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.


Introduction

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

[1]
Fundamentals of Multibody Dynamics. (2006). Birkhäuser Boston. https://doi.org/10.1007/b137682
[2]
Stewart, D. E. (2000). Rigid-Body Dynamics with Friction and Impact. SIAM Review, 42(1), 3–39. https://doi.org/10.1137/s0036144599360110
. 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
[3]
Garstenauer, H. (2006). A Unified Framework for Rigid Body Dynamics. A-4040 Linz - Altenberger Strasse 69.
. Prosthetic limbs and robotic devices are other important application areas
[4]
Burkett, B., Smeathers, J., & Barker, T. M. (2004). A Computer Model to Simulate the Swing Phase of a Transfemoral Prosthesis. Journal of Applied Biomechanics, 20(1), 25–37. https://doi.org/10.1123/jab.20.1.25
;
[5]
Alexander, J. C., & Maddocks, J. H. (1989). On the Kinematics of Wheeled Mobile Robots. The International Journal of Robotics Research, 8(5), 15–27. https://doi.org/10.1177/027836498900800502
.

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

[6]
Sultan, I. A. (2009). Introducing Differential Kinematics to Mechanical Engineering Students. International Journal of Mechanical Engineering Education, 37(3), 210–222. https://doi.org/10.7227/ijmee.37.3.5
. Use of commercial packages is also popular with the intent to offer an automated procedure for mechanism design
[7]
Marghitu, D. B., & Raju, P. K. (2009). Mechanism Analysis with Mathematica. International Journal of Mechanical Engineering Education, 37(2), 130–143. https://doi.org/10.7227/ijmee.37.2.5
.

The study of the motion of articulated devices and machines is a classic and well-established subject

[8]
Beatty, M. F. (1966). Kinematics of Finite, Rigid-Body Displacements. American Journal of Physics, 34(10).
;
[9]
Paul, B., & Krajcinovic, D. (n.d.). Computer Analysis of Machines With Planar Motion: Part I - Kinematics. Journal of Applied Mechanics, 37(3), 697–702.
.
[10]
S. W. Wolfe, K. H., & Duncan, J. S. (n.d.). Two-Dimensional Rigid-Body Kinematics Using Image Contour Registration. Journal of Biomechanics, 28(1), 119–124.
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
[11]
Hahn, J. K. (n.d.). Realistic Animation of Rigid Bodies. ACM SIGGRAPH Computer Graphics, 22(4), 299–308.
). 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.


Original mechanism

The analysis is traced back to

[12]
Shames, I. H. (n.d.). Engineering Mechanics: Statics and Dynamics.
, 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/s2rad/s
, or a variable
(0.10+2.3α)rad/s(0.10 + 2.3\alpha) rad/s
, with
α\alpha
being the turn angle of the disk (Fig. 1(b)).

Figure  1a: Articulated mechanism at its initial positions. Adapted from  SHAMES, IRVING H., ENGINEERING MECHANICS: STATICS AND DYNAMICS, 4th  Edition.
Figure 1a: Articulated mechanism at its initial positions. Adapted from SHAMES, IRVING H., ENGINEERING MECHANICS: STATICS AND DYNAMICS, 4th Edition.


Figure  1b: Articulated mechanism at its initial positions. Adapted from  SHAMES, IRVING H., ENGINEERING MECHANICS: STATICS AND DYNAMICS, 4th  Edition.
Figure 1b: Articulated mechanism at its initial positions. Adapted from SHAMES, IRVING H., ENGINEERING MECHANICS: STATICS AND DYNAMICS, 4th Edition.


Geometrical constraints

The three angles describing rotation of the articulated mechanism are

α\alpha
,
β\beta
, and
γ\gamma
. A one letter notation is introduced to write trigonometric functions as:

Since the length of the linking bar

ABAB
cannot change, the horizontal displacement of pin
AA
must be equal to the horizontal displacement of pin
BB
, or:

An overbar means the length of a given straight line segment. It can be inferred from Fig. 1(a) that

OA=BC\overline{OA}\,=\,\overline{BC}
​​​. Simplifying:

A right triangle with

AB\overline{AB}
being hypotenuse will have the following opposite side to angle
γ\gamma
[Fig. 1(b)]:

And the adjacent side:

Applying the Pitagorean theorem in Fig. 1(a):

These relations are needed to obtain

β\beta
,
γ\gamma
, and the equations for velocities and accelerations in the following sections.


