Example Analysis of MDOF Forced Damped Systems

Transcription

1 ASEN Structures Example Analysis of MDOF Forced Damped Systems ASEN 311 Lecture Slide 1

2 ASEN Structures Objective This Lecture introduces damping within the context of modal analysis. To keep the exposition focused we will primarily restrict the kind of damping considered to be linearly viscous, and light. Linearly viscous damping is proportional to the velocity. Light damping means a damping factor that is small compared to unity. In the terminology of Lecture 17, lightly damped mechanical systems are said to be underdamped. ASEN 311 Lecture Slide

3 Good and Bad News ASEN Structures Accounting for damping effects brings good and bad news. All real dynamical syste, experience damping because energy dissipation is like death and taxes: inevitable. Hence inclusion makes the dynamic model more physically realistic. The bad news is that it can seriously complicate the analysis process. Here the assumption of light viscous damping helps: it allows the reuse of major parts of the modal analysis techniques covered in the previous three Lectures. ASEN 311 Lecture Slide 3

4 What is Mechanical Damping? ASEN Structures Damping is the (generally irreversible) conversion of mechanical energy into heat as a result of motion. For example, as we scratch a match against a rough surface, its motion generates heat and ignites the sulphur content. When shivering under cold, we rub palms against each other to warm up. Those are two classical examples of the thermodynamic effect of friction. In structural systems, damping is more complex, appearing in several forms. These may be broadly categorized into internal versus external distributed versus localized ASEN 311 Lecture Slide 4

5 ASEN Structures Internal versus External Damping Internal damping is due to the structural material itself. Various sources: microstructural defects, crystal grain slip, eddy currents (in ferromagnetic materials), dislocations in metals, chain movements in polymers. Key macroscopic effect: a hysteresis loop. Loop area represents energy dissipated per unit volume of material and per stress cycle. Closely linked to cyclic motions. ASEN 311 Lecture Slide 5

6 Internal versus External Damping (cont'd) ASEN Structures External damping comes from boundary effects. An important form is structural damping, produced by rubbing friction, stick and slip contact or impact. May take place beween structural components such as joints, or between a structural surface and non-structural media such as soil. This form is often modeled as Coulomb damping, which describes the energy dissipation of rubbing dry friction. Another form of external damping is fluid damping. When a material is immersed in a fluid such as air or water and there is relative motion between the structure and the fluid, a drag force appears. This force causes energy dissipation through internal fluid mechanisms such as viscosity, convection or turbulence. A well known instance is a vehicle shock absorber: a fluid (liquid or air) is forced through a small opening by a piston. ASEN 311 Lecture Slide 6

7 ASEN Structures Distributed versus Localized Damping All damping ultimately comes from frictional effects, which may take place at several scales. If the effects are distributed over volumes or surfaces at macro scales, we speak of distributed damping. But occasionally the engineer uses damping devices intended to produce beneficial effects. For example: shock absorbers, airbags, parachutes motion mitigators for structures in seismic or hurricane zones active piezoelectric dampers for space structures These devices can be sometimes idealized as lumped objects, modeled as point forces or moments, and said to produce localized damping. The distinction beteen distributed and localized appears at the modeling level, since all motion-damper devices ultimately work as a result of some kind of internal energy dissipation at material micro scales. ASEN 311 Lecture Slide 7

8 ASEN Structures Distributed versus Localized Damping (cont') Localized damping devices may in turn be classified into passive: no feedback active: responding to motion feedback But this would take us too far into control systems, which are beyond the scope of the course. ASEN 311 Lecture Slide 8

9 Modeling Damping in Structures ASEN Structures In summary: damping is complicated business. It often is nonlinear, and level may depend on fabrication or construction details that are not easy to predict. Balancing those complications is the fact that damping in most structures is light. In addition the presence of damping is usually beneficial to safety in the sense that resonance effects are mitigated, This gives the structural engineer some leeway: o A simple model, such as linear viscous damping, can be assumed o Mode superposition is applicable because the EOM is linear. Frequencies and mode shapes for the undamped system can be reused if additional assumptions, such as Rayleigh damping or modal damping, are made ASEN 311 Lecture Slide 9

