07.3 - Kinematic Analysis
1. Kinematic Analysis
1.1 Jansen Leg Motion Analysis
1.1.1 Mobility of Jansen Mechanism
Below is a labeled diagram of the Jansen Leg Mechanism.
We have 8 links with 10 full joints and 0 half joints. It follows that by Gruebler’s Equation, we have the following:
In terms of mobility, the Jansen leg has 1 degree of freedom.
1.1.2 4 bar Linkages of Jansen Mechanism
To consider the position of the joints and links, we can divide the Jansen leg into 4 bar linkages which can be solved for position, velocity, and acceleration. Note: CDE is not a 4 bar linkage but a ternary link.
1.1.3 MATLAB Position Analysis
Using the 4 bar linkages from above, we used a MATLAB script to output the trajectory of the leg and model the motion of the joints/links.
The fixed crank pivot is placed at the global origin (0,0). While the base pin joint is located to the left of the fixed crank by 286 mm (-286,0).
The crank position can be modeled by a circle that rotates 360 deg about the crank pivot. The radius would be the length of the crank link: x_c
All the other joints, were found by conducting a position analysis on each of the 4 bar linkages from Section 1.1.2.
1.1.4 Jansen Leg Simulation on MotionGen Software for Velocity and Acceleration Analysis
We used a simulation from the MotionGen software to track the velocity and acceleration of the Jansen Leg. We were able to plot the displacement, linear velocity, and the linear acceleration. Below are the plots that were produced by the MotionGen software.
1.2 Planetary Gears Motion Analysis
We can consider the following values for the compounded planetary arrangement:
Number of Teeth on Sun Gear: 19
Number of Teeth on Planetary Gear: 11
Number of Teeth on Ring Gear:
We can assume the following input speeds:
Input Speed (speed of Sun Gear): 60 deg/s (for both modes)
Speed 1: Low Speed: ω_s = 0 deg/s (inner gear is held)
Speed 2: High Speed: ω_h = - 8 deg/s (inner gear is released)
We can compute the ring output speeds through Willis Equation:
where
ω_s = angular speed of sun gear
ω_c =angular speed of carrier gear
ω_r = angular speed of ring gear
N_r = number of teeth in ring gear
N_s = number of teeth in sun gear
We can solve for ω_R if we want to find the speeds for the low speed mode and the high speed mode.
Speed of Ring Gear R2:
Speed of Ring Gear R2:
These are the output speeds for each of the speed modes on our automatic transmission. The magnitude of the angular speed for when both gears are released are higher than the angular speed for when a singular gear is engaged.
1.3 Manual Transmission Motion Analysis
The manual transmission uses 2 gear pairs to vary the rotational speed delivered to the output shaft.
In the low-speed configuration, the 16-tooth gear on the input shaft drives a 17-tooth gear on the output shaft. This resulted in a speed ratio of 16/17 which is about 0.94. This means the output rotates slightly slower than the input. This configuration produces slower leg motion, making it easier to visualize the Jansen mechanism.
In the high-speed configuration, the dog clutch engages the second gear pair where the 23-tooth driving gear meshes with the 17-tooth driven gear. This produces a speed ratio of 23/17 which is about 1.35. This means that the output rotation speed rotates slightly faster than the input. This configuration increases the speed of the Jansen mechanism.
In terms of angular speed, the relationship between input and output angular speed of the gears can be modeled using the following equation:
Assuming we have a input speed (manual or from motor) of about 30 RPM. For the low speed configuration, we can say
If we use an input angular velocity of 3.14 rad/s (30 RPM), we get an output speed of about 2.953 rad/s. For the high speed configuration, we can say
We then get an output speed of about 4.2411 rad/s, which is faster than the low speed configuration by about 43.6%.