Vibrations of Structures

Transcription

1 Vibrations of Structures Module III: Vibrations of Beams Lesson 26: Special Topics in Beam Vibrations - II Contents: 1. Introduction 2. Influence of Axial Force on Dynamic Stability Keywords: Stability, Axial force, Follower force, Non-conservative force, Extended Hamilton s principle, Divergence instability, Flutter instability

2 Special Topics in Beam Vibrations - III 1 Introduction Instability may be broadly understood to be finite/large departures of a system from an equilibrium solution under an infinitesimally small disturbance. Under axial loads, beams can show instability behaviour which are relevant in certain applications such as stability of columns and slender missiles. Here, we study the stability problem using the dynamical formulation to illustrate two kinds of instabilities in beams. 2 Influence of Axial Force on Dynamic Stability Simply-supported Beam with Constant Axial Force The equation of motion of a simply-supported uniform Euler-Bernoulli beam z, w ρ, A, EI F x l Figure 1: A simply-supported beam with a constant axial force with a constant compressive axial force F, as shown in Fig. 1, is given by ρaw,tt + F w,xx + EIw,xxxx = 0. (1) 2

3 Assuming a modal expansion w(x, t) = j=1 p j (t) sin jπx l and taking the inner product with the k th mode yields ρa p k + k2 π 2 (EI k2 π 2 l 2 l 2 ) F p k = 0. (2) Setting p k (t) = Ce st, we have s = ±iω k, where k ω k = 2 π 2 (EIk 2 π 2 /l 2 F )/l 2 ρa (3) is the k th circular eigenfrequency. In a non-dimensional form, (3) may be represented as ω k = k 2 π 2 (k 2 π 2 S), where ω k = ω k ρal4 /EI, and S = F l 2 /EI. For k = 1, the variation of the first eigenfrequency with axial force is shown in Fig. 2. When the axial ω S = S c S Figure 2: Variation of the first non-dimensional circular eigenfrequency ω of a simplysupported beam with non-dimensional axial force S 3

4 I [s] S = S c ω(s < S c ) R[s] ω(s < S c ) Figure 3: Loci of eigenfrequencies of a simply-supported beam with variation of axial force z, w ρ, A, EI F x l Figure 4: A cantilever beam with a follower force force F crosses the critical value F c 1 = π 2 EI/l 2 (Euler buckling load), the first eigenfrequency becomes imaginary, implying a divergent solution of (2) (divergence instability). Thus, the undeformed equilibrium configuration of the beam becomes linearly unstable. Thus, the eigenvalues s go through zero before becoming real, as shown in Fig. 3. Cantilever Beam with Follower Force Consider a follower-force at the free-end of a cantilever Euler-Bernoulli beam, as shown in Fig. 4. The force has a component in the transverse direction given by F w,x (l, t) (assuming w,x (l, t) 1). This non-potential transverse 4

5 force does work on the system which needs to be accounted separately in the Hamilton s principle. The equation of motion is obtained using the extended Hamilton s principle t2 (δl + δw) dt = 0. (4) t 1 where L = 1 2 l 0 (ρaw 2,t EIw2,xx + F w2,x ) dx, and δw = F w,x(l, t)δw(l, t). The variational formulation leads to (5) t2 [( EIw,xx δw,x + (EIw,xxx + F w,x )δw ) l 0 F w,x(l, t)δw(l, t) ] dt t 1 t2 t 1 l 0 (ρaw,tt + EIw,xxxx + F w,xx ) δw dx dt. (6) Thus, the equation of motion, and boundary conditions for the problem are given by ρaw,tt + EIw,xxxx + F w,xx = 0, (7) w(0, t) = 0, w,x (0, t) = 0, w,xx (l, t) = 0, and w,xxx (l, t) = 0. (8) Assuming a modal solution of the form w(x, t) = W (x)e st for (7) leads to W + F EI W + ρas2 W = 0. (9) EI Writing the solution of (9) in the form W (x) = e βx, we get β 4 + F EI β2 + ρas2 EI = 0. (10) 5

6 Using the definitions one can rewrite (10) as S := F l2 EI, and ω2 := ρas2 l 4, (11) EI l 4 β 4 + Sl 2 β 2 ω 2 = 0. (12) Therefore, the general solution of W (x) can be represented as W (x) = B 1 cosh β 1 x + B 2 sinh β 1 x + B 3 cos β 2 x + B 4 sin β 2 x, (13) where β 1 and β 2 are obtained from (12) as β 1 = 1 2 l [ S + S ω 2 ] 1/2, and β 2 = 1 2 l [S + S ω 2 ] 1/2. Substituting the solution form (13) in the boundary conditions (8) yields B 1 + B 3 = 0, (14) B 2 + B 4 = 0, (15) B 1 cosh β 1 l + B 2 sinh β 1 l B 3 cos β 2 l B 4 sin β 2 l = 0, (16) and B 1 sinh β 1 l + B 2 cosh β 1 l + B 3 sin β 2 l B 4 cos β 2 l = 0. (17) These homogeneous equations can be represented as [A( ω, S)]b = 0, where b = (B 1, B 2, B 3, B 4 ) T. The condition of non-triviality of a solution of b yields the characteristic equation det[a( ω, S)] = 0, which can be solved for ω as a function of the axial force S. The first two eigenfrequencies of the beam as a function of the axial force are obtained numerically and presented in Fig. 5. For small values of S, the 6

7 ω 30 ω ω 1 S = S c S Figure 5: Variation of the first two non-dimensional eigenfrequencies of a cantilever beam with the non-dimensional follower force S I [s] ω 2 (S < S c ) S = S c ω 1 (S < S c ) R[s] Figure 6: Loci of natural frequencies of a beam with a follower force eigenfrequencies ω are purely real implying a harmonically oscillating solution. As the axial force is increased, the two eigenfrequencies come close, and they coalesce at a critical value of axial force given by S c , i.e., F c 20.05EI/l 2. Beyond this value of the axial force, we obtain two pairs of complex conjugate roots of ω that are opposite in sign, i.e., s, s, s, and s. The locus of s as a function of the axial force S is depicted in Fig. 6. When a complex eigenvalue s has a positive real part, the solution 7

8 form w(x, t) = W (x)e st implies oscillations with exponentially increasing amplitude. This phenomenon is known as flutter instability. The critical value of the follower force before the onset of flutter is obtained when the two eigenfrequencies coalesce. 8