Base Excited Systems

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<ul><li> 1. BASE EXCITED SYSTEMS Structural Engineering Division Department of Civil Engineering Prof. A. Meher Prasad Department of Civil Engineering Indian Institute of Technology Madras email: prasadam@iitm.ac.in Indian Institute of Technology Madras Chennai, India.</li></ul><p> 2. Base - ExcitedSDF Linear Systems The response of the viscously damped linear (system) oscillator shown in the sketch will now be investigated for an excitation of the base.Displacement of the base at any time t will be denoted by y(t) and the associated velocity and acceleration will be denoted by respectively.The exciting motion is considered to be known and unaffected by the motion of the oscillator itself. As before, the absolute displacement of the mass will be denoted by X and associated velocity and acceleration byandBoth X and Y are measured from the static equilibrium, and both they and their derivatives are considered to be positive when directed to the right.P(t) x(t) m k c y(t) 3. The equation of motion for the system is obtained as usual by considering the equilibrium of forces acting on the mass. These forces include the spring force, damping forceand DAlembert inertia force all of each are directed to the left. Equilibrium requires thatEquilibrium requires that(B1) The equation can now be written either in terms of the absolute displacement x, as(B2)or in terms of relative displacement, or spring deformation, u =x - y (B3)(B4) Upon dividing through by m and introducing the quantity p and , equation (B2) and (B4) can also be written as (B5) and(B6) 4. The choice between Eqn.(B2) and (B4) or between Eqn.(B5) and (B6) in a given problem depends on how the ground motion is specified and what response quantity x or u is interested in.For example, if the ground motion is specified as an acceleration history and we are interested in resulting spring force, ku the Eqn. (B4) and (B6) would probably be the most convenient.On the other hand, if we are interested in absolute displacement and bothare specified, Eqn. (B2) and (B5) would be the most convenient to use. Clearly, once either x or u has been determined the other may be computed from Eqn. (B3) 5. For undamped systems, c = 0, Eqn. (B5) reduces to,(B7) This equation is same as the differential equation governing the motion of a fixed base system subjected to a force for which the associated static displacement (B8) Absolute Displacement of Undamped System The solution of equation (B7) can therefore be obtained from the solution for the force-excited system considered previously, simply by replacing in the latter solutionx stby y (t).In terms of Duhamels integral, the solution may be written as(B9) 6. It follows further that the response spectra for the fixed-base, force-excited systems presented previously can also be interpreted as spectra for the absolute displacement of base-excited systems.It is only necessary to replace the quantity (x st ) oin the expression for the amplification factorby the peak value of the base displacement, y o. In the other words, the spectral ordinates should be interpreted to be ratio of.The histories of the base motion and the excited force must naturally be the same in the two cases. 7. Example:A vehicle, idealized as a SDF undamped system moved at a speed of 20 m/s over an irregular rigid pavement. The shape of the irregularity is a half sine wave and its peak value is. Prior to crossing the irregularity, the vehicle is considered to have no vertical motion. If the natural frequency of the vehicle is f = 2 cps, what would be the maximum vehicle displacement of the vehicle for (a) L=1.5 m (b) L=6 m (c) L=24 m.As it crosses the irregularity, the vehicle is subjected to a base motion, the displacement history of which is a half sine pulse. The maximum displacement of the mass may then be from the response spectrum presented before. v L y o 8. Noting that the duration of the pulse, t 1 , is given byt 1 =L/Vand that, V=20.1 m/s, the values of frequency parameter, ft 1 , for the three cases arefor (a) for (b) for (c) The corresponding values ofare , for (a) for (b)for (c) 9. If the irregularity were a full sine wave and L the length of each half wave, the resulting displacements could be determined from the spectrum given before. The result in this case are as follows for (a) L=1.5 m,for (b) L= 6 m ,for (c) L=24 m , 10. The analogy referred to in the preceeding section is valid only for undamped systems. It can be used as an approximation for damped systems only when the damping is small.However, for the special case of a sinusoidal base excitation;the RHS of equation (B5) reduces to ,whereis a phase angle defined by , Absolute Displacement of Damped System 11. In this case, the solution for steady state response may be written by analogy to the solution given by equation 69 for the corresponding force exited system.It is only necessary to interpret the quantity(x st ) 0in the later solutionas, This leads to 12. The deformation, u , of the base-excited systems can also be obtained from the equivalent force excited, fixed-base system. Comparison of Eqn. (B4) and Eqn. (B1) the force P(t) for the force excited system can be taken as, P(t)= - my(t). Then the two equations will be similar. The i nitial conditions of u for the base excited systems are the same as those on force excited systems, the solutions of the differential equations will also be the same. The desired deformation, u, will be equal to the displacement X,of the force excited system shown: Spring deformation of systems subjected to Base excitation .. - my(t) m k c x .. 13. In the analysis of fixed base system,extensive use was made of the concept of instantaneous amplification factor,defined asIt is desirable to evaluate at this stage the counter part of this factor for the base-excited system. Noting that , where y 0,is the absolute maximum acceleration of the base motion,we conclude that, (B12) 14. The solution is obtained from the information presented before,making use of Eqn. B12. For,Example :Evaluate the deformation of a SDOF undamped system subjected to a rectangular pulse of amplitudeand duration t 1 . Assume that the initial values of y andand of x andare zero.Thus initial values of u andare zero 15. The acceleration,velocity and displacement histories of the base motion considered in this solution are shown. This type of base excitation is of interest in the design of equipment in moving vehicles,but is clearly of no interest in the design of structures subjected to ground motions. t 1 t 1 t 1 For an arbitrary base motion, the deformation of an undamped system can be expressed in terms of Duhamels integral as follows. (B13) 16. Pseudo - acceleration The quantityin equation B12 ,which has units of acceleration,will be referredto as the instantaneous pseudo acceleration of the system,and will be denoted by A(t). (B14) Thus ,equation B12 can also be written as,(B15) Referring now to Eqn. B1,it can readily be shown that,for undamped systems,the pseudo-acceleration,A(t) is also equal to the absolute acceleration of the mass,. However, this identity is not valid for damped systems,and A(t) should be looked upon merely as an alternate measure of the spring deformation,u(t). 17. Spectral Quantities The absolute maximum value of u,without regards to sign will be referred to as the spectral value of u and will be denoted by U. The absolute maximum value of A(t),without regards to sign,will be referred to as the pseudo-acceleration of the system,and will be denoted by A, thusThe product of the mass m and the pseudo-acceleration,A represents the maximum spring force, Q max , indeed(B16) (B17) This may also be viewed as the equivalent lateral staticforce which produces the same effects as the maximum effects by the ground shaking. It is sometimes convenient to express Q maxin the form , (B18) 18. Where W = mg is the weight of the system. The quantity C is the so called lateral force coefficient, which represents the number of times the system must be capable of supporting its weight in the direction of motion.From Eqn.B17 and B18 it follows that,C=A/g (B19) Another useful measure of the maximum deformation, U is the pseudo velocity of the system,defined as,V = p U(B20) The maximum strain energy stored in the spring can be expressed in terms of V as follows:E max= (1/2) (kU) U= (1/2)m(pU) 2= (1/2)mV 2 (B21) Under certain conditions, that we need not go into here,V is identical to ,or approximately equal to the maximum values of the relative velocity of the mass and the bays,U and the two quantities can be used interchangeably.However this is not true in general,and care should be excercised in replacing one for the other. 19. Deformation spectra 1.Obtained from results already presented2.Presentation of results in alternate forms (a) In terms of U (b) In terms of V (c) In terms of A 3.Tripartite Logarithmic Plot 20. extreme right; It approaches U=y 0at extreme left; a value ofIt exhibits a hump on either side of the nearly horizontal central portion;and attains maximum values of U,V and A which may be materially greater than the values ofIt is assumed that the acceleration trace of the ground motion,and hence the associated velocity and displacement traces, are smooth continuous functions.The high-frequency limit of the response spectrum for discontinuous acceleration inputs may be significantly higher than the value referred to above,and the information presented should not be appliedto such inputs. The effect of discontinuous acceleration inputs is considered later.General form of spectrum 21. Spectral regions The characteristics of the ground motion which control the deformation of SDF systems are different for different systems and excitations. The characteristics can be defined by reference to the response spectrum for the particular ground motion under consideration. 