Mechanical Engineering

Graphical Representation of Displacement with Respect to Time The displacement of a moving body at different instants of time can be represented using a graph. Such a graph is drawn by taking displacement on the Y-axis and time on the X-axis. This curve is called the s–t curve . We consider two important cases: 1. When the body moves with uniform velocity When a body moves with constant velocity, it covers equal distances in equal intervals of time. Plotting displacement on the Y-axis and time on the X-axis produces a straight-line s–t graph , as shown in Fig. 2.1 (a). The equation of motion for uniform velocity is: s = u × t Thus, velocity at any instant is: Velocity at t₁ = s₁ / t₁ Velocity at t₂ = s₂ / t₂ Since velocity remains constant: s₂ – s₁ / t₂ – t₁ = tan θ Here, tan θ is the slope of the s–t graph. Therefore, the slope of the displacement-time curve gives the velocity of the body.              ...

Kinematics of Motion Introduction In the previous chapter, we studied that the Theory of Machines deals with the motion of machine parts and the forces acting on them. In this chapter, we focus only on the kinematics of motion , which covers the relative motion of bodies without considering the forces responsible for that motion. In simple terms, kinematics explains the geometry of motion and important concepts such as displacement, velocity, and acceleration. Plane Motion When a body moves in such a way that its motion is restricted to a single plane, it is known as plane motion . Such motion may be rectilinear or curvilinear . Rectilinear Motion Rectilinear motion is the simplest form of motion and occurs along a straight line. It is often referred to as translatory motion . Curvilinear Motion Curvilinear motion occurs along a curved path. When the motion is restricted to a plane but follows a curved trajectory, it is called plane curvilinear motion...

Force Force is a very important concept in engineering science. It may be described as an agent that produces motion, stops motion, or tends to change the existing state of motion of a body. In simple words, force can create, destroy, or modify motion. Resultant Force When multiple forces such as P, Q, R act on a particle at the same time, a single force that can replace all of them and produce the same overall effect is called the resultant force . The individual forces are known as component forces . The method of determining the resultant of these forces is known as the composition of forces . The resultant may be found analytically, graphically, or by using the following laws: Parallelogram law of forces: “If two forces act simultaneously on a particle, their resultant can be represented in magnitude and direction by the diagonal of a parallelogram formed using the two forces as adjacent sides.” Triangle law of forces: “If tw...

Fundamental Units Measuring physical quantities is one of the most essential tasks in science and engineering. Every measurement is expressed using standard and internationally accepted units known as fundamental units . In this chapter, all quantities are expressed using these three basic units: Length (L or l) Mass (M or m) Time (t) Derived Units Certain units are formed by combining fundamental units. These are called derived units . Examples include units of area, velocity, pressure, acceleration, and many more. Systems of Units Four systems of units are commonly used and globally recognized: C.G.S. System F.P.S. System M.K.S. System S.I. System C.G.S. Units In this system, the fundamental units are: centimetre for length, gram for mass, and second for time. This system is often referred to as the physicist’s unit system or absolute system. F.P.S. Units In the F.P.S. system, the basic units are: foot (length), pound (mass), and ...

Definition The subject Theory of Machines can be described as that branch of engineering science which focuses on the study of relative motion between different components of a machine and the forces acting on them . The understanding of this subject is extremely important for engineers while designing the various machine parts. Note: A machine is a device that receives energy in a usable form and converts it to perform a specific kind of work. Sub-divisions of Theory of Machines The Theory of Machines is generally divided into the following four major branches: Kinematics: This branch deals with the motion of machine parts without considering the forces causing that motion. It focuses only on relative movement between components. Dynamics: This part of Theory of Machines studies the forces and their effects acting on machine elements while they are in motion. Kinetics: This branch examines the inertia forces that arise due to the ...

ANALYSIS OF BARS OF COMPOSITE SECTIONS A composite bar is made up of two or more bars of different materials, of equal lengths, fixed rigidly so that they behave as one single unit. When such a bar is subjected to axial tensile or compressive load , the load is shared between each material according to its stiffness. Important Points Strain (extension per unit length) in each bar is the same , because bars are connected rigidly. Total load on the composite bar = Sum of loads carried by each material. Consider a Composite Bar Let the bar consist of two materials: L = Length of both bars A₁, A₂ = Cross-sectional areas of bar 1 and bar 2 E₁, E₂ = Young’s Modulus of materials 1 and 2 P = Total load P₁, P₂ = Load shared by bar 1 and bar 2 σ₁, σ₂ = Stress in bar 1 and bar 2 Total Load P = P₁ + P₂     ...(i) Stress in Bar 1 σ₁ = P₁ / A₁ Stress in Bar 2 σ₂ = P₂ / A₂ Strain Condition Because bars deform equally: σ₁ / ...

1- Extension of a Tapered Rectangular Steel Bar A rectangular steel bar is 2.8 m long and 15 mm thick . It carries an axial tensile load of 40 kN . The bar tapers in width from 75 mm at one end to 30 mm at the other end. If the modulus of elasticity of steel is E = 2 × 10 5 N/mm² , find the extension of the bar. Given: Length, L = 2.8 m = 2800 mm Thickness, t = 15 mm Load, P = 40 kN = 40,000 N Width at bigger end, a = 75 mm Width at smaller end, b = 30 mm Modulus of Elasticity, E = 2 × 10 5 N/mm² Formula Used: For a tapered rectangular bar, dL = (P × L) / (E × t × (a − b)) × ln(a/b) Substituting Values: dL = (40,000 × 2800) / (2 × 10 5 × 15 × (75 − 30)) × ln(75/30) = 0.8296 × 0.9163 Extension, dL = 0.76 mm ✔ Final Answer: 0.76 mm 2- Find Axial Load from Extension in a Tapered Bar A rectangular steel bar 400 mm long and 10 mm thick extends by 0.21 mm when loaded. The width tapers uniformly from 100 mm to 50 mm . Given E = 2...

