Solutions or Homewor #8 PROBEM. (P. 9 on page 78 in the note) An airplane is modeled as a beam with masses as shown below: m m m m π [rad/sec] anding gear m m.5 Find the stiness and mass matrices. Find all oscillator modes and requencies. I the landing gear has stiness such as it delects.m under it s own weight (when pared), determine the requencies and mode shapes o the plane ater landing. Explain how you would determine the maximum orce in the landing gear at the time o landing, assuming that the vertical descent rate at time o landing is m/sec. SOUION he system analyzed has rigid body motions (namely one translation and one rotation). By inspection, the modal shape or the translational rigid body motion is: { } φ Similarly, the modal shape or the rotational rigid body motion is: { } φ In order to determine the stiness matrix o the structure, we proceed as ollows: We calculate the stiness coeicients, and by calculating the orce needed to prevent translation at the st and rd degree o reedom, when the nd degree o reedom has a unit deormation, namely: u.
hereore, the stiness coeicients are calculated as ollows: 48 ( /) 4.5 ( /) 4.5 ( /) {} Due to symmetry o the stiness matrix, we also have: he stiness matrix is now: and {} K.5.5.5 where:.5 o calculate the remaining coeicients, we recall that the rigid body motions store no energy in the system, i.e.: K.5.5.5.5 +.5 K.5.5.5.5 {} hereore, rom the system o equations {}, we have:.5 he stiness and mass matrices o the structure are: K.5.5.5.5.5.5.5.5 M m m m he natural requencies associated with the rigid body motions are: o calculate the third modal shape, we recall that the modal shapes are orthogonal with respect to the mass and the stiness matrix, namely:
m + + m φ M { } m φ m φ m φ m m M { } m φ m φm m φ {4} hereore, the coeicients φ, φ are evaluated by solving the system o equations in {4} as ollows: m φ.5 {5} m φ o calculate the requency, we use the Rayleigh quotient, which leads to the exact value o the requency, when the exact modal shape is used: { } K.5.5.5.5 48.75 6 m M m m m { } m m.5.5.5.5.5.5.5.5 m.5 6 6 π rad /sec m When pared, the orce applied at the landing gear is the weight o the plane, namely: W m g + m g + m g {6} hereore, the stiness o the landing gear, g, is calculated as ollows: W m + m 6mg {7} g g 6 m ust.. he system is now modeled (when pared) as in the igure below, and there is only one rotational rigid body motion:
m m m anding gear he mass matrix o the structure remains the same, and the stiness matrix is now evaluated as ollows: 48 + + + 6m ( /) g g 4.5 ( /) 4.5 ( /) {8} he rotational rigid body motion stores no energy in the system, and the translational motion, where all the masses have the same displacement, stores energy only due to the presence o the lexible support, namely: K.5.5 + g.5 g.5 +.5 K.5.5 + g.5.5 {9} hereore, we have the stiness matrix and the mass matrix or the pared case:.5.5.5 m g K.5 +.5, M m {,}.5.5.5 m here are three requencies and mode shapes or this case. Again, by inspection, we can determine the irst requency and mode shape o the rotational rigid body motion as and { }, respectively. Next, we determine the second and third requencies, and mode shapes, by maing use o the orthogonality conditions in connection with both the stiness and mass matrices. o begin with, we assume the two mode shapes o the orm. { a }, { b } {,} hen, consider the ollowing two conditions. m M { a } m b m+ abm {4} m