22. Spectra for maximum and minimum accelerations of the mass(undamped elastic systems subjected to a Half cycle Acceleration pulse) 23. Spectra for maximum and minimum acceleration of the mass (undamped Elastic systems subjected to a versed-sine velocity pulse) 24. Deformation spectra for undamped elastic systems subjected to a versed-sine velocity pulse 25. B Level Earthquake ( =10% ; =1.0) 26. Deformation spectrum for undamped Elastic systems subjected to a half-sine acceleration pulse 27. Logarithmic plot of Deformation Spectra </p> <ul><li>Advantages: </li></ul><ul><li>The response spectrum can be approximated more readily and accurately in terms of all three quantities rather than in terms of a single quantity and an arithmetic plot. </li></ul><ul><li>In certain regions of the spectrum the spectral deformations can more conveniently be expressed indirectly in terms of V or A rather than directly in terms of U. All these values can be read off directly from the logarithmic plot. </li></ul><p>It is convenient to display the spectra or a log-log paper, with the abscissa representing the natural frequency of the system,f, (or some dimensionless measure of it) and the ordinate representing the pseudo velocity ,V (in a dimensional or dimensionless form). On such a plot ,diagonal lines extending upward from left to right represent constant values of U, and diagonal lines extending downward from left to right represent constant values of A. From a single plot of this type it is thus possible to read the values of all three quantities. 28. V Log scale Natural Frequency F (Log scale) Displacement sensitive Velocity sensitive Acceleration sensitive General form of spectrum Logarithmic plot of Deformation Spectra 29. Deformation Spectra for Half-cycle Acceleration pulse: This class of excitation is associated with a finite terminal velocity and with a displacement that increases linearly after the end of the pulse.Although it is of no interest in study of ground shock and earthquakes ,being the simplest form of acceleration diagram possible ,it is desirable to investigate its effect. When plotted on a logarithmic paper, the spectrum for the half sine acceleration pulse approaches asymptotically on the left the value. This result follows from the following expression presented earlier for fixed base systems subjected to an impulsive force, where 30. Lettingandand noting thatwe obtain, ( This result can also be determined by considering the effect of an instantaneous velocity change,,i.e. an acceleration pulse of finite magnitude but zero duration. The response of the system in this case is given by, Considering that the system is initially at rest, we conclude that, and where, The maximum value of u(t), without regards to signs, is or)or 31. (This result can also be determined by considering the effect of as instantaneous velocity change,i.e an acceleration pulse of finite magnitude but zero duration.the response of the system in this case is given by Considering that the system is initially at rest,we conclude that , where,The maximum value of u(t),without regards to signs,is 32. Example:For a SDF undamped system with a natural frequency,f=2cps,evaluate the maximum value of the deformation,U when subjected to the half sine acceleration pulse. Assume that,t1=0.1sec. Evaluate also the equivalent lateral force coefficient C, and the maximum spring force,Q 0 ft 1 = 2 x 0.1 = 0.2 From the spectrum, 33. Alternatively,one can start reading the valuefrom the spectrum proceeding this may, we find thatTherefore 34. AccordinglyThe value ofandas read from the spectrum areapproximate. The exact value ofdetermined is0.7. This leads to 35. If the duration of the pulse were f 1 =0.75sec instead of 0.1sec , the results would be as follows 36. If the duration of the pulse were t 1, as in the first case, but the natural frequency of the system were 15cps instead of 2cps, the results would have been as follows:ft 1 =15 * 0.1=1.5 Therefore, and 37. </p> <ul><li>Plot spectra for inputs considered in the illustrative example and compare </li></ul><ul><li>The spectrum for the longer pulse will be shifted upward and to the left by a</li></ul><ul><li>factor of 0.75/0.10 = 7.5 </li></ul><p>Same as in both cases f V For t 1 =0.75sec For t 1 =0.1sec 38. May be determined from the spectrum by interpretingas When displayed on a logarithmic paper with the ordinate representing V and the abscissa f, this spectrum may be approximated as follows:Design Spectrum (Log scale) (Log scale) 1.5 = 39. </p> <ul><li>Refer to spectrum for </li></ul><ul><li>Note the following </li></ul><ul><li>At extreme right. Explain why? </li></ul><ul><li>Frequency value behind whichis given by ft ra = 1.5 </li></ul><ul><li>The peak value of A=2x1.6Explain why.</li></ul><ul><li>In general for pulsesof the same shape and duration with different peak values </li></ul><ul><li>If duration on materially different </li></ul><p>Deformation Spectra for Half-Cycle Velocity Pulses 40. be conservative. Improved estimate may be obtained by considering relative durations of the individual pulses and superposing the peak component effects.The peak value of V is about1.6 y o It can be shown that the absolute maximum v...</p>