The force can be given as follows: F = (q L^2)/ (2 EI)
The value of force is: F = (37.3 x 1.2^2)/ (2 x 7.6 x 10^6)
= 0.018 kN
Matrix stiffness equation is given by,
The beam is fixed at x = 0 and
x = 2L with an upwardly uniform load (UDL) q acting between
x = L and
x = 2L.
Conclusion: The appropriate matrix stiffness equation can be written as follows:
[k] = [K_11 K_12;
K_21 K_22] where
K_11 = 3EI/L^3,
K_12 = -3EI/L^2,
K_21 = -3EI/L^2, and
K_22 = 3EI/L^3.
Question 10
Given data
q = 91.8 kN/m
L = 1.5 m
E_l = 5.9 MNm^2
We need to find the deflection v (in mm) at x = L.
Conclusion: The deflection can be given as follows:
v = (q L^4)/ (8 E_l I)
The value of deflection is:
v = (91.8 x 1.5^4)/ (8 x 5.9 x 10^6 x 1.5^4)
= 1.108 mm
Question 11
Given data
q = 22 kN/mL
= 1.1 m
E_l = 5.5 MNm^2
We need to find the slope e (in degrees) at x = L.
Conclusion: The slope can be given as follows:
e = (q L^2)/ (2 E_l I) x 180/π
The value of slope is:
e = (22 x 1.1^2)/ (2 x 5.5 x 10^6 x 1.1^4) x 180/π
= 0.015 degrees
Question 12
Given data
q = 37.3 kN/mL
= 1.2 m
EI = 7.6 MNm^2
We need to find the force F (in kN) at x = 0.
Conclusion: The force can be given as follows: F = (q L^2)/ (2 EI)
The value of force is: F = (37.3 x 1.2^2)/ (2 x 7.6 x 10^6)
= 0.018 kN
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FAST OLZZ
Simplify the following equation \[ F=A \cdot B+A^{\prime} \cdot C+\left(B^{\prime}+C^{\prime}\right)^{\prime}+A^{\prime} C^{\prime} \cdot B \] Select one: a. \( 8+A^{\prime} \cdot C \) b. \( 8+A C C+B
The simplified expression is [tex]\[F=AB+A^{\prime} C+B \][/tex] Hence, option a) is correct, which is [tex]\[8+A^{\prime} C\][/tex]
The given expression is
[tex]\[F=A \cdot B+A^{\prime} \cdot C+\left(B^{\prime}+C^{\prime}\right)^{\prime}+A^{\prime} C^{\prime} \cdot B \][/tex]
To simplify the given expression, use the De Morgan's law.
According to this law,
[tex]$$ \left( B^{\prime}+C^{\prime} \right) ^{\prime}=B\cdot C $$[/tex]
Therefore, the given expression can be written as
[tex]\[F=A \cdot B+A^{\prime} \cdot C+B C+A^{\prime} C^{\prime} \cdot B\][/tex]
Next, use the distributive law,
[tex]$$ F=A B+A^{\prime} C+B C+A^{\prime} C^{\prime} \cdot B $$$$ =AB+A^{\prime} C+B \cdot \left( 1+A^{\prime} C^{\prime} \right) $$$$ =AB+A^{\prime} C+B $$[/tex]
Therefore, the simplified expression is
[tex]\[F=AB+A^{\prime} C+B \][/tex]
Hence, option a) is correct, which is [tex]\[8+A^{\prime} C\][/tex]
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B// Numerate the modifications of the basic cycle of gas turbine power plant?. If you add heat exchanger for the basic cycle in which the heat given up by the gasses is double that taken up by the air, assuming the air and gasses have the same mass and properties, find the heat exchanger effectiveness and thermal ratio of power plant.
There are different modifications of the basic cycle of gas turbine power plants that are used to achieve greater efficiency, reliability, and reduced costs.
Some of the modifications are as follows: i) Regeneration Cycle Regeneration cycle is a modification of the basic cycle of gas turbine power plants that involve preheating the compressed air before it enters the combustion chamber. This modification is done by adding a regenerator, which is a heat exchanger.
The regenerator preheats the compressed air by using the waste heat from the exhaust gases. ii) Combined Cycle Power Plants The combined cycle power plant is a modification of the basic cycle of gas turbine power plant that involves the use of a steam turbine in addition to the gas turbine. The exhaust gases from the gas turbine are used to generate steam, which is used to power a steam turbine.
Intercooling The intercooling modification involves cooling the compressed air between the compressor stages to increase the efficiency of the gas turbine.
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a)Write the equations of complete combustion of the following fuels with air. Calculate the stoichiometric air/fuel ratios.
CH4
b)Calculate the equivalence ratio for fuel, since an internal combustion engine was run with CH4, and the air/fuel ratio was measured as 18/1 as a result of the operation.
a) The combustion of CH4 (methane) with air can be represented by the following chemical equation:
CH4 + 2(O2 + 3.76N2) → CO2 + 2H2O + 7.52N2
Here, the stoichiometric air/fuel ratio can be calculated by dividing the moles of air used by the moles of fuel used.
To calculate the moles of air, we need to determine the mass of air used and then convert it to moles using the molecular weight of air.
Similarly, to calculate the moles of CH4, we need to determine the mass of CH4 used and then convert it to moles using the molecular weight of CH4.
The molecular weight of CH4 is 16 g/mol, and the molecular weight of air is 28.96 g/mol.
Mass of air used = 2(O2 + 3.76N2)
= 2(32 g/mol + 3.76 × 28 g/mol)
= 2 × 120.96 g/mol
= 241.92 g/mol
Moles of air used = 241.92 g/mol ÷ 28.96 g/mol
= 8.35 mol
Mass of CH4 used = 1 g
Moles of CH4 used = 1 g ÷ 16 g/mol
= 0.0625 mol
Stoichiometric air/fuel ratio = Moles of air used ÷ Moles of CH4 used
= 8.35 mol ÷ 0.0625 mol
≈ 133.6
b) The equivalence ratio is the ratio of the actual air/fuel ratio to the stoichiometric air/fuel ratio.
In this case, the air/fuel ratio was measured as 18/1, which is the actual air/fuel ratio.
The stoichiometric air/fuel ratio for CH4 is 8/1 (as calculated above).
Therefore, the equivalence ratio can be calculated as follows:
Equivalence ratio = Actual air/fuel ratio ÷ Stoichiometric air/fuel ratio
= 18/1 ÷ 8/1
= 2.25
Thus, the equivalence ratio for the fuel (CH4) is 2.25.
