1. A PPLIED M ECHANICS Lecture 08 Slovak University of Technology Faculty of Material Science and Technology in Trnava

2. VIBRATIONS OF CONTINUOUS MECHNICAL SYSTEMS The representation of a physical system by a discrete model is usually an idealized view. In most cases, the main bodies, which compose a mechanical system are deformable, and the elastic elements which connect the main bodies has also their own inertia. Each constituent of a system possesses simultaneously inertia, stiffness and damping properties. The mathematical model of a continuous system undergoing time dependent deformation used in elastodynamics is then relevant. The governing equations of a continuous system we will resort to the theory of continuum mechanics. The equations of motion are expressed in terms of displacement field together with the boundary conditions to be satisfied. The space coordinates x, y, z being continuous, the system so described possesses an infinity of degrees of freedom.

3. VIBRATIONS OF CONTINUOUS MECHNICAL SYSTEMS Continuous structures - beams, rods, cables and plates can be modelled by discrete mass and stiffness parameters and analysed as multi-degree of freedom systems, but such a model is not sufficiently accurate for most purposes. Mass and elasticity have to be considered as distributed or continuous parameters. Analysis of structures with distributed mass and elasticity it is necessary to assume a homogeneous, isotropic material that follows Hooke’s law. Free vibration is the sum of the principal modes. However, in the unlikely event of the elastic curve of the body in which motion is excited coinciding exactly with one of the principal modes, only that mode will be excited.

4. VIBRATIONS OF STRINGS Strings are elastic elements that are subjected to tensile forces A(z) - area of cross-section of the string,  (z) - density, f(z, t) - vertical load.

5. VIBRATIONS OF STRINGS Element of the string - its position is determined by the coordinate z. Element mass dm = A(z)r(z)dz. Equation of motion of string element After arrangement, the equation of motion is where

6. VIBRATIONS OF RODS Element of the rods - longitudinal dimension is much more greater as the transverse size:  bar is axially symmetric,  sections perpendicular to the axis remain plane and perpendicular to the axis after deformation,  transverse deformations are neglected. Rods are elastic elements that are subjected to the axial forces:  cross-section A(z),  Young’s modulus E(z),  density r(z),  motion of the rod is excited by the axial force f(z, t).

7. VIBRATIONS OF RODS

8. Equation of motion of rod element After arrangement, the equation of motion is where

9. VIBRATIONS OF SHAFTS Shafts are elastic elements that are subjected to torques. Let us assume that the torque  (z, t) is distributed along the axis z and is a function of time t. The shaft has:  shear modulus G(z),  density  (z),  cross-section area A(z),  second moment of area J(z). Due to the moment  (z, t), the shaft performs the torsional vibrations and the instantaneous angular position of the cross-section at z is j(z, t).

10. VIBRATIONS OF SHAFTS

11. Let us consider the element dz of the shaft. Its moment of inertia about the axis z is defined by Equation of motion where

12. VIBRATIONS OF BEAMS Beams are elastic elements that are subjected to lateral loads (forces, moments that have their vectors perpendi- cular to the centre line of a beam). Let us consider a beam with:  second moment of area J(z),  cross-section A(z),  density  (z),  Young’s modulus E(z). The beam performs vibrations due to the external distri- buted unit load f(z, t).

13. VIBRATIONS OF BEAMS

14. The relationship between the bending moment M and the shearing force V The relationship between the deflection y(z, t) and the bending moment M(z, t). Equation of motion of beam element resp.

15. VIBRATIONS OF BEAMS If the following parameters of the beam A, J, E and  are constant, motion of the beam is governed by the following equation where