Vibration Analysis of a Single-Cylinder Reciprocating Compressor considering the Coupling Effects of Torsional Vibration

A piston slap is one of the main vibration sources of the reciprocating machinery. Much work has been done in this field, most of which was based on a constant rotating speed. However, in practice, the speed of a crankshaft may always fluctuate due to the uneven load or excitation. ,e inertia forces of moving components are much different at the fluctuating rotating speed comparing with that at a constant speed. In this paper, the piston slap and the induced vibration are analyzed based on the instantaneous angular speed measured on a single-cylinder reciprocating compressor. Firstly, the dynamics of a crankconnecting rod mechanism is analyzed based on the measured instantaneous angular speed which contains the torsional vibration of the air compressor. ,e time histories of piston slap impact forces considering and without considering torsional vibration are compared. ,en, in order to correlate the piston slap impact with the slap-induced vibration, the corresponding transfer functions between the middle stroke of the outer surface of the cylinder liner and the excitation points are measured. And the excitation force on the main bearing is also taken into account to bring the simulation closer to the experimental results. ,e effects of a torsional vibration on the vibration of the cylinder liner are analyzed, and the simulation results show that the torsional vibration is a factor that must be taken into account in the vibration analysis of the single-cylinder reciprocating compressor.


Introduction
A piston slap, the impact caused by the sidewise motions of pistons across the cylinder clearance spaces, has long been recognized to be a major source of noise and vibration in reciprocating machinery.And this phenomenon has been investigated from both analytical and experimental points of view for over fifty years.Ungar and Ross attempt to estimate the vibration and noise power levels induced by the piston slap for the first time [1].e piston slap at TDC has been simulated by Haddad and Fortescue in an analogue computer [2], and the effects of oil film on piston slap excitation were considered in his following studies [3].Ryan [4] and Rèpaci [5] analyzed the effect of the change in engine operating and geometric parameters on piston slap impact force through experiments and simulations.Simultaneous noise reduction in the conditions of both idling and high speed was achieved by Nakashima et al. [6].
ere have been many models developed to estimate the impact forces and the side-thrust force.A lumped parameter model of the piston and liner was developed to describe the dynamics of the piston slap, and the model parameters are obtained from mobility measurement.is model was used to estimate the overall vibration level at the engine block surface [7].
ere is another lumped parameter model with eight degrees of freedom which is developed using the dynamic characteristics of the piston, connecting rod, cylinder wall, and engine block.In [8], the time history of the piston slap force is determined, and the spectra of the force are compared for different operational regimes of the engine.A finite element method (FEM) has also been applied in order to analyze the impact forces [9].e models, as well as FEM, are not satisfactory to estimate the impact forces.
e relative motion between piston and cylinder liner is actually a lubrication process.So many mathematical models were based on the Reynolds equation [10][11][12].However, almost all models are difficult to analyze the exact frequency of piston-induced vibration and can only be used as an estimate of overall vibrational energy.
In addition, reciprocating machinery always has some degrees of torsional vibration (TV) during operation due to their reciprocating nature.But relatively few studies appear to have been done on the relations between torsional vibrations and linear or block vibration.In general, the effect of torsional vibration has been ignored in the recent piston slap studies.Whilst the crankshaft of an engine rotates with TV, the inertial forces of the piston, connecting rod, and crankshaft fluctuate, and the dynamics of the piston is affected both on reciprocating and secondary orientations.Many researchers have pointed it out that the combustion noise and the piston slap noise are found to be overlapped in the time domain and frequency domain [13][14][15].However, the exact frequency range of the vibration caused by the piston slap is hard to determine with the interference of combustion noise.In order to solve this problem, a single-cylinder air compressor where the combustion noise is absent is selected as the research object in this paper.
At the beginning of this paper, the instantaneous angular speed is measured on a single-cylinder reciprocating compressor whose medium-low-frequency vibration is mainly concentrated in 100-350 Hz.Based on the instantaneous angular speed, the dynamics of the crank-connecting rod mechanism is analyzed.
e time histories of piston slap impact forces considering and without considering torsional vibration are compared.
en, in order to correlate the piston slap impact with the slap-induced vibration, the corresponding transfer functions between the middle stroke of the outer surface of the cylinder liner and the excitation points are measured.And the excitation force on the main bearing is also taken into account to bring the simulation closer to the experimental results.e effects of the torsional vibration on the vibration of the cylinder liner are analyzed, and the simulation results show that the torsional vibration contributes greatly to the vibration of the cylinder liner vibration.

Experimental Configuration
e test bench is a motor-driven single-cylinder air compressor with a bearing in the middle of a rotating shaft and two inertia disks on each side of the bearing.Teeth are machined on the edge of the inertia disks for measuring the instantaneous angular speed.And an elastic coupling made of polyurethane material is mounted inside the bigger inertia disk.
e arrangement of the compressor is shown in Figure 1.
To measure the acceleration, pressure, and instantaneous angular speed signal of such a compressor, a B&K 4526 single-direction accelerometer, a Kisler 6125A pressure transducer, and a magnetoelectric sensor are used.
e signal is collected and processed in a B&K PULSE 3560B.In Figure 1, the gage map for the accelerometer (yellow star), the pressure sensor (green star), and the magnetoelectric sensor (red star) which is used to measure the instantaneous angular speed of the compressor are illustrated.

