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Dynamics of Aerospace Structures

by: Laila Windler

Dynamics of Aerospace Structures ASEN 5022

Laila Windler

GPA 3.96


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This 14 page Class Notes was uploaded by Laila Windler on Friday October 30, 2015. The Class Notes belongs to ASEN 5022 at University of Colorado at Boulder taught by Staff in Fall. Since its upload, it has received 37 views. For similar materials see /class/232177/asen-5022-university-of-colorado-at-boulder in Aerospace Engineering at University of Colorado at Boulder.

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Date Created: 10/30/15
18 Modeling for Structural Vibrations FEM Models Damping Similarity Laws Chapter 18 MODELING FOR STRUCTURAL VIBRATIONS FEM MODELS DAMPING SIMILMITY LAWS 181 FINITE ELEMENT MODELING OF VIBRATION PROBLEMS In modeling of cable Vibration problems by linear elements it was observed that one needs to model the cable by more than 50 elements if the fundamental frequency is to be computed within 4 digit accuracy What was not discussed therein is the accuracy of mode shapes To gain further insight into nite element modeling of Vibration problems let us consider the modeling of plane beam Vibration problems For simplicity a beam with simplesimple supports is used to model 5th modes Classical theory tells us that we have the following solution k E kth mode wk L P 181 knx kth mode shape Wx s1n T where E I is the bending rigidity 0 is the mass per unit beam length and L is the beam span 1 Frequency error vs elements for simple support beam 10 r r r r r r 100 7 Bth mode beta L 188497 7 g 103917 7 LE 3 x 3 e 103927 7 LL 103937 7 104 r r r r r r r r 10 15 20 25 30 35 40 45 50 Number of Beam Eleme t Figure 181 Frequency error vs number ofelements for sim ply Sim ply supported beam 1 8 2 18 3 181 FINITE ELEMENT MODELING OF VIBRATION PROBLEMS 1 Frequency error vs elements for xedsimple support beam 10 I I I I I 7 Bth mode beta L 196351 1d a 7 L E1047 3 ET 9 LL 103927 e 163 I I I I I I I I 10 15 0 25 30 35 40 45 50 Number of Beam Element Figure 182 Frequency error vs number of elements for xed simply supported beam In using the nite element method for modeling of beam vibrations the question arises how many elements does one need for an accurate computation of its kth mode and mode shape While there exists a vast amount of literature on the accuracy and convergence properties of various elements which one may employ to answer the question we will adopt a posteriori assessment approach To this end let us take a simply supported beam and concentrate on its 6th modes The frequency error vs the number of beam elements used are shown in g 181 As can be seen vedigit accuracy is achieved with about 45 elements If all one needs is the frequency with one percent accuracy one could use only 10 elements In Figure 182 the frequency error of a xedsimply supported beam is illustrated Although the error for this case is a little higher than that of the simply supported case the frequency error converges with the same trend let s now focus on the mode shape accuracy 1 SlmpIEr SlmpIE Suppun Beam Slmpler Slmple suppen Beam 1D elements 5m made beta L 1B 9243 Frer Ienz am 7825 5 elemems 5th made bela L I9 8825 FIeeHenz9IE 1EEI1 I Made shape Made shape I l I I mnveraeumuuemzpe umpuredmudemzpe WW D1 n2 n3 n4 a m us as 39 U1 n2 n3 n4 n5 n5 Eeamspan as u aeam span Figure 183 Frequency error vs number of elements for xed simply supported beam 1 8 3 Chapter 18 MODELING FOR STRUCTURAL VIBRATIONS FEM MODELS DAMPING SIMILMITY LAWS Supplier 51mp1e suppun Beam 51mph 51mp1e suppun Beam 715 mements sth Nude mam 18 8652 FreqHertz 826 5861 7 88 e1emems sth Nude beta L 18 8586 FreqHertz 825 8172 Made shape Made shape 81 82 88 84 85 DE 87 88 88 39 81 82 83 84 85 88 Beam spah Beam spah Figure 184 Frequency error vs number of elements for xed simply supported beam Supplier 51mp1e suppun Beam 51mph 51mp1e suppun Beam 7 48 mements 64h Nude mam 18 8488 FreqHertz 825 257 5D E Emems 5m We Mam 18 3497 quwmz 825 mm Made shape Made shape 87 88 88 39 81 82 83 84 87 88 88 81 82 83 84 85 85 Beamspan 85 88 Beam spah Figure 185 Frequency error vs number of elements for xed simply supported beam Figures 1835 illustrate how the mode shapes converge to the exact solution as the number of elements increases Also plotted is spline tted curves that utilize only the sample points or computed discrete mode shape points It is clear that curve tting in general enhances the discrete raw data points especially for the case of crude models Scanning over the siX mode shape plots Fig 