15.2 Kinematic Analysis
Head with Beak/Jaw Movement Mechanism
The dragon’s head mechanism functions as a four-bar linkage where the bottom jaw of the beak acts as the fixed ground (Link 1). When an input force is applied to Link 2, it rotates, causing Link 3 (the top jaw of the beak) to translate and rotate downward toward the fixed base. This motion effectively simulates a biting action, where the "beak" closes as the transmission changes, transferring the input rotation into a clamping force at the tip of the jaw.
Mobility Calculation
Gruebler Equation
M = 3(L - 1) - 2J1 - J2
L = 4, J1 = 4, J2 = 0
M = 3(4 - 1) - 2(4) - 0 = 1 DOF.
This confirms that the mechanism has one degree of freedom to produce the motion of the beak.
Grashof Condition
S + L < P + Q
S = 0.55 in , L = 1.8 in, P = 1.6 in, Q = 1.2 in
This inequality holds true (2.35 < 2.8) for Grashof’s condition. Because the shortest link (Link 1) is the ground, this specific configuration is a Double-Rocker, meaning both Link 2 and Link 4 can rock back and forth but cannot make a full circle.
Position Analysis
The position plot shows the relationship between the input crank angle theta_2 and the output rocker angle theta_4. We see that as the input moves through its range, the output jaw follows a predictable path when reaching its "closed" limit.
Velocity Analysis
The velocity plot illustrates how fast the beak closes relative to the input speed. From the plot, we observe that the velocity is non-linear; the beak starts closing slowly, reaches a peak speed mid-stroke, and decelerates as it approaches the final biting position, which is ideal for controlled movement.
Acceleration Analysis
The acceleration plot highlights the inertial loads on the popsicle stick links. Our plot shows a smooth transition, suggesting that the dragon's jaw will operate without excessive vibration or mechanical shock.
Force Analysis
Mechanical Advantage
Mechanical advantage (MA) is the ratio of output torque to input torque. In this mechanism, the MA increases significantly as the beak approaches the closed position. This means the dragon can apply the most "clamping force" right as the jaws meet, which is consistent with the functional requirements of a beak or jaw.
MotionGen Animation (extra)
The was also modeled in MotionGen to digitally visualize the kinematic behavior of all four links. This allowed the team to confirm the Grashof condition, observe the coupler curve traced by the head attachment point, and verify that the rocker swept through the intended range of motion, reducing the risk of geometric errors carrying into the physical build
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Animation
The animation demonstrates one full cycle of the "biting" motion. You can see the top jaw (Link 3) dipping down toward the fixed ground link, showcasing the synchronized movement of all four pins in the assembly.
Feather Movement Mechanism
The feather movement mechanism is a simple four bar slider mechanism. To make this mechanism unique, the slider is slotted to allow a set of followers to slide up and down as the slider moves horizontally. The followers will have a feather like geometry attached on top to make the body look like it has feathers ruffling in the wind. Vertical follower motion is achieved by using a grounded block with a unique surface that the followers slide against. There is no desired force output for this mechanism - the focus is on linear motion for the slider block.
Mobility Calculation
M = 3(L-1) - 2J1 - J2
L = 4, J1 = 4, J2 = 0
M = 3(4-1) - 2(4) - 0 = 1
Grashof Condition
The Grashof Condition for a crank-slider mechanism is slightly different than the usual S + L < P + Q, since one of the link lengths (link 1) is constantly changing. Instead, the Grashof condition equality is l + e < L, where l is the shortest link, e is the vertical offset, and L is the longest link. For this mechanism the geometry is:
l = link 2 = 63mm
e = 0
L = link 3 = 110 mm
From the given geometry, one can trivially tell that l + 0 < L, thus the mechanism is Grashof at link 2.
Kinematic Analysis
The only point of kinematic interest for this mechanism is the slider block, and due to this only link 1 kinematics will be analyzed. Matlab was used to create plots using position, velocity, and acceleration functions for a four bar slider. For this analysis, the input crank (link 2) angular velocity is set to 1 rad/s, and the angular acceleration is set to 0 rad/s^2.
Animation
Wing Movement Mechanism
The dragon wing mechanism functions as a combination of a cam-follower system and a four-bar linkage. The base of the mechanism acts as the fixed ground (Link 1). The cam follower is arranged approximately parallel to the four-bar rocker, with the cam positioned slightly lower. As the cam rotates, it displaces the follower, which pushes Link 2 (the input link). This induces a rocking motion in Link 2, oscillating back and forth over an 80-degree sweep. This motion is transmitted through Link 3 (the coupler) to Link 4, causing the "wing" to translate and rotate, effectively simulating a dragon's flapping motion.
Mobility Calculation
Gruebler Equation
M = 3(L-1) - 2J1 - J2
L = 4, J1 = 4, J2 = 0
M = 3(4-1) - 2(4) - 0 = 1
This confirms that the four-bar portion of the mechanism has exactly one degree of freedom to produce the flapping motion of the wing.
Grashof Condition
S + L < P + Q
S = 103
L = 45
P = 84
Q = 58
Because $S + L > P + Q$, this mechanism is a Non-Grashof (Class II) Triple-Rocker. This means that no single link is capable of making a full 360-degree rotation. This is perfectly suited for a flapping mechanism, as the input link is constrained by the cam-follower to an 80-degree operating window, avoiding the toggle points and ensuring smooth reversing motion.
Kinematic Analysis
The primary points of kinematic interest for this mechanism are the input link (Link 2) and the output wing (Link 4). Because Link 2 is driven by a cam, its angular position oscillates within an 80-degree range. MATLAB was used to create plots analyzing the angular position, velocity, and acceleration of the wing (Link 4) over one full cycle of the cam. For this simulation, the input rocker (Link 2) is modeled with a sinusoidal oscillation centered at 90 degrees to mimic the rise and fall of the cam follower.
Animation