Dynamic balancing apparatus and method using simple harmonic angular motion

US 20090255334A1
Filed: 03/24/2009
Published: 10/15/2009
Est. Priority Date: 03/25/2008
Status: Active Grant

First Claim

Patent Images

1. A dynamic balancing apparatus using a simple harmonic angular motion, comprising:

a rotational shaft on which a rotational body 1 having an unbalanced mass is installed;

a simple harmonic angular vibration generator which periodically excites the unbalanced mass;

a support for supporting both sides of the rotational shaft;

transducers 40 which is installed on the support 30 for measuring applied force or vibration generated by the unbalance having the simple harmonic angular motion; and

an angular vibration measuring sensor for measuring the amplitude generated by the simple harmonic angular motion of the rotational shaft.

View all claims

1 Assignment

Timeline View

Assignment View

0 Petitions

Accused Products

Abstract

Provided is a dynamic balancing apparatus using a simple harmonic angular motion, including a rotational shaft on which a rotational body having an unbalanced mass is installed; a simple angular motion generator for periodically vibrating the rotational shaft; a support for supporting both sides of the rotational shaft; an angular vibration measuring sensor for measuring the simple harmonic angular motion of the rotational shaft; and force or vibration transducers for measuring applied force acting on or vibration transferred to the support by means of the simple harmonic motion of the unbalance mass.

12 Citations

View as Search Results

8 Claims

1. A dynamic balancing apparatus using a simple harmonic angular motion, comprising:
- a rotational shaft on which a rotational body 1 having an unbalanced mass is installed;
  
  a simple harmonic angular vibration generator which periodically excites the unbalanced mass;
  
  a support for supporting both sides of the rotational shaft;
  
  transducers 40 which is installed on the support 30 for measuring applied force or vibration generated by the unbalance having the simple harmonic angular motion; and
  
  an angular vibration measuring sensor for measuring the amplitude generated by the simple harmonic angular motion of the rotational shaft.

2. A dynamic balancing method using a simple harmonic angular motion, comprising:
- periodically vibrating a rotational body so as to measure the tangential and centrifugal forces generated by the rotating unbalanced mass of the rotational body;
  
  measuring linear vibration generated by the tangential and centrifugal forces, or these forces generated by the rotating unbalanced mass of the rotational body acting on the support which supports the rotational shaft having the simple harmonic angular motion;
  
  estimating multiple Fourier coefficients of the measured tangential and the centrifugal forces; and
  
  estimating the angle and magnitude of the unbalanced mass of the rotational body by using multiple Fourier coefficients of the measured tangential and centrifugal forces generated from the simple harmonic motion of the unbalance.
- View Dependent Claims (3, 4, 5, 6, 7, 8)
- - 3. The dynamic balancing method of claim 2, wherein using the horizontal and/or vertical forces acting on the support is determined by the following equation:
    - $\begin{matrix} F_{X} (t) = - F_{t} \cdot \sin (θ (t)) + F_{n} \cdot \cos (θ (t)) \\ = - mr \cdot A_{θ} \cdot {(2 π f)}^{2} \cdot \sin (2 π ft) \cdot \sin {A_{θ} \cdot \sin (2 π ft) + θ_{0}} + \\ mr \cdot A_{θ}^{2} \cdot {(2 π f)}^{2} \cdot \frac{1}{2} {1 + \cos (4 π ft)} \cdot \cos {A_{θ} \cdot \sin (2 π ft) + θ_{0}} \end{matrix}$ $\begin{matrix} F_{Y} (t) = F_{t} \cdot \cos (θ (t)) + F_{n} \cdot \sin (θ (t)) \\ = mr \cdot A_{θ} \cdot {(2 π f)}^{2} \cdot \sin (2 π ft) \cdot \cos {A_{θ} \cdot \sin (2 π ft) + θ_{0}} + \\ mr \cdot A_{θ}^{2} \cdot {(2 π f)}^{2} \cdot \frac{1}{2} {1 + \cos (4 π ft)} \cdot \sin {A_{θ} \cdot \sin (2 π ft) + θ_{0}} \end{matrix}$ wherein F_x(t) is the horizontal component of each of the tangential force and the centrifugal force acting on the support by the rotating unbalanced mass of the rotational body;
      
      F_y(t) is the vertical component of each of the tangential force F_tand the centrifugal force F_nacting on the support by the rotating unbalanced mass of the rotational body;
      
      f[Hz] is the frequency of a simple harmonic angular vibration;
      
      A_θ[radians] is the amplitude of the simple harmonic angular motion;
      