1835 vs the increasing number of elements an acceptable number of element for capturing the siXth mode shape appears to be 30 Note that there are siX half sine waves in the 6th mode shape Hence for the 30element model each half sine wave is sampled by ve elements or siX nodal points meaning that ten elements span one full sine wave To conclude ten elements captures the siXth mode frequency with less than one percent error whereas for an adequate mode shape capturing three time of that elements 30 elements are needed This is often referred to as quotthreetoone rule Finally the case of tenelement model satis es the socalled Shannon s sampling criterion which state that for the minimum sampling number for a sinusoidal signal is three 18 5 182 CHARACTERIZATION OF LINEAR STRUCTURAL DYNAMICS EQUATIONS 182 CHARACTERIZATION OF LINEAR STRUCTURAL DYNAMICS EQUATIONS Let us now study the characteristics of the secondorder damped system MiiDuKuft D aM K 182 where a and 9 are constants The damping matrix D OtM K is called a Rayleigh damping as it is proportional both to mass and stiffness of the system The coupled equations of motion for a linear structure182 can be decoupled by the n gtlt n eigenvector matrix T which relates the displacement vector u to a generalized solution vector q via u Tq 183 Substituting 183 into 182 and premultiplying the resulting equations by TT one obtains IqaI AqAqfq fqTTff 184 in which TTMT 1 185 TTKT A diagwf 50 186 where wj is the jth undamped frequency component of 184 The solution of 184 for its homegeneous part can be expressed as q ces 187 where s is in general complex constants Substitution of 187 into 184 with fq 0 yields 52 a1 As Ac 0 188 Equation 188 has a nontrivial solution only if det521 a1 AsA 0 189 from which the kth solution component can be expressed as qk akeskt keskt 1810 where sk and 5 are the kth complex conjugate pairs of 189 and ak and in are arbitrary constants Two characterizations of 1 89 are possible frequency characterization Viz by xing the damping param eters a and 9 and varying the undamped frequency w and damping characterization Viz by xing the undamped frequency w and varying the damping parameters a and 9 1 8 5 Chapter 18 MODELING FOR STRUCTURAL VIBRATIONS FEM MODELS DAMPING SIMILMITY LAWS Root loci ofproponionally damped structural dynamics equation l l l l l l l B a 37 Increasingfrequency 7 3 E 39 E 27 m 7 lia z D 5 1e 5 7 c n 3 g 07c 0 S1 A 7 E flE 0 g D c 5 5 2 a D a E 3 Increasing frequency 8 k E B 4 l l l l l l l l 7 1 6 5 4 3 2 1 Real pan ofnormalized 39equency by sampling rate h Fig 186 Root Loci ofProportionally Damped Structural Dynamics System 1821 Frequency Characterization Based on the Root Locus Method The characteristic equation 189 represents n roots sflta w sjwfq j12n 1811 Since not only the distribution but also the maximum and minimum frequencies are not known apriori a complete range of the Rayleighdamped system can be characterized by the following expression s2a w2sw20 0350300 1812 In order for the subsequent characterization to remain valid for all sampling rates we normalize 1812 by the sampling rate 11 such that 52 ah 5wh2s39 60102 0 0 5 wk 5 00 ll 1813 E h 7 52a1 52 a20 0700 E 060 Note rst that the rigidbody motion ie o wh 0 corresponds from 1813 to 51 0 0 and s2 1111 0 11 0 1814 1 8 6 18 7 182 CHARACTERIZATION OF LINEAR STRUCTURAL DYNAMICS EQUATIONS These two roots are marked as s1 and s2 in Fig 186 As the undamped frequency w is increased the root locus approaches the branch point at A in Fig 186 A h1 v1 Dim197 0 1 1 073mg 0 1815 As an is further increased the root locus follows the half circle with its center at 118 0 and with its radius 1 64818 viz 5B 137 in ldEE 1816 The two complex roots merge at 5C 2 410 W7 0 1817 For a gt sc one locus branches out toward the negative in nite aXis while the other approaches the origin 0 0 This is illustrated in Fig 186 For the special case 19 0 Viz mass proportional damping the branch occurs at 5 0 when w 112 Hence the solution components with frequency w lt eXhibit overcritically damped responses as the locus becomes a straight line ERG In particular if a 0 the root loci coincide with the imaginary aXis whose response is characterized by purely oscillatory components From the physical viewpoint the case of massproportional damping introduces higher modal damping for lower frequency solution components and the degree of damping decreases as the frequency increases This does not however necessarily mean that the response components of the highfrequency modes will decay slower than those of the lowfrequency modes within a time period As a matter of