Applied Structural and Mechanical Vibrations 2009 Part 4 pdf

displacement of the mass relative to a fixed frame of reference. Equation (4.55) can be rearranged to (4.56) Using complex algebra, we let the support motion be of the form and the complex motion transmissibility is found to be (4.57) with magnitude and phase angle φ xy given by which are exactly the same as the magnitude and phase angle of the force transmissibility T of equations (4.53) and (4.54). Graphs of eqs (4.53) and (4.54) are shown in Figs 4.10 and 4.11 for the two values of damping and Fig. 4.10 Transmissibility (magnitude) versus β . Copyright © 2003 Taylor & Francis Group LLC By subtracting the quantity mÿ from both sides of eq (4.55) we obtain (4.59) where we can see that the base motion has the effect of adding a reversed inertia force to the equation of relative motion. Equation (4.59) is useful because in many applications relative motion is more important than absolute motion and also because the base acceleration is relatively easy to measure. The quantity f eff (t) is the effective support excitation loading, i.e. the system responds to the base acceleration as it would to an external load equal to the product (mass) × (base acceleration). The minus sign indicates that the effective force opposes the direction of the base acceleration; this has little importance in practice, since the base motion is generally assumed to act in an arbitrary direction. 4.3.3 Resonant response of damped and undamped SDOF systems Immediately after the driving force is turned on it is not reasonable to expect that the oscillator response is given by eq (4.43). In fact, the force has not been acting long enough even to establish what its frequency is and it takes a while for the motion to settle into the steady state. The mathematical counterpart to this statement is, as we have seen before, that the general solution is the sum of two parts: the transient complementary function and a particular integral which represents the steady-state term. Explicitly, we can write (4.60) The amplitude and phase angle of the steady-state term are still given by eqs (4.44) and (4.47), but the initial conditions and now lead to (4.61) With the intention to investigate what happens in resonance conditions (i.e. when ), we assume that the system starts from rest and we get for A and B the values Copyright © 2003 Taylor & Francis Group LLC since With the further assumption of small damping, the damped frequency is nearly equal to the undamped frequency, we can write and eq (4.60) becomes (4.62) from which it is evident that the response rapidly builds up asymptotically to its maximum value It is left to the reader to draw a graph of eq (4.62), and also to determine, for different values of ζ , how many cycles are needed to practically reach the maximum response amplitude. The case of an undamped SDOF system can be easily worked out from the considerations of the preceding sections by letting In this case the magnitude of the response is given by (eq (4.44)) the phase angle is given by and eq (4.60) becomes (4.63) Again, we assume for simplicity that our system starts from rest and from the initial conditions we get Substituting in eq (4.63), the displacement response is which becomes indeterminate at resonance, i.e. when we let Using L’Hospital’s rule we finally obtain (4.64) Copyright © 2003 Taylor & Francis Group LLC The response builds up linearly with time and it is evident that after a few cycles the linear equations considered so far are no longer valid: the amplitude of oscillation increases indefinitely until disruptive effects ensue. Figure 4.12 illustrates the undamped resonant response given by eq (4.64). 4.3.4 Energy considerations In the case of forced vibrations of a viscously damped system energy is dissipated because of damping and energy is supplied to the system by the driving force. The energy input per cycle can be obtained considering the infinitesimal work dW f done by f(t) as the system moves through a small distance dx and integrating over one cycle, i.e. where we have taken the harmonically varying force in the form and the displacement in the form from which it follows The integration gives after a few passages (4.65) It is instructive to see how the same result can be obtained by using phasors if we remember the convention explained in Section 1.3 (eq (1.12)). We have now to consider the product force times velocity (which is the input supplied Fig. 4.12 Undamped resonant response. Copyright © 2003 Taylor & Francis Group LLC power) where force and velocity are in the complex form respectively. Note that we have temporarily dropped the notation |X| for the magnitude of the complex displacement and we are using X throughout this section to be consistent with the ‘sinusoidal’ notation of eq (4.65). The integrated value over one cycle is exactly W f and is obtained by calculating the quantity where we had to multiply by the period because eq (1.12) gives the average over one cycle and incorporates the division by T. The same procedure can be used to calculate the work done by the damping force per cycle of motion. We have now in sinusoidal notation (4.66) or, using phasors, It is not difficult at this point to show that since we get from eqs (4.44), (4.46) and (4.47) that must be substituted in eq (4.65) to give (4.67) Copyright © 2003 Taylor & Francis Group LLC The known relations and can be used to rearrange the result of eq (4.66) to which is the same as eq (4.67) and proves that the energy delivered by the driving force just equals the energy lost by friction. This fact implies that the work done per cycle by the spring and