Search results for: differential actuated joints

Authors: Qinghua Zhang, Yanhe Zhu, Xiang Zhao, Yeqin Yang, Hongwei Jing, Guoan Zhang, Jie Zhao

Abstract:

This paper presents a wearable reconfigurable supernumerary robotic limb with differential actuated joints, which is lightweight, compact and comfortable for the wearers. Compared to the existing supernumerary robotic limbs which mostly adopted series structure with large movement space but poor carrying capacity, a prototype with the series-parallel configuration to better adapt to different task requirements has been developed in this design. To achieve a compact structure, two kinds of cable-driven mechanical structures based on guide pulleys and differential actuated joints were designed. Moreover, two different tension devices were also designed to ensure the reliability and accuracy of the cable-driven transmission. The proposed device also employed self-designed bearings which greatly simplified the structure and reduced the cost.

Keywords: cable-driven, differential actuated joints, reconfigurable, supernumerary robotic limb

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Authors: Mansour Mohieddin Ghomshei, Reza Shahi

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Laminated composite tubes with adhesively bonded joints are widely used in aerospace and automotive industries as well as oil and gas industries. In this research, adhesively tubular single lap joints subjected to torsional and hygrothermal loadings are studied using the differential quadrature method (DQM). The analysis is based on the classical shell theory. At first, an approximate closed form solution is developed by omitting the lateral deflections in the connecting tubes. Using the analytical model, the circumferential displacements in tubes and the shear stresses in the interfacing adhesive layer are determined. Then, a numerical formulation is presented using DQM in which the lateral deflections are taken into account. By using the DQM formulation, the circumferential and radial displacements in tubes as well as shear and peel stresses in the adhesive layer are calculated. Results obtained from the proposed DQM solutions are compared well with those of the approximate analytical model and those of some published references. Finally using the DQM model, parametric studies are carried out to investigate the influence of various parameters such as adhesive layer thickness, torsional loading, overlap length, tubes radii, relative humidity, and temperature.

Keywords: adhesively bonded joint, differential quadrature method (DQM), hygrothermal, laminated composite tube

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Authors: Chaitanya Sharma, Vikas Upadhyay, A. Tripathi

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Friction stir welding and tungsten inert gas welding techniques were employed to weld armor grade aluminum alloy to investigate the effect of welding processes on tensile behavior of weld joints. Tensile tests, Vicker microhardness tests and optical microscopy were performed on developed weld joints and base metal. Welding process influenced tensile behavior and microstructure of weld joints. Friction stir welded joints showed tensile behavior better than tungsten inert gas weld joints.

Keywords: friction stir welding, microstructure, tensile properties, fracture locations

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Authors: B. Paygozar, S.A. Dizaji, A.C. Kandemir

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In this study, the effects of the modified adhesive including different concentrations of Nano-silica are surveyed on the bonding strength of the adhesive scarf joints. The nanoparticles are added in two different concentrations, to an epoxy-based two-component structural adhesive, Araldite 2011, to survey the influences of the nanoparticle weight percentage on the failure load of the joints compared to that of the joints manufactured by the neat adhesive. The effects of being exposure to a moist ambience on the joint strength are also investigated for the joints produced of both neat and modified adhesives. For this purpose, an ageing process was carried out on the joints of both neat and improved kinds with variable immersion periods (20, 40 and 60 days). All the specimens were tested under a quasi-static tensile loading of 2 mm/min speed so as to find the quantities of the failure loads. Outcomes indicate that the failure loads of the joints with modified adhesives are measurably higher than that of the joint with neat adhesive, even while being put for a while under a moist condition. Another result points out that humidity lessens the bonding strength of all the joints of both types as the exposure time increases, which can be attributed to the change in the failure mode.

Keywords: bonding strength, humidity, nano-silica, scarf joint

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Authors: Ibtisam Masmali, Edwin Beggs

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In this paper, we take example of differential calculi, on the finite group A4. Then, we apply methods of non-commutative of non-commutative differential geometry to this example, and see how similar the results are to those of classical differential geometry.

Keywords: differential calculi, finite group A4, Christoffel symbols, covariant derivative, torsion compatible

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Authors: Hamid Ahmadi, Neda Azari Dodaran

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This paper aims to study the structural behavior of tubular KT-joints subjected to axial loading at fire induced elevated temperatures. At first, a finite element (FE) model was developed and validated against the data available from experimental tests. Then, a set of 810 FE analyses were performed to study the influence of temperature and dimensionless geometrical parameters (β, γ, θ, and τ) on the ultimate strength and initial stiffness. The joints were analyzed under two types of axial loading and five different temperatures (20 ºC, 200 ºC, 400 ºC, 550 ºC, and 700 ºC). Results show that the ultimate strength and initial stiffness of KT-joints decrease considerably by increasing the temperature. In the joints having bigger values of the β, the temperature elevation leads to less reduction in ultimate strength; while in the joints with bigger values of the γ, the temperature elevation results in more reduction in ultimate strength. The influence of the θ on the ultimate strength is independent from the temperature. To our knowledge, there is no design formula available for determining the ultimate strength of KT-joints at elevated temperatures. Hence, after parametric study, two equations were developed through nonlinear regression, for calculating the ultimate strength of KT-joints at elevated temperatures.

Keywords: axial loads, fire condition, parametric formula, static strength, tubular KT-joint

Procedia PDF Downloads 155

Authors: N. Zhang, J. S. Kuang, S. Mogili

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To investigate seismic performance of beam-column knee joints, four full-scale reinforced concrete beam-column knee joints, which were fabricated to simulate those in as-built RC frame buildings designed to ACI 318-14 and ACI-ASCE 352R-02, were tested under reversed cyclic loading. In the experimental programme, particular emphasis was given to the effect of horizontal reinforcement (in format of inverted U-shape bars) on the shear strength and ductility capacity of knee joints. Test results are compared with those predicted by four seismic design codes, including ACI 318-14, EC8, NZS3101 and GB50010. It is seen that the current design codes of practice cannot accurately predict the shear strength of seismically designed knee joints.

Keywords: large-scale tests, RC beam-column knee joints, seismic performance, shear strength

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Authors: Seyyed Mohammad Hasheminia, Heoung Jae Chun, Jong Chan Park, Hong Suk Chang

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Composite joints have been getting attention recently due to their high specific mechanical strength to weight ratio that is crucial for structures such as aircrafts and automobiles. In this study on hybrid joints, quasi-static experiments and finite element analysis were performed to investigate the failure mechanism of hybrid composite joint with respect to the joint properties such as the adhesive material, clamping force, and joint geometry. The outcomes demonstrated that the stiffness of the adhesive is the most imperative design parameter. In this investigation, two adhesives with various stiffness values were utilized. Regarding the joints utilizing the adhesive with the lower stiffness modulus, it was observed that the load was exchanged promptly through the adhesive since it was shared more proficiently between the bolt and adhesive. This phenomenon permitted the hybrid joints with low-modulus adhesive to support more prominent loads before failure when contrasted with the joints that utilize the stiffer adhesive. In the next step, the stress share between the bond and bolt as a function of various design parameters was studied using a finite element model in which it was understood that the geometrical parameters such as joint overlap and width have a significant influence on the load sharing between the bolt and the adhesive.

Keywords: composite joints, composite materials, hybrid joints, single-lap joint

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Authors: Orkun Yavuz, Ferhat Kadioğlu, M. Emin Ercan

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This study aims to investigate the effects of repeated high loadings on the performance of adhesively-bonded Single Lap Joints (SLJs) by employing both experimental and numerical approaches. A projectile with a mass of 1.25 gr and density of 11.3 gr/cm3 was fired at the joints with a velocity of about 280 m/s using a specially designed experimental set-up, and the impact was recorded via a high-speed camera. The SLJs were manufactured from 6061 aluminum adherend (AA6061) material and an adhesive film. The joints, which have an adherend thickness of 4 mm and overlap length of 15 mm, were subjected to up to 3 shots for the ballistic test, followed by quasi-static tensile testing. The impacted joints, then, were compared to the non-impacted and one-shot impacted ones, which was a subject of investigation carried out before. It was found that while the joints subjected to 2 shots mechanically deteriorated, those subjected to 3 shots experienced a complete failure at the end of the experiment. A numerical study was also conducted using an ABAQUS package program. While the adherends were modelled using the Johnson-Cook deformation parameters, an elastoplastic behavior of the adhesive was used as input data in the analyses. It seems the experimental results confirm the numerical ones.

