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:
2.3 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.
3. 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.
3.1 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.
3.2 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:
4. Steering Mechanism
In the final project demo, we designed a 3D spiral cam that autonomously guides the tricycle car through a “Figure-8” trajectory. Through the application of pure cycloidal motion profiles and numerical optimization algorithms, a continuous, jerk-free 3D spatial spline was generated. The resulting exported coordinates are scaled and formatted for direct integration into CAD software for subsequent 3d printing.
4.1 System Parameters
The design bridges the vehicle's dynamic path planning with the physical constraints of the mechanical steering linkage.
Vehicle Dynamics:
Wheelbase (L): 0.325 m
Constant Velocity (v): 0.25 m/s
Linkage Geometry (Pre-Scaled to mm):
Crank Length (R): 35.35 mm
Connecting Rod Length (L_c): 79.05 mm
Ground Joint Offset (X): 100.00 mm
Cam Design Parameters:
Thread Pitch: 20.0 mm per revolution
Base Radius (R_b): 25.0 mm
Cam Angular Velocity (ω_cam): π/4 rad/s
4.2 Path Planning and Steering Synthesis
To achieve the Figure-8 trajectory, the vehicle maneuvers were discretized into a 40-second continuous sequence:
Initial Approach: Straight driving for 2.0 s.
First Lobe (Left Sweep): A -225° heading change executed over 16.0 s.
Diagonal Crossover: Straight driving for 4.0 s to connect the two lobes.
Second Lobe (Right Sweep): A +225° heading change executed over 16.0 s.
Exit: Straight driving for 2.0 s to complete the path.
To eliminate mechanical shock within the cam groove, the steering angle (δ) transitions follow a pure cycloidal rise-and-return profile. Because the yaw rate (ψ') is defined by the kinematic bicycle model as:
The peak steering angle required to achieve exactly 225° of rotation is non-linear.
A numerical root-finding optimizer (scipy.optimize.fsolve) was deployed to integrate the theoretical yaw rate and solve for the exact cycloidal amplitude. The solver successfully converged on optimal peak steering angles for the 16-second durations, ensuring the vehicle hits the precise 225° orientation required to close the Figure-8 loops without oversteering.
4.3 Mechanism Kinematics
The angular position of the crank (θ) is directly coupled to the optimized steering angle requirement:
To determine the corresponding radial displacement required by the cam track, the fundamental position loop equation for the inverted slider-crank was utilized. The geometric formulation was updated to reflect the specific orientation of the ground joint relative to the crank center:
This modification correctly models the physical assembly limits of the mechanism, allowing the slider to oscillate safely within the linkage's real-world workspace.
4.4 Cam 3D Spatial Mapping
The time-domain kinematic data were mapped onto a cylindrical coordinate system (r, ϕ, z) to generate the physical cam groove:
Radial Position (r):
Angular Position (ϕ):
Axial Position (z): Driven by the 20.0 mm thread pitch, accommodating the longer 40-second sequence without self-intersection of the groove.
The cylindrical coordinates were converted back to Cartesian space $(X, Y, Z)$. Before the final CSV export, a uniform scaling factor of $\frac{1}{10}$ was applied to the coordinate arrays (X/10.0, Y/10.0, Z/10.0). This conversion bridges the native millimeter mathematical model to centimeter-scale units, streamlining the import process for the downstream Fusion 360 CAD environment and preparing the geometric spline for precision 3D printing.