10 Modeling Damping in Structures (cont') ASEN Structures It should be stressed that the foregoing simplifications are not recommended if precise modeling of damping effects is important to safety and performance. This occurs in the following scenarios: o o Damping is crucial to function or operation. Think, for instance, of a shock absorber, airbag, or parachute. Damping may destabilize the system by feeding energy instead of removing it. This can happen in active control systems and aeroelasticity. The last two scenarios are beyond the scope of this course. In this Lecture we focus attention on linear viscous damping, which usually will be assumed to be light. ASEN 311 Lecture Slide 10

11 ASEN Structures Matrix EOM of Two-DOF Example Consider again the two-dof mass-spring-dashpot example system of Lecture 19. This is reproduced on the right for convenience. The physical-coordinate EOM derived in that Lecture are, in detailed matrix notation: Static equilibrium position u 1 = u 1(t) ;; ;; m m ü 1 ü + c 1 + c c c c u 1 u + k 1 + k k k k u 1 u = p 1 p p (t) 1 Static equilibrium position u = u (t) k 1 k p (t) c 1 Mass m 1 c Mass m ASEN 311 Lecture Slide 11

12 ASEN Structures Matrix EOM of Two-DOF Example (cont') Passing to compact matrix notation,... M u + C u + K u = p Here M, C and K denote the mass, damping and stiffness matrices,... respectively, p, u, u and u are the force, displacement, velocity and acceleration vectors, respectively. The latter four are functions of time: u = u(t), etc., but the time argument will be often omitted for brevity. As previously noted, matrices M, C and K are symmetric, whereas M is diagonal. In addition we will assume that M is positive definite (PD) whereas K is nonnegative definite (NND). ASEN 311 Lecture Slide 1

13 EOM Using Undamped Modes ASEN Structures This technique attempts to reuse modal analysis methods covered in Lecture Suppose that damping is removed so that C = 0. Get the natural frequencies and mode shapes of the undamped and.. unforced system governed by M u + K u = 0, by solving the eigenproblem K U i = ω im U i. Normalize the vibration mode shapes U into φ so that they are orthonormal wrt M: i i φ T i M φ j = δ ij in which δ ij denotes the Kronecker delta. Let Φ be the modal matrix constructed with the orthonormalized mode shapes as columns, and denote by η the array of modal amplitudes, also called generalied coordinates. As before, assume modal superposition is valid, so that physical DOF are linked to mode amplitudes via u = Φ η ASEN 311 Lecture Slide 13

14 Matrix EOM of Damped System (cont') ASEN Structures Following the same scheme as in previous Lectures, the transformed EOM in modal coordinates are: Φ T M Φ ü + Φ T C Φ u + Φ T K Φ u = Φ T p(t) Define the generalized mass, damping, stiffness and forces as M g = Φ T M Φ C g = Φ T C Φ Kg = Φ T K Φ f = Φ T p Of these the generalized mass matrix M g and the generalized stiffness matrix K g were introduced in Lecture 0. If Φ contains mode shapes orthonormalized wrt M, it was shown there that M g = Iγ K g = diag[ω i ]γ are diagonal matrices. The generalized forces f were introduced in the previous Lecture. ASEN 311 Lecture Slide 14

15 Matrix EOM of Damped System (cont') ASEN Structures The new term in the modal EOM is the generalized damping matrix C g, also called the modal damping in the literature. Substituting the definitions we arrive at the modal EOM for the damped system: η(t) + C g η (t) + diag[ ωi ] η ( t) = f(t) Here we run into a major difficuty: C g generally will not be diagonal. If that happens, the above modal EOM will not decouple. We seem to have taken a promising path, but hit a dead end. ASEN 311 Lecture Slide 15

