Mechanical Vibration - APS

五维生物

2015-11-30

APS, ME

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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