2. Chain Rule (to introduce time derivatives into path (x-y or r-q) equations):
Curvilinear
Motion: x-y Coordinates
1.
Dot Notation (for time derivatives)
Often
in engineering a single and double dot shorthand notation is used to signify
(usually time) derivatives.
Because velocities and accelerations are time derivatives, we will often
use dot notation to simplify writing derivatives.
2.
Chain Rule (to introduce time derivatives into path (x-y or r-q)
equations):
The
chain rule, an important topic in calculus, is also important in engineering
dynamics. It
is used frequently to introduce time derivatives into
equations written in position coordinates (for example, x-y, r-q,
etc.) only.
|
Example
2-1: Use
of the Chain Rule |
|
|
We
wish to take the time derivatives of the “path” equation: |
|
|
This
is of the form: |
|
|
The
chain rule states that the time derivatives are found by taking the y
derivative of f(y) and multiplying it by dy/dt, and by
taking the x derivative of g(x) and multiplying it by dx/dt |
|
|
Applying
this to our example, using the dot notation, we get: |
|
|
To
take another derivative in time, note that we now have a product
the
product rule [ d(uv) = (du)v + u(dv) ].
This yields: |
|
|
Example
2-2: Use
of the Chain Rule |
|
|
We
wish to take the time derivatives of the hyperbolic “path” equation
consisting of the product of x and y: |
|
|
Using
the product rule and the chain rule yields: |
|
|
Using
the product and chain rule again yields: |
|
|
This
simplifies to: |
|
