Zadatci Iz Satatike

47
In the given four-bar linkage find the linear velocity of point C in link 3 and the angular velocity of link 3 when link 2 rotates at a constant angular velocity. The linear velocity of point A is given to scale on the mechanism. V c = __________in/sec.; ù 3 = __________rad/sec. Lay off the velocity vector, V c , on the mechanism. 4 0 3 C A K = 4 in/in S 4 2 0 V A 2 V 0 K = 10 in/sec-in V

description

statika

Transcript of Zadatci Iz Satatike

In the given four-bar linkage find the linear velocity of point C in link 3 and the angular velocity of link 3 when link 2 rotates at a constant angular velocity. The linear velocity of point A is given to scale on the mechanism. Vc = __________in/sec.; ù3 = __________rad/sec. Lay off the velocity vector, Vc , on the mechanism.

4

0

3C

A

K = 4 in/inS

4

20

VA

2

V0

K = 10 in/sec-inV

The wheel in the figure rolls without slipping on the horizontal surface; using the velocity of point A as shown, determine: a) the velocity of point B; VB = __________ft/sec. b) the velocity of point D; VD = __________ft/sec. c) the angular velocity of link 3: ù3 = ________ rad/sec.________ .

C

4

3

2

AB

K = 2 ft/inS

K = 25 ft/sec-in

0V

V

a

The velocity diagram for the mechanism is given below. (Link 2 rotates at a constant angular velocity.) Point G is the center of mass of link 3.

a) Determine AG and á3.

b) From point G on the mechanism, lay off to scale the acceleration of point G as a vector.

c) Find the point in link 3 that has zero acceleration and show on the mechanism.

3 AB

G

42K = 1/3 ft/inS

b

K = 7 fps/inV

a

0V

a0

K = 147 fps /ina2

Problems in Biaxial Stresses

In each of the following biaxial stress situations determine whether or not failure should occur. In these problems a brittle material will be defined as one having a percentage elongation in 2” of 5% or less. Materials having an elongation of greater than 5% will be assumed to be ductile. Properties of materials are listed in Chapter 14 in Spotts. If two values are given for one property use an average value. Gray cast irons are brittle materials.

If a material is ductile as defined above, apply both the maximum shear stress theory of failure and the distortion energy theory failure. If a material is brittle, apply the maximum normal theory of failure.

SAE

1020CR steel

42,000

1.

5,000

SAE

5,000

CR steel1020

2.

42,000

5,000

3.

1020CR steel

SAE42,000

4. (a) For a ductile material does an added tensile stress at 90° weaken or strengthen the element?

(b) For a ductile material does an added compressive stress at 90° strengthen or weaken the element?

(c) What is the significance of applying a failure theory to the element of #3?

6.

5,000

21,000

ASTM 40

5,000

irongray cast

5.

21,000

Average

irongray cast

Average7.

gray castiron

21,000

8. (a) For a brittle material does an added tensile stress at 90° weaken or strengthen the element?

(b) For a brittle material does an added compressive stress at 90° weaken or strengthen the element?

(c) What is the significance of applying a failure theory to the element of #7?

11.

SAE

103545,000

9.

5,000

10.

5,000steelcast

medium

30,000

40,00060,000

AluminumBronze(drawn) 20,000

12.

SteelStainless

18-850,000

13. Would the maximum normal stress theory give the same result in #10 as the Maximum shear theory?

5,000

14.

irongray cast

Aluminum16,000

15.SAE

1025 10,00050,000

2330

90,000

16.SAE

20,000mediumhard

30,000

The steel part in the figure is subjected to a static bending moment of 20,000 in- lbf. Using a factor of safety of 3.5 determine the minimum value of D for safe operation. The yield strength of the material is 30,000 psi.

7"

1" THICK

D

h

r = 1/2"

20,000 in-lb f20,000 in-lb f

A steel member 12 feet long is subjected to an axial tensile load of 10,000 lb, and an operating temperature of 400°C. The creep constants in the relation ε = Bt (S/S0)n for the material at 400°C are B = 4.52 x 10-16 in/in/day, and n = 6.80. Find the cross-sectional area required based on

a) a yield stress of 25 ksi at 400°C and nFs = 2.0;

b) an allowable creep of 0.020 in/in. in 20 years;

c) a rupture stress of 20,000 at 400°C in 20 years using nFs = 2.5.

