#### Abstract

The interpolation method of discrete spectrum is applied to the engine excitation force identification. The frequency, amplitude, and phase of the vibration response of each measuring point on cylinder surface are obtained accurately based on the interpolation method. Combination with the inertia parameters and the dynamic properties of the mounts, the excitation force, and moment at the center of gravity of engine can be exactly retrieved. The nonlinear problem caused by the imprecise phase of the measurement points is avoided, and the identified method is simplified. Simulation results confirm the importance of the interpolation method on the accurate identification of the excitation force. Influences on the excitation force identification are analyzed quantitatively. The reasons causing the errors are analyzed and the avoidance methods are given too. Through the multi-rigid-body dynamics model simulation, the excitation force identification method is confirmed. Then, the proposed method is carried out on an engine.

#### 1. Introduction

The exact value of engine excitation force is indispensable to the study of engine kinematics and dynamics, mounting system optimization, and vibration attenuation and isolation. It is normally calculated based on the engine parameters [1, 2], the gas explosion force, reciprocating inertial force, and rotating inertia force, for instance. The shortcoming is that it requires substantial data of engine that is sometimes difficult to be exactly measured, especially the rolling moment caused by a gas explosion. The alternative may rely on the use of experimental method.

The excitation force identification is a typical inverse problem in vibration control [3]; usually, the inverse of matrix of frequency response function (FRF) method is used to identify the excitation force [4, 5]. The response of a structure to an artificial excitation is measured at different locations on the structure so as to build the matrix of FRF. This method is useful for a linear mechanical system subject to multipoint excitation. However, the generalized inverse matrix is inevitable for the FRF matrix inversion method, which may involve morbid matrix inversion problem, and the morbid matrix inversion may lead to error amplification [6]. Tao et al. [7] proposed a method to measure the excitation force at the center of gravity (c.g.) of engine, with the full information of velocity (amplitude and phase) at all mounting points; the excitation force and moment at the c.g. can be exactly retrieved; due to the replacement of absolute values of phases by phase difference between three directions at each mounting point, a group of overdetermined equations are involved. In addition, there was no consideration about noise and experiment. To improve the matrix inversion condition, literature [8] introduced the singular value decomposition (SVD) method to abandon the information seriously polluted by noise by eliminating relatively small singular values. The shortcoming is that the results are quite sensitive to the threshold of the singular value; a small change of the threshold may result in a big change of the results. A similar method [9] to that of literature [7] was used to identify the powertrain excitation force; however, the method ignores the phases of each response and assuming they are zero, and obviously it will lead to a big error. Otsuka et al. [10] presented a method to identify vibration force of a single cylinder outboard engine by linear structural modeling and equivalent force transformation. First, a set of equivalent force vectors that can express actual vibration force generated by firing and the inertia due to dynamic motion of piston-crank mechanism and others were determined by FRF matrix inversion; then the equivalent force was transformed to the point at the intersection between the center line of the crankshaft and the connection rod by linear structural modeling. The problem is that the measuring accuracy of the FRF was not verified, and larger error may be caused by noise. In essence, there are two causes of the formation of morbid matrix: one is the measurement error, and the other is the analysis error of the amplitude and phase due to the discrete spectrum analysis method.

The main contributions of the paper include the following: (1) the interpolation method of discrete spectrum is applied to the engine excitation force identification; the nonlinear problem caused by the imprecise phase of the measurement points is avoided, and the identified method is greatly simplified; (2) the measuring influences on the excitation force identification are analyzed quantitatively; (3) experiment is carried out on an engine.

The rest of the paper is organized as follows. In Section 2, the theory fundaments are derived in detail; in Section 3 simulation results confirm the importance of the interpolation method on the accurate identification of the excitation force; the effects on the excitation force identified are investigated carefully and suggestions to reduce errors are given in Section 4; in Section 5 the accuracy of the method is verified by multi-rigid-body dynamics model simulation experiment; the engine experiment is carried out in Section 6. Finally, the conclusions are summarized in Section 7.

#### 2. Theoretical Fundaments

The natural frequency of an engine-mount system is usually within 30 Hz, and the first elastic resonance frequency is located in more than 100 Hz; that is, the first elastic resonance frequency is located far from the rigid body resonances; thus, the vibration of the engine can be considered as rigid body motion. Meanwhile, the support platform is treated as rigid body, and complex spring stiffness is used to model the dynamic behavior of the mount [11]. The engine-mount system can be simplified as a six-degree-of-freedom vibration model shown in Figure 1.