fact the decay rate is uniform for all frequency components since we have e5 9 for all to 1818 The root locus of the stiffnessproportional damping case a 0 touches the origin and forms the half circle as w is increased As an is further increased the locus branches out into the negative real aXis Therefore the decay rate increases as w increases As such this representation of system damping is often used in the modeling of structural damping due to joint effects acoustic noise and internal material friction 1822 Damping Characterization for Constant Frequency The conventional characterization of damping into the equations of motion for structures is to eXpress each component of189 sg2swskw2o 1819 so that the characteristic roots can be eXpressed as 5 550 I 1 25u 1820 where 5 is termed the damping ratio The equivalent damping ratio for 1811 therefore is 1 a seq 1850 lg21 2 w 1 8 7 Chapter 18 MODELING FOR STRUCTURAL VIBRATIONS FEM MODELS DAMPING SIMILMITY LAWS Now we like to examine for a xed so what happens when 5 is varied For the undamped case viz 3 0 the roots lie on the imaginary axis 5 0 2150 As 5 is increased the roots rotate toward the lefthand complex plane with the constant magnitude of a wh since wh 1822 and the shift angle being 0 tan 1 N1 525 The two complex roots merge on the real axis when 5 1 thus the root locus is a halfcircle with radius wh For 5 gt 1 one root branches out to the in nite negative axis while the other approaches the origin see Figure 186 This invariance property that is the magnitude of the complex root remains the same for all damping ratios plays an important role both for controller synthesis and computational algorithms 183 VARIOUS DAMPING MODELS Modeling of damping remains a challenge in structural dynamics Over the years various damping models have been proposed Below lists a sample of parameterized damping models 1831 Mass Proportional Damping Model If the damping can be made proportional to the mass such as dynamic friction cases the damping matrix can be parameterized according to D 1M 1823 as already discussed in the previous section The root loci are a special case of 182 governed by 52a 562 0 1824 which is plotted in Fig 187 as method A with its parameter a 1 Note that the root loci form a straight vertical line when w gt 112 1832 Viscous Damping Model One of the most widely used damping models is the viscous damping characterization This has been adopted for modeling of coated damping layers of lubricants in rotating machines and of a plethora of unknown sources Mathematically the viscous damping model can be expressed as D 2T wp TT W diag a wf 52 w 5N 601 1825 TTMT I TTKT A Therefore the characteristic equation of a viscous damped system is given by s39 25 of 5 of 0 1826 Notethatthetwocases p 0 5152 5N 11 and p 2 5152 5N have been studied in the preceding section 1 8 8 18 9 183 VARIOUS DAMPING MODELS Root loci of various proportional damping models Imaginary part of normalized frequency by sampling rate h 72 713 716 714 711 71 708 706 704 702 0 Real part of normalized frequency by sampling rate h Fig 187 Root Loci ofProportionally Damped Structural Dynamics System The root loci of a Viscous damped case when combined with a massproportional damping are obtained by 171 51525Ns gt 52a25a a20 1827 which is plotted in Fig 187 labeled as model C with its parameters a 10 5 0025 Note that in this model the modal damping ratio is the same 5 const for all the frequencies greater than an gt a21 5 5 lt 10 1833 Intermediate frequency damped case In practice such as acoustically treated structural systems both the low and high frequencies are associ ated with very low damping While intermediate frequencies are damped depending on damping treatment employed Such damping models may be parameterized according to D aMT 8 1 tanhywh 56TT 1828 which results in the following root loci equation 52a31 tanhywh WNW 0 1829 The root loci of this model are plotted in Fig 187 as method D1 D2 D3 with its parameters p 2 a 10 h 10 0025 y 0025 005 01 1830 1 8 9 Chapter 18 MODELING FOR STRUCTURAL VIBRATIONS FEM MODELS DAMPING SIMIKAIRITY LAWS Note that as w becomes large the real part of the root loci approaches 5 gt a as w gt 00 1831 Hence if a 0 the root loci will lie on the imaginary axis indicating no damping for those frequencies 184 USE OF SHVIILARITY LAWS IN MODELING Dimensional analysis has been widely used in experiment design and scalemodel construction The task in dimensional analysis is to identify physical quantities that in uence the physical phenomena on hand then deduce a independent set of dimensionless products These dimensionless products are then utilized for experiment design and scalemodel construction The underlying principle is Buckingham s theorem that enables a systematic construction of linearly independent dimensionless products It should be pointed out that dimensional analysis is applicable both to statics and dynamics In dynamics a useful dimensional analysis may be canied out which can provide insight into dynamic behavior It is referred to mechanical similarity after Landau and Lifshits The basic idea is as follow The starting point of mechanical similarity is that multiplication of kinetic and potential energy expression or the Lagrangian by any constant does not affect the equations of motion It is this observation that can yield the dynamic behavior of a system without actually solving the equations of motion For example we may pose the question Does the amplitude of string vibration affect the frequency assuming its potential energy is a quadratic function of its amplitudes To answer this question let us consider the simplest case ie a single mass and a single spring case given by Tlmx2 Ulkx2 gt T Y L 1832 mJEkx0 x0x0 x0x0 where m and k are the mass and spring constant x is the vibration amplitude and x0 and x0 are the initial amplitude and velocity of the mass Of course we all know that the frequency of the above system is given by w xkm 1833 which clearly shows that the frequency w is independent of the amplitude x t Let us scale the new amplitude x by x X g 1834 so that the potential energy U changes to Ux kx 2 58 1835 Now we introduce the time scale t r 1836 t where 1 characterizes the ratio of periods of motion or time durations of the two systems 18 10 18 11 184 USE OF SIMILARITY LAWS IN MODELING The kinetic energy fo the system is now changed to dx X 1 2 1 2 2 Kx t mx 1837 Therefore the Lagrangian of the new system becomes 1 E T U 2 mxz X2x2 X2 g me kx2 1838 The above equation states that the only way the resulting equation of mition is unaltered is to choose 1839 H X2 kx2 XZE Since the time scale ratio is unity it means that the frequencies for both systems are the same In other words the frequencies of a lumped springmass system is unaffected by their vibration amplitudes Comparing 1833 and 1839 one may argue that the mechanical similarity method is somewhat more complicated than what could be obtained by the frequency equation 1833 Its chief advantage is that one needs not solve the governing equation 1832 for its frequency eXpression 1841 Kepler s Third Law Let us now apply the mechanical similarity to the motions of two satellites around a heavenly body Newton s law of universal gravitation states GMm GMm gt F 2 U 1840 where G is the universal gravitational constant M and m are the masses of the heavenly body and the satellite and r is the distance between the two bodies The Lagrangian of the two satellites may be eXpressed as GM d 1m1V139V1 m1 V1i 39 l d 18 41 I 1 GMWIZ drz 39 m v v v 2 2 2 2 2 lrzl 2 db Let us transform 2 by introducing t2 rtl r2 Xrl 1842 to obtain 2 1 GMm m 2cm1V1V1 3 1 c 212 1843 X 1 1 m1 139 Note that the above scaling on 2 amounts to a thought experiment in which the satellite m2 is brought to the orbit of satellite m 1 For this thought experiment to be true its corresponding equation of motion should be the same as that obtained by 1 This can be the case only if the following condition is satis ed t2 1 2 2 3 1844 t1 1 1 which states that square of the revolution ratio of the two satellites is proportional to the cube of the ratio of the orbital sizes Kepler s third law r2 2C 1 gt F1 gt 18 11 Chapter 18 MODELING FOR STRUCTURAL VIBRATIONS FEM MODELS DAMPING SIMIKAIEITY LAWS 1842 The Race between a Sliding Block and a Rolling Cylinder on an Inclined Slope Consider a cube sliding on an inclined plane without friction whose Lagrangian can be expressed as rm ms2 mgs sin0 1845 where m is the mass of the cube s is the distance along the slope measured from the bottom of the slope g is the gravity 0 is the inclined angle of the slope and s is the speed of the cube along the slope If a cylinder of the same mass m is to roll along the slope without slip its Lagrangian can be obtained as 5y1ms2 mr2w2 mgs sin0 1846 where r is the radius of the cylinder and w is the angular velocity of the cylinder during rolling without slip Since the angular velocity can be eXpressed in terms of squot as a w i y 1847 1846 can be simpli ed to 3 4 Tm s 2 mgs sine 1848 Transforming the above by the scaling t t 1 ss X 1849 yields 3 2 3 2 212 cm 2w mng sine E gms z E mgssine 1850 Comparing the above with 1845 we nd for the resulting equation of motion from 1850 to be the same