Parameter optimization design method for power transmission system additionally provided with centrifugal pendulum damper

Abstract

The invention provides a parameter optimization design method for a power transmission system additionally provided with a centrifugal pendulum type shock absorber. The method comprises the following steps that 1, a power transmission system model additionally provided with the centrifugal pendulum type shock absorber is established; wherein the power transmission system model comprises an engine, a dual mass flywheel DMF and a transmission input shaft which are connected in sequence; a centrifugal pendulum damper CPVA is arranged on the dual-mass flywheel DMF; step 2, constructing thermodynamic and dynamic models of the engine; step 3, constructing a DMF kinetic model; 4, constructing a CPVA kinetic model and a pendulum knocking power analysis model; step 5, constructing a dynamic model of the transmission input shaft; step 6, defining decision variables and objective functions in multi-objective optimization; and step 7, optimizing the decision variables by adopting MNSGA-II. The aim of reducing the swing knocking power and the torsional amplitude value of the transmission system can be achieved, and the dynamic performance and durability of the system are improved.

Description

Technical Field

[0001] The invention belongs to the technical field of automobile vibration control, and in particular relates to a parameter optimization design method for a power transmission system equipped with a centrifugal pendulum shock absorber. Background Art

[0002] With the continuous development of the automobile industry, the vibration problem of vehicle power transmission system has gradually attracted widespread attention. The vibration generated by the main components such as the engine and transmission during operation not only affects the comfort and noise level of the vehicle, but may also have an adverse effect on the reliability and service life of the power transmission system. Especially under low-speed conditions, the torque fluctuations generated by the engine are large, which can easily cause resonance or unstable vibration in the power transmission system, which puts higher requirements on driving experience and vehicle performance.

[0003] In order to effectively suppress these vibrations, researchers have proposed a variety of vibration reduction technologies. Among them, the centrifugal pendulum vibration absorber (CPVA), as a device that reduces torsional vibrations through the inertial force of the pendulum mass, has been widely used in rotating systems, especially in the aerospace and automotive industries. In traditional engines and their supporting power transmission systems, the addition of CPVA helps to reduce the vibrations generated by irregular engine operation and improve the stability of the system.

[0004] However, existing CPVA applications are mainly concentrated in medium and high speed conditions. In low speed conditions, such as engine start-up, idling and low-speed driving, the vibration reduction effect of CPVA is often limited due to insufficient centrifugal force. In addition, traditional designs usually fail to fully consider the complex working conditions of the automotive powertrain system, resulting in the inability to effectively exert the vibration reduction performance of CPVA under certain working conditions. Therefore, how to optimize the CPVA parameters in the powertrain system so that it can work efficiently under different speeds and working conditions is an urgent problem to be solved in current technology. Summary of the invention

[0005] The purpose of the present invention is to provide a method for optimizing the design of power transmission system parameters with a centrifugal pendulum shock absorber, which can achieve the goal of reducing the impact power of the pendulum and the torsional amplitude of the transmission system, and improve the dynamic performance and durability of the system. The technical solution adopted is as follows:

[0006] A method for optimizing the parameters of a power transmission system equipped with a centrifugal pendulum shock absorber comprises the following steps:

[0007] Step 1: Establish a power transmission system model with a centrifugal pendulum shock absorber installed;

[0008] The power transmission system model includes an engine, a dual-mass flywheel DMF, and a transmission input shaft connected in sequence; a centrifugal pendulum shock absorber CPVA is arranged on the dual-mass flywheel DMF;

[0009] Step 2: Construct the engine thermodynamics and kinetics model;

[0010] Step 3, constructing a DMF kinetic model;

[0011] Step 4, constructing the CPVA dynamics model and the pendulum striking power analysis model;

[0012] Step 5: construct a dynamic model of the transmission input shaft;

[0013] Step 6: Define the decision variables and objective function in multi-objective optimization:

[0014] The decision variables include: several parameters in the engine thermodynamics and kinetics model and the CPVA kinetics model;

[0015] The objective function includes objective function one and objective function two;

[0016] Objective function 1 is to minimize the peak factor CF of the transmission input shaft angular acceleration; objective function 2 is to minimize the knock index KI of CPVA;

[0017] Among them, the knocking index KI of CPVA is generated based on the pendulum knocking power analysis model;

[0018] Step 7: Use MNSGA-II to optimize decision variables.