Let be a general reference frame, the crossing of the back-end surface of engine block, the jointing surface of the clutch and flywheel shell and the crankshaft, the same direction as the crankshaft, and the moving direction of the cylinder and piston (inline engines). Under the assumption of small motion, the engine-mount system equations can be expressed aswhere is the engine rigid mass matrix, is the damping matrix, is the stiffness matrix, is the generalized displacement vector at the c.g. of the engine, and is the generalized force vector, respectively:Taking Fourier Transformation on both sides, then (1) becomes

The majority of mounts used in the engine mounting are of a rubber bonded to metal or elastomeric construction. The complex stiffness of a mount in the three directions of its local coordinate system is defined by the equationwhere and is the loss stiffness.

The complex engine vibration model can be expressed as

From (5), it can be seen that once the engine inertia parameters and the mount complex stiffness are known, combination with the acceleration at the c.g. of the engine, the excitation force can be derived.

Consider ( is required to be at least 6 [8]) acceleration measurement points are applied. The coordinates of the th () point relative to the c.g. are . Under the assumption of “small” motion, we can getThen can be determined in a least-square sense aswhere is the acceleration vector of all points with three orthogonal directions and is the transpose matrix:Assuming the engine is supported by mounts, the local reference frame is defined for each mount, and the three orthogonal coordinates , , and are the three lord stiffness directions of the mount, respectively. And is the elastic center, that is, the intersection of the three elastic axes. The complex stiffness matrix can be expressed as

Substituting (7) and (9) into (5), the excitation force can be calculated as

As we can see from (13), with the knowledge of the engine’s rigid mass matrix and the complex stiffness matrix of the mounting system, as well as the frequency, amplitude, and phase accurately extracted from more than 3 points on the surface of the cylinder block (usually on the mounts), the corresponding order excitation force can be easily obtained.

Non integer period sampling leads to errors for the discrete spectrum analysis [12], which would decrease the accuracy of engine excitation force identification. The interpolation method for discrete spectrum with a Hanning window has good correction accuracy even when the signals are heavily corrupted with noise [13, 14]. Thus, the interpolation method with a Hanning window is applied to the extraction of frequency, amplitude, and phase, and the engine excitation force can be accurately identified consequently.

#### 3. Computer Simulations

Assuming the engine is supported by three rubber mounts, the engine mass kg, and the inertia parameters The stiffness and damping ratio of the mounts are shown in Table 1. The c.g. of the engine and the position of the mounts are listed in Table 2. The three lord directions of each mount are consistent with the three axes of the general reference frame, respectively. The running speed of the engine is 1695 rpm which means the rotational frequency is 28.25 Hz. Table 3 listed the amplitude and phase of the excitation force. The acceleration at the mounting points can be calculated based on (13) and the given parameters. With the sampling frequency of 512 Hz and the sampling points of 512, the acceleration curves are shown in Figure 2. The corrected and uncorrected amplitude, frequency, and phase, extracted from the acceleration by the interpolation method for discrete spectrum with a Hanning window and the traditional FFT (Fast Fourier Transform Algorithm) method, respectively, are used to identify the engine excitation force. The identified amplitude and phase of the excitation force are listed in Tables 4 and 5.

It is found from Tables 4 and 5 that, comparing with the traditional FFT method, the errors of the amplitude and the phase of the excitation force identified based on interpolation method with Hanning window are negligible. The big errors in amplitude, frequency, and phase extracted by the traditional FFT will result in the big errors and even totally wrong estimated results for the excitation force. For example, the amplitude errors for the excitation forces of the first harmonic are all nearly 4%, and the phase errors are about 45°, and the results of the second harmonic are totally wrong. Since the absolute phase errors of the first harmonic of the acceleration are all about 45° before correction, thus their relative phases between acceleration responses are all right, and the identified errors are around 4%, while, for the second harmonic, the maximum spectrum lines are not at the same frequency, which causes different phase errors for the second harmonic and the relative phases are changed; thus the identified excitation force is totally wrong.

#### 4. Error Analyses

##### 4.1. The Measurement Errors

###### 4.1.1. Sensitivity of Method to Errors in Mount Locations

As shown in (13), the mount locations, that is, matrix , are important input for the algorithm, so it is essential to understand how sensitive the method is to errors in this step. Random errors (random numbers in a certain range, e.g., random numbers in the range −2.56, 2.56) were added to the mount locations. Simulations are performed 200 times for each point. The results are shown in Figures 3 and 4. It can be seen that the amplitude and phase errors of the identified excitation force increase with the increasement of location errors, and the mount location errors have a relatively larger effect on the second harmonic of excitation force . For example, when the mount location errors are at the range −23.3, 23.3 mm, the amplitude error is 27% and the phase error is 30° for the second-harmonic , while it is only less than 2% for the main vibration . The reason is that for the 4-cylinder engine the main vibration of the second harmonic is vertical, and is relatively small and is greatly influenced by errors compared with . In practice, we should pay much attention to measuring the geometry of the system to reduce errors.