16 Three Ways Out ASEN Structures There are three ways out of the dead end: Diagonalization. Stay with the modal EOM, but make C diagonal through some artifice Complex Eigensystem. Set up and solve a different eigenproblem that diagonalizes two matrices that comprise M, C and K as submtarices. [The name comes from the fact that it generally leads to frequencies and mode shapes that are complex numbers.] Direct Time Integration, or DTI. Integrate numerically the EOM in physical coordinates. Each approach has strengths and weaknesses. (Obviously, else we would mention only one.) g ASEN 311 Lecture Slide 16

17 Diagonalization ASEN Structures Advantages Diagonalization allow straightforward reuse of undamped frequencies and mode shapes, which are fairly easy to obtain with standard eigensolver software. The uncoupled modeal equations often have straightforward physical interpretation, allowing comparison with experiments. Only real arithmetic is necessary. Disadvantages We don't solve the original EOM, so some form of approximation is generally inevitable. This is counteracted by the fact that structural damping is often difficult to quantify since it can come from many sources. Thus the approximation in solving the EOM may be tolerable in view of modeling uncertainties. This is particularly true if damping is light. ASEN 311 Lecture Slide 17

18 ASEN Structures When Diagonalization Fails There are problems, however, in which diagonalization cannot adequately represent damping effects within engineering accuracy. Three such scenarios: (1) Structures with localized damper devices: shock absorbers, piezoelectric dampers,... () Structure-media interaction: building foundations, tunnels, aeroelasticity, parachutes, marine structures, surface ships,... (3) Active control systems In those situations one of the two remaining approaches: complex arithmetic or direct time integration (DTI) must be taken. ASEN 311 Lecture Slide 18

19 ASEN Structures Complex Eigensystem The complex eigensystem approach is mathematically irreproachable and can solve the original EOM in physical coordinates without additional approximations. No assumptions as to light versus heavy damping are needed. However, it involves a substantial amount of preparatory work because the EOM must be transformed to the so-called state-space form. For a large number of DOF, solving complex eigensystems is unwieldy. Physical interpretation of complex frequencies and modes is less immediate and may require substantial expertise in math as well as engineering experience. Finally, it is restricted in scope to linear dynamic systems unless some convenient form of linearization is available. ASEN 311 Lecture Slide 19

20 ASEN Structures Direct Time Integration (DTI) DTI has the advantages of being completely general. Numerical time integration can in fact handle not only the linear EOM, but nonlinear systems, which occur for other types of damping (e.g. Coulomb friction, turbulent fluid drag). No transformation to mode coordinates is necessary and no complex arithmetic emerges. The main disadvantage is that requires substantial expertise in computational handling of ODE, which is a hairy topic onto itself. Since DTI can only handle numerically specified models, the approach is not particularly useful during preliminary design stages, when many design parameters float around. Beacuse the last two approcahes (complex arithmetic and DTI) lie outside the scope of an introductory course (they are usually taught at the graduate level) our choice is easy: diagonalization it is. ASEN 311 Lecture Slide 0

21 ASEN Structures Example System With Specific Data & 3 Free Parameters [ ] [ ][ ] [ m1 0 ][ü1 c1 + c + c u1 k1 + k + k 0 m ü c c u k k ][ u1 u ] = [ p1 p ] m 1 =, m = 1, k 1 = 6, k = 3, c = c 1 = c, p 1 = 0, p = F cos t [ ] 0 ][ü ü [ ][ ] c c u1 + c c u [ ][ ] [ 9 3 u1 = 3 3 u ] 0 F cos t M = [ ] 0, C = 0 1 [ c c c c ], K = [ ] [ 9 3, p = 3 3 ] 0 F cos t ASEN 311 Lecture Slide 1

22 ASEN Structures Generalized Mass, Damping, Stiffness and Forces In Terms of The Undamped Modal Matrix Φ ω1 = 3, ω = 6, Φ = [ φ 1 φ ] = = [ ] [ ] c M g = Φ T M Φ = = I, C 0 1 g = Φ T C Φ = c 3 c 3 5c K g = Φ T K Φ = [ ] 3/ [ 6 ] = diag[3/, 6], f (t) = Φ T p (t) = F cos t 1 3 ASEN 311 Lecture Slide