inertial forces is zero. In fact, the inertia and spring forces are related to the displacement by and a plot of f I (or f S ) versus x over one cycle is a straight line enclosing a zero area. If we remember that the area enclosed in a graph of this kind represents the work done by the force over one cycle, we have justified the statement above. Obviously, this same statement can be proven by performing the calculations in sinusoidal or phasor notation. With regard to the damping force we have already determined that the work done by f D over one cycle is different from zero (eq (4.66)), but a graphical representation may also be useful. We can write squaring and rearranging leads to i.e. (4.68) which relates force and displacement and is the equation of an ellipse with area equal to We note that at resonance the phase angle φ is π /2 radians and eq (4.65) reduces to (4.69) 4.4 Damping in real systems, equivalent viscous damping Damping is an inherent property of every real system; its effect is to remove energy from the system by dissipating it as heat or by radiating it away. There are many mechanisms which can cause damping in materials and Copyright © 2003 Taylor & Francis Group LLC structures: internal friction, fluid resistance, sliding friction at joints and interfaces within a structure and at its connections and supports. Therefore, the basic physical characteristics of damping are seldom fully understood and many different types—besides viscous damping which we have considered so far—can be encountered in practice. One often finds reference to structural (hysteretic), Coulomb (dry-friction) or velocity-squared (aerodynamic drag) damping. They are all damping mechanisms based on some modelling assumptions that try to explain and fit the experimental data from vibration analysis. Unfortunately, in real systems damping is rarely of a viscous nature even if, on the other hand, most systems are lightly damped and the difference is insignificant in regions away from resonance. It is then possible to obtain approximate models of nonviscous damping in terms of equivalent viscous dampers and exploit this simple vibration model in different situations. The concept of equivalent viscous damping is based on the equivalence of energy dissipated per cycle by a viscous damping mechanism and by the given nonviscous real situation. We have seen in the preceding section that the energy loss per cycle (eq (4.66)) is directly proportional to the frequency of motion, the damping coefficient c and the square of the amplitude. However, experimental tests show that the actual energy loss per cycle of stress is directly proportional to the square of the amplitude, but independent of frequency over wide ranges of frequency and temperature; this suggests a relation of the type (4.70) where α is a constant for a given frequency and temperature range. This type of damping is called structural (or hysteretic) damping and is attributed to the hysteresis phenomenon observed in cyclic stress of elastic materials, where the energy loss per cycle is equal to the area inside the hysteresis loop. Equating eqs (4.66) and (4.70) we get The equivalent viscous damping coefficient can be defined in this case as (4.71) Our structurally damped system subjected to harmonic excitation can thus be treated as if it were viscously damped with a coefficient given by eq (4.71). By introducing this result in the equation of motion we obtain the complex amplitude response (4.72) Copyright © 2003 Taylor & Francis Group LLC with magnitude, real and imaginary parts and phase angle given by (4.73) (4.74) (4.75) (4.76) where we have defined the structural damping factor (or loss factor) A plot of eqs (4.73) and (4.76) is similar to Figs 4.8 and 4.9 but there are differences worthy of note: • The amplitude response is always maximum at (irrespective of the value of γ ) and for very low values of β the response depends on γ . • The phase angle tends to the value arctan γ for while for viscous damping. By comparing the denominators of eqs (4.43) and (4.72) we can see that γ. corresponds to the quantity 2 ζβ of the viscous case and since damping factors are usually small and are effective only in the vicinity of resonance, we have (4.77) Another equivalent way to introduce structural damping is to incorporate in the complex equation of motion a term which is proportional to displacement but in phase with velocity, i.e. (4.78) where γ is as before; (or if we adopt the positive exponential form ) is called complex stiffness and was introduced for the calculation of the flutter speeds of airplane wings and tail surfaces [3]. One word of caution is necessary: the analogy between structural and viscous damping is valid only for harmonic excitation, because a driving force at frequency ω is implied in the foregoing discussion. Copyright © 2003 Taylor & Francis Group LLC Other damping models that are frequently used and encountered in practice are Coulomb and velocity-squared damping. We limit the discussion to some fundamental results. Coulomb damping arises from sliding of two dry surfaces; to start the motion the force must overcome the resistance due to friction, i.e. it must be greater than µ s mg, where 0<µ s <1 is the static friction coefficient and mg is the weight of the sliding mass. When this happens, the resistance force suddenly drops to µ k mg, where µ k is the kinetic friction coefficient and is generally smaller than µ s . The friction force opposes velocity (i.e. with the appropriate sign for every half cycle of motion) and remains constant as long as the forces acting on m are sufficient to overcome the dry friction. The motion stops when this is no longer the case. It is easy to see that a graph of force versus displacement is a rectangle in this case (with sides 2X and 2µ k mg) and the energy lost per cycle is given by so that the following equivalent damping coefficient is obtained: (4.79) Other characteristics of Coulomb damping are: 1. The free vibration decay still occurs at the ‘frequency’ ω n (remember, however, that this is an improper term because the decaying motion is not strictly periodic) but is linear (and not exponential) in time with an amplitude reduction of 4µ k mg/k per cycle of motion (this can be easily verified by solving the homogeneous equation of motion). 