Motion analysis of a device including a disk and two slender bars with a design change for full disk revolution
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 2rad/s, or a variable (0.10+2.3α)rad/s , with α being the turn angle of the disk (Fig. 1(b)).
The three angles describing rotation of the articulated mechanism are α , β , and γ . A one letter notation is introduced to write trigonometric functions as:
Since the length of the linking bar AB cannot change, the horizontal displacement of pin A must be equal to the horizontal displacement of pin B, or:
An overbar means the length of a given straight line segment. It can be inferred from Fig. 1(a) that OA=BC. Simplifying:
A right triangle with AB being hypotenuse will have the following opposite side to angle γ [Fig. 1(b)]:
And the adjacent side:
Applying the Pitagorean theorem in Fig. 1(a):
These relations are needed to obtain β, γ, and the equations for velocities and accelerations in the following sections.
With pin A viewed as part of the disk, its velocity is:
Where ω⃗=2k^rad/s for constant rotation, or ω⃗=(0.10+2.3α)k^rad/s for variable rotation (k^ is the unit cartesian vector in the z -direction). For α going from 0o to 90o counterclockwise starting at the lowest position, ω varies from 0.10rad/s to 3.7rad/s, giving 1.9rad/s for the average angular velocity. We can thus write:
Taking the time derivative of ω we get:
With pin B as part of the output bar BC, its velocity is:
Where ω⃗BC is the still unknown angular velocity vector of bar BC. The motion of linking bar AB can be described using Chasles theorem :
Where ω⃗AB is the still to be determined angular velocity vector. V⃗A and V⃗B are always tangential to the circular trajectories of pins A and B. For the mechanism’s intermediate position shown in Fig. 1(b), x and y components for both velocities can be written as:
The fixed vectors in the rigid bodies going from O to A (ρ⃗OA), C to B (ρ⃗CB), and B to A (ρ⃗BA) can also be expressed using their x and y components. An inspection of Fig. 1(b) allows one to write:
Solving the vector algebra of Eq. (14) using Eqs. (15) – (19):
Having obtained V⃗B, Eq. (13) can be used to determine ω⃗BC.
Here we need to distinguish between the constant and the variable ω scenarios. Pin A only has radial acceleration with constant ω, but it has both radial and tangential components otherwise. Pin B will have both radial and tangential acceleration components.
Constant input rotation rate
Taking the derivative of Eq. (14) with respect to time:
The linear accelerations a⃗A, a⃗BR, and a⃗BT can be expressed in terms of unit vectors i^ and j^:
Substituting back into Eq. (23) and working the vector algebra, we obtain expressions for angular accelerations of bars AB and BC. Since the expressions are long, they are grouped into four different terms as AB1 through AB5 for ωAB, and BC1 through BC5 for ωBC (see the appendix for details):
Variable input rotation rate
Since the disk now has a variable rate of rotation ω˙, pin A’s acceleration has both radial and tangential components:
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 AB and BC are written as:
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 x and positive y components respectively. Pin B’s velocity will have both negative components:
The fixed vectors in the rigid bodies from O to A (ρ⃗OA), C to B (ρ⃗CB), and B to A (ρ⃗BA) can be expressed using their x and y components for an intermediate position:
Similarly to what was done earlier, we find VB and ωAB as functions of VA:
The linear accelerations a⃗A, a⃗BR and a⃗BT can be expressed in terms of unit vectors i^ and j^:
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 90o turn of the disk. The inversion of the sense of rotation for the linking bar AB takes place at 30o (top two panels): γ reaches a maximum and then starts to decrease, and ωAB changes from counterclockwise to clockwise (positive to negative). VB remains nearly equal to the constant VA during the change (lower left panel). Angular accelerations, despite their intricate expressions (Eqs. 27 through 32), are slowly varying functions after the start-up period. Both linear and angular velocities for the initial and final positions are shown in table 3. Angular accelerations are shown in table 4. The values for ωAB, ωBC, ω˙AB and ω˙BC at α=0o match the solution presented in , validating our implementation.
