# Comparison of Assemblies of Four-Link Structural Groups of 3 rd Class on the Transmission Angle

I.N. Matsyuk, I.I. Morozova, E. Shlyakhov

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I.N. Matsyuk, I.I. Morozova, E. Shlyakhov. Comparison of Assemblies of Four-Link Structural Groups of 3 rd Class on the Transmission Angle. Mechanics, Materials Science & Engineering Journal, 2016, 10.13140/RG.2.1.2348.6969 . hal-01367496

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<figure><img src="image_1.png" /></figure>
# Comparison of Assemblies of Four-Link Structural Groups of 3rd Class on the Transmission Angle

Matsyuk I.N.¹, Morozova T.I.¹,ᵃ, Shlyakhov E.M.¹,ᵇ

1 – State Higher Educational Institution “National Mining University”, city of Dnipropetrovsk, Ukraine,

a – tanyaDP@mail.ru

b – shlyahove@nmu.org.ua

<figure><img src="image_2.png" /></figure>

DOI 10.13140/RG.2.1.2348.6969

**Keywords:** Mathcad, planar mechanism, vector, complex number, group assemblies, transmission angle, structural group of third class.

**ABSTRACT.** Comparison of various assemblies of four-link structural group of 3rd class with revolute joints on the transmission angle is performed. Equations of the trajectories of plug points of one of the groups of joint are obtained to determine transmission angles. Derived functions of these equations enable to determine the values of transmission angles for each assembly group. It is shown that only two assemblies of maximum possible assembling number of such group (six) have practical value. The solution of this problem was performed with the help of Mathcad program.

**Introduction.** One of the criteria determined synthesis quality of planar linkage is a parameter, which characterizes the quality of power transmission by one link to another one.

Historically, pressure angle according to the theory of mechanism in Russian scientific literature is taken as this parameter. It is the angle between the vector of power transmission and the velocity vector of application point of this power.

The term “transmission angle” is often used in English literature. Transmission angle is the angle between the vectors of absolute and relative velocities of the common link points. For example, as for crank-rocker mechanism it is the angle between coupler and rocker. It is more difficult to determine this angle within mechanisms with the great number of links (for instance, between two couplers).

It is well known, that these angles complement each other up to 0,5 π radian.

**Analysis of the recent research.** In classical problem setting the task of synthesis of the planar crank-rocker mechanism was formulated as follows – to determine link lengths in terms of the given rocker rotation angle and to provide for minimal deviation of the transmission angle from 90°.

In other words, the first synthesis criterion is to provide for the given angle of rocker rotation when crank makes a complete turn. This task will be solved geometrically in [1] while designing the drive of ratchet-wheel where desired turn angle was formed by the end points of the rocker. Analytical definition of the dead rocker positions is given in [2].

The problem of determining of the transmission angle for crank-rocker mechanism was set and solved in 1972 in [2], however, the papers devoted to various aspects of this problem for relatively simple mechanism appears up to the present days [3, 4, 5].

**Formulation of work objectives.** Therefore, the problem of estimation of this parameter for mechanisms with the complex structure especially for mechanisms of 3rd class is of great interest.
E.E. Peisakh in the paper [6] gives an example of determining six variants of four-link structural group of 3rd class assemblies containing revolute joints.

Visualization of this example was performed by one of the authors of this paper in [7] with the help of Mathcad program. It enables to evaluate approximately the values of the transmission angle from the coupler to the basic link within various assemblies.

**Statement of the basic material.** Let's take as an example mechanism (Fig. 1) containing four-link structural group of 3rd class. This group was taken from [6]. It is shown that it has six variants of assemblies under the fixed position of its external joints.

<figure><img src="image_3.png" /><figcaption>Fig. 1. Diagram of 3rd class mechanism.</figcaption></figure>

Research task is to determine the values of the transmission angles between coupler BC and basic link CDF for each of the possible assemblies which are the part of the given mechanism.

Coordinates of the external linkage joint of the group: $x_B = -10$; $y_B = 0$; $x_E = 19,5$; $y_E = -122$; $x_G = 91,5$; $y_G = -122$. Other parameters of the group have following values: $l_2 = 78$; $l_3 = 70$; $l_4 = 78$; $l_5 = 70$; $l_{31} = 135$; $l_{32} = 70$.

Vector interpretation of the group links is given in Fig. 2.

<figure><img src="image_4.png" /><figcaption>Fig. 2. Vector image of the 3rd class group.</figcaption></figure>
To determine the angles of the transmission angle let's disconnect joint C. For each point of the plugs (C₂ within the link 2 and C₃ within the link 3) we can derive an equation of their trajectories.

Radius circle l₂ with the centre in the point B will be the trajectory of the point C₂. The trajectory of the point C₃ is determined by the four-link chain geometry EDFG.

