978-0073380292 Chapter 3 Part 4

Problem 3.50

Repeat Part (b) of Example 3.4 on p. 140, using optimization methods of calculus. Hint: Solve Eqs. (4) and

(5) in Example 3.4 for

FBD

as a function of

˛

. Then solve for the value of

˛

that makes

dFBD=d˛ D0

(this equation is difficult to solve analytically, and you may need to solve it graphically or by other

approximate means). This problem can also be effectively solved using computer mathematics programs

such as Mathematica or Maple.

Problem 3.51

A device for tensioning recording tape in a video cassette recorder is shown.

The tape wraps around small pins at

A

,

B

, and

C

. The pins at

A

and

C

are fixed,

and the pin at

B

is supported by a spring and can undergo vertical motion in the

frictionless slot. Friction between the tape and pins is negligible. The spring

has stiffness

kD0:5 N=mm

and is unstretched when

hD25 mm

. Neglecting

the size of the pins, determine the tension in the tape when

(a) hD18 mm.

(b) hD10 mm.

Problem 3.52

The brake linkage for a vehicle is actuated by pneumatic cylinder

AB

.

Cylinder

AB

, springs

CD

,

EF

, and

GH

, and the slotted tracks at

A

,

C

,

E

, and

G

are all parallel. If the slotted tracks are frictionless, and cylinder

AB

produces a tensile force of

12 kip

, determine the deﬂection

ı

and forces

in members AC ,AE, and AG. All springs are unstretched when ıD0.

Problem 3.53

The system shown may undergo vertical motion only. For the values of

mA

,

mB

,

k1

, and

k2

given below, determine the displacements ıAand ıB.

mAD100 kg, mBD80 kg, k1D200 N=mm, and k2D300 N=mm.

Problem 3.54

The system shown may undergo vertical motion only. For the values of

mA

,

mB

,

k1

, and

k2

given below, determine the displacements ıAand ıB.

mAD5kg, mBD4kg, k1D6N=mm, and k2D3N=mm.

Problem 3.55

In Fig. P3.55(a), four identical crates weighing

2000 lb

each are stacked one

on top of another, and in Fig. P3.55(b), a simple model for determining the

deformation of the stack of crates is shown. In this model, each spring has

the same stiffness

kD4500 lb=in:

and the forces are determined using the

following idealization. Half the weight of the top crate is applied to point

A

and

half to point

B

. Similarly, half the weight of the next crate is applied to point

B

and half to point C, and so on. Determine the deﬂections ıA,ıB,ıC, and ıD.

Problem 3.56

In Prob. 3.55, let the top two crates weigh

3000 lb

each, and the bottom two crates weigh

4000 lb

each, and

all springs have the same stiffness

kD5000 lb=in

. Use the scheme described in Prob. 3.55 to determine

the forces FA–FDand determine the deﬂections ıA,ıB,ıC, and ıD.

Problem 3.57

Collar

A

has negligible weight and slides without friction on the vertical bar

CD. Determine the vertical force Fthat will produce D30ıif

(a) Spring AB is unstretched when D0ı.

(b) Spring AB has an unstretched length of 2ft.

(c) Spring AB has an unstretched length of 4ft.

Problem 3.58

Point

A

is supported by springs

BC

and

DE

and cable segments

AB

and

AD

.

The springs and cables have negligible weight, and the springs have identical

stiffness

kD10 N=cm

and

20 cm

unstretched length. The structure has the

geometry shown when FD225 N.

(a) Determine the lengths of cables AB and AD.

(b) Determine the coordinates of point Awhen FD0.

334 Solutions Manual

Problem 3.59

Spring

AB

is supported by a frictionless roller at

B

so that it is always vertical.

If the spring is unstretched when

D0ı

, determine



and the forces in spring

AB

and bar

AC

when

F

has the value indicated. Hint: The force supported by

bar AC is zero when  < 90ı, and may be nonzero when D90ı.

(a) FD25 lb.

(b) FD50 lb.

Problem 3.60

A collar with a pulley slides on a frictionless vertical bar

GH

. A string

ABCD

is wrapped around the pulley, where portion

AB

of the string is horizontal. A

spring with

2lb=in:

stiffness is placed between the collar and point

H

. The

spring has

8in:

unstretched length and

6in:

final length. The collar and pulley

have a combined weight of

5lb

, and the weights of all other components are

negligible. Determine the value of

P

required for equilibrium, and the reaction

between the collar and bar GH .

Problem 3.61

A collar with a pulley slides on a frictionless vertical bar

GH

. A string

ABCD

is wrapped around the pulley, where portion

AB

of the string is horizontal. A

spring with

2lb=in:

stiffness is placed between the collar and point

H

. The

spring has

8in:

unstretched length and

6in:

final length. The collar and pulley

have a combined weight of

5lb

, and the weights of all other components are

negligible. Determine the value of

P

required for equilibrium, and the reaction

between the collar and bar GH .

Problem 3.62

The machine shown is used for compacting powder. Collar

C

slides on plunger

AB

and is driven by a motor (not shown in the sketch) so that it has the oscillatory vertical

motion

ıD.10 mm/.sin t/

where

t

is time in seconds. Plunger

AB

weighs

4

N

and is pressed against the powder by the spring whose end is driven by the motion of

collar

C

. The spring has stiffness

kD0:1 N=mm

and

100 mm

unstretched length.

The

50 mm

dimension shown is the spring length when the machine is started (i.e.,

ıD0

and the powder column is at its initial height

hD110 mm

); this dimension

changes with the oscillation of collar

C

and as the plunger

A

moves down. Assume

there is no friction in the system (other than friction between individual grains of

powder), and assume the motion of

C

is slow enough that there are no dynamic

effects. Determine the largest and smallest forces the plunger applies to the powder

over a full cycle of motion of Cif

(a) The powder column is at its initial height hD110 mm.

(b) The powder column is at its compacted height hD80 mm.

Problem 3.63

A fuel pump is driven by a motor-powered ﬂywheel. The pump behaves as a

spring with stiffness

2N=mm

that is unstretched when

˛D ˙60ı

. Neglect any

possible dynamic effects.

(a)

Determine the largest tensile and compressive forces spring

AB

expe-

riences during one revolution of the ﬂywheel, and state the positions

˛

where these occur.

(b)

Without further analysis, is it certain that the largest tensile and compres-

sive forces in crank arm

BC

occur at the same positions

˛

determined in

Part (a)? Explain.

Problem 3.64

The suspension for the landing gear of an aircraft is shown. The wheel is attached

to bar

BC

, which slides vertically without friction in housing

A

, which is fixed to

the frame of the aircraft. The spring is precompressed so that it does not undergo

additional deﬂection until the force supported by the landing gear is sufficiently large.

Further, if the force supported by the landing gear exceeds a limit, the suspension

“bottoms out” and deﬂects no more. Specify the spring stiffness and initial length if

hD18 in:for F500 lb and hD12 in:for F1500 lb.

Problem 3.65

A hanger is made of cord-reinforced rubber. It is used as a spring support

with limited travel for a wide variety of applications. If prototype samples

are available, an effective means to characterize its nonlinear stiffness is by

testing in a laboratory, where forces of known magnitude are applied and the

deﬂections that result are measured. Imagine this produces the load-deﬂection

data provided in the table of Fig. P3.65.

(a)

Determine the constants

a

,

b

, and

c

that will fit the general quadratic

equation FDaCbı Ccı2to the load-deﬂection data for this hanger.

(b) Plot the load-deﬂection relation determined in Part (a).

(c)

Speculate on the range of values for

F

for which the relation obtained in

Part (a) will be reasonably accurate.