Analysis of Uniformly Tapering Circular Rod A circular bar whose diameter gradually reduces from D₁ at one end to D₂ at the other end is shown in Fig. 1.13. Such a bar is said to be uniformly tapering . Let: P = Axial tensile load acting on the bar L = Total length of the bar E = Young’s Modulus of the material Diameter at any Section Consider a very small element of the rod at a distance x from the left end. The diameter at this section may be written as: Dₓ = D₁ − kx where k = (D₁ − D₂) / L Area of Cross-Section at Distance x Aₓ = (π/4) × (D₁ − kx)² Stress at Distance x The stress in the infinitesimal section is: σₓ = P / Aₓ σₓ = 4P / [π (D₁ − kx)²] Strain at Distance x eₓ = σₓ / E eₓ = 4P / [πE (D₁ − kx)²] Extension of a Small Element dx dL = eₓ × dx dL = [4P dx] / [πE (D₁ − kx)²] Total Extension of the Rod To get the overall elongation, integrate from x = 0 to x = L : L_total = ∫ (4P / πE (D₁ − kx)² ) dx Performing the inte...

Analysis of Bars of Varying Sections Many practical bars are made of different lengths and with different diameters, so their cross-sectional areas are not the same along the length. Figure 1.6(a) shows a bar composed of three segments, each with its own area and length. The entire bar is subjected to an axial load P . Even though the same load acts through all sections, the stress, strain and extension in each part will be different because the areas (and possibly material properties) are different. The total change in length of the bar is obtained by adding the change in length of each segment. Let: P = axial load on the bar L₁, L₂, L₃ = lengths of sections 1, 2 and 3 A₁, A₂, A₃ = cross-sectional areas of sections 1, 2 and 3 E = Young’s Modulus for the material of the bar (same for all sections here) Stress in each section: σ₁ = P / A₁     (section 1) σ₂ = P / A₂     (section 2) σ₃ = P / A₃     (section 3...

Problem 1.1 – Stress, Strain and Elongation of a Rod A rod of 150 cm length and 2.0 cm diameter is subjected to an axial tensile force of 20 kN . The modulus of elasticity of the rod material is: E = 2 × 10 5 N/mm² Determine: (i) The stress (ii) The strain (iii) The elongation of the rod Given Data Length , L = 150 cm Diameter , D = 2.0 cm = 20 mm Load , P = 20 kN = 20,000 N Modulus of Elasticity , E = 2 × 10 5 N/mm² 1. Calculate Cross-Sectional Area For a circular rod: A = (π/4) × D² A = (π/4) × (20)² = 100π mm² 2. Calculate Stress (σ) Using the formula: σ = P / A σ = 20,000 / 100π = 63.662 N/mm² Answer: Stress = 63.662 N/mm² 3. Calculate Strain (e) Using Hooke’s Law ratio: e = σ / E e = 63.662 / (2 × 10 5 ) = 0.000318 Answer: Strain = 0.000318 4. Calculate Elongation (ΔL) Using the strain formula: e = ΔL / L Therefore: ΔL = e × L ΔL = 0.000318 × 150 = 0.0477 cm Answer: Elongation = 0.0477 cm ...

Constitutive Relationship Between Stress and Strain For One-Dimensional Stress System For a uniaxial or one-directional stress condition (i.e., normal stress acting only in one direction), the relationship between stress and strain follows Hooke’s Law . According to this law, when a material is loaded within its elastic range, the normal stress developed in the body is directly proportional to the strain it produces. This implies that the ratio of normal stress to corresponding strain remains a constant as long as the material stays within the elastic limit. This constant is known as the Modulus of Elasticity or Young’s Modulus . Normal stress / Corresponding strain = Constant or σ / e = E Where: σ = Normal stress e = Strain E = Young’s Modulus e = σ / E     ...(1.7 A) The above expression provides the stress–strain relation for normal stress acting in a single direction. For Two-Dimensional Stress System Before establishing th...

Elasticity and Elastic Limit Whenever an external load acts on a material, the body experiences deformation. If the applied load is removed and the body fully regains its original size and shape (with all deformation disappearing), the material is called an elastic body . This characteristic — the ability of a material to return to its original configuration after the external force is withdrawn — is known as elasticity . A body will return to its initial form only when the deformation produced by the force remains within a certain allowable range. There exists a specific maximum force up to which the deformation will completely vanish after unloading. This limiting value of force is termed the elastic limit . If the applied stress goes beyond this elastic limit, the material will lose part of its elastic behavior. Even after removing the force, the body will not fully return to its original shape — a permanent deformation will remain. Hooke’s Law and Elastic...

Types of Stresses In engineering mechanics, a body may experience two primary types of stresses: normal stress or shear stress . Normal stress is the stress that acts perpendicular to the surface area. It is commonly represented by the symbol σ (sigma) . Normal stress is further divided into tensile and compressive stresses. Tensile Stress Tensile stress develops in a material when it is pulled by two equal and opposite forces. As shown in Fig. 1.1(a) , when a bar is stretched, its length increases. This induced stress is known as tensile stress . The relative increase in length compared to the original length is termed as tensile strain . Tensile stress acts perpendicular to the area and tends to pull the section apart. Let: P = Applied pull (force) A = Cross-sectional area L = Initial length dL = Increment in length σ = Tensile stress ε = Tensile strain When a section x–x divides the bar, the internal resisting force balances the applied...