[0019] Preferably, in step 6, the decision variables include: ignition advance angle, injection advance angle, average throttle opening, combustible mixture formation time in the engine thermodynamics and dynamics model, and the pendulum transient position when the vehicle starts in the CPVA dynamics model.

[0020] Preferably, in step 6, the first objective function is:

[0021]

[0022] in, is the peak value of the transmission input shaft angular acceleration;

[0023] is the RMS value of the transmission input shaft angular acceleration.

[0024] Preferably, in step 6, the second objective function is:

[0025]

[0026]

[0027]

[0028] and are the peak tapping power ω2 and the average tapping power The normalized value of

[0029] ω1 and ω2 represent the weights of the peak value and the average value, respectively.

[0030] Compared with the prior art, the advantages of the present invention are:

[0031] 1. Through precise dynamic modeling and optimized design, the knocking power and torsional vibration generated by the CPVA pendulum during engine starting are effectively reduced.

[0032] 2. The improved MNSGA-II algorithm is used to optimize key parameters and improve the accuracy of vibration control and system stability. The overall solution not only improves the performance of the powertrain system, but also significantly improves the NVH performance and vehicle comfort. BRIEF DESCRIPTION OF THE DRAWINGS

[0033] Figure 1 It is a structural diagram of the power transmission system.

[0034] Figure 2 This is a schematic diagram of the engine model.

[0035] Figure 3 It is the dynamic model diagram of the secondary flywheel in DMF;

[0036] Figure 4 It is a structural schematic diagram of a centrifugal pendulum shock absorber;

[0037] Figure 5 Flowchart of the multi-objective optimization algorithm.

[0038] Figure 6 is the optimization result after multiple iterations. DETAILED DESCRIPTION

[0039] The following will be described in more detail with reference to the schematic diagram of the power transmission system parameter optimization design method with the centrifugal pendulum shock absorber of the present invention, wherein the preferred embodiment of the present invention is shown, and it should be understood that the present invention described herein can be modified by those skilled in the art, while still achieving the advantageous effects of the present invention. Therefore, the following description should be understood as being widely known to those skilled in the art, and not as a limitation of the present invention.

[0040] The present invention provides a parameter optimization design method for a whole vehicle power transmission system equipped with CPVA, aiming to significantly improve the vibration reduction performance of CPVA under vehicle starting conditions.

[0041] Step 1: Establish a power transmission system model with a centrifugal pendulum shock absorber installed.

[0042] like Figure 1 As shown, the method is based on the torque transmission path of each component in the vehicle powertrain. Starting from the engine output torque, it is transmitted to the CPVA rotor through the primary flywheel of the DMF. After inertia tuning is achieved inside the CPVA to weaken the torsional vibration, the torque is further transmitted to the secondary flywheel of the DMF and finally acts on the transmission input shaft to complete the efficient power transmission and vibration reduction functions.

[0043] This optimization design method first requires: building an accurate powertrain system model, including dynamic models of key components such as the engine, DMF, CPVA and transmission input shaft model, to ensure that the dynamic characteristics and mutual coupling of each component can be fully reflected, laying the foundation for subsequent parameter optimization.

[0044] like Figure 1 As shown in the figure, the powertrain system model mainly includes the following key components and their interconnections:

[0045] 1. Engine

[0046] The engine provides torque to the entire powertrain through its output shaft and is the power source of the system.

[0047] 2. Dual Mass Flywheel (DMF)

[0048] DMF consists of a primary flywheel, annular long-arc coil spring and a secondary flywheel:

[0049] (1) Primary flywheel: directly connected to the engine output shaft and receives the engine's torque input.

[0050] (2) Annular long-arc helical spring: It is the core elastic element of the DMF and is arranged in a channel opened on the circumference of the secondary flywheel.

[0051] (3) Secondary flywheel: Receives the smooth torque transmitted by the primary flywheel through an annular long-arc coil spring and is fixedly connected to the rotor of the centrifugal pendulum vibration absorber (CPVA).

[0052] 3. Centrifugal Pendulum Vibration Absorber (CPVA)

[0053] CPVA is installed on the DMF secondary flywheel. Its main components include rotor, pendulum and limit mechanism:

[0054] (1) Rotor: Fixed on the DMF secondary flywheel, rotating synchronously with the secondary flywheel, providing installation and motion support for the pendulum.