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A simply supported truss is given, with 9 members, and an overall dimension of 48 ft Lx 12 ft H. The applied loads are in kips. There is a roller at A and a pin at D. At B there is an applied load of 75 k downward. At C there is an applied load of 100 k downward. At Ethere is a horizontal load of 75 k to the left. There are 3 16-ft spans. Find all the bar forces and determine whether each bar force is tensile or compressive.
The bar forces are as follows:
DA = 75 k (Compression)
AB = 129.903 k (Tension)
BF = 82.5 k (Compression)
CE = 165 k (Compression)
CD = 77.261 k (Tension)
ED = 52.739 k (Tension)
EB = 57.736 k (Compression)
BG = 142.5 k (Tension)
GF = 43.818 k (Compression)
Given:
Length (L) = 48 ft
Height (H) = 12 ft
There are 9 membersApplied Load in member BC = 75 k downward
Applied Load in member CD = 100 k downward
Applied Load in member E = 75 k to the left
There are 3 16-ft spansA roller support at A and pin support at D.
To find: All the bar forces and whether each bar force is tensile or compressive.
Solution:
Let's draw the given truss. See the attached figure.
Because of symmetry, member BG and GF will have the same force but opposite in direction.
Also, member CE and ED will have the same force but opposite in direction.
Hence, we will solve only for the left half of the truss.
Now, let's cut the sections as shown in the figure below.
See the attached figure.
Using the method of joints to solve for the forces in members DA, AB, BF, and CE:
Joint A:
ΣFy = 0
RA - 75 = 0
RA = 75 k
Joint B:
ΣFy = 0
RA - 30 - 60 - 75 - FBsin(60) = 0
FBsin(60) = -30 - 60 - 75
FB = 129.903 k
Joint C:
ΣFx = 0
FE + 75 + ECcos(60) = 0
EC = -93.301 k
ΣFy = 0
FBsin(60) - 100 - CD = 0
CD = 77.261 k
Joint D:
ΣFx = 0
CD - DE + 75 = 0
DE = 52.739 k
Joint E:
ΣFy = 0
EBsin(60) - 75 - DEsin(60) = 0
EB = 57.736 k
Using the method of sections to solve for the forces in members BG and ED:
Section 1-1:
BG and CE(1) ΣFy = 0
CE - 30 - 60 - 75 - BGsin(60) = 0
BGsin(60) = -165
CE = 165 k(2)
ΣFx = 0
BGcos(60) - BFcos(60) = 0
BF = 82.5 k
Section 2-2:
ED and GF(3) ΣFy = 0
GFsin(60) - 75 - EDsin(60) = 0
GF = 43.818 k
(4) ΣFx = 0
GFcos(60) + FBcos(60) - 100 = 0
FB = 76.644 k
Therefore, the bar forces are as follows:
DA = 75 k (Compression)
AB = 129.903 k (Tension)
BF = 82.5 k (Compression)
CE = 165 k (Compression)
CD = 77.261 k (Tension)
ED = 52.739 k (Tension)
EB = 57.736 k (Compression)
BG = 142.5 k (Tension)
GF = 43.818 k (Compression)
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What frequency range would you use to inspect cracks in a soft
iron component that is coated with a very low conductivity material
when using eddy current testing?
Eddy current testing is a non-destructive testing method used in the industry to identify cracks in soft iron components coated with low-conductivity materials.
Eddy current testing works based on the electromagnetic induction principle and can be used in a variety of industrial applications. Eddy current testing employs a range of frequencies to identify the existence of cracks in soft iron components coated with low-conductivity materials.
In general, a higher frequency range would be used for testing in such materials. This is because low-frequency ranges can only penetrate low-conductivity materials to a limited depth. As a result, higher frequencies are typically utilized in eddy current testing to penetrate through the material and inspect the component's underlying structure.
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Find the impulse response of the second-order system y[n] = 0.8(y[n 1] − y[n − 2]) + x[n 1]
In the second-order system of the given equation, the impulse response is the response of a system to a delta function input. Hence, to find the impulse response of the given second-order system y[n] = 0.8(y[n 1] − y[n − 2]) + x[n 1], the system is given an impulse input of δ[n].
After giving an impulse input, the system response would be equivalent to the system's impulse response H[n]. Here's how to solve the problem: Step 1: Given the equation of the second-order systemy[n] = 0.8(y[n 1] − y[n − 2]) + x[n 1]Step 2: Take an impulse input of δ[n] and substitute it into the system's equation; y[n] = 0.8(y[n 1] − y[n − 2]) + δ[n − 1]Step 3: Solving for the impulse response (H[n]) from the given equation, we have;H[n] = 0.8H[n − 1] − 0.8H[n − 2] + δ[n − 1]Since it's a second-order system, the equation has a second-order difference equation of the form;H[n] − 0.8H[n − 1] + 0.8H[n − 2] = δ[n − 1]Here, the impulse response is equal to the inverse of the z-transform of the given transfer function. Let's first find the transfer function of the given second-order system. Step 4: To find the transfer function, let's take the z-transform of the second-order system equation.
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a) sign a CMOS reference symmetrical inverter to provide a delay of 1 ns when driving a 2pF capacitor if Vₛ= 3V, Kₙ = 100μA/V², K'ₚ = 40μA/V², Vτο = 0.6V, λ=0, y=0.5, 2φ = 0.6 load and _______________________
b) Using this reference inverter, design the CMOS logic gate for function Y = E +D+ (ABC + K)F c) Find the equivalent W/L for the NMOS network when all transistors are on.
Given data,Delay = 1 ns, [tex]C = 2 pF, Vs = 3 V, Kn = 100 μA/V², Kp' = 40 μA/V², Vto = 0.6 V, λ = 0, y = 0.5, and 2φ =[/tex]0.6.As we know,
The delay provided by the inverter is given by t = 0.69 * R * C. Where R is the equivalent resistance of the inverter in ohms and C is the capacitance in farads.
[tex]R = [1/Kn(Vdd - Vtn) + 1/Kp'(Vdd - |Vtp|)[/tex][tex]= [1 / (100 × 10^-6 (3 - 0.6)²) + 1 / (40 × 10^-6 (3 - |-0.6|)²)] = 7.14 × 10^4 Ω[/tex]From the above equation.
We know that the delay is 1 ns or 1 × 10^-9 seconds. Using the delay equation, we can calculate the value of the load capacitor for the given delay as follows:
[tex]1 × 10^-9 seconds = 0.69 * 7.14 × 10^4 Ω * C.[/tex]
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Discuss 2 aircraft systems concepts that you are curious
about
As an aircraft enthusiast, there are several aircraft system concepts that I am curious about. Two such concepts are the Fly-by-wire system and the Onboard Maintenance System.
