Abstract
The problem of forced nonlinear periodic vibrations of an internal combustion engine (ICE) is considered (Fridman et al. in Nonlinear chaotic and periodic vibrations in an internal combustion engine, pp. 403–406, [2]). The widespread design of the piston ICE involves a system of three shafts interconnected by gear wheels. These shafts are the crankshaft and two camshafts. In stationary operation mode the ICE shafts and gear wheels perform periodic torsional vibrations. The vibrations arise under the action of periodic torques that are generated by crank mechanisms. In turn, these mechanisms are driven by reciprocating motion of the pistons inside the ICE cylinders.
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1 Statement of the Problem (Original Model of an Internal Combustion Engine)
The problem of forced nonlinear periodic vibrations of an internal combustion engine (ICE) is considered [1]. The widespread design of the piston ICE involves a system of three shafts interconnected by gear wheels. These shafts are the crankshaft and two camshafts. In stationary operation mode the ICE shafts and gear wheels perform periodic torsional vibrations. The vibrations arise under the action of periodic torques that are generated by crank mechanisms. In turn, these mechanisms are driven by reciprocating motion of the pistons inside the ICE cylinders.
Let us begin with that part of an ICE consisting of shafts and gears, where the acting torques are assumed to be those given. The initial structure with distributed parameters can be substituted by the finite-dimensional system. For this purpose each of the shafts can be represented as a system of absolutely hard disks interconnected by torsion bars subject to torsion. The gear wheels are also assumed as disks, the resilient connection between which is determined by the elasticity of the teeth. The damper bodies and the object (i.e. the reactive load) driven by a motor are also considered additional disks. The total mechanical system can be conventionally shown as in Fig. 12.1. For descriptive reasons the image is given in planform. The hard disks are system nodes, while the torsion bars are replaced by weightless elastic springs. A scheme of the gearwheels is shown in Fig. 12.2.
There are two reasons for the difficulty of calculation of ICE oscillations. First, the oscillatory system under consideration has a complicated branched structure. Second, and very importantly, there are backlashes in the connections between the gears without which it is impossible to collect the system of gearwheels. In other words, the connections between the gears are discontinuous. The presence of backlashes in the tooth connections restricts the problem of ICE vibrations to being a nonlinear one.
2 The Method Used for Solution
Initially, the main backbone chain is set apart from the complex ramified elastic-mass system. It is indicated in Fig. 12.3 by black circles, whereas the lateral branches of the basic chain are indicated by white circles.
I
The nodes to which the branches are attached are set apart in the main chain. The “fictitious” nodes with very small mass are attached to these nodes. Further, very soft springs (shown in Fig. 12.3 by dashed lines) are entered into the diagram. These springs connect the fictitious nodes to the utmost nodes of lateral branches. Thus, the ramified system is transformed into a tape system. Sequential numbering is assigned to its nodes including the fictitious ones. The node-disk number is denoted by \( s \) and the total number of nodes by \( S \)
where the designation \( s\, - 1,s \) is given to the spring connecting the (\( s\, - 1 \))th and the \( s \) th disks.
Let us introduce some notation: \( j_{s} \), \( \vartheta_{s} (t) \) is the moment of inertia and rotation angle of the \( s \) th disk; \( c_{s - 1,s} \) is the torsional stiffness of the torsion bar connecting the (\( s\, - \,1 \))th and \( s \)th disks; \( g_{s} \) is the coefficient of viscous damping applied to the \( s \)th disk; \( \upmu_{s} (t) \) is the external periodic torque applied to the \( s \)th disk; \( m_{s - 1,s} (t) \) is the moment acting on the \( s \)th disk from the (\( s\, - 1,\,s \))th spring; and \( 0 \le t \le T \) is time.
The equation of oscillations of the \( s \)th disk is
In this equation the linear relationship \( m_{s - 1,s} = c_{s - 1,s}\uppsi_{s - 1,s} \) holds for the torsion bar, where
The function \( m = c(\uppsi) \) is nonlinear regarding the connection of gearwheels. An ideal graph of this connection is shown in Fig. 12.4. The connection is bilateral because a tooth of one contact gearwheel is located between two teeth of the other adjacent gearwheel.
Let us introduce the vector functions \( \upvartheta = (\upvartheta_{s} ),\,\upmu = (\upmu_{s} ) \), which allows us to write concisely
instead of the system of Eq. (12.2).
