Motion and Deformation of Mechanical Systems with 1 Degree of Freedom (1DOF) – combination translation and rotational displacements


Textbook Chapter 5 (Lectures 24-31)

Acronyms: M: mass, K: stiffness, C: damping, EOM: equation of motion, SEP: static equilibrium position, DOF: degree of freedom, FBD: free body diagram, CME: Principle of Conservation of Mechanical Energy



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Recommended  homework problems


What is a rigid body? Properties of inertias (mass, center of mass and mass moments of inertia): development of equations. Parallel axis formula and restrictions to its application.

Example: determine mass center and moment of inertia of a physical assembly. Steps in procedure. Use published Tables containing useful mass properties

Appendix C.2, C.4, C.6



Motion of a rigid body on a plane: derivation of force and moment EOMs in Cartesian coordinates. Vector analysis. Reduced forms for the moment EOM: about mass center, about fixed point in inertial space.

Applications. 1-DOF torsional vibrations: definition of torsional stiffness, natural frequency, and similitude to response of 1-DOF M-K-C system. Fixed axis rotation: simple rotor on bearings, pulleys connected by belts, gear transmission

5.1, 5.3, 5.4, 5.5



Kinetic energy of rigid body in planar motion (translation and rotation). Examples of fixed axis rotation: derivation of equations from CME: rotor on bearings, torsional vibrations, pulleys connected by belts.

Same examples as seen from conservation of mechanical energy

Work 5.1, 5.2 (without viscous damping)

and derive the EOM using CME. Rework 5.4, 5.5 using CME


Nonlinear EOMs for compound pendulum connected to (linear) spring and viscous damper: FBDs, application of force and moment equations, geometric nonlinearities at linear element (K, C) connections, linearization of EOM about SEP. EOM derived from CME. Preload in spring elements. Finding natural frequencies and motions for bars and plates connected to springs and dampers.

5.7, 5.9, 5.10


More compound pendulum applications: nonlinear spring force –displacement relations and linearization of EOMS for motions about SEP.

Rigid body motion with prescribed acceleration of pivot support: FBDs, application of force and moment equations. Equations for reaction forces, interpretation of results

5.21, 5.22, 5.24



5.28, 5.30, 5.31


Motion of cylinders rolling w/o slipping. FBDs, identification of forces and rolling constraint, derivation of EOM. Definition of Coulomb (dry friction) forces. When will the cylinder slip and not roll? EOM derived from CME. EOM for imbalanced cylinder rolling down an inclined plane, oscillations of a half cylinder on a flat plane: prediction and measurement of natural frequency.

5.33, 5.34, 5.36


More applications of rolling w/o slipping motion: cylinder restrained by spring, cylinder rolling inside a concave surface: FBDs, identification of forces and rolling constraint, derivation of EOM, linearization and identification of natural frequency. EOM derived from CME. Example: pulley assembly connected to spring element

5.32, 5.37, 5.38


Examples of 1-DOF mechanical systems combining rotation and translation. Using Newton’s laws and Conservation of Mechanical Energy (CME) to derive EOMs. When to linearize an EOM and when not?

5.41, 5.42, 5.69, 5.70


Get all: Lectures 24-31


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