.5.5.5 g K { a }.5 +.5 b [ b+ ( + / g ) ab a].5.5.5 From equations {4} and {5}, we obtain two constants a and b as ollows. A+ A 4B A A 4B a, b {6,7} m where A +, and B m. hen, we can calculate the two requencies, by g m m using Rayleigh quotient as ollows. K, K M M {5} {8,9} he orce at the landing gear is given by: F g g u. hereore, we need to determine the displacement time history o u, and calculate the moment when the displacement is maximized. For the response u o the nd degree o reedom, we use modal superposition, namely: ( ) φ ( ) ( ) + ( ) + ( ) u t q t q t a q t b q t j j j For initial velocity o the system, u { } ollows: β Mu Mu β M M {}, we evaluate the modal components β and β as {,} and the response o the system would be: u( t) β β a sin( t) b sin ( t) + {}
PROBEM. (P. on page 78 in the note) Consider a -story rame with rigid girders (beams) and mass lumped at the level o each loor, which is supported on a sloping ground. It is nown that m m 5 g m / m, 5 N / m m / he deormed coniguration and the equivalent -DOF model are m m Formulate the equations o motion or the system. Estimate the undamental requency, using Rayleigh s quotient. Find the exact modal shapes and requencies What is the proportional damping matrix that gives uniorm modal damping ξ ξ.5? his is a -degree o reedom system, with lateral displacements at the level o the loors, u and u. he equation o motion or this system is ormulated as ollows: 4 4 m u u m + 4 6 u + u ( /) u 4 4 u m + u 4 76.5 u {} u 4 4 u + u m 4 76.5 u
In order to estimate the undamental requency o the system, we use the Rayleigh quotient, estimating that the undamental shape o the structure will be a straight line, i.e.: v { } 4 4 4 4 { } { } v K v 4 76.5 4 76.5.5 6.77 [ rad /sec ] v M v { } m { } he eigenvalue problem is ormulated as ollows: ( ) For the solution o {} to be non-trivial, we demand: K M {} i K M 4 4 m 4 76.5 4 4 m m 4 76.5 m m {} 5 For.5 m 5, equation {} is now evaluated as ollows: λ 8.5 λ where λ. {4} he solution o the quadratic equation in {4}, has the ollowing solutions: [ rad ] [ rad ] 6.6 /sec 4.765 /sec o deine the modal shapes, we set φ n., and solve or φ n, using the eigenvalues calculated above. We thereore have: n. 8.5 φ n n {5} From the irst equation in {5}, we evaluate φ n as ollows: φ n n n {6} Substituting in {6} n,, we evaluate: φ.47 m, φ -.6 m
he Rayleigh damping matrix or the -degree o reedom system, is ormulated as ollows: C α M + α K We now rom the class notes that the coeicients α and α are evaluated as ollows: ξ α ξ α {7} Setting in equation {6} ξ ξ.5, we evaluate the coeicients as ollows: α 6.6.5 6.6.56 α 4.765.5 4.765. {8} hereore, the damping matrix is now: 4 4 C α M + α K.56 m. + 4 76.5.7 9.78 C 9.78 5.764 {9}
PROBEM. (P. on page 79 in the note) Consider a set o coupled penduli, as shown in the igure below (gravity is on ). θ θ m m Find the equations o motion. Specialize the result o (a) or small angles θ and θ. Find the requencies and model shapes o the system (in terms o / m. SOUION o determine the equations o motion or the system above, we irst draw the ree body diagram (centripidal orces are not shown): θ θ m θ (sin θ - sin θ ) ) m g θ θ θ m θ θ m g We ormulate the equations o motion, by taing moment equilibrium around the pivot o each pendulum: ( ) mθ + mgsinθ sinθ sinθ cosθ ( ) mθ + mgsinθ + sinθ sinθ cosθ {} We linearize equation {}, assuming small displacements as ollows: Small displacement assumption: sinθ θ, cosθ.
( ) mθ + mg θ θ θ ( ) mθ + mgθ + θ θ {} Equation {} is written in matrix orm as ollows: mg + m θ θ m + θ mg θ + {} he eigenvalue problem is ormulated as ollows: ( ) For the solution o {4} to be non-trivial, we demand: K M {4} i K M mg + m mg m + {5} We deine s / m, and equation {5} is transormed as ollows: g + s g s + s 4 g g g s+ 4 + + s {6} he solutions o equation {6} (only positive solutions have physical meaning) are: g p g +.5 p +.5 m m {7} where p is the natural requency o the simple pendulum. o deine the modal shapes, we set φ n., and solve or φ n, using the eigenvalues calculated above. We thereore have:
g + s s. g φn + s s {8} From the irst equation in {8}, we evaluate φ n as ollows: g φ n + {9} s s Substituting in {9} n,, we evaluate: φ φ.5