In Table 3, cells with an asterisk highlight the same numerical values. For example, the calculated value for ωAB is equal to ω and VB at α=0o. Solving for this instantaneous position only, as is typical in a routine paper and pencil exercise, may lead to the conclusion that these quantities remain equal at other positions. This is clearly not true as an inspection of Fig. 3 reveals. Values for VA, ωAB and ωBC also have the same absolute values at α=90o. Note that the linking bar AB stops at the same inclination of its initial position (γ=16.7o).
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 α=30o . Note that VB again matches VB (linearly increasing) at this position (lower left panel). Angular accelerations reach higher values than the constant rotation case, and they change sign (zero crossings at lower right panel), indicating rotation slowdown for both bars. The linking bar AB starts decelerating at α=15o and the output bar BC starts at α=50o. Initial and final values of angles, velocities and accelerations are also shown in Tables 5 and 6. Cells with * in Table 5 have the same meaning of the constant ω case. Here, for example, a calculation for α=90o yields equal values for ωAB and ωBC. But an inspection of the top right panel of Figure 4 clearly shows that it is only at this particular position these values are equal.
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 ωAB is always negative (clockwise rotation). This can also be understood by noting that γ is always decreasing (top left panel). Bar AB has a zero-crossing in its angular acceleration (bottom left panel) indicating the bar starts decelerating at α≈54o. Bar BC is always turning counterclockwise since ωBC>0, but continually decelerating (ω˙BC<0). Shaded cells in Table 7 have the same meaning as before.
The motion of the modified mechanism has one peculiarity not present in the previous two cases. At the end of the 90o turn of the disk, both bars will be horizontal, triggering a division by zero in the calculations of ωAB˙ and ω˙BC. The computation is thus stopped at α=89o . Angular velocities are not affected as both numerator and denominator of the f parameter approach zero (Eq. 53). Scripts and data files to make Figures 3, 4 and 5 are available in .
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 90o turn of the disk. The input rotation had two options: either a constant or a linearly varying rotation rate. Due to the geometrical configuration of the mechanism, the disk could only complete half a revolution. A design modification was implemented to allow full revolution, making the third option of the device.
Results of the simulation runs were presented using line plots for the variables of interest as functions of the disk’s turn angle α, complemented with tables of the initial and final positions. The analysis showed various aspects of the motions that are usually not accessible by only performing calculations for instantaneous values. It was possible to infer where a bar’s rotation change from clockwise to counterclockwise during its planar motion, as well as where it starts decelerating during the 90o turn of the disk. The new modified design was analyzed in a similar fashion, and important changes on the mechanism’s behavior were observed.
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 ρ⃗BA and performing the vector products:
Now we separate Eq. (64) into its x- and y-components, obtaining a system of two equations with two unknowns (ω˙AB and ω˙BC).
Multiplying the x-component by o, the y-component by −n and summing up, the result will have only ω˙AB as the unknown:
Solving Eq. (66) for ω˙AB:
Eq. (67) can be compared to Eq. (29) to identify terms AB1 through AB5. To get ω˙BC, we now multiply the x-component by q, and the y-component by p. Summing the two component equations will result in a single equation with ω˙BC as the only unknown:
Finally, solving it for ω˙BC :
And Eq. (69) can be compared to Eq. (32) to identify terms BC1 through BC5.
After a review of your revised document submitted in response to the reviewers comments, we have decided to accept your article for publication in the Journal of Open Engineering.
Thank you for your contribution!
Thank you for your submission to the Journal of Open Engineering. Please consider the two reviews that have been completed. In particular, there are three main points that I'd like you to address.
1) While TJOE does not screen submissions on the basis of novelty or impact, both reviewers have noted that it is not clear what the main thrust of your manuscript is. Can you please clarify the main point that you are trying to make with this article? From your manuscript you address both numerical evaluation of the kinematic equations as well as the project-based engineering pedagogy. Perhaps the manuscript could be reframed to make it more clear what the implications are in one of these areas.