Experimental Results
e rotating speed is set to 400 rpm, and the instantaneous angular speed within one cycle is shown in Figure 2. Figure 2(a) demonstrates the raw signal of the electromagnetic transducer, and it can be used to calculate the instantaneous angular speed.As it is shown in Figure 2(b), the fluctuation range is about 5%.e variation of the pressure inside the cylinder during an intake and exhaust cycle is shown in Figure 3. e piston is at the top dead center in the initial position.And Figure 4 shows the acceleration response of the cylinder liner outer wall.It can be seen that the medium-low-frequency vibration of the cylinder liner is mainly concentrated in 100-350 Hz.In this paper, the influence of torsional vibration on the piston slap-induced vibration in this frequency band will be analyzed.
Based on the above measurements, the dynamics of the crank-connecting rod mechanism is analyzed in the following part.

Primary Motions and Inertia Forces of Machine Components considering TV
e forces acting within the compressor may most readily be studied with the aid of a diagram shown in Figure 5. e left part of this figure describes the geometry of a typical crank-connecting rod mechanism, defines the sign conventions used in this paper, and serves as a basis for the determination of the motions and inertia forces of the moving components.ese forces and the pertinent interaction forces are demonstrated in the "free body" diagrams of the crank, connecting rod, and piston which appear in the right portion of the figure.Shock and Vibration α stands for the crank angle, and if the rotating speed is constant without any uctuation, _ α const, and € α 0. While considering the torsional vibration, the angular velocity is not a constant any more and angular acceleration € α ≠ 0.
e piston motion is expressed as the crankshaft angle function as follows: ( Take the measured instantaneous angular velocity into the formula, and the angular acceleration can be obtained by calculating the rst derivative of the velocity to time.e di erence in the piston motion considering and without considering TV is shown in Figure 6.

Piston Side-rust Force and Extra Piston Slaps due to the Torsional Vibration.
e piston side-thrust force is derived from the forces and moment of the crankshaft, connecting rod, and piston.It can be deduced from the equations of the system as shown below:   Shock and Vibration e clearance between the piston skirt and the cylinder liner is only 10 μm, which makes the inertial force of the piston in the y direction neglectable.e main driving force of the piston slap-induced vibration is the piston side-thrust force which acts on the piston skirt imparted by the piston connecting rod.From the equations, one may solve for the side-thrust force F N which takes into account the effect of the variation of the system inertia with respect to the rotation angle of the crankshaft:

Shock and Vibration
where To analyze the effect of the torsional vibration on each part of the side-thrust force, a dimensionless side-thrust force parameter ψ s is divided into three parts based on the method reported in [1].ψ g , ψ p , and ψ c represent the contributions to the value of this parameter made by the gas force, the piston inertia, and the connecting rod inertia, respectively.
e term m p Rω 2 λ, which has been used as a normalization factor in equation (3), may be treated as representing the centrifugal force that the piston would exert if it were a point mass attached to the rotating crankshaft at the crank radius.
Secondary motion of the piston across the cylinder clearance could be recognized to be initiated if the sidethrust force F N acting on the piston changes the direction.erefore, when the right-hand side of equation ( 3) changes its algebraic sign or equals zero, the crank angle α is the initiated position of the piston slap.It can easily be seen that the initiated positions are for ψ g � (ψ p + ψ c ). Figure 7 shows the condition of the piston slap when the rotating speed is constant.e left portion of this figure demonstrates the contribution of the piston and connecting rod inertia to the piston slap.And the right portion shows the initiated position of secondary motion of the piston.e curve of ψ g intersects the curve of (ψ p + ψ c ), not only at the top dead center (α � 0 °) and bottom dead center (α � 180 °), but also at a mid-stroke" position during the intake stroke for the selected compressor in this paper.
When TV is considered, the conditions will vary enormously as is shown in Figure 8. e inertial force of the piston slap and connecting rod fluctuate greatly because of the nonzero rotational acceleration of the crankshaft.ere are more intersections around the three positions than those at the constant speed.e number of times F N changes sign increased from 3 to 22. erefore, the torsional vibration will definitely affect the piston slap and its induced vibration.Figure 9 compares the side-thrust force F N with and without TV.

Main Bearing Reacting Force.
e main bearing reacting force is numerically equal to the normal force exerted by the connecting rod on the crank, which can be obtained by solving equation ( 2): where m r and r represent the equivalent rotational mass and radius of the crank, respectively.e results of the main bearing reacting force with and without torsional vibration are shown in Figure 10.

