Structural Dynamic Response to Excitations
Understanding the dynamic behaviour of a structure, may it be rotating or static, can be very complicated. Most analyst would look at the problem by first transforming the system from spatial to modal coordinates. Mathematically, it involves uncoupling of the system mass, stiffness, damping and force matrices into sets of Single Degree of Freedom (SDOF) containing modal parameters, namely modal frequencies, modal shapes and modal damping. Subsequently, one assesses the problem by expressing that any dynamic response of a structure is a superimposition of various modes of vibration, and which of the modes would contribute most to the final response of the structure under investigation.
It can be stated that the dynamic response of a structure depends upon the following factors:
a) Closeness of the modal frequencies to the excitation frequencies
b) Amount of damping of the modes close to excitation frequencies
c) Interaction between the distribution of excitation forces and the mode shapes
All the factors above are summarily contained in the equation stated below. If one can dwell deep into this equation, then one should be able to systematically analyse any linear structural dynamics problems.
However, to be able to do this it is crucial that one has to first evaluate the dynamic characteristics of the given structure, namely the natural(modal) frequencies, mode shapes and modal damping.
Closeness of the natural frequencies to the excitation frequencies
Any given structure has a lot of natural frequencies and their corresponding modes. In fact, for a continuous system the number is infinite. This is inherent in any structure that possesses stiffness and mass, hence the name ‘natural’. However, we would only be interested in those modal frequencies that are within a certain range, namely those that contribute and dominate the response of the structure. The contribution of the modes depends on the closeness of the modal frequencies to the excitation frequencies. This is contained in the magnification factor, Br, stated as
For a given damping, w~pr will cause the first part of the denominator to disappear leaving Br as
This results in the amplitude response for this mode to be large and only limited by the amount of the modal damping in the system.
Amount of modal damping
Damping does not play an important role in the region away from the natural frequencies. In fact, in this situation, we can treat the system as undamped. However, close to the natural frequency a small change in damping would cause a large change in the response. Hence, obtaining accurate modal damping value may be crucial. If we examine the magnification factor again, for w~pr,
This implies that for region close to resonance, the amplitude is very much dependable on the damping of the system. Away from resonance, difference in damping does not cause much change in the response and depends mainly on the following
It is worthwhile to mention here that at resonance, the structure’s internal resistances, namely the stiffness (k) and inertia (mw2), cancel each other out. In a totally undamped structure, all the excitation energy that enters the structure accumulates within, resulting in escalation of vibration amplitude eventually leading to structural destruction. The presence of damping in the structure causes it to acts as an energy dissipater. The higher the damping the more of this energy is dissipated, implying lower vibration amplitude. If we assume that damping is related to amount of mass and stiffness in the structure, we can say, in general that, for a given amount of excitation energy at resonance, the amplitude of vibration is reduced for a structure of higher mass or stiffness.
Interaction between the excitation force vector and the mode shape vector
The third factor contributing to the response of the structure is the generalised force qpr. The generalised force for mode r is the dot product of the mode shape vector and the excitation force vector.
The strength of the generalised force depends on whether the force vector is symmetrical or anti-symmetrical with the shape vector. Let’s take an example of two modes, one anti-symmetrical (mode 1) and the other symmetrical (mode 2) with a unity force vector.
For anti-symmetrical interaction of mode 1,
For symmetrical interaction of mode 2,
Hence, it can be said that even if the excitation force has a frequency close to the natural frequency of mode 1, the contribution of mode one is zero due to it being anti-symmetrical with the force vector.
In summary, the contribution of mode r to the final response of a given structure depends on the generalised force and the magnification factor. The generalised force, in turn, depends on the interaction between the force vector and the shape vector while the magnification factor depends on the closeness of excitation frequency to the natural frequency and damping. Hence, before one embarks on the investigation of vibration problem in structure, it is obligatory to first determine the dynamic characteristics of the structure, namely, the natural frequencies, the corresponding mode shapes and modal damping.
People of www.mdt-quadrant2.com.my and http://asivr.ump.edu.my are heavily involved in this study.
by Prof Dr Abdul Ghaffar Abdul Rahman