Keywords: ballistic tests, adhesive joints, numerical analysis, SLJ

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Authors: Chang He, Hiroshi Tamura, Hiroshi Katsuchi, Jiaqi Wang

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In this study, the aim is to infer the slip limit (static friction limit) of contact interfaces in bolt friction joints by analyzing other bolt friction joints with the same contact surface but in a different shape. By using the Weibull distribution to deal with microelements on the contact surface statistically, the slip limit of a certain type of bolt joint was predicted from other types of bolt joint with the same contact surface. As a result, this research succeeded in predicting the slip limit of bolt joins with different numbers of contact surfaces and with different numbers of bolt rows.

Keywords: bolt joints, slip coefficient, finite element method, Weibull distribution

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Authors: Hamid Ahmadi, Shadi Asoodeh

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The fatigue life of tubular joints commonly found in offshore industry is not only dependent on the value of hot-spot stress (HSS), but is also significantly influenced by the through-the-thickness stress distribution characterized by the degree of bending (DoB). The determination of DoB values in a tubular joint is essential for improving the accuracy of fatigue life estimation using the stress-life (S–N) method and particularly for predicting the fatigue crack growth based on the fracture mechanics (FM) approach. In the present paper, data extracted from finite element (FE) analyses of tubular KT-joints, verified against experimental data and parametric equations, was used to investigate the effects of geometrical parameters on DoB values at the crown 0˚, saddle, and crown 180˚ positions along the weld toe of central brace in tubular KT-joints subjected to axial loading. Parametric study was followed by a set of nonlinear regression analyses to derive DoB parametric formulas for the fatigue analysis of KT-joints under axial loads. The tubular KT-joint is a quite common joint type found in steel offshore structures. However, despite the crucial role of the DoB in evaluating the fatigue performance of tubular joints, this paper is the first attempt to study and formulate the DoB values in KT-joints.

Keywords: tubular KT-joint, fatigue, degree of bending (DoB), axial loading, parametric formula

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Authors: Krolo Paulina

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The seismic response of moment-resisting steel frames depends on the behavior of the joints, especially when they are considered as ductile zones. The aim of this research is to provide a realistic assessment of the moment-resisting steel frame behavior under seismic loading using nonlinear static pushover analysis (N2 method). The hysteresis behavior of the joints in the frame model was described using a new hysteresis envelope model. The obtained seismic response was compared with the results of the seismic analysis obtained for the same steel frame that takes into account the monotonic model of the joints.

Keywords: beam-to-column joints, hysteresis envelope model, moment-resisting frame, nonlinear static pushover analysis, N2 method

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Authors: Khalifa Kherafa

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This east project consists in studying beam–column joints concrete subjected to seismic loads. A bibliographical study was introduced to clarify the work undertaken by the researchers in the field during the three last decades and especially the two last year’s results which were to study for the determination of the method of calculating of transverse reinforcement in the various nodes of a structure. For application, the efforts in the posts el the beams of a building in R+4 in zone 3 were calculate according to the finite element method through the software .

Keywords: beam–column joints, cyclic loading, shearing force, damaged joint

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Authors: F. Ghasim-akbari, A. M. Hadian, A. M. Aminazad

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Fluid transmission pipes made of aluminum are widely use in petrochemical industries. For many applications they have to be brazed to each other. The brazed joints, in many cases, are encountered with corrosive medias. This paper reports a part of a work to investigate the corrosion behavior of brazed Al6061 using Al4047 as filler metal with and without the use of flux to discover the effect of different brazing atmospheres. The samples brazed under air, vacuum, argon, and hydrogen atmospheres. The interfacial area of the joints was examined to ensure being free of any defects. The sides of each test piece were covered with insulator and the surface of the joint was encountered to polarization test. The results revealed a significant difference of corrosion resistance. The samples that brazed under argon and hydrogen atmospheres had better corrosion resistance than other samples. Microstructure of the corroded joints revealed that the amount of the filler metal is a critical parameter on corrosion resistance of the joints.

Keywords: brazing, corrosion behavior, Al6061, polarization

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Authors: K. R. Tshunza, A. Elvin, A. Gabremmeskel

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Final results of an extensive laboratory research program on “lapped gusset joints in compression” are presented. The investigation was carried out at the Heavy structures laboratory at the University of the Witwatersrand in Johannesburg, South Africa. A proposed, relatively easy to use analytical equation was found to be reasonably adequate in determining the global compressive capacity of lapped gussets joints under compressive load. A wide range of lapped mild steel plates of varying slenderness, welded on 219*10 and 127*6 Mild steel circular hollow sections of 1m length were tested in compression and the formula was validated with experimental results. The investigation show that the connection’s capacity is controlled by flexure due to the eccentricity between the plates that are connected side to side.