**(a) Amplitude error**

**(b) Phase error**

**(a) Amplitude error**

**(b) Phase error**

###### 4.1.2. Sensitivity of Method to Variation in Mount Stiffness

Mounts are known to vary greatly in stiffness. From manufacturing variability, temperature, and displacement the dynamic stiffness of each mount changes greatly. Similar to the mount locations, the excitation method takes into account the stiffness of each mount; it is also very important to understand how variability in stiffness affects the results. The stiffness changes were obtained by adding the random errors in a certain range, that is, 5% to 40% of the maximum value of the three main directions of stiffness of each mount. Results are shown in Figures 5 and 6.

**(a) Amplitude error**

**(b) Phase error**

**(a) Amplitude error**

**(b) Phase error**

From Figure 5 there is a small change in the calculated second-harmonic force as the mount stiffness varies; the calculated only changes about 0.1% for the amplitude and 0.1° for the phase even when the mount stiffness changes 40%. From Figures 5 and 6 it is shown that variability in the mount stiffness causes little change in the other calculated excitation force. These results show that a good nominal estimate of the dynamic mount stiffness is enough to get good results from the engine excitation method.

###### 4.1.3. Sensitivity of Method to Errors in Inertia Parameters

Another important input to the algorithm is inertia parameters of the engine. Owing to the engine’s complexity, its inertia parameters cannot be easily determined by using theoretical tools. The alternative is to use the experimental method. However, the inertia tensor, especially the product of inertia, can be difficult to measure correctly. Therefore it is important to understand how sensitive the method is to errors in inertia tensor. Similar to that of the mount stiffness, 5% to 40% random errors of the maximum value of the six inertia tensors were added to measure the sensitivity. The results are shown in Figures 7 and 8.

**(a) Amplitude error**

**(b) Phase error**

**(a) Amplitude error**

**(b) Phase error**

From Figures 7 and 8 it can be seen that the inertia tensor errors caused relative big errors in the force moment and the corresponding phase. When the inertia tensor has errors of 40%, the amplitude of the force moment changes about 5% and the phase changes about 5°. Errors in inertia tensor cause no changes in the three transitional forces; the reason is that the inertia tensors are only spread over the last three lines in the matrix , which have no effect on the first three rows.

##### 4.2. Other Error Factors

###### 4.2.1. The Influence of Elastic Deformation

In the excitation force method the engine is modeled as a rigid body; thus, the elastic deformation of the engine may decrease the identification accuracy.

###### 4.2.2. The Influence of Rigid Body Model Simplification

The engine was considered to be a rigid body; thus the influences of the accessories connecting to the engine are neglected. Under the assumption of “small” motion, the engine-mount nonlinear model was transformed into a linear model. In addition, complex spring stiffness is used to model the dynamic behavior of the mount; this also can lead to excitation force errors. Meanwhile, the engine assembly platform is supposed to be completely stationary; in practice this hypothesis may affect the results.

##### 4.3. Measurements to Improve the Identification Accuracy

(1)Avoid the elastic deformation effect on the rigid body model. Modal experiment can be carried out to find the engine rigidity modal frequency range. Otherwise, the excitation force identification frequency range should be within 100 Hz.(2)During the vibration measuring, to make sure the nonlinear model was transformed into a linear model, the vibration should be small and the engine should be installed on solid platform. And the direction in the mounting of the accelerometers should ensure that the three directions of each accelerometer are consistent with the three axes of the general reference frame.(3)The relative parameters such as the mount locations, mount stiffness, and the inertia parameters should be tested carefully.

#### 5. Calibration

The excitation force identification method is calibrated by a multi-rigid-body dynamics model simulation. The model is shown in Figure 9; the engine is supported by three mounts. The engine was loaded at the center of the flywheel, and the explosion pressure of each cylinder was provided by cylinder pressure curve. The running speed of the engine is 1695 rpm.

The inertia parameters of the powertrain except for crankshaft, connecting nod, and piston are listed in Table 6, and the c.g. of the powertrain and the position of the mounts are listed in Table 7.

The three-mount static stiffness is N/m, and the damping is 200 . The identified and computed results are shown in Figure 10. The computed excitation force is the gas force minus the inertia force of the crankshaft, the connection nod, and the piston. From Figure 10 we can see that the second, the fourth, and the sixth harmonic are dominant in the excitation force. Due to small values and the powertrain rigid model simplification, the excitation forces and have relatively big errors. Since the vertical load is dominant for the inline 4-cylinder engine, the simulation result fits well with the computed results for the excitation force and verified the accuracy of the method.

