Structural Dynamics

Lecture 5

Outline of Lecture 5

� Multi-Degree-of-Freedom Systems (cont.)

� Rigid Body Motions.

� Solution to Undamped Eigenvibration Problems.

� Modal Analysis.

� Orthogonality Conditions.

1

� Orthogonality Conditions.

� Modal Decoupling.

� Modal Coordinate Equations.

� Mechanical Energy in Modal Coordinates.

Structural Dynamics

Lecture 5

Equations of Motion:

� Multi-Degreee-of-Freedom Systems (cont.)

2

� , : Mass matrix.

� , : Stiffness matrix.

� , : Dissipative damping matrix.

Undamped eigenvibrations ( , ):

Structural Dynamics

Lecture 5

� Rigid Body Motions

3

If , then . indicates an unsupported structure with one or more independent rigid body motions.

Rigid body motions are related with the eigenvalue 0.

Structural Dynamics

Lecture 5

If the body has independent rigid body modes , all related with the eigenvalue (multiplicity ), then any rigid body motion can be written as a linear combination of these:

Since, , , it follows that . The characteristic polynomium has the form:

j = 1, . . . , nr

4

characteristic polynomium has the form:

Obviously, is an -multiple root of (4).

Structural Dynamics

Lecture 5

� Example 1 : Eigenvibrations of a flexible drive train

5

, : Torsional stiffness of rotor and generator shaft.

Structural Dynamics

Lecture 5

� : Degrees of freedom.� : Mass moment of inertia of rotor and generator rotor, [kgm2].� : Auxiliary degrees of freedom.

Kinematical constraint: No slip between the gear wheels:

6

Analytical dynamics:

Structural Dynamics

Lecture 5

7

Structural Dynamics

Lecture 5

Lagrange’s equations of motion:

Elimination of by the last equation:

8

Insertion of (12) into the first two equations of (11):

Structural Dynamics

Lecture 5

Matrix formulation of the equations of motion:

9

Characteristic equation:

existence of a rigid body motion.

Structural Dynamics

Lecture 5

10

Eigenmodes:

Structural Dynamics

Lecture 5

� :

(rigid mode, )

11

� :

(elastic mode, )

Structural Dynamics

Lecture 5

� Solution to Undamped Eigenvibration Problems

is a solution to (2) for an arbitrary phase angle , if fulfils the generalized eigenvalue problem. Hence, two linear

independent solutions can be obtained for and , i.e.

12

The general eigenvibration solution of (2) is a linear combination of the indicated linear independent solutions cf. Box 1:

Structural Dynamics

Lecture 5

The expansion coefficients and are determined from the initial values:

13

Structural Dynamics

Lecture 5

: Modal matrix (Dim : )n× n

14

Structural Dynamics

Lecture 5

Box 1 : Solution theory for homogeneous linear matrix differential equations

Complete solution:

where and are arbitrary constants, and and are linear

15

where is the dimension of , are arbitrary constants, and are linear independent solutions to (30).

are determined from the initial values.

Structural Dynamics

Lecture 5

� Example 2 : Undamped eigenvibrations of 2DOF system

16

Initial conditions:

Structural Dynamics

Lecture 5

From Eq. (86), Lecture 4:

17

Structural Dynamics

Lecture 5

From Eq. (29):

18

From Eq. (25):

Structural Dynamics

Lecture 5

� Modal Analysis

SDOF System:

Solution, cf. Lecture 3, Eq. (13):

19

Solution, cf. Lecture 3, Eq. (13):

� : Undamped angular eigenfrequency.

� : Damping ratio.

� : Damped angular eigenfrequency.

Structural Dynamics

Lecture 5

� : Impulse response function.

� : Frequency response function.

20

Modal analysis:

Reduction of Eq. (1) to independent SDOF equations of motion.

Requirements:

� Linear systems.

� Special properties must be fulfilled by the damping matrix .

Structural Dynamics

Lecture 5

Generalized eigenvalue problem:

rigid body modes : .elastic modes : .

The total motion of the system defined by Eq. (1) may be decomposed

21

The total motion of the system defined by Eq. (1) may be decomposed into a rigid body component and an elastic component :

form a vector base for the rigid body motion :

Structural Dynamics

Lecture 5

form a vector base for the elastic motion :

: Modal matrix for rigid body motions.

: Rigid body modal coordinates.

22

form a vector base for the elastic motion :

: Elastic modal coordinates.

: Modal matrix for elastic motions.

Structural Dynamics

Lecture 5

Combined coordinate transformation:

� : Modal coordinate vector.

� : Modal matrix.

23

� : Modal matrix.

is a non-singular coordinate transformation matrix relating the physical coordinates with the modal coordinates .

(46) is inserted into Eq. (1), and the resulting equation is pre-multiplied with :

Structural Dynamics

Lecture 5

Initial values of modal coordinates:

: Modal mass matrix.

: Modal damping matrix.

: Modal stiffness matrix.

: Modal load vector.

24

Initial values of modal coordinates:

(49), (51) specify the equations of motion in modal coordinates. Next, it will be shown that and are diagonal matrices, and that under certain conditions can be assumed to be diagonal, too. Only in this case is Eq. (49) of any use.

Structural Dynamics

Lecture 5

� Orthogonality Conditions

Theorem 1:

Let the eigenmodes and be associated with different eigenvalues and , i.e. . Then, the following orthogonality properties prevail:

25

where:

: Modal stiffness.

