Kinetic theory of liquid j. frenkel 1955 free download

Next SlideShares. Download Now Download to read offline and view in fullscreen. Download Now Download Download to read offline. Applications of fluid mechanics Download Now Download Download to read offline. Vishu Sharma Follow. Ex Associate Consultant, Wipro Technologies. Applications of Fluid Mechanics. Fluid mechanics. Control of spring mass damper system using Xcos Scilab. Fundamental concepts of fluid mechanics. Introduction to refrigeration systems. Composite materials.

Shear Force and Bending Moment Diagrams. Measuring instruments. Related Books Free with a 30 day trial from Scribd. Germany, September Elsevier Books Reference. Elsevier Books Reference. Related Audiobooks Free with a 30 day trial from Scribd. Elizabeth Howell. Mariam Magdy. Leonora Fuentes-Iglesias. Babar Ali Muhammad Ashraf. Sudhir Nikam.

Saurabh Pathoi. Prabhu Natarajan. Melissa Marie Dimaculangan. Show More. Views Total views. Actions Shares. No notes for slide. Applications of fluid mechanics 1. It is the study of fluids at rest or in motion. It is a branch of continuum mechanics, a subject which models matter without using the information that it is made out of atoms; that is, it models matter from a macroscopic viewpoint rather than from microscopic.

A fluid is a substance that deforms continuously under the application of a shear tangential stress no matter how small the shear stress may be.

Fluids tend to flow when we interact with them while solids tend to deform or bend. We can also define a fluid as any substance that cannot sustain a shear stress when at rest. We refer to solids as being elastic and fluids as being viscous. Substances which exhibits both springiness and friction; they are viscoelastic.

Many biological tissues are viscoelastic. The basic laws, which are applicable to any fluid, are- 1. The conservation of mass 2. The principle of angular momentum 4. The first law of thermodynamics 5. In the first case the resulting equations are differential equations; solution of the differential equations of motion provides a means of determining the detailed behavior of the flow.

We often are interested in the gross behavior of a device; in such cases it is more appropriate to use integral formulations of the basic laws. B-Lagrangian versus Eulerian Approach Method of description that follows the particle is referred to as the Lagrangian method of description. In Lagrangian approach we analyze a fluid flow by assuming the fluid to be composed of a very large number of particles whose motion must be described.

In control volume analyses, it is convenient to use the field, or Eulerian, method of description, which focuses attention on the properties of a flow at a given point in space as a function of time. In the Eulerian method of description, the properties of a flow field are described as functions of space coordinates and time. Many exciting areas have developed in the last quarter-century. Some examples include environmental and energy issues e.

We will discuss these areas in details. Weather forecasts are made by collecting quantitative data about the current state of the atmosphere at a given place and using scientific understanding of atmospheric processes to project how the atmosphere will change. Once an all-human endeavor based mainly upon changes in barometric pressure, current weather conditions, and sky condition, weather forecasting now relies on computer-based models that take many atmospheric factors into account.

There are a variety of end uses to weather forecasts. Weather warnings are important forecasts because they are used to protect life and property. Forecasts based on temperature and precipitations are important to agriculture, and therefore to traders within commodity forecasts can be used to plan activities around these events, and to plan ahead and survive them.

It was not until the 20th century that advances in the understanding of atmospheric physics led to the foundation of modern numerical weather prediction. Practical use of numerical weather prediction began in , spurred by the development of programmable electronic computers.

The basic idea of numerical weather prediction is to sample the state of the fluid at a given time and use the equations of fluid dynamics and thermodynamics to estimate the state of the fluid at some time in the future.

Humans are required to interpret the model data into weather forecasts that are understandable to the end user. Humans can use knowledge of local effects which may be too small in size to be resolved by the model to add information to the forecast.

While increasing accuracy of forecast models implies that humans may no longer be needed in the forecast process at some point in the future, there is currently still a need for human intervention. Aerodynamics is a sub-field of fluid dynamics and gas dynamics, and many aspects of aerodynamics theory are common to these fields. Most of the early efforts in aerodynamics worked towards achieving heavier-than air flight, which was first demonstrated by Wilbur and Orville Wright in Since then, the use of aerodynamics through mathematical analysis, empirical approximations, wind tunnel experimentation, and computer simulations has formed the scientific basis for ongoing developments in heavier-than-air flight and a number of other technologies.

