Vibration Tutorial.ppt

1、Elementary Tutorial,Prepared byDr. An Tran in collaboration withProfessor P. R. Heyliger Department of Civil Engineering Colorado State University Fort Collins, Colorado June 2003,Fundamentals of Linear Vibrations,Developed as part of the Research Experiences of Undergraduates Program on “Studies of

2、 Vibration and Sound” , sponsored by National Science Foundation and Army Research Office (Award # EEC-0241979). This support is gratefully acknowledged.,Fundamentals of Linear Vibrations,Single Degree-of-Freedom Systems Two Degree-of-Freedom Systems Multi-DOF Systems Continuous Systems,Single Degre

3、e-of-Freedom Systems,A spring-mass system General solution for any simple oscillator General approach Examples Equivalent springs Spring in series and in parallel Examples Energy Methods Strain energy C = damping matrix; K = stiffness matrix;P = force vector Note: Matrices have positive diagonals an

4、d are symmetric.,Undamped free vibrations,Zero damping matrix C and force vector P,Assumed general solutions:,Characteristic polynomial (for det =0):,Eigenvalues (characteristic values):,Characteristic equation:,Undamped free vibrations,Special case when k1=k2=k and m1=m2=m,Eigenvalues and frequenci

5、es:,Two mode shapes (relative participation of each mass in the motion):,The two eigenvectors are orthogonal:,Eigenvector (1) =,Eigenvector (2) =,Undamped free vibrations (UFV),For any set of initial conditions: We know A(1) and A(2), 1 and 2 Must find C1, C2, 1, and 2 Need 4 I.C.s,Single-DOF:,For t

6、wo-DOF:,UFV Example 1,Given:,No phase angle since initial velocity is 0:,From the initial displacement:,UFV Example 2,Now both modes are involved:,Solve for C1 and C2:,From the given initial displacement:,Hence,or,Note: More contribution from mode 1,Transformation of coordinates,Introduce a new pair

7、 of coordinates that represents spring stretch:,UFV model problem:,“inertially uncoupled”,“elastically coupled”,z1(t) = x1(t) = stretch of spring 1 z2(t) = x2(t) - x1(t) = stretch of spring 2 orx1(t) = z1(t) x2(t) = z1(t) + z2(t),Substituting maintains symmetry:,“inertially coupled”,“elastically unc

8、oupled”,Transformation of coordinates,We have found that we can select coordinates so that: Inertially coupled, elastically uncoupled, or Inertially uncoupled, elastically coupled. Big question: Can we select coordinates so that both are uncoupled?,Notes in natural coordinates:,The eigenvectors are

9、orthogonal w.r.t M: The modal vectors are orthogonal w.r.t K: Algebraic eigenvalue problem:,Transformation of coordinates,Governing equation:,Modal equations:,Solve for these using initial conditions then substitute into (*).,General approach for solution,We were calling “A” - Change to u to match M

10、eirovitch,Substitution:,Let,or,Known solutions,Transformation - Example,2)Transformation:,1)Solve eigenvalue problem:,So,As we had before. More general procedure: “Modal analysis” do a bit later.,Model problem with:,Response to harmonic forces,Model equation:,M, C, and K are full but symmetric.,F no

11、t function of time,Assume:,Substituting gives:,Hence:,Special case: Undamped system,Zero damping matrix C,Entries of impedance matrix Z:,For our model problem (k1=k2=k and m1=m2=m), let F2 =0:,Notes: 1) Denominator originally (-)(-) = (+). As it passes through w1, changes sign. 2) The plots give bot

12、h amplitude and phase angle (either 0o or 180o),Substituting for X1 and X2:,Multi-DOF Systems,Model Equation Notes on matrices Undamped free vibration: the eigenvalue problem Normalization of modal matrix U General solution procedure Initial conditions Applied harmonic force,Multi-DOF model equation

13、,Model equation:,Notes on matrices:,They are square and symmetric. M is positive definite (since T is always positive) K is positive semi-definite: all positive eigenvalues, except for some potentially 0-eigenvalues which occur during a rigid-body motion. If restrained/tied down positive-definite. A

14、ll positive.,Vector mechanics (Newton or D Alembert) Hamiltons principles Lagranges equations,We derive using:,Multi-DOF systems are so similar to two-DOF.,UFV: the eigenvalue problem,Matrix eigenvalue problem,Equation of motion:,Substitution of,in terms of the generalized D.O.F. qi,leads to,For mor

15、e than 2x2, we usually solve using computational techniques. Total motion for any problem is a linear combination of the natural modes contained in u (i.e. the eigenvectors).,Normalization of modal matrix U,Do this a row at a time to form U.,This is a common technique for us to use after we have sol

16、ved the eigenvalue problem.,We know that:,So far, we pick our eigenvectors to look like:,Instead, let us try to pick so that:,Then:,and, Let the 1st entry be 1,General solution procedure,For all 3 problems: Form Ku = w2 Mu(nxn system) Solve for all w2 and u U. Normalize the eigenvectors w.r.t. mass

17、matrix (optional).,Consider the cases of: Initial excitation Harmonic applied force Arbitrary applied force,Initial conditions,2n constants that we need to determine by 2n conditions,General solution for any D.O.F.:,Alternative: modal analysis,Displacement vectors:,UFV model equation:,n modal equati

18、ons:, Need initial conditions on h, not q.,Initial conditions - Modal analysis,Using displacement vectors:,As a result, initial conditions:,Since the solution of,And then solve,hence we can easily solve for,is:,Applied harmonic force,Driving force Q = Qocos(wt),Equation of motion:,Substitution of,le

19、ads to,Hence,then,Continuous Systems,The axial bar Displacement field Energy approach Equation of motion Examples General solution - Free vibration Initial conditions Applied force Motion of the base Ritz method Free vibration Approximate solution One-term Ritz approximation Two-term Ritz approximat

20、ion,The axial bar,Main objectives: Use Hamiltons Principle to derive the equations of motion. Use HP to construct variational methods of solution.,A = cross-sectional area = uniform E = modulus of elasticity (MOE) u = axial displacement r = mass per volume,Energy approach,For the axial bar:,Hamilton

21、s principle:,Axial bar - Equation of motion,Hamiltons principle leads to:,If area A = constant ,Since x and t are independent, must have both sides equal to a constant.,Separation of variables:,Hence,Fixed-free bar General solution,= wave speed,For any time dependent problem:,Free vibration:,EBC:,NB

22、C:,General solution:,EBC ,NBC ,Fixed-free bar Free vibration,are the eigenfunctions,For free vibration:,General solution:,Hence,are the frequencies (eigenvalues),Fixed-free bar Initial conditions,or,Give entire bar an initial stretch. Release and compute u(x, t).,Initial conditions:,Initial velocity

23、:,Initial displacement:,Hence,Fixed-free bar Applied force,or,Now, B.Cs:,From,B.C. at x = 0:,B.C. at x = L:,Hence,we assume:,Substituting:,Fixed-free bar Motion of the base,Using our approach from before:,Resonance at:,Hence,From,B.C. at x = 0:,B.C. at x = L:,Ritz method Free vibration,Start with Hamiltons principle after I.B.P. in time:,Seek an approximate solution to u(x, t): In time: harmonic function cos(wt) (w = wn) In space: X(x) = a1f1(x) where:a1 = constant to be determin