1D Mechanical Systems In this lecture, we shall deal with 1D mechanical systems that can either ih translate or rotate in a one-dimensional i space. We shall demonstrate the similarities betweeneen 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.
Table of Contents Linear components of translation Linear components of rotation The D Alembert principle Example of a translational system Horizontal sorting
Linear Components of Translation Mass f 1 m f 2 f 3 f v 1 v 2 f B m a = (f i ) i dv = a dx Friction f = B (v B(v 1 v 2 ) B Spring f x 1 k x 2 f f = k (x 1 x 2 )
Linear Components of Rotation Inertia 1 J 2 J = ( i ) i d = d = Friction B = B ( B( 1 2 ) k Spring = k ( 1 2 )
Joints without Degrees of Freedom Node (Translation) x a f a f b f c x b x a = x b = x c v a b c a a = a b = a c x c f a + f b + f c =0 Node (Rotation) ti a a b c b a = b = c a = b = c c a = b = c a + b + c = 0
Joints with One Degree of Freedom Prismatic x 1 x 2 x 1 x 2 y 1 = y 2 1 = 2 Cylindrical 1 2 Scissors 1 2 x 1 =x 2 y 1 = y 2 1 2
The D Alembert Principle By introduction of an inertial force: f m = - m a the second law of Newton: m a = (f i ) i can be converted to a law of the form: (f i ) = 0 i
f m x m Sign Conventions d(m v) f m = + f k = + k (x x Neighbor ) f k f B =+B (v v Neighbor ) f B m f m x f k f B d(m v) f m = - f k =-k (x k(x x Neighbor ) f B = - B (v v Neighbor )
1. Example (Translation) Network View Topological View
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.
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 dv 1 1 1 F I2 = m dv 2 2 2 F I3 = m dv 3 3 3 F Ba = B 1 (v 3 v 2 ) F Bb = B 1 (v 3 v 1 ) F Bc = B 1 v 2 F B2 =B 2 (v 2 v 1 ) 3 F k2 =k x 2 2
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(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 = dv 1 F I1 m 1 1 F Ba = B 1 1 (v 3 v 2 ) = dv 1 F I1 m 1 1 F Ba = B 1 1 (v 3 v 2 ) 1 =m dv 2 F I2 2 F Bb = B 1 (v 3 v 1 ) F Bc = B 1 1 v 2 1 =m dv 2 F I2 2 F Bb = B 1 (v 3 v 1 ) F Bc = B 1 1 v 2 2 F I3 = m dv 3 3 3 F B2 = B 2 (v 2 v 1 ) B2 2 2 1 F k2 = k 2 x 2 2 F I3 = m dv 3 3 3 F B2 = B 2 (v 2 v 1 ) B2 2 2 1 F k2 = k 2 x 2
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(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 = dv 1 F I1 m 1 1 F Ba = B 1 1 (v 3 v 2 ) = dv 1 F I1 m 1 1 F Ba = B 1 1 (v 3 v 2 ) 1 =m dv 2 F I2 2 F Bb = B 1 (v 3 v 1 ) F Bc = B 1 1 v 2 1 =m dv 2 F I2 2 F Bb = B 1 (v 3 v 1 ) F Bc = B 1 1 v 2 2 F I3 = m dv 3 3 3 F B2 = B 2 (v 2 v 1 ) B2 2 2 1 F k2 = k 2 x 2 2 F I3 = m dv 3 3 3 F B2 = B 2 (v 2 v 1 ) B2 2 2 1 F k2 = k 2 x 2
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(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 = dv 1 F I1 m 1 1 F Ba = B 1 1 (v 3 v 2 ) = dv 1 F I1 m 1 1 F Ba = B 1 1 (v 3 v 2 ) 1 =m dv 2 F I2 2 F Bb = B 1 (v 3 v 1 ) F Bc = B 1 1 v 2 1 =m dv 2 F I2 2 F Bb = B 1 (v 3 v 1 ) F Bc = B 1 1 v 2 2 F I3 = m dv 3 3 3 F B2 = B 2 (v 2 v 1 ) B2 2 2 1 F k2 = k 2 x 2 2 F I3 = m dv 3 3 3 F B2 = B 2 (v 2 v 1 ) B2 2 2 1 F k2 = k 2 x 2
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 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 = dv 1 F I1 m 1 1 1 =m dv 2 F I2 2 2 F I3 = m dv 3 3 3 F Ba = B 1 1 (v 3 v 2 ) F Bb = B 1 (v 3 v 1 ) F Bc = B 1 1 v 2 F B2 = B 2 (v 2 v 1 ) B2 2 2 1 F k2 = k 2 x 2 dv 1 dv 2 dv 3 = F I1 / m 1 F Ba = B 1 (v 3 v 2 ) 1 F Bb = B 1 (v 3 v 1 ) = F I2 /m 2 2 = F I3 / m 3 3 F Bc = B 1 v 2 F B2 = B 2 (v 2 v 1 ) F k2 = k 2 x 2
Sorting Algorithm The sorting algorithm operates exactly in the same way as for electrical circuits. It is totally independent of the application domain.
References Cellier, F.E. (1991), Continuous System Modeling, Springer-Verlag, New York, Chapter 4.