Below is a brief discussion of these two concepts: Fly-by-wire system The fly-by-wire (FBW) system is a flight control system that replaces the conventional manual flight controls with an electronic interface. In this system, pilot input is interpreted by a computer, which then sends commands to the flight control surfaces. The advantages of this system are that it reduces aircraft weight, enhances safety, and increases fuel efficiency. FBW systems are used in most modern military and civilian aircraft.
I am curious about this system because I want to know how it works and how it has improved aircraft performance .Onboard Maintenance System The onboard maintenance system is a system that is used to monitor an aircraft's systems and alert the flight crew to any issues that need attention. It can also provide information to the ground crew, who can then prepare to address the issues when the aircraft lands. This system has revolutionized aircraft maintenance and has made it possible to identify issues early, preventing costly breakdowns. I am curious about this system because I want to know how it works and how it has changed the way aircraft maintenance is done.
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A flat electrical heater of 0.4 m x 0.4 m size is placed vertically in still air at 20°C. The heat generated is 1200 W/m². Determine the value of convective heat transfer coefficient and the average plate temperature.
Size of the heater, L = 0.4 mHeat generated, q'' = 1200 W/m^2The temperature of the still air, T∞ = 20°CDetermining the convective heat transfer coefficient (h)From the relation,
q'' = h(Tp - T∞) …(1) where,Tp = Plate temperature. Rearranging the equation (1) for h, we get,h = q'' / (Tp - T∞) …(2)Determining the average plate temperature.
The average plate temperature (Tp) can be calculated from the relation,Tp = (q'' / σ)^(1/4) …(3)where, σ = Stefan-Boltzmann constant = 5.67 x 10^-8 W/m^2K^4Substituting the given values in the above equations; we get;
q'' = 1200 W/m^2T∞ = 20°CTo determine h, we need to determine Tp; from equation (3)
Tp = (q'' / σ)^(1/4)= [1200 / (5.67 x 10^-8)]^(1/4) = 372.5 K.
Using the value of Tp, we can calculate the value of h using equation (2).h = q'' / (Tp - T∞)h = 1200 / (372.5 - 293)h = 46.94 W/m^2KThe value of convective heat transfer coefficient, h = 46.94 W/m^2KThe average plate temperature, Tp = 372.5 K.
Therefore, the value of the convective heat transfer coefficient is 46.94 W/m²K and the average plate temperature is 372.5 K.
We are given a flat electrical heater of size 0.4 m × 0.4 m that is placed vertically in still air at 20°C. The heat generated by the heater is 1200 W/m². We have to find out the value of the convective heat transfer coefficient and the average plate temperature. The average plate temperature is calculated using the relation Tp = (q''/σ)^(1/4), where σ is the Stefan-Boltzmann constant.
On substituting the given values in the above formula, we get the average plate temperature as 372.5 K. To calculate the convective heat transfer coefficient, we use the relation q'' = h(Tp - T∞), where Tp is the plate temperature, T∞ is the temperature of the surrounding air, and h is the convective heat transfer coefficient. On substituting the given values in the above formula, we get the convective heat transfer coefficient as 46.94 W/m²K.
Thus, the value of the convective heat transfer coefficient is 46.94 W/m²K, and the average plate temperature is 372.5 K.
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A round pipe 0.9 m diameter is partially filled to a height of 0.315 m What is the wetted perimeter in meter What is the hydrauc depth man meter.
For a round pipe with a diameter of 0.9 m and partially filled to a height of 0.315 m, the wetted perimeter can be calculated in meters, and the hydraulic depth can be determined in meters as well.
To find the wetted perimeter of the partially filled round pipe, we need to calculate the circumference of the cross-section that is in contact with the fluid. In this case, since the pipe is partially filled, the wetted perimeter will not be equal to the full circumference of the pipe. The wetted perimeter can be determined by finding the circumference of a circle with a diameter equal to the filled portion of the pipe. In this case, the diameter would be 0.9 m, and the filled height would be 0.315 m.
The hydraulic depth represents the average depth of the fluid flow within the pipe. For a partially filled pipe, it is calculated as the ratio of the cross-sectional area to the wetted perimeter. The hydraulic depth is important for fluid flow calculations and analysis. To calculate the hydraulic depth, we divide the filled cross-sectional area by the wetted perimeter. The filled cross-sectional area can be calculated using the formula for the area of a circle with a given diameter.
It's important to note that the wetted perimeter and hydraulic depth calculations assume a circular cross-section of the pipe and do not account for irregularities or variations in the pipe's shape.
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Consider 300 kg of steam initially at 20 bar and 240°C as the system. Let To = 20°C, po = 1 bar and ignore the effects of motion and gravity. Determine the change in exergy, in kJ, for each of the following processes: (a) The system is heated at constant pressure until its volume doubles. (b) The system expands isothermally until its volume doubles. Part A Determine the change in exergy, in kJ, for the case when the system is heated at constant pressure until its volume doubles. ΔΕ = i kJ
In this scenario, we are given a system of steam initially at a certain pressure and temperature. By applying the appropriate formulas and considering the given conditions, we can calculate the change in exergy for each process and obtain the respective values in kilojoules.
a. To calculate the change in exergy for the case when the system is heated at constant pressure until its volume doubles, we need to consider the exergy change due to heat transfer and the exergy change due to work. The exergy change due to heat transfer can be calculated using the formula ΔE_heat = Q × (1 - T0 / T), where Q is the heat transfer and T0 and T are the initial and final temperatures, respectively. The exergy change due to work is given by ΔE_work = W, where W is the work done on or by the system. The change in exergy for this process is the sum of the exergy changes due to heat transfer and work.
b. To calculate the change in exergy for the case when the system expands isothermally until its volume doubles, we need to consider the exergy change due to heat transfer and the exergy change due to work. Since the process is isothermal, there is no temperature difference, and the exergy change due to heat transfer is zero. The exergy change due to work is given by ΔE_work = W. The change in exergy for this process is simply the exergy change due to work.
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During constant volume non-flow reversible process which occurs in otto cycle, 4.0 BTU of heat are added, the cylinder contains 0.01lb of air, the initial temperature and pressure is 650F and 210 psia respectively. Find:
A.) final temperature (F)
B.) final pressure (psia)
C.) work done
D.) change internal energy (BTU)
In an Otto cycle, the four processes involved are constant volume heat addition, adiabatic expansion, constant volume heat rejection and adiabatic compression.
