## 2013年9月4日水曜日

### Contact Mechanics - Motion and forces at a point of contact, Surface traction

1 Motion and forces at point contact
1.4 Surface tractions

The forces and moments which we have just been discussing are transmitted across the contact interface by surface traction at the interface. The normal traction (pressure) is denoted by p and the tangential traction (due to friction) by q, shown acting positively on the lower surface in Fig. 1.2. While nothing can be said at this stage about the distribution of p and q over the area of contact S, for overall equilibrium:

(1.10)
(1.11)

With non-conforming contacts (including cylinders have parallel axes) the contact area lies approximately in the $x$-$y$ plane and slight warping is neglected, whence
(1.12)
and
(1.13)

When the bodies have closely conforming curved surface, as for example in a deep-groove ball-bearing, the contact area is warped appreciably out of the tangent plane and the expressions for Mx and My (1.12) have to be modified to be modified to include terms involving the shear tractions qx and qy.
Examples of the treatment of such problems are given later in § 8.5.

To illustrate the approach to contact kinematics and statics presented in this chapter, two examples from engineering practice will be considered briefly.

## 2013年9月3日火曜日

### Contact Mechanics - Motion and forces at a point of contact, Frame of reference, Forces transmitted at a point of contact

1 Motion and forces at point contact
1.3 Forces transmitted at a point of contact

The resultant force transmitted from one surface to another through a point of contact is resolved into a normal force P acting along the common normal, which generally must be compressive, and a tangental force Q in the tangent plane sustained by friction. The magnitude of Q must be less than or, on the limit, equal to the force of limiting friction, i.e.

where μ is the coefficient of limitting friction. Q is resolved components Qx and Qy parallel to axes Ox and Oy. In a purely sliding contact the tangential force reaches its limiting value in a direction opposed to the sliding velocity, from which:

The forces transmitted at a nominal point of contact has the effect of compressing deformable solids so that they make contact over an area of finite size. As a result it becomes possible for the contact to transmit a resultant moment in addition to a force (Fig. 1.2). The components of this moment Mx and My are defined as rolling moments. They provide the resistance to a rolling motion commonly called 'rolling friction' and in most practical problems are small enough to be ignored.
Fig. 1.2 Forces and moments acting on contact area S.

The third component Mz, acting about the common normal, arises from friction within the contact area and is referred to as the spin moment. When spin accompanies rolling the energy dissipated by the spin moment is combined with that dissipated by the rolling moments to make up the overall rolling resistance.

At this point it is appropriate to define free rolling ('inertia rolling' in the Russian literature). We shall use this term to describe a rolling motion in which spin is absent and where the tangential force Q at the contact point is zero. This is the condition of the unpowered and unbraked wheels of a vehicle if rolling resistance and bearing wheels which transmit sizable tangential forces at their points of contact with the road or rail.

## 2013年9月2日月曜日

### Contact Mechanics - Motion and forces at a point of contact, Relative motion of the surfaces - sliding, rolling and spin

1 Motion and forces at point contact
1.2 Relative motion of the surfaces - sliding, rolling and spin

The motion of a body at any instant of time may be defined by the linear velocity vector of an arbitrily chosen point of reference in the body together with the angular velocity vector of the body. If we now take reference points in each body coincident with the point of contact O at the given instant, body (1) has linear velocity V1 and granular velosity Ω1, and body (2) has linear velocity V2 and angular velocity Ω2. The frame of reference defined above moves with the linear velocity of the contact point  VO and rotates with angular velocity ΩO in order to maintain its orientation relative to the common normal and tangent plane at the contact point.

Within the frame of reference the two bodies have linear velocities at O:
(1.2)
and angular velocities:
(1.3)

We now consider the cartesian components of v1v2ω1 and ω2.  If contact is continuous, so the surfaces are neither separating nor overlapping, their velocity components along the comon normal must be equal, viz:
(1.4)

We now define sliding as the relative linear velocity between the two surfaces at O and denote it by Δv.
The sliding velocity has components:
(1.5)

Rolling is defined as a relative angular velocity between the two bodies about an axis lying in the tangent plane. The angular velocity of roll has components:
(1.6)
Finally spin motion is defined as a relative angular velocity about the common normal, viz.:
(1.7)
Any motion of contacting surfaces must satisfy the condition of continuous contact (1.4) and can be regarded as the combination of sliding, rolling and spin. For example, the wheels of a vehicle normally roll without slide or spin. When it turns a corner spin is introduced; if it skids with the wheels lockedm it slides without rolling.

## 2013年9月1日日曜日

### Contact Mechanics - Motion and forces at a point of contact, Frame of reference

1 Motion and forces at point contact
1.1 Frame of reference

Non-conforming surfaces brought into contact by a negligibly small force touch at a single point.
We take this point O as origin of rectangular coodinate axes Oxyz. The two bodies, lower and upper as shown in Fig. 1.1, are denoted by suffixes 1 and 2 respectively. The Oz axis is chosen to coincide with the common normal to the two surfaces at O. Thus the x-y plane is the tangent plane to the two surfaces, sometimes called the osculating plane. The directions of the axes Ox and Oy are chosen for convenience to coincide, where possible, with axes of symmetry of the surface profiles.

Fig. 1.1 Non-conforming surfaces in contact at O

Line contact, which arises when two cylindrical bodies are brought into contact with their axes parallel, appears to constitute a special case. Their profiles are non-confirming in the plane of cross-section, but they do confirm along a line of contact in the plane containing the axes of the cylinders.
Neverthless this important case is covered by the general treatment as follows: we choose the x-axis to lie in the plane of cross-section and the y-axis parallel to the axes of the cylinders.
The undeformed shapes of two surfaces are specified in this frame by the functions:
(1.1)