Velocities

With pin

AA
viewed as part of the disk, its velocity is:

Where

ω=2k^rad/s\vec{\omega}\,=\,2\,\hat{k}\,rad/s
for constant rotation, or
ω=(0.10+2.3α)k^rad/s\vec{\omega}\,=\,\left(0.10\,+\,2.3\,\alpha \right) \hat{k}\, rad/s
for variable rotation (
k^\hat{k}
is the unit cartesian vector in the
zz
-direction). For
α\alpha
going from
0o0^{o}
​to
90o90^{o}
​​ counterclockwise starting at the lowest position,
ω\omega
varies from
0.10rad/s0.10\,rad/s
to
3.7rad/s3.7\,rad/s
, giving
1.9rad/s1.9\,rad/s
for the average angular velocity. We can thus write:

Taking the time derivative of

ω\omega
we get:

And

With pin

BB
as part of the output bar
BCBC
, its velocity is:

Where

ωBC\vec{\omega}_{BC}
is the still unknown angular velocity vector of bar
BCBC
. The motion of linking bar
ABAB
can be described using Chasles theorem
[12]
Shames, I. H. (n.d.). Engineering Mechanics: Statics and Dynamics.
:

Where

ωAB\vec{\omega}_{AB}
​​is the still to be determined angular velocity vector.
VA\vec{V}_A
​​and
VB\vec{V}_B
are always tangential to the circular trajectories of pins
AA
and
BB
. For the mechanism’s intermediate position shown in Fig. 1(b),
xx
and
yy
components for both velocities can be written as:

The fixed vectors in the rigid bodies going from

OO
to
AA
(
ρOA\vec{\rho}_{OA}
),
CC
to
BB
(
ρCB\vec{\rho}_{CB}
​​), and
BB
to
AA
(
ρBA\vec{\rho}_{BA}
) can also be expressed using their
xx
and
yy
components. An inspection of Fig. 1(b) allows one to write:

Solving the vector algebra of Eq. (14) using Eqs. (15) – (19):

and

Where

Having obtained

VB\vec{V}_B
, Eq. (13) can be used to determine
ωBC\vec{\omega}_{BC}
.


Accelerations

Here we need to distinguish between the constant and the variable

ω\omega
scenarios. Pin A only has radial acceleration with constant
ω\omega
, 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

aA\vec{a}_{A}
,
aBR\vec{a}_{BR}
, and
aBT\vec{a}_{BT}
can be expressed in terms of unit vectors
i^\hat{i}
and
j^\hat{j}
​​:

Substituting back into Eq. (23) and working the vector algebra, we obtain expressions for angular accelerations of bars

ABAB
and
BCBC
. Since the expressions are long, they are grouped into four different terms as
AB1AB_1
through
AB5AB_5
​​ for
ωAB\omega_{AB}
, and
BC1BC_1
through
BC5BC_5
for
ωBC\omega_{BC}
(see the appendix for details):

Variable input rotation rate

Since the disk now has a variable rate of rotation

ω˙\dot{\omega}
, 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

ABAB
and
BCBC
are written as:

And,

Modified mechanism

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.

[13]
Yan, H.-T. (n.d.). Innovative design of a variable intermittent slider-crank mechanism. International Journal of Mechanical Engineering Education, 34(4), 201–214.
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
[14]
Sinatra, R., & Angeles, J. (n.d.). A Novel Approach to the Teaching of Planar Mechanism Dynamics - A Case Study. International Journal of Mechanical Engineering Education, 31(3), 349–363.
.

Figure  2: Design modification to allow full disk revolution. The initial  position is shown and the disk rotates 90 degrees clockwise.
Figure 2: Design modification to allow full disk revolution. The initial position is shown and the disk rotates 90 degrees clockwise.

