The resulting eigenvalue equation becomes The force in the spring is assumed to be out of phase with the displacement, resulting in a complex-valued stiffness. ![]() This model cannot be explicitly described in terms of time derivatives, but is expressed directly in terms of complex numbers in the frequency domain. Another common damping model is hysteretic damping or loss factor damping. The viscous damping used above is popular because of its mathematical simplicity. An overdamped system will not resonate at any natural frequency.ĭamping processes are, in general, difficult to characterize. Oscillating solutions can exist only when. Thus, in a damped system, the free vibrations will die out.ĭecaying free vibrations for different damping ratios. In front of the harmonic part, there is an exponentially decaying multiplier. The periodic part of this expression has the damped natural (angular) frequency. Inserting this value of, the complex-valued displacement is The eigenvalues, which are the solutions to the quadratic equation above, are Here, is called the undamped natural (angular) frequency and is called the damping ratio. The eigenvalue equation can be written as This equation can only be fulfilled for certain values of (for the nontrivial case ), given by The equation of motion can, in the absence of any external forces, then be transformed into In this notation, each time derivative gives a factor. In complex notation, the displacement can be written as, where is a complex-valued amplitude. Such a notation will be used in the following equation. When analyzing harmonic vibrations, it is convenient to employ a complex notation, where the harmonic functions are represented by. Damped SystemsĪssuming that there is also viscous damping in the system, then the equation of motion is The kinetic energy of the mass is transformed into strain energy in the spring, and vice versa. Under a free vibration, the energy in the system is conserved. The expression for the eigenfrequency above exhibits very general behavior in terms of how stiffness and mass influence eigenfrequencies: ![]() In real life, there is always some damping, so ultimately the vibrations would fade away. If, for example, you stretched the spring and then let go, the mass would vibrate forever at this frequency. We can interpret the solution above as: Once the process has started, a free vibration can exist at exactly this frequency without any external excitation. As long as there is no risk of confusion, a less stringent language is sometimes used, in which is called the natural frequency. It is related to the natural frequency (unit: Hz) by. Here, is the natural angular frequency, having the unit rad/s. It can be immediately verified thatįulfills the homogeneous equation of motion if If, however, no external force is acting on the mass, nonzero solutions may still exist. Undamped system with a single degree of freedom (DOF). Provide insight into how design changes can affect a certain eigenfrequency by studying its mode shape.Provide eigenmodes for a subsequent analysis based on mode superposition.Investigate suitable choices of time steps or frequencies for a subsequent dynamic response analysis.Check if a quasistatic analysis of a structure is appropriate based on the fact that all natural frequencies are high when compared to the frequency content of the loading.Ascertain that a periodic excitation causes a resonance in, for example, a piezoelectric vibrator.Ascertain that a periodic excitation does not cause a resonance that may lead to excessive stresses or noise emission. ![]() Some objectives of such an analysis are to: The true size of the deformation can only be determined if an actual excitation is known together with damping properties.ĭetermining the eigenfrequencies of a structure is an important part of structural engineering. An eigenfrequency analysis can only provide the shape of the mode, not the amplitude of any physical vibration. When vibrating at a certain eigenfrequency, a structure deforms into a corresponding shape, the eigenmode. Here, we mainly describe the study of eigenfrequencies in mechanical structures, but many of the concepts are generally applicable. Natural frequencies appear in many types of systems, for example, as standing waves in a musical instrument or in an electrical RLC circuit. Proof (short version).Structural Mechanics Eigenfrequency Analysis Introduction to Eigenfrequency AnalysisĮigenfrequencies or natural frequencies are certain discrete frequencies at which a system is prone to vibrate. The second proof explains more details and give proofs of the facts which are not proved in the first proof. The first version is a short proof and uses some facts without proving.
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