P. M. ALABUZHEV, O. D. ALIMOV, B. A. SHEKHOVTSEV

APPLICATION OF THE THEORY OF SIMILARITY AND DIMENSIONS TO THE SELECTION OF MOTOR-VEHICLE PARAMETERS

The theory of similarity and modeling is widely used in engineering [3, 6, 9, 11, 12]; however, we know of no works that apply this theory to the dynamics of motor vehicles. Meanwhile, having a tested specimen of a motor vehicle (a model), one can, by the formulas of the theory of similarity and dimensions, within the limits of experimental data and within known boundaries, draw a conclusion about the selection of parameters for a dynamically similar motor vehicle.

The table gives the principal parameters of widely used passenger cars and trucks. We shall show that, within the limits of this table, dynamic similarity exists among certain types of motor vehicles.

Let us agree to denote the coefficients of similarity by the same letter as that used to denote the given quantity [1], but with the subscript “c.” The basic specimen adopted as the model will be marked with the subscript “1,” and each subsequent specimen similar to the model will be marked by the same letter but without a subscript. Then, assuming that the forces are oriented identically at corresponding points of the model and of the full-scale object, we have, under similarity:

\[ \begin{array}{ll} \text{geometric} & t_c = \dfrac{t}{t_1}; \quad l_c = \dfrac{l}{l_1}; \\[1.0em] \text{material} & m_c = \dfrac{m}{m_1} = \dfrac{P}{P_1} = \rho_c \cdot l_c^3; \\[1.0em] \text{dynamic} & F_c = \dfrac{F}{F_1} = m_c \cdot \dfrac{l_c}{t_c^2}; \end{array} \]

whence follows the relation of velocities

\[ v_c = \dfrac{v}{v_1} = \dfrac{l_c}{t_c}; \]

of accelerations

\[ a_c = \dfrac{a}{a_1} = \dfrac{l_c}{t_c^2}; \]

of numbers of revolutions

\[ n_c = \dfrac{n}{n_1} = \dfrac{\omega}{\omega_1} = \omega_c = t_c^{-1}; \]

of works

\[ A_c = \dfrac{A}{A_1} = m_c \cdot v_c^2 = m_c \cdot \dfrac{l_c^2}{t_c^2}; \]

of powers

\[ N = \dfrac{N}{N_1} = \dfrac{A_c}{t_c} = m_c \cdot \dfrac{l_c^2}{t_c^3} = m_c \cdot v_c^2 \cdot n_c. \]

When one and the same machine operates under a different regime of inertial loading (i.e., when $\rho_c = l_c = 1$), the ratio of the forces developed is $F_c = A_c = n_c^2$, and the ratio of powers $N_c = n_c^3$ is proportional to the cube of the ratio of the numbers of revolutions of the driven shaft. Thus, in order to double the speed of one and the same machine, its power must be increased 8-fold. “The part of indicated power that is expended on losses is called friction power” [7] and amounts to 10–15% of the indicated power. The friction power varies in proportion to the square of the crankshaft speed, $N_{\text{fr}} = n_c^2$. The different character of the resistance forces during the motion of a machine has a substantial effect on the dynamic similarity of operating regimes. The magnitude of the deviation from dynamic similarity shows the degree to which the resistance forces of the motor vehicle are taken into account. It is not difficult to show that dynamic similarity exists among the motor vehicles included in Table 1. In fact, taking the Volga automobile as the model, for the Moskvich automobile we obtain the similarity coefficients

Table 1

Note: in the denominator is the maximum useful load $P_g$.

\[ l_c=\frac{4055}{4830}=0.84;\qquad m_c=\frac{990}{1460}=0.68;\qquad v_c=\frac{115}{130}=0.885; \]

\[ q_c=\frac{6.5}{13.5}=0.5;\qquad N_c=\frac{45}{70}=0.64;\qquad t_c=\frac{0.84}{0.885}=0.95;\qquad n_c=\frac{3600}{3600}=1.0. \]

According to the theory of similarity,

\[ N_c=m_c\cdot\frac{l_c^2}{t_c^3} =0.68\cdot\frac{0.84^2}{0.95^3} =0.56;\qquad \Delta N_c=\frac{0.64-0.56}{0.64}\cdot 100=12.5\%. \]

Since the length of the motor vehicle $l$ is an external parameter (one that does not directly affect operation), in this case it is better to use, for verification, the formula $N_c=m_c\cdot v_c^2\cdot n_c$ through the data of experiments:

\[ N_c=0.68\cdot 0.885^2\cdot 1.0=0.535;\qquad \Delta N_c=\frac{0.11\cdot 100}{0.64}=17\%. \]

Then it may be assumed that \(q_c = N_c\) with a maximum error of

\[ \frac{N_c - q_c}{N_c}\cdot 100 = \frac{0.64 - 0.5}{0.64}\cdot 100 = 21\%. \]

For the “Chaika” passenger car, the similarity coefficients are

\[ l_c = \frac{5600}{4830} = 1.16;\qquad m_c = \frac{2500}{1460} = 1.71;\qquad v_c = \frac{160}{130} = 1.23; \]

\[ q_c = \frac{15}{13.5} = 1.1;\qquad N_c = \frac{195}{70} = 2.78;\qquad t_c = \frac{1.16}{1.23} = 0.945;\qquad n_c = 1. \]

According to the theory of similarity,

\[ N_c = m_c \cdot v_c^2 \cdot n_c = 1.71 \cdot 1.23^2 \cdot 1 = 2.58. \]

Here there is a discrepancy in the values of \(N_c\) from experimental data and from the calculation according to the theory of similarity,

\[ \Delta N_c = \frac{2.78 - 2.58}{2.78}\,100 = 7.2\%. \]

The discrepancy in the values of \(q_c\) and \(N_c\) (of the order of 35–60%) is explained by the more advanced fuel-utilization system of the “Chaika” passenger car. It is of interest to compare passenger cars of the “Volga” make and trucks with carburetor engines. For example, for the UAZ-450 the similarity coefficients are

\[ l_c = \frac{4350}{4830} = 0.9;\qquad m_c = \frac{1745}{1460} = 1.195;\qquad v_c = \frac{90}{130} = 0.69 \]

\[ q_c = \frac{14}{13.5} \simeq 1.0;\qquad N_c = \frac{65}{70} = 0.93;\qquad n_c = \frac{3860}{3600} = 1.08. \]

According to the theory of similarity,

\[ N_c = 1.195 \cdot 0.69^2 \cdot 1.08 = 0.59;\qquad \Delta N_c = 36.5\%. \]

If the carrying capacity of the UAZ-450 is taken into account, then, in comparison with the preceding passenger cars, there is a larger deviation from geometric, kinematic, and dynamic similarity. Here

\[ m_c = 1.72;\qquad N_c = 1.72 \cdot 0.69^2 \cdot 1.08 = 0.85;\qquad \Delta N_c = 3.6\%. \]

For the GAZ-51 truck,

\[ N_c = m_c \cdot v_c^2 \cdot n_c = \frac{5200}{1460}\cdot \left(\frac{70}{130}\right)^2 \cdot \frac{2800}{3600} = 0.82;\qquad \Delta N_c = 12\% \]

For the ZIL-164,

\[ N_c = \frac{104}{70} = 1.48;\qquad N_c = \frac{8100}{1460}\cdot \left(\frac{65}{130}\right)^2 \cdot \frac{2600}{3600} = 1.04; \]

\[ \Delta N_c = 48\%;\qquad q_c = \frac{42}{13.5} = 3.1;\qquad q_c \ne N_c. \]

Finally, let us consider the similarity between the GAZ-51 and ZIL-164 trucks. Taking the ZIL-164 as the model, the similarity coefficients will have the values

\[ l_c = \frac{5725}{6725} = 0.85;\qquad m_c = \frac{2710}{4100} = 0.65;\qquad v_c = \frac{70}{65} = 1.08; \]

\[ q_c = \frac{20}{42} = 0.48;\qquad N_c = \frac{70}{104} = 0.67;\qquad n_c = \frac{2800}{2600} = 1.08; \]

\[ N_c = 0.65 \cdot 1.08^2 \cdot 1.08 = 0.82; \qquad \Delta N_c = \frac{0.15 \cdot 100}{0.67} = 22.4\%. \]

A considerable deviation from dynamic similarity also occurs between the ZIL-164 and UAZ-450 trucks. Taking the ZIL-164 as the model, we have

\[ N_c = \frac{65}{104} = 0.62; \qquad q_c = \frac{14}{42} = 0.33; \]

\[ N_c = \frac{2495}{8100} \cdot \left(\frac{90}{65}\right)^2 \cdot \frac{3860}{2600} = 0.86; \qquad \Delta N_c = 28\%. \]

Here, apparently, there is a difference in the resistance to motion of the vehicles or inaccurate initial data. Thus, for example, with the indicated values of payload capacity, speeds, and rotational speeds \(N_c = 0.86\), then the power of the UAZ-450 truck should be \(N = 0.86 \cdot N_1 = 0.86 \cdot 104 = 89.3\) hp, or, for a specified value \(N = 65\) hp, the ZIL-164 requires a power of

\[ N_1 = \frac{65}{0.86} = 76 \text{ hp}. \]

If the actual values of powers, payload capacities, and rotational speeds are retained, then

\[ v_c = \sqrt{\frac{N_c}{m_c \cdot n_c}} = \sqrt{\frac{0.62}{0.31 \cdot 1.46}} = 1.17, \]

whence, at the ZIL-164 speed \(v_1 = 65\) km/h, the speed of the UAZ-450 truck should be \(v = 76\) km/h, instead of the tabulated 90 km/h. Or, conversely, if the speed \(v = 90\) km/h, then the speed \(v_1\) for the ZIL-164 should be increased to 77 km/h instead of the indicated 65 km/h.

Choosing the GAZ-51 truck as a prototype, we find the value of the power \(N\) for a machine dynamically similar to it at \(m_c = 8;\ v_c = 2;\ n_c = t_c = 1;\ l_c = 2\), then \(N_c = 8 \cdot 2^2 \cdot 1 = 32\); whence \(N = N_c \cdot 70 = 224\) hp. The load on the parts in the new machine will increase by

\[ F_c = \frac{m_c l_c}{t_c^2} = 8 \cdot 2 \cdot 1 = 16 \text{ times}. \]

The theory of similarity and dimensions gives a clear idea of the direction of design work in designing motor vehicles and in selecting their technical and operating characteristics. For comparison, the values of the principal parameters of 59 motor vehicles (passenger cars, trucks, buses), according to literature data [2, 8, 10], are presented in Fig. 1 in the coordinates

\[ \frac{P_m \cdot v}{N}; \qquad \frac{q \cdot l}{V}; \qquad \frac{P_g}{P_m}, \]

where
\(P_g\) — payload capacity, kg;
\(v\) — maximum speed of motion, m/sec;
\(N\) — minimum engine power, kgm/sec;
\(q\) — control fuel consumption, cm\(^3\)/m;
\(l\) — length of the motor vehicle, mm;
\(V\) — working volume, cm\(^3\);
\(P_m\) — weight of the motor vehicle in running order, kg.

Each point in the coordinates \(\dfrac{P_m \cdot v}{N};\ \dfrac{q \cdot l}{V}\) corresponds simultaneously to six quantities \((P_m, v, N, q, l, V)\), and in the coordinates \(\dfrac{q \cdot l}{V};\ \dfrac{P_g}{P_m}\) it corresponds to the ratio of the payload capacity of the motor vehicle to its own weight. For comparison, the most characteristic quantities determining the efficiency of a motor vehicle were selected: power, payload capacity, travel speed, dimensions, and economy.

In Fig. 1, two regions stand out: passenger cars and light trucks built on the basis of passenger cars (for example, point 8 corresponds—

Passenger cars

Buses and trucks

Passenger cars

Buses and trucks

Note. The arrow indicates the direction of development and improvement of the vehicles.

Fig. 1.

—corresponds to the UAZ-450 vehicle). Buses of the Bussing firm (FRG), type 13U7H, are more economical than domestic buses of the ZIL-158 type similar to them, since, with a lower fuel consumption per 100 km of travel, they have a higher payload capacity. From operational—

From the operational and economic point of view, domestic and foreign passenger cars are at the same level. In modern vehicles, especially trucks and buses, designers are following the path of reducing fuel consumption and increasing load capacity, as is evidenced in Fig. 1 by the curve in the coordinates

\[ \frac{q \cdot l}{V}; \qquad \frac{P_g}{P_m}. \]

In Fig. 1, and subsequently, the following designations were introduced.

Fig. 2.

9 — GAZ-51
10 — GAZ-63
11 — “Ural-ZIS-355M”
12 — ZIL-164

22 — “Dauphine”
23 — “Anglia”
24 — “Herald”
25 — “Record-1700”

35 — “Jaguar-2.4 standard”
36 — “Mercedes-300L”
37 — Facel Vega “Excellence”

Fig. 3.

13 — ZIL-157
38 — “Major”
39 — “Austin Princess”
40 — “Sapphire”

26 — 356 B/1600
46 — FK 4500
47 — UAZ-450V
48 — PAZ-652

54 — FK4500 tractor
55 — 12R/U54
56 — 13U7H
57 — 13R/U10

41 — L 319
42 — L 311
43 — L 322
44 — L 337
45 — FK 3500 LBAC

49 — LAZ-695B
50 — ZIL-158
51 — LK 311
52 — L 312
53 — 321

58 — TU10
59 — 14R/U10

Figure 2 presents curves in relative units

\[ N_c;\ q_c;\ V_c;\ P_c;\ n_c;\ v_c . \]

As the specimen (the standard for comparison), the widely used Volga automobile was taken.

Fig. 4.

Fig. 5.

The curves in Fig. 2 show that with an increase in engine power, \(v\), \(q\), \(V\), \(P_m\) increase. The crankshaft speed \(n\), with increasing \(N\), remains almost unchanged. An increase in the weight of the automo-

Fig. 6.

Axes visible in the figure: \(q\); \(N\); [[unclear: right-hand horizontal-axis label and units]].

tires as compared with the “Volga” proceeds intensively up to a value of 1.6, and then ceases. The weight of “heavy” passenger cars remains almost unchanged. The speed of motion, with an increase in \(N\), increases insignificantly; but at the same time the engine displacement \(V\) (litre capacity) increases sharply.

Figure 3 shows curves in relative units for trucks with a carburetor engine. The GAZ-51 was taken as the standard of comparison. For trucks, with an increase in \(N\), \(v\) and \(n\) decrease, but \(V\) increases sharply. In terms of fuel consumption per 100 km of travel, trucks are divided into two groups: a) domestic, b) foreign. For foreign vehicles, with an increase in \(N\), \(q\) increases only slightly, which cannot be said of domestic vehicles.

Figure 4 presents curves in relative units for buses, which were compared with the LAZ-695B. The character of the curves in Fig. 4 is analogous to the curves of Fig. 3, since many buses have a truck engine.

Figure 5 shows curves for motor vehicles with a diesel engine, which were compared with the YaAZ-214. For diesel vehicles, an increase in \(N\) reduces \(v\), \(n\), but at the same time \(P_z\), \(q\), and \(V\) increase sharply.

Figure 6 presents curves in the coordinates \(q\), \(N\) and the dimensionless complex of quantities

\[ \frac{P_m \cdot v}{N}. \]

Here two regions were distinguished: those of trucks and passenger cars. For passenger cars an increase in \(N\) raises \(q\) only slightly, whereas for trucks \(q\) increases sharply. In Fig. 6, in the coordinates \(q\);

\[ \frac{P_m \cdot v}{N} \]

two regions were also distinguished: those of trucks and passenger cars.

Conclusion

Dynamic similarity holds for motor vehicles built according to one and the same design scheme. The methods of the theory of similarity and dimensions can be successfully applied to the selection and calculation of the basic parameters of motor vehicles, as well as to the study of regimes of motion and the loads arising in components. In the absence of dynamic similarity among motor vehicles built according to different principles, in dimensionless coordinates it is possible to encompass the regions of parameters of existing vehicles and the direction of historical development in their design. Thus, for example, from Fig. 1 it is evident:

a) the region of parameters of passenger cars is quite limited and has stabilized over time, which indicates that designers have approached a certain limit of the possibilities for choosing more optimal parameters for motor vehicles operating on known principles. To create a new quality of motor vehicles, new principles are needed;

b) buses and trucks show a tendency (historically over time) toward increasing load capacity, increasing engine displacement (litre capacity), and reducing fuel consumption.

Graphs constructed in relative coordinates possess the greatest clarity and universality.

The methodology set forth by us can be extended to various types of transport.

LITERATURE

1. P. M. Alabuzhev, Application of the theory of similarity and dimensions to the study (modeling) of impact-action machines, Izvestiya TPI, vol. 73, Tomsk, 1952.
2. V. I. Anokhin, Domestic Motor Vehicles, Mashgiz, 1961.
3. G. A. Barsov, L. B. Bezmenov, L. G. Grozdenskaya, A. V. Zhiligovskaya, G. A. Kuvshinnikov, I. O. Kulbachevsky, S. I. Pantele-

[[unclear: beginning of reference 3]]ev, Zh. I. Zhevets, V. V. Yudenich. Theory of Plane Mechanisms and Dynamics of Machines. Izvestiya Vysshei Shkoly, Moscow, 1961.

1. V. A. Venikov. Problems of modeling and the theory of similarity, journal Vestnik Vysshei Shkoly, No. 2, Izvestiya Vysshei Shkoly, Moscow, 1963.

2. V. L. Kirpichev. Conversations on Mechanics. GITTL, Moscow—Leningrad, 1957.

3. M. V. Kirpichev. Theory of Similarity. Academy of Sciences of the USSR, Moscow, 1953.

4. A. V. Maksai, N. I. Polyansky. Theory of Aircraft Engines. Military Publishing House of the Ministry of Defense of the USSR, Moscow, 1950.

5. I. P. Plekhanov, V. A. Chernyakin, S. V. Pampel. Driver’s Handbook. Avtotransizdat, 1956.

6. A. V. Reznyakov. A Short Handbook on the Theory of Similarity. Publishing House of the Academy of Sciences of the Kazakh SSR, Alma-Ata, 1950.

7. G. I. Samal. Foreign Motor Vehicles, Transactions of NAMI, 1960.

8. L. I. Sedov. Methods of the Theory of Similarity and Dimensions in Mechanics. GITTL, Moscow, 1957.

9. Collection Theory of Similarity and Modeling. Publishing House of the Academy of Sciences of the USSR, Moscow, 1951.