In Chapter 2, we studied techniques for the derivation of the equations of motion. The next task is to learn as much as possible about the behavior of the system from these equations, which involves the study of the stability characteristics, as well as the derivation of the system response. The task involves subjects such as input-output relations, state equations, equilibrium positions, linearization about equilibrium, the transition matrix, the eigenvalue problem, controllability and observability and control sensitivity.

A system is defined as an assemblage of components acting together as a whole.

## Dynamics and Control of Structures - A Modal Approach | Wodek K. Gawronski | Springer

A system acted upon by a given excitation exhibits a certain response. Dynamics is the study of this cause-and-effect relation. In order to study the dynamic behavior of a system, it is necessary to construct a mathematical model of the system. This amounts to identifying essential components of the system, establishing their excitation-response characteristics, perhaps experimentally, as well as considering the manner in which they work together.

Throughout this process, appropriate physical laws must be invoked. The mathematical model permits the derivation of the equations governing the behavior of the system. For the most part, the equations have the form of differential equations, but at times they have the form of integral equations. In effect, we have already derived such differential equations in Chapter 1 by means of Newton's second law and in Chapte. Before a mathematical model can be accepted with confidence, it must be ascertained that it is capable of predicting the observed behavior of the actual system.

## Non-Linearity in Structural Dynamics and Experimental Modal Analysis

In system analysis terminology, systems are often referred to as plants, or processes. Moreover, the excitation is known as the input signal, or simply the input , and the response as the output signal, or simply the output. It is convenient to represent the relation between the input and output schematically in terms of a block diagram, as shown in Fig. Quite frequently , the plant characteristics and the input are known and the object is to determine the output.

### Description:

This task is known as analysis. At times, however, the plant characteristics are not known and the object is to determine them.

To this end, it is necessary to acquire information about the input and output , very likely through measurement, and to infer the system characteristics from this information. The task of determining the system properties from known inputs and outputs is referred to as system identification. In engineering, the functions a system must perform and the conditions under which it must operate are well defined, but the system itself does not exist yet. In such cases, one must design a system capable of performing satisfactorily under the expected conditions.

System control generally involves all three tasks to some degree, although for the most part the plant characteristics can be regarded as known , albeit not exactly.

Moreover, reference here is not to the design of the plant itself but of a mechanism guaranteeing satisfactory performance in the presence of uncertainties. Figure 3. The figure is typical of an uncontrolled system and it describes a situation arising ordinarily in the physical world. For example, in the case of structures during earthquakes, the structure represents the plant and the motion of the foundation represents the input. Of course, the output is the motion of the structure itself. In controlled systems, on the other hand, the object is to elicit a satisfactory response.

This desired output is preprogrammed into the controller and remains the same irrespective of the actual output, so that the output does not affect the input. The type of control depicted in Fig. An example of open-loop control is a heating system in a building, where the heating unit is set to start working at a given time independently of the temperature inside the building. Of course, in. Indeed, in the case of the heating system, it makes more sense to design the system so as to actuate the heating unit when the temperature drops to a preset level and to shut it off when it reaches a higher temperature level, which requires a temperature-sensing device.

This type of control is shown in Fig. This also explains the term open-loop for the control shown in the block diagram of Fig. Modern heating systems are of the closed-loop type.

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- Dynamics and Control of Structures: A Modal Approach - MATLAB & Simulink Books.
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Feedback control tends to be more stable than open-loop control and is capable of compensating for unexpected disturbances, uncertainties in the plant model, sensor measurements and actuator outputs. However, in addition to being more expensive, feedback control tends to be more complicated than open-loop control and is hard to justify when the disturbances and uncertainties are sufficiently small that open-loop control can perform satisfactorily. In developing a mathematical model for a structure, it is necessary to identify the various members in the structure and ascertain their excitationresponse characteristics.

These characteristics arc governed by given physical Jaws, such as laws of motion or force-deformation relations, and are described in terms of system parameters, such as mass and stiffness. These parameters are of two types, namely, lumped and distributed. Lumped parameters depend on the spatial position only implicitly. On the other hand, distributed parameters depend explicitly on the spatial coordinates. Consistent with this, in lumped-parameter systems the input and output arc functions of time alone and in distributed-parameter systems the input and Error detector Error.

As a result , the behavior of lumped-parameter systems is governed by ordinary differential equations and that of distributed-parameter systems by partial differential equations. In both cases, the parameters appear in the form of coefficients. For the most part, in structures the coefficients do not depend on time, and such systems are said to be time-invariant. If the coefficients do depend on time, then the system is known as time-varying.

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In this text , we study both lumped-parameter and distributed-parameter time-invariant systems. Note that lumped-parameter systems are also known as discrete systems. Designing controls for distributed structures is quite often very difficult, so that in such cases it is necessary to construct a discrete model to represent the behavior of the distributed-parameter system. This tends to introduce uncertainties in the plant model, as no discrete model is capable of representing a distributed model exactly.

The system of Fig. If the parameters are distributed nonuniformly, then the construction of a discrete model, perhaps of the type shown in Fig. One property of structures with profound implications in mathematical analysis is linearity. To illustrate the concept, let us consider a system characterized by the single input f t and single output y t , as shown in Fig.

## Modal Analysis

Then , if the excitation f t is the sum of two different inputs, or 3. If, on the other h and,. Equation 3. Indeed, it is possible to state whether a system is linear or nonlinear by mere inspection of the system differential equation. Because the order of the highest derivative in Eq.

### Dynamics and Control of Structures: A Modal Approach (Mechanical Engineering Series)

The reason for this can be traced to the fact that the dependent variable y t and its time derivatives appear in Eq. Note that linearity rules out not only higher powers and fraction al powers of y t and its derivatives but also mixed products thereof. Of course, some derivatives of y t , or even y t itself, can be absent from the differential equation. The above criterion is valid whether the coefficients a 1 are constant or time-dependent. The only restriction on the coefficients a; is that they do not depend on y t or its derivatives. This is not necessarily the case in closedloop control.

In many cases, whether the system is linear or nonlinear depends on the magnitude of the output y t and its derivatives. Indeed, when y t and the -time derivatives of y t are sufficiently small that the nonlinear terms can be ignored, the system behaves like a linear one. On the other hand , when y t and its time derivatives are so large that some of the nonlinear terms cannot be ignored, the same system acts as a nonlinear system.

We shall return to this subject later in this chapter.