      θ
      
      (t) is the angular displacement of the rotational shaft;
      
      ω
      
      (t) is the angular velocity;
      
      α
      
      (t) is the angular acceleration;
      
      m[kg] is the unbalanced mass of the rotational body;
      
      r[m] is the distance of the rotating unbalanced mass of the rotational body, which is spaced apart from the rotational shaft; and
      
      θ
      
      ₀[radians] is the angular position in reference to the rotational shaft.
  - 4. The dynamic balancing method of claim 3, wherein the angular position and magnitude of the rotating unbalanced mass of the rotational body are determined from multiple Fourier coefficients of the tangential and centrifugal forces generated by the rotating unbalanced mass of the rotational body, and their multiple Fourier coefficients are calculated by the following equations:
    - $\begin{matrix} C_{F_{X}} (k) = \frac{2}{T} \int_{0}^{T} F_{X} (t) \cdot \cos (k \cdot 2 π ft) \partial t \\ = {\begin{matrix} \begin{matrix} mr \cdot A_{θ} \cdot {(2 π f)}^{2} \cdot \\ {\begin{matrix} k \cdot S_{\sin} (A_{θ}; k - 1) + \\ k \cdot S_{\sin} (A_{θ}; k + 1) \end{matrix}} \cdot \cos (θ_{0}) \end{matrix} & for k = 2, 4, 6, \dots \\ 0 & for k = 1, 3, 5, \dots \end{matrix} \end{matrix}$ $\begin{matrix} S_{F_{X}} (k) = \frac{2}{T} \int_{0}^{T} F_{X} (t) \cdot \sin (k \cdot 2 π ft) \partial t \\ = {\begin{matrix} \begin{matrix} mr \cdot A_{θ} \cdot {(2 π f)}^{2} \cdot \\ {\begin{matrix} - k \cdot C_{\cos} (A_{θ}; k - 1) - \\ k \cdot C_{\cos} (A_{θ}; k + 1) \end{matrix}} \cdot \sin (θ_{0}) \end{matrix} & for k = 1, 3, 5, \dots \\ 0 & for k = 2, 4, 6, \dots \end{matrix} \end{matrix}$ $\begin{matrix} C_{F_{Y}} (k) = \frac{2}{T} \int_{0}^{T} F_{Y} (t) \cdot \cos (k \cdot 2 π ft) \partial t \\ = {\begin{matrix} \begin{matrix} mr \cdot A_{θ} \cdot {(2 π f)}^{2} \cdot \\ {\begin{matrix} k \cdot S_{\sin} (A_{θ}; k - 1) + \\ k \cdot S_{\sin} (A_{θ}; k + 1) \end{matrix}} \cdot \sin (θ_{0}) \end{matrix} & for k = 2, 4, 6, \dots \\ 0 & for k = 1, 3, 5, \dots \end{matrix} \end{matrix}$ $\begin{matrix} S_{F_{Y}} (k) = \frac{2}{T} \int_{0}^{T} F_{Y} (t) \cdot \sin (k \cdot 2 π ft) \partial t \\ = {\begin{matrix} \begin{matrix} mr \cdot A_{θ} \cdot {(2 π f)}^{2} \cdot \\ {\begin{matrix} - k \cdot C_{\cos} (A_{θ}; k - 1) + \\ k \cdot C_{\cos} (A_{θ}; k + 1) \end{matrix}} \cdot \cos (θ_{0}) \end{matrix} & for k = 1, 3, 5, \dots \\ 0 & for k = 2, 4, 6, \dots \end{matrix} \end{matrix}$ wherein C_Fxis the cosine component corresponding to Fourier coefficients of the horizontal force;
      
      S_Fxis the sine component corresponding to Fourier coefficients of the horizontal force;
      
      C_Fyis the cosine component corresponding to Fourier coefficients of the vertical force;
      
      S_Fyis the sine component corresponding to Fourier coefficients of the vertical force;
      
      T is the period of the simple harmonic angular motion of frequency f [Hz] (T=1/f);
      
      m[kg] is the unbalanced mass of the rotational body;
      
      r[m] is the distance of the unbalanced mass, which is spaced apart from the rotational shaft;
      
      θ
      
      ₀[radians] is the angular position in reference to the rotational shaft;
      
      A_θ[radians] is the amplitude of the periodic angular motion;
      
      θ
      
      (t) is the angular displacement of the rotational shaft; and
      
      k is an integer indicating the order of Fourier coefficients;
      
