Camshaft Loosening Diagnosis on the Basis of Generalised Force Recognition at the Centre of Gravity of an Engine

A valve mechanism supports the working process of an engine cylinder, and a camshaft is a key component required to open and close a valve. When a camshaft loosens, the balance of the engine disrupts. In the meanwhile, the generalised force at its centre of gravity (CG) alters. This study proposed a novel technique to detect camshaft loosening based on recognising the generalised force at the CG of the engine. We conducted Hanning windowed interpolation of discrete spectra to extract the precise phase and amplitude by utilising the acceleration signals at the engine cylinder and mounts and cylinder head. We then accurately computed the generalised force at the CG. Finally, we accurately extracted the camshaft loosening features by analysing the main harmonic orders for the generalised force. As indicated by simulations, our method can be used to eﬀectively detect combustion engine faults involving camshaft loosening.


Introduction
A camshaft is a key component required to open and close the valve in an engine valve distribution mechanism. Camshaft projections are in contact with a tappet. Excessive stress due to contact between the camshaft projection and the tappet roller may cause wear failure [1]. Additionally, under long service time, a camshaft is subject to alternating load torque, bending moment, and impact load. It possibly operates with an angular vibration along with a bending vibration. Bearing and bush wear and camshaft loosening can lead to overall wear and unpleasant noise. e efficiency and reliability of camshafts decrease because of inaccurate movement, eventually resulting in force variations at the centre of gravity (CG). An injection pump camshaft of a truck diesel engine failed after a mere 13,000 km run because fatigue cracks were initiated at the keyway root through circumferential tangential stress because the stationary frictional force between the camshaft cone and timer working on keyway edges was inadequate [2]. A diesel engine camshaft underwent instantaneous circumferential cracking that propagated brittlely due to tensile stress generated at the site of camshaft straightening [3]. Du and Yu [4] developed an engine power assembly and accompanying valve train. ey compared the vibration acceleration signals of the engine cylinder head in normal and fault states to determine the loosening fault features of camshaft bearings. Diagnosis using mechanical systems is a key subject in modern industry. In recent years, intelligent methods have been widely used in structural diagnosis to detect faults in 3D printers. Li et al. introduced an extremelearning-machine-based intelligent solution for fault diagnosis that used a low-cost and precise attitude sensor and helped diagnostic measure studies for determining faults in delta 3D printers [5,6]. Kumar et al. developed a diagnostic technique based on deep learning to recognise defects in two-wheeler vehicle engines [7]. Ting et al. [8] determined the cluster centre in a radial basis function neural network by using the K-means clustering algorithm to study faults in camshaft grinders. Fault samples are critical to initiate an artificial intelligence (AI) mode. However, achieving convenient and thorough sampling in actual applications is difficult because mostly mechanical systems are healthy and normal. us, it becomes difficult to attain AI model training even with sufficient fault samples [9]. e diagnosis of camshaft faults must be effectual and accurate to ensure that the engine operates as usual [10]. Vibration signals, containing a large amount of fault information, are extensively utilised to diagnose faults [11][12][13][14]. e vibration pattern of an engine can offer diverse information about its state. In most cases, engine faults are manifested directly as the fluctuation of generalised force at the engine CG, achieved by utilising the acceleration signals at the engine mounts. Xu et al. computed generalised force at the engine CG and extracted features to detect single and continuous/intermittent double cylinder misfires [15]. e primary vertical force for a four-cylinder four-stroke motor refers to the second-order force, and the half-and first-order forces are nearly zero [16]. However, the force at the CG varies with camshaft faults.
Our study proposes a novel technique to detect camshaft loosening based on recognising the generalised force at the engine CG. We extracted the precise phase, frequency, and amplitude through Hanning windowed interpolation. Two prime advantages of our detection method are its high fault sensitivity and less subjectivity to external factors and experimental conditions. e following presents the organization of the rest of the paper. Section 2 introduces the identification principles for generalised force at the CG. Section 3 describes the diagnostic procedure for camshaft loosening based on the computed generalised force and illustrates the simulation results. e conclusions are made in Section 4. Figure 1, a powertrain mounting system is simplified into a six-degree-of-freedom vibration model to calculate the engine excitation force.

