AMA GEMERY A3 implifications of matrices A3 olid ( ) has a plane of symmetry (,x,y) any point of co-ordinate + z, can be associated with a point of co-ordinate - z, hence: x y yzdm ( ) ( ) D E xzdm ( ) INA-LYN / CAN nd er Cycle/ Mechanics, nd semester 3
AMA GEMERY A3 solid ( ) has an axis of symmetry z he axis of symmetry can be viewed as the intersection of two planes of symmetry and the simplification above can be extended, i e, yxdm ( ) yzdm ( ) ( ) x y xydm xzdm ( ) ( ) and the matrix of inertia in becomes diagonal when expressed in ( R) (,x,y,z) INA-LYN / CAN nd er Cycle/ Mechanics, nd semester 4
AMA GEMERY A33 solid ( ) has an axis of revolution axis of revolution axis of symmetry + x and y play the same role A he matrix of inertia in reads, when (,z ) is the axis of revolution: [ I ] o,s A I (,) B A, i A C / R A I (,) B B, i INA-LYN / CAN nd er Cycle/ Mechanics, nd semester 5
AMA GEMERY IMRAN ALICAIN **** Being ( ), a solid of revolution, show that the matrix of inertia in is the same in ( R ) and ( *) R,,,!"²+!"² ( ) z, z * θ x x * θ y * y INA-LYN / CAN nd er Cycle/ Mechanics, nd semester 6
AMA GEMERY A4 Change of basis,%&,%',%' *,%& ( Note that ( is orthonormal: * + (,,,-,,- /,,,,-,,- /,,,, / 3, 3 4, 4 3, 4, 4 -, 3 Remark: It is possible to find a basis R* where the extra-diagonal terms (product of inertia) are nil, it is the principal basis of inertia (A34) INA-LYN / CAN nd er Cycle/ Mechanics, nd semester 7
AMA GEMERY A5 Examples of Inertia Matrices Fill the matrices using Binet s notation and according to the symmetry properties of each solid z y, x,5 x z,,5 y INA-LYN / CAN nd er Cycle/ Mechanics, nd semester 8
AMA GEMERY,,5,,5 INA-LYN / CAN nd er Cycle/ Mechanics, nd semester 9
B KINEIC B KINEIC urpose: Kinetics relates Kinematics to Mass Geometry B Kinetic or Momentum wrench: B Definition: From the velocity field, let us define the following vector field (elemental vectors), dk ( ) ( ) dm where dm is the elemental mass associated with Consider a solid ( ) so that the following global quantities can be defined INA-LYN / CAN nd er Cycle/ Mechanics, nd semester
B KINEIC a) um: σ dk ( ) ( ) ( ) ( ) dm which is known as the linear momentum of solid ( ) b) Moment at C C C dk dm ( ) ( ) C ( ) ( ) which is the angular momentum at C INA-LYN / CAN nd er Cycle/ Mechanics, nd semester
B KINEIC c) Momentum wrench { K }: he moment at one point L and at another point M are related by the equation: proof: L M + LM σ ( L) ( ) ( L) LM ( ) ( ) L ( ) ( LM + M) ( ) ( ) dm + ( L) LM σ + ( M ) ( ) M ( ) dm dm LM ( ) ( ) dm + ( M ) his expression corresponds to the shifting (change of point) formula for the moment field of a wrench INA-LYN / CAN nd er Cycle/ Mechanics, nd semester
B KINEIC B Expression of the linear momentum: By definition σ ( ) ( ) dm hence by introducing the velocity at the centre of mass G: σ σ ( G) + G + G d G d d dm dm dm + M M σ ( G) ( ) G dm INA-LYN / CAN nd er Cycle/ Mechanics, nd semester 3
B KINEIC B3 Expression of the angular momentum: B3 Angular momentum of a solid in its centre of inertia: ( G) ( ) G ( G) ( G) ( ) G ( G) ( ) ( ) ( ) ( G) ( G) G dm + G dm + ( ) ( ) G ( G) + L [ I ] ( G, ) G G + G G G, G dm dm dm ( G ) [ I ] G, INA-LYN / CAN nd er Cycle/ Mechanics, nd semester 4
B KINEIC B3 Angular momentum of a solid in a point C By using the change of point formula, one gets: G, ( C ) [ I ] { } + CG M ( G ) C G + CG M ( G) KENIG' formula INA-LYN / CAN nd er Cycle/ Mechanics, nd semester 5