Understanding the motion of air around an object often called a flow field enables the calculation of forces and moments acting on the object. In many aerodynamics problems, the forces of interest are the fundamental forces of flight: lift, drag, thrust, and weight.

Of these, lift and drag are aerodynamic forces, i. As aircraft speed increased, designers began to encounter challenges associated with air compressibility at speeds near or greater than the speed of sound. The differences in air flows under these conditions led to problems in aircraft control, increased drag due to shock waves, and structural dangers due to aeroelastic flutter.

Stress distribution on fast moving train Turbomachinery, in mechanical engineering, describes machines that transfer energy between a rotor and a fluid, including both turbines and compressors.

While a turbine transfers energy from a fluid to a rotor, compressor transfers energy from a rotor to a fluid In general, the two kinds of turbomachines encountered in practice are open and closed turbomachines. Open machines such as propellers, windmills, and unshrouded fans act on an infinite extent of fluid, whereas, closed machines operate on a finite quantity of fluid as it passes through housing or casing.

At a very basic level, hydraulics is the liquid version of pneumatics. Fluid mechanics provides the theoretical foundation for hydraulics, which focuses on the engineering uses of fluid properties. In fluid power, hydraulics are used for the generation, control, and transmission of power by the use of pressurized liquids. Hydraulic topics range through some part of science and most of engineering modules, and cover concepts such as pipe flow, dam design, An ideal coolant has high thermal capacity, low viscosity, is low-cost, non-toxic, and chemically inert, neither causing nor promoting corrosion of the cooling system.

Some applications also require the coolant to be an electrical insulator. While the term coolant is commonly used in automotive and HVAC applications, in industrial processing, heat transfer fluid is one technical term more often used, in high temperature as well as low temperature manufacturing applications.

Another industrial sense of the word covers cutting fluids. Cooling effect can be achieved either through free or forced convection.

Its high heat capacity and low cost makes it a suitable heat-transfer medium. It is usually used with additives, like corrosion inhibitors and antifreeze. Air cooling uses either convective airflow passive cooling , or a forced circulation using fans in electronics. Its thermal conductivity is higher than all other gases, it has high specific heat capacity, low density and therefore low viscosity, which is an advantage for rotary machines susceptible to windage losses.

Hydrogen-cooled turbogenerators are currently the most common electrical generators in large power plants. Helium has a low tendency to absorb neutrons and become radioactive.

Oils are used for applications where water is unsuitable. With higher boiling points than water, oils can be raised to considerably higher temperatures above degrees Celsius without introducing high pressures within the container or loop system in question.

Mineral oils serve as both coolants and lubricants. Purpose-designed nanoparticles of e. CuO, alumina titanium dioxide, carbon nanotubes, silica, or metals e. Copper, or silver nanorods dispersed into the carrier liquid enhance the heat transfer capabilities of the resulting coolant compared to the carrier liquid alone.

The experiments however did not prove so high thermal conductivity improvements, but found significant increase of the critical heat flux of the coolants. The particles form rough porous surface on the cooled object, which encourages formation of new bubbles, and their hydrophilic nature then helps pushing them away, hindering the formation of the steam layer.

These fluids are engineered colloidal suspensions of nanoparticles in a base fluid. The nanoparticles used in nanofluids are typically made of metals, oxides, carbides, or carbon nanotubes. Common base fluids include water, ethylene glycol and oil. They exhibit enhanced thermal conductivity and the convective heat transfer coefficient compared to the base fluid.

Knowledge of the rheological behavior of nanofluids is found to be very critical in deciding their suitability for convective heat transfer applications Process engineering encompasses a vast range of industries, such as chemical, petrochemical, agriculture, mineral processing, advanced material, food, pharmaceutical, software development and biotechnological industries.

Several accomplishments have been made in Process Systems Engineering- Process design: synthesis of energy recovery networks, synthesis of distillation systems azeotropic , synthesis of reactor networks, hierarchical decomposition flowsheets, design multiproduct batch plants. Design of the production reactors for the production of plutonium, design of nuclear submarines. Process control: model predictive control, controllability measures, robust control, nonlinear control, statistical process control, process monitoring, thermodynamics-based control Process operations: scheduling process networks, multiperiod planning and optimization, data reconciliation, real-time optimization, flexibility measures, fault diagnosis.

Refrigeration is a process of moving heat from one location to another in controlled conditions. The work of heat transport is traditionally driven by mechanical work, but can also be driven by heat, magnetism, electricity, laser, or other means. Refrigeration has many applications, including, but not limited to: household refrigerators, industrial freezers, cryogenics, and air conditioning.