A turbine rotor is being designed for a commercial aircraft engine and the material for the turbine blades has to be determined. Each blade on this rotor is 6 inches long and has a cross-sectional area of 0.5 square inches. The most critical load on the blades will be the centrifugal tension, and the maximum stress along the blade is 10,000 psi. To inhibit leakage, the actual amount of radial clearance when the blades are newly installed is to be 0.030 inches, but to prevent catastrophic failures, the blades will be replaced after they have elongated only 0.010 inches. If the engine is to run 1000 hours on one set of blades, select the proper material from the creep data in the table below. Assume that when the engine stops and starts again, the creep curve continues where it left off. The creep relation ε = Bt (S/S0)n is valid.

Creep Constants for Possible Materials ________________________________________________________________________

Material B, in/in-day n S0, psi A 1.0 x 10-12 7 1000

B 2.0 x 10-12 7 1000

C 3.5 x 10-12 7 1000

D 5.0 x 10-12 7 1000

E 6.0 x 10-12 7 1000

Brand X 7.5 x 10-12 7 1000

In the figure is shown a rotating shaft machined from bar stock and with load P varying from 1000 to 3000 lb. The material is 1020 cold drawn steel. The factor of safety is equal to 2.0. Find the largest value of D for continuous operation.

24" 15" 15" 24"

3" D

P R = 0.45"

The shaft S in the figure is rotating 1200 rpm with the unbalanced weight W = 10 lbf. The center of mass of W is 2 inches from the centerline of the shaft. The static force P = 2000 lbf is applied to the shaft through the yoke and the anti- friction bearings B and C. The shaft material had the following properties: in uniaxial tension: Su = 75,000 psi, Sy = 60,000 psi, and Se = 40,000 psi. For a factor of safety of 2.0, determine the shaft diameter of the basis of an analysis at the section A-A.

10"

W2

10" 10" 10"

B CA

A

P

5" 15"

Determine the small diameter of the axially loaded round bar in the figure subjected to a cyclic load. The maximum tensile load 100,000 lbf and the minimum tensile load is 90,000 lbf. The form stress concentration factor at the step is 2.5 and the factor of safety is to be 2.0. A ductile material of the following properties will be used: Su = 100,000 psi

Sy = 85,000 psi

Se = 65,000 psi

q = 0.40

P

P

d

A structural steel part (SAE 1035 hot rolled) is subjected to the following load spectrum: Pl max = 5000 lbf SAE 1035 Pl ave = 2500 lbf SYP = 54 ksi STP = 85 ksi P2 max = 3000 lbf P2 ave = 0 The part is loaded in simple tension and compression, and a cross-sectional area of 0.12 in2 has been specified. If the factor of safety has been included in the load estimates, and a fatigue stress concentration factor of 1.2 has been selected, compute the and determine whether or not the part is satisfactory.

}

}

n2 = 106 cycles

n1 = 65,000 cycles

∑i iN

in

In most crankshafts the cheeks and bearing journals are so short compared with their cross-sectional dimensions that the entire crankshaft exhibits an extremely complicated stress distribution. It is still not completely understood, and the simplified formulas for calculating stress from strength of materials are not at all valid. However, one can examine a crankshaft approximately with the simple strength formulae if the cheeks and bearing journals are long relative to their cross-sectional dimensions. A force analysis on one end of one such crankshaft reveals that it is in equilibrium under the loads shown in Figure 1 and that F = 2270 lbf. P is a force but M1 and M2 are torques.

a) Determine the critical section in the cheek. Explain your reasoning. b) Determine the points in the critical section where you would calculate the

stresses for design purposes (critical points). There may be more than one of them.

(Hint: Torsional stress on a shaft of rectangular cross-section is much higher at the midpoints of the sides verses the corners. In Figure 2 the torsional stress is high at B and it is slightly higher at A.) Explain your reasoning.

2 1/2 2 1/22 1/2F

FIGURE 1

P

FIGURE 2

B

A

B

A

RECTANGULARSHAFT

CROSS-SECTION

When a harpoon is fired horizontally from the dock of a whaling ship the harpoon of weight W travels essentially horizontally for the early part of the harpoon and is allowed to uncoil from the dock. If the rope is considered weightless relative to the harpoon (and therefore the force in the rope is zero as it is being uncoiled), determine the force P in the rope when the coil suddenly becomes jammed and the rope ceases to feed. Let the velocity of the weight be v and the length of the rope be L at the time of jamming. Your answer should be a function of v, L, .., A, and E where A is the cross-sectional area of the rope and K is the modulus of elasticity of the rope. Hint: A model of this system might be as shown in the figure.

v

W

L

COIL OFROPE

The overhauling of the Wizard-of-Oz Flying Machine showed that members were subjected to the following stresses: Sx = 20,000 psi

Sy = 4,000 psi

τwy = 6,000 psi

and the material tests at Syp = 40,000 psi in tension. Find the minimum value for the Factor-of-Safety via the maximum shear theory of failure.

