Find the resonant frequencies of the following structures,...

Find the resonant frequencies of the following structures, assuming the nodes all have unit mass. Then find a means of forcing the structure at one of the resonant frequencies, and yet not exciting the resonance. Can you also force the structure without exciting any mechanism or rigid motion? (a) The square truss of Exercise 6.3 .5 ; (b) the joined square truss of Exercise 6.3.6; (c) the house of Exercise 6.3.8; (d) the triangular space station of Example 6.6; (e) the triatomic molecule of Example 10.41; $(f)$ the water molecule of Exercise 10.5.30.

Key Concepts

Resonant Frequencies

In structural dynamics, resonant frequencies are the natural frequencies at which a system tends to oscillate when disturbed. When a structure is forced at one of these frequencies, even a small periodic force can produce large amplitude oscillations due to the cumulative effect of energy input into the specific mode of vibration.

Modal Analysis

Modal analysis is the study of the different vibration modes (patterns of deformation) of a structure. It involves determining the natural frequencies and associated mode shapes by solving an eigenvalue problem, which is critical for predicting how the structure will respond to dynamic loading.

Forced Vibrations

Forced vibrations occur when an external time-dependent force is applied to a structure. The response of the structure depends on the frequency of the applied force relative to its natural frequencies. When the forcing frequency coincides with a resonant frequency, the amplitude of vibrations can become very large unless the force is applied in a way that avoids exciting that particular mode.

Selective Force Application

Selective force application involves designing and placing external forces so that they excite only certain modes of the structure or avoid exciting undesired resonant modes. By tuning the magnitude, direction, or phase of the forces, one can isolate the response to specific vibration modes, potentially even forcing the structure at a resonant frequency while not exciting the associated mode.

Rigid Body Modes and Mechanisms

Rigid body modes refer to the motions of a structure where displacement occurs without any relative deformation, such as translations and rotations. In the context of dynamic forcing, understanding rigid body modes is essential because forces might inadvertently excite these motions or internal mechanisms, leading to large, uncontrollable responses. The problem of designing forcing functions often includes ensuring that these unwanted modes are not excited.

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