Kinematic Analysis & Synthesis - FC

Position Analysis

First, initial prototype was measured to start kinematic analysis. Afterwards the mechanism was broken into two separate four bar linkages and MATLAB was used to conduct the remainder of the kinematic analysis. Positional Analysis was conducted in two parts: position analysis for first four bar mechanism with input angle as θ₂, and position analysis for second four bar mechanism. From position analysis we get:

Figure 1: Position Analysis for four bar 1

Figure 2: Position Analysis for four bar 2

e = √(c² + d²)

θ_2,1 = arcsin(^c⁄_e)                                                                      

θ_2,2 = θ₂ - θ_2,1                                                                                                                

b₁ = √(a² + e² - 2ae*cos(θ₂))

θ₃ = arctan(^{(a*sin(θ₂)-c)} ⁄_{(a* cos(θ2) -d)})

θ₅ = arccos(^{(d₂ - a₂*cos(θ₃))}⁄_b2 )

c₂ = -a₂*sin(θ₃) - b₂*sin(θ₅)

Velocity Analysis

To get the velocity analysis, I took the time derivative of the position analysis. From velocity analysis, we get:

V_b = ω₂∗a∗(sin(θ₂)cos(θ₃) - cos(θ₂)sin(θ₃))

ω₃ = ^{(V_bcos(θ₃) - ω₂∗a∗sin(θ₂))}⁄(b∗sin(θ₃))

ω₅ = ^{(-V_b∗cos(θ₃) - ω₃∗a₂∗sin(θ₃))}⁄_{(b2∗sin(θ5))}

V_c2 = -(ω₃∗a₂∗cos(θ₃) - V_b∗sin(θ₃) + ω₅∗b₂∗cos(θ₅))

Acceleration Analysis

For acceleration analysis, the time derivative of velocity was taken. From acceleration analysis, we get:

A_b = b∗ω₃² + a∗ω₂²(cos(θ₂)sin(θ₃)-sin(θ₂)cos(θ₃))

α₃ = -V_b∗ω₃ + a∗ω₂²(cos(θ₂)sin(θ₃)-sin(θ₂)cos(θ₃))

α₅ = ^{-A_b∗cos(θ₃) - a₂*∗α₃sin(θ₃) - a₂∗ω₃²cos(θ₃) + 2∗V_b∗ω₃sin(θ₂) - b₂∗ω₅²cos(θ₅)}⁄_(b2sin(θ5))

A_c = -a₂∗α₃cos(θ₃) + a₂∗ω₃²sin(θ₃) + 2∗V_b∗ω₃cos(θ₃) + A_bsin(θ₃) - b₂∗α₅cos(θ₅) + b₂∗ω₅²sin(θ₅)

Results

From the kinematic analysis we can animate the motion of the link with help from MATLAB. We can also relate variations in the position, velocity, and acceleration of the slider as a function of θ₂

Figure 3: Results from Kinematic Analysis

.