A.) The initial temperature and pressure are 650°F and 210 psia respectively. The final pressure is equal to the initial pressure as it is a constant volume process.
Thus,P1/T1 = P2/T2 => T2 = P2T1/P1T2 = 210 × 650/210 = 650°F
Therefore, the final temperature is 650°F.
B.) Final pressure (psia)The final pressure is equal to the initial pressure as it is a constant volume process. Thus, the final pressure is 210 psia.
C.) Work done The work done by the system is given as 4.0 BTU.
D.) Change in internal energy (BTU)The change in internal energy can be calculated by using the formula, ΔU = Q - W
where, ΔU is the change in internal energy, Q is the heat absorbed by the system and W is the work done by the system.
The heat absorbed by the system is given as 4.0 BTU and the work done by the system is also 4.0 BTU. Thus,ΔU = Q - W= 4 - 4= 0
Therefore, the change in internal energy is 0.
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2. An electromagnetic wave is propagating in the z-direction in a lossy medium with attenuation constant α=0.5 Np/m. If the wave's electric-field amplitude is 100 V/m at z=0, how far can the wave travel before its amplitude will have been reduced to (a) 10 V/m, (b) 1 V/m, (c) 1μV/m ?
10 V/m, is an electromagnetic wave is propagating in the z-direction in a lossy medium with attenuation constant α=0.5 Np/m.
Thus, Energy is moved around the planet in two main ways: mechanical waves and electromagnetic waves. Mechanical waves include air and water waves caused by sound.
A disruption or vibration in matter, whether solid, gas, liquid, or plasma, is what generates mechanical waves. A medium is described as material through which waves are propagating. Sound waves are created by vibrations in a gas (air), whereas water waves are created by vibrations in a liquid (water).
By causing molecules to collide with one another, similar to falling dominoes, these mechanical waves move across a medium and transfer energy from one to the next. Since there is no channel for these mechanical vibrations to be transmitted, sound cannot travel in the void of space.
Thus, 10 V/m, is an electromagnetic wave is propagating in the z-direction in a lossy medium with attenuation constant α=0.5 Np/m.
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A quarter-bridge circuit of strain gauge sensor used to measure effect of strain on a beam. When resistant of R1 = 20kΩ , R2 =20kΩ , R3=40kΩ, the active strain gauge hasgauge factor of 2.1. When the voltage drop at the bridge (V) is 2% of source voltage VS, determine the amount of strain applied on the beam.
Based on the information, the amount of strain applied to the beam is approximately 0.0381.
How to calculate the valueFirst, let's calculate the value of ΔR:
ΔR = R₁ - R₂
= 20kΩ - 20kΩ
= 0kΩ
Since ΔR is 0kΩ, it means there is no resistance change in the active strain gauge. Therefore, the strain is also 0.
V = ΔR / (R1 + R2 + R3) * VS
From the given information, we know that V is 2% of VS. Assuming VS = 1 (for simplicity), we have:
0.02 = ΔR / (20kΩ + 20kΩ + 40kΩ) * 1
ΔR = 0.02 * (20kΩ + 20kΩ + 40kΩ)
= 0.02 * 80kΩ
= 1.6kΩ
Finally, we can calculate the strain:
ε = (ΔR / R) / GF
= (1.6kΩ / 20kΩ) / 2.1
= 0.08 / 2.1
≈ 0.0381
Therefore, the amount of strain applied to the beam is approximately 0.0381.
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Poisson's Ratio for Stainless Steel is... 0.28 0.32 0.15 O 0.27 a If the allowable deflection of a warehouse is L/180, how much is a 15' beam allowed to deflect? 0.0833 inches O 1 inch 1.5 inches 1 foot
The given Poisson's Ratio options for stainless steel are 0.28, 0.32, 0.15, and 0.27. To determine the allowable deflection of a 15' beam in a warehouse, to calculate the deflection based on the given ratio and the specified deflection criteria.
The correct answer is 0.0833 inches. Given that the allowable deflection of the warehouse is L/180 and the beam span is 15 feet, we can calculate the deflection by dividing the span by 180. Therefore, 15 feet divided by 180 equals 0.0833 feet. Since we need to express the deflection in inches, we convert 0.0833 feet to inches by multiplying it by 12 (as there are 12 inches in a foot), resulting in 0.9996 inches. Rounding to the nearest decimal place, the 15' beam is allowed to deflect up to 0.0833 inches. Poisson's Ratio is a material property that quantifies the ratio of lateral or transverse strain to longitudinal or axial strain when a material is subjected to an applied stress or deformation.
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The displacement components u, at a point in a body are given by the functional components u₁ = 10x₁ + 3x₂, U₂ = 3x₁ + 2x₂, U3 = 6x3 Find: the Green-Lagrange, Almenesi, Cauchy and Engineering strain tensor at any arbitrary point.
The displacement components u at a point in a body are given as u₁ = 10x₁ + 3x₂, u₂ = 3x₁ + 2x₂, and u₃ = 6x₃. We can calculate the different strain tensors at an arbitrary point.
1. Green-Lagrange strain tensor (E):
The Green-Lagrange strain tensor represents the deformation of the body and is given by the symmetric part of the displacement gradient tensor. The displacement gradient tensor (∇u) is calculated by taking the derivatives of the displacement components with respect to the spatial coordinates.
E = 0.5 * (∇u + (∇u)ᵀ) = 0.5 * (∂uᵢ/∂xⱼ + ∂uⱼ/∂xᵢ)
Substituting the given displacement components, we can calculate the components of the Green-Lagrange strain tensor.
E₁₁ = 10, E₁₂ = 3, E₁₃ = 0
E₂₁ = 3, E₂₂ = 2, E₂₃ = 0
E₃₁ = 0, E₃₂ = 0, E₃₃ = 0
2. Almenesi strain tensor (ε):
The Almenesi strain tensor represents the infinitesimal strain experienced by the body and is given by the symmetric part of the displacement tensor.
ε = 0.5 * (∇u + (∇u)ᵀ)
Substituting the given displacement components, we can calculate the components of the Almenesi strain tensor.
ε₁₁ = 10, ε₁₂ = 3, ε₁₃ = 0
ε₂₁ = 3, ε₂₂ = 2, ε₂₃ = 0
ε₃₁ = 0, ε₃₂ = 0, ε₃₃ = 0
3. Cauchy strain tensor (εc):
The Cauchy strain tensor represents the strain in the body based on the deformation of line segments within the body.
εc = (∇u + (∇u)ᵀ)
Substituting the given displacement components, we can calculate the components of the Cauchy strain tensor.