      $C_{\cos} (A_{θ}; k) = \frac{1}{T} \int_{0}^{T} \cos (k \cdot 2 π ft) \cdot \cos {A_{θ} \cdot \sin (2 π ft)} \partial t, C_{\cos} (A_{θ}; k) \neq 0 for k = 2, 4, 6, \dots$ $S_{\sin} (A_{θ}; k) = \frac{1}{T} \int_{0}^{T} \sin (k \cdot 2 π ft) \cdot \sin {A_{θ} \cdot \sin (2 π ft)} \partial t, S_{\sin} (A_{θ}; k) \neq 0 for k = 1, 3, 5, \dots$ $C_{\sin} (A_{θ}; k) = \frac{1}{T} \int_{0}^{T} \sin (k \cdot 2 π ft) \cdot \cos {A_{θ} \cdot \sin (2 π ft)} \partial t = 0, for k = 1, 2, 3, \dots$ $S_{\cos} (A_{θ}; k) = \frac{1}{T} \int_{0}^{T} \cos (k \cdot 2 π ft) \cdot \sin {A_{θ} \cdot \sin (2 π ft)} \partial t = 0, for k = 1, 2, 3, \dots$ $C_{\cos} (A_{θ}; k) = \frac{1}{T} \int_{0}^{T} \cos {A_{θ} \cdot \sin (2 π ft)} \partial t, for k = 0$ $C_{\sin} (A_{θ}; 0) = S_{\cos} (A_{θ}; 0) = S_{\sin} (A_{θ}; 0) = 0$
  - 5. The dynamic balancing method of claim 4, wherein the angular position and unbalanced amount (m·
    - r) of the unbalanced mass of the rotational body are determined by the following equation using multiple Fourier cosine and sine components of both horizontal and vertical forces measured from the horizontal and vertical directions of the support;
      
      $[\begin{matrix} S_{F_{X}} (1) \\ S_{F_{Y}} (1) \\ C_{F_{X}} (2) \\ C_{F_{Y}} (2) \\ S_{F_{X}} (3) \\ S_{F_{Y}} (3) \\ C_{F_{X}} (4) \\ C_{F_{Y}} (4) \\ • \\ • \end{matrix}] = A_{θ} \cdot {(2 π f)}^{2} [\begin{matrix} 0 & - C_{\cos} (A_{θ}, 0) - C_{\cos} (A_{θ}, 2) \\ C_{\cos} (A_{θ}, 1) + C_{\cos} (A_{θ}, 3) & 0 \\ 2 S_{\sin} (A_{θ}, 1) + 2 S_{\sin} (A_{θ}, 3) & 0 \\ 0 & 2 S_{\sin} (A_{θ}, 1) + 2 S_{\sin} (A_{θ}, 3) \\ 0 & - 3 C_{\cos} (A_{θ}, 2) - 3 C_{\cos} (A_{θ}, 4) \\ 3 C_{\cos} (A_{θ}, 2) + 3 C_{\cos} (A_{θ}, 4) & 0 \\ 4 S_{\sin} (A_{θ}, 3) + 4 S_{\sin} (A_{θ}, 5) & 0 \\ 0 & 4 S_{\sin} (A_{θ}, 3) + 4 S_{\sin} (A_{θ}, 5) \\ • & • \\ • & • \end{matrix}] [\begin{matrix} U_{\cos} \\ U_{\sin} \end{matrix}]$ wherein U_cosis the cosine component of the unbalanced amount (U_cos=mr·
      
      cos(θ
      
      ₀)) and U_sinis the sine component (U_sin=mr·
      
      sin(θ
      
      ₀)), and $U = mr = \sqrt{U_{\cos}^{2} + U_{\sin}^{2}}, θ_{0} = \tan^{- 1} (\frac{U_{\sin}}{U_{\cos}}) .$
  - 6. The dynamic balancing method of claim 4, wherein the angular position and unbalanced amount of the unbalanced mass of the rotational body are determined by the following equation using multiple Fourier cosine and sine components of the horizontal force measured from the horizontal direction of the support, $[\begin{matrix} S_{F_{X}} \end{matrix}$
    - ( 1 ) C F X 
      
      ( 2 ) S F X 
      
      ( 3 ) C F X 
      
      ( 4 ) •
      
      •
      
      ] = A θ
      
      ·
      
      ( 2 
      
      
      
      π
      
      
      
      
      
      f ) 2 
      
      [ 0 - C cos 
      
      ( A θ
      
      , 0 ) - C cos 
      
      ( A θ
      
      , 2 ) C cos 
      
      ( A θ
      
      , 0 ) + C cos 
      
      ( A θ
      
      , 2 ) - 3 
      
      
      
      C cos 
      
      ( A θ
      
      , 2 ) - 3 
      
      C cos 
      
      ( A θ
      
      , 4 ) 3 
      
      
      
      C cos 
      
      ( A θ
      
      , 2 ) + 3 
      
      C cos 
      
      ( A θ
      
      , 4 ) 0 •
      
      •
      
      •
      
      •
      
      ] 
      
      [ U cos U sin ] wherein U_cosis the cosine component of the unbalanced amount (U_cos=mr·
      
      cos(θ
      
      ₀)) and U_sinis the sine component (U_sin=mr·
      
      sin(θ
      
      ₀)), and $U = mr = \sqrt{U_{\cos}^{2} + U_{\sin}^{2}}, θ_{0} = \tan^{- 1} (\frac{U_{\sin}}{U_{\cos}}) .$
  - 7. The dynamic balancing method of claim 4, wherein the angular position and unbalanced amount of the unbalanced mass of the rotational body are determined by the following equation using multiple Fourier cosine and sine components of the vertical force measured from the vertical direction of the support, $[\begin{matrix} S_{F_{Y}} \end{matrix}$
    - ( 1 ) C F Y 
      
      ( 2 ) S F Y 
      
      ( 3 ) C F Y 
      
      ( 4 ) ⋮
      
      ⋮
      
      ] = A θ
      
      ·
      
      ( 2 
      
      
      
      π
      
      
      
      
      
      f ) 2 
      
      [ C cos 
      
      ( A θ
      
      , 1 ) + C cos 
      
      ( A θ
      
      , 3 ) 0 0 
      
      2 
      
      S sin 
      
      ( A θ
      
      , 1 ) + 2 
      
      
      
      S sin 
      
      ( A θ
      
      , 3 ) 3 
      
      
      
      C cos 
      
      ( A θ
      
      , 2 ) + 3 
      
      C cos 
      
      ( A θ
      
      , 4 ) 
      
      0 0 4 
      
      S sin 
      
      ( A θ
      
      , 3 ) + 4 
      
      
      
      S sin 
      
      ( A θ
      
      , 5 ) ⋮
      
      ⋮
      
      ⋮
      
      ⋮
      
      ] 
      
      [ U cos U sin ] wherein U_cosis the cosine component of the unbalanced amount (U_cos=mr·
      
      cos(θ
      
      ₀)) and U_sinis the sine component of the unbalanced amount (U_sin=mr·
      
      sin(θ
      
      ₀)), and $U = mr = \sqrt{U_{\cos}^{2} + U_{\sin}^{2}}, θ_{0} = \tan^{- 1} (\frac{U_{\sin}}{U_{\cos}}) .$
  - 8. The dynamic balancing method of claim 2, wherein it exploits horizontal and/or vertical vibration signals that linearly depend on the horizontal and/or vertical components of the tangential and centrifugal forces acting on the support when the unbalanced mass of the rotational body is running, whose outputs of vibration measurement sensors are characterized by the following equation:
    - F_X≈
      
      m_eff,X·
      
      a_X(t), F_Y≈
      
      m_eff,Y·
      
      a_Y(t)m_effis an effective mass of each vibration measurement direction, and a_X(t) and a_Y(t) are the measured vibration acceleration signals that satisfy a linear relationship between the measured vibration accelerations and the tangential and centrifugal forces acting on the support and the Measured vibration acceleration signals from the support are used to determine the angular position and magnitude of an unbalanced mass by installing vibration transducers in the horizontal and/or vertical movement directions of the support, and Fourier coefficients of the horizontal and vertical force components are evaluated for implementing vibration measurement-based dynamic balancing.

Specification

Resources

Litigation Campaign Assessment

Current Assignee
Korea Research Institute of Standards and Science
Original Assignee
Korea Research Institute of Standards and Science
Inventors
Lim, Jong Yeon, Cheung, Wan Sup, Yoon, Sae Won

Granted Patent

US 8,171,789 B2
Time in Patent Office

Days
Field of Search
US Class Current

73/460
CPC Class Codes

G01M 1/24 Performing balancing on ela...

Dynamic balancing apparatus and method using simple harmonic angular motion

First Claim

1 Assignment

0 Petitions

Accused Products

Abstract

12 Citations

8 Claims

Specification

Use Cases

Quick Links

Others

Dynamic balancing apparatus and method using simple harmonic angular motion

First Claim

1 Assignment

Subscription Required

Subscription Required

0 Petitions

Subscription Required

Accused Products

Subscription Required

Abstract

12 Citations

8 Claims

Specification

Subscription Required

Use Cases

Quick Links

Others