2.2 Analytical Position Analysis of Mechanisms#
Analytical analysis offers a precise and systematic algebraic solution, ideal for comprehensive analysis and automation via computers.
Principles and General Procedure#
Analytical analysis can encode the same geometric procedure used in the graphical method, but transforms it into a set of equations.
The position of the links is determined by solving trigonometric equations (involving sine and cosine functions) for the \(x\) and \(y\) components of key points. For example, the coordinates of a point A on a driving link of length \(a\) can be expressed as \(A_x = a \cdot \cos(q_2)\) and \(A_y = a \cdot \sin(q_2)\).
A powerful method for formulating these equations is the vector loop configuration of mechanisms, proposed by Raven. In this method, the links are represented as position vectors that form a closed loop, where the sum of the vectors around the loop is zero. Although the vector formulation uses complex numbers as vectors, solving the resulting equations involves manipulating their real and imaginary parts, which translates directly into trigonometric expressions.
The solution of these algebraic equations, as in the graphical method, will generally yield two possible solutions for the output link angles, corresponding to the open and crossed configurations of the mechanism.
Advantages#
Once the analytical solution has been derived, it can be solved very quickly (by a computer) for all positions. This contrasts with the repetitive and tedious nature of the graphical method for multiple positions.
It allows for a high degree of precision in calculations, overcoming the inherent limitations of graphical measurement.
It is the basis for calculating velocities and accelerations in later stages of kinematic analysis, as the position equations can be differentiated with respect to time to obtain these quantities.
Computation allows for “almost instantaneous visual feedback on the quality of the design.”
Applicable Mechanisms#
Four-bar linkage: The most studied mechanism, and the position equations for its angles \(q_3\) and \(q_4\) are derived extensively.
Slider-crank mechanism (standard and offset inversion): The derivation of position equations for this mechanism is covered, including its standard and offset versions.
Inverted slider-crank mechanism: Its position analysis is also covered.
Geared five-bar mechanism: The formulation of its vector loop is also mentioned, implying its analytical solution.
Python and Matplotlib Integration#
Using Python and Matplotlib to generate animations is an excellent idea. These animations will serve to dynamically visualize how the trigonometric solutions (derived from the position equations) translate numbers into the real motion of the mechanism.