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Stiffness Calculation of Gear Hydraulic System Based on the Modeling of Nonlinear Dynamics Differential Equations in the Progressive Method

Published Online: 15 Jul 2022
Volume & Issue: AHEAD OF PRINT
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Received: 06 Jan 2022
Accepted: 14 Mar 2022
Journal Details
License
Format
Journal
eISSN
2444-8656
First Published
01 Jan 2016
Publication timeframe
2 times per year
Languages
English
Abstract

The paper calculates the nonlinear dynamic differential equation model based on the stiffness of the gear teeth and gives the calculation method of the spring stiffness of the transmission system. Choose the Lyapunov energy function and derive the adaptive law that can make the system asymptotically stable globally. At the same time, we discussed the influence of the phase combination of the coupling shaft's torsional stiffness and the gears' meshing stiffness in the multi-stage gear transmission system on the system dynamics. The example calculation shows that the asymptotic method has higher solution accuracy and higher calculation efficiency. This algorithm is a highly versatile analytical solution method.

Keywords

MSC 2010

Introduction

As the core part of the electro-hydraulic load simulator, the electro-hydraulic loading system is used to generate the desired loading force accurately and apply it to the load object. The system is widely used in aerospace, navigation, and military industries. Therefore, under the premise of ensuring the system's stability, there are also high requirements for its bandwidth and accuracy. The most commonly used in electro-hydraulic servo loading systems is PID control. It has the advantages of reliable work and easy implementation, but it is only suitable for linear systems with simple structure and constant parameters [1]. The loading system is a complex nonlinear system. Factors such as load, pressure, and external interference are constantly changing during the working process, and the system parameters are time-varying within a relatively large range. The current research focus is to ensure the stability, high response, and high precision of the system under this condition.

Model reference adaptive control (MRAC) is a better control method for time-varying parameter systems. It is mainly based on the output error of the controlled object and the reference model to continuously adjust the controller's parameters to suppress the adverse effects of the controlled object's parameter changes and external interference. In recent years, the MRAC method has been widely used in loading systems [2]. The nonlinear dynamic differential equation describing the dynamic characteristics of the gear system is very complicated, and it is impossible to obtain its exact solution directly. Therefore, only high-precision numerical solutions can be obtained for complex high-order nonlinear dynamic equations. However, numerical solutions cannot replace approximate analytical solutions. The perturbation method is the most important tool for studying nonlinear complex differential equations [3]. At present, the perturbation methods mainly include the direct expansion method, Lindstedt-Poincare method, re-normalization method, multi-scale method, parameter variation method, and progressive method. Certain results have been achieved in the research of strongly nonlinear vibration systems. Some scholars have used the phase plane method combined with digital simulation to study the dynamic stability of a single degree of freedom gear transmission system. Some scholars have proposed a vibration equation with a single degree of freedom, constant stiffness, clearance, and parameter excitation of transmission errors. They found the transition frequency, sub-harmonic resonance, and chaos after solving the problem with the harmonic balance method. However, all these methods are approximate solutions proposed for a particular vibration system and are not universal [4]. The author uses the progressive method to solve the nonlinear dynamic differential equations of the gear system. The example calculation shows that the asymptotic method has higher solution accuracy and higher calculation efficiency. This algorithm is a highly versatile analytical solution method.

Nonlinear dynamic differential equations of the gear transmission system

We assume that the transmission shaft and the support shaft are rigid in the gear system with backlash, and the dynamic model of a pair of spur gear pairs is shown in Figure 1.

Figure 1

The dynamic model of the gear pair

We assume that the damping coefficient is ce. The base circle radii of the main and driven gears are rp, rg respectively. The torsional vibration analysis model of the gear pair is Ipd2θpdt2+rpce(rpdθgdtrgdθgdtdedt)+rpke(t)δ(x)=Tp(t) {I_p}{{{d^2}{\theta _p}} \over {d{t^2}}} + {r_p}{c_e}\left( {{r_p}{{d{\theta _g}} \over {dt}} - {r_g}{{d{\theta _g}} \over {dt}} - {{de} \over {dt}}} \right) + {r_p}{k_e}\left( t \right)\delta \left( x \right) = {T_p}\left( t \right) Igd2θpdt2+rgce(rpdθpdtrgdθgdtdedt)+rgke(t)δ(x)=Tg(t) {I_g}{{{d^2}{\theta _p}} \over {d{t^2}}} + {r_g}{c_e}\left( {{r_p}{{d{\theta _p}} \over {dt}} - {r_g}{{d{\theta _g}} \over {dt}} - {{de} \over {dt}}} \right) + {r_g}{k_e}\left( t \right)\delta \left( x \right) = - {T_g}\left( t \right)

