Mechanical Engineers Handbook Episode 8 potx

m 3.5.2 APPLICATION OF THE HOLZER METHOD TO A MECHANICAL MODEL WITH ROTORS AND HOOKE MODELS In this case, vectors of state attached to a shaft section are used.. and with ji t† ˆ jisin o

recurrence relations under the condition that a displacement or a force must

be zero

EXAMPLE 3.14 Using the Holzer method, determine the natural frequencies for the

mechan-ical model in Fig 3.18

SolutionWrite successive Eqs (3.41):

EXAMPLE 3.15 Using the Holzer method, determine the natural frequencies of the

mechan-ical model in Fig 3.19

SolutionWrite successive Eqs (3.41) to obtain

F3ˆ F2‡ m3o2q3:Replacing F2 as a function of q1and q3 as a function of q1 in the expressionfor F3, and given that F3 ˆ 0, we obtain the expression for the naturalfrequencies of the mechanical system m

3.5.2 APPLICATION OF THE HOLZER METHOD TO A MECHANICAL MODEL WITH ROTORS AND HOOKE MODELS

In this case, vectors of state attached to a shaft section are used One vector

of state is ‰zŠ ˆ ‰j; M ŠT, where j is the angle of rotation of the section and M

is the torsion moment (torque) in some section

Because in the zone of the rotor the diagram of moments is uous, Ml6ˆ Mr

discontin-i, where Ml is the torsion moment at left and Mr

i at right (Fig

3.20) Because the rotor is rigid, jl ˆ jr

i ˆ ji (the rotation at left is equal tothe rotation at right)

Vectors of state at left and right of section i are

‰zŠliˆ jMll

; and ‰zŠri ˆ jri

Mr i

:The equation of motion of rotor i is

and with ji…t† ˆ jisin…ot ‡ j†, Mlˆ Mlsin…ot ‡ j†, Mr

‰zŠri ˆ ‰AiŠ‰zŠli; …3:45†where

‰AiŠ ˆ ÿo12J 0

:For the zone between two successive rotors,

Ml ˆ Mr

iÿ1; jl

iÿ jr iÿ1ˆkMl

iÿ1ˆMiÿ1r

kiÿ1: …3:46†There results the following recurrence relation between vectors of state at theextremity of a shaft section between two successive rotors:

‰zŠli ˆ ‰BiŠ‰zŠriÿ1; …3:47†where

35:

In conclusion, the recurrence relations between vectors of state are

‰zŠri ˆ ‰AiŠ‰zŠli in a second of rigid rotors

‰zŠliˆ ‰BiŠ‰zŠriÿ1in a section between two successive rotors:

EXAMPLE 3.16 With the Holzer method, determine the natural frequencies for the

mechan-ical model from Fig 3.21

35; ‰B2Š ˆ 1

1

k2

0 1

24

35; ‰B1Š ˆ 1

1

k1

0 1

24

35

‰zŠl

3ˆ ‰B3Š‰A2Š‰B2Š‰A1Š‰B1Š‰zŠr

0;but

0

Ml 3

0 ˆ 0 one can obtain the equation for natural frequencies m

where mik are called the coef®cients of distribution.

The vectors of distribution can be introduced:

‰mŠ1ˆ

1

m21

mn1

266

377; ‰mŠ2 ˆ

1

m22

mn2

266

377; ; ‰mŠnˆ

1

m2n

mnn

266

377:

It easy to demonstrate that the vectors of distribution ‰mŠk verify the system

‰‰RŠ ÿ o2

k‰M ŠŠ‰mŠk ˆ ‰0Š: …3:50†The natural modes of vibration associated with the natural frequency are thecolumn vectors

37

2‰qŠTk‰RŠ‰qŠk:Replacing ‰qŠk with Eq (3.51), the following results are obtained:

T…k†ˆ 12‰mŠT

k‰M Š‰mŠkA2

1ko2

kcos2…okt ‡ jk† …3:54†and

V…k†ˆ1

2‰mŠT

k‰RŠ‰mŠkA2

1ksin2…okt ‡ jk†: …3:55†The system is undamped; therefore, Tmax…k† ˆ Vmax…k† , and there results therelation of calculus for the natural frequency,

The following is the methodology for working with the Rayleigh method:

j Adopt an expression for ‰mŠk and determine, using Eq (3.56), a ®rstvalue for ok

j With the natural frequency calculated in this way, introduce thetorsors of inertia (forces or torques) and determine the new displace-ment, namely a new expression for ‰mŠk, which is reintroduced in Eq.

(3.56) The Rayleigh method gives the minimum natural frequencysuperior to the real value, and maximum natural frequency inferior tothe real value

EXAMPLE 3.17 Using the Rayleigh method, determine one natural frequency for the

mechanical model from Fig 3.22, where k1ˆ k, k2ˆ 2k, k3ˆ k, m1ˆ2m, m2ˆ 3m, m3ˆ m

SolutionThe mathematical model is

35;

and the matrix of rigidity (stiffness) is

‰RŠ ˆ ÿkk11 k1ÿk‡ k12 ÿk02

0 ÿk2 k2‡ k3

24

35:

The vectors of distribution is ‰mŠk ˆ ‰1; m2k; m3kŠT

Replacing this in Eq (3.49), one ®nds

okˆ

‰1; m2k; m3kŠ ÿkk11 k1ÿk‡ k12 ÿk02

0 ÿk2 k2‡ k3

24

3

5 m12k

m3k

24

35

‰1; m2k; m3kŠ m01 m02 00

24

r

: m

3.5.4 ANALYSIS OF STABILITY OF VIBRANT SYSTEM

A vibrant system with more than one degree of freedom is a multivariableopen linear system with [F ] or ‰fŠ as inputs and [q] as output The matrix oftransfer for the open system is

‰H Š ˆ ‰C ŠT‰s‰I Š ÿ ‰AŠŠÿ1‰BŠ; …3:57†where matrices [A], [B], [C ] are given as functions of the inertia matrix [M],damping matrix [D], and stiffness matrix [R]

The transfer matrix becomes

‰H Š ˆ ‰‰I Š; ‰0ŠŠ
‰s‰I Š ÿ ‰AŠŠÿ1 ‰0Š ‰0Š

which becomes

P…s† ˆ det‰s2‰M Š ‡ s‰DŠ ‡ ‰RŠŠ ˆ 0: …3:61†The stability of vibrant systems with n degrees of freedom depends on theposition of the roots of the polynomial Eq (3.61) The stability criteria arealgebraic (Routh, Hurwitz), or grapho-analytical criteria (Cramer, Leonhard)

The use of polar diagrams (Nyquist) or Bode diagram requests a procedure toreduce the multivariable system to a monovariable system

EXAMPLE 3.18 Determine the conditions of stability of motion for the vibrant system

presented in Example 3.8

SolutionWith

375; ‰DŠ ˆ

c1‡ c2 ÿc2 0

ÿc2 c2‡ c3 ÿc3

0 ÿc3 c3‡ c4

26

375;

375;

the characteristic polynomial becomes

a1 ÿc2s ÿ k2 0

ÿc2s ÿ k2 a2 ÿc3s ÿ k3

0 ÿc3s ÿ k3 a3

...