}}\).
The phase difference between force and motion is:
\(\varphi_{0}=\arctan \left[-\omega c /\left(k-\omega^{2} m_{A}\right)\right]\).
- Solutions \(y_{e, h}(t)\) of the homogeneous EoM \(m_{e} \ddot{y}_{e, h}+c_{e} \dot{e}_{e, h}+k_{e} y_{e, h}=0\) for free vibrations can be added to the steady-state solution \(y_{e, f}(t)\) to obtain new solutions of the EoM. For damped vibrations \((c>0)\), the function \(y_{e, h}(t)\) reduces to zero after sufficient time, it is therefore called the transient part of the solution. In general, any solution of the forced EoM can always be expressed as \(y_{e}(t)=y_{e, f}(t)+y_{e, h}(t)\).