Ip, Ig represents the moment of inertia of the driving and driven gears, and (kg/m)2. θp, θg, represents the angular displacement of the main and driven gears. Tp(t), Tg(t) represents the external torque borne by the driving and driven gears, (N/n). e Represents the static transmission error of meshing gear teeth, (m). ke(t) Represents the time-varying meshing stiffness of the gear pair, (N/m), which can be developed into the form of the 5th harmonic. ke(t)=km+j=15Ajcos(jω¯et+Φj) {k_e}\left( t \right) = {k_m} + \sum\limits_{j = 1}^5 {{A_j}\cos \left( {j{{\bar \omega }_e}t + {\Phi _j}} \right)} . δ(x) It represents the descriptive function of gear tooth deformation under the action of gear dynamic meshing force when there is tooth backlash [5]. We assume that the difference between the dynamic transmission error xd(t) and the static transmission error e(t) of the gear system is x(t), then x=xd(t)e(t)=rpθp(t)rgθg(t)e(t) x = {x_d}\left( t \right) - e\left( t \right) = {r_p}{\theta _p}\left( t \right) - {r_g}{\theta _g}\left( t \right) - e\left( t \right)

We can use equation (3) to transform the equations described by equations (1) and (2) into a degree of freedom differential equation, namely med2xdt2+cedxdt¯+ke(t)δ(x)=FaT(t)+Fah(t) {m_e}{{{d^2}x} \over {d{t^2}}} + {c_e}{{dx} \over {d\bar t}} + {k_e}\left( t \right)\delta \left( x \right) = {F_{aT}}\left( t \right) + {F_{ah}}\left( t \right) me represents the equivalent mass of the gear pair, me=IpIgIprg2+Igrp2 {m_e} = {{{I_p}{I_g}} \over {{I_p}r_g^2 + {I_g}r_p^2}} . FaT (t) means external stimulus, FaT(t)=merpTp(t)Ip+mergTg(t)Ig {F_{aT}}\left( t \right) = {{{m_e}{r_p}{T_p}\left( t \right)} \over {{I_p}}} + {{{m_e}{r_g}{T_g}\left( t \right)} \over {{I_g}}} . Fah (t) represents the internal excitation of the system, Fah(t)=med2xdt2 {F_{ah}}\left( t \right) = - {m_e}{{{d^2}x} \over {d{t^2}}} . We use simple harmonic function as the external excitation, namely FaT(t)=FaTcos(ωeTt+ΦT) {F_{aT}}\left( t \right) = {F_{aT}}\cos \left( {{\omega _{eT}}t + {\Phi _T}} \right)

FaT, ωeT represents the amplitude and fundamental frequency of the external excitation. We also express the internal excitation as a simple harmonic function. We get e(t)=ecos(ωeh+ΦT) e\left( t \right) = e\cos \left( {{\omega _{eh}} + {\Phi _T}} \right) Fah(t)=meeωeh2cos(ωeh+ΦT) {F_{ah}}\left( t \right) = {m_e}e\omega _{eh}^2\cos \left( {{\omega _{eh}} + {\Phi _T}} \right) e, ωeh represents the amplitude and fundamental frequency of the internal excitation. According to equation (4), we can describe the differential equation of the gear system's nonlinear dynamics as: med2xdt2+cedxdt+ke(t)x=FaTcos(ωeTt+ΦT)+meeωeh2cos(ωeht+Φh) {m_e}{{{d^2}x} \over {d{t^2}}} + {c_e}{{dx} \over {dt}} + {k_e}\left( t \right)x = {F_{aT}}\cos \left( {{\omega _{eT}}t + {\Phi _T}} \right) + {m_e}e\omega _{eh}^2\cos \left( {{\omega _{eh}}t + {\Phi _h}} \right)