Keywords: compression, eccentricity, lapped gussets joints, moment resistance

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Authors: Doo W. Lee, Soo J. Lee, Bye R. Yoon, Jae Y. Jho, Kyehan Rhee

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Hand exoskeletons have been developed in order to assist daily activities for disabled and elder people. A figure exoskeleton was developed using ionic polymer metal composite (IPMC) actuators, and the performance of it was evaluated in this study. In order to study dynamic performance of a finger dummy performing pinching motion, force generating characteristics of an IPMC actuator and pinching motion of a thumb and index finger dummy actuated by IMPC actuators were analyzed. The blocking force of 1.54 N was achieved under 4 V of DC. A thumb and index finger dummy, which has one degree of freedom at the proximal joint of each figure, was manufactured by a three dimensional rapid prototyping. Each figure was actuated by an IPMC actuator, and the maximum fingertip force was 1.18 N. Pinching motion of a dummy was analyzed by two video cameras in vertical top and horizontal left end view planes. A figure dummy powered by IPMC actuators could perform flexion and extension motion of an index figure and a thumb.

Keywords: finger exoskeleton, ionic polymer metal composite, flexion and extension, motion analysis

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Authors: G. Hemath Kumar, H. Mohit, K. Karthick

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One of the most important issues in the composite technology is the repairing of parts of aircraft structures which is manufactured from composite materials. In such applications and also for joining various composite parts together, they are fastened together either using adhesives or mechanical fasteners. The tensile strength of these joints was carried out using Universal Testing Machine (UTM). A parametric study was also conducted to compare the performance of the hybrid joint with varying adherent thickness, adhesive thickness and overlap length. The composition of the material is combination of epoxy resin and carbon fibre under the method of reinforcement. To utilize the full potential of composite materials as structural elements, the strength and stress distribution of these joints must be understood. The study of tensile strength in the members involved under various design conditions and various joints were took place.

Keywords: carbon fiber, FRP composite, MMC, automotive

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Authors: Fuziyah Ishak, Siti Norazura Ahmad

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Volterra integro-differential equations appear in many models for real life phenomena. Since analytical solutions for this type of differential equations are hard and at times impossible to attain, engineers and scientists resort to numerical solutions that can be made as accurately as possible. Conventionally, numerical methods for ordinary differential equations are adapted to solve Volterra integro-differential equations. In this paper, numerical solution for solving Volterra integro-differential equation using extended trapezoidal method is described. Formulae for the integral and differential parts of the equation are presented. Numerical results show that the extended method is suitable for solving first order Volterra integro-differential equations.

Keywords: accuracy, extended trapezoidal method, numerical solution, Volterra integro-differential equations

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Authors: M. R. Abusrea, Shiyi Jiang, Dingding Chen, Kazuo Arakawa

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Novel wind-lens turbine designs can augment power output. Vacuum-Assisted Resin Transfer Molding (VARTM) is used to form large and complex structures from a Carbon Fiber Reinforced Polymer (CFRP) composite. Typically, wind-lens turbine structures are fabricated in segments, and then bonded to form the final structure. This paper introduces five new adhesive joints, divided into two groups: One is constructed between dry carbon and CFRP fabrics, and the other is constructed with two dry carbon fibers. All joints and CFRP fabrics were made in our laboratory using VARTM manufacturing techniques. Specimens were prepared for tensile testing to measure joint performance. The results showed that the second group of joints achieved a higher tensile strength than the first group. On the other hand, the tensile fracture behavior of the two groups showed the same pattern of crack originating near the joint ends followed by crack propagation until fracture.

Keywords: adhesive joints, CFRP, VARTM, resin transfer molding

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Authors: A. Elrefaei

Abstract:

The possibility of having sheets with different thicknesses and materials in one assembly facilitates the optimal material distribution within the final product and reduces the weight of the structure. Ability of joining process to assembly these different material combinations is always a challenge to the designer. In this study, 0.6 mm thick 6061-T6 and 2 mm thick 5182-O were robot CMT welded using ER5356 and ER4043 filler metals. The thermal effect of welding resulted in a loss of hardness in the 6061 HAZ. Joints welded by ER5356 filler metal were much higher in fracture load than joints welded by ER4043 and the elongation of joints welded by ER5356 was almost double its corresponding joints welded by ER4043 filler. Owing to the big difference in formability and thickness of base metals, the fracture in forming test occurred in the softened 6061 HAZ out from the weld centerline.

Keywords: aluminum, CMT, mechanical, welding

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Authors: Chinmay Shah

Abstract:

The project is all about development of a close loop traffic light control system using PLC (Programmable Logic Controller). This project is divided into two parts which are hardware and software. The hardware part for this project is a model of four way junction of a traffic light. Three indicator lamps (Red, Yellow and Green) are installed at each lane for represents as traffic light signal. This traffic control model is a replica of actuated traffic control. Actuated traffic control system is a close loop traffic control system which controls the timing of the indicator lamps depending on the fluidity of traffic for a particular lane. To make it autonomous, in each lane three IR sensors are placed which helps to sense the percentage of traffic present on any particular lane. The IR Sensors and Indicator lamps are connected to LG PLC XGB series. The PLC controls every signal which is coming from the inputs (IR Sensors) to software and display to the outputs (Indicator lamps). Default timing for the indicator lamps is 30 seconds for each lane. But depending on the percentage of traffic present, if the traffic is nearly 30-35%, green lamp will be on for 10 seconds, for 65-70% traffic it will be 20 seconds, for full 100% traffic it will be on for full 30 seconds. The software part that operates with LG PLC is “XG 5000” Programmer. Using this software, the ladder logic diagram is programmed to control the traffic light base on the flow chart. At the end of this project, the traffic light system is actuated successfully by PLC.

Keywords: close loop, IR sensor, PLC, light control system

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Authors: Ferhat Kadioglu, Hasan Puskul

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A composite material with carbon fiber and polymer matrix has been used as adherent for manufacturing adhesive joints. In order to evaluate different fiber orientations on joint performance, the adherents with the 0°, ±15°, ±30°, ±45° fiber orientations were used in the single lap joint configuration. The joints with an overlap length of 25 mm were prepared according to the ASTM 1002 specifications and subjected to tensile loadings. The structural adhesive used was a two-part epoxy to be cured at 70°C for an hour. First, mechanical behaviors of the adherents were measured using three point bending test. In the test, considerations were given to stress to failure and elastic modulus. The results were compared with theoretical ones using rule of mixture. Then, the joints were manufactured in a specially prepared jig, after a proper surface preparation. Experimental results showed that the fiber orientations of the adherents affected the joint performance considerably; the joints with ±45° adherents experienced the worst shear strength, half of those with 0° adherents, and in general, there was a great relationship between the fiber orientations and failure mechanisms. Delamination problems were observed for many joints, which were thought to be due to peel effects at the ends of the overlap. It was proved that the surface preparation applied to the adherent surface was adequate. For further explanation of the results, a numerical work should be carried out using a possible non-linear analysis.

Keywords: composite materials, adhesive bonding, bonding strength, lap joint, tensile strength

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Authors: Sh. Foroughi, V. Karamzadeh, M. Packirisamy

Abstract:

This paper presents design and analysis of an electrothermally symmetrical actuated microgripper applicable for performing micro assembly or biological cell manipulation. Integration of micro-optics with microdevice leads to achieve extremely precise control over the operation of the device. Geometry, material, actuation, control, accuracy in measurement and temperature distribution are important factors which have to be taken into account for designing the efficient microgripper device. In this work, analyses of four different geometries are performed by means of COMSOL Multiphysics 5.2 with implementing Finite Element Methods. Then, temperature distribution along the fingertip, displacement of gripper site as well as optical efficiency vs. displacement and electrical potential are illustrated. Results show in addition to the industrial application of this device, the usage of that as a cell manipulator is possible.

