ARO 406 MDOF - trylam.comMULTIEDOFSYSTEMS$ Week7 Winter!2016!...

ARO 406: Advance Dynamics and Vibra7ons of Aerospace Systems

Winter 2016

Try Lam Department of Aerospace Engineering

California State Polytechnic University, Pomona

MULTI-‐DOF SYSTEMS Week 7

Winter 2016 ARO 406: Advance Dynamics of Aerospace Systems 2

Content

•  IntroducKons •  EquaKons of MoKon for MDOF •  Matrix Method •  Undamped Normal Modes •  Damping with MDOF •  Response of MDOF


Introduc7ons

•  MulK-‐Degree of Freedom (MDOF) system are systems where you have at least 2 DOF.

•  The method describe here is for simple MDOF system (i.e., no FEM and a few DOF)


Equa7ons of Mo7on (Current Method)

•  For the system on the right we can use a number of current methods we already steady to derive the EOM. –  Using Newton’s Law –  Using Energy Method –  Using SKffness Matrix –  Using Lagrange’s EquaKon


Equa7ons of Mo7on (Newton’s Law)

•  EquaKons of MoKon


Matrix Method

•  For the analysis of the MDOF responses it is essenKal to use Matrix methods

•  The basic ideas are –  Coordinate transformaKon of basic mass and sKffness elements to equaKons in the global coordinates (usually by some FE method).

–  TransformaKon of the global coordinates to modal coordinates (matrix method).


Mass Transforma7on to Global Coordinate

•  Assume some linear relaKon between the actual posiKon of the mass element and the global coordinate given by

where {r} is the vector displacement and {z} is the global displacement

•  If given a kineKc energy of the form


Mass Transforma7on to Global Coordinate

•  Since we have and

•  Then


S7ffness Transforma7on to Global Coord.

•  Now if we look at the potenKal energy we see that

•  Now assuming a linear relaKon of the form •  The potenKal energy can now be wri`en as


Example 6.1


Example 6.1

•  Since we have no external forces or dampers the EOM is of the form

•  The mass matrix is


Example 6.1

•  The transformaKon, [Xm] is found by geometry, and is the transformaKon matrix for small angles between the 5 displacements of the 5 ‘mass’ elements (m, I, m1, m2, m3) to the 2 global coordinates (z, θ).

•  The matrix is therefore a 5 x 2 matrix –  1st column is the transformaKon due to the displacement (rm, rI, rm1, rm2,

rm3) when z is increased form 0 to 1 unit (displaced 1 unit upward) –  2nd column is the transformaKon due to the displacement (rm, rI, rm1, rm2,

rm3) when θ is increased by 1 unit clockwise (1 radian)


r{ }= Xm[ ] z{ }

rm rI rm1 rm2 rm3{ }T= Xm[ ] z

θ

!"#

$#

%&#

'#

Example 6.1


•  Therefore

Example 6.1


•  Therefore for the mass matrix we have

Example 6.1


•  The sKffness matrix from is

•  To find [Xs], we must again look at the global displacement of 1 unit for z and θ. NoKng that sn = kn (displacement). Therefore:

K{ }= XS[ ]T k!" #$ XS[ ] = XS[ ]Tk1 00 k2

!

"%%

#

$&&XS[ ]

Example 6.1


•  The sKffness matrix is now

•  So finally we have:

Modal Coordinates

•  So far we have considered coordinates that are easily to recognize (i.e., along an axis from a mass). –  In general, for larger systems, we idenKfy displacements (translaKons and rotaKons) of global coordinates at nodes, and look at displacements only at the points of interests.

•  For MDOF system, however, it is possible to define modal coordinates –  Represents moKon of a system by coordinates –  A mode is a set of displacements for the enKre structure –  Changing a single modal coordinate can affect the displacement over all or part of the structure


Modal Coordinates

•  There are two types (classes) of modes –  Normal modes: special modes where the mass and sKffness matrices are diagonal

•  MoKon of each mode can be treated independently of other modes •  Each mode can then be treated as a SDOF

–  Assumed (or arbitrary) shapes. Combined with normal modes and the displacement funcKons (FEM) are assumed modes defined between nodes.


Global to Modal Coordinates

•  StarKng with the equaKons for the global coordinates

•  We want an alternate form in modal coordinates

•  Define a transformaKon

–  If the number of {q} << number of {z} then high frequency informaKon is lost. IF number {q] = number {z}, then no loss.



•  TransformaKon of the Mass Matrix –  From the KineKc Energy we have

where

•  TransformaKon of the SKffness Matrix –  From the PotenKal Energy

where Winter 2016 ARO 406: Advance Dynamics of Aerospace Systems 21


•  TransformaKon of the External Forces –  From {F} to {Q} using virtual work

•  Thus we have

where

–  These relaKonships apply to any transformaKon from global into modal coordinates, whether the modes are assumed or normal modes.

, , Winter 2016 ARO 406: Advance Dynamics of Aerospace Systems 22

Example 6.2


Example 6.2

•  Using the following transformaKon

–  Arbitrary decided that the 1st mode is [1, 1]T and 2nd mode is [0, 1]T

–  1st mode: 2 masses move 1 unit upward together •  Compresses k1 but not k2

–  2nd mode: m1 doesn’t move and m2 moves 1 unit upward •  Compresses k2 but not k1


Part (a)

Therefore, no coupling of the stiffness

Example 6.2

•  Using the transformaKon we get


Part (a)

Example 6.2

•  Form the force we have

plug this into the previous equaKon •  Local Kme history can be found from


Part (b)

Undamped Normal Modes

•  A special set of coordinates will diagonalize [M] and [K]. This happens if the columns of the transformaKon maKx X are the “eigenvectors” or the natural mode shape of the system.

•  Eigenvectors means the characterisKc moKon of the system, and the eigenvalues are the characterisKc frequencies of vibraKon.

•  Eigenvectors and eigenvalues can be real or complex: –  If no damping, they are both real –  Most cases, with damping, eigenvalues and eigenvectors are complex –  If proporKonal damping, eigenvalues are complex but vectors are real –  If damping is small, they are “almost” real, and works well for many

structures –  If gyroscopic coupling (i.e., turntable), get complex modes.


Finding Eigenvector and Eigenvalues (undamped system)

•  Start with the physical system (F=0)

•  Assume soluKon


Example 6.3


Recall:

Example 6.3


•  Omimng the applied force (homogenous equaKons)

(Eq. 6.3.1)

Example 6.3


•  Solve for the two eigenvalues

Find the eigenvector by substituting the eigenvalues back into Eq. 6.3.1, one at a time.

Example 6.3


•  Solve for the two eigenvectors

Example 6.3


•  Modes

0.56

1.00

1.00

0.28

MODE 1 MODE 2

Example 6.3


•  TransformaKon EquaKon: transformaKon matrix [X] is formed using the eigenvectors.

Example 6.3


•  EquaKon in normal coordinates

Both mass and stiffness matrices are diagonal (no cross coupling)

NOTE: magnitude of generalized modal mass and stiffness are arbitrary since they depend on how the eigenvectors are scaled.

Example 6.3


Example 6.3


Method 1: On Scaling

•  If the diagonal mass and sKffness modal coordinates are known the rescale by α.

Example 6.3



•  Then

Example 6.3



•  More directly

Therefore you get the same udpated eigenvectors

ARO 406 MDOF - trylam.comMULTIEDOFSYSTEMS$ Week7 Winter!2016!...

Documents

Transcript of ARO 406 MDOF - trylam.comMULTIEDOFSYSTEMS$ Week7 Winter!2016!...