[0055] (2) Pendulum: It is mounted on the rotor through a hinge point and can swing around the center of the rotor under the action of centrifugal force. The periodic motion of the pendulum generates a reverse inertial force to offset the torsional vibration of a specific order, thereby optimizing the dynamic performance of the system.

[0056] (3) Limiting mechanism: It is installed at the end of the pendulum's travel and is used to limit the pendulum's swing amplitude to avoid mechanical shock or damage caused by excessive swing, while ensuring the reliability and durability of the system.

[0057] 4. Transmission input shaft

[0058] The DMF secondary flywheel is connected to the transmission input shaft via a rigid connection, transmitting smooth torque that has been vibration-suppressed and dynamically optimized to the transmission, supporting subsequent power transmission and vehicle drive.

[0059] The overall structure of the powertrain is:

[0060] Engine output shaft → DMF primary flywheel → DMF secondary flywheel → CPVA rotor → CPVA pendulum → transmission input shaft.

[0061] In step 1, the dynamic parameters of each component, such as inertia, stiffness, damping coefficient, etc., are determined to provide a basic mathematical model for subsequent optimization analysis.

[0062] Step 2: Engine thermodynamics and kinetics modeling.

[0063] A zero-dimensional transient model of a four-cylinder four-stroke engine is established. By assuming that the working medium is an ideal gas, combining the mass conservation equation in the cylinder and the first law of thermodynamics, the mathematical expression of the transient torque is derived. Based on the transient torque, the effects of the ignition advance angle, injection advance angle, throttle opening and the formation time of the combustible mixture on the transient torque output are analyzed, and they are used as decision variables for multi-objective optimization.

[0064] like Figure 2 As shown in Figure 1, the torque output of the engine is closely related to the thermodynamic process of the working medium in its cylinder. In order to accurately describe the state evolution in the cylinder, it is assumed that the working medium is an ideal gas and its state parameters (temperature, pressure, etc.) are uniformly distributed in the cylinder.

[0065] First, from the perspective of mass conservation, the mass balance equation of the medium in the cylinder is established:

[0066]

[0067] dm e =τ(dms +g(φ ia )dx)

[0068] Among them, m c is the mass of the working medium in the cylinder;

[0069] m s is the mass of air flowing into the cylinder;

[0070] m e is the mass of exhaust gas leaving the cylinder;

[0071] τ is the correction coefficient considering the residual exhaust gas, which is the set value;

[0072] φ ia is the injection advance angle, is the set value, and is also one of the decision variables for multi-objective optimization;

[0073] g(φ ia ) is the cycle injection volume, is the set value;

[0074] is the percentage of the fuel mixture, which is a table value calibrated according to the instantaneous crankshaft angle;

[0075] φ is the instantaneous crankshaft angle, φ=0 represents the top dead center, φ=π represents the bottom dead center, which is the simulation output value.

[0076] Due to the volume lag effect in the intake system, the intake flow is affected by the periodic changes in the crankshaft angle.

[0077] Therefore, using the volume method to calculate the state parameters of the intake and exhaust system, the rate of change of the intake air mass is as follows:

[0078]

[0079] ω is the crankshaft angular velocity, and is the simulation output value;

[0080] μ is the exhaust flow coefficient, which characterizes the formation time of the combustible mixture and is a set value. It is also one of the decision variables for multi-objective optimization.

[0081] A s is the flow area of ​​the intake valve, is the set value;

[0082] p s is the pressure of the intake gas before the valve, which is a table value calibrated according to the instantaneous crankshaft angle;

[0083] χ to is the average opening of the throttle, is the set value, and is also one of the decision variables for multi-objective optimization; v s is the mass volume of the intake gas, and is a table value calibrated according to the instantaneous crankshaft angle;

[0084] In order to describe the dynamic characteristics of heat release in the cylinder during combustion, the combustion heat release rate equation is introduced:

[0085]

[0086] Among them, Q B The heat released by combustion;

[0087] η u is the combustion efficiency, is the set value;

[0088] m B0 The gas mass supplied for each cycle is the set value;

[0089] H u is the lower heating value of the fuel, is the set value;

[0090] Δφ is the crankshaft angle of the combustion process, which is the simulation output value;

[0091] φ VB is the ignition advance angle, is the set value, and is also one of the decision variables for multi-objective optimization;

[0092] z is the combustion quality index, which is the set value;

[0093] It can be seen that It depends on key factors such as combustion efficiency, lower calorific value of fuel, and ignition advance angle.