Geometrical constraints

Trigonometric functions follow the same short-hand notation described earlier. The relation between horizontal displacements for pins A and B are now given by:

Simplifying :

A right triangle with AB as the hypotenuse gives:

Velocities

An intermediate position will have a velocity vector for pin A with negative

xx
and positive
yy
components respectively. Pin B’s velocity will have both negative components:

The fixed vectors in the rigid bodies from O to A (

ρOA\vec{\rho}_{OA}
), C to B (
ρCB\vec{\rho}_{CB}
), and B to A (
ρBA\vec{\rho}_{BA}
) can be expressed using their
xx
and
yy
components for an intermediate position:

Similarly to what was done earlier, we find

VBV_B
​​ and
ωAB\omega_{AB}
as functions of
VAV_A
:

and

Where

Accelerations

The linear accelerations

aA\vec{a}_A
,
aBR\vec{a}_{BR}
and
aBT\vec{a}_{BT}
can be expressed in terms of unit vectors
i^\hat{i}
and
j^\hat{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

[15]
Fleischfresser, L. (n.d.). Uso de Programa Computacional no Ensino de Dinâmica para Engenharia. In Anais do XLII Congresso Brasileiro de Educação em Engenharia. Retrieved from http://198.136.59.239/ abengeorg/cobenge-2014/Artigos/128970.pdf
and the Fortran code is available
[16]
Fleischfresser, L. (n.d.). Scaling-Octo-Tribble.
.

Table 1: Angular acceleration terms of linking bar AB for the three configurations.
Table 1: Angular acceleration terms of linking bar AB for the three configurations.


Table 2: Angular acceleration terms of output bar BC for the three configurations.
Table 2: Angular acceleration terms of output bar BC for the three configurations.

Numerical experiments


Original mechanism: constant angular velocity

In figure 3 we plot angles, velocities and accelerations for a

90o90^{o}
turn of the disk. The inversion of the sense of rotation for the linking bar
ABAB
takes place at
30o30^{o}
(top two panels):
γ\gamma
reaches a maximum and then starts to decrease, and
ωAB\omega_ {AB}
changes from counterclockwise to clockwise (positive to negative).
VBV_{B}
remains nearly equal to the constant
VAV_{A}
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\omega_{AB}
,
ωBC\omega_{BC}
,
ω˙AB\dot{\omega}_{AB}
and
ω˙BC\dot{\omega}_{BC}
at
α=0o\alpha\;=\;0^{o}
match the solution presented in
[12]
Shames, I. H. (n.d.). Engineering Mechanics: Statics and Dynamics.
, validating our implementation.

Figure  3: Top left: β and γ vs. α. Bottom left: linear velocities of pins A  and B vs. α. Top right: angular velocities of bars AB and BC vs. α .  Bottom right: angular accelerations of bars AB and BC vs. α. Angular  step size ∆α = 1 deg.
Figure 3: Top left: β and γ vs. α. Bottom left: linear velocities of pins A and B vs. α. Top right: angular velocities of bars AB and BC vs. α . Bottom right: angular accelerations of bars AB and BC vs. α. Angular step size ∆α = 1 deg.


Table 3: Head and tail of linear and angular velocities. See text for remarks on cells with *.
Table 3: Head and tail of linear and angular velocities. See text for remarks on cells with *.


In Table 3, cells with an asterisk highlight the same numerical values. For example, the calculated value for

ωAB\omega_{AB}
is equal to
ω\omega
and
VBV_{B}
at
α=0o\alpha\;=\;0^{o}
. 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
VAV_{A}
,
ωAB\omega_{AB}
and
ωBC\omega_{BC}
also have the same absolute values at
α=90o\alpha\;=\;90^{o}
​. Note that the linking bar
ABAB
stops at the same inclination of its initial position (​
γ=16.7o\gamma\;=\;16.7^{o}
).

Table 4: Head and tail of angular accelerations.
Table 4: Head and tail of angular accelerations.

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\alpha\;=\;30^{o}
​​. Note that
VBV_B
again matches
VBV_B
(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
ABAB
starts decelerating at
α=15o\alpha\;=\;15^{o}
and the output bar
BCBC
starts at
α=50o\alpha\;=\;50^{o}
. 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\omega
case. Here, for example, a calculation for
α=90o\alpha\;=\;90^{o}
yields equal values for
ωAB\omega_{AB}
and
ωBC\omega_{BC}
​​. 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.

Figure  4: Top left: β and γ vs. α. Bottom left: linear velocities of pins A  and B vs. α. Top right: angular velocities of bars AB and BC vs. α.  Bottom right: angular accelerations of bars AB and BC vs. α. Angular  step size ∆α = 1 deg.
Figure 4: Top left: β and γ vs. α. Bottom left: linear velocities of pins A and B vs. α. Top right: angular velocities of bars AB and BC vs. α. Bottom right: angular accelerations of bars AB and BC vs. α. Angular step size ∆α = 1 deg.


