9.1 UNDAMPED SINGLE-DEGREE-OF-FREEDOM (SDOF) SYSTEM

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Many aspects of vibration isolation may be understood by examination of

an SDOF system consisting of a mass and a linear spring, as shown in Fig.

9-1. For a more extensive treatment of mechanical vibrations, there are

several references available in the literature (Rao, 1986; Tongue, 1996;

Thomson and Dahleh, 1998). Let us denote the mechanical mass of the

system by M and the spring constant (force per unit displacement) by KS.

Using Newton’s second law of motion рFnet ј MaЮ, the equation of motion

for the system may be written in the following form:

M

d2y

dt2 ю KSy ј 0 (9-1)

The displacement of the mass from its equilibrium position is denoted by y

and the symbol t denotes time.

Let us define the following parameter:

!2

n ј

KS

M

(9-2)

This quantity is called the undamped natural frequency of the system. As will

be shown in this section, !n is the frequency at which the system will oscillate

after being disturbed from its static equilibrium position by an initial

displacement or an initial velocity.

Making the substitution from Eq. (9-2) into Eq. (9-1), we obtain the

following result:

d2y

dt2 ю !2

ny ј 0 (9-3)

Vibration Isolation for Noise Control 407

FIGURE 9-1 Undamped SDOF vibrating system.

Copyright © 2003 Marcel Dekker, Inc.

The general solution of Eq. (9-3) may be written in either complex notation

or in terms of trigonometric functions directly:

yрtЮ ј Ae j!nt ю Be_j!nt ј C1 cosр!ntЮ ю C2 sinр!ntЮ (9-4)

The constants of integration in Eq. (9-4) are determined by the initial

conditions for the system. For example, suppose we have the following

conditions at the initial time t ј 0:

(a) initial displacement: yр0Ю ј yo

(b) initial velocity:

vр0Ю ј

dyр0Ю

dt ј vo

If we make these substitutions into Eq. (9-4), the following results are

obtained:

C1 ј yo and C2 ј vo=!n (9-5)

A ј 1

2 Ѕ yo _ jрvo=!nЮ_ and B ј 1

2 Ѕ yo ю jрvo=!nЮ_ (9-6)

The motion of a free undamped SDOF system may be found by substituting

the expressions for the constants of integration into Eq. (9-4):

yрtЮ ј yo cosр!ntЮ ю рvo=!nЮ sinр!ntЮ (9-7)

yрtЮ ј 1

2 yoрe j!nt ю e_j!ntЮ _ 1

2 jрvo=!nЮрe j!nt _ e_j!ntЮ (9-8)

We note that the two expressions are identical because of the following

identities:

cosр!ntЮ ј 1

2 рe j!nt ю e_j!ntЮ and sinр!ntЮ ј _1

2 jрe j!nt _ e_j!ntЮ

We may write the expression for the motion of the undamped SDOF

system in an alternative form:

yрtЮ ј C cosр!nt ю     Ю (9-9)

Using the trigonometric identity for the cosine of the sum of two angles, Eq.

(9-9) may be written as follows:

yрtЮ ј CЅcos  cosр!ntЮ _ sin            sinр!ntЮ_ (9-10)

By comparing Eqs (9-10) and (9-7), we note that the following relations

exist:

C cos    ј yo and C sin  ј _vo=!n (9-11)

408 Chapter 9

Copyright © 2003 Marcel Dekker, Inc.

The values for the constants C and        may be obtained as follows:

C2рcos2           ю sin2             Ю ј C2 ј y2

o ю рvo=!nЮ2 (9-12)

tan        ј

sin       

cos        ј _

vo

!nyo

(9-13)

The final expression for the displacement of the system is as follows:

yрtЮ ј Ѕy2

o ю рvo=!nЮ2_1=2 cosр!nt ю          Ю (9-14)

We observe from Eq. (9-14) that the motion is sinusoidal or simple

harmonic with a frequency (in radians/second, for example) of !n. The

undamped natural frequency fn may also be expressed in Hz units:

fn ј

!n

2_ ј рKS=MЮ1=2

2_

(9-15)

The static deflection of the system (denoted by d) is the deflection of

the spring due to the weight of the attached mass:

Mg ј KSd or d ј Mg=KS (9-16)

The quantity g is the acceleration due to gravity. At standard conditions, g

ј 9:806m=s2 (32.174 ft/sec2 or 386.1 in/sec2). By combining Eqs (9-15) and

(9-16), we find the relationship between the static deflection and the

undamped natural frequency for the system:

fn ј рg=dЮ1=2

2_

(9-17)

In practice, the static deflection may be easily measured, and the undamped

natural frequency may then be determined from experimental measurements

of the static deflection.

Example 9-1. A machine has a mass of 50 kg (110.2 lbm). It is desired to

design the support system such that the undamped natural frequency is

5 Hz. Determine the required spring constant for the support system and

the corresponding value of the static deflection.

The undamped natural frequency is given by Eq. (9-15). The required

spring constant is found as follows:

KS ј Mр2_fnЮ2 ј р50ЮЅр2_Юр5Ю_2

KS ј 49,350N=m ј 49:35kN=m р281:8 lbm=inЮ

Vibration Isolation for Noise Control 409

Copyright © 2003 Marcel Dekker, Inc.

The static deflection is found from Eq. (9-16):

d ј

Mg

KS ј р50Юр9:806Ю

р49:35Юр103Ю ј 9:93 _ 10_3 m ј 9:93mm р0:391 inЮ

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