Generalised Force Estimation Method. As shown in
Assuming a slight movement of the engine mount system, the kinematic equation for the power assembly mounting system is where M denotes the 6 × 6 rigid mass matrix of the engine, C denotes the 6 × 6 damping matrix, K denotes the 6 × 6 stiffness matrix, and Q and F are the generalised displacements at the engine CG and 6 × 1 generalised force vector, respectively.
where m is the engine mass; x c , y c , and z c are the CG coordinates in the OXYZ reference system; J xxo , J yyo , and J zzo are the inertial moments for X, Y, Z coordinates, separately; and J xyo , J yzo , and J xzo are the cross inertial moments.
K 6×6 � k xx k xy k xz k xα k xβ k xc k yx k yy k yz k yα k yβ k yc k zx k zy k zz k zα k zβ k zc where k xx , k yy , and k zz are the total reciprocating stiffnesses of elastic support; k xy , k xz , and k zy are the coupling stiffnesses of elastic support; and k αα , k ββ , and k cc are the rotary stiffnesses about the coordinate axis.
where F x , F y , and F z represent transitional applied forces; Considering s points at which acceleration can be determined, where the coordinates of the i th point (i � 1, . . . , s) for the CG are x i y i z i , the following equation can be derived under a "slight" motion hypothesis: Accordingly, € Q can be computed using the least-squares method: where E is the transpose matrix and A denotes the acceleration vectors in three orthogonal directions for all the points.
We can precisely extract the acceleration signal phases and amplitudes by exploiting the engine acceleration signals at the mounts and block by using discrete spectrum interpolation. e generalised force at the CG [17] is given by At local coordinates, the mount complex stiffnesses in the three directions are given as where j � �� � −1 √ and K ′ � ωC � 2πfC are the loss stiffnesses. e phase difference of the gas-pressure torque for each cylinder of an i-cylinder engine is 4π/i, which varies with the engine ignition order. e highest amplitude of the first harmonic order k � i/2 appears in the resultant torque spectrum. e primary harmonic orders of a 4-cylinder 4stroke 4-cylinder 4-cylinder engine include k � i/2, i, 3i/2. Figure 2 shows the phase angle graphs for a typical 4-cylinder 4-stroke 4-cylinder 4-cylinder engine, the firing order of which is1-3-4-2.
Considering the same contribution by each cylinder to the engine torque, primary orders in the torque's structure are merely present due to gas forces, which are multiples of half the cylinder quantity for the 4-cylinder 4-stroke engine ( Figure 2). Besides, the primary harmonic orders for the torque are the 2nd, 4th, and 6th orders, when the engine runs as usual. Primary harmonic orders change in the event of a fault. (9) requires accurate extraction of the amplitudes, frequencies, and phases of the engine mounts and block. However, actual test signals contain noise. In discrete spectral analyses, no-integer period sampling leads to erroneous amplitude, frequency, and phase [18][19][20][21]. Conventional FFT extracts a highly errored generalised force, especially for the phase. Before the correction, the first harmonic of the acceleration shows similar errors to those for absolute phases.

Interpolation Method for Discrete Spectrum. Equation
us, attaining accurate relative phases between acceleration responses leads to a decrease in the error estimates of the generalised force. However, in the 2nd harmonic, phase errors vary, provided the failure of maximum spectral lines is within different frequencies, and alterations are present in the relative phases [17]. e interpolation method is used to estimate frequency bias based on the amplitude ratio of the first two maximum spectral lines. Consider x(t) as a single sequence of the harmonic signal and its amplitude, frequency, and phase as f 0 , A, and θ 0 , respectively. en, rough equispaced sampling using N sampling points at sampling frequency f 0 , we get a new sampling sequence: Applying Fourier transformation to equation (12), it becomes where w(n) is the window function. After correction, the amplitude and phase are given by where X w (k) is the amplitude of the maximum spectral line of the discrete spectrum of harmonic signals, that is, the greatest amplitude in the major lobe, and W(∇f 1 ) is the spectral mode function. e rectangle window is expressed as e Hanning window is expressed as Highly accurate amplitudes can be attained even at a considerably low sound-to-noise ratio (SNR) of −2.33 dB. e maximum error rates for the rectangle and Hanning windows are 4.5% and 6%, respectively. e phase precision for the Hanning window is <10°; however, due to an improper interpolation direction in the interpolation technique that is multiplied with the rectangular window when SNR � 2.33 dB and ∇f 1 ≤ 0.25, the resulting error is Mathematical Problems in Engineering 3 relatively large. e Hanning window is twice in width as the rectangle window, leading to a significant reduction in the possibility of interpolation direction error, and there is no arch exception [18]. Hence, we applied Hanning windowed interpolation discrete spectrum in order to extract the phase, amplitude, and frequency of the engine acceleration signals.

