Download presentation

Presentation is loading. Please wait.

Published byDeanna Comer Modified over 3 years ago

1
Start of Presentation Mathematical Modeling of Physical Systems © Prof. Dr. François E. Cellier September 20, 2012 1D Mechanical Systems In this lecture, we shall deal with 1D mechanical systems that can either translate or rotate in a one-dimensional space. We shall demonstrate the similarities between the mathematical descriptions of these systems and the electrical circuits discussed in the previous lecture. In particular, it will be shown that the symbolic formulae manipulation algorithms (sorting algorithms) that were introduced in the previous lecture can be applied to these systems just as easily and without any modification.

2
Start of Presentation Mathematical Modeling of Physical Systems © Prof. Dr. François E. Cellier Table of Contents Linear components of translation Linear components of rotation The D’Alembert principle Example of a translational system Horizontal sorting September 20, 2012

3
Start of Presentation Mathematical Modeling of Physical Systems © Prof. Dr. François E. Cellier Linear Components of Translation Mass Friction Spring m·a = (f i ) ii dv dt = a dx dt = v B f v 1 f v 2 f = B·(v 1 – v 2 ) B f x 1 f x 2 k f = k·(x 1 – x 2 ) m f 1 f 2 f 3 September 20, 2012

4
Start of Presentation Mathematical Modeling of Physical Systems © Prof. Dr. François E. Cellier Linear Components of Rotation Inertia Friction Spring J· = ( i ) ii dd dt = dd dt = J 1 2 B = B·( 1 – 2 ) k = k·( 1 – 2 ) September 20, 2012

5
Start of Presentation Mathematical Modeling of Physical Systems © Prof. Dr. François E. Cellier Joints without Degrees of Freedom Node (Translation) Node (Rotation) x a = x b = x c f a + f b + f c = 0 v a = v b = v c a a = a b = a c a = b = c a + b + c = 0 a = b = c a = b = c x a x b f a f b x c f c a b a b c c September 20, 2012

6
Start of Presentation Mathematical Modeling of Physical Systems © Prof. Dr. François E. Cellier Joints with One Degree of Freedom Prismatic Cylindrical Scissors x 1 x 2 x 1 x 2 y 1 = y 2 1 = 2 1 2 x 1 = x 2 y 1 = y 2 1 21 2 1 2 September 20, 2012

7
Start of Presentation Mathematical Modeling of Physical Systems © Prof. Dr. François E. Cellier The D’Alembert Principle By introduction of an inertial force: the second law of Newton: can be converted to a law of the form: f m = - m·a m·a = (f i ) ii (f i ) = 0 ii September 20, 2012

8
Start of Presentation Mathematical Modeling of Physical Systems © Prof. Dr. François E. Cellier Sign Conventions d(m·v) dt f m = + f k = + k·(x – x Neighbor ) f B = + B·(v – v Neighbor ) x m f m f k f B x m f m f k f B d(m·v) dt f m = - f k = - k·(x – x Neighbor ) f B = - B·(v – v Neighbor ) September 20, 2012

9
Start of Presentation Mathematical Modeling of Physical Systems © Prof. Dr. François E. Cellier 1. Example (Translation) Topological View Network View September 20, 2012

10
Start of Presentation Mathematical Modeling of Physical Systems © Prof. Dr. François E. Cellier 1. Example (Translation) II The system is being cut open between the individual masses, and cutting forces are introduced. The D’Alembert principle can now be applied to each body separately. September 20, 2012

11
Start of Presentation Mathematical Modeling of Physical Systems © Prof. Dr. François E. Cellier 1. Example (continued) F(t) = F I3 + F Ba + F Bb F Ba = F I2 + F Bc + F B2 + F k2 F Bb + F B2 = F I1 + F Bd + F k1 F I1 = m 1 · dv 1 dt dx 1 dt = v 1 F I2 = m 2 · dv 2 dt dx 2 dt = v 2 F I3 = m 3 · dv 3 dt dx 3 dt = v 3 F Ba = B 1 · (v 3 – v 2 ) F Bb = B 1 · (v 3 – v 1 ) F Bc = B 1 · v 2 F Bd = B 1 · v 1 F B2 = B 2 · (v 2 – v 1 ) F k1 = k 1 · x 1 F k2 = k 2 · x 2 September 20, 2012

12
Start of Presentation Mathematical Modeling of Physical Systems © Prof. Dr. François E. Cellier Horizontal Sorting I F(t) = F I3 + F Ba + F Bb F Ba = F I2 + F Bc + F B2 + F k2 F Bb + F B2 = F I1 + F Bd + F k1 F I1 = m 1 · dv 1 dt dx 1 dt = v 1 F I2 = m 2 · dv 2 dt dx 2 dt = v 2 F I3 = m 3 · dv 3 dt dx 3 dt = v 3 F Ba = B 1 · (v 3 – v 2 ) F Bb = B 1 · (v 3 – v 1 ) F Bc = B 1 · v 2 F Bd = B 1 · v 1 F B2 = B 2 · (v 2 – v 1 ) F k1 = k 1 · x 1 F k2 = k 2 · x 2 F(t) = F I3 + F Ba + F Bb F Ba = F I2 + F Bc + F B2 + F k2 F Bb + F B2 = F I1 + F Bd + F k1 F I1 = m 1 · dv 1 dt dx 1 dt = v 1 F I2 = m 2 · dv 2 dt dx 2 dt = v 2 F I3 = m 3 · dv 3 dt dx 3 dt = v 3 F Ba = B 1 · (v 3 – v 2 ) F Bb = B 1 · (v 3 – v 1 ) F Bc = B 1 · v 2 F Bd = B 1 · v 1 F B2 = B 2 · (v 2 – v 1 ) F k1 = k 1 · x 1 F k2 = k 2 · x 2 September 20, 2012

