MECHATRONICS

1. Slovak University of Technology Faculty of Material Science and Technology in Trnava MECHATRONICS Lecture 08

2. VIBRATION OF THE SYSTEMS WITH MORE DOF Oscillation of the model systems with more degrees of freedom is described by the matrix equation where M, K, B are mass, stiffness, damping matrices gained by discreditation of a technological system by “lumping” the masses and formulation of nonmass linkings or by the final element method (FEM). To model the damping matrix, the next formula is used (proportional model) First member of this matrix equation righ side describes the ”external” damping, the second one dscribes the ”internal” damping by the mass. Spectrum of natural frequencies and the form of the oscillation period of the free oscillating model system describes by the solution of eigenvalues

3. Algorithm of solution is the standard procedure of any software for solving dynamical problems. The results are: • spectral (diagonal) matrix with quadrats of all naturalangular frequencies on the main diagonal • modal matrix with all the forms of own oscillation period where vi = [vi1 , vi2 , …..vin]T. Solving the complex problem of the eigenvalues defined by the equation

4. Various modifications and transforms are needed, e.g. the homogenous part of the equation is transformed into the form where The own values of the complex problem λi represent the solution of the characteristic equation

5. NONLINEAR MODEL SYSTEMS Description of properties of mechanical systems using linear mathematical models is merely the very first approximation to the behaviour of real machines and equipments. Any real mechanical system is more or less nonlinear. Linear relations between among deformations, speeds, accelerations are valid within a (often very) narrow zone. Nonlinearities of the technological systems can be found mainly as • nonlinear elastic springs • nonlinear damping forces - but also nonlinear relations of so-called mixed types Some of those phenomenae are visible in courses of amplitude and phase resonance curves of the simple example of „solidifying“ elastic characteristics, applied in motion equation where ε is much smaller number than │b│ and │k│.

7. Examples of resonance curves deformations in model systems with nonlinear restoring force are in the Tab.

8. Whirling vibrations of shafts, critical angular speed A symetrical shaft with a single wedged disc for undamped example the next values are valid

9. From the dynamical equilibrum m(y+e)ω2 – ky = 0 comes out the shaft sag y = e/(wkr2/w2 – 1) = eη2/(1 – η2), where η=ω/ωkr. Graphically |y/e| vs. tuning coefficient η see Fig. together with the typical configuration for η → 0 and η → ∞ (the phase change from 0 to π when passing the critical speed). Under statical shaft sag due to the weight of disc yst = G/k. Critical angular frequency is

10. For simple shafts with two bearings on their ends the critical speed can be known from: Where ymaxis the maximal static shaft sag and coefficient  = 1 for the single lumped mass,  1,27 for continuously distributed mass along the shaft,  1,8 for rotors in turbines compressors, centrifugal pumps,  1,20 for rotors of turbogenerators and electromotors. For shafts with relative damping bpthe amplitude and phase characteristics is equal With courses analogical to Fig. but the curves y/e begin for η = 0 from zero and run to 1 for η  ∞.