Make X=xb,ωn=kmme,ζ=ce2ωnme,τ=ωnt,F¯aT=FaTbkm,F¯ah=Fahbkm,ω¯e=ωeωn,ω¯eT=ωeTωn,ω¯eh=ωehωn. X = {x \over b},{\omega _n} = \sqrt {{{{k_m}} \over {{m_e}}}} ,\zeta = {{{c_e}} \over {2{\omega _n}{m_e}}},\tau = {\omega _n}t,{\bar F_{aT}} = {{{F_{aT}}} \over {b{k_m}}},{\bar F_{ah}} = {{{F_{ah}}} \over {b{k_m}}},{\bar \omega _e} = {{{\omega _e}} \over {{\omega _n}}},{\bar \omega _{eT}} = {{{\omega _{eT}}} \over {{\omega _n}}},{\bar \omega _{eh}} = {{{\omega _{eh}}} \over {{\omega _n}}}.

Equation (8) can be transformed into a dimensionless equation X(τ)+2ζbX(τ)+1bke(τ)X=1bF¯aTcos(ω¯eTτ+Φj)+1bF¯eh2cos(ω¯ehτ+Φh) X\left( \tau \right) + {{2\zeta } \over b}X\left( \tau \right) + {1 \over b}{k_e}\left( \tau \right)X = {1 \over b}{\bar F_{aT}}\cos \left( {{{\bar \omega }_{eT}}\tau + {\Phi _j}} \right) + {1 \over b}\bar F_{eh}^2\cos \left( {{{\bar \omega }_{eh}}\tau + {\Phi _h}} \right)

Asymptotic method for solving approximate solutions of nonlinear differential equations

The differential equation of gear system dynamics described by equation (9) is very complicated, and it is impossible to find its exact solution directly. Therefore, only approximate solutions can be obtained [6]. The author introduces the asymptotic method for solving differential equations with a single degree of freedom, also known as the three-series method or the KBM method. This method expresses the solution of the equation and the amplitude and phase angle as a power series function of a small parameter ɛ. It then uses the method of separating variables to find the unknown coefficients of these power series functions. If there is the following nonlinear differential equation d2xdt2+ω02x=εfk(x,dxdt)d2xdt2+ω02x=εfM(d2xdt2,dxdt) \matrix{ {{{{d^2}x} \over {d{t^2}}} + \omega _0^2x = \varepsilon {f_k}\left( {x,{{dx} \over {dt}}} \right)} \hfill \cr {{{{d^2}x} \over {d{t^2}}} + \omega _0^2x = \varepsilon {f_M}\left( {{{{d^2}x} \over {d{t^2}}},\,{{dx} \over {dt}}} \right)} \hfill \cr }

ɛ means small parameter. If ɛfk or ɛfM can be written as ɛfk = ɛFk + kcx, ɛFM + Mcx. kc and Mc represent linear average stiffness (or equivalent stiffness) and linear average mass (or equivalent average mass). Fk(x,dxdt) {F_k}\left( {x,{{dx} \over {dt}}} \right) , FM(d2xdt2,dxdt) {F_M}\left( {{{{d^2}x} \over {d{t^2}}},{{dx} \over {dt}}} \right) , fk(x,dxdt) {f_k}\left( {x,{{dx} \over {dt}}} \right) and fM(d2xdt2,dxdt) {f_M}\left( {{{{d^2}x} \over {d{t^2}}},{{dx} \over {dt}}} \right) represent both nonlinear functions. That is, because the linear average stiffness or average mass is taken out of the nonlinear function in the above formula, the following analytical approximation is closer to reality [7]. Taking into account the influence of nonlinear forces, we assume that the solution of the equation has the following form x=acosφ+εu1(a,φ)+ε2u2(a,φ)+ x = a\cos \varphi + \varepsilon {u_1}\left( {a,\varphi } \right) + {\varepsilon ^2}{u_2}\left( {a,\varphi } \right) + \cdots a and φ represent the periodic function of time t. u1(a, φ), u2(a, φ), … and so on represent the power series function of ɛ. We can make the following assumptions [8]. The equivalent damping ratio δe and the equivalent natural frequency ωe of the nonlinear coefficient can be expressed in the form of a small parameter power series: δe(a)=εδ1(a)+ε2δ2(a)+,ωe(a)=ω0+εω1(a)+ε2ω2(a)+ {\delta _e}\left( a \right) = \varepsilon {\delta _1}\left( a \right) + {\varepsilon ^2}{\delta _2}\left( a \right) + \cdots ,{\omega _e}\left( a \right) = {\omega _0} + \varepsilon {\omega _1}\left( a \right) + {\varepsilon ^2}{\omega _2}\left( a \right) + \cdots

