Frame Of Reference Definition

Frame of reference is a system which is used as measuring rods by an observer for measuring the surrounding space provided by the coordinates. The coordinates are measured using fixed points, lines, or planes. The frame of reference is very much essential in modeling rotation. Frame of reference definition also states as a set of coordinate axes in which position or movements are specified with reference to the physical laws which are mathematically stated.


Frame of reference definition is almost used in geometrical rotations. Here we consider two terms of frame of reference during modeling the rotations namely global frame-of-reference and local frame-of-reference.

Global frame-of-reference

Global frame-of-reference definition states that rotations in terms of global coordinates, here we use 3 global co-ordinates,



z=toward viewer.

The matrix representations of the individual rotations is in the following way,

[resulting transformation] = [second transformation] * [first rotation]

Multiply vector Vin using matrix [A] to produce an intermediate vector Vmid, next we multiply this using matrix [B]. Then substitute [A]Vin in the place of Vmid, so that we can get overall matrix = [B][A].

Local frame-of-reference

Local frame-of-references definition states that rotations in terms of local coordinates, other definition states that the coordinate system that rotates along with the object.

The matrix representations of the individual rotations is in the following way,

[resulting transformation] = [second transformation] * [first rotation]

This is the easiest way to work, in this situation, instead of working with object position transforms. Result of consecutive transformations is obtained by multiplying the individual transformation.


Example 1:

Rotation of an aero plane using global frame of reference

1. Initial Orientation is carried as usual.

2. First rotate+90 degrees about y axis.

3. Second rotate +90 degrees about z axis.

4. Third rotate-90 degrees about y axis. (Resemble z axis)

Example 2:

Consider two peoples facing each other standing on either side of a North-South street.

1. A car drives past them moving towards south.

2. The person facing east seems that the car was moving towards right.

3. For the person facing west seems the car was moving towards left.

4. This is due to two people uses two different frames of reference.

The transfer function is a ratio of the output Laplace Transform to the input Laplace Transform assuming zero initial conditions. It is used in dynamic or control systems
The procedure to find the transfer function of linear differential equation from input to the output is to use the Laplace Transforms of both sides assuming zero condition

Find the frequency transfer function

How will we find the transfer function:
In most cases the equation will be linear, consisting of a variable and it derivatives. to find the Laplace Transform most useful transfer function in differentiation theorem. Several properties are shown below:

Time Domain Frequency Domain

Linearity f(t) + g(t) F(S) +G(S)
Function x(t)
1st Derivative x'(t)
2nd Derivative x”(t)
nth Derivative xn(t)

We will now enter this transfer function into MATLAB. Because MATLAB cannot change symbolic variables, we will now assign numerical values to each variable.
m = 20,000kg
b = 500kg/s
>> m = 20000;
>> b = 500;
if we want to enter the transfer function MATLAB, we must separate the numerator and denominator, this example are denoted by ‘num’ and ‘den’ respectively. The formats for either matrix is to enter the coefficients of sn.
>> num = [ 1 ];
>> den = [ m b ];
>> first_tf = tf(num, den)
This will assign first transfer function as the name of the transfer function as well as yield the following output:
Transfer function:
20000 s + 500

frequency transfer function

The transfer function is used in the single-input single-output filters for instance. mostly used in signal processing, communication & control theory. The term is often used to refer to linear, time-invariant systems (LTI), Most real systems have non-linear input/output characteristics.

continuous time input signal x(t) & output y(t), the transfer function is the linear mapping of the Laplace transform of the input, X(s), to the output Y(s):transfer function:

Y(s) = H(s);X(s)


H(s) = frac{Y(s)}{X(s)} = frac{ mathcal{L}left{y(t)right} }{ mathcal{L}left{x(t)right} }
where H(s) is the transfer function of the LTI system.

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