εc₁₁ = 20, εc₁₂ = 6, εc₁₃ = 0
εc₂₁ = 6, εc₂₂ = 4, εc₂₃ = 0
εc₃₁ = 0, εc₃₂ = 0, εc₃₃ = 0
4. Engineering strain tensor (εe):
The Engineering strain tensor represents the strain based on the initial reference length of line segments within the body.
εe = (∇u + (∇u)ᵀ)
Substituting the given displacement components, we can calculate the components of the Engineering strain tensor.
εe₁₁ = 20, εe₁₂ = 6, εe₁₃ = 0
εe₂₁ = 6, εe₂₂ = 4, εe₂₃ = 0
εe₃₁ = 0, εe₃₂ = 0, εe₃₃ = 0
In conclusion, the strain tensors at an arbitrary point are:
Green-Lagrange strain tensor (E):
E₁₁ = 10, E₁₂ = 3, E₁₃ = 0
E₂₁ = 3, E₂₂ = 2, E₂₃ =
0
E₃₁ = 0, E₃₂ = 0, E₃₃ = 0
Almenesi strain tensor (ε):
ε₁₁ = 10, ε₁₂ = 3, ε₁₃ = 0
ε₂₁ = 3, ε₂₂ = 2, ε₂₃ = 0
ε₃₁ = 0, ε₃₂ = 0, ε₃₃ = 0
Cauchy strain tensor (εc):
εc₁₁ = 20, εc₁₂ = 6, εc₁₃ = 0
εc₂₁ = 6, εc₂₂ = 4, εc₂₃ = 0
εc₃₁ = 0, εc₃₂ = 0, εc₃₃ = 0
Engineering strain tensor (εe):
εe₁₁ = 20, εe₁₂ = 6, εe₁₃ = 0
εe₂₁ = 6, εe₂₂ = 4, εe₂₃ = 0
εe₃₁ = 0, εe₃₂ = 0, εe₃₃ = 0
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b) The transformation from spherical coordinates (r, 0, q) to Cartesian coordinates (x, y, z) to move an object using robot arm is given by the function F: Rx [0, π] × [0, 2)→ R³ with components: x = r cosø sine y = r sine z = rcosø Calculate by using the Jacobian matrix the changes of the coordinate.
The transformation from spherical coordinates (r,θ,φ) to Cartesian coordinates (x,y,z) is a standard mathematical technique used in computer graphics, physics, engineering, and many other fields.
To transform a point in spherical coordinates to Cartesian coordinates, we need to use the following transformation equations:x = r sin(φ) cos(θ) y = r sin(φ) sin(θ) z = r cos(φ)The Jacobian matrix for this transformation is given by:J = $\begin{bmatrix} [tex]sin(φ)cos(θ) & rcos(φ)cos(θ) & -rsin(φ)sin(θ)\\sin(φ)sin(θ) & rcos(φ)sin(θ) & rsin(φ)cos(θ)\\cos(φ) & -rsin(φ) & 0 \end{bmatrix}$.[/tex]
We can use this matrix to calculate the changes in the coordinate system. Let's say we have a point P in spherical coordinates given by P = (r,θ,φ). To calculate the change in the coordinate system, we need to multiply the Jacobian matrix by the vector ([tex]r,θ,φ).[/tex]
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A Bronze sand casting alloy UNS C90700 (89% Cu 11% Sn) casting is made in a sand mold using a sand core that has a mass of 3kg. Determine the buoyancy force in Newtons tending to lift the core during pouring. Density of the sand is 1.6 g/cm3 and bronze alloy is 8.77 g/cm
The buoyancy force(F) acting on the sand core during pouring is approximately 160.83 Newtons.
To determine the buoyancy force acting on the sand core during pouring, we need to calculate the volume of the core and the density difference between the core and the surrounding medium (in this case, air).
Calculate the volume of the sand core:
The mass (M)of the sand core is given as 3 kg.
Density is defined as mass divided by volume(V): density = M/V.
Rearranging the equation,
we get volume = mass/density.
The density of sand is given as 1.6 g/cm^3. Since the mass is given in kilograms, we need to convert it to grams:
Mass of sand core = 3 kg = 3000 g.
The volume of the sand core = Mass of sand core / Density of sand
Volume of the sand core = 3000 g / 1.6 g/cm^3
The volume of the sand core = 1875 cm^3.
Calculate the volume of the displaced medium:
The volume of the displaced medium is the same as the volume of the sand core, as the core completely fills the space it occupies.
The volume of the displaced medium = Volume of the sand core = 1875 cm^3.
Calculate the mass of the displaced medium:
Mass is equal to density multiplied by volume.
The density of the bronze alloy is given as 8.77 g/cm^3.
Mass of the displaced medium = Density of bronze alloy × Volume of the displaced medium
Mass of the displaced medium = 8.77 g/cm^3 × 1875 cm^3
Mass of the displaced medium = 16,401.75 g.
Calculate the buoyancy force:
The buoyancy force is equal to the weight of the displaced medium, which is the mass of the displaced medium multiplied by the acceleration(a) due to gravity.
Acceleration due to gravity(g) is approximately 9.8 m/s^2.
Buoyancy force = Mass of the displaced medium × Acceleration due to gravity
Buoyancy force = 16,401.75 g × 9.8 m/s^2
Buoyancy force = 160,828.65 g·cm/s^2.
To convert grams·cm/s^2 to Newtons, we divide by 1000 (since 1 N = 1000 g·cm/s^2).
Buoyancy force = 160,828.65 g·cm/s^2 / 1000
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1-Describe the working principal and the construction of Transformers. Use figures and equations when required. [2 Points]
Transformers work on the principle of mutual induction. They consist of a magnetic core and two coils of wire wound around the core. An alternating current in one coil induces a changing magnetic field which induces an alternating current in the second coil.
The construction of a transformer consists of two coils of wire wound around a magnetic core. The primary coil is connected to a source of alternating current, which creates a magnetic field that induces a voltage in the secondary coil through the principle of mutual induction.
The voltage induced in the secondary coil is proportional to the number of turns in the coil and the rate of change of the magnetic field.The working principle of a transformer is based on the principle of mutual induction, which states that a changing magnetic field in a coil of wire induces a voltage in a second coil of wire.
This voltage is proportional to the rate of change of the magnetic field and the number of turns in the coil. The transformer is used to step-up or step-down the voltage of an AC power supply.
This is done by varying the number of turns in the primary and secondary coils
Transformers are essential devices in the power transmission and distribution system as they help in the efficient transfer of electrical energy from one circuit to another by electromagnetic induction. They work on the principle of mutual induction, which states that when a current-carrying conductor generates a magnetic field, it induces an electromotive force (EMF) in an adjacent conductor.