This operator equation can be solved using the Newton–Kantorovich method. The iterative algorithm can be written as
where
and \( n \) is the number of approximation.
Otherwise
where
At the (\( n\, + 1 \))th step of the iterative process (12.7) we consider the functions \( \upvartheta^{n} \) and \( q(\upvartheta^{n} ) \) to be known. There is a need to determine \( \upvartheta^{n + 1} \) (the desired function in the (\( n\, + 1 \))th approximation) by the recurrent formula (12.7). To do this the index \( n \) can be omitted since the algorithm for solution of Eq. (12.7) does not depend on this index
or, in more detail, taking Eq. (12.2) into account
If the (\( s - 1 \))th and \( s \)th disks are connected by a torsion bar, then \( c_{s - 1,s} \) represents the torsional stiffness. Linearized stiffness for the connection of gearwheels is
Approximate calculation of this derivative can be done by smoothing the function depicted in Fig. 12.4 and then replacing it with the function given in Fig. 12.5. This should be done because the Newton–Kantorovich method does not allow the use of functions whose first derivative is zero and whose second derivative is not bounded, as is the case for the ideal function shown in Fig. 12.4. Smoothing can partly be justified by the fact that there is always a layer of oil between the teeth. A graph of function \( c = c(\uppsi) \) is plotted in Fig. 12.5.
Equation (12.10) can be solved using the projection method again. The projection conditions can be written as
where
Calculating the integrals by parts and taking the periodicity of the harmonic functions and their derivatives into account, we can write that
The desired solution can be approximately represented as a finite harmonic series
Substituting this series into the projection conditions (12.12) and formulas (12.14) and (12.15), followed by changing the order of integration and summation, we obtain a system of linear algebraic equations for the coefficients \( \upvartheta_{s}^{k} \)
where, in particular, the equations
are taken into account and the orthonormality property of the functions \( {\text{e}}^{im\lambda t} \) (5.4) can also be used.
Entering the vectors \( m_{s} \, = \,(q_{s}^{k} ) \), \( \uptheta_{s} = (\upvartheta_{s}^{k} ) \) and the Toeplitz matrices \( C_{s - 1,s} \, = \,(c_{s - 1,s}^{km} ) \), we can write the system of Eq. (12.17) in vector-matrix form
where
In the band system considered the first and last nodes are not fixed against rotation; hence, the static component of the angular position of system nodes is uncertain. To avoid this, we adjoin “soft” springs (fastened at the other end) to the extreme nodes indicated, such that instead of (12.1) there will be \( s = 0,1, \ldots ,,S,S + 1 \) and hence
Solving the system of linear Eqs. (12.20) and (12.21) with a three-diagonal matrix can be made easier using the matrix sweep method which involves recursion formulas of the form (4.86). Recall that the band mechanical system considered was obtained by transforming the original branched system. This should be taken into account when drawing up the matrix sweep method algorithm. Necessary modification of the method of calculation can be carried out as follows. The nodes of the main chain, to which the lateral branches have been attached, are extracted. The force exerted by the side branch is added to these nodes during direct sweep. During reverse sweep the equality of displacements of the main chain node and attached fictitious node is taken into account.
Using this method we can calculate the vibrations of a real ICE (see Fig. 12.3 for a conditional scheme). The displacements of all units and forces in all connections for one oscillation period have been determined. For example, Figs. 12.6 and 12.7 show graphs of calculated moments acting on gear wheel 11 from the tooth connection of wheels 11 and 14 in both the absence and presence of backlashes.
Comparison of the graphs shows that forces increase about 4.8 times as a result of backlashes in tooth connection. Calculation using a personal computer lasted just a few seconds.
References
Fridman, V., Tibbetts, D., & Piraner, I. (1997). Nonlinear chaotic and periodic vibrations in an internal combustion engine. In Proceedings First International Conference on Control of Oscillations and Chaos (Vol. 3, pp. 403–406).
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Fridman, V. (2018). Vibrations of an Internal Combustion Engine. In: Theory of Elastic Oscillations. Foundations of Engineering Mechanics. Springer, Singapore. https://doi.org/10.1007/978-981-10-4786-2_12
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DOI: https://doi.org/10.1007/978-981-10-4786-2_12
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