自由振动的特征值问题

Solution of eigenvalue problemFree vibration equation can be expressed as0Mu Ku +=(1)Which represent a set of n simultaneous homogeneous differential equations of the type()()0,1,2,n nijjijjjjm u t k u t i n +==∑∑ (2)We are interested in a special type of solution of Eqs. (2), namely, that in which all the coordinates ()()1,2,,j u t j n =execute synchronous motion. Physically, thisimplies a motion in which all the coordinates have the same dependence, and the general configuration of the motion does not change, except for the amplitude, so that the ratio between any two coordinate ()j u t and ()i u t , i j ≠, remains constant during the motion. Mathematically, this type of motion can be represented by[1]()()j j u t q f t =(3)Where ()1,2,,j q n are constant amplitudes and ()f t is a function of time, thesame function for all the coordinate ()j u t . The interest lies in the case in which the coordinates ()j u t represent stable oscillation, which implies the ()f t must be bounded.Inserting Eq. (3) into Eq. (2) and recognizing that the function ()f t does not depend on the index j , we obtain()()110,1,2,,n nij j ij j j j f t m q f t k q i n ==+==∑∑ (4)Eq. (4) can be rewritten in the form()()11,1,2,,nij jj nij jj k qf t i n f t m q==-==∑∑ (5)Using the standard argument, we obtain that the left side of Eq. (5) does not dependent on the index i ,whereas the right side does not depend on time, so that two ratios must be equal to a constant, and in particular to the sameconstant. Assuming that ()f t is a real function, the constant must be a real number. Denoting the constant by λ, Eqs. (5) yield ()()0f t f t λ+=(6)And()0,1,2,,nijij j jkm q n n λ-==∑ (7)We consider a solution of Eq. (6) in the exponential form()st f t Ae =(8)Introducing solution (8) in Eq. (6) and dividing through by st Ae , we conclude that s must satisfy the equation20s λ+=(9)Which has two roots12s s = (10)If λ is a negative number, then 1s and 2s are real number, equal in magnitude but opposite in sign. In this case, Eq. (6) has two solutions, however, are inconsistent with stable bounded motion, so that the possibility that λis negative must be discarded and the one that λis positive must be retained. Letting 2λω=, where ωis real, Eq. (10) yields12s i s ω=± (11)So that the solution of Eq. (8) becomes()12i t i tf t Ae A e ωω=+ (12)Because ()f t is a real function, 2A must be conjugate of 1A . Then solution (12) can be expressed in the form()()cos f t C t ωφ=-(13)Where C is an arbitrary constant, ωthe frequency of harmonic motion and φits phase angle, all three quantities being the same for every coordinate ()j u t . To complete the solution of Eqs. (2), we must determine the amplitude i q . Its convenient to write Eqs.(7) in the matrix for2ω=Kq Mq(14)Eq. (14) represents the eigenvalue problem associated with matrixs M and K and it possesses nontrivial solutions if and only of the determinant of the coefficients vanishes. This can be expressed in the form()22det 0ωω⎡⎤∆=-=⎣⎦K M(15)Where ()2ω∆is called the characteristic determinant or charasteristic polynomial. Associated with every one of frequencies r ωthere is a certain nontrivial vectorr u whose component ir u are real number, where r u is a solution of theeigenvalue problem, and hence it satisfies2,1,2,,r r r r n ω==Kq Mq (16)The vector r q are known as characteristic vectors , or eigenvector . The eigenvector are also referred to as modal vector and represent physically the so-called natural modes . These vectors are unique only in the sense that the ratiobetween any two components ir q and jr q is constant. Because Eq. (14) is homogeneous, if r q is a solution of the equation, then r r αq is also a solution, where r αis an arbitrary constant. It follow that the shape of the natural modes is unique but the amplitude is not .If the magnitude of the eigenvector r q is assigned a certain value, then the eigenvector is rendered unique. The process of adjusting the magnitude of the natural modes to render them unique is called normalization, and the resulting eigenvectors are referred to as normal modes . A very convenient normalization scheme consist of setting1,1,2,,T r r r n ==q Mq(17)Which has the advantage that2,1,2,,T r r r r n ω==q Kq (18)Clearly, the normalization process is devoid of physical significance and shouldbe regarded as a mere convenience .In view of Eqs. (3) and(13), we conclude that Eq.(1) has the solutions ()()r r r t f t =u q (19)Where()()cos r r r r f t C t ωφ=-(20)In which r C and r φ are constants of integration representing amplitudes and phase angles, respectively. Hence, the free vibration problem admits special independent solutions in which the system vibrates in any one of the natural modes. These solutions are referred to as natural motions . Then, invoking the superposition principle, we can write the general solution of Eq. (1) as a linear combination of the natural motions, or ()()()()11n nr r r r r t t f t U t =====∑∑u u q f(21)Where []12n U u u u =is the modal matrix and ()()()()12Tn f t f t f t f t =⎡⎤⎣⎦is avector with components given by Eqs. (20). The constant r C and r φ entering into ()t f depend on the initial condition ()0q and ()0q , as well as on the normalizaton scheme used for ()1,2,,r q r n =.1. Meirovitch, L., Fundamentals of Vibrations . 2001: McGraw-Hill. 806.。