Mechanical Engineering Test - 5

Explanation:

Variables in welding:

Important welding variables apart from flux

Electrode wire size, welding voltage, current, and speed.

Welding speed:

• If the welding speed is increased, power or heat input per unit length of the weld is decreased, less welding material is applied per unit length of the weld, and consequently less weld reinforcement results and penetration decreases.
• Travel speed is used primarily to control bead size and penetration. It is interdependent with the current.
• Excessive high travel speed decreases wetting action, increases the tendency for undercut, arc blow, porosity, and uneven bead shapes while slower travel speed reduces the tendency to porosity and slag inclusion.

∴ too fast welding speed in arc welding would result in the too-small bead.

Welding current:

• Welding current is the most influential variable as it controls electrode melting rate, depth of penetration and the amount of base metal fused.
• Very high current shall lead to too much penetration resulting in burn through in the metal being joined, excessive reinforcement and increased weld shrinkage and, therefore, a large amount of distortion.
• A Low current shall lead to insufficient penetration, lack of fusion and unstable arc.
• Too low welding current in arc welding would result in excessive piling up of weld metal.

∴ too high welding current in arc welding would result in excessive spatter.

Concept:

Independent events: If occurrence of one event do not depend on the occurrence of other event, then both the events are called independent events.

In case of independent events,

P(A/B) = P(B)

∴ P(A ∩ B) = P(A) ⋅ P(B)

Addition theorem of probability:

P(A ∪ B) = P(A) + P(B) – P(A ∩ B)

Where, A ∪ B = Atleast one of A or B

Multiplication theorem of probability:

P(A ∩ B) = P(A/B) ⋅ P(B)

Where, A ∩ B = Simultaneous occurrence of A and B

Calculation:

Concept:

Streamline and equipotential lines are orthogonal to each other.

Slope of the Stream line is calculated as 

xdy + ydx = 0

Concept:

Properties of Eigenvalues:

• The sum of Eigenvalues of a matrix A is equal to the trace of that matrix A
• The product of Eigenvalues of a matrix A is equal to the determinant of that matrix A
• If λ is an eigenvalue of a matrix A, then λn will be an eigenvalue of a matrix An.
• If λ is an eigenvalue of a matrix A, then kλ will be an eigenvalue of a matrix kA where k is a scalar

Calculation:

Let the other two eigenvalues be λ₁ and λ₂

Since the sum of Eigenvalues of a matrix A is equal to the trace of that matrix A, we can write:

1 + λ₁ + λ₂ = 3

λ₁ + λ₂ = 2   ---(1)

Now, the product of eigenvalues = det A, i.e.

1.λ₁λ₂ = 2

Using Equation (1), we get:

λ₁ (2 - λ₁) = 2

2λ₁ – λ²₁ = 2

λ²₁ - 2 λ₁ + 2 = 0

λ₁ = 1 + i

λ₂ = 1 – i

So, the eigenvalues of A are 1, 1 + i, 1 – i

Now, the eigenvalues of A² will be:

= 1², (1 + i)², (1 - i)²

= 1, 2i, -2i

(A² – 2I) X = A²X – 2IX

According to the Cayley-Hamilton theorem, every matrix 'A' satisfies its own characteristic equation, i.e.

= λ²X – 2X

= (λ² - 2) X

∴ The Eigenvalues of (A² – 2I) is λ² – 2

So, eigenvalues are -1, 2i – 2, -2i – 2

Concept:

Linear Differential Equation: Linear differential equations are those in which the dependent variable, its derivatives occur only in the first degree and they are not multiplied together. It is of the form:

Concept:

To solve this, impulse momentum equation is used.

First consider the control volume as shown in the figure.

Let the force on plate exerted by jet is F_x and force on Control Volume is R_x

∑F_x = 0 ⇒ F_x = R_x

Apply Impulse momentum equation on control volume,