Applied Structural and Mechanical Vibrations Theory, Methods and Measuring Instrumentation 5 More SDOF—transient response and approximate methods 5 1 Introduction The harmonic excitation considered so.
5 More SDOF—transient response and approximate methods 5.1 Introduction The harmonic excitation considered so far is a special kind of deterministic dynamic loading that only in a few cases can approximate a real situation Nevertheless, its consideration is a necessary prerequisite for any further analysis, and not only for didactical purposes If we remember that for linear systems the principle of superposition holds, from the fact that any reasonably wellbehaved excitation function can be written as the sum or integral of a series of simple functions, it follows that the total response is then the sum (or integral) of the individual responses So, in principle, the complications seem to be more of a mathematical nature rather than a physical nature, and such a statement of the problem does not seem to add anything substantial to the understanding of the behaviour of a linear SDOF system under the action of a complex exciting load However, things are not so simple; a number of different approaches and techniques are available to deal with this problem and the choice is partly a subjective matter and partly dictated by the complexity of the situation, the final results that one wants to achieve and the mathematical tractability of the calculations by means of analytical or computer-based methods The first and main distinction can be made between: • • time-domain techniques frequency-domain techniques As the name itself implies, the first approach relies on the manipulation of the functions involved (generally the loading and the response functions) in the domain of time as the independent variable of interest The important concepts are ultimately the impulse response function and the convolution, or Duhamel’s, integral On the other hand, the second technique is based on the powerful tool of mathematical transforms: the manipulations are made in the domain of an appropriate independent variable (frequency, for example, hence the name) and then, if necessary, the result is transformed back to the domain of the original variable Copyright © 2003 Taylor & Francis Group LLC The two approaches, as one might expect, are strictly connected and the result does not depend on the particular technique adopted for the problem at hand Unfortunately, except for simple cases, both techniques involve evaluations of integrals that are not always easy to solve and their practical application must often rely heavily on numerical methods which, in their turn, require the relevant functions to be ‘sampled’ at regular intervals of the independent variable This ‘sampling’ procedure introduces further complications that belong to the specific field of digital signal analysis, but they cannot be ignored when measurements are taken and computations via electronic instrumentation are performed Their basic aspects will be dealt in future chapters Until a few decades ago the computations involved in frequency-domain techniques were no less than those in a direct evaluation of the discrete convolution in the time domain The development of a special algorithm called the fast Fourier transform [1] has completely changed this situation, cutting down computational time of orders of magnitude and making frequency techniques more effective Both the convolution integral and the transform methods apply when linearity holds; for nonlinear systems recourse must be made to a direct numerical integration of the equations of motion, a technique which, obviously, applies to linear systems as well When the predominant frequency of vibration is the most important parameter and the system is relatively complex, the Rayleigh ‘energy method’ and other techniques with a similar approach turn out to be useful to obtain such a parameter The simplest application represents a multiple- (or infinite-) degree-of-freedom system as a ‘generalized’ SDOF system after an educated guess of the vibration pattern has been made The method is approximate (but so are numerical methods, and in general are much more time consuming) and its accuracy depends on how well the estimated vibration pattern matches the true one Its utility lies in the fact that even a crude but reasonable guess often results in a frequency estimate which is good enough for most practical purposes 5.2 Time domain—impulse response, step response and convolution integral Let us refer back to Fig 4.7 The SDOF system considered so far is a particular case of the situation that this figure illustrates, i.e a single-input single-output linear system where the output x(t) and the input f(t) (written as a force for simplicity, but it need not necessarily be so) are related through a linear differential equation of the general form (5.1) Copyright © 2003 Taylor & Francis Group LLC The coefficients and bj (i=1, 2, 3, ···, n; j=1, 2, 3, ···, r), that is the parameters of the problem, may also be functions of time, but in general we shall consider only cases when they are constants On physical grounds, this assumption means that the system’s parameters (mass, stiffness and damping) not change, or change very slowly, during the time of occurrence of the vibration phenomenon This is the case, for example, for our spring—mass-damper SDOF system whose equation of motion is eq (4.13), which is just a particular case of eq (5.1) Very common sources of excitation are transient phenomena and mechanical shocks, both of which are obviously nonperiodic and are characterized by an energy release of short duration and sudden occurrence Broadly speaking, we can define a mechanical shock as a transmission of energy to a system which takes place in a short time compared with the natural period of oscillation of the system, while transient phenomena may last for several periods of vibration of the system An impulse disturbance, or shock loading, may be for example a ‘hammer blow’: a force of large magnitude which acts for a very short time Mathematically, the Dirac delta function (Chapter 2) can be used to represent such a disturbance as (5.2) where has the dimensions newton-seconds and describes an impulse (time integral of the force) of magnitude (5.3) One generally speaks of unit impulse when From Newton’s second law fdt=mdv, assuming the system at rest before the application of the impulse, the result on our system will be a sudden change in velocity equal to without an appreciable change in its displacement Physically, it is the same as applying to the free system the initial conditions x(0)=0 and The response can thus be written (eq (4.8)) (5.4) for an undamped system, and (eq (4.27)) (5.5) Copyright © 2003 Taylor & Francis Group LLC for a damped system In both cases it is convenient to write the response as (5.6) where h(t) is called the unit impulse response (some authors also call it the weighting function) and is given by (5.7a) for the undamped and damped case, respectively Equations (5.7a) represent the response to an impulse applied at time t=0; if the impulse is applied at time they become (5.7b) for and zero for since the change of variable from t to is geometrically a simple translation of the coordinate axes to the right by an amount seconds In practice, an impact of duration ∆t of the order of 10–3 s (and ∆ t is short compared to the system’s period) is a common occurrence in vibration testing of structures In these cases, the considerations above apply Figure 5.1 illustrates a graph of for a damped system with unit mass, damping ratio and damped natural frequency Example 5.1 Let us consider the response of an undamped system to an impulse of a constant force f0 that acts for the short (compared to the system’s period) interval of time 0