Introduction to Machinery Principles

Rotational motion, Newton's law, and Power relationships

One dimensional linear velocity along a line is defined as the rate of change of the displacement along the line(r) with respect to time.

线速度：

Similarly, angular velocity ω is defined as the rate of change of the angular θ displacement with respect to time.

角速度：

If the units of angular position are radians, then angular velocity is measured in radians per second.

The following symbols are used to describe angular velocity:

 

ω_m        angular velocity expressed in radians per second

f_m        angular velocity expressed in revolutions per second

n_m        angular velocity expressed in revolutions per minute

 

These measures of shaft speed are related to each other by the following equations:

 

One-dimensional liner acceleration is defined by the equation

线加速度：

Angular acceleration is the rate of change in angular velocity with respect to time, defined by

角加速度：

 

The torque on an object is defined as the product of the force applied to the object and the smallest distance between the line of action of the force and the object's axis of rotation. If r is a vector pointing from the axis of rotation to the point of application of the force, and if F is the applied force, then the torque can be described as

转矩：    τ = (force applied) (perpendicular distance)

= (F) (r sin θ)

=r F sin θ

Where θ is the angle between the vector r and the vector F.The units of torque are Newton-meters (N·M) in SI.

 

Newton's law for object moving along a straight line describes the relationship between the force applied to an object and its resulting acceleration. This relationship is given by the equation

牛顿定律：

Where

    F = net force applied to an object

    m = mass of the object

    a = resulting acceleration

In SI units, force is measured in new-tons, mass in kilograms, and acceleration in meters per second squared (m/s2).

A similar equation describes the relationship between the torque applied to an object and its resulting angular acceleration. This relationship, called Newton's law of rotation, is given by the equation

牛顿旋转定律：τ = J α

Where τ is the net applied torque in Newton-meters and α is the resulting angular acceleration in radians per second squared (rad/s²). The term J serves the same purpose as object's mass in linear motion. It called the moment of inertia (转动惯量) of the object and is measured in kilogram-meters squared (kg·m²) .

 

For linear motion, work is defined as the application of a force through a distance. In equation form,

功：

Where it is assumed that the force is collinear with the direction of motion.

同一直线方向时做功：W = F r

The units of work are joules (焦耳) in SI.

For rotational motion, work is the application of a torque through an angle. Here the equation for work is

If the torque is constant,

W = τ θ

 

Power is the rate of doing work, or the increase in work per unit time. The equation for work is

功率：

It is usually measured in joules per second (watts瓦特),but also can be measured in horsepower(马力).

If force is constant and collinear with the direction of motion, power is given by

功率=力*线速度

Similarly, assuming constant torque, power in rotational motion is given by

功率=转矩*角速度

 

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20140606