J. Eng. Technol. Sci., Vol. 54, No. 2, 2022, 220209 Received November 25 th , 2020, Revised August 12 th , 2021, Accepted for publication December 12 th , 2021. Copyright ©2022 Published by ITB Institute for Research and Community Services, ISSN: 2337-5779, DOI: 10.5614/j.eng.technol.sci.2022.54.2.9 A New Mechanical Analysis of a Crankshaft-connecting Rod Dynamics Using Lagrange’s Trigonometric Identities Hasan H. Ali 1, * , Firas M. Abdulsattar 2 & Ahmed W. Mustafa 3 1 Directorate of Reconstruction and Projects, Ministry of Higher Education and Scientific Research, Baghdad, Iraq 2 Department of Refrigeration and Air Conditioning Techniques Engineering, Dijlah University College, Baghdad, Iraq 3 Department of Mechanical Engineering, Al-Nahrain University, Baghdad, Iraq *E-mail: hha2mf@mail.missouri.edu Highlights:  Lagrange’s trigonometric identities were used to simplify an actual and an equivalent connecting rod by minimizing the problem to an algebraic form with good accuracy.  The largest external forces resultant acting in the x-direction was exerted by the single- cylinder engine, while the smallest external forces resultant acting in the x-direction was exerted by the six-cylinder engine in the studied cases.  The largest resultant of torque imbalance acting on the crankshaft was found to be associated with two-cylinders engines and was negligible for engines with 5 and 6 cylinders.  The external forces resultant acting in the x-direction and the torque imbalance acting on the crankshaft can be eliminated by using a seven- or more cylinder engine.  The results also revealed that the values of all external forces resultants acting in the y- direction were equal to zero for multi-cylinder engines. Abstract: The main objective of this study was to conduct a new and simple but accurate analysis of the dynamics of a crankshaft-connecting rod system based on Lagrange’s trigonometric identities. Actual and equivalent connecting rod mass approximations of single- and multi-cylinder reciprocating engines were studied. Several examples were studied to demonstrate the dynamics of the system. Lagrange’s trigonometric identities were used to simplify the model, while MATLAB was used to obtain the results. For both the proposed reduced model and the full model, the resultant forces and torques of an actual and an equivalent connecting rod mass were compared. The results showed that the proposed reduced model gives force and torque results that match the results of the full model very well. It was shown that the largest torque imbalance resultant on the crankshaft was exerted by the two-cylinder engine. In addition, it was shown that the largest external forces resultant acting in the x- direction was exerted by the one-cylinder engine. The results also revealed that the resultant of external forces acting in the y-direction was zero for multi-cylinder engines. The relative error, which mainly occurred at the points of maximum force and torque, ranged from about 1% to about 15%. Keywords: crankshaft; dynamics; equivalent mass; internal combustion engine; Lagrange’s trigonometric identities; torque imbalance.