The crank and slider mechanism's equation describes the position of the slider as a function of the
crank angle. For an in-line slider-crank, the slider's horizontal displacement (x) can be expressed
as: x = L2 * cos(θ) + L3 * cos(φ), where L2 is the crank length, L3 is the connecting rod length,
θ is the crank angle, and φ is the connecting rod angle. This equation can be further refined
using trigonometric relationships to express φ in terms of θ and the link lengths, leading to a
single equation for x as a function of θ. 
Elaboration:
The crank and slider mechanism is a
common linkage used to convert rotary motion into linear motion. It consists of a rotating crank, a
connecting rod, and a slider that moves along a straight path. 
Key Components and
Parameters:
Crank (L2): The rotating link connected to the driving shaft. 
Connecting Rod (L3):
The link connecting the crank to the slider. 
Slider: The link that moves linearly along a guide.

θ (Crank Angle): The angle between the crank and a reference line (usually the horizontal). 
φ
(Connecting Rod Angle): The angle between the connecting rod and the reference line. 
Displacement
Equation:
Basic Relationship: The horizontal position of the slider (x) can be initially expressed
as the sum of the horizontal projections of the crank and the connecting rod: 
x = L2 * cos(θ) +
L3 * cos(φ) 
Relating φ to θ: To get a single equation, we need to express φ in terms of θ.
This can be achieved using the law of sines and considering the geometry of the mechanism. A common
approximation, particularly for mechanisms where L3 is significantly larger than L2, is to use a
Taylor series expansion for cos(φ) and truncate it after the second term, according to
alistairstutorials.co.uk:
cos(φ) ≈ cos(θ) + (L2/L3) * sin^2(θ) 
Substituting and Simplifying:
Substituting this approximation back into the initial equation for x, we get:
x = L2 * cos(θ) + L3
* [cos(θ) + (L2/L3) * sin^2(θ)]
x = (L2 + L3) * cos(θ) + L2 * (L2/L3) * sin^2(θ) 
In-line
Slider-Crank: For an in-line slider-crank mechanism, the stroke (maximum slider displacement) is
simply twice the crank length (2 * L2), says Wikipedia.
Velocity and Acceleration:
The velocity
and acceleration of the slider can be found by differentiating the displacement equation with
respect to time. This involves using the chain rule and trigonometric derivatives. 
This video
demonstrates the velocity and acceleration analysis of a slider-crank mechanism: