On the Equation of Motion of a Crank-and-Rod Mechanism

P. M. Alabuzhev

The application of Lagrange equations of the second kind to problems of determining inertia forces or the moments due to them in general form has been considered by Academician I. I. Artobolevsky in the university course Theory of Mechanisms and Machines [1]. However, very often engineers, and even researchers, are not familiar with the equation of motion of a machine (or mechanism) in general form, when the moment of inertia is not a constant quantity. Evidently, this is explained by the fact that in courses on theoretical mechanics the equation of rotational motion of a rigid body about a fixed axis is given in the form:

\[ I \cdot \varepsilon = M, \tag{1} \]

where \(I = \mathrm{const}\) is the moment of inertia of the body about the fixed axis of rotation;

\[ \varepsilon = \frac{d\omega}{dt} = \frac{d^2 \varphi}{dt^2} \]

is the angular acceleration of the rotating rigid body;

\(M\) is the resultant torque on the shaft due to external forces \((M = M_d - M_c,\) where \(M_d\) is the moment of the driving forces, \(M_c\) is the moment of the resistance forces).

In most courses on applied mechanics and internal-combustion engines, the kinetic-energy theorem is usually used to express the law of motion of a machine. Nevertheless, sometimes in the study of machines with a crank-and-rod mechanism one may also encounter formula (1), when there are sufficient grounds for approximately considering the reduced moment of inertia \(I\) to be constant (for example, in the case of a large flywheel).

However, in practice there are high-speed machines with a crank-and-rod mechanism (for example, an electric percussion hammer) for which the moment of inertia \(I_n\) of the moving masses is a variable quantity, depending on the angle of rotation of the crank \(\varphi\) and varying within considerable limits. Mechanical application of formula (1) may lead to significant errors.

The purpose of the present work is to derive the equation of motion of the principal link (shaft) of a machine with a crank-and-rod mechanism \((I_n \ne \mathrm{const})\), relying here on propositions that do not go beyond the framework of a university course in theoretical mechanics. In addition, other methods of proving the general formula are presented, and remarks are given concerning certain results.

Let us first consider the present question on the basis of energy considerations: the kinetic energy of the system of a crank-and-rod mechanism may be determined by the formula

\[ T = T_1 + T_2 + T_3 = \frac{I_n \cdot \omega^2}{2}, \tag{2} \]

where \(T_1=\dfrac{1}{2} I_1\omega^2\) is the kinetic energy of the flywheel and crankshaft; here \(I_1=\mathrm{const}\) and \(\omega=\omega_1\);

\[ T_2=\frac{1}{2}m_2 v_c^2+\frac{1}{2}I_c\omega_2^2 \]

is the kinetic energy of the connecting rod, determined by König’s theorem;

\[ T_3=\frac{1}{2}m_3v_3^2 \]

is the kinetic energy of the reciprocating mass reduced to the piston;

\[ I_n=I_1+I_2+I_3 =I_1+m_2\left(\frac{v_c}{\omega}\right)^2 +m_2 k^2\left(\frac{\omega_2}{\omega}\right)^2 +m_3\left(\frac{v_3}{\omega}\right)^2 \]

is the reduced moment of inertia with respect to the shaft and is a function of the angle \(\varphi\), i.e. \(I_n=f(\varphi)\).

Let us use the equation of motion in Lagrange form:

\[ \frac{d}{dt}\left(\frac{\partial T}{\partial \dot q}\right) -\frac{\partial T}{\partial q}=Q . \tag{3} \]

Since we have a system with one degree of freedom, the generalized coordinate \(q\) here is the angle \(\varphi\); then \(\dot q=\dot\varphi=\omega\), the angular velocity. The generalized force corresponding to the coordinate \(\varphi\) is the moment \(M\). Having the expression for the kinetic energy

\[ T=\frac{\omega^2}{2}\cdot I_n, \]

we compose:

\[ \frac{\partial T}{\partial q} = \frac{\partial T}{\partial \varphi} = \frac{\omega^2}{2}\cdot \frac{dI_n}{d\varphi}; \qquad \frac{\partial T}{\partial \dot q} = \frac{\partial T}{\partial \omega} = \omega\cdot I_n \]

\[ \frac{d}{dt}\left(\frac{\partial T}{\partial \dot q}\right) = \frac{d}{dt}\left(\omega\cdot I_n\right) = \frac{d\omega}{dt}\cdot I_n + \omega\frac{dI_n}{dt} = \varepsilon\cdot I_n + \omega^2\cdot \frac{dI_n}{d\varphi}. \]

Substituting in (3), after replacement we obtain:

\[ \varepsilon\cdot I_n+\omega^2\cdot\frac{dI_n}{d\varphi} -\frac{\omega^2}{2}\cdot\frac{dI_n}{d\varphi} =M \]

or

\[ I_n\cdot\varepsilon = M-\frac{\omega^2}{2}\cdot\frac{dI_n}{d\varphi}. \tag{4} \]

This is the general form of the equation of motion of the machine for a variable value of the reduced moment of inertia \(I_n\). Equation (4) is valid for any position of the machine and any instant of time.

For \(I_n=\mathrm{const}\), equation (4) becomes equation (1).

The same result can be obtained from the kinetic-energy theorem written in differential form:

\[ d\left(I_n\cdot\frac{\omega^2}{2}\right)=M\,d\varphi, \]

where

\[ I_n=f(\varphi)\quad\text{and}\quad \omega=f_1(\varphi), \]

whence

\[ \frac{d I_n}{d \varphi}\cdot \frac{\omega^2}{2}+I_n\omega\,\frac{d\omega}{d\varphi}=M. \]

Since

\[ \omega\cdot \frac{d\omega}{d\varphi} = \frac{d\varphi}{dt}\cdot \frac{d\omega}{d\varphi} = \frac{d\omega}{dt} = \varepsilon, \]

then

\[ I_n\varepsilon=M-\frac{\omega^2}{2}\cdot \frac{d I_n}{d\varphi}. \tag{4'} \]

The same result could have been obtained from d’Alembert’s principle and, conversely, on the basis of (4) it can be shown that the equation of motion can be given the form of an equilibrium equation.

Indeed, substituting \(I_n=I_1+I_2+I_3\), where \(I_1=\mathrm{const}\), from (4) we obtain:

\[ 0=M-\frac{\omega^2}{2}\cdot \frac{d}{d\varphi}\left(I_1+I_2+I_3\right) -\varepsilon\left(I_1+I_2+I_3\right)= \]

\[ =M+\left(-\varepsilon I_1\right) +\left(-\varepsilon I_2-\frac{\omega^2}{2}\cdot \frac{d I_2}{d\varphi}\right) +\left(-\varepsilon I_3-\frac{\omega^2}{2}\cdot \frac{d I_3}{d\varphi}\right), \]

or

\[ M+M_1+M_2+M_3=0. \tag{5} \]

Here

\(M_1=-\varepsilon I_1\) — the moment of the inertia forces of the flywheel, shaft, and crank relative to the axis of rotation;

\[ M_2=-\left(\varepsilon I_2+\frac{\omega^2}{2}\cdot \frac{d I_2}{d\varphi}\right) \]

— the moment of the inertia forces of the connecting rod relative to the axis of rotation; moreover, by this formula it is considerably simpler to find the moment of the inertia forces of the connecting rod than to compose it directly, especially if its mass is not reduced only to two points;

\[ M_3=-\left(\varepsilon I_3+\frac{\omega^2}{2}\cdot \frac{d I_3}{d\varphi}\right) \]

— the moment of the inertia forces of the translationally moving masses relative to the axis of rotation.

From (5) it follows that

\[ I_1\varepsilon=M+M_2+M_3. \tag{6} \]

With a variable moment of inertia \(I_n=I_1+I_2+I_3\), we can write the equation of motion in the form (1), i.e., just as in the case when the moment of inertia \(I=I_1=\mathrm{const}\), but to the right-hand side of the equation, to the resultant moment of the external forces, one must add the moments of the inertia forces of the piston and the connecting rod.

In equation (4), the second term on the right-hand side represents the moment on the crankshaft from the inertia forces of the moving masses at the same angular velocity \(\omega\); the moment from the inertia forces of the same masses, caused by the presence of nonuniform rotation \((\varepsilon\ne 0)\), is contained in the left-hand side of this equation.