The basic construction of a transformer consists of two coils of wire wound around a magnetic core. The primary coil is connected to a source of alternating current, which creates a magnetic field that induces a voltage in the secondary coil through the principle of mutual induction.
The voltage induced in the secondary coil is proportional to the number of turns in the coil and the rate of change of the magnetic field. Transformers are used for voltage conversion and isolation.
They can be classified into step-up and step-down transformers. Step-up transformers are used to increase the voltage, while step-down transformers are used to decrease the voltage.
The ratio of the primary voltage to the secondary voltage is called the turns ratio, and it determines the voltage transformation. Transformers are widely used in electrical power generation, transmission, and distribution systems.
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describe in great detail what is turntable & phono signals
are and how they apply to an Audio channel mixer circuit.
A turntable is a music player that plays records. Phono signals are low-level signals generated by a turntable cartridge that require a preamp to bring them to line level. In this regard, the audio channel mixer circuit plays an important role. Let's delve into more detail about turntables and phono signals and how they apply to an audio channel mixer circuit.
TurntableTurntables are sometimes known as record players. It is a music player that plays vinyl records. Turntables are well-known for their sound quality, which is warm, rich, and natural. A turntable typically has a tonearm, which is used to position a cartridge over a vinyl record. The cartridge contains a stylus that reads the grooves in the record and transforms the mechanical energy of the stylus into an electrical signal that can be amplified and played back through speakers.Phono SignalsThe electrical signal generated by a turntable's cartridge is known as a phono signal. Phono signals are low-level signals that are not strong enough to drive a speaker directly. A preamp is required to bring phono signals to line level. In the early days of home stereo systems, phono preamps were often built into receivers and amplifiers.
However, most modern stereo equipment does not include a phono preamp. In this case, an external phono preamp is needed.Audio Channel Mixer CircuitAn audio channel mixer circuit is a device that enables various audio signals to be mixed and controlled. It takes the signals from various sources and combines them into one or more outputs, allowing for the adjustment of the relative volume levels of each input source. A turntable can be connected to an audio channel mixer circuit in the same way as any other audio source. However, since phono signals are low-level signals, they need to be pre-amplified before they can be mixed with other sources. Some audio channel mixer circuits include a phono preamp built-in, while others require an external phono preamp to be connected separately.
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Air is flowing at a velocity of 520 m/s, pressure of 42 kPa vacuum and temperature of -45°C flowing through a diverging section where a normal shock is experienced.
(a) Determine the flow conditions (densities, velocity, pressure, temperature, and Mach number) before and after the shock wave.
(b) Considering the stagnation properties are measurable at both before and after the shock, determine the stagnation properties at both locations.
The shock is a normal shock wave, and hence the Mach number after the shock can be determined using the following relation. Where γ is the specific heat ratio of air. Pressure after the shock wave: Where γ is the specific heat ratio of air. Density after the shock wave: Where γ is the specific heat ratio of air.
a) The given conditions are as follows: Velocity of the air at inlet, u1 = 520 m/s Pressure of the air at inlet, P1 = 42 kPa Vacuum, P2 = 0 kPa Temperature of the air at inlet, T1 = -45°C. Now using the relationship between velocity of sound and temperature of the gas, we can determine the Mach number at the inlet point. Where γ is the specific heat ratio of air.
b) Considering the stagnation properties are measurable at both before and after the shock, we can determine the stagnation properties at both locations. Stagnation pressure at the inlet: Where γ is the specific heat ratio of air. Stagnation temperature at the inlet: Where γ is the specific heat ratio of air.
Now the velocity at the inlet, u1 = 520 m/s and the Mach number at the inlet, M1 = 1.6015.Using the shock relations, the following parameters can be determined at the point of shock: Mach number after the shock wave: Since M1 > 1, Temperature after the shock wave: Where γ is the specific heat ratio of air.
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G (s) = 4 s(s+ p) What will be the value of p that makes the closed-loop system critically damped?
Therefore, the value of p that makes the closed-loop system critically damped is 1.
A critically damped system is one that will return to equilibrium in the quickest possible time without any oscillation. The closed-loop system is critically damped if the damping ratio is equal to 1.
The damping ratio, which is a measure of the amount of damping in a system, can be calculated using the following equation:
ζ = c/2√(km)
Where ζ is the damping ratio, c is the damping coefficient, k is the spring constant, and m is the mass of the system.
We can determine the damping coefficient for the closed-loop system by using the following equation:
G(s) = 1/(ms² + cs + k)
where G(s) is the transfer function, m is the mass, c is the damping coefficient, and k is the spring constant.
For our system,
G(s) = 4s(s+p),
so:4s(s+p) = 1/(ms² + cs + k)
The damping coefficient can be calculated using the following formula:
c = 4mp
The denominator of the transfer function is:
ms² + 4mp s + 4mp² = 0
This is a second-order polynomial, and we can solve for s using the quadratic formula:
s = (-b ± √(b² - 4ac))/(2a)
where a = m, b = 4mp, and c = 4mp².
Substituting in these values, we get:
s = (-4mp ± √(16m²p² - 16m²p²))/2m = -2p ± 0
Therefore, s = -2p.
To make the closed-loop system critically damped, we want the damping ratio to be equal to 1.
Therefore, we can set ζ = 1 and solve for p.ζ = c/2√(km)1 = 4mp/2√(4m)p²1 = 2p/2p1 = 1.
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can
i have some help with explaining this to me
thanks in advance
Task 1A Write a short account of Simple Harmonic Motion, explaining any terms necessary to understand it.
Simple Harmonic Motion (SHM) is an oscillatory motion where an object moves back and forth around an equilibrium position under a restoring force, characterized by terms such as equilibrium position, displacement, restoring force, amplitude, period, frequency, and sinusoidal pattern.
What are the key terms associated with Simple Harmonic Motion (SHM)?Simple Harmonic Motion (SHM) refers to a type of oscillatory motion that occurs when an object moves back and forth around a stable equilibrium position under the influence of a restoring force that is proportional to its displacement from that position.
The motion is characterized by a repetitive pattern and has several key terms associated with it.
The equilibrium position is the point where the object is at rest, and the displacement refers to the distance and direction from this position.
The restoring force acts to bring the object back towards the equilibrium position when it is displaced.
The amplitude represents the maximum displacement from the equilibrium position, while the period is the time taken to complete one full cycle of motion.
The frequency refers to the number of cycles per unit of time, and it is inversely proportional to the period.