[0094] Combining equations (1) and (2), it can be seen that the heat released by combustion has a direct effect on the change of the thermodynamic state of the gas in the cylinder.

[0095] Based on the ideal gas state equation, the instantaneous pressure in the cylinder can be expressed as:

[0096]

[0097]

[0098] Among them, V c is the working volume of the cylinder, is the set value;

[0099] c v is the specific heat capacity in the cylinder, which is the set value;

[0100] R is the gas constant, which is the set value;

[0101] is the heat dissipation rate of the cylinder;

[0102] Q w is the heat dissipation in the cylinder;

[0103] a gis the average heat dissipation coefficient, is the set value;

[0104] A w Heat dissipation area, which is the set value;

[0105] ΔT is the temperature difference between the working fluid in the cylinder and the wall, which is the set value;

[0106] The above formula dynamically relates the combustion heat release to the cylinder pressure and is used to calculate the gas pressure torque T g key items.

[0107] In order to obtain the power output of the engine, a dynamic model is further introduced.

[0108] The instantaneous torque is determined by the reciprocating inertia torque M m and gas pressure torque M g It consists of two parts, which are respectively expressed as:

[0109]

[0110]

[0111] Among them, m p is the equivalent reciprocating mass of the piston, piston ring, piston pin and connecting rod, which is the set value;

[0112] r is the crank radius, which is the set value;

[0113] λ s is the ratio of crank radius to connecting rod length, which is the set value;

[0114] t is the rotation time of the crank, which is the set value;

[0115] d p is the piston diameter, is the set value;

[0116] Combined with the cylinder pressure calculated in equation (4), equation (6) describes the direct effect of gas pressure on torque.

[0117] Finally, the total torque output by the engine is:

[0118] M e =M m +M g (7)

[0119] Formula (7) is the instantaneous engine output torque M e The final expression of , fully combines the thermodynamic and kinetic models.

[0120] Through the logical association of the above formulas, the influence of multiple parameters such as injection advance angle, formation time of combustible mixture, throttle opening, etc. on the engine output torque can be quantitatively analyzed.

[0121] Step 3: Establish the DMF kinetic model.

[0122] like Figure 3 As shown in the figure, the discretization method is used to model the annular long arc helical spring, which is decomposed into an equivalent system consisting of multiple mass blocks and short arc springs. Combining the torsional dynamics equations of the primary and secondary flywheels, the torsional response characteristics of the DMF are derived, and its regulating effect on the vibration of the power transmission system is analyzed.

[0123] The dual mass flywheel (DMF) model with annular long-arc coil spring consists of three main parts: primary flywheel, annular long-arc spring and secondary flywheel. The annular long-arc spring on the circumference of the flywheel is continuously compressed and released in the channel, thereby transmitting torque from the primary flywheel to the secondary flywheel.

[0124] The dynamic differential equations of the primary flywheel and secondary flywheel are as follows:

[0125]

[0126] Among them, J d1 and J d2 They are the rotational inertia of the primary flywheel and the secondary flywheel respectively;

[0127] θ1 is the torsion angle of the primary flywheel, θ2 is the torsion angle of the secondary flywheel, and is the simulation output value;

[0128] k d is the equivalent spring stiffness, is the set value;

[0129] γ0, γ1, γ n are the center angle of the discrete short spring when not compressed, the center angle of the first equivalent spring deformation, and the center angle of the nth equivalent spring deformation, which are simulation output values;

[0130] M d2 is the torque output by the secondary flywheel, and is the simulation output value;

[0131] M seal is the damping force caused by the sealing element between the primary flywheel and the secondary flywheel, and is a look-up table value calibrated according to the instantaneous crankshaft angle;

[0132] Step 4: Construct the CPVA dynamics model and the pendulum striking power analysis model.

[0133] like Figure 4 As shown in the figure, consider that there are two identical pendulums installed on the rotor in CPVA, and their motion trajectory is an epicycloid, where XY is a fixed coordinate system on the rotor axis, x i -y i is a non-fixed coordinate system that rotates with the pendulum.