In the figure shown an element with its component stresses at a critical point in a part.

a) Using the stresses in the figure and the equation below, write out the cubic equation used to determine principal stresses.

b) Check if –3000 psi is a principal stress. If so, divide your equation by S+(-3000).

c) Solve the remaining quadratic fo rthe other principal stresses.

d) Why was –3000 psi a principal stress?

Helpful formula: S3 - (σx + σy + σz)S2 + (σx σy + σy σz + σz σx - τ2

xy - τ2yz - τ2

zx)S - (σx σy σz + 2 τxy τyz τzx - σx τ2

yz - τy τ2zx - σz τ2

xy) = 0

3,000 psi

10,000 psi

5,000 psi

6,000 psi

A triaxial static state of stress at a point is indicated in the figure.

a) For this state of stress, determine the cubic equation from which the principal normal stresses can be obtained.

Helpful equation:

S3 - (σx + σy + σz)S2

+ (σx σy + σy σz + σz σx - σ2xy - σ2

yz - σ2xz)S

- (σx σy σz + 2 σxy σyz σzx - σx σ2yz - σy σ2

zx - σz σ2xy) = 0.

b) Is a compressive stress of 40,000 psi a principal normal stress? Show your reasoning. (The statement in part c is not valid reasoning.)

c) A graphical plot of this cubic equation shows one principal normal stress to be

40,000 psi compressive. Determine the other two principal normal stresses. Hint: Regardless of what equation you obtained in part a. use k3 – 3.3k2 – 17.7k + 46.0 = 0 for part c. where k = S/10,000.

29,650 psi

13,100 psi

30,000 psi

30,000 psi

27,000 psi

A certain state of stress is described by the six stress components Sx = 35,000 psi, Sy = 20,000 psi, Sz = 9,500 psi,

Sxy = 8,670 psi, Syz = 15,700 psi, Sxz = 0

a) Set up the cubic equation from which the principal stresses could be calculated.

b) Solve the cubic equation to obtain numerical values for the principal normal stresses.

The part in the figure is subjected to a steady load P = 10,000 lbf and a release load of Q = 2,000 lbf as shown. Determine numerical values for sa, saz, smax, and smin at the critical point in section A-A’. Area = b2, I = bh3/12.

20"3"A'

AP = 10,000 lb

Q = 2,000 lb f

f

1" x 1"

All the discussion below centers on the circular shaft with loads as shown in the figure.

a) Determine the location and magnitude of the maximum bending moment.

b) Clearly indicate the location of the critical point(s) in the figure. Justify your selection.

c) Regardless of what you chose as critical point(s) in part (b), it is now desired to

determine the components of stress at the point on the bottom of the shaft right at the built- in end. Determine the magnitudes of all the six independent components of stress shown on the cube. Also indicate the directions of the normal stresses. The diameter of the shaft is 2.0 inches.

y

xz

xz

xy

zx

5"

20"

5"

x

y

z 500 lbf

6,000 lbf

List five distinct types of failure stress.

a) In determining a factor of safety when no other information is available, a designer must exercise judgment. To do this carefully he must evaluate the effects of several factors that would influence that factor of safety. List 4 of these factors.

b) List three general manufacturing processes and give one example of each.

List five common modes of failure of mechanical equipment.

a) List four modes of failure of steel machine parts.

b) Name two types of metallic-arc welds and make a sketch of the weld material and parent metal configuration.

c) List three classifications of vibration problems.

List four general manufacturing processes and give examples of each.

The ends of the shaft in the figure are simply supported but keyed against rotation. The stress concentration factors at the junction of the bracket and each piece of shaft are 2.0 in bending and 1.6 in torsion.

a) If the shaft is to be uniform in diameter along its entire length, where are the critical point(s)? Clearly show them (it) in the sketch. Explain the reasoning for your selection.

b) If Region A and Region B of the shaft were each made a different diameter, in

which region would the larger diameter occur? Why?

c) Determine the bending moment at the critical point(s) that you selected in part (a).

6"

f250 lb

20"

20" 16"

The only time that Mohr’s circle can be used to obtain principal normal stress in a triaxial state of stress is when one the normal stress components is already a normal stress (Fig. B) or when all stress components on one surface are zero (biaxial state of stress, Fig.A). To convince yourself of this try reducing the cubic equation for principal normal stresses to equation (1) below which was derived for a biaxial state of stress and for which the Mohr’s circle construction is applicable:

a) For Figure A, make appropriate substitutions for stress components into the cubic equation. One root should be zero. Divide this root out and solve the remaining quadratic. It should look like equation (1).

b) For Figure B, substitute stress component into the cubic equation. The one normal stress component is a principal stress. Divide the cubic by (S – Si) and solve the remaining quadratic for the other two principal stresses. It should look like equation (1).