B KINEIC B33 Angular momentum in a point s fixed in a frame ( R ): tarting from the general definition, s s dm s ( ) ( ) one obtains: s s s ( s) ( ) ( ) ( s) ( ) ( s) + Hence: ( ) + ( ) ( ) ( ) dm + dm dm s { } ( ) [ I ], dm INA-LYN / CAN nd er Cycle/ Mechanics, nd semester 6
B KINEIC / : B34 Angular momentum relative to the instantaneous axis of rotation s n ( I ) n I ( ) Let s calculate ( ) dm n ( ) s / I ( ) ( I ) n I ( I ) dm n + I ( I) dm n ( ) ( ) I n L n +, ( ) I t ( I ) n n [ I ], { } t ( I ) n n [ I ] n w, as, w n nn I nn ω INA-LYN / CAN nd er Cycle/ Mechanics, nd semester 7
B KINEIC B Dynamic wrench: B Definition: From the acceleration field and the mass of a material system, two vectors can be set up: a) um: b) Moment in C Σ δ ( ) J dm ( ) C dm C J ( ) ( ) INA-LYN / CAN nd er Cycle/ Mechanics, nd semester 8
B KINEIC c)change of point formula: δ δ δ δ ( L) L J ( ) ( ) ( L) LM J ( ) ( ) dm dm + ( L) LM Σ + δ ( M ) ( L) δ ( M ) + LM Σ ( ) ( LM + M) J ( ) ( ) M J ( ) dm dm δ ( L) δ ( M ) + LM Σ d) consequence: δ L G + LG Σ δ INA-LYN / CAN nd er Cycle/ Mechanics, nd semester 9
B KINEIC B Expression of the sum: Σ Σ J ( ) ( ) ( ) M J ( G) dm d dm d ( ) d ( ) dm σ M ( G) d Σ M J ( G) INA-LYN / CAN nd er Cycle/ Mechanics, nd semester 3
B KINEIC B3 Expression of the moment: d d d ( C) C ( ) dm ( C ( ) ) dm ( ) ( ) d d d C C dm C ( ) dm d d d + d C C dm + C J dm ( ) ( ) ( ) + δ ( ) d C C dm C ( ) C δ C C dm d δ ( C) ( C) + ( C) σ δ d ( C) ( C) + ( C) σ INA-LYN / CAN nd er Cycle/ Mechanics, nd semester 3
B KINEIC implifications: a) if C G b) or if C is fixed in ( R ) c) or if ( C) // ( G) (translation for example) d then δ ( C ) ( C ) In these three cases, the dynamic wrench { } wrench { K } D is the time derivative of the momentum INA-LYN / CAN nd er Cycle/ Mechanics, nd semester 3
B KINEIC B3 Kinetic Energy: B3 Definition for a material point of mass dm, d ( ) dm by extension, it comes for a solid ( ), ( ) dm Caution: Kinetic energy is a CALAR but it depends on a frame of reference INA-LYN / CAN nd er Cycle/ Mechanics, nd semester 33
B KINEIC INA-LYN / CAN nd er Cycle/ Mechanics, nd semester 34 B3 Expression for a solid tarting from the general definition, kinetic energy reads: [ ] dm G G + [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ],,,,, G L G M dm G G G M G dm G G M G dm G G M G dm G G M dm G dm G + + + + + + [ ] [ ], G t I G M +
B KINEIC INA-LYN / CAN nd er Cycle/ Mechanics, nd semester 35 B33 Expression for a solid with one point s fixed in the frame of reference R : imilarly, it comes: [ ] dm s s +, with s [ ] [ ] [ ] [ ],,,,, L dm dm dm dm dm [ ], t I if s fixed in the frame of reference R
B KINEIC B34 imple applications Rotation of a solid around a fixed axis ranslation of a solid along a fixed axis { } E F G ψɺ y ( s ), s { } E F G ( s ) yɺ y, INA-LYN / CAN nd er Cycle/ Mechanics, nd semester 36
B KINEIC B35 Examples B35 Gyroscope - Using Binet s notations, give the simplest form of [I],, [I], Calculate the dynamic moments in with respect to (R ) for,, U 3 Determine the total kinetic energy with respect to (R ) INA-LYN / CAN nd er Cycle/ Mechanics, nd semester 37
B KINEIC B35 Euler's pendulum: x z z z θ l G x, mass of solid : m mass of solid : m - Determine the momentum wrench with respect to (R ) for, in - Calculate the total kinetic energy with respect to (R ) INA-LYN / CAN nd er Cycle/ Mechanics, nd semester 38
B KINEIC B353 Gyroscopic pendulum and are massless, 3 mass m 3 Calculate for 3 in : - the angular momentum, - the dynamic moment, - the kinetic energy INA-LYN / CAN nd er Cycle/ Mechanics, nd semester 39