13
Start of Presentation Mathematical Modeling of Physical Systems © Prof. Dr. François E. Cellier Horizontal Sorting II F(t) = F I3 + F Ba + F Bb F Ba = F I2 + F Bc + F B2 + F k2 F Bb + F B2 = F I1 + F Bd + F k1 F I1 = m 1 · dv 1 dt dx 1 dt = v 1 F I2 = m 2 · dv 2 dt dx 2 dt = v 2 F I3 = m 3 · dv 3 dt dx 3 dt = v 3 F Ba = B 1 · (v 3 – v 2 ) F Bb = B 1 · (v 3 – v 1 ) F Bc = B 1 · v 2 F Bd = B 1 · v 1 F B2 = B 2 · (v 2 – v 1 ) F k1 = k 1 · x 1 F k2 = k 2 · x 2 F(t) = F I3 + F Ba + F Bb F Ba = F I2 + F Bc + F B2 + F k2 F Bb + F B2 = F I1 + F Bd + F k1 F I1 = m 1 · dv 1 dt dx 1 dt = v 1 F I2 = m 2 · dv 2 dt dx 2 dt = v 2 F I3 = m 3 · dv 3 dt dx 3 dt = v 3 F Ba = B 1 · (v 3 – v 2 ) F Bb = B 1 · (v 3 – v 1 ) F Bc = B 1 · v 2 F Bd = B 1 · v 1 F B2 = B 2 · (v 2 – v 1 ) F k1 = k 1 · x 1 F k2 = k 2 · x 2 September 20, 2012

14
Start of Presentation Mathematical Modeling of Physical Systems © Prof. Dr. François E. Cellier Horizontal Sorting III F(t) = F I3 + F Ba + F Bb F Ba = F I2 + F Bc + F B2 + F k2 F Bb + F B2 = F I1 + F Bd + F k1 F I1 = m 1 · dv 1 dt dx 1 dt = v 1 F I2 = m 2 · dv 2 dt dx 2 dt = v 2 F I3 = m 3 · dv 3 dt dx 3 dt = v 3 F Ba = B 1 · (v 3 – v 2 ) F Bb = B 1 · (v 3 – v 1 ) F Bc = B 1 · v 2 F Bd = B 1 · v 1 F B2 = B 2 · (v 2 – v 1 ) F k1 = k 1 · x 1 F k2 = k 2 · x 2 F(t) = F I3 + F Ba + F Bb F Ba = F I2 + F Bc + F B2 + F k2 F Bb + F B2 = F I1 + F Bd + F k1 F I1 = m 1 · dv 1 dt dx 1 dt = v 1 F I2 = m 2 · dv 2 dt dx 2 dt = v 2 F I3 = m 3 · dv 3 dt dx 3 dt = v 3 F Ba = B 1 · (v 3 – v 2 ) F Bb = B 1 · (v 3 – v 1 ) F Bc = B 1 · v 2 F Bd = B 1 · v 1 F B2 = B 2 · (v 2 – v 1 ) F k1 = k 1 · x 1 F k2 = k 2 · x 2 September 20, 2012

15
Start of Presentation Mathematical Modeling of Physical Systems © Prof. Dr. François E. Cellier Horizontal Sorting IV F(t) = F I3 + F Ba + F Bb F Ba = F I2 + F Bc + F B2 + F k2 F Bb + F B2 = F I1 + F Bd + F k1 F I1 = m 1 · dv 1 dt dx 1 dt = v 1 F I2 = m 2 · dv 2 dt dx 2 dt = v 2 F I3 = m 3 · dv 3 dt dx 3 dt = v 3 F Ba = B 1 · (v 3 – v 2 ) F Bb = B 1 · (v 3 – v 1 ) F Bc = B 1 · v 2 F Bd = B 1 · v 1 F B2 = B 2 · (v 2 – v 1 ) F k1 = k 1 · x 1 F k2 = k 2 · x 2 F I3 = F(t) - F Ba - F Bb F I2 = F Ba - F Bc - F B2 - F k2 F I1 = F Bb + F B2 - F Bd - F k1 = F I1 / m 1 dv 1 dt dx 1 dt = v 1 F Ba = B 1 · (v 3 – v 2 ) F Bb = B 1 · (v 3 – v 1 ) F Bc = B 1 · v 2 F Bd = B 1 · v 1 F B2 = B 2 · (v 2 – v 1 ) F k1 = k 1 · x 1 F k2 = k 2 · x 2 = F I2 / m 2 dv 2 dt dx 2 dt = v 2 = F I3 / m 3 dv 3 dt dx 3 dt = v 3 September 20, 2012

16
Start of Presentation Mathematical Modeling of Physical Systems © Prof. Dr. François E. Cellier Sorting Algorithm The sorting algorithm operates exactly in the same way as for electrical circuits. It is totally independent of the application domain. September 20, 2012

17
Start of Presentation Mathematical Modeling of Physical Systems © Prof. Dr. François E. Cellier References Cellier, F.E. (1991), Continuous System Modeling, Springer-Verlag, New York, Chapter 4.Continuous System ModelingChapter 4 September 20, 2012

Similar presentations

OK

Start Presentation November 15, 2012 8 th Homework - Solution In this homework, we shall model and simulate a mechanical system as well as exercise the.

Start Presentation November 15, 2012 8 th Homework - Solution In this homework, we shall model and simulate a mechanical system as well as exercise the.

© 2018 SlidePlayer.com Inc.

All rights reserved.

To make this website work, we log user data and share it with processors. To use this website, you must agree to our Privacy Policy, including cookie policy.

Ads by Google