Table 5: Head and tail of linear and angular velocities. See text for remarks on cells with *.
Table 5: Head and tail of linear and angular velocities. See text for remarks on cells with *.
Table 6: Head and tail of angular accelerations.
Table 6: Head and tail of angular accelerations.

Modified mechanism

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\omega_{AB}
is always negative (clockwise rotation). This can also be understood by noting that
γ\gamma
is always decreasing (top left panel). Bar
ABAB
has a zero-crossing in its angular acceleration (bottom left panel) indicating the bar starts decelerating at
α54o\alpha\;\approx\;54^{o}
. Bar
BCBC
is always turning counterclockwise since
ωBC>0\omega_{BC}\;>\;0
, but continually decelerating
(ω˙BC<0)\left(\dot{\omega}_{BC}\;<\;0 \right)
. Shaded cells in Table 7 have the same meaning as before.

Figure  5: Top left: β and γ vs. α. Bottom left: linear velocities of pins A  and B vs. α. Top right: angular velocities of bars AB and BC vs. α.  Bottom right: angular accelerations of bars AB and BC vs. α. Angular  step size ∆α = 1 deg.
Figure 5: Top left: β and γ vs. α. Bottom left: linear velocities of pins A and B vs. α. Top right: angular velocities of bars AB and BC vs. α. Bottom right: angular accelerations of bars AB and BC vs. α. Angular step size ∆α = 1 deg.


Table 7: Head and tail of linear and angular velocities. See text for remarks on cells with *.
Table 7: Head and tail of linear and angular velocities. See text for remarks on cells with *.
Table 8: Head and tail of angular accelerations.
Table 8: Head and tail of angular accelerations.

The motion of the modified mechanism has one peculiarity not present in the previous two cases. At the end of the

90o90^{o}
turn of the disk, both bars will be horizontal, triggering a division by zero in the calculations of
ωAB˙\dot{\omega_{AB}}
and
ω˙BC\dot{\omega}_{BC}
. The computation is thus stopped at
α=89o\alpha\;=\;89^{o}
​. Angular velocities are not affected as both numerator and denominator of the
ff
parameter approach zero (Eq. 53). Scripts and data files to make Figures 3, 4 and 5 are available in
[17]
Fleischfresser, L. (2016). Plot Files for Motion Analysis of a Device Including a Disk and Two Slender Bars With a Design Change for Full Disk Revolution. Figshare. https://doi.org/10.6084/m9.figshare.3840429
.


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

90o90^{o}
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\alpha
, 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
90o90^{o}
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 (

[2]
Stewart, D. E. (2000). Rigid-Body Dynamics with Friction and Impact. SIAM Review, 42(1), 3–39. https://doi.org/10.1137/s0036144599360110
,
[3]
Garstenauer, H. (2006). A Unified Framework for Rigid Body Dynamics. A-4040 Linz - Altenberger Strasse 69.
,
[18]
Chatterjee, A. (n.d.). Rigid Body Collisions: Some General Considerations, New Collision Laws, and Some Experimental Data.
).

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

ρBA\vec{\rho}_{BA}
and performing the vector products:

Now we separate Eq. (64) into its

xx
- and
yy
-components, obtaining a system of two equations with two unknowns (
ω˙AB\dot{\omega}_{AB}
and
ω˙BC\dot{\omega}_{BC}
).

Multiplying the

xx
-component by
oo
, the
yy
-component by
n-n
and summing up, the result will have only
ω˙AB\dot{\omega}_{AB}
as the unknown:

Solving Eq. (66) for

ω˙AB\dot{\omega}_{AB}
:

Eq. (67) can be compared to Eq. (29) to identify terms

AB1AB_1
through
AB5AB_5
. To get
ω˙BC\dot{\omega}_{BC}
, we now multiply the
xx
-component by
qq
, and the
yy
-component by
pp
. Summing the two component equations will result in a single equation with
ω˙BC\dot{\omega}_{BC}
as the only unknown:

Finally, solving it for

ω˙BC\dot{\omega}_{BC}
​:

And Eq. (69) can be compared to Eq. (32) to identify terms

BC1BC_1
through
BC5BC_5
.

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