Utente:Grasso Luigi/sandbox2/Centro di rotazione istantaneo

Fig 1: Centro istantaneo P di figura piama in movimento

Il centro di rotazione istantaneo, anche detto centro di velocità istantaneo, oppure centro istantaneo o centro all'istante, è il punto fissato a un corpo sottoposto a movimento planare che ha velocità zero in un particolare istante di tempo. A questo istante, i vettori di velocità delle traiettorie di altri punti nel corpo generano un campo circolare attorno a questo punto che è identico a ciò che è generato da una rotazione pura.

Il moto planare di un corpo rigido spesso viene descritto da una figura piana in movimento in un piano bidimensionale. Il centro istantaneo è il punto nel piano in movimento attorno al quale ruotano tutti gli altri punti in un determinato istante di tempo. Il movimento continuo di un piano ha un centro istantaneo per qualsiasi valore del parametro t. Questo genera una curva chiamata lo spostamento del centroide. I punti nel piano fisso corrispondenti a questi centri istantanei formano il centroide fisso.

La generalizzazione di questo concetto allo spazio tridimensionale è quello di una torsione attorno a una vite. La vite ha un asse che è una linea nello spazio 3D (non necessariamente attraverso l'origine), e la vite ha anche un passo finito (una traslazione fissa lungo il suo asse corrispondente ad una rotazione attorno all'asse della vite).

Polo di uno spostamento planare

Sketch 2: Pole of a planar displacement

The instant center can be considered the limiting case of the pole of a planar displacement.

The planar displacement of a body from position 1 to position 2 is defined by the combination of a planar rotation and planar translation. For any planar displacement there is a point in the moving body that is in the same place before and after the displacement. This point is the pole of the planar displacement, and the displacement can be viewed as a rotation around this pole.

Construction for the pole of a planar displacement: First, select two points A and B in the moving body and locate the corresponding points in the two positions; see the illustration. Construct the perpendicular bisectors to the two segments A¹A² and B¹B². The intersection P of these two bisectors is the pole of the planar displacement. Notice that A¹ and A² lie on a circle around P. This is true for the corresponding positions of every point in the body.

If the two positions of a body are separated by an instant of time in a planar movement, then the pole of a displacement becomes the instant center. In this case, the segments constructed between the instantaneous positions of the points A and B become the velocity vectors V_A and V_B. The lines perpendicular to these velocity vectors intersect in the instant center.

The algebraic construction of the Cartesian coordinates $\left(P_{x},P_{y}\right)$ can be arranged as follows: The midpoint between $A^{1}$ and $A^{2}$ has the Cartesian coordinates

$A_{x}^{m}={\frac {1}{2}}\left(A_{x}^{1}+A_{x}^{2}\right);\quad A_{y}^{m}={\frac {1}{2}}\left(A_{y}^{1}+A_{y}^{2}\right),$

and the midpoint between $B^{1}$ and $B^{2}$ has the Cartesian coordinates

$B_{x}^{m}={\frac {1}{2}}\left(B_{x}^{1}+B_{x}^{2}\right);\quad B_{y}^{m}={\frac {1}{2}}\left(B_{y}^{1}+B_{y}^{2}\right).$

The two angles from $A^{1}$ to $A^{2}$ and from $B^{1}$ to $B^{2}$ measured counter-clockwise relative to the horizontal are determined by

$\tan \tau _{A}={\frac {A_{y}^{2}-A_{y}^{1}}{A_{x}^{2}-A_{x}^{1}}},\quad \tan \tau _{B}={\frac {B_{y}^{2}-B_{y}^{1}}{B_{x}^{2}-B_{x}^{1}}},$

taking the correct branches of the tangent. Let the center $\left(P_{x},P_{y}\right)$ of the rotation have distances $d_{A}$ and $d_{B}$ to the two midpoints. Assuming clockwise rotation (otherwise switch the sign of $\pi /2$):

