1. DIMENSIONAL ANALYSIS OF CONSTRUCTIVE UNITS ( Basic Equations of Dimension Chains. Definition of Dimension Chains) Lecturer Egor Efremenkov Tomsk - 2015

2. Basic Equations of Dimension Chains There is Dependency between basic performances of closing link and simple links of plane dimensional chain with dimension links. Determinate this dependency for simple dimensional chain

3. Basic Equations of Dimension Chains You see that nominal meaning of closing link А  is In common case there is equation which called Nominal Equation (1)(1)

4. Using transfer ratio  i which is equal to +1 for increase links and -1 for decrease links that the Nominal Equation was transformed Basic Equations of Dimension Chains Maximum and minimum meanings of closing link is equal (2)(2)

5. Basic Equations of Dimension Chains In common case After transformation we obtained So closing link’s tolerance is equal sum of tolerances of formative links (3)(3) (4)(4)

6. wher Δ в А i, Δ в А Δ – is top deviations of formative links and closing link; Δ н А i, Δ н А Δ – low deviations of them The Dependency is found between limit deviations of closing link and deviations of formative links Basic Equations of Dimension Chains

7. If equations put into (2) we obtain Basic Equations of Dimension Chains After transformation we obtained So top deviation of closing link is diminution of sums of top deviations of increase links and low deviations of decrease links. But for low deviation of closing link vice versa. (5)(5)

8. These equation put into (5) and after transformation we obtained Basic Equations of Dimension Chains So middle coordinate of tolerance zone of closing link is diminution of sums of middle coordinate of tolerance zone of increase links and decrease links. (6)(6) The Dependency is found between middle coordinate of tolerance zone of closing link (Δ 0 А Δ ) and middle coordinate of tolerance zone of formative links (Δ 0 А i )

9. Make sum (1) and (6) as a result: Basic Equations of Dimension Chains But we now that Then obtained So middle meaning of closing link is diminution of sums of middle meaning of increase links and decrease links.

10. The method is using these equations (3) – (5) we calling The Method of Maximum and Minimum Basic Equations of Dimension Chains or (7)(7)

11. Basic Equations of Dimension Chains By theorem where t Δ – risk factor, find by Laplace’s function Ф(t); λ i – relative standard deviation. And remember about (7) we obtain in common case

12. Basic Equations of Dimension Chains or Risk Р,% 32231694,62,10,940,510,270,1 Risk factor t  11,21,41,722,32,62,833,3 Таблица 1 relative standard deviation find like Risk (P, %) correspond to Ф(t) by equation Risk (P, %) is took from row

13. Basic Equations of Dimension Chains Таблица 1

14. Definition of Dimension Chains Assembly example

15. Definition of Dimension Chains Temperature elongation of the shaft may be find like where t 1 –temperature of environment; t 2 – working temperature of the shaft; l – length of the shaft.

16. Thanks for Your attention