Simulation Models.
e engine valve mechanism comprises a cam, a lifter, valves, valve springs, and a camshaft. Each component has a certain mass and elasticity to produce an elastic deformation during motion. e valve train produces high acceleration when running at a high speed and is subject to a specific dynamic load. In the dynamic analysis, the components of the valve mechanism are simplified into an equivalent lumped mass and an equivalent spring, respectively. us, the lumped mass-spring vibration model can be applied to perform the dynamic analysis of the valve mechanism. e dynamic model of the valve train has been designed based on AVL Excite Timing Drive and includes double overhead camshafts for a 4-cylinder 4-stroke engine. e component unit of the valve train contains parts such as a valve stem, valve seat, camshaft bearing, cam, lifter, and spring and phase units, in sync with the composition of the valve mechanism. Figure 3 presents the logic schematic diagram of the component connections of the valve train.
We performed a high-quality FEA using AVL Excite and multibody dynamics modelling for an in-line 4-cylinder 4stroke engine installed with mounts ( Figure 4). First, an engine crankshaft and powertrain finite element model is established. e finite element has an enormous degree of freedom; thus, its matrix is simplified using MSC Nastran to improve the computational efficiency. Next, the coupled multibody dynamics model is set up based on the AVL Excite. e excitation force includes the force of a gas explosion and the inertial forces of reciprocation and rotation. All excitation forces are included in our model to ensure that the vibration is similar to that of an actual engine. e explosion pressure for each cylinder is set based on the cylinder pressure plot. e cylinder pressure curve of an engine running at a speed of 2000 rpm at full load condition is shown in Figure 5. e piston load of each cylinder at regular intervals, the pressure load of each cylinder wall, and the load torque at the flywheel output are generated automatically by the software as per the firing order. e same is applied to the corresponding nodes. e valve mechanism force can be considered an extended load of the powertrain, which contains valve seat force, valve spring force, and camshaft support force. e other force is the piston striking force produced due to the gap between the piston and the cylinder liner.
For determining the valve train excitation force, we used the valve train model based on the Excite Timing Drive and loaded the calculated force into the Power Unit model. e data connection interface is preferred for both models. e piston striking force is calculated using excited Piston and Rings, and the dynamic striking force of five nodes is distributed in the height direction of the main and vice thrust side of the cylinder liner. e calculated piston striking force is also considered an extended force loaded into the corresponding nodes of the powertrain model. Table 1 presents the CG of the power assembly and the mount locations. Table 2 presents the power assembly's inertia parameters excluding those of the piston, connecting rod, and crankshaft. e three mounts have a stiffness of 3 × 10 5 N/m and damping of 200 N s/m.

Camshaft Bearing Loosening Fault Setting.
e simulation of loosening bearing faults is created by removing the SRBS unit from the model, provided that fixation of intake and exhaust camshafts at the cylinder head is achieved using five sliding bearings. No effect is exerted by the loosening bearing on the pressing camshaft. In the present simulation, the bearing of the intake camshaft of cylinder-3 (SRBS int4) is removed so that bearing-4 has a zero acting force throughout the work cycle, and the dynamic driving force of cylinder-3's intake and exhaust valves is transmitted to the cylinder through the adjacent bearings 3 and 5 only. Figures 6 and 7 display the bearing forces of bearing 3 and 5, respectively, at 2,000 rpm under normal and faulty scenarios. When the bearing of cylinder-3 loosens, the supporting force of the adjacent bearing changes to a large extent. e excitation force of the valve mechanism is taken as the input force for the engine powertrain. e acting force of the camshaft on the cylinder head is mainly transmitted through the bearing. e valve train excitation force is calculated based on the valve train model, followed by introducing force into the Power Unit model through the loading points as shown in Figure 8.

Camshaft Loose Diagnosis.
e loosening of the camshaft bearing changes the generalised force at the CG. us, the fault features of camshaft bearing loosening can be determined by calculating the generalised force based on the vibration signals of the engine system. e Hanning windowed interpolation, as mentioned in Section 2.2, is used for the precise phase, frequency, and amplitude extraction of the three engine acceleration signals at the cylinder block, head, and mounts, respectively. e generalised force computation at the CG is determined using equation (9). e engine has a firing order of 1-3-4-2. e frequency of the excitation signal of the 4-cylinder 4-stroke engine is twice that of the engine RPM. e vibration signals produced by the engine are the superposition of the fundamental frequency signal and its higher frequency signal, based on the harmonic signal characteristics. Under normal conditions, the primary harmonic orders of engine vibration signal are the 2nd, 4th, and 6th orders, respectively, whereas the 0.5th order is nearly zero. e highest amplitude of the 2nd harmonic appears on the resultant torque spectrum. Additionally, the signal of the engine block vibration conforms to the same law. e vertically generalised force, identified under normal conditions, is shown in Figure 9. e amplitude of the 2nd harmonic is the highest, whereas the amplitude of the 0.5th harmonic is closer to zero, which is consistent with the theoretical analysis. Camshaft loosening was induced in cylinder-3 by removing the intake camshaft bearing SRBS int4 (Figure 1). In this case, a large impact is observed after each alternate revolution by the engine, and its frequency is half of the engine RPM. e loosening of the camshaft leads to the formation of another excitation force cycle, and the crankshaft rotates twice per cycle. It also leads to an increase in the amplitude of the 0.5th harmonic and its multiples increases considerably ( Figure 10).

Conclusions
is study proposed a novel technique for detecting camshaft loosening, which was used for analysing the order of the harmonics of the generalised force at the CG. Precise phase, frequency, and amplitude extraction were conducted using the Hanning windowed interpolation, thereby achieving the generalised force accurately at the CG. e proposed method is effective in diagnosing the fault induced due to camshaft loosening as demonstrated by the simulation results based on the AVL software. e loosening of the camshaft changes the generalised force at the CG. e amplitude of the 0.5th harmonic and its multiples increase sharply because the crankshaft rotates twice per cycle. e 0.5th harmonic and its multiples increase when the rate of change reaches half of the engine speed.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication.