14.3 Kinematic Analysis
1. Mobility Calculation (Degrees of Freedom)
The system has exactly 1 Degree of Freedom, meaning a single input (the cam's rotation) fully defines the motion of all other links.
2. Determination of Link Lengths
Before simulating the dynamic motion of the mechanism, it is necessary to synthesize the exact dimensions of the crank (R) and the connecting rod (L_c) to satisfy the prescribed boundary conditions of our design problem.
The mechanism is defined as an in-line reversed slider-crank, where the ground joint of the crank is offset along the x-axis at x = 10 cm. We define the mechanism to have two precision positions:
Position 1: Slider is at x = 0 cm, Crank angle is 135°
Position 2: Slider is at x = 5 cm, Crank angle is 45°
2.1 Geometric Formulation
Let the coordinates of the crank ground joint be O(10, 0), and let the position of the slider be denoted by S(x, 0). The coordinates of the moving crank pin (P), for any given crank angle (θ), are given by:
The primary geometric constraint of the system is the connecting rod, which acts as a rigid link of constant length (L_c) between the slider (S) and the crank pin (P). Applying the Euclidean distance formula yields the fundamental position equation for the mechanism:
2.2 Analytical Solution
To find R and L_c, we evaluate the fundamental position equation at both prescribed states.
State 1 (x = 0, θ = 135°):
State 2 (x = 0, θ = 45°):
Since the connecting rod is a rigid body, L_c2 must be equal in both states. Therefore, we can solve for the value of R:
With the crank length R determined, we substitute it back to solve for the connecting rod length L_c:
Conclusion
The required mechanism dimensions to achieve the prescribed boundary states are a crank radius of R = 3.535 units and a connecting rod length of L_c = 7.905 units. These values are treated as the static geometric constants for all subsequent position, velocity, and acceleration analyses.
2. Ideal Motion Profiles
To move the slider from x = 0 to x = 5 without generating infinite jerk (which causes vibration and wear), we will apply Cycloidal Motion to the cam profile. It guarantees zero acceleration at the start and end of the stroke, making it ideal for the follower driving our slider-crank.
As the slider moves from x = 0 to x = 5, the reversed slider-crank mechanism drives the crank to rotate from 45° to 135°.
Assume that the cam is rotating at π/8 rad/s (16s per full revolution). The mechanism is animated as follows.
Position, Velocity, and Acceleration of the Mechanism
With the slider's state (position, velocity, acceleration) defined by the cam, we map this to the crank.
Position (θ): Derived via the Law of Cosines based on the connecting rod constraint.
Velocity (ω): Derived by differentiating the position loop equations with respect to time.
Acceleration (α): Derived by differentiating the velocity equations.
Force & Mechanical Advantage
We calculate the Mechanical Advantage (MA) using the principle of virtual work (Conservation of Power). Assuming a frictionless system, the mechanical advantage is:
Assume there is a constant resisting torque of 10 Nm on the crank. The required cam force is therefore:
Calculate and plot MA and F_cam: