英语翻译geometry of shallow curvesto establish the geometric rel

英语翻译
geometry of shallow curves
to establish the geometric relationships required to derive the differential equation of the elastic curve,we will consider the deformations of the cantilever beam in Fugure9.1a.
the deflected shape is represented in Figure9.1b by the displaced position of the longitudinal axis(also called the elastic curve).as reference axes,we estabilsh an x-y coordinate system whose origin is located at the fixed end.for clarity,vertical distances in this figure are greatly exaggerated.slopes,for example,are typically very small--on the order of a few tenths of a degree.if we were to show the deflected shape to scale,it would appear as a straight line.
to establish the geometry of a curved element,we will consider an infinitesimal element of length ds located a distance x from the fixed end.as shown in Figure9.1c,we denote the radius of the curved segment by p.angle between these tangents is denoted by dФ.since the tangents to the curve are perpendicular to the radii at points A and B,it follows that the angle between the radii is also dФ.the slope of the curve at point A equals
if the angles are small(tanФ≈Ф radians),the slope can be written
from the geometry of the triangular segment ABO in figure9.1c,we can write
dividing each side of the equation above by ds and rearranging terms give
where dФ/ds,representing the change in slope per unit length of distance along the curve,is called the curvature and denoted by the symbol Ф.since slopes are small in actual beams,ds≈dx,and we can express the curvature in equation9.3 as

浅几何曲线
建立所需的几何关系,求出微分方程的弹性曲线,我们将考虑变形的悬臂梁在Fugure9.1a .
在偏转形状是代表Figure9.1b的流离失所者的立场纵轴（也称为弹性曲线） .作为参考轴线,我们estabilsh的XY坐标系统,其原产地是位于固定结束.为明确起见,垂直距离在这个数字是大大夸大了.斜坡,例如,通常是非常小的-关于秩序的零点几秒的程度.如果我们要显示偏转形成规模,它将显示为一条直线.
建立的几何形状弯曲的元素,我们会考虑的一个微元长度双链位于距离x的固定结束.所示Figure9.1c ,我们指的半径曲线段之间的p.angle这些切线是指由dФ .因为切线的曲线是垂直的半径点A和B ,所以之间的夹角半径也dФ .斜坡曲线在A点等于
如果角度小（ tanФ ≈ Ф弧度） ,坡度可以这样写：
从几何三角部分ABO血型在figure9.1c ,我们可以写
除以每一方的上述方程DS和重新安排方面给予
在dФ /双链,代表坡度变化单位长度的距离沿曲线,被称为曲率和命名的象征Ф .自从小斜坡在实际光束,双链≈型,我们可以表达的曲率作为在equation9.3