Mathematical formulations and numerical experiments for the kinematics of an articulated mechanism are presented. The system of connected rigid bodies is configured in three different ways. For the first two configurations (double rocker), the input motion is either a constant counter-clockwise rotation rate for the disk, or a linearly increasing one. A design modification is implemented to allow full disk revolution with a constant and clockwise input rotation (crank-rocker). Angles, velocities and accelerations are computed and analyzed for a quarter revolution of the disk. The results corroborate the inversion of the linking bar sense of rotation in planar motion and its deceleration. This sense of rotation is always clockwise in the modified design. We note that some quantities have instantaneous equal values at particular positions. This fact may preclude a better understanding of the motion if only instantaneous calculations are made. The approach illustrates how applied rigid body mechanics can be used with profit to describe the motion of articulated devices.
The rigid body is an important idealization of moving and interacting parts and beings. The key feature is the fixed distance between two points in the body, since forces cannot cause deformation of its shape . The subject is relevant in the development of video-game physics engines since the simulation of motions and interactions of rigid bodies approximates reality fairly well . Prosthetic limbs and robotic devices are other important application areas ; .
Here we offer a tool to simulate the motion of an articulated mechanism made with a disk and two slender bars. An input rotation is imposed on the disk, triggering the planar motion of the long intermediate link, and the partial rotation of the short output bar. We develop the kinematics in detail using applied vector algebra, calculus and descriptive geometry. An approach based on differential modeling has been reported in the literature as a simplified alternative based on kinematic Jacobians . Use of commercial packages is also popular with the intent to offer an automated procedure for mechanism design .
The study of the motion of articulated devices and machines is a classic and well-established subject ; .  have used alternative methods to describe the two-dimensional motion of rigid bodies in biomechanics applications. Our approach is based on first principles carefully accounting for the physical constraints of the articulated mechanism. The main thrust is the numerical evaluation of the kinematic equations and its reproducibility. We display results using line plots that show how key parameters evolve during the simulation (this is in contrast to visualization systems employed for the realistic animation of articulated devices ). The interest for this work originated from the impetus to develop alternative learning tools for undergraduate vector mechanics. Some pedagogical considerations of using these tools have been addressed elsewhere.
The outline of the article is as follows. The original mechanism is presented and its two configurations are formulated. Geometrical constraints are introduced, as well as formulations to obtain various velocities and accelerations, paying particular attention to the proper use of Chasles’ theorem. Then the modified design is explained, and a similar sequence of formulations for the geometry, the velocities, and the accelerations is presented. The numerical experiments properly said are then described. Results are presented for each one of the three configurations, where various aspects of the motions are demonstrated. Final remarks are then laid out, summarizing the important results of this investigation. At the end, an appendix with the details of the angular accelerations’ calculations for the original mechanism is given.
The analysis is traced back to , where there is a solved example for the instantaneous position as shown in Fig. 1(a), and general guidelines for a computer project are proposed. In what follows, equations for the kinematics are derived showing angle and length constraints, as well as velocity and acceleration relations while the disk undergoes either a constant
The three angles describing rotation of the articulated mechanism are
Since the length of the linking bar
An overbar means the length of a given straight line segment. It can be inferred from Fig. 1(a) that
A right triangle with
And the adjacent side:
Applying the Pitagorean theorem in Fig. 1(a):
These relations are needed to obtain
Taking the time derivative of
The fixed vectors in the rigid bodies going from
Solving the vector algebra of Eq. (14) using Eqs. (15) – (19):
Here we need to distinguish between the constant and the variable
Constant input rotation rate
Taking the derivative of Eq. (14) with respect to time:
The linear accelerations
Substituting back into Eq. (23) and working the vector algebra, we obtain expressions for angular accelerations of bars
Variable input rotation rate
Since the disk now has a variable rate of rotation
Referring back to Fig. 1(b), the tangential component can be written as:
The radial component is still given by Eq. (24). Following a similar procedure as for the constant case, angular accelerations for bars
A design modification of the original mechanism is shown in Fig. 2. The distance between fixed centers O and C is made shorter than the original to allow full revolution of the disk while keeping the size of the linking bar AB unchanged. The full revolution is not possible in the original mechanism in Fig. 1.  proposed an innovative design approach for a slider-crank mechanism using pneumatic cylinders. His design could be an alternative to allow full disk revolution for the 2-D device investigated here, but it would require replacing the linking bar with a pneumatic cylinder without the need to reduce the distance between centers O and C. Other approaches to study the kinematics and dynamics of slider-crank mechanisms have been proposed and their benefits in learning theoretical aspects of mechanism design have been highlighted .
Trigonometric functions follow the same short-hand notation described earlier. The relation between horizontal displacements for pins A and B are now given by:
A right triangle with AB as the hypotenuse gives:
An intermediate position will have a velocity vector for pin A with negative
The fixed vectors in the rigid bodies from O to A (
Similarly to what was done earlier, we find
The linear accelerations
Substituting back into Eq. (23) and working the vector algebra:
Tables 1 and 2 present a summary of all acceleration terms for the three configurations of the articulated mechanism. The reader is referred to the appendix for a derivation of these terms. These equations have been implemented numerically  and the Fortran code is available .
Original mechanism: constant angular velocity
In figure 3 we plot angles, velocities and accelerations for a
In Table 3, cells with an asterisk highlight the same numerical values. For example, the calculated value for
Original mechanism: variable angular velocity
The results for the original mechanism with an increasing rotation rate for the disk are presented in Fig. 4. Since the mechanism’s geometry is the same, the change in sense of rotation for the linking bar still takes place at
Results for the modified design are shown in Fig. 5. In this scenario there is no change in rotation direction for linking bar AB. This can be seen in the top right panel, where
The motion of the modified mechanism has one peculiarity not present in the previous two cases. At the end of the
Discussion and conclusions
The relations needed to simulate the motion of a two-dimensional device were presented and the kinematics of three variants of this device were analyzed through a
Results of the simulation runs were presented using line plots for the variables of interest as functions of the disk’s turn angle
Despite the complexities of the rigid bodies’ formulations described here, one should keep in mind that this is still a relatively simple mechanics problem. All principles applied are exact and the numerical implementations only required algebraic expressions. Complications arise if one attempts to formulate and model collisions between rigid bodies. Impulse-momentum principles are needed compared to the limited kinematics toolbox used in this investigaton (, , ).
This work was motivated by the desire to introduce project-based activities in the undergraduate engineering mechanics’ classroom. Without showing how the motion of articulated mechanisms is obtained, one may not appreciate the intricacies of the kinematics of rigid bodies in key applications such as video-game physics engines, and the design of prosthetic limbs or robotic devices.
Appendix: Derivation of angular acceleration terms
Details of the procedure to obtain the angular acceleration terms for the original mechanism are shown below. Similar procedures apply for the other two configurations. We start by substituting Eqs. (24) through (26) into Eq. (23):
Using Eqs. (19) for
Now we separate Eq. (64) into its
Solving Eq. (66) for
Eq. (67) can be compared to Eq. (29) to identify terms
Finally, solving it for
And Eq. (69) can be compared to Eq. (32) to identify terms
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