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Design of springs and levers
The document discusses problems related to the design of helical springs, leaf springs, and coned disc springs. It provides examples of spring design problems involving the calculation of spring dimensions, loads, stresses, and deflections given various input parameters such as material properties, load ranges, spring indices, and dimensional constraints. The problems cover topics such as determining wire diameter, number of coils, spring rates, thicknesses of leaf springs, and stresses induced in coned disc springs. Solutions to the problems are presented using relevant spring design equations.
Design of couplings
The document discusses the design of various types of rigid and flexible couplings. It provides steps to design a flange coupling connecting two shafts transmitting 37.5 kW power at 180 rpm. Key details include calculating torque from power, selecting shaft diameter, coupling dimensions based on standards, and checking design of key and bolts for shearing and crushing. The document also provides problems and solutions for designing flange, muff, and clamp couplings for given power and speed conditions.
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Design of Temporary and Permanent Joints
This document discusses different types of temporary and permanent joints used in structures, including bolted joints, knuckle joints, cotter joints, welded joints, and riveted joints. It specifically focuses on eccentrically loaded riveted joints and provides examples of calculating the size of rivets needed based on the load and permissible stresses.
Balancing of reciprocating masses
The document discusses balancing of reciprocating masses in engines. It describes:
1. The various forces acting on reciprocating parts and how the inertia force is balanced by an opposing force on the crankshaft, leaving an unbalanced force.
2. Methods to partially balance the primary unbalanced force using a balancing mass on the crank, which changes the direction of the maximum unbalanced force.
3. How balancing is applied to two-cylinder locomotives, reducing variation in tractive force, swaying couple, and hammer blow.
Balancing of rotating masses
This document discusses balancing rotating masses on a shaft. It provides examples of balancing a single mass, balancing multiple masses in the same plane, and balancing masses in different planes. It then gives two example problems:
1) A shaft carries four masses rotating at different radii and angles, and asks to find the balancing mass and its position if placed at a given radius.
2) A shaft carries four masses at different radii and angles across three planes, and asks to find the magnitudes and positions of balancing masses placed at a given radius in two other planes.
Design of fasteners and welded joints
The document contains several examples of calculating weld sizes and bolt sizes for different welded and bolted joints. The examples involve calculating shear stresses in welds due to direct loads and bending moments and determining necessary weld sizes. They also involve calculating bolt sizes for different bolted joints based on allowable shear stresses and safety factors. The final example calculates the necessary bolt size for connecting a steam engine cylinder head based on the steam pressure, cylinder dimensions, bolt material properties, and a preload on the bolts.
Design of springs and levers
The document discusses problems related to the design of helical springs, leaf springs, and coned disc springs. It provides examples of spring design problems involving the calculation of spring dimensions, loads, stresses, and deflections given various input parameters such as material properties, load ranges, spring indices, and dimensional constraints. The problems cover topics such as determining wire diameter, number of coils, spring rates, thicknesses of leaf springs, and stresses induced in coned disc springs. Solutions to the problems are presented using relevant spring design equations.
Design of couplings
The document discusses the design of various types of rigid and flexible couplings. It provides steps to design a flange coupling connecting two shafts transmitting 37.5 kW power at 180 rpm. Key details include calculating torque from power, selecting shaft diameter, coupling dimensions based on standards, and checking design of key and bolts for shearing and crushing. The document also provides problems and solutions for designing flange, muff, and clamp couplings for given power and speed conditions.
Design of bearings and flywheel
Here are the steps to design and draw a flywheel for a four stroke four cylinder 133 kW engine running at 375 rpm with a diameter not exceeding 1 m:
Given:
Power of engine, P = 133 kW = 133000 W
Number of cylinders, n = 4
Speed of engine, N = 375 rpm
Maximum diameter, Dmax = 1 m
Step 1) Calculate the mean effective pressure (p):
p = (2*P)/(n*π*D^2*N)
p = (2*133000)/(4*π*(0.5)^2*375) = 7 bar
Step 2) Calculate the mass moment of inertia (I) required:
I
Forced vibrations unit 4
1. The document discusses forced vibrations of mechanical systems subjected to periodic external forces such as harmonic, stepped, or periodic disturbances. It provides examples and discusses the amplitude of forced vibrations.
2. Vibration isolation techniques are introduced to minimize the transmission of vibrations from machines to foundations using springs and dampers. The transmissibility ratio, which is the ratio of transmitted force to applied force, is defined.