The introduction of refrigeration allowed for the hygienic handling and storage of perishables, and as such, promoted output growth, consumption, and nutrition. The change in our method of food preservation moved us away from salts to a more manageable sodium level.

Probably the most widely used current applications of refrigeration are for air conditioning of private homes and public buildings, and refrigerating foodstuffs in homes, restaurants and large storage warehouses. The use of refrigerators in kitchens for storing fruits and vegetables has allowed adding fresh salads to the modern diet year round, and storing fish and meats safely for long periods. HVAC heating,ventilating, and air conditioning; also heating, ventilation, and air conditioning is the technology of indoor and vehicular environmental comfort.

Its goal is to provide thermal comfort and acceptable indoor air quality. HVAC system design is a sub-discipline of mechanical engineering, based on the principles of thermodynamics, fluid mechanics, and heat transfer. HVAC is important in the design of medium to large industrial and office buildings such as skyscrapers, on board vessels, and in marine environments such as aquariums, where safe and healthy building conditions are regulated with respect to temperature and humidity, using fresh air from outdoors.

Rapid deployment of renewable energy and energy efficiency, and technological diversification of energy sources, would result in significant energy security and economic benefits. It would also reduce environmental pollution such as air pollution caused by burning of fossil fuels and improve public health, reduce premature mortalities due to pollution and save associated health costs.

Determine the resultant of the following force and its location 2. Replace the two forces acting on the post by a resultant force and couple moment at point O. A rectangular concrete slab support loads at its four corners as shown in figure. Determine the resultant of the forces and the point of application of the resultant. A rectangular block is subjected to three forces as shown in figure. Reduce them into an equivalent force couple system acting at A.

Equilibrium of Non-Coplanar Concurrent Forces In vector notation, the equation of equilibrium can be summarized as A balloon is moored by three cables as shown. The lift on the balloon is N. Draw the FBD of the balloon and write a vector expression for the tension in each cord. The square steel plate has a mass of kg withmass center at its center G. Calculate the tension ineach of the three cables with which the plate islifted while remaining horizontal. All dimensions are in meters.

A tripod supports a load of 2 kN as shown in figure. The ends A, B and C in the x-z plane. Find the forces in the three legs of the tripod 2. Equilibrium of Non-Coplanar Non-Concurrent Forces This following question is just to rise your curiosity regarding equilibrium of non-coplanar non concurrent systems.

The vertical mast supports the 4-kN force and is constrained by the two fixed cables BC and BD and by a ball-and-socket connection at A. Calculate the tension in BD. FRICTION In the preceding chapters we have usually assumed that the forces of action and reaction between contacting surfaces act normal to the surfaces.

This assumption is valid only between smooth surfaces. When two surfaces are moving against each other, these bumps interlock against each other and oppose the motion. Friction originates from this opposition.

Types of Friction a Dry Friction : Dry friction occurs when the unlubricated surfaces of two solids are in contact under a condition of sliding or a tendency to slide. For highly elastic materials the recovery from deformation occurs with very little loss of energy due to internal friction. Types of Dry Friction 1. Static Friction : It is the friction experienced by a body when it is at rest or on the verge of motion. Eg — Body resting on an ramp with small inclination 2.

Dynamic Friction : It is the friction experienced by a body when it is in motion. Static friction is the frictional force acting between two surfaces which are attempting to move, but are not moving.

The body remains in equilibrium 2. Static friction is proportional to the external forces and increases linearly with the force applied until it reaches a maximum value. Static friction could have a value less or greater than the value for kinetic friction. But it cannot increase beyond a maximum value called the limiting value of frictional force.

The coefficient of static friction is less than the dynamic friction 5. Eg : Body resting on an ramp with small inclination 1 Kinetic friction is the frictional force acting between two surfaces which are in motion against each other 2. Kinetic friction remains constant regardless of the force applied. It is independent of mass and acceleration. Kinetic friction has a value less than the limiting value of static friction.

Eg : Resisting force experienced by a rolling skater When two bodies are in contact, the direction of force of friction on one of them is opposite to the direction in which this body has a tendency to move tangent to the surface. The frictional force is independent of the area of contact of the surfaces. The force of friction is dependent upon the types of materials of the two bodies in contact. The limiting frictional force bears a constant ratio with the normal reaction.