If the law of variation of the angle \(\varphi\) with time is known, then after substituting \(d\varphi=\omega\,dt\), equation (4) takes the form

\[ I_n\varepsilon=M-\frac{\omega}{2}\cdot \frac{d I_n}{dt}. \tag{4''} \]

Here

\[ M=f(t)\quad \text{and}\quad I_n=f_1(t). \]

This form of the equation of motion \((4'')\) can be found in the literature on electric drives \([6,20]\).

Let us consider an example.

The kinetic energy of the crank-and-connecting-rod mechanism system (with sufficient accuracy for a number of practical cases) can be determined by the formula:

\[ T = I_1 \cdot \frac{\omega^2}{2} + \frac{m_B \cdot v_B^2}{2} = \frac{\omega^2}{2}\left[I_1 + m_B \cdot r^2\left(\sin\varphi + \frac{\lambda}{2}\sin 2\varphi\right)^2\right] = \frac{\omega^2}{2}\cdot I_n . \]

Here

\[ I_n = I_1 + m_B \cdot r^2\left(\sin\varphi + \frac{\lambda}{2}\sin 2\varphi\right) \]

is the moment of inertia reduced to the shaft;

\[ I_1 = \text{const} \]

is the moment of inertia of the shaft, the crank, and \(2/3\) of the mass of the connecting rod;

\[ r \]

is the crank radius;

\[ \lambda = \frac{r}{l} \]

is the ratio of the crank radius to the length of the connecting rod;

\[ m_B \]

is the mass of the piston plus \(1/3\) of the mass of the connecting rod;

\[ v_B = \frac{dx}{dt} = r \cdot \omega\left(\sin\varphi + \frac{\lambda}{2}\sin 2\varphi\right) \]

is the piston velocity;

\[ B \]

is the piston pin (Fig. 1).

Differentiating \(I_n\) with respect to \(\varphi\) and forming equation (4), we obtain:

\[ I_n \cdot \varepsilon = M - m_B \cdot r^2 \cdot \omega^2 \left(\sin\varphi + \frac{\lambda}{2}\sin 2\varphi\right) \cdot \left(\cos\varphi + \lambda\cos 2\varphi\right) \]

Fig. 1

or

\[ I_1 \cdot \varepsilon = M - m_B \cdot r^2 \cdot \omega^2 \left(\sin\varphi + \frac{\lambda}{2}\sin 2\varphi\right) \cdot \left(\cos\varphi + \lambda\cos 2\varphi\right) - \]

\[ - m_B r^2 \left(\sin\varphi + \frac{\lambda}{2}\sin 2\varphi\right)^2 \cdot \varepsilon = M - m_B \left[ r\omega^2 \cdot \left(\cos\varphi + \lambda\cos 2\varphi\right) + \right. \]

\[ \left. + r \cdot \varepsilon \left(\sin\varphi + \frac{\lambda}{2}\sin 2\varphi\right) \right] \cdot \left(\sin\varphi + \frac{\lambda}{2}\sin 2\varphi\right) \cdot r = \]

\[ = M - m_B \cdot \frac{dv_B}{dt} \cdot \left(\sin\varphi + \frac{\lambda}{2}\sin 2\varphi\right) \cdot r = M + F_{\mathrm{in}} \cdot r, \tag{7} \]

where \(F_{\text{in}}\) is the reduced force of inertia of the reciprocating masses at the crank pin \(\left(\dfrac{dv_B}{dt}\right.\) is the acceleration of the piston\(\left.\vphantom{\dfrac{dv_B}{dt}}\right)\).

In fact, as is seen from the figure, the ratio of the velocities of points \(A\) and \(B\) is found by determining the position of the instantaneous center of rotation and the angular velocity \(\Omega\) about it:

\[ \frac{V_B}{V_A}=\frac{\Omega \cdot BC}{\Omega \cdot AC}=\frac{BC}{AC}. \]

The forces acting on the crank-and-connecting-rod mechanism, when the latter is in equilibrium, are in the same ratio:

\[ P_1 \cdot AC=P_2 \cdot BC, \]

whence

\[ P_1=P_2 \cdot \frac{BC}{AC}. \]

Since in our case

\[ \frac{BC}{AC}\simeq \left(\sin \varphi+\frac{\lambda}{2}\sin 2\varphi\right), \]

then

\[ P_1=P_2 \cdot \left(\sin \varphi+\frac{\lambda}{2}\sin 2\varphi\right), \]

which also demonstrates the validity of expression (7), which could have been obtained directly from d’Alembert’s principle.