3. Several examples are provided to calculate the natural frequency, stiffness, amplitude of vibrations, damping coefficient, and transmissibility ratio of systems undergoing forced vibrations. Resonance conditions and their effects are also considered.
Governors
Here are the key steps to solve this problem:
1) Calculate the centrifugal forces at the minimum and maximum radii using Fc = mω2r
2) Use the lever equation to relate the centrifugal forces to the spring forces:
Fc1/S1 = r1/x
Fc2/S2 = r2/x
3) The initial compression of the spring is r2 - r1
4) Use the relation between spring force and compression to find the spring constant:
ΔS/Δx = s
Where ΔS is the change in spring force (S2 - S1) and Δx is the change in compression (r2 - r1
Gyroscopic couple
1. The document discusses gyroscopic couple, which acts on a spinning object that is rotating about another axis.
2. It provides examples of gyroscopic couple in naval ships, where the spinning of propeller shafts affects steering, pitching, and rolling.
3. The document also examines the gyroscopic couple and centrifugal couple in vehicles like cars and motorcycles taking turns, and how this affects their stability.
Equivalent
The document discusses equivalent dynamical systems used to model the motion of rigid bodies acted on by external forces. It defines an equivalent dynamical system as one where: 1) the sum of the masses equals the total body mass, 2) the center of gravity of the masses matches the body's, and 3) the sum of the mass moments of inertia about the center of gravity equals the body's. It then gives an example of calculating the equivalent dynamical system for a connecting rod suspended and oscillating, determining the radius of gyration and placing one mass at the small end center and the other a distance l2 from the center of gravity.
flywheel
A punching press punches 38mm holes in 32mm thick plates, requiring 7 N-m of energy per square mm of sheared area. It punches one hole every 10 seconds. The mean flywheel speed is 25 m/s.
To calculate the motor power required, the energy per hole is calculated based on the sheared area of the hole. The mass of the flywheel required to limit speed fluctuations to 3% of the mean is then calculated using the energy variation and flywheel properties.
Unit 3 free vibration
This document covers free and damped vibrations. It defines key terms like natural frequency, damping, and damping ratio. It describes the equations of motion for an undamped single degree of freedom system and how to calculate the natural frequency. It also covers calculating the natural frequency of damped systems and defines types of damping like overdamped, underdamped, and critically damped systems. Formulas are provided for damped vibration frequency, logarithmic decrement, and damping ratio. Examples are given on calculating natural frequency, damping coefficient, and damping ratio from data provided on an oscillating system.
Transverse vibrations
This document provides instructions on how to calculate the natural frequency of vibration of a shaft that is carrying multiple loads at different points. It describes calculating: 1) the moment of inertia of the shaft, 2) the static deflection caused by each point load individually, 3) the total static deflection by summing the individual deflections, and 4) the natural frequency of transverse vibration of the shaft based on the total deflection. It then gives two example problems involving shafts of different sizes, materials, and load configurations to calculate the natural frequency.
Torsional vibrations
This document discusses torsional vibrations in shafts. It provides equations to calculate the natural frequency of torsional vibrations based on the shaft's torsional stiffness, mass moment of inertia, and material properties. As an example, it calculates the natural frequency of a flywheel mounted on a vertical shaft. It then discusses multi-rotor shaft systems and how to determine the location of nodes. Finally, it provides methods to calculate the natural frequency and node locations of stepped shafts with varying diameters connecting multiple flywheels.
Free vibrations
The document provides information on vibrations including definitions of key terms like natural frequency, forced vibrations, and damped vibrations. It also discusses single degree of freedom systems and how to calculate the natural frequency of longitudinal and transverse vibrations in beams and shafts using different methods. An example is provided to show how to calculate the natural frequencies of longitudinal and transverse vibrations for a cantilever shaft with a mass at the free end. Formulas are given for stiffness, static deflection, and natural frequency of beams and shafts.
Damped vibrations
The document discusses different types of damping in vibrating systems, including viscous, Coulomb, and critical damping. It provides equations to calculate damping coefficient, logarithmic decrement, damping ratio, natural frequency, and damped vibration frequency. Examples are given to show how to determine damping coefficient, critical damping coefficient, damping factor, logarithmic decrement, and ratio of damped to undamped frequencies based on given mass, spring constant, amplitude decay between cycles.
