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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.

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1D Mechanical Systems

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  1. 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. 1D Mechanical Systems

  2. Linear components of translation Linear components of rotation The D’Alembert principle Example of a translational system Horizontal sorting Table of Contents

  3. Mass Friction Spring m·a = S (fi ) i m dv dx = v = a dt dt f f f f f f f B 1 2 3 B k v x x v 2 2 1 1 Linear Components of Translation f = B·(v1 – v2 ) f = k·(x1 – x2 )

  4. Inertia Friction Spring J·a = S (ti ) d i = a J dt d =  dt     B 2 1 k   Linear Components of Rotation  = B·(1 – 2 )  = k·(1 – 2 )

  5. Node (Translation) Node (Rotation) f xa = xb = xc a = b = c  f  f  b a c a c b x x   a va = vb = vc a = b = c b a b aa = ab = ac aa = ab = ac x  c c fa+fb + fc = 0 a+b + c = 0 Joints without Degrees of Freedom

  6. Prismatic Cylindrical Scissors x1 x2 x1= x2 y1 = y2 y1 = y2 1 = 2 12    x x  1 2 1 2 1 2 Joints with One Degree of Freedom

  7. By introduction of an inertial force: m·a = S (fi ) i S (fi ) = 0 i The D’Alembert Principle fm = - m·a • the second law of Newton: can be converted to a law of the form:

  8. x d(m·v) d(m·v) fm = - fm = + dt dt m fk = + k·(x – xNeighbor ) fk = - k·(x – xNeighbor ) fB = - B·(v – vNeighbor ) fB = + B·(v – vNeighbor ) f f f f f f B B m k m k Sign Conventions x m

  9. 1. Example (Translation) Network View Topological View

  10. 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.

  11. F(t) = FI3+ FBa + FBb FBa = FI2 + FBc + FB2 + Fk2 FBb + FB2 = FI1 + FBd + Fk1 FBa = B1· (v3 – v2 ) FBb = B1· (v3 – v1 ) FBc = B1· v2 FBd = B1· v1 FB2 = B2· (v2 – v1 ) Fk1 = k1· x1 Fk2 = k2· x2 FI1 = m1· FI2 = m2· FI3 = m3· = v1 = v3 = v2 dv3 dx2 dv2 dx3 dx1 dv1 dt dt dt dt dt dt 1. Example (continued)

  12. F(t) = FI3+ FBa + FBb FBa = FI2 + FBc + FB2 + Fk2 FBb + FB2 = FI1 + FBd + Fk1 F(t) = FI3+ FBa + FBb FBa = FI2 + FBc + FB2 + Fk2 FBb + FB2 = FI1 + FBd + Fk1 dv1 FI1 = m1· FBa = B1· (v3 – v2 ) FBb = B1· (v3 – v1 ) FBc = B1· v2 FBd = B1· v1 FB2 = B2· (v2 – v1 ) Fk1 = k1· x1 Fk2 = k2· x2 dt FBa = B1· (v3 – v2 ) FBb = B1· (v3 – v1 ) FBc = B1· v2 FBd = B1· v1 FB2 = B2· (v2 – v1 ) Fk1 = k1· x1 Fk2 = k2· x2 FI1 = m1· FI2 = m2· FI3 = m3· dx1 = v1 dt dv2 = v3 = v1 = v2 FI2 = m2· dt dx1 dv2 dx2 dx3 dv3 dv1 dx2 = v2 dt dt dt dt dt dt dt dv3 FI3 = m3· dt dx3 = v3 dt Horizontal Sorting I 

  13. F(t) = FI3+ FBa + FBb FBa = FI2 + FBc + FB2 + Fk2 FBb + FB2 = FI1 + FBd + Fk1 F(t) = FI3+ FBa + FBb FBa = FI2 + FBc + FB2 + Fk2 FBb + FB2 = FI1 + FBd + Fk1 dv1 dv1 FI1 = m1· FI1 = m1· FBa = B1· (v3 – v2 ) FBb = B1· (v3 – v1 ) FBc = B1· v2 FBd = B1· v1 FB2 = B2· (v2 – v1 ) Fk1 = k1· x1 Fk2 = k2· x2 FBa = B1· (v3 – v2 ) FBb = B1· (v3 – v1 ) FBc = B1· v2 FBd = B1· v1 FB2 = B2· (v2 – v1 ) Fk1 = k1· x1 Fk2 = k2· x2 dt dt dx1 dx1 = v1 = v1 dt dt dv2 dv2 FI2= m2· FI2 = m2· dt dt dx2 dx2 = v2 = v2 dt dt dv3 dv3 FI3 = m3· FI3 = m3· dt dt dx3 dx3 = v3 = v3 dt dt Horizontal Sorting II 

  14. F(t) = FI3+ FBa + FBb FBa = FI2 + FBc + FB2 + Fk2 FBb + FB2 = FI1 + FBd + Fk1 F(t) = FI3+ FBa + FBb FBa = FI2 + FBc + FB2 + Fk2 FBb + FB2 = FI1 + FBd + Fk1 dv1 dv1 FI1= m1· FI1 = m1· FBa = B1· (v3 – v2 ) FBb = B1· (v3 – v1 ) FBc = B1· v2 FBd = B1· v1 FB2 = B2· (v2 – v1 ) Fk1 = k1· x1 Fk2 = k2· x2 FBa = B1· (v3 – v2 ) FBb = B1· (v3 – v1 ) FBc = B1· v2 FBd = B1· v1 FB2 = B2· (v2 – v1 ) Fk1 = k1· x1 Fk2 = k2· x2 dt dt dx1 dx1 = v1 = v1 dt dt dv2 dv2 FI2 = m2· FI2= m2· dt dt dx2 dx2 = v2 = v2 dt dt dv3 dv3 FI3 = m3· FI3 = m3· dt dt dx3 dx3 = v3 = v3 dt dt Horizontal Sorting III 

  15. FI3= F(t)- FBa - FBb FI2= FBa - FBc - FB2 - Fk2 FI1= FBb + FB2 - FBd - Fk1 F(t) = FI3+ FBa + FBb FBa = FI2 + FBc + FB2 + Fk2 FBb + FB2 = FI1 + FBd + Fk1 dv1 dv1 dv2 dv3 FBa = B1· (v3 – v2 ) FBb = B1· (v3 – v1 ) FBc = B1· v2 FBd = B1· v1 FB2 = B2· (v2 – v1 ) Fk1 = k1· x1 Fk2 = k2· x2 = FI2 / m2 = FI3 / m3 = FI1 / m1 FI1 = m1· FBa = B1· (v3 – v2 ) FBb = B1· (v3 – v1 ) FBc = B1· v2 FBd = B1· v1 FB2 = B2· (v2 – v1 ) Fk1 = k1· x1 Fk2 = k2· x2 dt dt dt dt dx3 dx1 dx1 dx2 = v1 = v3 = v1 = v2 dt dt dt dt dv2 FI2 = m2· dt dx2 = v2 dt dv3 FI3 = m3· dt dx3 = v3 dt Horizontal Sorting IV 

  16. The sorting algorithm operates exactly in the same way as for electrical circuits. It is totally independent of the application domain. Sorting Algorithm

  17. Cellier, F.E. (1991), Continuous System Modeling, Springer-Verlag, New York, Chapter 4. References

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