To obtain the trajectory of the point C₃ we use Freudenstein equation for jointed four-link chain [2]. For four-link chain EDFG it will be:

$$ \cos(\varphi_5 - \varphi_4) = k_1 \cos \varphi_5 - k_2 \cos \varphi_4 + k_3. \quad (1) $$

Coefficients of this equation have the values:

$$ k_1 = \frac{l_0}{l_4}, \quad k_2 = \frac{l_0}{l_5}, \quad k_3 = \frac{l_0^2 + l_4^2 + l_5^2 - l_3^2}{2l_4l_5}. \quad (2) $$

Using universal trigonometric substitution:

$$ \sin \varphi_4 = \frac{2tg \frac{\varphi_4}{2}}{1 + tg^2 \frac{\varphi_4}{2}}, \quad \text{and} \quad \cos \varphi_4 = \frac{1 - tg^2 \frac{\varphi_4}{2}}{1 + tg^2 \frac{\varphi_4}{2}}, \quad (3) $$

And replaced:

$$ x = tg \frac{\varphi_4}{2}, \quad (4) $$

We obtain quadratic equation:

$$ A(\varphi_5)x^2 + B(\varphi_5)x + C(\varphi_5) = 0. \quad (5) $$

Coefficients of this equation are:

$$ A(\varphi_5) = -(1 + k_1)\cos \varphi_5 - k_2 - k_3, \quad (6) $$

$$ B(\varphi_5) = 2\sin \varphi_5, \quad (7) $$

$$ C(\varphi_5) = (1 - k_1)\cos \varphi_5 + k_2 - k_3. \quad (8) $$
Solution of quadratic equation enables to find out two values of the angle $\phi_4$. From now on mathematical expressions are given in the form of Mathcad - 11 program fragments.

$$ \phi_{4a}(\phi_5) := 2 \operatorname{atan}\left(\frac{-B(\phi_5) + \sqrt{B(\phi_5)^2 - 4A(\phi_5)C(\phi_5)}}{2A(\phi_5)}\right) \\ \phi_{4b}(\phi_5) := 2 \operatorname{atan}\left(\frac{-B(\phi_5) - \sqrt{B(\phi_5)^2 - 4A(\phi_5)C(\phi_5)}}{2A(\phi_5)}\right) \qquad (9) $$

After that, we can obtain two vector values of the point C₃.

$$ \begin{align*} l_{3a}(\phi_5) &:= l_4 \cdot \exp(\phi_{4a}(\phi_5) \cdot i) - l_0 \cdot \exp(i \cdot \beta) - l_5 \cdot \exp(i \cdot \phi_5) \\ l_{3b}(\phi_5) &:= l_4 \cdot \exp(\phi_{4b}(\phi_5) \cdot i) - l_0 \cdot \exp(i \cdot \beta) - l_5 \cdot \exp(i \cdot \phi_5) \\ \phi_{3a}(\phi_5) &:= \arg(l_{3a}(\phi_5)) \qquad \phi_{3a}(\phi_5) = -80 \, \text{deg} \\ \phi_{3b}(\phi_5) &:= \arg(l_{3b}(\phi_5)) \qquad \phi_{3b}(\phi_5) = 161.509 \, \text{deg} \\ l_{Ca}(\phi_5) &:= z_E + l_0 \cdot \exp(i \cdot \beta) + l_5 \cdot \exp(\phi_5 \cdot i) + l_{31} \cdot \exp[(\phi_{3a}(\phi_5) + \alpha) \cdot i] \\ l_{Cb}(\phi_5) &:= z_E + l_0 \cdot \exp(i \cdot \beta) + l_5 \cdot \exp(\phi_5 \cdot i) + l_{31} \cdot \exp[(\phi_{3b}(\phi_5) + \alpha) \cdot i] \end{align*} \qquad (10) $$

Two closed trajectories of the point C₃ are as follows (Fig. 3).

<figure><img src="image_5.png" /><figcaption>Fig. 3. Two possible trajectories of the point C₃.</figcaption></figure>

Equation of the point C₂ trajectory
$$z(\phi_2) := 1_2(\cos(\phi_2) + 1i \cdot \sin(\phi_2)). \qquad (11)$$

It's easy to see that the circle with the centre in the point B (-10; 0) and radius $l_2 = 78$ cannot have common points with the low trajectory. Thus, possible assemblies can exist only on the upper trajectory. Let's draw the trajectory of the point C₂.

<figure><img src="image_6.png" /><figcaption>Fig. 4. On determining possible assemblies of 3rd class group</figcaption></figure>

Therefore, there are four common points of trajectories in Fig. 4. They confirm the existence of six assemblies of the given group of 3rd class. The same result was obtained earlier by one of the authors of this paper and described in [7].