Three rotor system
The document describes a three rotor system with rotors A, B, and C connected by a shaft. Rotor A has an inertia of 0.15 kg-m2, rotor B has an inertia of 0.30 kg-m2, and rotor C has an inertia of 0.09 kg-m2. The system can vibrate with nodes forming between rotors C and A or between rotors C and B depending on the direction of rotation. The task is to find the natural frequency of the torsional vibrations using the given inertias, shaft dimensions, and modulus of rigidity.
Free torsional vibrations of a geared system
The document discusses the benefits of exercise for mental health. Regular physical activity can help reduce anxiety and depression and improve mood and cognitive functioning. Exercise causes chemical changes in the brain that may help protect against mental illness and improve symptoms.
Critical speed of shaft
1) The shaft carries a disc that is eccentrically mounted, subjecting the shaft to bending stresses from centrifugal force as it rotates.
2) The critical speed is the speed at which the additional deflection from bending becomes infinite, causing high vibrations and potential failure.
3) Other factors that determine the critical speed include the distance between the center of gravity of components and the axis of rotation, as well as the speed of rotation.
flywheel
1. The document contains information about flywheel design including formulas for centrifugal stress, energy variation, and determining the necessary mass and dimensions of a flywheel.
2. Key parameters that must be determined from the problem information include the energy variation, mean resisting torque, angular velocity, and coefficient of fluctuation of speed.
3. The mass and dimensions of the flywheel can then be calculated using the density of the material, maximum safe centrifugal stress, energy variation, and other specified parameters such as diameter.
Equivalent Dynamics system
The document discusses equivalent dynamical systems used to model the motion of rigid bodies acted on by external forces. It defines an equivalent dynamical system as one where: 1) the sum of the masses equals the total body mass, 2) the center of gravity of the masses matches the body's, and 3) the sum of the mass moments of inertia about the center of gravity equals the body's. It then gives an example of calculating the equivalent dynamical system for a connecting rod suspended and oscillating, determining the radius of gyration and placing one mass at the small end center and solving to find the location of the other mass.
Correction couple
The document discusses determining a correction couple to make a two-mass system dynamically equivalent. The two-mass system is a connecting rod in an internal combustion engine, with one mass of 2 kg at the gudgeon pin and another implied mass at the crank pin 250 mm away. If the connecting rod is replaced by just the 2 kg mass at the gudgeon pin, a correction couple must be applied to account for the removed mass and maintain dynamic equivalence.
Velocity and acceleration
The document discusses methods for determining the velocity and acceleration of reciprocating parts in engines, such as pistons. It presents both graphical and analytical methods. The graphical methods covered are Klein's, Ritterhaus's, and Bennett's constructions. The analytical method involves modeling the motion of a crank and connecting rod system. Equations are developed to calculate the displacement, velocity, acceleration, angular velocity, and angular acceleration of the piston and connecting rod at different positions of the crank. Examples of problems applying these equations are also provided.
TURNING MOMENT DIAGRAM
1) A turning moment diagram (TMD) graphically represents torque over crank angle and is used to calculate work done per cycle and mean torque.
2) A flywheel stores excess energy from the engine during the power stroke and releases it during other strokes to reduce fluctuations in speed. There are disc and rim types of flywheels.
3) To calculate the energy stored in a flywheel, formulas use mass, radius of gyration, angular velocity, and the coefficient of fluctuation of speed. The problem gives values for an engine to calculate the required flywheel's mass moment of inertia.
Enigne forces analysis
The document discusses formulas for calculating various forces acting on components of internal combustion engines, including:
1. Piston effort (FP), which is affected by gas forces (FL), inertia forces (FI), and friction forces (RF).
2. Force along the connecting rod (FQ), thrust on the cylinder walls (FN), thrust on the crankshaft bearings, and crank-pin effort (FT).
3. Torque (T) on the crankshaft, which is calculated as the product of the crank-pin effort (FT) and the crank radius (r).
The document also provides example calculations using the formulas to determine pressure on slide bars,
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Design of bearings and flywheel
Here are the steps to design and draw a flywheel for a four stroke four cylinder 133 kW engine running at 375 rpm with a diameter not exceeding 1 m:
Given:
Power of engine, P = 133 kW = 133000 W
Number of cylinders, n = 4
Speed of engine, N = 375 rpm
Maximum diameter, Dmax = 1 m
Step 1) Calculate the mean effective pressure (p):
p = (2*P)/(n*π*D^2*N)
p = (2*133000)/(4*π*(0.5)^2*375) = 7 bar
Step 2) Calculate the mass moment of inertia (I) required:
I
Forced vibrations unit 4
1. The document discusses forced vibrations of mechanical systems subjected to periodic external forces such as harmonic, stepped, or periodic disturbances. It provides examples and discusses the amplitude of forced vibrations.