[0134] Considering the superposition of the centrifugal field and the gravitational field, the differential equations of motion of the rotor and the pendulum obtained based on the second-kind Lagrangian equation are as follows:

[0135]

[0136] Among them, α is the rotation angle of the rotor, and is the simulation output value;

[0137] S i is the transient displacement of the pendulum, and it is stipulated that the pendulum is in y i On-axis time S i =0, which is the simulation output value; in addition, S at t=0 i As one of the decision variables in multi-objective optimization;

[0138] E is the kinetic energy of the system, which is the calculated value obtained according to the existing formula;

[0139] V is the potential energy of the system, which is the calculated value obtained according to the existing formula;

[0140] U p is the elastic potential energy of the limiting mechanism, which is the calculated value obtained according to the existing formula;

[0141] i is the number of the pendulum;

[0142] μ α is the viscous damping coefficient of the rotor, which is the set value;

[0143] M d1 is the excitation torque acting on the rotor, is the simulation output value;

[0144] μ si is the viscous damping coefficient of the pendulum, which is the set value;

[0145] μ ri is the rolling damping coefficient of the pendulum, which is the set value;

[0146] F ni is the positive pressure on the pendulum, which is the calculated value obtained according to the existing formula;

[0147] In order to compare the performance of the pendulum under starting conditions, the striking power of the pendulum is defined as an indicator of the strength of the pendulum's impact on the limit mechanism.

[0148] Construct a pendulum strike power analysis model:

[0149] The mathematical expression of the pendulum striking power is defined to describe its dynamic response under different starting conditions. The average striking power and peak striking power of the pendulum are included in the evaluation indicators of multi-objective optimization through normalization.

[0150] Pendulum peak striking power ω2 and average striking power The calculation formula is as follows:

[0151]

[0152]

[0153] Among them, P ij is the power of the i-th pendulum at the j-th strike;

[0154] S ij is the displacement of the i-th pendulum at the j-th strike, which is the simulation output value;

[0155] S pi is the position of the pendulum limit mechanism, is the set value, and its reference coordinate system is the same as S i Be consistent.

[0156] k si is the stiffness of the pendulum limit mechanism, is the set value;

[0157] t start and t end They are the start and end moments of the tapping, which are the set values;

[0158] Q is the sum of the number of times all pendulums hit the limit mechanism over time, and N is the number of pendulums.

[0159] Step 5: Build a dynamic model of the transmission input shaft.

[0160] The end of the powertrain is the transmission input shaft.

[0161] The differential equation of motion for the input shaft is given by:

[0162]

[0163] Among them, J s is the moment of inertia, is the set value;

[0164] is the angular velocity of the input shaft, is the simulation output value;

[0165] k s is the equivalent stiffness, is the set value;

[0166] c s is the equivalent damping coefficient, is the set value;

[0167] M r is the load torque, is the set value;

[0168] That is, the torsional dynamic equation of the transmission shaft is established, the transient torsional vibration response during the engine startup phase is analyzed, the inertia parameters, equivalent damping coefficient and load torque of the input shaft are defined, and the impact of its dynamic characteristics on the overall transmission system performance is quantified.

[0169] Step 6: Define the parameters in multi-objective optimization.

[0170] The peak factor (CF) of the transmission input shaft angular acceleration and the knock index (KI) of the CPVA are minimized as the objective function, and the five parameters of ignition advance angle, injection advance angle, idle throttle opening and pendulum starting position are used as decision variables to construct a multi-objective optimization problem.

[0171] This embodiment focuses on the dynamic behavior of the powertrain system during engine startup, and aims to minimize the knock power generated by the CPVA pendulum and the torsional vibration of the input shaft of the drive system by optimizing the system parameter combination.

[0172] The optimized initial pendulum position is determined by numerical simulation of the system model. The angular acceleration amplitude spectrum of different initial positions is calculated, and the position with the largest angular acceleration amplitude is optimized first.

[0173] The first objective function is to minimize the peak factor CF of the transmission input shaft angular acceleration, which is calculated as follows:

[0174]

[0175] in, is the peak value of the transmission input shaft angular acceleration;

[0176] is the RMS value of the transmission input shaft angular acceleration.

[0177] The second objective function is to minimize the knock index (KI) considering the dynamic characteristics of CPVA, and its calculation formula is as follows:

[0178]

[0179] in, and are the normalized values ​​of the peak tapping power and average tapping power in equation (12) and (13), respectively.

