Mechanical Vibration - APS

1. Basic Concept

1.1 Definition

• Mechanical Vibration: system moving forward and backward between the Balanced position
• system
• input
• response

1.2 Basic question

• vibration analysis[this] : system,input $\rightarrow$ response
• system identification : i. r$\rightarrow$s
• Load identification : s,r $\rightarrow$ i
  • Steps for vibration analysis
• Mechanical model
• Mathematical model : Motion differential equation
• solve Motion differential equation
• get engineering information
• test for verification

1.3 Elements

• Inertial component : mass,m,J
• Elastic component : stiffness, k
• Damping element : Damping coefficient,c

1.4 Classification

• Component

  • Discrete component: Discrete system

    • 1 dof : $f(x,\dot{x},\ddot{x},t)=0$
    • 2 dof
    • multiple dof

  • Linear Component : Linear System

    • m : $F=m\ddot{x}$

    • k : $F=k(x_1-x_2)$

    • c : $F=c(\dot{x_1}-\dot{x_2})$

  • Time-varying system : parameter change with time

  • Time invariant system
    • Constant coefficient differential equation

• Force

  • free vibration
    • Homogeneous equation
  • forced vibration
    • Nonhomogeneous equation

1.5 Simple harmonic vibration

$x=x_u\sin(wt\pm \theta)=Im[x_ue^{j(wt+\theta)}]$

2. 1 DOF

Generalized equation : at any time

$m\ddot{x}+c\dot{x}+kx=f(t)$

• Generalized coordinates
• Initial conditions

2.1 Free vibration

$m\ddot{x}+kx=0$

• set $x=x_u\sin(wt+ \theta)$, generalized equation
• $w_n=\sqrt{\dfrac{k}{m}}$ : Natural frequency

2.2 Damping vibration

$m\ddot{x}+c\dot{x}+kx=0$

• Characteristic equation : $m\lambda^2+c\lambda+k=0$

• Characteristic root : $\lambda_{1,2}=-\dfrac{c}{2m}\pm\sqrt{(\dfrac{c}{2m}^2)-\dfrac{k}{m}}\=(-\zeta\pm\sqrt{\zeta^2-1})w_n$

  • =0, $c=2\sqrt{mk}=c_e$ , Critical damping

  • $\zeta=\dfrac{c}{c_e}=\dfrac{c}{2\sqrt{mk}}$,Damping ratio

• Underdamped system : $\zeta<1$

  • Real solution
  • $x(t)=Ae^{-\zeta w_nt}\sin(w_d+\theta)$
  • $w_d=\sqrt{1-\zeta^2}w_n$
• Overdamped system : $\zeta>1$
• Critical damping system : $\zeta=1$
  • $x(t)=(c_1+c_2t)e^{-w_nt}$

2.3 Forced damping vibration

$m\ddot{x}+c\dot{x}+kx=f(t)\ \Downarrow\ [m(jw)^2+cjw+k]X=P \\Downarrow\ Z(w)X=P\\Downarrow\X=H(w)P​$

$H(w)=h_ue^{j\theta}=\dfrac{1}{Z(w)}=\dfrac{1}{(k-w^2m)+jwc}$

• $h_u=\dfrac{1}{\sqrt{(k-w^2m)^2+(wc)^2}}$
• $h(w)$ : Amplitude-frequency characteristic
• $\theta(w)$ : Phase frequency characteristic

$\beta=\dfrac{x_u}{x_{st}}=h_uk=\dfrac{1}{\sqrt{(1-\lambda^2)^2+(2\zeta\lambda)^2}}$

• $\lambda=\dfrac{w}{w_c}$
• $Re[\beta]$ :real number frequency characteristics
• $Im[\beta]$ : Imaginary number frequency characteristics

• Resonance : $\zeta=0.707$

3. Non-harmonic forced vibration

• Frequency domain analysis

3.1 Fourier

• Fourier expanation

  $x(t)=\dfrac{1}{2}a_0+\sum\limits_1^\infty(a_k\cos w_kt+b_k\sin w_k t)$

• Fourier transform

3.2 Steady state response

superposition of linear system

3.3 Transient response

• Unit pulse function

  

  

• Duhamel’s integral

  

• $x(t)=x_{free}(t)+x_d(t)$

4.Application for 1dof system

4.1 Vibration isolation

• double direction effect

• Passive : stop the transmission

• Active : stop the base

Vibration isolation coefficient

$R_{TR}=|H_{f,f_T}(w)|=\dfrac{|k+jwc|}{|k-w^2m+jwc|}$

Active and passive are same with same H(w)

4.2 Vibration absorber

• absorb at some particular frequency

4.3 Inertial force excitation

u(t)=esinwt

5. 2 DOF

• Mass matrix : [m]
• Damping matrix : [c]
• Stiffness matrix : [k]

5.1 No damping vibration

• Impedance matrix : [Z]

• Frequency equation : no zero in generalized function

  

  

• 2dof : 2 natural frequency

  • first : lower , basic frequency
  • second : higher

• Mode shape

  

  

• mode shape matrix

  

• solution

  

• EXAMPLE

  

5.2 Steady state response

• Frequency response matrix

6.Multiple DOF

Lagrange equation

• Generalized coordinates

• Generalized force : Virtual displacement method, virtual work,$p_i=\dfrac{\delta W_i}{\delta q_i}$

• kinetic energy : $E_k=\frac{1}{2}{\dot{q}}^T[m]{\dot{q}}$

• Potential energy : $E_p=\frac{1}{2}{q}^T[k]{q}$

• Dissipative function : $E_D=\frac{1}{2}{\dot{q}}^T[c]{\dot{q}}$

• example

  

6.1 No damping system

$[m]{\ddot{x}}+[k]{x}={0}\\Downarrow {x}={u}\sin(w_nt+\theta)\ [k]{u}=w^2_n[m]{u}$

• Generalized eigenvalue problem : $[A]{u}=\lambda[B]{u}$
• Frequency function : $\Delta (w_n)=\det ([Z(w_n)])$
• Impedance function : $[Z(w_n)]=-w_n^2[m]+[k]$

• Feature vector : ${u}r$,$[Z(w{nr})]{u}_r={0}$

  • $w_{nr}$ : natural frequency
• Modal matrix : $[u]=[{u}_i]$

• General solution : ${x(t)}=[u]{a_r\cos w_{nr}t+b_r\sin w_{nr}t}$

• example

  

• Initial conditions : ${a}=[u]^{-1}{x_0}$

6.2 Primary coordinate

• Master mass matrix
• Master stiffness matrix

• Primary coordinate : ${y}=[u]{x}$

• Regular vibration vector : ${\mu}_r=\dfrac{1}{M_r}{u}_r$

• Regular coordinate : ${x}=[\mu]{\eta}$

• example

  

6.3 Multiple DOF forced steady state response

Modal method