2) One of the reviewers noted that the notation presented in equations 1 through 3 made following the rest of the mathematics more difficult. Please consider eliminating this notation and instead writing out the trigonometric expressions fully in the subsequent equations.
3) In the numerical experiments portion of the manuscript you include some figures. In the spirit of the journal, it would be desirable to provide the code used to generate those figures as part of your submission.
Thank you again for your submission and please consider the reviewer comments as well as my comments above in your revision.
1. The main thrust is the numerical evaluation of the kinematic equations and its reproducibility. The interest to develop this work originated from my daily routine of teaching vector mechanics to undergraduates, and a couple of years ago the pedagogical side of using computer projects was the main interest.
But the focus is different here.
2. An argument for keeping the notation was made in my reply to the reviewer.
3. Direct link:
REVIEW FOR THE JOURNAL OF OPEN ENGINEERING
The manuscript presents an kinematics analysis of a mechanism denoted by two slender bars with a design revolution. This mechanism is a typical four-bar linkage analyzed in a wide variety of books and papers about mechanism and dynamics of machinery. Numerical simulations are performed to compare two different configurations.
Technical formulation seems to be correct. However, an approach using generalized coordinates or dimensionless variables might be more interesting.
Some typos can be found in the paper:
- vector notation: omega, V_A, V_B, ,rho_OA, rho_CB, rho_BA,....
- units in mathematical mode
It is not clear the main contribution of the manuscript: the kinematic analysis or numeric simulations?
In spite of technically correct formulation of the kinematics equations, the notation and abbreviations used makes it difficult to read. It might be preferable to use throughout the manuscript the notation sin(alpha) instead l, cos(alpha) instead, and so on.
It is not clear why the kinematic analysis was separated in original mechanism and modified mechanism. That seems the original mechanism is a particular configuration of the modified mechanism.
Author does not explain why they employed the constant rotation w= 2 rad/s and the variable rotation w = 0,1+2,3alpha rad/s for the kinematics analysis. The selected simulated rotation constitute two really simple examples to evaluate the mechanism. It would be more appropriate to use more complex simulations. For instance, harmonic and exponential rotation.
The state-of-the-art section is weak. Taking a look at the literature, it can check that there are many works about kinematics and kinetics analysis of four-bar mechanisms.
The citations section is incomplete, no reference has date.
+ 1 more...
REVIEW FOR THE JOURNAL OF OPEN ENGINEERING
In this paper the author derives the kinematic equations governing the four bar linkage, a common dynamics benchmark problem. The author then compares different linkage lengths and two driver input angular velocities by numerically evaluating the equations.
This article is something that may belong well in an introduction to kinematics course notes, but does not really belong in a modern technical journal. The four bar linkage is one of the most common and widely studied linkages there are. There are many thousands and thousands of derivations of this out in the wild. It isn't clear to me that this derivation offers anything new to this classic problem.
At the end of the article the author states "This work was motivated by the desire to introduce project-based activities in the undergraduate engineering mechanics’ classroom." This article could possibly be framed in this light and offer something useful to the journal's audience. Maybe this derivation method is an exceptional way to enlighten students and that would be worth telling a story about. But as it stands this article doesn't have a story worth telling in this journal.
I think this article is likely technically sound. If the author compared his results with other benchmark results of the four bar linkage then we could feel fully confident of the soundness.
The writing is fine but doesn't use common nomenclature and descriptions for kinematics. For example, it is odd that the author never mentions the phrase "four bar linkage". The introduction cites a number of frivolous references that do not seem related to the article in any substantial way.
This is fine.
OPENNESS AND REPRODUCIBILITY
This article over presents the derivation of this common problem, which is good for reproducibility but simply showing the final kinematic equations would be more than sufficient. The author evaluates the equations numerically and it isn't apparent if software was used to do this. If it was used, there is no code or data artifact cited.
One additional comment. It is not acceptable to use copyrighted figures from a textbook in this venue.