A solution carried out on the basis of energy considerations often leads to finding the result more quickly and simply (especially when there is a mechanical system with several degrees of freedom).

In later works, Academician I. I. Artobolevsky and B. M. Abramov [2] and [3] showed that, for a number of cases, the equation of motion of machines (4), even with a variable moment of inertia \(I_n\), can be represented in the following form:

\[ \Delta M_o = I_o \cdot \omega_o \cdot \frac{d\omega_o}{d\varphi_o} = I_o \cdot \frac{d^2\varphi_o}{dt^2}; \tag{8} \]

here:

\[ \begin{aligned} \Delta M_o &\text{— fictitious excess moment applied to the reduction link;}\\ I_o &\text{— fictitious, arbitrarily chosen reduced moment of inertia of the reduction link;}\\ \varphi_o &\text{— fictitious angle of rotation of the reduction link;}\\ \omega_o &\text{— fictitious angular velocity of the reduction link.} \end{aligned} \]

The quantities \(\Delta M_o\) and \(I_o\) are related to the actual values \(\Delta M=M_d-M\) and \(I_n\) by the conditions:

\[ \Delta M_o \cdot \omega_o = \Delta M \cdot \omega \quad \text{and} \quad \frac{I_o \omega_o^2}{2}=\frac{I_n \omega^2}{2}, \]

i.e., for the general case of nonuniform motion, with a variable moment of inertia, equation (8), which represents fictitious motion, has the same form as equation (1), which is valid only for the case \(I=\mathrm{const}\).

In the cited literature [2] and [3], examples are given of the practical application of the equation in the form (8) to the investigation of mechanisms with one degree of mobility.

An equation in the form (8) can also be extended to systems having several degrees of freedom; for example, in the case of a crank-and-rod mechanism, if to the right of piston \(B\) (Fig. 1) we attach a load by means of a spring, then we obtain a system with two degrees of freedom. For a given value of the driving moment, on the basis of equation (8) one can find the angular velocity of the mechanism by a grapho-analytical method.

In conclusion, let us note that for dynamically similar systems equation (4) is easily and simply reduced in form to equation (8)—in the form of the generalized (criterion) relation:

\[ J_c \cdot \varepsilon_c = M_c = \omega_c^2 \cdot J_c , \]

where the subscript \(c\) denotes the similarity coefficients of the corresponding homogeneous quantities (for example, \(J_c = \dfrac{J_1}{J_2}\), etc.). Equation (4) may be found in the literature [7, 8].

LITERATURE

1. Artobolevskii I. I. Theory of Mechanisms and Machines, Moscow–Leningrad, 1940.
2. Artobolevskii I. I. and Abramov B. M. On one form of the equation of motion of machines, DAN, 59, No. 7, 1948.
3. Artobolevskii I. I. and Abramov B. M. Solution of the equation of motion of a machine for the case of forces depending on velocity, DAN, 59, No. 9, 1948.
4. Zinoviev V. A. Investigation of the motion of a mechanical aggregate by the energy method. Proceedings of the Seminar on the Theory of Machines and Mechanisms, IV, Academy of Sciences of the USSR, 1948.
5. Alabuzhev P. M. Investigation of the working process of an electric breaker hammer with an elastic (air) connection, “Izvestiya TPI,” vol. 61, issue 1, 1947.
6. Golovan A. T. Electric Drive, GEI, Moscow–Leningrad, 1948.
7. Popov V. K. Fundamentals of Electric Drive, Gosenergoizdat, Moscow–Leningrad, 1951.
8. Kosmodemyanskii A. A. General theorems of the dynamics of a body of variable mass, Scientific Notes of Moscow State University, issue 152, vol. III, Moscow, 1951.