2. Vibration isolation techniques are introduced to minimize the transmission of vibrations from machines to foundations using springs and dampers. The transmissibility ratio, which is the ratio of transmitted force to applied force, is defined.
3. Several examples are provided to calculate the natural frequency, stiffness, amplitude of vibrations, damping coefficient, and transmissibility ratio of systems undergoing forced vibrations. Resonance conditions and their effects are also considered.
Governors
Here are the key steps to solve this problem:
1) Calculate the centrifugal forces at the minimum and maximum radii using Fc = mω2r
2) Use the lever equation to relate the centrifugal forces to the spring forces:
Fc1/S1 = r1/x
Fc2/S2 = r2/x
3) The initial compression of the spring is r2 - r1
4) Use the relation between spring force and compression to find the spring constant:
ΔS/Δx = s
Where ΔS is the change in spring force (S2 - S1) and Δx is the change in compression (r2 - r1
Gyroscopic couple
1. The document discusses gyroscopic couple, which acts on a spinning object that is rotating about another axis.
2. It provides examples of gyroscopic couple in naval ships, where the spinning of propeller shafts affects steering, pitching, and rolling.
3. The document also examines the gyroscopic couple and centrifugal couple in vehicles like cars and motorcycles taking turns, and how this affects their stability.
Equivalent
The document discusses equivalent dynamical systems used to model the motion of rigid bodies acted on by external forces. It defines an equivalent dynamical system as one where: 1) the sum of the masses equals the total body mass, 2) the center of gravity of the masses matches the body's, and 3) the sum of the mass moments of inertia about the center of gravity equals the body's. It then gives an example of calculating the equivalent dynamical system for a connecting rod suspended and oscillating, determining the radius of gyration and placing one mass at the small end center and the other a distance l2 from the center of gravity.
flywheel
A punching press punches 38mm holes in 32mm thick plates, requiring 7 N-m of energy per square mm of sheared area. It punches one hole every 10 seconds. The mean flywheel speed is 25 m/s.
To calculate the motor power required, the energy per hole is calculated based on the sheared area of the hole. The mass of the flywheel required to limit speed fluctuations to 3% of the mean is then calculated using the energy variation and flywheel properties.
Unit 3 free vibration
This document covers free and damped vibrations. It defines key terms like natural frequency, damping, and damping ratio. It describes the equations of motion for an undamped single degree of freedom system and how to calculate the natural frequency. It also covers calculating the natural frequency of damped systems and defines types of damping like overdamped, underdamped, and critically damped systems. Formulas are provided for damped vibration frequency, logarithmic decrement, and damping ratio. Examples are given on calculating natural frequency, damping coefficient, and damping ratio from data provided on an oscillating system.
Transverse vibrations
This document provides instructions on how to calculate the natural frequency of vibration of a shaft that is carrying multiple loads at different points. It describes calculating: 1) the moment of inertia of the shaft, 2) the static deflection caused by each point load individually, 3) the total static deflection by summing the individual deflections, and 4) the natural frequency of transverse vibration of the shaft based on the total deflection. It then gives two example problems involving shafts of different sizes, materials, and load configurations to calculate the natural frequency.
Torsional vibrations
This document discusses torsional vibrations in shafts. It provides equations to calculate the natural frequency of torsional vibrations based on the shaft's torsional stiffness, mass moment of inertia, and material properties. As an example, it calculates the natural frequency of a flywheel mounted on a vertical shaft. It then discusses multi-rotor shaft systems and how to determine the location of nodes. Finally, it provides methods to calculate the natural frequency and node locations of stepped shafts with varying diameters connecting multiple flywheels.
Free vibrations
The document provides information on vibrations including definitions of key terms like natural frequency, forced vibrations, and damped vibrations. It also discusses single degree of freedom systems and how to calculate the natural frequency of longitudinal and transverse vibrations in beams and shafts using different methods. An example is provided to show how to calculate the natural frequencies of longitudinal and transverse vibrations for a cantilever shaft with a mass at the free end. Formulas are given for stiffness, static deflection, and natural frequency of beams and shafts.