: Modal mass.

Structural Dynamics

Lecture 5

Proof:

The generalized eigenvalue problems for the jth and the kth mode read:

The 1st equation is pre-multiplied by . Use of and

26

The 1st equation is pre-multiplied by . Use of and provides:

Structural Dynamics

Lecture 5

The 2nd equation of Eq. (55) is pre-multiplied by :

Withdrawal of (57) from (56) gives:

27

Since , (58) can only be fulfilled, if (52) is valid. Then, the left-hand side of (57) also becomes 0 for . Hence, (53) follows immediately.

Structural Dynamics

Lecture 5

� Example 3 : 2DOF system

28

For the 2DOF system considered in Example 2 we have, cf. Eqs. (85), (86), Lecture 4:

Structural Dynamics

Lecture 5

The modal masses and stiffnesses become:

As seen and . Further, it follows that:

29

Obviously, and are not orthogonal by themselves. Actually,

Structural Dynamics

Lecture 5

Multiple eigenvalues:

The linear eigenmodes fulfill the GEVP with the same eigenvalue . This may occur for both rigid body modes

, and for elastic modes .

Any linear combination of is also an eigenmode associated with the eigenvalue :

30

associated with the eigenvalue :

Structural Dynamics

Lecture 5

Theorem 2:

Linear independent eigenmodes associated with the same eigenvalue can be determined as a linear combinations of arbitrary linear independent eigenmodes which are also all associated with , by a special procedure (Gram-Schmidt orthogonalization), so become mutually

-orthogonal (and hence -orthogonal).

,

31

-orthogonal (and hence -orthogonal).

From the theorem follows that, (52) and (53) may be made valid for all rigid body or elastic modes. Then, the modal mass and modal stiffness matrices become diagonal:

Structural Dynamics

Lecture 5

32

Structural Dynamics

Lecture 5

� Example 4 : 2 DOF system with two identical eigenvalues

33

Structural Dynamics

Lecture 5

34

Obviously, the eigenmodes fulfill (52), (53), even though . Hence,and in (66) correspond to and in Theorem 2.

Structural Dynamics

Lecture 5

� Modal decoupling

Modal analysis is only of importance, if it can be assumed that also the modal damping matrix becomes diagonal:

35

This may be presumed, if the damping is low and the eigenvalues are well-separated.

(67) can be questioned for wind turbine rotors under normal (unstalled) operation due to the substantial aerodynamic damping. It may also be quistioned for jacket offshore structures due to the relative high fluid damping.

Structural Dynamics

Lecture 5

� Modal Coordinate Equations

Modal load vector:

36

Then, the component form of (49) reads:

: Modal damping ratio.

: Modal load. Scalar product of and .

Structural Dynamics

Lecture 5

For rigid body modes ( ), (71) reduces to:

Initial values of modal coordinate vector is given by (51). Inversion ofat the calculation of these can be omitted by the use of the orthogonality

properties:

37

Inversion of is almost free (diagonal matrix). Similarly,

Structural Dynamics

Lecture 5

With (70), (73), (74) the modal coordinate can be determined by an uncoupled SDOF equation of motion similar to (1). Next, is determined from (46).

Comparison of (51) and (73) provides the following result for the inverse modal matrix:

38

modal matrix:

Structural Dynamics

Lecture 5

The solution of (70) with the related initial values becomes, cf. Lecture 3, Eqs. (9), (10), (13):

39

� : th damped angular eigenfrequency.

� : th impulse response function

Structural Dynamics

Lecture 5

� Mechanical Energy in Modal Coordinates

The mechanical energy of the system follows from Lecture 4, Eqs. (xx), (yy):

40

The mechanical energy is distributed on the various modes. If no damping and external loading are present, the mechanical energy in each mode is preserved. For nonlinear system this is not the case. Energy flow between the modes may occur.

: Mechanical energy in mode .

Structural Dynamics

Lecture 5

� Example 5 : Modal coordinate differential equations for a flexible drive train

41

Structural Dynamics

Lecture 5

Equation of motion, Eqs. (15), (16):

42

Eigenfrequencies and eigenmodes, Eqs. (17), (21), (23):

Structural Dynamics

Lecture 5

Modal matrix:

Modal equations of motion:

43

Structural Dynamics

Lecture 5

The component modal equations of motion become:

controls the rigid body mode. If , the rigid body

44

controls the rigid body mode. If , the rigid body mode starts to accelerate.

Structural Dynamics

Lecture 5

Summary of Lecture 5

� Rigid Body Motions.

� Up to 6 rigid body motions. Aeroplanes and ships: 6. Wind turbine rotors: 1.

� (zero angular eigenfrequency).

� Solution to Undamped Eigenvibration Problems.

45

� Solution to Undamped Eigenvibration Problems.

� linear independent solutions:

, where

� Complete solution to the homogeneous undamped matrix differential equation:

and are determined from the initial values

and .

Structural Dynamics

Lecture 5

� Modal Analysis.

� Coordinate transformation . :

Modal coordinate vector. Coordinate transformation matrix : Modal matrix.

� Under certain conditions to be fulfilled by the damping matrix,

may be obtained as the solution to independent SDOF systems.

� Orthogonality Conditions.

46

�

�

Inversion of modal matrix: .

� Modal Decoupling. Lightly damped structures with well separated eigenfrequencies.

� Mechanical Energy. No flow of energy between modes during undamped eigenvibrations.