Tài liệu Handbook of Machine Foundations P2 pdf

@m=Gpp, 4,=0 In other words, when the values my and k, are such that lo is equal to the fre- Tạ quency m of exciting force acting on mass mm, then the amplitude of mass m, will be zero.

GENERAL THEORY 17 Eqs 2.34a and 2.34b may be written in the form of expressions for dynamic factors

#, and p, as follows with appropriate substitutions

and

where

P

đạt = Ee

_— 0m _

m V Km and

tìm

t V K,/m,

For the particular case when z= (considered in Section 2.3a), that is, n,=7,

(= say) the Eqs 2.34c and 2.34d may be further simplified as

and

1

(1-7?) (1+ a7?) —«

Fig 2.5 shows the variation of and gy (given by Eqs 2.34e and 2.34f) with for

the case when a=0.2

Two points worth noting from this figure are:

1, There are two values of y at which p, or py is oo The values of wm corresponding

to these infinite ordinates are the natural frequencies wp, and ong

2 When y=1, i.e @m=Gpp, 4,=0

In other words, when the values my and k, are such that lo is equal to the fre-

Tạ quency (m) of exciting force acting on mass mm, then the amplitude of mass m, will be zero

When Gn2=©m, while a4,=0, the amplitude of mass m, may be obtained (from Eq 2.34b) as

(2.352)

The amplitude of mass m, is thus equal to its static displacement (displacement of m,

under the static influence of P,)

Application: The above theoretical treatment will be useful in the application of an

undamped vibration neutralizer for a rigid block foundation as explained in Section 7.3c

18 HANDBOOK OF MACHINE FOUNDATIONS

1 ‡

I it

|

Fig 2.5: Response Curves for an Undamped Two-Degree Freedom System

for the Case when «=0.2 and J = & `

m 2.3.2 Damped Case

a Frec Vibrations

Consider the system shown in Fig 2.3b Viscous dampers with damping coefficients C, and C, are additionally introduced here It is difficult to precisely assess the values of C, and C, in practice and consequently they are not generally considered in practical designs

based on multiple degree freedom systems However, the following theoretical treatment

will be helpful in cases where the influence of damping cannot be neglected, and this data can be obtained from field measurements or otherwise The equations of motion for the

system shown in Fig 2.3b may be written as under:

m &+C,%4+ 4 Z¡ + Ky (21-22) + Cy (41%) = 0 (2.36a)

Both z, and z, are harmonic functions and can be represented by vectors Writing the vec-

tors as complex numbers and substituting

in Eqs 2.37a and 2.37b, and solving, the following governing equation is obtained for the natural frequencies of the system

( F(o2,) }?4-402,(G Gar(@2,—08) o/ TF at Gatna(@e— Om) (Ita) }?=0 (2.38)

GENERAL THEORY l9”

where,

F(a, )=of, a m(1-†-ø) (9 2 +o? „+# & Cefn ®4T+ & ato, a? x(1-E#) (2.39)

where @ ta, ®,, and @ are already defined in Eqs, 2.23a, 2.23b and 2.24 respectively; %,

and ¢, are damping ratios defined by

Corollary: When (=0, and €,=0, Eq 2.39 reduces to the form given by Eq 2.22

for the undamped case

b Forced Vibrations

Case 1: When the harmonic force P, sin wmf acts on mass mj

The equations of motion for the system’ may be written as

my 2+ Cy & + Kyat Ky (21-22) + G (4—-%&) = P, sin amt (2.41a)

Since the system moves at the frequency of the exciting force under steady-state conditions,

the solution may be assumed in the form:

Substituting these relations in Eqs 2.41a and 2.41b and solving, the following relations

are obtained for a and a,

my (e2) +2i@n[ em (6ã—0m) 4/T-+a+be One (2,—a2) (1a) fy”

d

where F(w2.) is given by Eq 2.39

Using the principles of complex algebra, the modulus of a, and a, may be written as

(ope Om)? +403 Why

aN (oa) Paar ea, (02,— vet eat 2g)0+aj t5

“ND {F(w2) 4-402, Sale ath eats Ges ee oF) (1+0) }? (2.44b) ‘

20 HANDBOOK OF MACHINE FOUNDATIONS

Particular Case: When %=0 (i.e., the damping in the lower system is neglected) and

¢,=6, the amplitude of mass m, subjected to a harmonic force P, sin Ont is given by

= mL TOF 1-4 o2 Cos on (Fe) (a R0] vl (2.45a)