**(a)**

**(b)**

**(c)**#### 6. Experiment

The experimental study was carried out on the Austria AVL engine test bed. The inline 4-cylinder engine is attached to the test bed through four rubber mounts and operated by the AVL engine control system. Four triaxial acceleration sensors are used to measure the vibration above the mount, and four single-axis acceleration sensors are used to measure the vibration under the mount. The acceleration signals were measured and processed by a multichannel BBM analyzer allowing for simultaneous acquisition of 18 responses and one force in a conventional test. The test engine and the AVL console are shown in Figure 11.

**(a) The test engine**

**(b) AVL console**

The three lord directions, that is, , , and , of each mount are consistent with the three axes of the general reference frame, that is, , , and , respectively. Inertial characteristics of the engine have been determined by a method which uses residual inertia method [15]. From Figures 4 and 5 in Section 4.1 we can see that variability in the mount stiffness causes little change in the identification of excitation force results; a good nominal estimate of the dynamic mount stiffness is enough; thus the static stiffness provided by the manufacturer was used to estimate the dynamic stiffness, and the dynamic to static ratio is 1.4. The running speeds of the engine are 1200 rpm, 1600 rpm, and 1800 rpm with the load 55 Nm. Four triaxial acceleration sensors are attached on the four mounts, respectively, to measure the acceleration signals. The sampling frequency is 4096 Hz and the sampling points are 8192. The discrete spectrum analysis time is 2 s.

##### 6.1. The Hypothesis Test

The engine excitation force identified method supposes that the engine is a linear rigid body; thus the two conditions should be met; that is, the support platform is stationary and the deformation model locates far from the rigid model. To test the two hypotheses, a modal test of the engine on the test bed is needed to check the frequency range of the rigid model.

With the sampling frequency of 512 Hz and the sampling points of 1024, the frequency function and the coherent function along the direction of the rear left mount are shown in Figure 12. As can be seen from Figure 12(a) the rigid model is within 50 Hz, and the deformation model has little influences on the rigid model, which indicates that the main vibration of the engine within 100 Hz is a rigid body vibration; thus, the engine force identification is carried out within 100 Hz. From Figure 12(b), we can see that the values of the coherent coefficients are close to 1, which indicates that the experiment is reliable. According to the acceleration signals above and under the mount in Figure 13, the amplitude of the acceleration signal under the mount is nearly zero compared to that above the mount; the platform of the engine can be considered to be stationary.

**(a) Frequency function**

**(b) Coherent function**

##### 6.2. Identification Results and Discussion

The interpolation method for discrete spectrum with a Hanning window is used to extract the frequency, amplitude, and the phase of the acceleration signals from the four triaxial acceleration sensors attached to the four mounts. The identified results are shown in Figure 14; since only within 100 Hz is the excitation force analyzed, from 0.5 to 5 harmonics, 0.5 to 3.5 harmonics, and 0.5 to 3 harmonics are analyzed for 1200 rpm, 1600 rpm, and 1800 rpm, respectively.

For inline 4-cylinder engine vibration, the second, fourth, and sixth harmonic are dominant; see Figure 15. The force and moment of 1200 rpm are larger than those of 1600 rpm and 1800 rpm. As shown in Figure 12(a), the engine nature frequency is around 40 Hz. Since the rotational frequency of the engine is 40 Hz at the speed of 1200 rpm which is consistent with the engine nature frequency, resonance occurs.

**(a) 1200 rpm**

**(b) 1800 rpm**

Without resonance, according to [16], the second harmonic is dominant for vertical vibration; the value of the vibration is proportional to the square of engine speed; that is, the excitation force identified at speed 1800 rpm is (1800/1600)^{2} = 1.26 times that at speed 1600 rpm. The identified results are 1053 N and 899.9 N; the ratio is 1.17. For the measurement errors analyzed in Section 4, the identified result is close to the computed ratio.

The engine works focus on a moment of working stroke, which leads to the output torque fluctuating greatly. Since the measured acceleration cannot reflect the static deformation of the engine, the identified excitation torque is the varying component of the practice torque.

#### 7. Conclusions

The interpolation method with a Hanning window is applied to the excitation force identification, accurate phase is extracted, and the nonlinear problem caused by the imprecise phase of the measurement points is avoided. Simulation results verified that engine excitation force can be identified precisely based on the interpolation method with a Hanning window. Further investigation into the robustness of this method shows mount variation causes little error in results and nominal mount stiffness can be used. Mount locations and engine inertia parameters must be measured carefully as errors in these parameters can cause errors in results. Refined measurement means are offered to improve the identification precision. Engine experiment gives good estimations of the excitation force.

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgments

This work described in this paper is supported by the National Natural Science Foundation of China (Grant nos. 51405272, 51475169) and also was partly supported by the Ministry of Transport Project (2013319817190) and Fundamental Research Funds for the Central Universities (310822151117).