Keywords: electro thermal actuator, MEMS, microgripper, MOEMS

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Authors: Javad Yarmahmoudi, Alireza Mirzaee

Abstract:

Many active and passive structural health monitoring (SHM) techniques have been developed for detection of the defects of plates. Generally, riveted joints hold the plates together and their failure may create accidents. In this study, well known active and passive methods were modified for the evaluation of the health of the riveted joints between the plates. The active method generated Lamb waves and monitored their propagation by using lead zirconate titanate (PZT) disks. The signal was analyzed by using the wavelet transformations. The passive method used the Fiber Bragg Grating (FBG) sensors and evaluated the spectral characteristics of the signals by using Fast Fourier Transformation (FFT). The results indicated that the existing methods designed for the evaluation of the health of individual plates may be used for inspection of riveted joints with software modifications.

Keywords: structural health monitoring, SHM, active SHM, passive SHM, fiber bragg grating sensor, lead zirconate titanate, PZT

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Authors: D. S. Palimkar

Abstract:

The FRIGDI (functional random integrodifferential inclusion) seems to be new and includes several known random differential inclusions already studied in the literature as special cases have been discussed in the literature for various aspects of the solutions. In this paper, we prove the existence result for FIGDI under the non-convex case of multi-valued function involved in it.Using random fixed point theorem of B. C. Dhage and caratheodory condition. This result is new to the theory of differential inclusion.

Keywords: caratheodory condition, random differential inclusion, random solution, integro-differential inclusion

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Authors: Gurmeet Singh Cheema, Navjotinder Singh, Gurjinder Singh, Amardeep Singh

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Aluminium and its alloys play have excellent corrosion resistant properties, ease of fabrication and high specific strength to weight ratio. In this investigation an attempt has been made to study the effect of different post weld heat treatment methods on the mechanical and metallurgical properties of TIG welded joints of the commercial aluminium alloy. Three different methods of post weld heat treatments are, solution heat treatment, artificial aged and combination of solution heat treatment and artificial aging are given to TIG welded aluminium joints. Mechanical and metallurgical properties of as welded and post weld treated joints of the aluminium alloys was examined.

Keywords: aluminium alloys, TIG welding, post weld heat treatment

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Authors: Kohei Inoue, Kenji Hara, Kiichi Urahama

Abstract:

We describe a relationship between integral images and differential images. First, we derive a simple difference filter from conventional integral image. In the derivation, we show that an integral image and the corresponding differential image are related to each other by simultaneous linear equations, where the numbers of unknowns and equations are the same, and therefore, we can execute the integration and differentiation by solving the simultaneous equations. We applied the relationship to an image fusion problem, and experimentally verified the effectiveness of the proposed method.

Keywords: integral images, differential images, differential filters, image fusion

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Authors: Teoman Ozer, Ozlem Orhan

Abstract:

This study deals with symmetry group properties and conservation laws of partial differential equations. We give a geometrical interpretation of notion of μ-prolongations of vector fields and of the related concept of μ-symmetry for partial differential equations. We show that these are in providing symmetry reduction of partial differential equations and systems and invariant solutions.

Keywords: λ-symmetry, μ-symmetry, classification, invariant solution

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Authors: Abdallah H. Ramini, Alwathiqbellah I. Ibrahim, Mohammad I. Younis

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We investigate experimentally and theoretically the dynamics of a capacitive resonator under mixed frequency excitation of two AC harmonic signals. The resonator is composed of a proof mass suspended by two cantilever beams. Experimental measurements are conducted using a laser Doppler Vibrometer to reveal the interesting dynamics of the system when subjected to two-source excitation. A nonlinear single-degree-of-freedom model is used for the theoretical investigation. The results reveal combination resonances of additive and subtractive type, which are shown to be promising to increase the bandwidth of the resonator near primary resonance frequency. Our results also demonstrate the ability to shift the combination resonances to much lower or much higher frequency ranges. We also demonstrate the dynamic pull-in instability under mixed frequency excitation.

Keywords: electrostatically actuated resonator, multi-frequency excitation, nonlinear dynamics, AC harmonic signals

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Authors: Yildiray Keskin, Omer Acan, Murat Akkus

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In this paper, the solution of fractional diffusion equations is presented by means of the reduced differential transform method. Fractional partial differential equations have special importance in engineering and sciences. Application of reduced differential transform method to this problem shows the rapid convergence of the sequence constructed by this method to the exact solution. The numerical results show that the approach is easy to implement and accurate when applied to fractional diffusion equations. The method introduces a promising tool for solving many fractional partial differential equations.

Keywords: fractional diffusion equations, Caputo fractional derivative, reduced differential transform method, partial

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