A four-bar linkage, also known as a quadric cycle or four-link mechanism, is a planar mechanism with
four links and four joints. The loop equations are used to analyze the linkage's position, velocity,
and acceleration. These equations are based on vector analysis and complex numbers to describe the
relationships between the link lengths and angles. 
Here's a breakdown of the key concepts and
equations:
1. Vector Loop Equation:
A four-bar linkage forms a closed loop, meaning the vector sum
of the links must be zero. 
Representing each link as a vector with a magnitude (link length) and
an angle (relative to a reference axis), the loop equation can be written as: r1 * e^(iθ1) + r2 *
e^(iθ2) = r3 * e^(iθ3) + r4 * e^(iθ4)**, where:
r1, r2, r3, and r4 are the lengths of the links.

θ1, θ2, θ3, and θ4 are the angles of the links with respect to a reference axis. 
e^(i*θ) is
the complex exponential form (Euler's formula) representing a vector in the complex plane. 
2.
Grashof Condition:
This condition determines the type of motion a four-bar linkage can exhibit. 
S
+ L ≤ P + Q, where S is the shortest link, L is the longest link, and P and Q are the other two
links. 
If the inequality holds, the shortest link can rotate fully with respect to a neighboring
link (a crank). 
3. Position Analysis:
The goal is to determine the unknown link angles (usually
θ3 and θ4) given the input angle (usually θ2) and link lengths. 
The vector loop equation can be
rearranged to solve for the unknown angles. 
For example, solving for θ3 and θ4 involves:

Rearranging the loop equation to isolate the terms with unknown angles.
Expressing the equation
in terms of its real and imaginary components (using Euler's formula). 
Applying trigonometric
identities to solve for the unknown angles. 
4. Velocity and Acceleration Analysis: 
Velocity and
acceleration analysis builds upon the position analysis. 
The time derivatives of the loop equation
(using complex numbers and the chain rule) are used to find the angular velocities and accelerations
of the links. 
Example:
Let's say you have a four-bar linkage with link lengths r1, r2, r3, and
r4, and you want to find θ3 and θ4 given θ2. You would: 
Vector Loop Equation: Write the
equation r1e^(iθ1) + r2e^(iθ2) = r3e^(iθ3) + r4e^(iθ4).
Isolate Unknowns: r3e^(iθ3) -
r4e^(iθ4) = r1e^(iθ1) - r2e^(iθ2).
Complex Form: Let z = r1e^(iθ1) - r2e^(iθ2). Then,
r3e^(iθ3) - r4e^(iθ4) = z.
Solve: Use equations involving sin and cos to solve for θ3 and θ4.

In essence, the four-bar linkage equations allow engineers to: 
Analyze the motion of the
linkage.
Design linkages for specific motion requirements.
Understand the relationships between
link lengths, angles, velocities, and accelerations.