Damped vibrations
The document discusses different types of damping in vibrating systems, including viscous, Coulomb, and critical damping. It provides equations to calculate damping coefficient, logarithmic decrement, damping ratio, natural frequency, and damped vibration frequency. Examples are given to show how to determine damping coefficient, critical damping coefficient, damping factor, logarithmic decrement, and ratio of damped to undamped frequencies based on given mass, spring constant, amplitude decay between cycles.
Three rotor system
The document describes a three rotor system with rotors A, B, and C connected by a shaft. Rotor A has an inertia of 0.15 kg-m2, rotor B has an inertia of 0.30 kg-m2, and rotor C has an inertia of 0.09 kg-m2. The system can vibrate with nodes forming between rotors C and A or between rotors C and B depending on the direction of rotation. The task is to find the natural frequency of the torsional vibrations using the given inertias, shaft dimensions, and modulus of rigidity.
Free torsional vibrations of a geared system
The document discusses the benefits of exercise for mental health. Regular physical activity can help reduce anxiety and depression and improve mood and cognitive functioning. Exercise causes chemical changes in the brain that may help protect against mental illness and improve symptoms.
Critical speed of shaft
1) The shaft carries a disc that is eccentrically mounted, subjecting the shaft to bending stresses from centrifugal force as it rotates.
2) The critical speed is the speed at which the additional deflection from bending becomes infinite, causing high vibrations and potential failure.
3) Other factors that determine the critical speed include the distance between the center of gravity of components and the axis of rotation, as well as the speed of rotation.
flywheel
1. The document contains information about flywheel design including formulas for centrifugal stress, energy variation, and determining the necessary mass and dimensions of a flywheel.
2. Key parameters that must be determined from the problem information include the energy variation, mean resisting torque, angular velocity, and coefficient of fluctuation of speed.
3. The mass and dimensions of the flywheel can then be calculated using the density of the material, maximum safe centrifugal stress, energy variation, and other specified parameters such as diameter.
Equivalent Dynamics system
The document discusses equivalent dynamical systems used to model the motion of rigid bodies acted on by external forces. It defines an equivalent dynamical system as one where: 1) the sum of the masses equals the total body mass, 2) the center of gravity of the masses matches the body's, and 3) the sum of the mass moments of inertia about the center of gravity equals the body's. It then gives an example of calculating the equivalent dynamical system for a connecting rod suspended and oscillating, determining the radius of gyration and placing one mass at the small end center and solving to find the location of the other mass.
Correction couple
The document discusses determining a correction couple to make a two-mass system dynamically equivalent. The two-mass system is a connecting rod in an internal combustion engine, with one mass of 2 kg at the gudgeon pin and another implied mass at the crank pin 250 mm away. If the connecting rod is replaced by just the 2 kg mass at the gudgeon pin, a correction couple must be applied to account for the removed mass and maintain dynamic equivalence.
Velocity and acceleration
The document discusses methods for determining the velocity and acceleration of reciprocating parts in engines, such as pistons. It presents both graphical and analytical methods. The graphical methods covered are Klein's, Ritterhaus's, and Bennett's constructions. The analytical method involves modeling the motion of a crank and connecting rod system. Equations are developed to calculate the displacement, velocity, acceleration, angular velocity, and angular acceleration of the piston and connecting rod at different positions of the crank. Examples of problems applying these equations are also provided.
TURNING MOMENT DIAGRAM
1) A turning moment diagram (TMD) graphically represents torque over crank angle and is used to calculate work done per cycle and mean torque.
2) A flywheel stores excess energy from the engine during the power stroke and releases it during other strokes to reduce fluctuations in speed. There are disc and rim types of flywheels.
3) To calculate the energy stored in a flywheel, formulas use mass, radius of gyration, angular velocity, and the coefficient of fluctuation of speed. The problem gives values for an engine to calculate the required flywheel's mass moment of inertia.
Enigne forces analysis
The document discusses formulas for calculating various forces acting on components of internal combustion engines, including:
1. Piston effort (FP), which is affected by gas forces (FL), inertia forces (FI), and friction forces (RF).
2. Force along the connecting rod (FQ), thrust on the cylinder walls (FN), thrust on the crankshaft bearings, and crank-pin effort (FT).
3. Torque (T) on the crankshaft, which is calculated as the product of the crank-pin effort (FT) and the crank radius (r).
The document also provides example calculations using the formulas to determine pressure on slide bars,
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