When one body is just on the verge of sliding over another body, the force of friction is maximum and this maximum frictional force is called Limiting Static Frictional Force 6. Limiting static frictional force is greater than kinetic frictional force for any two surfaces of contact. The kinetic frictional force is independent of the relative velocities of the bodies in contact.

Limiting Force of Friction The maximum value of frictional force at the surfaces of contact when the body is at the verge of motion is called limiting static frictional force Fs. When this value is reached, the block is in unstable equilibrium since any further increase in P will cause the block to move. It is experimentally found that friction directly varies as the applied force until the movement produces in the body. A pull of 20 N, inclined at to the horizontal plane is required to just move a body placed on a rough horizontal plane.

But the push required to move the body is 25 N inclined at Find the weight of the body and the coefficient of friction. Determine the coefficient of static friction between the plane and the crate, and find the mass of the crate. Crates A and B weigh N and respectively.

They are connected together with a cable and placed on the inclined plane. If the angle is gradually increased, determine when the crates begin The coefficient of friction between the inclined plane and the block A is 0. Two blocks A and B, connected by a horizontal rod and frictionless hinges are supported on two rough planes as shown in Figure.

The coefficients of friction are 0. If the block B weighs N, what is the smallest weight of block A, that will hold the system in equilibrium? An inclined plane as shown in Figure is used to unload slowly a body weighing N from a truck 1. The coefficient of friction between the underside of the body and the plank is 0. State whether it is necessary to push the body down the plane or hold it back from sliding down.

What minimum force is required parallel to the plane for this purpose? Two blocks A and B of weights 1 kN and 2 kN respectively are in equilibrium position as shown in Figure.

If the coefficient of friction between the two blocks as well as the block B and the floor is 0. Determine the least value of the force P to cause motion to impend rightwards. Assume the co- efficient of friction under the blocks to be 0. A uniform ladder of weight kN rests against a smooth wall at B and the end A rests on the rough horizontal plane for which the coefficient of static friction is 0.

Determine the angle of inclination of the ladder, the frictional force at A and the normal reactions at A and B 2. A uniform ladder 3 m long weighs N. The coefficient of friction between the wall and the ladder is 0. The ladder, in addition to its own weight, has to support a man of N at its top at B.

Calculate: a. The horizontal force P to be applied to ladder at the floor level to prevent slipping. If the force P is not applied, what should be the minimum inclination of the ladder with the horizontal, so that there is no slipping of it with the man at its top.

A ladder of length 5m and weight N is placed on a flat floor against a vertical wall. If the coefficient of friction are 0. Determine the distance s to which the kg painter can climb without causing the 4-m ladder to slip at its lower end A. The top of the kg ladder and at the ground the coefficient of static friction is 0. The coefficient of friction between the ladder and the wall is 0. If a man, whose weight is one-half of that of the ladder ascends it, how high will it be when the ladder slips?

A ladder shown in Figure is 4 m long and is supported by a horizontal floor and vertical wall. The coefficient of friction at the wall is 0.

The weight of the ladder is 30 The ladder also supports a vertical load of at C. The weight of the ladder is N and acts at its middle. The ladder is at the point of sliding, when a man weighing N stands on a rung 1. Calculate the coefficient of friction between the ladder and the floor.

Assuming the coefficient of friction between all the surfaces in contact to be 0. Determine the smallest horizontal force P required to pull out wedge A. The crate has a weight of N and the coefficient of static friction at all contacting surfaces is 0. Neglect the weight of the wedge. Two Blocks are resting on the wall as shown in figure.

Find the range of value of force P applied to the lower block for which the system remains in equilibrium. Coefficient of friction is 0. But if we assume that a body in equilibrium undergoes small imaginary displacements known as virtual displacements consistent with the geometrical conditions, imaginary work is said to be done by the system of forces.

This imaginary work done is called virtual work. The body A shown below is in equilibrium under the action of forces. However it assumed that it undergoes an imaginary displacement in the direction of force F. Principle of virtual work If a system of forces acting on a body or a system of bodies is in equilibrium, the total virtual work done by the forces acting on the body or system of bodies is zero for any virtual displacement consistent with geometrical conditions.

For example, consider the free-body diagram of the particle ball that rests on the floor. Find support reactions of the following beams using virtual work It is denoted by G A.

Locate the centroid of the following areas 3. Determine mathematically the position of centre of gravity of the section. Find centre of gravity of the following lamina. Determine the coordinates of the centroid of the given shaded area Problems for Practice 1. Find centroid of the shaded area. First Theorem The surface area A of a surface of revolution generated by rotating a plane curve about an axis external to it and on the same plane is equal to the product of the arc length of the curve and the distance y traveled by its geometric centroid.

Examples Second Theorem The second theorem states that the volume V of a solid of revolution generated by rotating a plane area about an external axis is equal to the product of the area A and the distance y traveled by its geometric centroid.

Obtain the expression for the area of a surface generated by rotation of a semicircular arc of radius r about an axis passing through its end points. Obtain the expression for the volume of a body generated by revolution of a triangle when its base h is in touch with the axis of rotation and the length of the other side is r. Moment of Inertia is given as the sum of the products of each area in the lamina with the square of its distance from the axis of rotation.

It is denoted with I for an axis that lies in the plane or with a J for an axis perpendicular to the plane referred as polar moment of inertia. Parallel Axis Theorem Parallel Axis Theorem states that the moment of inertia of a plane lamina about any axis is equal to the sum of moment of inertia about a parallel axis passing through the centroid and the product of area and square of the distance between the two parallel axes. Proof In Figure, the x0-y0 axes pass through the centroid C of the area.

OR Perpendicular Axis Theorem states that the polar moment of inertia is equal to the sum of moment of inertias of the two centroidal axis in the plane of the lamina Proof In Figure, the x0-y0 axes pass through the centroid C of the area.

Divide the given area into its simpler shaped parts. Locate the centroid of each part and indicate the perpendicular distance from each centroid to the desired reference axis. Determine moment of inertia about the x and y axis. Find the moment of inertia of the section about the centroidal axis Determine the moments of inertia of the cast-iron beam section about horizontal and vertical axes passing through the centroid of the section.

Find moments of inertia about the centroidal axes of the section shown in figure 5. Find the moment of inertia the beam section about an axis passing through its centre of gravity and parallel to X-X axis. Find the value of x so that the centre of gravity of the uniform lamina shown in figure remains at the centre of the rectangle ABCD Problems for Practice 1.

Find the MI of the lamina with a circular hole of 30 mm diameter about the axis AB 2. A rectangular hole is made in a triangular section as shown in figure. Determine the moment of inertia of the section abut X-X axis passing through the centre of gravity of the section and parallel to BC. Also find moment of inertia through BC Find moment of inertia with respect to centroidal axes Polar Moment of Inertia The moment of inertia of a plane lamina about an axis perpendicular to the lamina and passing through the centroid is called polar moment of inertia Radius of Gyration The radius of gyration of an area about an axis is the distance to a long narrow strip whose area is equal to the area of the lamina and whose moment of inertia remain the same as that of the original area Consider a lamina of area A whose moment of Inertia is Ix, Iy and Iz with respect to x, y and z axis.

Imagine the area is concentrated into a thin strip parallel to the axis and has the same moment of inertia as that of the lamina The moments of inertia about the principal axes are called Principal Moment of Inertia and yield the maximum and minimum values of moment of inertia of the lamina. Determine the moments of inertia and the radius of gyration of the shaded area with respect to the x and y axes. For the given section, find i. Radius of gyration about x, y and z axes iii.

Mass Moment of Inertia is the sum of the products of the mass of each particle in the body with the square of its distance from the axis of rotation. Dynamics has two distinct parts 1. Kinematics : Kinematics is the branch of dynamics which describes the motion of bodies without reference to the forces which either cause the motion or are generated as a result of the motion.

Kinematics deals only with the geometric aspects of the motion. Some engineering applications of kinematics include calculation of flight trajectories for aircraft, rockets, and spacecraft 2.

Kinetics : Kinetics is the study of the relationships between motion and the corresponding forces which cause or accompany the motion. Translation Motion : Any motion in which every line in the body remains parallel to its original position at all times. In translation there is no rotation of any line in the body. When the paths of motion for any two points on the body are parallel lines, the motion is called rectilinear translation.

If the paths of motion are along curved lines which are equidistant, the motion is called curvilinear translation, 2. Rotation about a fixed axis : When a rigid body rotates about a fixed axis, all the particles of the body, except those which lie on the axis of rotation, move along circular paths.

All lines in the body which are perpendicular to the axis of rotation including those which do not pass through the axis rotate through the same angle in the same time.