as obtained by 1845 we must have M 1 gt t 3X 18 51 1 3 X t 2 Now if the distance is the same ie s s we have X 1 Therefore the cylinder will arrive at the bottom of the inclined slope by a factor of g longer over the arrival time of the cube In other words the cube will arrive at the bottom faster than the cylinder and the arrival time ratio of the two bodies will be proportional to the ratio of the square root of the kinetic energies Physically the reason for the slower average speed of the cylinder is because for the case of cylinder the same potential energy change is translated into both the translational and rotational kinetic energy while the entire change energy gets into the translational kinetic energy for the cube 1843 Mechanical Similarity in String Rod and Shaft Vibrations The Lagrangian of strings rods and shafts can be eXpressed as T U Z a T0 mltxgt1 w 12dx 1852 E U 0 kx 12k 18 12 18 13 184 USE OF SIMILARITY LAWS IN MODELING where mx is the mass per unit length for string and rods and the mass moment of inertia for shafts respectively and kx denotes the tension force the product of Young s modulus and the cross section area E Ax and the shear rigidity G x for strings rods and shafts respectively Let us consider the following scaling transformations m k x E t p 7 7 r 1853 m M k K x X E t p which when substituted into 1852 yields MX K MX K r2 E 2 T U 2 T 2 U 1854 I X r M In order for the transformations not to affect the resulting equations of motion we must have E K152 CE gt 21 gt 1 E 1 K 3 x it 1855 k Period ratio i i p k m E where C E is the energy ratio whose role we will discuss shortly The preceding similarity law is specialized for strings rods and shafts in Table 81 Note that if the frequency to be the same from one structure to another eg in laboratory experiment compared to the real system then the experiment design must employ E K k m km 1856 Table 81 Similarty Laws in Strings Rods and Shafts Strings Rods Shafts 7 T A 2 E 2 GI A 2 Period Ratio p p TZA 7 E Z 7 01 7 On the other hand if the frequency ratio is to be proportional to the dimension ratio then one must see to it that the following is observed p E k k gt 1857 p E m m Other scaling can be similarly considered from Table 81 It should be noted that the preceding similarity laws are valid when the scaled and original systems have the same boundary conditions 18 13 Chapter 18 MODELING FOR STRUCTURAL VIBRATIONS FEM MODELS DAMPING SIMIKAIHITY LAWS Observe that the similarity laws summarized in Table 81 can be derived if one solves for each of the three vibration problems The emphasis is to utilize the respective energy expressions rather than the solutions of the governing differential equations This is because it is often considerably easier to obtain the energy expressions of a system than obtain the vibration characteristics or quasistatic deformations Finally let us consider the role of the energy ratio MX C E 7 1858 which plays a pivotal role in sizing up the experiment design as it offers the experiment energy requirements relative to the actual system In practice it is often the energy requirement that dictates the scaling than the geometrical considerations Notice that since we have only one constraint for four parameters the remaining three parameters can be used in meeting practical considerations in experiment design 1844 Mechanical Similarity in Euler Bernoulli Beam The Lagrangian of a continuous EulerBemoulli beam can be expressed as E T Um Uggt Z T 0 0Ax M12 dx 8t 5 82wxt 1859 Um 0 EIx dex E U 10 PM awx t2 dx 2 Bx where wx t is the transverse displacement of the neutral axis of the beam m x is the mass per unit beam length E I x denotes the bending rigidity of the beam and P x denotes the prestressed axial force Introducing the scaling transformations given by 1853 as modi ed to the case of beam the Lagrangian of the new system can be expressed as LX K f E I P T U U r2 X3 m X g K E17 f P 2 2 x Kr fx T U U r2 WM m K g 1860 U 2 2 L L if K r4 1 and fi 1 1 LX K Therefore the similarity law for a beam is expressed as E A E P E I 82 5 lt gt2 provided x 2 ifPx0 1861 p EIpA z Px E1 5 Notice that the appendage arms for a spinning satellite helicopter rotor blades and turbine blades all experi ence axial tensions Hence the scaling of beam sizes must consider the axial force scaling as well One important application of the preceding scaling law is for the experiment design of rotating members for micromachine members as the rotating speed is very high and experiment design often necessitates a scaling up rather than scaling down 18 14


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