2. In forced vibration conditions the damping does not limit the amplitude at resonance and the quantity tan φ is independent of β , but its sign changes abruptly as β passes through 1. Bodies moving with moderate speed in a fluid (for example air, water or oil) experience a resisting force that is proportional to the square of the speed (aerodynamic damping), i.e. where the minus sign is used when is positive and vice versa. It is left to the reader to determine that the energy lost per cycle is given by (4.80) and the equivalent viscous damping is given in this case by (4.81) Copyright © 2003 Taylor & Francis Group LLC [...]... the system’s response for which is (5 .45 ) The response in the ‘free vibration era’ (t>t1) is obtained from eq (4. 8) with initial conditions that are determined by the state of the system at t=t1: this is (5 .46 ) where now (5 .47 ) Copyright © 2003 Taylor & Francis Group LLC Substitution of eqs (5 .47 ) into eq (5 .46 )—considering that —yields (5 .48 ) Figures 5 .4( a) and 5 .4( b) illustrate how the same undamped... Example 5 .4 The response to this kind of pulse is given by eqs (5 .45 ) and (5 .48 ) for the forced and free vibration era, respectively and, once again, the maximum value can be attained for or t>t1, depending on the value of t1 The two cases are illustrated in Figs 5 .4( a) and 5 .4( b) Let us suppose that the maximum response occurs in the forced vibration era; equating the derivative of eq (5 .45 ) to zero... thus giving (4. 93) where we recognize eq (4. 42) and we understand that the FRF representation gives information on both the magnitude of response and the phase angle of lag between response and excitation This particular form of FRF (displacement response-driving force) is called receptance; other forms can be obtained by considering velocity or acceleration as the measured response Table 4. 3 gives a... corresponding names that are commonly used The real part of the function in eq (4. 93) is obtained from eq (4. 45) simply dividing by the force and is plotted in Fig 4. 13 for two different values of ζ It is not difficult to determine that the two extrema occur at the values so that the damping ratio is usually calculated by (4. 94) The same expressions on the right-hand side give the value of γ in the case of... of the body and to the density of the fluid in which the body is immersed It should be noted that both coefficients (4. 79) and (4. 81) are nonlinear, since they are functions of the amplitude of the vibration 4. 4.1 Measurement of damping There are many ways to quantify and measure the damping of a system The ideal situation would be that all these quantities were consistent with each other and that a... (or) From eq (4. 47) we get the two values of β from the conditions which give (4. 87a) (4. 87b) and the result of eq (4. 86) can be obtained by subtracting eq (4. 87a) from eq (4. 87b) Copyright © 2003 Taylor & Francis Group LLC This latter procedure relies again on the possibility to measure phase angles between input force and output displacement which, as we said before, may not be an easy task Energy... harmonically excited SDOF system considered so far Free vibration decay This method has already been explained in Section 4. 2.3 (eqs (4. 34) and (4. 36)) In practice, the system is excited by any convenient means and then allowed to vibrate freely The time history of the free vibration is recorded and the displacement amplitudes of successive peaks can be used for the calculation of damping For instance, with... displacement is then given by which is eq (4. 44) with ; it follows that (4. 82) However, it may not be easy to apply the exact resonance frequency and the measurement of the phase may also be somewhat difficult An alternative is to obtain the amplitude response curve in the vicinity of resonance and measure the peak value; for viscous damping this is given by (eq (4. 50)) from which ζ can be easily obtained... where the small crosses are equal increments in frequency Finally, Fig 4. 15 [6] is a useful schematic representation of the different types and sources of damping encountered in structural dynamics 4. 5 Summary and comments In Chapter 4 the general model of single-degree-of-freedom (SDOF) system is considered A single mass with a spring and a viscous damper—the socalled harmonic oscillator—is the simplest... 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 . magnitude and phase angle of the force transmissibility T of equations (4. 53) and (4. 54) . Graphs of eqs (4. 53) and (4. 54) are shown in Figs 4. 10 and 4. 11 for the two values of damping and Fig. 4. 10. imaginary parts and phase angle given by (4. 73) (4. 74) (4. 75) (4. 76) where we have defined the structural damping factor (or loss factor) A plot of eqs (4. 73) and (4. 76) is similar to Figs 4. 8 and. since we get from eqs (4. 44) , (4. 46) and (4. 47) that must be substituted in eq (4. 65) to give (4. 67) Copyright © 2003 Taylor & Francis Group LLC The known relations and can be used to rearrange