FIGURE A

FIGURE B

(1) xyyxyx σ

σσσσ+

±+

=2

22 S

An element of 1045 hot-rolled steel (su = 98,000 psi and sy = 59,000 psi) has a steady axial stress of 10,000 psi. Find the permissible shearing stress that can be superposed if the factor of safety is to be 2.5 by the maximum principal normal stress theory of failure. Helpful Equations:

S3 –(σx + σy + σz)S2 + (σx σy +σy σz +σz σx - τ2xy - τ2

yz - τ2zx)S

- (σx σy σz + 2 τxy τyz τzx - σx τ2yz - σy τ2

zx - σx τ2xy) = 0

S1,2 = 2xy

2

yxyx ô2

óó

2

óó+

−±

+

( ) ( ) ( )[ ]212

232

221 SSSSSS

6Åì1

V −+−+−+=

A 15 x 20x 0.5 inch plate of 1045 steel (su = 98,000 psi and sy = 59,000 psi) carries a uniformly distributed tension of 90,000 lb on the 15- inch edge. Find the compressive stress on the 20- inch edge permissible if the factor of safety is 2.0 by the maximum principal shear stress theory of failure.

y (COMPRESSIVE)

The shaft in an aircraft landing gear is made of 4140 steel, annealed and cold-drawn. The part had a solid circular cross-section with a transverse oil hole and is loaded by a torque of 5000 in- lbf. If the part has the dimensions below, is it safe based on yielding? What is the safety factor? Shaft diameter 1 inch

Hole diameter 1/16 inch

Length 10 inches

10"

1" DIA1/16" DIA

T= 5,000

A vehicle is designed to operate at the deepest point in the Pacific Ocean is spherical in shape and is made of stainless steel. A stress analysis has been performed on the structure, and the circumferential stress at the outer surface is found to be where p is the pressure and r1 and r2 are the inner and outer radii respectively. If the sphere is built with r1 = 33.5”, r2 = 36”, and the expected pressure is 15,000 psi compare the margin of safety based on the maximum shear stress failure theory with that obtained from the maximum distortion energy theory. Use a design stress of 100,000 psi.

3

2

1

rr

1

1.05pcs

−=

The cylinder shown below is subjected to an internal pressure p and a torque T. From the loads and the dimensions the following stresses have been computed. Axial stress = Sa = 15,000 psi = Sx Tangential stress = St = 30,000 psi = Sy Torsional stress = τat= 10,000 psi = τxy A design stress, SD, of 30,000 psi has been chosen for the tank material. Using the maximum shear stress theory and neglecting the compression on the inside surface, determine whether of not the tank is properly designed.

T

P

In a certain pipe internal pressure causes both an axial stress of 10,000 psi and a tangential stress of 15,000 psi at a critical point. Because of the support arrangement fo r the pipe, the same critical point is subjected to a torsional stress of 4,000 psi. Find the factor of safety on the basis of yielding at the critical point using a) the distortion energy theory of failure, and

b) the maximum shear stress theory of failure. The yield stress of the material is 40,000 psi.

Determine the small diameter of the axially loaded round bar in the figure subjected to a minimum tensile load of 60,000 lbf and a 10,000 lbf amplitude of cyclic load. The stress concentration at the step is 1.75 and the factor of safety is to be 2.5. Material properties are: Su = 80,000 psi

Se = 50,000 psi

Syp = 35,000 psi

d K = 1.75

P

P

f

The round bar shown is subjected to a varying tensile load. For the first 3,000 cycles the maximum value of the tensile force is 285,000 lbf and the minimum value is 75,000 lb compressive. For the next 50,000 cycles the load varies from 60,000 lb in tension to 200,000 lb in compression. Determine the remaining cycles to failure for a completely reversed load of 150,000 lb. Material properties: Su 140,000 psi

Sy 120,000 psi

q 0.6

Fatigue data: stress cycles 100,000 104

75,000 30,000 65,000 105 60,000 106

60,000 107

P

P

3"

2" R= 0.2"

The bar shown in the figure is subjected to a varying bending moment. It is subjected to a completely reversed loading of 680,000 in- lbf for 15,000 cycles and then a released moment is applied to the part for 5700 cycles after which failure occurs. Determine the magnitude of the released bending moment. Material properties are: Su = 100,000 psi q = 0.80

Sy = 85,000 psi See curve

Se = 40,000 psi

MM 2" THICK 10"

K= 2.5

104

70 6080

510

S-N

CU

RV

E

90

100

1

2

3

4

9

5

6

7

8

1

2

3

4

5

610

710

8

7

5

4

3

2

1

6

7

89

109

6

The shaft pictured below rotates but does not carry any torque load. It is machined from 1020 cold drawn steel for which q=1. Calculate the factor of safety for the shaft.