Table 1 below shows angle values, which determine the position of group links in various assemblies.

<table><caption>Table 1. Angle values determining group assemblies (radian)</caption><thead><tr><th>Assembly</th><th>1</th><th>2</th><th>3</th><th>4</th><th>5</th><th>6</th></tr></thead><tbody><tr><td>φ<sub>2</sub></td><td>3.72</td><td>4.254</td><td>4.646</td><td>5.148</td><td>5.82</td><td>6.15</td></tr><tr><td>φ<sub>5</sub></td><td>2.791</td><td>1.592</td><td>0.97</td><td>5.438</td><td>4.367</td><td>3.591</td></tr></tbody></table>

Let us put

$$x(\phi_5) := \operatorname{Re}(l_{\mathrm{Cb}}(\phi_5)) \quad y(\phi_5) := \operatorname{Im}(l_{\mathrm{Cb}}(\phi_5)). \qquad (12)$$

These expressions are parametric equations of the trajectory point C₃ which parameter is the turn angle of the link 5. For the trajectory of the point C₂ analogue equations will be:
$$x(\phi_2) := \mathrm{Re}(z(\phi_2)) \quad y(\phi_2) := \mathrm{Im}(z(\phi_2)), \qquad (13)$$

where the angle of the link turn 2 is a parameter.

Let us determine the derived functions described the trajectories of the points C₂ and C₃.

$$g_1(\phi_2) := \mathrm{atan} \left( \frac{\frac{d}{d\phi_2} \mathrm{Im}(z(\phi_2))}{\frac{d}{d\phi_2} \mathrm{Re}(z(\phi_2))} \right) \qquad g_2(\phi_5) := \mathrm{atan} \left( \frac{\frac{d}{d\phi_5} \mathrm{Im}(1_{\mathrm{Cb}}(\phi_5))}{\frac{d}{d\phi_5} \mathrm{Re}(1_{\mathrm{Cb}}(\phi_5))} \right). \qquad (14)$$

As is known, their values in the intersection points of trajectories are tangents of slope angles concerning trajectories. Tangent as for the trajectory of the point C₃ characterizes the line of action of the absolute velocity of point C. Tangent as for trajectory of the point C₂ is the line of action of the relative velocity of point C. The angle between these tangents is the transmission angle between couplers 2 and 3 of the mechanism. Its diagram is given in Fig. 1.

Values of the acute transmission angles determined for six mechanism assemblies are given in the table 2.

Rational values of the transmission angles [2]:

$$40^{\circ} < \gamma < 140^{\circ}.$$

<table><caption>Table2. Transmission angles γ (degree)</caption><tr><td>Assembly</td><td>1</td><td>2</td><td>3</td><td>4</td><td>5</td><td>6</td></tr><tr><td>γ</td><td>63.652</td><td>18.886</td><td>19.581</td><td>32.703</td><td>28.821</td><td>57.241</td></tr></table>

**Summary.** On the basis of the study we can draw the following conclusions:

- Only the first and the sixth assemblies of possible ones can be adaptable for given mechanism;

- The four rest assemblies have only theoretical interest;

- While synthesizing complex mechanisms, which can have various assemblies it is required to approach minimal number of assemblies where the probability of obtaining rational values of the transmission angle is higher;

- If four-link group of 3rd class has maximum number of assemblies (six) it is required to be at least one of the work drivers of the crank.

**References**

[1] H. Alt. Ueber die Totlagen des Gelenkvierecks, Z.A.M.M. 1925, Vol. 5 (No. 1), 347-354.

[2] F. Freudenstein, E.J. Primrose, The classical transmission-angle problem, The Institution of Mechanical Engineers, C96/72, Mechanisms 1972, London.
[3] Eres Söylemez, Classical transmission-angle problem for slider-crank mechanism, Mechanism and Machine Theory, Vol. 37 (2002), 419-425, doi: 10.1016/S0094-114X(01)00083-0.

[4] S. Youliang, Optimization design of crank rocker mechanism based on maximum of minimum transmission angle, Journal of machine design, Vol. 31, Jul. 2014, pp. 29-33.

[5] Y. Jinhu, Research of the Transmission Angle Function of Offset Crank Rocker Mechanism, Journal of Mechanical, Vol. 38, Aug. 2014, pp. 43-46.

[6] E.E. Peisakh, Determining link position of three and two work drivers of four-link Assura with turning pairs // Mechanical engineering. – 1985. -No 5. – P. 55-61.

[7] I.N. Matsyuk, Geometrical analysis of two work driver structural group of 3rd class in the Mathcard program. Bulletin of Kryvyi Rih Technical University – Kryvyi Rih, 2013. – Vol. 35. – P. 209-213.