= Po _ ni +4 mã Còng tân, :

where f (w2)is given by Eq 2.32

or in terms of basic parameters, substituting Cj=0 and C2=C

and

Expressed in non-dimensional form, Eqs 2.46a and 2.46b may be further written as

By [ [xn? —(nï—] (䧗1)]? + 2E nề (ni—1+ sua | (2.47a) and

where

1

`

For the case = = AL (considered in preceding case), y,=7, (=, say), Fig 2.6

shows the variation of u, with q for various damping values (€¢)

It is interesting to note from Fig 2:6 that irrespective of the degree of damping, all the response curves pass through two fixed points S, and S,, the abscissa of which may be obtained as roots of the following equation

GENERAL THEORY 21

ale 8

: [ 3.9 I i 7 N \ ] | | \

i Nị

——p.=m om

Qn ong Fig 2.6: Response of Mass m, for Various Damping Ratios (€)

where B = min;

Substituting, the abscissae of the fixed points S, and S, in Fig 2.6 are given by

2

Tables 2.2 and 2.3 give the values of ụạ and y, for various values of frequency ratio a

mass ratio « and damping ratio { for the particular case when y,=7, or the relation

— =—+ is satisfied

Ms my

Application: The theory explained in the above particular case is used in the design of

auxiliary mass-vibration dampers, which will be explained in Section 7.3c The data con-

tained in Tables 2.2 and 2.3 will be useful in the choice of appropriate parameters for the

design of auxiliary mass-vibration dampers for a rigid block foundation

Although the vibration analysis of a multiple-degree freedom system is relatively more

complicated and often necessitates the use of a digital computer, the theoretical approach

for the analysis of such a system for the undamped case is given in this section for the benefit

of interested readers Matrix notation* is used here for a concise presentation

*Readers not familiar with this notation may refer to standard books on matrix algebra or Section 28,

Vol 2 of Ref G 1.6.

KT

24 HANDBOOK OF MACHINE FOUNDATIONS

"24.1 Free Vibrations

Consider the system shown in Fig 2.7 ‘Under free vibration, there is no exciting force on any of the masses The system is said to possess n degrees of freedom leading to 2 : equations of motion which may be written in the form

My Zq-+ Ky (Z2—21) + Kg (Za—za) =0 wenn ener nate eee e nee eeeeeteees (2.51)

2 h Mn Zn + i (Zn—2Zn-1) = 0

Benak The equation system (2.51) may be written in matrix form as

where (M) is the diagonal mass matrix given by

mg Ù cv 0

[KX] is the stiffness matrix which, in its general form, is repre-

TKuÍla «Kin

KniẤng:- - Kan

Ky are the stiffness coefficients which can be evaluated for a given structural system,

For the chain-like system shown in Fig 2.7,

K,+ K, —Ky 0

—K, K,+Ñy —;.0

km nh hy HH Hinh vã Ky Substituting z1=a sin af, z=, sin wt and so on, in Eq 2.52 and simplifying,

* ay

ay

GENERAL THEORY 25 The algebraic problem represented by Eq 2.56 is called the ““matrix cigen value problem."

It is also called the “real eigen value problem” to distinguish it from the complex eigen

value problem obtained when the damping matrix is also considered in the equations of

motion (Eq 2.51) Eq 2.56 represents a set of homogeneous equations (right-hand side

equal to zero), the condition for obtaining a non-trivial solution being that the determinant

formed by the coefficients of the left-hand side of the equation system should vanish This

gives the relation in its general form as

Kay —Cmó3 Ấqy, Tân

Eq 2.58 on expansion gives n roots for w%, say œñŸ, ö$ cŸ such that öŸ<<oöŸ<à oŸ

The fundamental natural frequency is w, and w,, wy w, are the higher-order fre-

quencies of the multiple degree freedom system The terms «,, @, @n are also called

the “eigen values” of the system

Substituting cach value of w? at a time in the equation system, one can evaluate the rel-

ative values of a), ay ap It may be noted that the absolute values of a,, ay @, cannot

be obtained since the equations are homogeneous ‘There are numerous methods avail-

able for the solution of cigen value problems Standard computer programmes are also

available for solving the eigen value problem involving large matrices, as in the case

when the number of degrees of freedom is too large to be handled by manual calculation

If {V;} denotes the column vector with relative components đị, độ, đa Corresponding to

a value œr (tt eigen value) then {V;} is called the eigen vector (also called modal vector

or mode shape) corresponding to the eigen value wr

The following important relations, known as “orthogonality conditions of eigen vectors,”

will be useful:

where r and s are two distinct modes,

The superscript T denotes the transpose of the matrix contained in flower brackets

To obtain the displacement matrix {Z:} at any instant ¢ after the free motion is set in,

the appropriate initial conditions are to be applied

Let {2%} and {Z,} denote the initial displacement and velocity vectors at time t=0

