In two previous tutorials in this series we obtained values for velocities and accelerations in the crank mechanism by the following methods:

- from general expressions obtained by differentiating linear displacement of slider point B with respect to time (differentiating once for velocity and twice for acceleration).
- from velocity diagrams, velocity pole diagrams and acceleration diagrams, by scaling directly or by calculation from the geometry.

A disadvantage of velocity and acceleration diagrams is that each "snapshot" of the mechanism at a specific crank angle requires a unique diagram. By using vectors we can develop general equations to overcome this limitation.

To recap from a previous tutorial we show below how relative velocities (and accelerations) are expressed as vectors. This relationship is the basis of the vector equations that follow.

Note that in this tutorial we change the
convention previously used to designate quantities. Quantities
denoted by a single subscript are assumed to be relative to
"ground" frame of reference point O. Thus in this example v_{a/o} is
designated v_{a}. Relative quantities are denoted as before
with a dual subscript, for example v_{b/a} is the velocity
of point B relative to point A.

We now develop vector equations for velocity and acceleration for a generic crank mechanism shown below. In both cases the angular velocity ω of the crank arm is constant.

The diagram below shows the velocity vectors with known
directions. If derivation of these vectors is unclear refer to the
previous tutorial. Note the
directions chosen for unit
vectors* i* and

We can express* v*_{A},
*v*_{B}
and *v*_{B/A} in terms of unit vectors ** i**
and

From the above we now develop equations for scalar values of velocities **
v**

We now apply these equations to the crank mechanism with parameters stated below (we used this example in previous tutorials).

||v_{A}|| = ω (OA) = (2π)(1) =
2π m/s

n = AB/OA = 3

The plot below shows ||v_{B}||
against crank angle θ.

It can be shown that the expression for v_{B}
obtained above is equivalent to the one
obtained in a previous tutorial by differentiation of displacement
of point B with respect to time,
viz.

The plot below shows the angular velocity ω_{AB} of
the connecting rod AB against crank angle θ from the relation ω_{AB}
= ||v_{B/A}||/(AB)
with ||v_{B/A}||
computed from the above expression.

This plot shows the sign of ω_{AB} changes at θ = 90° and θ
= 270° during one
revolution of the crank arm. The sign convention follows the
right-hand rule for vector cross-product** v_{t}
**=

The diagram below shows the acceleration vectors with known
directions. Note that *a*_{B/A}
has two components, radial component *a*_{rB/A}
directed towards centre of rotation A and tangential component
*a*_{tB/A}. If derivation of these vectors is unclear refer to the
previous tutorial. Note the
directions chosen for unit
vectors* i* and

To determine direction of vector ** a_{tB/A}**
we examine the plot for ω

However, we cannot assign the direction of *a*_{B
}with certainty. As this is the only
acceleration vector in the diagram with its direction undetermined
we assign an arbitrary direction of positive unit vector i and allow
the equations to determine direction as seen below.

It should be noted that the vector diagram above can be constructed for any crank angle θ crank cycle with corresponding lines of action and directions of vectors.

We can express* a*_{A},
*a*_{B}
and *a*_{B/A} in terms of unit vectors ** i**
and

It can be shown that the expression for ||a_{B }|| above is equivalent to one
obtained in a previous tutorial by differentiation of
velocity v_{B},
viz.

We compute ||a_{B} || and ||a_{tB/A }|| for
the example mechanism below expressing sin φ and cos φ in
terms of sin θ and cos θ as before, viz.

The plot below shows ||a_{B}||. It is identical to the plot in a previous tutorial
obtained by sequential differentiation of displacement of point B with respect to
time.

The two plots below show respectively ||a_{tB/A}|| the
tangential acceleration of point B relative to point A and angular
acceleration α_{AB} the relationship between the
two quantities being ||a_{tB/A}|| = α_{AB}
|AB| This relationship in general terms is the vector
cross-product *a*_{t} = **
α** x

It is instructive to compare the plots of α_{AB} above
and ω_{AB} below. The relationships are best
illustrated in the second group of diagrams below where validity of vector
cross-products ** v_{t}
**=

The table below provides an overview of crank angles θ where values of kinematic parameters are zero, positive maximum and negative maximum.

Value
of parameter |
v_{B } |
a_{B} |
ω_{AB} |
α_{AB} |

zero | θ = 0 ° and 180° | θ identical to maximum values of v_{B} |
θ = 90° and 270° | θ = 0° and 180° |

maximum (positive) | θ < 90° dependent on ratio n | θ = 0 ° | θ = 180° | θ = 90° |

maximum (negative) | θ > 270° dependent on ratio n | two symmetrical points θ < 180° and θ > 180° dependent on ratio n | θ = 0° | θ = 270° |

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