Advanced Mechanical Vibrations

The main purpose of advanced mechanical vibrations is to transfer the modal analysis approach in discrete systems to vibration problems of continuous systems. Influence coefficients, stiffness matrix, mass matrix, generalized coordinates and dependency concepts are discussed. By establishing the matrix form of the equations of motion in discrete systems, preparation is made for the concepts of eigenvalue problem, mode shape and orthogonality to be encountered in continuous systems.

In discrete multi-degree of freedom systems, free vibration equations are treated as eigenvalue problems. Natural frequencies, mode shapes, modal orthogonality, modal normalization and modal coordinates are examined. Semi-definite systems, zero-frequency rigid body modes and physical freedoms of the system are interpreted through worked examples and problem solving.

Forced and damped vibrations in discrete systems are summarized within the framework of modal analysis. The concepts of proportional damping, modal damping, steady-state response under harmonic force and frequency response are discussed. In this section, the discrete system approach forms the basis for the transition to modal expansion and modal superposition methods to be used in continuous systems.

Vibrating motion in continuous systems is handled with the infinite degrees of freedom system approach. The concepts of field variable, governing equation, boundary condition, initial condition and boundary value problem are introduced. How the solution turns into an eigenvalue problem in continuous systems with the separation of variables method is explained through the physical meaning of natural frequency and mode shapes.

Transverse vibrations of wires and longitudinal vibrations of rods are modeled using the differential element approach. The equations of motion are obtained by Newton's second law. The separation of variables method is applied under fixed, free, forced and mixed boundary conditions. Natural frequencies, mode shapes, nodal points, orthogonality and establishing motion from initial conditions are studied through worked examples and problem solving.

Torsional vibrations of continuous shafts are examined through the angular displacement field. The equation of motion is established using torsional rigidity, polar moment of inertia, mass inertia effect and torque boundary conditions. Natural frequencies and mode shapes are obtained under free, fixed and elastic boundary conditions. In this section, the engineering significance of torsional vibrations in rotors, shafts and power transmission systems is interpreted.

Hamilton's principle is used to obtain the equation of motion and boundary conditions in continuous systems using the energy approach. By writing kinetic energy, potential energy and virtual work expressions, it is shown how the governing equation and natural boundary conditions arise in systems such as wire, rod, shaft and beam. In this section, model building skills are deepened by comparing the Newtonian approach with the variational approach.

Using the Euler-Bernoulli beam theory, the transverse vibration equation of motion of the beams is obtained. The eigenvalue problem is established under anchored, simply supported, free and free-ended boundary conditions. Natural frequencies, mode shapes and characteristic equations are obtained with the separation of variables method. The effect of boundary conditions on frequencies and mode shapes is examined with engineering interpretation.

In beam vibrations, non-homogeneous or mixed boundary conditions such as elastic support, end mass, end spring, moment input and force input are considered. The emphasis is on writing the boundary conditions correctly, transferring the physical support behavior to the mathematical model, and selecting the solution strategy. In this section, the effects of deviations from ideal boundary conditions in real engineering systems on natural frequencies and mode shapes are interpreted through worked examples and problem solving.

Forced vibrations in continuous systems are examined with the modal expansion approach. The equation of motion is solved under distributed force, point force, harmonic excitation and base excitation. The concepts of excitation level of modes, generalized force, modal coordinates, resonance and frequency response are discussed. In this section, how the continuous system response is represented by the superposition of modes is studied by solving the problem.

In continuous systems, the damping effect is handled through viscous damping, proportional damping and modal damping approaches. The effect of damping on natural frequency, mode shape, transient response and frequency response is examined. In this section, how the concept of damping in discrete systems is transferred to continuous systems and what assumptions are made in engineering calculations are interpreted.

Approximate methods are examined for natural frequency estimation in vibration problems that are difficult to solve analytically. Rayleigh ratio, Dunkerley formula, Rayleigh-Ritz method and Galerkin approximation are discussed. Trial function selection, boundary conditions, energy approximation and physical meaning of approximate mode shapes are studied through worked examples and problem solving.