The following expression for {Z;} may be derived in terms of the eigen values and eigen

vectors of the system

n

{Zi} = > See [ «zor COS @yt + ` {2p} sin os | (2.60)

r=l

lq 2.60 gives the displacements z¡, z; Zn at any tỉme ( It may be noted that

the matrix product {V,}r[Af]{V;} in the denominator is a scalar quantity

26_ HANDBOOK OF MACHINE EOUNDATIONS

A useful check on the calculation is provided by the following identity,

Ằ ƒVJ(Vj*r[m] —- mm

The right-hand side is the identity matrix, also called the “unit matrix”

2.4.2 Forced Vibrations

" Consider the system shown in Fig 2.7 with harmonic exciting forces P,-sin wnt, P, sin wnt

Py sin wmt acting on masses m,, m, m, respectively The amplitudes of exciting

| P,

| ¬

Pn |

The equation of motion of the system may be written in matrix form thus:

The steady-state solution of Eq 2.63 may be expressed in the form

where {a} is the unknown column vector of amplitudes ;

Substituting Eq 2.64 in Eq 2.63 and simplifying, ‘the following set of equations is

obtained:

{[4] —«? [41]} {a} = {F} (2.65)

where, the superscript ~1 denotes the inversion of the square matrix contained in the flower brackets of Eq 2.66

Nore: Since damping has not been considered in equation system 2.63, if wm is equal to one

of the natural frequencies of the system, the matrix {1 —w*{M]} becomes a singular matrix (value of its determinant becomes zero) and therefore cannot be inverted

Alternative solution: The natural frequencies w, (r==1, 2, 2) and natural modes

{Vr} are first determined as explained in the preceding section The amplitudes can be obtained from the following relation

la) (co2— 2) Fx IM] {Vr} | {F} (2.67)

GENERAL THEORY 27

2.5.1 Response of Single-Degree Freedom System

Consider the motion of a spring mass system (Fig 2.1) under the influence of a general

transient force F(z) shown in Fig 2.8 The variation of force with the time as shown

in the diagram may be considered to be made up of pulses of short duration At

Fig 2.8: A General Force-Time Relationship

The response Az of the system subjected to a pulse having a momentum A, may be written

as

where @n is the natural frequency of the system and + is the period upto which the system

has been at rest before the action of the pulse

given by

‡

0

fig 2.70 is called the ‘“Duhamel’s integral” or “‘convolution integral”

also be added to the right-hand side of Eq 2.70 to obtain the total displacement at any, time ft

‘Thus in general

(2.71)

Z=Asin woyt+ Boos wo wf 28

0

Particular case: Consider the response of a single- degree undamped system subjected

to a rectangular pulse shown in Fig 2.9 The load P, is suddenly applied and kept on the

eystem for a duration T,

28 HANDBOOK: OF MACHINE FOUNDATIONS

Ay

Po

Fig 2.9: A Rectangular Pulse -

Fig 2.10 shows the variation of dynamic factor p-(=2/2at, Where Ze is the static displacement,

P,/K) with the period ratio T/T, where Ty is the natural period of the system

Fig 2.10: Transient Response for a Single-Degree

System Duc to Rectangular Pulse

PERIOD RATIOZ- Tn

Application: The foregoing theoretical treatment will be useful for the dynamic analysis

of block foundations supporting impact causing machinery such as hammers, presses, etc

(See Example 3 in Section 4.5.7)

2.5.2 Response of Multiple-Degree Freedom System Response of a multiple-degree freedom system subjected to a transient force vector {F()} may be obtained as follows, Let the matrix of initial displacements and velocities

at the time =0 be denoted by the {Z}, and {2} If Vr is the eigen vector corresponding

to the eigen value (or circular natural frequency @,r,) then the column matrix [2]; contain- ing displacements of the system at any time # is given by the following general relation

{Zhi = 2 'W.r[MI{(VJ {z}ocos wrt + —— {Z}q sin ox |

+ > ar VAT LA {Va} au (t) sin wy (f—1) dt (2.72)

ˆ_ GENERAL THEORY 29

It may be noted that the first part of the right-hand side of Eq 2.72 denotes the displace-

ment under free vibration (Eq 2.60) and the second part is the response due to the transient

force (Eq 2.70.)

Eq 2.72 will be useful only if the integrals involving the forcing function can be evaluated

Numerical integration using a digital computer will be necessary if the force-time relation

is of a random nature For methods of numerical integration, the reader may refer to

*S.H Crandall, Engineering Analysis, McGraw-Hill, New York, 1956.