10"

12"

0.5"

7.5"

0.8"

0.5"0.6"

0.1" R= 0.06"

16 lbf24 lbf

40 lbf

Determine the small diameter of the torsionally loaded round bar in the figure subjected to a minimum torque load of 60,000 in- lbf and a 10,000 in- lbf amplitude of cyclic torque load. The stress concentration at the step is 1.65 in torsion and the factor of safety is to be 2.5. Material properties are: Su = 80,000 psi

Se = 50,000 psi

Syp = 35,000 psi

K = 1.65

T

d f

T

A cantilever beam, shown in the figure, is machined from cold drawn AISI 1030 steel of circular cross section and is subjected to a load which varies from –F to 3F. Determine the maximum load that this member can withstand for an indefinite life using a factor of safety of 2. The notch sensitivity for the material at the 1/8 inch radius is 0.9. Analyze at the change of cross section only.

6

7

3/4" DIA

1" DIA

3F

F1/8" R

Creep Problem

The following data have been obtained for 4140 steel:

T(°F) S(psi) t(hrs.) εc

−6o x10

hr%

dtdå

εo

1000 3000 250 0.037 77 0.017 500 0.049 30 0.034 750 0.054 10 0.047 10,000 250 0.19 376 0.100 500 0.261 204 0.160 750 0.311 204 0.160 1000 0.362 204 0.160 20,000 250 1.1 3010 0.35 500 1.85 3010 0.35 750 2.57 3010 0.35 1000 3.36 3010 0.35 You have been asked to check the design of a turbine part what is to operate at a stress of 30,000 psi. What total creep would you predict in the part at 1000°F in 600 hours?

FORCE ANALYSIS The mechanism shown is to be used to measure the belt tensions T1 and T2. Derive expressions for these tensions in terms of the torque τ, the force F, and any dimensions needed. Indicate all chosen dimensions on the sketch. Also derive an expression for the force of the lower shaft on the rear mount.

F

T2

T1

τ

1. A three inch diameter cold rolled steel shaft is loaded in reversed torsion. Your boss has asked you to determine the effect of drilling a 0.40 inch diameter transverse hole through the shaft. What increase in stress will you tell him the hole will cause?

2. Another engineer had designed a simple tension member as shown below based on a design stress of 64,000 psi. Check the design.

2"1/16" DIA

1/8"

5,000 lb f5,000 lbf

1. State at least five important reasons why a high factor of safety might be used in a machine part. 2. A cylinder head is to be built for a small experimental gasoline engine. Use your Manufacturing Processes notes to help choose a manufacturing method. 3. A simply supported beam is to be of circular cross-section, and will be centrally loaded. Determine a merit index to be used in selecting a material for the application if deflection is to be minimized.

ME 561 DESIGN ANALYSIS OF REFRIGERATION COMPRESSOR

Data: Running speed = 3600 RPM Piston Weight = 260 grams Slider Weight = 106.8 grams Crank Pin Weight = 81.2 grams Counterbalance Weight = 185.6 grams Coefficient of Friction = 0.20 Operating Temperature = 250°F You are asked to make an analysis of the refrigeration compressor mechanism in order to determine the cause of premature crankshaft failures. Consider the position where maximum gas force exists on the cylinder.

1. Complete a motion analysis for the mechanism using the full size drawing provided.

2. Complete a force analysis for the mechanism. 3. Determine the nominal stress at the failure location. Material is gray cast iron. 4. For a stress concentration of 3.5, determine the actual stress in the crankshaft. 5. Make conclusions concerning the failure. 6. Redesign the mechanism to prevent the failure, i.e. discuss all possibilities and

choose the best solution.

300

010

020

0C

RA

NK

AN

GL

E, 0

(DE

GR

EE

S)

100

200

300

FORCE, P (lbs)

DY

NA

MIC

AN

AL

YSI

S O

FR

EFI

GE

RA

TIO

N C

OM

PRE

SSO

RG

AS

FOR

CE

P A

S A

FU

NC

TIO

NO

F C

RA

NK

AN

GL

E