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2 The Science and Art of Structural Dynamics
P
Distributed
inertia force
(b)
Figure 1.1 Cantilever beam under (a) static loading and (6) dynamic loading.
A structural dynamics problem differs from the corresponding static problem in two
important respects. The first has been noted above: namely, the time-varying nature of
the excitation. Of equal importance in a structural dynamics problem, however, is the
role played by acceleration. Figure 1.1a shows a cantilever beam under static loading.
The deflection and internal stresses depend directly on the static load P. On the other
hand, Fig. 1.1 b shows a similar cantilever beam subjected to a time-varying load P{t).
The acceleration of the beam gives rise to a distributed inertia force, as indicated in
the figure. If the inertia force contributes significantly to the deflection of the structure
and the internal stresses in the structure, a dynamical investigation is required.
Figure 1.2 shows the typical steps in a complete dynamical investigation. The three
major steps, which are outlined by dashed-line boxes, are: design, analysis, and testing.
The engineer is generally required to perform only one, or possibly two, of these steps.
For example, a civil engineer might be asked to perform a dynamical analysis of an
existing building and to confirm the analysis by performing specific dynamic testing of
the building. The results of the analysis and testing might lead to criteria for retrofitting
the building with additional bracing or damping to ensure safety against failure due to
specified earthquake excitation.[U,L2] Automotive engineers perform extensive analysis
and vibration testing to determine the dynamical behavior of new car designs.1 L3,l‘4i
Results of this analysis and testing frequently lead to design changes that will improve
the ride quality, economy, or safety of the vehicle.
In Section 1.2 we introduce the topic of mathematical models. In Section 1.3 we
introduce the prototype single-degree-of-freedom model and indicate how to analyze the
dynamic response of this model when it is subjected to certain simple inputs. Finally,
in Section 1.4 we indicate some of the types of vibration tests that are performed on
structures.
1.2 MODELING OF STRUCTURAL COMPONENTS AND SYSTEMS
Perhaps the most demanding step in any dynamical analysis is the creation of a mathe
matical model of the structure. This process is illustrated by steps 2a and 2b of Fig. 1.2.
In step 2a you must contrive an idealized model of the structural system to be studied, a
model essentially like the real system (which may already exist or may merely be in the
design stages) but easier to analyze mathematically. This analytical model consists of:
1. A list of the simplifying assumptions made in reducing the real system to the
analytical model. _ „
2. Drawings that depict the analytical model (e.g., see Fig. 1.3)
3. A list of the design parameters (i.e., sizes, materials, etc.)