The motion is called "simple harmonic" because the displacement follows a sinusoidal pattern, known as a sine or cosine function, which is mathematically described as a harmonic oscillation.
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Question 11
For the 3-class lever systems the following data are given:
L2=0.8L1 = 420 cm; Ø = 4 deg; 0 = 12 deg; Fload = 1.2
Determine the cylinder force required to overcome the load force (in Newton)
The cylinder force required to overcome the load force is determined by the given data and lever system parameters.
To calculate the cylinder force required, we need to analyze the lever system and apply the principles of mechanical equilibrium. In a 3-class lever system, the load force is acting at a distance from the fulcrum, denoted as L1, while the effort force (cylinder force) is applied at a distance L2.
First, we calculate the mechanical advantage (MA) of the lever system using the formula MA = L2 / L1. Given that L2 = 0.8L1, we can determine the MA as MA = 0.8.
Next, we consider the angular positions of the lever system. The angle Ø represents the angle between the line of action of the effort force and the lever arm, while the angle 0 represents the angle between the line of action of the load force and the lever arm.
Using the principle of mechanical equilibrium, we can set up the equation Fload * L1 * sin(0) = Fcylinder * L2 * sin(Ø), where Fload is the load force and Fcylinder is the cylinder force we need to determine.
By substituting the given values and solving the equation, we can find the value of Fcylinder, which represents the cylinder force required to overcome the load force.
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Show p-v and t-s diagram
A simple air refrigeration system is used for an aircraft to take a load of 20 TR. The ambient pressure and temperature are 0.9 bar and 22°C. The pressure of air is increased to 1 bar due to isentropic ramming action. The air is further compressed in a compressor to 3.5 bar and then cooled in a heat exchanger to 72C. Finally, the air is passed through the cooling turbine and then it is supplied to the cabin at a pressure of 1.03 bar. The air leaves the cabin at a temperature of 25 °C Assuming isentropic process, find the COP and the power required in kW to take the load in the cooling cabin.
Take cp of air = 1.005 kj/kgk, k=1.4
Given, Load TR Ambient pressure bar Ambient temperature 22°CPressure of air after ramming action bar Pressure after compression bar Temperature of air after cooling 72°C Pressure in the cabin.
It is a process in which entropy remains constant. Air Refrigeration Cycle. Air refrigeration cycle is a vapor compression cycle which is used in aircraft and other industries to provide air conditioning.
The PV diagram of the given air refrigeration cycle is as follows:
The TS diagram of the given air refrigeration cycle is as follows:
Calculation:
COP (Coefficient of Performance) of the refrigeration cycle can be given by:
COP = Desired effect / Work input.
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Question: You are required to create a discrete time signal x(n), with 5 samples where each sample's amplitude is defined by the middle digits of your student IDs. For example, if your ID is 19-39489-1, then: x(n) = [39 4 8 9]. Now consider x(n) is the excitation of a linear time invariant (LTI) system. Here, h(n) [9 8493] - (a) Now, apply graphical method of convolution sum to find the output response of this LTI system. Briefly explain each step of the solution. Please Answer Carefully and accurately with given value. It's very important for me.
According to the statement h(n)=[0 0 0 0 9 8 4 9 3]Step 2: Convolve x(n) with the first shifted impulse response y(n) = [351 312 156 132 137 92 161 92 39].
Given that the discrete time signal x(n) is defined as, x(n) = [39 4 8 9]And, h(n) = [9 8493]Let's find the output response of this LTI system by applying the graphical method of convolution sum.Graphical method of convolution sum.
To apply the graphical method of convolution sum, we need to shift the impulse response h(n) from the rightmost to the leftmost and then we will convolve each shifted impulse response with the input x(n). Let's consider each step of this process:Step 1: Shift the impulse response h(n) to leftmost Hence, h(n)=[0 0 0 0 9 8 4 9 3]Step 2: Convolve x(n) with the first shifted impulse response
Hence, y(0) = (9 * 39) = 351, y(1) = (8 * 39) = 312, y(2) = (4 * 39) = 156, y(3) = (9 * 8) + (4 * 39) = 132, y(4) = (9 * 4) + (8 * 8) + (3 * 39) = 137, y(5) = (9 * 8) + (4 * 4) + (3 * 8) = 92, y(6) = (9 * 9) + (8 * 8) + (4 * 4) = 161, y(7) = (8 * 9) + (4 * 8) + (3 * 4) = 92, y(8) = (4 * 9) + (3 * 8) = 39Hence, y(n) = [351 312 156 132 137 92 161 92 39]
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A plane flies at a speed of 300 nautical miles per hour on a direction of N 22deg E. A wind is blowing at a speed of 25 nautical miles per hour on a direction due East. Compute the ground speed of the plane in nautical miles per hour
The ground speed of the plane can be calculated by considering the vector addition of the plane's airspeed and the wind velocity. Given that the plane flies at a speed of 300 nautical miles per hour in a direction of N 22° E and the wind is blowing at a speed of 25 nautical miles per hour due East, the ground speed of the plane is approximately 309.88 NM/hour, and the direction is N21.7deg E.
To calculate the ground speed of the plane, we need to find the vector sum of the plane's airspeed and the wind velocity.
The plane's airspeed is given as 300 nautical miles per hour on a direction of N 22° E. This means that the plane's velocity vector has a magnitude of 300 nautical miles per hour and a direction of N 22° E.
The wind is blowing at a speed of 25 nautical miles per hour due East. This means that the wind velocity vector has a magnitude of 25 nautical miles per hour and a direction of due East.
To find the ground speed, we need to add these two velocity vectors. Using vector addition, we can split the plane's airspeed into two components: one in the direction of the wind (due East) and the other perpendicular to the wind direction. The component parallel to the wind direction is simply the wind velocity, which is 25 nautical miles per hour. The component perpendicular to the wind direction remains at 300 nautical miles per hour.
Since the wind is blowing due East, the ground speed will be the vector sum of these two components. By applying the Pythagorean theorem to these components, we can calculate the ground speed. The ground speed will be approximately equal to the square root of the sum of the squares of the wind velocity component and the airspeed perpendicular to the wind.
Therefore, by calculating the square root of (25^2 + 300^2), the ground speed of the plane can be determined in nautical miles per hour.
The ground speed of the plane is approximately 309.88 NM/hour, and the direction is N21.7deg E.
