The two slider-crank mechanisms in the panda model are analyzed individually as two separate fourbar non-offset inversion #1 slider-crank linkage mechanisms in the XY Plane (2-Dimensional). The two mechanisms are analyzed separately, since it makes it easier to perform the vector loop position, velocity, and acceleration approach analysis on mechanisms that each have one input, one output, and one degree-of-freedom. Thus, we will first perform position, velocity, and acceleration analysis on the fourbar horizontal non-offset inversion #1 slider-crank mechanism that is comprised of four-bar linkages and has one degree of freedom. Then, we will perform position, velocity, and acceleration analysis on the fourbar vertical non-offset inversion #1 slider-crank mechanism. Kinematic still drawings of both slider-crank mechanisms are illustrated below.
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Kinematic drawing of horizontal slider-crank mechanism (for bamboo stick)
Kinematic drawing of vertical slider-crank mechanism (for lower jaw)
3.1 Given/measured parameters for both slider-crank mechanisms
Fig.
The known parameters prior to performing kinematic analysis of each fourbar slider-crank mechanism are shown in Fig.1. Both fourbar slider-crank mechanisms have the same link length values and configuration (open). The constant link length values (input crank length (a), coupler rod length(b), and ground offset (c)), were obtained by measuring each link with a ruler in units of millimeters. Link c (offset ground length) is 0mm because the slider axis extended passes through the crank pivot. Thus, this means both slider-crank mechanisms are fourbar non-offset inversion #1 slider-crank mechanisms, where the slider block translates (Norton).
The ground link/slider position angle for the horizontal slider-crank mechanism, theta1, has a constant angle of 0 degrees from the global X-axis. The origin of the global coordinate system is situated on crank pivot/axle. The ground link/slider position angle for the vertical slider-crank mechanism, theta1m, has a constant angle of 90 degrees from the global X-axis, which indicates the slider position/ground is positioned vertically.
The angular velocity of both input cranks (omega2) are the same for both mechanisms because both cranks are pivoted on the same axle. The angular velocity was measured by counting the number of revolutions the crank axle disk made in one minute in the counterclockwise direction. As a result, the crank axle disk made 78 RPM, which converted to 8.1681 rad/s. For both mechanisms, the input crank angular velocity is assumed to be constant. Thus, it is not dependent on time or input crank angle. Also, since the input crank angular velocity is constant, then its angular acceleration is 0 rad/s2. The time for both mechanisms to complete one revolution was found by dividing converting 78 RPM to revolutions per second. Thus, the time for both mechanisms to complete one revolution is 1.3 seconds.
The input crank angle of a fourbar non-offset inversion #1 slider-crank mechanism is dependent on time. Thus, the input crank angle for the horizontal crank-slider mechanism, theta2, initially is positioned at 0 degrees from the global X-axis when the time is at 0 seconds. As time goes from 0 to 1.3 seconds the crank angle, theta2, rotates 360 degrees from 0 to 360 degrees in increments of 1 degree every 0.00361 seconds. The input crank angle for the vertical crank-slider mechanism, theta2m, initially is positioned at 90 degrees from the global X-axis when the time is at 0 seconds. As time goes from 0 to 1.3 seconds the crank angle, theta2m, rotates 360 degrees from 90 to 450 degrees in increments of 1 degree every 0.00361 seconds.
3.2 Position Analysis/Results
3.2.1 Position Analysis for both Horizontal and Vertical Slider-Crank Mechanisms
Given all the measured parameters that we know, shown in fig. ### we are able to perform a position analysis on each slider-crank mechanism, in order to find its coupler bar angle (theta3) , and its ground slider position length from the crank pivot (d). To begin the analysis of each slider-crank mechanism, the linkages of the mechanism are represented as position vectors as shown in Fig. ### below. The position vector R4 is 0, since the mechanism is a fourbar non-offset inversion #1 slider-crank. A closed vector loop equation (1) shown below represents the vector positions in the mechanism.
R2 –R3 – 0 – R1 = 0 (1)
The vector position loop equation is then separated into real (X-components) and imaginary (Y-components), which are then rearranged to find the angle of the coupler rod (theta3) and the ground slider position length from the crank pivot (d) as shown in the MATLAB slider-crank position function shown below.
3.2.2 Position Analysis Results for both Horizontal and Vertical Slider-Crank Mechanisms Key Result
The slider length position vs. input crank angle plot shown below depicts the slider/ground link positioned at an offset of 90 degrees from the global X-axis for the vertical mechanism, and 0 degrees for the horizontal mechanism. The figure also illustrates the total displacement of both the lower jaw and bamboo stick. The difference between the max peak value (116mm) and the min peak value (68mm) on the plot is 48mm. This 48mm difference on the plot illustrates both the maximum position displacement of the bottom jaw in the vertical Y-direction and of the bamboo stick in the horizontal X-direction.
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The slider position from pivot vs time plot shown above illustrates that despite the vertical ground/slider link and input crank angle being offset by 90 degrees, both the vertical and horizontal mechanisms make a complete revolution in the same amount of time. Therefore, it is shown in time plot that in one revolution the panda’s mouth opens and closes in the Y-direction one time, while the bamboo stick simultaneously gets 48mm closer to the panda in the horizontal direction when the lower jaw is opened 48mm vertically and then returns back to its maximum distance away from the panda the same time the panda mouth returns to its closed mouth position.
3.3 Velocity Analysis/Results
3.3.1 Velocity Analysis for both Horizontal and Vertical Slider-Crank Mechanisms
After the position analysis has been done, the velocity of the slider for both mechanisms can be found since the link lengths and link angles and input crank angular velocity are known for both mechanisms. Thus, the velocity analysis is done just by taking the time derivative of the position vector, illustrated in the matlab function shown below.
3.3.2 Velocity Analysis Results for both Horizontal and Vertical Slider-Crank Mechanisms
The slider velocity vs input crank angle plot and slider velocity vs time plot both show that as the lower jaw and bamboo stick both hit their maximum displacements, their velocities are zero and while the jaw and bamboo stick are at distance halfway (24mm) between the maximum displacement and zero displacement, their velocities are at their maximum. The slider velocity vs time plot also illustrates that despite the vertical ground/slider link and input crank angle being offset by 90 degrees, both the vertical and horizontal mechanisms make a complete revolution in the same amount of time. Therefore, it is shown that both the panda’s lower jaw (vertically) and bamboo stick (horizontally) have the same velocities at the exact same time.
3.4 Acceleration Analysis/Results
3.3.1 Acceleration Analysis for both Horizontal and Vertical Slider-Crank Mechanisms
After the position analysis and velocity analysis have been done, the acceleration of the slider for both mechanisms can be found since the position and velocity parameters and the input crank angular acceleration are known for both mechanisms. Thus, the acceleration analysis is done just by taking the time derivative of the velocity vector, illustrated in the matlab function shown below.
3.3.2 Acceleration Analysis Results for both Horizontal and Vertical Slider-Crank Mechanisms
The slider acceleration vs input crank angle plot and slider acceleration vs time both shown above, illustrate that as the lower jaw and bamboo stick are at positions displaced between maximum and minimum displacements (48mm and 0 mm) the acceleration is constant since the velocities of both slider are changing speeds. The slider acceleration vs time plot also illustrates that despite the vertical ground/slider link and input crank angle being offset by 90 degrees, both the vertical and horizontal mechanisms make a complete revolution in the same amount of time. Therefore, it is shown from the time plot that both the panda’s lower jaw (vertically) and bamboo stick (horizontally) have the same accelerations at the exact same times.