Start of Presentation Mathematical Modeling of Physical Systems © Prof. Dr. François E. Cellier September 20, D 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.

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

Start of Presentation Mathematical Modeling of Physical Systems © Prof. Dr. François E. Cellier Linear Components of Translation Mass Friction Spring m·a =  (f i ) ii 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

Start of Presentation Mathematical Modeling of Physical Systems © Prof. Dr. François E. Cellier Linear Components of Rotation Inertia Friction Spring J·  =  (  i ) ii dd dt =  dd dt =  J  1  2  B  = B·(  1 –  2 )  k  = k·(  1 –  2 ) September 20, 2012

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

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   21   2  1  2 September 20, 2012

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 ) ii  (f i ) = 0 ii September 20, 2012

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

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

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

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

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

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

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

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

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

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