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0542-4220

Theory of vibrations

Also listed as: תורת התנודות

From one degree of freedom to continuous systems, with modal analysis in the middle. It is the course that turns "the machine shakes" into a number you can design against.

Semester
Semester A
Weekly hours
4h
Counts as
Core: solids
Interest areas
Structures and computational mechanics · Miniaturised systems and materials · Aeronautics and space

What it covers

Single degree of freedom

  • Free vibration: undamped, under-, critically and over-damped, logarithmic decrement
  • Harmonic excitation, resonance, magnification factor, phase
  • Rotating unbalance, base excitation, transmissibility, vibration isolation
  • Duhamel's integral for arbitrary and transient forcing

Multiple degrees of freedom

  • Equations of motion by Newton and by Lagrange, mass and stiffness matrices
  • The eigenvalue problem, natural frequencies, mode shapes, orthogonality
  • Modal analysis and the vibration absorber
  • Numerical methods: Rayleigh, Dunkerley, Holzer, matrix iteration

Continuous systems

  • Strings, axial bars, torsional shafts, Euler–Bernoulli beams
  • Boundary conditions and their effect on the spectrum
  • Damping models, instrumentation and measurement
  • Introduction to nonlinear and random vibration

Results worth carrying out

  • Single degree of freedom

    mx¨+cx˙+kx=F(t),ωn=km,ζ=c2kmm\ddot{x} + c\dot{x} + kx = F(t), \qquad \omega_n = \sqrt{\frac{k}{m}}, \qquad \zeta = \frac{c}{2\sqrt{km}}
  • Eigenvalue problem

    [M]{x¨}+[K]{x}=0    det ⁣([K]ω2[M])=0[M]\{\ddot{x}\} + [K]\{x\} = 0 \;\Rightarrow\; \det\!\left([K] - \omega^2[M]\right) = 0
  • Euler–Bernoulli beam

    EI4wx4+ρA2wt2=0EI\,\frac{\partial^4 w}{\partial x^4} + \rho A\,\frac{\partial^2 w}{\partial t^2} = 0

Figures worth knowing

  • Magnification and transmissibility against frequency ratio
  • Mode-shape diagrams
  • Campbell diagram
  • Beam mode shapes

Related courses

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