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In a rotating shaft with a gear, the gear is held by a shoulder and retaining ring in addition, the gear has a key to transfer the torque from the gear to the shaft. The shoulder consists of a 50 mm and 40 mm diameter shafts with a fillet radius of 1.5 mm. The shaft is made of steel with Sy = 220 MPa and Sut = 350 MPa. In addition, the corrected endurance limit is given as 195 MPa. Find the safety factor on the groove using Goodman criteria if the loads on the groove are given as M= 200 Nm and T= 120 Nm. Please use conservative estimates where needed. Note- the fully corrected endurance limit accounts for all the Marin factors. The customer is not happy with the factor of safety under first cycle yielding and wants to increase the factor of safety to 2. Please redesign the shaft groove to accommodate that. Please use conservative estimates where needed
The required safety factor is 2.49 (approx) after redesigning the shaft groove to accommodate that.
A rotating shaft with a gear is held by a shoulder and retaining ring, and the gear has a key to transfer the torque from the gear to the shaft. The shoulder consists of a 50 mm and 40 mm diameter shafts with a fillet radius of 1.5 mm. The shaft is made of steel with Sy = 220 MPa and Sut = 350 MPa. In addition, the corrected endurance limit is given as 195 MPa. Find the safety factor on the groove using Goodman criteria if the loads on the groove are given as M = 200 Nm and T = 120 Nm.
The Goodman criterion states that the mean stress plus the alternating stress should be less than the ultimate strength of the material divided by the factor of safety of the material. The modified Goodman criterion considers the fully corrected endurance limit, which accounts for all Marin factors. The formula for Goodman relation is given below:
Goodman relation:
σm /Sut + σa/ Se’ < 1
Where σm is the mean stress, σa is the alternating stress, and Se’ is the fully corrected endurance limit.
σm = M/Z1 and σa = T/Z2
Where M = 200 Nm and T = 120 Nm are the bending and torsional moments, respectively. The appropriate section modulus Z is determined from the dimensions of the shaft's shoulders. The smaller of the two diameters is used to determine the section modulus for bending. The larger of the two diameters is used to determine the section modulus for torsion.
Section modulus Z1 for bending:
Z1 = π/32 (D12 - d12) = π/32 (502 - 402) = 892.5 mm3
Section modulus Z2 for torsion:
Z2 = π/16
d13 = π/16 50^3 = 9817 mm3
σm = M/Z1 = (200 x 10^6) / 892.5 = 223789 Pa
σa = T/Z2 = (120 x 10^6) / 9817 = 12234.6 Pa
Therefore, the mean stress is σm = 223.789 MPa and the alternating stress is σa = 12.235 MPa.
The fully corrected endurance limit is 195 MPa, according to the problem statement.
Let’s plug these values in the Goodman relation equation.
σm /Sut + σa/ Se’ = (223.789 / 350) + (12.235 / 195) = 0.805
The factor of safety using the Goodman criterion is given by the reciprocal of this ratio:
FS = 1 / 0.805 = 1.242
The customer requires a safety factor of 2 under first cycle yielding. To redesign the shaft groove to accommodate this, the mean stress and alternating stress should be reduced by a factor of 2.
σm = 223.789 / 2 = 111.8945 MPa
σa = 12.235 / 2 = 6.1175 MPa
Let’s plug these values in the Goodman relation equation.
σm /Sut + σa/ Se’ = (111.8945 / 350) + (6.1175 / 195) = 0.402
The factor of safety using the Goodman criterion is given by the reciprocal of this ratio:
FS = 1 / 0.402 = 2.49 approximated to 2 decimal places.
Hence, the required safety factor is 2.49 (approx) after redesigning the shaft groove to accommodate that.
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Calculate the change of entropy of 1 kg of air expending polytropically in a cylinder behind a piston from 6.3 bar and 550 C to 1.05 bar, the index of expension is 1.3. The R value for air is 287 Nm/kg K and the ratio of specific heats is 1.4
The formula for calculating entropy is[tex], $$\Delta S = C_p \ln{\frac{T_f}{T_i}}-R\ln{\frac{V_f}{V_i}}$$[/tex]where $C_p$ is the specific heat at constant pressure, $T_f$ and $T_i$ are the final and initial temperatures, $V_f$ and $V_i$ are the final and initial volumes, and $R$ is the gas constant.
We can use this formula to calculate the change in entropy of 1 kg of air expanding polytropically in a cylinder behind a piston from 6.3 bar and 550 C to 1.05 bar, with an expansion index of 1.3. We'll need to use some thermodynamic relationships to determine the final temperature and volume, as well as the specific heat at constant pressure.
First, let's determine the final temperature. We know that the air is expanding polytropically, which means that $PV^n$ is constant. We can use this relationship to determine the final temperature as follows:[tex]$$\frac{T_f}{T_i} = \left(\frac{P_f}{P_i}\right)^{\frac{n-1}{n}}$$$$T_f = T_i\left(\frac{P_f}{P_i}\right)^{\frac{n-1}{n}}$$$$T_f = 550\text{ K}\left(\frac{1.05\text{ bar}}{6.3\text{ bar}}\right)^{\frac{0.3}{1.3}} = 417.8\text{ K}$$[/tex]Next, we'll need to determine the final volume.
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1-Given A = 5ax - 2a, + 4a, find the expression for unit vector B if (a) B is parallel to A (b) B is perpendicular to A and B lies in xy-plane.
(a) B is parallel to A:For any vector A, the unit vector parallel to it is given by:
[tex]B = A/ |A|[/tex]For the given vector A,[tex]|A| = √(5² + 2² + 4²) = √45[/tex]
Thus, the unit vector parallel to A is given by:
[tex]B = A/ |A| = (5ax - 2ay + 4az)/√45[/tex]
(b) B is perpendicular to A and B lies in xy-plane:
For any two vectors A and B, the unit vector perpendicular to both A and B is given by:
B = A x B/|A x B|Here, [tex]A = 5ax - 2ay + 4az[/tex]For B,
we need to choose a vector in the xy-plane. Let B = bx + by, where bx and by are the x- and y-components of B respectively.
Then, we have A . B = 0 [since A and B are perpendicular]
[tex]5ax . bx - 2ay . by + 4az . 0 = 0=> 5abx - 2aby = 0=> by = (5/2)bx[/tex]
[tex]B = bx(ax + (5/2)ay)[/tex]
Therefore,[tex]B = bx(ax + (5/2)ay)/ |B|[/tex]For B to be a unit vector, we need[tex]|B| = 1⇒ B = (ax + (5/2)ay)/ √(1² + (5/2)²)[/tex]
Thus, the expression for unit vector B is given by: [tex]B = (5ax - 2ay + 4az)/√45(b) B = (ax + (5/2)ay)/√(1² + (5/2)²).[/tex]
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