Kinematics and Dynamics of Machines (EE 521)

2021 Fall
Faculty of Engineering and Natural Sciences
Electronics Engineering(EE)
Volkan Pato─člu,
Doctoral, Master
Formal lecture
Project based learning,Simulation,Case Study
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Introduction to mechanisms, kinematics of mechanisms, displacement analysis, kinematics velocity analysis, acceleration analysis, forces in mechanisms, work, energy and power, momentum and impact, geometry of mechanisms, synthesis of mechanisms, transmission mechanisms, vibration, multi-body dynamics.


This course is designed to equip students with fundamental theories and computational methodologies that are used in (computer aided) analysis of multibody systems. Students will learn how to analytically formulate dynamics equations for multibody systems as well as how to utilize numerical algorithms to simulate such systems. Computational mechanics is of high value for the purposes of performance evaluation, sensitivity studies, control system design, model based monitoring and so on.

Students will be introduced to generalized coordinates and speeds, analytical and computational determination of inertia properties, generalized forces, Kane's method, Lagrange's equations, holonomic and nonholonomic constraints. Computerized symbolic manipulation and time integration methods for dynamic analysis will be exercised.

Of the available techniques for formulating equations of motion for multibody systems, symbolic formulation and Kanes method will be emphasized. Being a vector based approach and making optimal use of generalized coordinates and speeds, Kane's method is preferred for its relative ease of computerization and its computational efficiency. Efficiency may be interpreted here both as producing equations efficiently (with the fewest symbolic operations) and producing efficient equations (which require the fewest numerical operations
for their solution). Also, Kanes method produces equations in ordinary differential form (ODEs) even for nonholonomically constrained systems, which can be accommodated using (stabilized) standard solvers.

The emphasis in this course is not on the excessive mathematical abstraction but rather on an integrated understanding of modeling, equation derivation and numerical solution. A solid understanding of the principles of dynamics in the context of modern analytical and computational methods is aimed.


Identify relevant points, bodies, and bases; choose generalized coordinates to represent a multibody
Relate several frames through rigid body rotations.
Form the required position vectors.
Differentiate relevant vectors to form required velocities and accelerations.
Select generalized speeds and formulate kinematical differential equations.
Formulate equations of motion for unconstrained systems.
Form constraint equations and solve for independent variables.
Formulate equations of motion for systems with constrains.
Numerically integrate resulting equations of motion even for systems with changing kinematic constraints.
Check validity of numerical integration of equations of motion.
Linearize equations of motion.
Form work functions, calculate kinetic and potential energy of the system.
Formulate the Lagrangian and express the equations of motion in a DAE form.
Numerically solve for the DAEs using stabilized integration methods.


  Percentage (%)
Final 15
Exam 20
Individual Project 35
Homework 30



Kane and Levison, Dynamics Online: Theory and Implementation with Autolev,, 2000.

Kane and Levison, Dynamics: Theory and Applications, Mcgraw Hill, 1985.

Kane, Spacecraft Dynamics, Mcgraw-Hill College, 1981.

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