978-0073380308 Chapter 9 Solution Manual Part 8

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Dynamics 2e 1991

Problem 9.45

For the system in Prob. 9.44, deter-

mine the maximum forcing frequency

!0

for steady-state motion, such that

the cord does not go slack.

2FsLmBgL

2DIO˛B;(2)

where

T

is the cord tension,

˛B

is the angular acceleration of the bar,

Fs

is the force in the spring, and

IOD1

3mBL2is the mass moment of inertia of the bar Babout point O.

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Canceling an Land noting that 1

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Dynamics 2e 1993

Problem 9.46

One of the propellers on the Beech King Air 200 is unbalanced, such that the unbalanced weight

Wu

is a distance

R

from the spin axis of the propeller. The propellers spin at a constant rate

!p

, and the

weight of each engine is

We

(this includes the mass of the propeller). Assume that the wing is a uniform

beam that is cantilevered at

A

, has weight

Ww

and bending stiffness

EI

, and whose mass center is at

G

. Treat the engine as a point mass and evaluate your answers for

WuD3oz

,

WeD450 lb

,

RD5:1 ft

,

!pD2000 rpm

,

EI D1:13 1011 lbin:2

,

dD8:7 ft

, and

hD10:9 ft

. In addition, ignore the angular

motion of the wing in computing the angular velocity and angular acceleration of the propeller, and ignore

the time-dependent inertia term in the final equation of motion.

Neglect the mass of the wing, and model the wing as was done in Example 9.2. Determine the

resonance frequency of the system, and find the MF for the given parameters.

gd2

have treated the engine as a point mass.

Force Laws. The force law for the torsional stiffness is given by

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permission of McGraw-Hill, is prohibited.

Kinematic Equations. Relating the acceleration of the unbalanced mass to that of the engine, we obtain

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To find the

MF

, we need to find a particular solution

p

to the equation of motion in Eq. (4), and then use

its amplitude to determine the

MF

. The most direct way to find a particular solution is to assume a solution

of the form pDDsin !ptand then substitute it into the equation of motion. Doing so, we obtain

where we have used the expression for !nfound above and !pD2000 rpm D2000 

30 rad=s.

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permission of McGraw-Hill, is prohibited.

Problem 9.47

One of the propellers on the Beech King Air 200 is unbalanced, such that the unbalanced weight

Wu

is a distance

R

from the spin axis of the propeller. The propellers spin at a constant rate

!p

, and the

weight of each engine is

We

(this includes the mass of the propeller). Assume that the wing is a uniform

beam that is cantilevered at

A

, has weight

Ww

and bending stiffness

EI

, and whose mass center is at

G

. Treat the engine as a point mass and evaluate your answers for

WuD3oz

,

WeD450 lb

,

RD5:1 ft

,

!pD2000 rpm

,

EI D1:13 1011 lbin:2

,

dD8:7 ft

, and

hD10:9 ft

. In addition, ignore the angular

motion of the wing in computing the angular velocity and angular acceleration of the propeller, and ignore

the time-dependent inertia term in the final equation of motion.

Let the mass of the wing be

mwD350 lb=g

, and model the wing as was done in Example 9.2.

Determine the resonance frequency of the system, and find the MF for the given parameters.

gd2C1

12

g.2h/2CWw

gh2DWe

gd2C4

3

gh2;

where we have assumed that the wing is a uniform beam of length 2h.

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permission of McGraw-Hill, is prohibited.

Dynamics 2e 1997

Force Laws. The force law for the torsional stiffness is given by

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and therefore, the inertia term is dominated by the constant part, which means that we can approximate the

gsin !ptC3EI

dDsin !ptD Wu

gdR!2

which, upon canceling sin !ptand solving for D, gives

p

p=.3gEI /

Therefore,

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Dynamics 2e 1999

Problem 9.48

In the design of a MacPherson strut suspension, what would you choose for

the damping ratio



? Explain your answer in terms of automotive ride and

comfort.

shock

absorber

and spring

steering

link

car frame

lower

control arm

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