23 ASEN Structures Modal EOMs Are Coupled Through C g η (t) + C g η (t) + diag[3/, 6] η (t) = f (t) c 3 C g = Φ T C Φ = c 3 c 3 5c ASEN 311 Lecture Slide 3

24 Diagonalization Device: Rayleigh Damping ASEN Structures Often used in Civil Engineering structures. Assume that the damping matrix is a linear combination of the mass and stiffness matrices: assume C = a M + a K 0 in which a 0 and a 1 are numerical coefficients with physical dimensions of (1/T) and T, respectively, T being a time unit. Transforming to modal coordinates gives the generalized damping matrix (a.k.a. modal matrix): T g 0 g 1 g C = Φ C Φ = a M + a K which is a diagonal matrix. If the modes in Φ are orthonormal with respect to the mass matrix, the diagonal entries of C are 1 C gii = a + a ω 0 1 i ASEN 311 Lecture Slide 4

25 ASEN Structures Choosing the Rayleigh Damping Coefficients Recall that for the single-dof oscillator with viscous damping, the coefficient of the velocity term in the canonical form is ξ ω where ξ is the dimensionless damping ratio and ω the natural frequency. By analogy the i th diagonal entry of the Rayleigh-damping diagonalized C g can be taken to be ξ i ω i. Equating that to C gii = a 0 + a 1 ω i (from previous slide) shows that the i th damping ratio is ( ) 1 a0 ξ i = 1 i ω + a ω i The assignment of values to a 0 and a 1 is often done by matching the damping ratios of two modes. For example, matching ξ 1 for mode 1 and ξ for mode gives two linear equations ( ) ( ) 1 a ω + a 1 ω 1 ξ = + a 1 ω 1 ω ξ = from which a and a can be determined. 0 1 a ASEN 311 Lecture Slide 5

26 ASEN Structures A Second Diagonalization Device Also Bears The Name Rayleigh (but for a different purpose) The Rayleigh Quotient (RQ) was introduced by Lord Rayleigh as a device to approximate the fundamental frequency of a linear acoustic system if an approximate mode shape is known. The RQ is frequently used in linear algebra just for eigenvalue calculations. Here we will use it in another context: conservation of dissipation energy over a cycle when the damping matrix is diagonalized ASEN 311 Lecture Slide 6

27 "Rayleigh Quotient" (abbrev. RQ) Diagonalization Of Damped Matrix C ASEN Structures C RQ g [ ] = diag[c RQ 1, C RQ C R Q ] = 1 0 Q 0 C R C RQ 1 = φt 1 Cφ 1 φ T 1 φ 1 = c 5, C RQ = φt Cφ φ T φ = 5c The effective modal damping factors are ξ RQ 1 = c 5(ω 1 ) = c ξ RQ = 5c (ω ) = c ASEN 311 Lecture Slide 7

28 ASEN Structures RQ-Damped Modal EOM Now Decouple η 1 (t) η (t) + c 5 η 1 (t) + 3 η 1 (t) = F cos t 6 + 5c η (t) + 6η (t) = 1 F cos t 3 But an approximation has been introduced (Is it serious?) ASEN 311 Lecture Slide 8

29 ASEN Structures We Will Investigate This Issue By Comparing Two Solutions ;; Exact responses of original EOM of example system in physical coordinates (obtained by Direct Time Integration, aka DTI) Approximate responses obtained by solving the RQ modal EOM and transforming to physical coordinates by the undamped modal matrix Φ u (t) 1 u (t) p (t) 1 k 1 k c 1 Mass m 1 c Mass m p (t) m 1 =, m = 1, k 1 = 6, k = 3, c = c 1 = c, p 1 = 0, p = F cos t ASEN 311 Lecture Slide 9