General plane motion : When a body is subjected to general plane motion, it undergoes a combination of translation and rotation Displacement: The displacement of the particle is defined as the change in its position.

Velocity: If the particle moves through a displacement during the time interval the average velocity of the particle during this time interval is Consequently, the instantaneous velocity is a vector defined as Acceleration: Provided the velocity of the particle is known at two points, the average acceleration of the particle during the time interval is defined as The instantaneous acceleration at time t is a vector that is found by B.

Here the coordinates x and y are independently as functions of time This vector has a magnitude of measured in degrees, radians, or revolutions per minute rpm 3. Since occurs during an instant of time dt, then, 4. This state of equilibrium is called dynamic or kinetic equilibrium. In other words, the body is in equilibrium under the action of the real force F and the fictitious force called Inertia force acting in the opposite direction of applied force and resisting acceleration.

D'Alembert's principle is merely another way of writing Newton's second law, it has the advantage of changing a problem in kinetics into a problem in statics. Inertia Force Inertia force is fictitious force acting in the opposite direction of applied force and resisting acceleration.

It is denoted by Fi is equal to the product of the mass m and acceleration a of the body. Determine the tension in the strings and accelerations of two blocks of mass kg and 50 kg connected by a string and a frictionless and weightless pulley as shown in Figure.

Find the acceleration of a solid body A of mass 10 kg, when it is being pulled by another body B of mass 5 kg along a smooth horizontal plane as shown in Figure. Also find the tension in the string, assuming the string to be inextensible.

The system of bodies shown in Figure starts from rest. Determine the acceleration of body B and the tension in the string supporting body A. Two smooth inclined planes whose inclinations with the horizontal are and are placed back to back. Two bodies of mass 10 kg and 6 kg are placed on them and are connected by a light inextensible string passing over a smooth pulley as shown in Figure. Find the tension in the string and acceleration of the masses. The coefficient of friction between the upper block A and the plane is 0.

In what time block A reach block B? After they touch and move as a single unit, what will be their combined acceleration? A lift carries a weight of Nandi is moving with a uniform acceleration of 2. Determine the tension in the cables supporting the lift, when i lift is moving upwards, and ii lift is moving downwards. Take g - 9. A life has an upward acceleration of 1. What pressure will a man weighing N exert on the floor of the lift? What pressure would he exert if the lift had an acceleration of 1.

What upward acceleration would cause his weight to exert a pressure of N on the floor? An elevator weighs N and is moving vertically downwards with a constant acceleration.

Write the equation for the elevator cable tension. Starting from rest it travels a distance of 35 metres during an interval of 10 seconds. Find the cable tension during this time. Neglect all other resistances to motion. What are the limits of cable tension? A cage, carrying 10 men each weighing N, starts moving downwards from rest in a mine vertical shaft. Find the pressure exerted by each man on the floor of the cage. During this ascent its operator whose weight is N is standing on the scales placed on the floor.

What is the scale reading? What will he the total tension in the cables of the elevator during this motion? This centre point which is instantaneously at rest and has zero velocity is called the instantaneous centre. Characteristics of I-Centre 1. The instantaneous centre is changing every instant and is not a fixed point 2. The velocity of the instantaneous centre is zero Method of locating instantaneous centre 1. The lines of action of two non-parallel velocities vA and vB are known Draw perpendiculars to velocity at the points A and B whose point of intersection will yield IC at the instant considered.

The lines of action of two parallel velocities vA and vB are known Here the location of the IC is determined by proportional triangles. A cylinder of radius 1 m rolls without slipping along a horizontal plane. Find the velocities of the points D and E on the circumference of the cylinder as shown in Figure 2. The diameter of the roller is 40 cm. Find the velocities of points A and B 4. Find the angular velocity of crank about the dead centre C and linear velocity of crank at B ii.

Draw perpendiculars to velocity at the two ends of the connecting crank, one can locate the instantaneous centre at O. Find AO and BO by sine rule v. In a reciprocating pump, the lengths of connecting rod and crank are mm and mm respectively. The crank is rotating at r. A reciprocating engine mechanism has its crank 10 cm long and the length of the connecting rod is 40 cm.

The constraints to the motion reduce the degee of freedom of the system. Single Degree of Freedom System 2. Multiple Degree of Freedom system A. Forced Vibration : When a body vibrates under the influence of a continuous periodic disturbing internal or external force, then the body is said to be under forced vibration Eg : vibration in rotating engines Longitudinal Vibration: When the particles of bar or disc move parallel to the axis of the shaft, then the vibrations are called longitudinal vibrations.