${\begin{aligned}P_{x}&=A_{x}^{m}+d_{A}\cos \left(\tau _{A}-{\frac {\pi }{2}}\right);\\P_{x}&=B_{x}^{m}+d_{B}\cos \left(\tau _{B}-{\frac {\pi }{2}}\right);\\P_{y}&=A_{y}^{m}+d_{A}\sin \left(\tau _{A}-{\frac {\pi }{2}}\right);\\P_{y}&=B_{y}^{m}+d_{B}\sin \left(\tau _{B}-{\frac {\pi }{2}}\right).\end{aligned}}$

Rewrite this as a $4\times 4$ inhomogeneous system of linear equations with 4 unknowns (the two distances $d$ and the two coordinates $P$ of the center):

${\begin{pmatrix}1&0&-\sin \tau _{A}&0\\1&0&0&-\sin \tau _{B}\\0&1&\cos \tau _{A}&0\\0&1&0&\cos \tau _{B}\end{pmatrix}}\cdot {\begin{pmatrix}P_{x}\\P_{y}\\d_{A}\\d_{B}\end{pmatrix}}={\begin{pmatrix}A_{x}^{m}\\B_{x}^{m}\\A_{y}^{m}\\B_{y}^{m}\end{pmatrix}}.$

The coordinates of the center of the rotation are the first two components of the solution vector

${\begin{pmatrix}P_{x}\\P_{y}\\d_{A}\\d_{B}\end{pmatrix}}={\frac {1}{\sin \left(\tau _{A}-\tau _{B}\right)}}{\begin{pmatrix}\left(B_{y}^{m}-A_{y}^{m}\right)\sin \tau _{A}\sin \tau _{B}+B_{x}^{m}\sin \tau _{A}\cos \tau _{B}-A_{x}^{m}\cos \tau _{A}\sin \tau _{B}\\\left(A_{x}^{m}-B_{x}^{m}\right)\cos \tau _{A}\cos \tau _{B}+A_{y}^{m}\sin \tau _{A}\cos \tau _{B}-B_{y}^{m}\cos \tau _{A}\sin \tau _{B}\\\left(B_{x}^{m}-A_{x}^{m}\right)\cos \tau _{B}+\left(B_{y}^{m}-A_{y}^{m}\right)\sin \tau _{B}\\\left(B_{x}^{m}-A_{x}^{m}\right)\cos \tau _{A}+\left(B_{y}^{m}-A_{y}^{m}\right)\sin \tau _{A}\\\end{pmatrix}}.$

Traslazione pura

If the displacement between two positions is a pure translation, then the perpendicular bisectors of the segments A¹B¹ and A²B² form parallel lines. These lines are considered to intersect at a point on the line at infinity, thus the pole of this planar displacement is said to "lie at infinity" in the direction of the perpendicular bisectors.

In the limit, pure translation becomes planar movement with point velocity vectors that are parallel. In this case, the instant center is said to lie at infinity in the direction perpendicular to the velocity vectors.

Centro istantaneo di una ruota che rotola senza scivolare

Fig 3: Rotella.

Consider the planar movement of a circular wheel rolling without slipping on a linear road; see sketch 3. The wheel rotates around its axis M, which translates in a direction parallel to the road. The point of contact P of the wheel with road does not slip, which means the point P has zero velocity with respect to the road. Thus, at the instant the point P on the wheel comes in contact with the road it becomes an instant center.

The set of points of the moving wheel that become instant centers is the circle itself, which defines the moving centrode. The points in the fixed plane that correspond to these instant centers is the line of the road, which defines the fixed centrode.

The velocity vector of a point A in the wheel is perpendicular to the segment AP and is proportional to the length of this segment. In particular, the velocities of points in the wheel are determined by the angular velocity of the wheel in rotation around P. The velocity vectors of a number of points are illustrated in sketch 3.

The further a point in the wheel is from the instant center P, the proportionally larger its speed. Therefore, the point at the top of the wheel moves in the same direction as the center M of the wheel, but twice as fast, since it is twice the distance away from P. All points that are a distance equal to the radius of the wheel 'r' from point P move at the same speed as the point M but in different directions. This is shown for a point on the wheel that has the same speed as M but moves in the direction tangent to the circle around P.