[0180] The normalization method used is the maximum-minimum method.

[0181] ω1 and ω2 represent the weights of the peak value and the average value, respectively, which are set values.

[0182] In the multi-objective optimization, five key parameters of the engine and CPVA are selected as decision variables. Table 1 shows all decision variables and the value ranges of the defined decision variables.

[0183] Table 1 All decision variables and the value range of defined decision variables

[0184]

[0185] Step 7: Design of target optimization algorithm.

[0186] An improved non-dominated sorting genetic algorithm (MNSGA-II) is used to find a better Pareto solution set in multi-objective optimization problems by dynamically adjusting the population size and diversity maintenance strategy, thus avoiding the problems of premature convergence of the population and uneven distribution of the optimized solutions.

[0187] like Figure 5 As shown in the figure, the optimization process starts with the initial parameter setting, generates an initial population and uses crossover and mutation operations to achieve population diversity. Subsequently, fitness evaluation is performed, and the selection and retention of the optimal individuals are determined by using the non-dominated sorting and crowding calculation of the merged population. After the initial optimization, local searches are further performed for the five decision variables in Table 1 to improve the globality and stability of the solution. Finally, after multiple rounds of iterations, the optimization results are output until the convergence conditions are met. The entire process fully combines the global search capability of the genetic algorithm with the engineering constraint characteristics of the problem itself, effectively improving the efficiency and accuracy of parameter optimization.

[0188] Step 8: Numerical verification of optimization results.

[0189] Based on multi-body dynamics simulation, the effect of optimization design on improving the vibration characteristics of the system is verified. Figure 6 The Pareto front solution in the paper is substituted into the power transmission system model, focusing on analyzing the improvement of the optimized pendulum striking power and the torsional vibration response of the transmission system to prove the effectiveness and feasibility of the optimization model.

[0190] The above is only a preferred embodiment of the present invention and does not limit the present invention in any way. Any technician in the relevant technical field, without departing from the scope of the technical solution of the present invention, makes any form of equivalent replacement or modification to the technical solution and technical content disclosed in the present invention, which does not depart from the content of the technical solution of the present invention and still falls within the protection scope of the present invention.

Claims

1. A method for optimizing the parameters of a power transmission system equipped with a centrifugal pendulum shock absorber, characterized in that: The following steps are involved: Step 1: Establish a power transmission system model with a centrifugal pendulum shock absorber installed; The power transmission system model includes an engine, a dual-mass flywheel DMF, and a transmission input shaft connected in sequence; a centrifugal pendulum shock absorber CPVA is arranged on the dual-mass flywheel DMF; Step 2: Construct the engine thermodynamics and kinetics model; Step 3, constructing a DMF kinetic model; Step 4, constructing the CPVA dynamics model and the pendulum striking power analysis model; Step 5: construct a dynamic model of the transmission input shaft; Step 6: Define the decision variables and objective function in multi-objective optimization: The decision variables include: several parameters in the engine thermodynamics and kinetics model and the CPVA kinetics model; The objective function includes objective function one and objective function two; Objective function 1 is to minimize the peak factor CF of the transmission input shaft angular acceleration; objective function 2 is to minimize the knock index KI of CPVA; Among them, the knocking index KI of CPVA is generated based on the pendulum knocking power analysis model; Step 7: Use MNSGA-II to optimize decision variables.

2. The method for optimizing the parameters of a power transmission system equipped with a centrifugal pendulum shock absorber according to claim 1, characterized in that: In step 6, the decision variables include: the ignition advance angle, the injection advance angle, the average throttle opening, the combustible mixture formation time in the engine thermodynamics and dynamics model, and the pendulum transient position when the vehicle starts in the CPVA dynamics model.

3. The method for optimizing the parameters of a power transmission system equipped with a centrifugal pendulum shock absorber according to claim 2, characterized in that: In step 6, the objective function 1 is: in, is the peak value of the transmission input shaft angular acceleration; is the RMS value of the transmission input shaft angular acceleration.

4. The method for optimizing the parameters of a power transmission system equipped with a centrifugal pendulum shock absorber according to claim 2, characterized in that: In step 6, the second objective function is: and are the peak tapping power ω2 and the average tapping power The normalized value of ω1 and ω2 represent the weights of the peak value and the average value, respectively.

Images