Transverse Vibration: When the particles of the bar or disc move approximately perpendicular to the axis of the shaft on either side in the transverse direction, then the vibrations are known as transverse vibrations. Torsional Vibration: When the particles of the bar or disc get alternately twisted and untwisted on account of vibratory motion of suspended body, it is said to be undergoing torsional vibrations.

Damped vibrations and resonance 1. Damped Vibrations :The vibrations of a body whose amplitude goes on reducing over every cycle of vibrations are known as damped vibrations. This is due to the fact that a certain amount of energy possessed by the vibrating body is always dissipated in overcoming frictional resistance to the motion.

Resonance : When the frequency of external force is equal to the natural frequency of the vibrations, resonance takes place, amplitude or deformation or displacement will reach to its maximum at resonance and the system will fail due to breakdown. This state of disturbing force on the vibrating body is known as the state of resonance.

Periodic and Oscillatory motion 1. Periodic Motion : A motion which repeats itself at regular intervals of time is called a periodic motion.

Eg — motion of pendulum, motion of planets around sun 2. Oscillatory Motion :If a body is moving back and forth repeatedly about a mean position, it is said to possess oscillatory motion. Eg - motion of the pendulum, vibrations of the string An oscillatory motion is always periodic.

A periodic motion may or may not be oscillatory. For example, the motion of planets around the Sun is always periodic but not oscillatory. The motion of the pendulum of a clock is periodic as well as oscillatory.

Time Period :The time interval after which the motion is repeated itself is called time period. It is usually expressed in seconds. Cycle :The motion completed during one time period is called cycle. Frequency :The number of cycles executed in one second is called frequency. It is usually expressed in hertz Hz. Amplitude : The maximum displacement from the mean position is called the amplitude B. Therefore, angle turned by the particle Projection of point P about the y-axis is given as Differentiating with respect to time t Expressing the acceleration in terms of x.

Expressing it in the standard format General Equations 1. Time Period 2. Frequency C. Consider a compact mass m that slides over a frictionless horizontal surface. Suppose that the mass is attached to one end of a light horizontal spring of stiffness k whose other end is anchored in an immovable wall. Restoring Force of the spring due to its stiffness 2. The weight of the body 3. Hence, spring mass model can be considered as a body executing SHM with a natural frequency given by Natural Angular Frequency 2.

Time Period 3. A 80 N weight is hung on the end of a helical spring and is set vibrating vertically. The weight makes 4 oscillations per second. Determine the stiffness of the spring 2. A weight of 50 N suspended from a spring vibrates vertically with an amplitude of 8 cm and a frequency of 1 oscillation per second. Find a the stiffness of the spring, b the maximum tension induced in the spring and c the maximum velocity of the weight.

The body is pulled 50 mm down from its equilibrium position and then released. Calculate the i. A particle moving with SHM has an amplitude of 4. Find the time required by the particle to pass two points which are at a distance of 3. A helical spring under a weight of 20 N extends 0. A weight of N is supported on the same spring. Determine the period and frequency of vibration of the weight and spring when they are displaced vertically by a distance of 0.

Find the velocity of the weight when the weight is 4 mm below its equilibrium position. Take the weight of spring as negligible. A body, moving with simple harmonic motion, has an amplitude of 1 m and period of oscillation is 2 seconds. The weight of an empty railway wagon is 24 kN. On loading it with goods weighing 32 kN, its springs get compressed by 8 cm. Calculate its natural period of vibrations when : i empty and ii loaded as above.

It is set into natural vibrations with an amplitude of 10 cm when empty. Calculate the velocity when its displacement is 4 cm from statical equilibrium position. A particle is moving with simple harmonic motion and performs 8 complete oscillations per minute.

If the particle is 5 cm from the centre of the oscillation, determine the amplitude, the velocity of the particle and maximum acceleration. Given that the velocity of the particle at a distance of 7 cm from the centre of oscillation in 0. Calculate the frequency and amplitude of the motion. What is the acceleration when the displacement is 75 mm?

The piston of an IC engine moves with simple harmonic motion. The crank rotates at length is 40 cm. Find the velocity and acceleration of the piston when is at a distance of 10 cm from the mean position. Total views 23, On Slideshare 0. From embeds 0. Number of embeds 6. Downloads Shares 0. Comments 0. Likes You just clipped your first slide!