X-direction angle

Figure 11-2 Two illustrations of the x-direction angle (a) and y-direction angle (/3) for a two-dimensional coordinate system.

Figure 11-3 The x-direction angle a and j-direction angle /3 for a line segment, where A is (3, 5) and B is (2, 7) (see example in text).

This well-known expression is also equivalent to the tangent of the x direction angle for the line segment defined by the two points on the line. Thus the slope of the line given in Figure 11-4 is tan(120°) = -1.74. 

Figure 11-4 Illustration of the slope of a line given an x-direction angle of 120°.

Therefore, we can project the AB vector in 3-D space onto 2-D space by using a projection onto the (x, z) plane, resulting in a point on a vector (on the 2-D (x, z) plane) the vector being 6.33 units in length and having an X-direction angle equal to 71.57° (as in Figure 13-4). 

Strength in X direction—angle between fibers and stress load ... 

Now, in order to determine the stiffness in the global (jc, y) directions for a lamina in which the fibres are aligned at an angle 6 to the global x direction, it is necessary to go through the following three steps ... 

Consider an angle-ply laminate composed of orthotropic laminae that are symmetrically arranged about the middle surface as shown in Figure 4-48. Because of the symmetry of both material properties and geometry, there is no coupling between bending and extension. That is, the laminate in Figure 4-48 can be subjected to and will only extend in the x-direction and contract in the y- and z-directions, but will not bend. 

It should be noted in this respect that a torque is obtained as a product of a force in the X-direction and an arm of length Ly, say, in a direction at right-angles to the 7-direction. Thus, the dimensions of torque are MLxLyT 2, which distinguish it from energy. 

In the present discussion only the problem of steady flow will be considered in which the time average velocity in the main stream direction X is constant and equal to ux. in laminar flow, the instantaneous velocity at any point then has a steady value of ux and does not fluctuate. In turbulent flow the instantaneous velocity at a point will vary about the mean value of ux. It is convenient to consider the components of the eddy velocities in two directions—one along the main stream direction X and the other at right angles to the stream flow Y. Since the net flow in the X-direction is steady, the instantaneous velocity w, may be imagined as being made up of a steady velocity ux and a fluctuating velocity ut, . so that ... 

For the flow of liquid of depth s down a plane surface of width w inclined at an angle 0 to the horizontal, as shown in Figure 3.19, a force balance in the X-direction (parallel to the surface) may be written. In an element of length dr the gravitational force acting on that part of the liquid which is at a distance greater than y from the surface is given by ... 

Consider the fluid to be flowing in a direction parallel to the surface (X-direction) and for momentum and heat transfer to be taking place in a direction at right angles to the surface (T-direction positive away from surface). 

Solid contacts are incommensurate in most cases, except for two crystals with the same lattice constant in perfect alignment. That is to say, a commensurate contact will become incommensurate if one of the objects is turned by a certain angle. This is illustrated in Fig. 30, where open and solid circles represent the top-layer atoms at the upper and lower solids, respectively. The left sector shows two surfaces in commensurate contact while the right one shows the same solids in contact but with the upper surface turned by 90 degrees. Since the lattice period on the two surfaces, when measured in the x direction, are 5 3 A and 5 A, respectively, which gives a ratio of irrational value, the contact becomes incommensurate. 

Exact Treatment for the Freely Jointed Chain (or Equivalent Chain).4 >5— Consider one of the bonds of a freely jointed chain acted upon by a tensile force r in the x direction. Letting xpi represent the angle between the bond and the o -axis, its component on the x-axis is Xi = l cos pi. The orientation energy of the bond is —rXi, and the probability that its x component has a value between Xi and Xi- -dxi therefore is proportional to... 

The second important spin-angular operation is the 90° rotation where the polarization is transformed from the z to the x direction or vice versa. A Mezei coil in the x,z plane is adjusted such that the resultant field points exactly in the direction of the bisection of the angle between x and z. A 180° rotation around this axis transforms the z component of polarization to the x direction. At the same time, the sign of the y component is inverted (Fig. lc). 

Because the third dimension is represented by the z axis, we calculate the z-direction angle on the (x, z) plane as y ... 

Considering the motion of a particle of mass m in the earth s gravitational field, at some time t the particle will be moving at an angle a to the horizontal with a velocity u as shown in Figure 3.7. The velocity u may then be resolved into two components, x and y, in the horizontal and vertical directions, i and x will be taken to denote the first and second derivatives of the displacement x in the X-direction with respect to time, and v and y the corresponding derivatives of v. 

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