Centro di rotazione relativo per due corpi planari in contatto

Fig 4: Esempio di centro rotazione relativo di due corpi in contatto in C, ruotanti in A e D.

If two planar rigid bodies are in contact, and each body has its own distinct center of rotation, then the relative center of rotation between the bodies has to lie somewhere on the line connecting the two centers. As a result, since pure rolling can only exist when the center of rotation is at the point of contact (as seen above with the wheel on the road), it is only when the point of contact goes through the line connecting the two rotation centers that pure rolling can be achieved.

This is known in involute gear design as the pitch point, where there is no relative sliding between the gears. In fact, the gearing ratio between the two rotating parts is found by the ratio of the two distances to the relative center. Nell'esempio di Fig.4 osserviamo due corpi in contatto in C, uno ruota attorno A e l'altro attorno B deve avere un centro di rotazione relativo da qualche parte lungo la linea "AB". Essendo che i due corpi non possono compenetrarsi il centro di rotazione relativo deve stare lungo la direzione normale rispetto moto del contatto e attraverso la C . L'unica soluzione è che stia nel punto D. Il rapporto the gearing ratio is $\gamma ={\frac {AD}{BD}}$

Centro di rotazione instantaneo e meccanismi

Sketch 1 above shows a four-bar linkage where a number of instant centers of rotation are illustrated. The rigid body noted by the letters BAC is connected with links P₁-A and P₂-B to a base or frame.

The three moving parts of this mechanism (the base is not moving) are: link P₁-A, link P₂-B, and body BAC. For each of these three parts an instant center of rotation may be determined.

Considering first link P₁-A: all points on this link, including point A, rotate around point P₁. Since P₁ is the only point not moving in the given plane it may be called the instant center of rotation for this link. Point A, at distance P₁-A from P₁, moves in a circular motion in a direction perpendicular to the link P₁-A, as indicated by vector V_A.

The same applies to link P₂-B: point P₂ is the instant center of rotation for this link and point B moves in the direction as indicated by vector V_B.

For determining the instant center of rotation of the third element of the linkage, the body BAC, the two points A and B are used because its moving characteristics are known, as derived from the information about the links P₁-A and P₂-B.

The direction of speed of point A is indicated by vector V_A. Its instant center of rotation must be perpendicular to this vector (as V_A is tangentially located on the circumference of a circle). The only line that fills the requirement is a line colinear with link P₁-A. Somewhere on this line there is a point P, the instant center of rotation for the body BAC.

What applies to point A also applies to point B, therefore this instant center of rotation P is located on a line perpendicular to vector V_B, a line colinear with link P₂-B. Therefore, the instant center of rotation P of body BAC is the point where the lines through P₁-A and P₂-B cross.

Since this instant center of rotation P is the center for all points on the body BAC for any random point, say point C, the speed and direction of movement may be determined: connect P to C. The direction of movement of point C is perpendicular to this connection. The speed is proportional to the distance to point P.

Continuing this approach with the two links P₁-A and P₂-B rotating around their own instant centers of rotation the centrode for instant center of rotation P may be determined. From this the path of movement for C or any other point on body BAC may be determined.

Esempi di applicazione

In biomechanical research the instant center of rotation is observed for the functioning of the joints in the upper and lower extremities. For example, in analysing the knee, ankle, or shoulder joints. Such knowledge assists in developing artificial joints and prosthesis, such as elbow or finger joints.

Study of the joints of horses: "...velocity vectors determined from the instant centers of rotation indicated that the joint surfaces slide on each other.".

Studies on turning a vessel moving through water.

The braking characteristics of a car may be improved by varying the design of a brake pedal mechanism.

Designing the suspension of a bicycle, or of a car.

In the case of the coupler link in a four-bar linkage, such as a double wishbone suspension in front view, the perpendiculars to the velocity lie along the links joining the grounded link to the coupler link. This construction is used to establish the kinematic Roll centre of the suspension.

Voci correlate

• Centroide
• Centro di equilibrio
• Asse elicoidale
• Corpo rigido
• Rotazione attorno asse fisso
• Velocità angolare