When δ1, δ2, ω1, ω2 is found, the sum of a can be found. a and φ satisfy the following equation dxdt=[εδ1(a)+ε2δ2(a)+ε3]a,dφdt=ω0+εω1(a)+ε2ω2(a)+ {{dx} \over {dt}} = \left[ {\varepsilon {\delta _1}\left( a \right) + {\varepsilon ^2}{\delta _2}\left( a \right) + {\varepsilon ^3} \cdots } \right]a,\,{{d\varphi } \over {dt}} = {\omega _0} + \varepsilon {\omega _1}\left( a \right) + {\varepsilon ^2}{\omega _2}\left( a \right) + \cdots

The applicability of the asymptotic method does not depend on the convergence of the above two equations, but on the asymptoticity of the equation solution when ɛ approaches zero. As long as the value of ɛ is small and the time interval is sufficiently long, the above two expressions can give a sufficiently accurate solution [9]. Since u1, u2 does not contain the first harmonic, u1, u2 should satisfy the following formula 02πu1(a,φ)cosφdφ=002πu2(a,φ)cosφdφ=002πu1(a,φ)sinφdφ=002πu2(a,φ)sinφdφ=0 \matrix{ {\int\limits_0^{2\pi } {{u_1}\left( {a,\,\varphi } \right)\cos \varphi d\varphi = 0} } & {\int\limits_0^{2\pi } {{u_2}\left( {a,\,\varphi } \right)\cos \varphi d\varphi = 0} } \cr {\int\limits_0^{2\pi } {{u_1}\left( {a,\,\varphi } \right)\sin \varphi d\varphi = 0} } & {\int\limits_0^{2\pi } {{u_2}\left( {a,\,\varphi } \right)\sin \varphi d\varphi = 0} } \cr }

At this time, we can develop the nonlinear force as a Fourier series and substitute it into the equation (10) to eliminate the first harmonic term to obtain the expression of u1, u2. The expression for the first approximate solution of the equation is x = a cos φ + ɛu1(a, φ) and the expression for the second approximate solution is x=acosφ+εu1(a,φ)+ε2u2(a,φ) x = a\cos \,\varphi + \varepsilon {u_1}\left( {a,\varphi } \right) + {\varepsilon ^2}{u_2}\left( {a,\varphi } \right)

According to the above method, solutions with different accuracy requirements can be obtained. The accuracy of the equation solution is determined by the order n.

Calculation examples

We use Matlab / Simulink software for numerical simulation. Use ode23s variable step size algorithm and take the maximum step size as 1×10−5 s,, and calculate the relative error as 1×10−6. We simulate the step signal with an input amplitude of 6kN when R is 10, 50, 500, and 1000 respectively [10]. The results are shown in Figure 2 and Figure 3.

Figure 2

Step response of the system at different R values after adding adaptive control

Figure 3

The error between the system and the reference model at different values of R after adding adaptive control

It can be seen from Figure 2 and Figure 3 that the system with adaptive control can follow the reference model well. The following error e eventually converges to zero and remains stable. From the error indicators shown in Table 1, it can be seen that as the adaptive coefficient R increases, the convergence speed is faster, and the maximum error value is also smaller. But compared with R=500, the maximum error and the decrease in convergence time when R=1000 are already very small [11]. When R>1000, the follow-up effect of the system will not be greatly improved since the maximum error and convergence time at R=1000 can meet the purpose of accurately and quickly following the reference model, the adaptive coefficient R is taken as 1000 in the subsequent simulation analysis.

Error indicators for different R values.

R Maximum error/kN Error convergence time/ms
10 2.62 6
50 1.08 4.4
500 0.34 2.4
1000 0.28 2.1

The hydraulic stiffness Kh corresponding to the piston displacement x at 50, 75, and 99 mm is 56.3, 75.0, and 1420 MN/m, respectively. At the same time, we consider three kinds of load stiffness K and select K and Kh values as shown in Table 2 for simulation based on Kh=56.3 MN/m and K=51.0 MN/m.

Simulation parameters (unit: MN/M).

Serial number K Kh
1 5.1 56.3
2 25.5 56.3
3 51 56.3
4 51 75
5 51 1420

We take R=1000, and the simulation result of the stop signal with input amplitude of 6kN before and after the adaptive control is added shown in Fig. 4. The dynamic performance index of the system is shown in Table 3.

Figure 4

Step response at different values of K and Kh before and after adaptive control is added to the system

Dynamic performance indicators of the system before and after adaptive control.

Original system System after adding adaptive control
Serial number tr/ms ts/ms ess/% tr/ms ts/ms ess/%
1 54.6 85.8 8.33 4.8 14.6 0.03
2 10.7 43.6 8.33 4.8 14.6 0.03
3 8 38.7 8.33 4.8 14.6 0.03
4 8.7 56.3 8.33 4.8 14.6 0.03
5 10.4 59.5 8.33 4.8 14.6 0.03

It can be seen from Table 3 that the dynamic response of the original system changes greatly when K and Kh change. When Kh is constant, the smaller K is, the larger tr and ts are. When K is constant, the greater the Kh, the greater the tr and ts. The dynamic performance of the original system is relatively best when K=51.0 MN/m and Kh=56.3 MN/m. The steady-state error of the original system under the five sets of parameters is 8.33%, which indicates that the system accuracy is very poor. After adding adaptive control, the response of the system under 5 sets of parameters shows excellent consistency. Compared with the situation when the dynamic performance of the original system is at its best, it is greatly reduced, and the steady-state error is very small.

Conclusion

Compared with the existing methods for solving nonlinear differential equations, the asymptotic method has higher solving accuracy and higher efficiency. This algorithm is an effective method for solving complex nonlinear differential equations.

Figure 1

The dynamic model of the gear pair
The dynamic model of the gear pair

Figure 2

Step response of the system at different R values after adding adaptive control
Step response of the system at different R values after adding adaptive control

Figure 3

The error between the system and the reference model at different values of R after adding adaptive control
The error between the system and the reference model at different values of R after adding adaptive control

Figure 4

Step response at different values of K and Kh before and after adaptive control is added to the system
Step response at different values of K and Kh before and after adaptive control is added to the system

Dynamic performance indicators of the system before and after adaptive control.

Original system System after adding adaptive control
Serial number tr/ms ts/ms ess/% tr/ms ts/ms ess/%
1 54.6 85.8 8.33 4.8 14.6 0.03
2 10.7 43.6 8.33 4.8 14.6 0.03
3 8 38.7 8.33 4.8 14.6 0.03
4 8.7 56.3 8.33 4.8 14.6 0.03
5 10.4 59.5 8.33 4.8 14.6 0.03

Simulation parameters (unit: MN/M).

Serial number K Kh
1 5.1 56.3
2 25.5 56.3
3 51 56.3
4 51 75
5 51 1420

Error indicators for different R values.

R Maximum error/kN Error convergence time/ms
10 2.62 6
50 1.08 4.4
500 0.34 2.4
1000 0.28 2.1

Rahaman, H., Kamrul Hasan, M., Ali, A. & Shamsul Alam, M. Implicit Methods for Numerical Solution of Singular Initial Value Problems. Applied Mathematics and Nonlinear Sciences., 2021; 6(1): 1–8 RahamanH. Kamrul HasanM. AliA. Shamsul AlamM. Implicit Methods for Numerical Solution of Singular Initial Value Problems Applied Mathematics and Nonlinear Sciences 2021 6 1 1 8 10.2478/amns.2020.2.00001 Search in Google Scholar

Hassan, S., Reddy, M. & Rout, R. Dynamics of the Modified n-Degree Lorenz System. Applied Mathematics and Nonlinear Sciences., 2019; 4(2): 315–330 HassanS. ReddyM. RoutR. Dynamics of the Modified n-Degree Lorenz System Applied Mathematics and Nonlinear Sciences 2019 4 2 315 330 10.2478/AMNS.2019.2.00028 Search in Google Scholar

Mikhailov, A. A., Keresten, I. A., Nikitin, M. A., Voinov, I. B., & Morozov, D. I. About the Landing Gear Design Experience for a Light Aircraft Based on Multidisciplinary Modeling. Russian Aeronautics., 2020;63(4): 586–593 MikhailovA. A. KerestenI. A. NikitinM. A. VoinovI. B. MorozovD. I. About the Landing Gear Design Experience for a Light Aircraft Based on Multidisciplinary Modeling Russian Aeronautics 2020 63 4 586 593 10.3103/S1068799820040042 Search in Google Scholar

Ma, X. J., Zhao, X., Huang, B., Fu, X. Y., & Wang, G. Y. On study of non-spherical bubble collapse near a rigid boundary. Journal of Hydrodynamics., 2020; 32(3): 523–535 MaX. J. ZhaoX. HuangB. FuX. Y. WangG. Y. On study of non-spherical bubble collapse near a rigid boundary Journal of Hydrodynamics 2020 32 3 523 535 10.1007/s42241-019-0056-7 Search in Google Scholar

Svanberg, A., Larsson, S., Mäki, R., & Jonsén, P. Full-scale simulation and validation of bucket filling for a mining rope shovel by using a combined rigid FE-DEM granular material model. Computational Particle Mechanics., 2021; 8(4): 825–843 SvanbergA. LarssonS. MäkiR. JonsénP. Full-scale simulation and validation of bucket filling for a mining rope shovel by using a combined rigid FE-DEM granular material model Computational Particle Mechanics 2021 8 4 825 843 10.1007/s40571-020-00372-z Search in Google Scholar

Perera-Castro, A. V., Nadal, M., & Flexas, J. What drives photosynthesis during desiccation? Mosses and other outliers from the photosynthesis–elasticity trade-off. Journal of Experimental Botany., 2020; 71(20): 6460–6470 Perera-CastroA. V. NadalM. FlexasJ. What drives photosynthesis during desiccation? Mosses and other outliers from the photosynthesis–elasticity trade-off Journal of Experimental Botany 2020 71 20 6460 6470 10.1093/jxb/eraa32832686831 Search in Google Scholar

Yang, Y., Wan, L., & Liu, P. Analysis on the Effect of Slideway Friction to the Slider-Type Hydraulic Powered Support. Tehnički vjesnik., 2019; 26(6): 1593–1605 YangY. WanL. LiuP. Analysis on the Effect of Slideway Friction to the Slider-Type Hydraulic Powered Support Tehnički vjesnik 2019 26 6 1593 1605 Search in Google Scholar

Kim, D. J., Oh, J. Y., Cho, J. W., Kim, J., Chung, J., & Song, C. Design study of impact performance of a DTH hammer using PQRSM and numerical simulation. Journal of Mechanical Science and Technology., 2019; 33(11): 5589–5602 KimD. J. OhJ. Y. ChoJ. W. KimJ. ChungJ. SongC. Design study of impact performance of a DTH hammer using PQRSM and numerical simulation Journal of Mechanical Science and Technology 2019 33 11 5589 5602 10.1007/s12206-019-1052-0 Search in Google Scholar

Shavazov, K., Berdimuratov, P., Abdulmajidov, X., Telovov, N., Murtazayev, N., & Razikov, N. The Performance Of The Dredger With The Movement Of The Bucket According To Strict Guidelines. International Journal of Progressive Sciences and Technologies., 2021; 26(1): 515–521 ShavazovK. BerdimuratovP. AbdulmajidovX. TelovovN. MurtazayevN. RazikovN. The Performance Of The Dredger With The Movement Of The Bucket According To Strict Guidelines International Journal of Progressive Sciences and Technologies 2021 26 1 515 521 Search in Google Scholar

Fragassa, C., Minak, G., & Pavlovic, A. Measuring deformations in the telescopic boom under static and dynamic load conditions. Facta Universitatis, Series: Mechanical Engineering., 2020; 18(2): 315–328 FragassaC. MinakG. PavlovicA. Measuring deformations in the telescopic boom under static and dynamic load conditions Facta Universitatis, Series: Mechanical Engineering 2020 18 2 315 328 10.22190/FUME181201001F Search in Google Scholar

Zheng, M., Szabo, T. L., Mohamadi, A., & Snyder, B. D. Long-duration tracking of cervical-spine kinematics with ultrasound. IEEE transactions on ultrasonics, ferroelectrics, and frequency control., 2019; 66(11): 1699–1707 ZhengM. SzaboT. L. MohamadiA. SnyderB. D. Long-duration tracking of cervical-spine kinematics with ultrasound IEEE transactions on ultrasonics, ferroelectrics, and frequency control 2019 66 11 1699 1707 10.1109/TUFFC.2019.292818431484114 Search in Google Scholar

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