The Effect of TV on the Vibration of the Cylinder Liner
In order to calculate the vibration of the cylinder liner, three transfer functions are measured.One of them is measured between the middle stroke of the inner and outer walls of the cylinder liner.e other two are measured between the outer wall and the main bearing excitation point in horizontal and vertical directions.ese transfer functions are used to calculate the vibration of the cylinder liner.In the measurement of these transfer functions, the cylinder head is removed and one accelerometer is fixed on the outside of the cylinder wall at the middle stroke, and the transfer function between inside and outside the cylinder liner is obtained by hitting the inner wall of the cylinder liner at the middle stroke with an impact hammer.
e other two transfer functions are measured by hitting the horizontal and vertical directions of the shaft section at the joint of the main bearing.In [7], the transfer functions are used to predict the vibratory response in the frequency range below about 500 Hz.And it will also be used to calculate the vibration of the cylinder liner in the frequency bands of interest in this paper, which is 100-350 Hz, the most intense frequency band of the given compressor.
e measured transfer functions between excitation forces and the vibration acceleration are expressed in Figure 11.In this figure, H1 represents the transfer function of the inner wall of the cylinder liner and the response point.H2 represents the transfer function of the horizontal direction of the main bearing position and the response point, and H3 represents the transfer function of the vertical direction of the main bearing position and the response point.
e vibration of the cylinder liner can be obtained by adding the calculated force with and without TV into the measured transfer functions.is system can be defined by the basic transform relation [16]:

Shock and Vibration
For arbitrary n(t), which may be correlated with x 1 (t) and/or x 2 (t) and x 3 (t), the one-sided autospectral density function of y(t) for a very long but nite record length T is given by When n(t) is uncorrelated with x i (t), equation ( 6) becomes Figure 12 compares the estimated vibration and measured vibration.e main frequency interval of vibration is 6.67 Hz, which is the fundamental frequency of the shaft.As shown in the gure, without considering the torsional    Shock and Vibration vibration, the shaft rotates at a steady and constant speed, and the piston slap force and the main bearing force only contribute to the vibration in the frequency range of 100-150 Hz. ere is almost no contribution between 200 and 350 Hz, where the vibration is more intense.However, if torsional vibration is taken into account, it can be seen that these forces not only contributes between 100 and 150 Hz but also within 150-350 Hz.
In order to make the contrast clearer, the overall acceleration level is calculated within each 50 Hz band as shown in Table 1.
Figure 13 shows the corresponding bar graph of the comparison between estimated and measured vibration acceleration level at the cylinder liner outer wall in the selected motoring condition.As shown in Figure 13, if the TV is not considered, the smallest error is 6.7 dB which is between 100 and 150 Hz and all the rest errors are larger than 20 dB. is is unacceptable.In contrast, considering TV's response is far superior to not considering TV and closer to the measured response.In 4 of the 5 frequency bands, the calculation error is within 2.5 dB, and only 250-300 Hz has a larger error.e reason is that the two largest line spectra, 260 Hz and 300 Hz, are not caused by the forces considered and may be caused by other excitation, such as the slap of the intake and exhaust valves; the relevant work will be veri ed in subsequent studies.

Shock and Vibration
It can be seen that the torsional vibration is a factor that must be taken into account when calculating the vibration of the cylinder liner, as the reciprocating air compressors or internal-combustion engines are often accompanied by violent speed uctuations due to uneven load or excitation.

Conclusions
In this paper, a torsional vibration coupled dynamics analysis of a crank-connecting rod mechanism has been presented.e result shows that when the torsional vibration is considered, the corresponding uctuations in the inertia forces of the piston and connecting rod have a great in uence on piston slap force.en the vibration of the cylinder liner is estimated approximately based on the measured transfer functions.
e estimated vibrations demonstrate that the piston slap force and the main bearing force contribute not only to the vibration between 100 and 150 Hz but also to the vibration in the higher frequency bands, and this can be observed only when the torsional vibration is considered.Finally, the prediction of overall vibration level shows the necessity of torsional vibration in the vibration analysis of the singlecylinder reciprocating compressor.
is result is also applicable to vibration prediction of other reciprocating mechanical equipment.What needs special attention is that the frequency bands being analyzed can be di erent according to the characteristics of the given engine or compressor, and

Figure 1 :
Figure 1: Arrangement of the compressor.

Figure 5 :Figure 6 :
Figure 5: Piston and connecting rod geometry and forces.

Figure 7 :
Figure 7: Inertia and gas force contributions to the piston slap without TV.

Figure 8 :
Figure 8: Inertia and gas force contributions to the piston slap with TV.

Figure 10 :Figure 9 :
Figure 10: Main bearing reacting force with and without TV.

Figure 11 :
Figure 11: Measured transfer functions of the cylinder.

Figure 12 :
Figure 12: Comparison between estimated and measured vibration.

Table 1 :
Overall acceleration level between estimated and measured vibration.Hz was just for the speci c compressor in this paper.