EXERCISES -- Torsional Vibrations

EXERCISES

1. Derive an expression for the frequency of free torsional vibrations for a shaft fixed at one end and carrying a load on the free end.
2. Discuss the effect of inertia of a shaft on the free torsional vibrations.
3. How the natural frequency of torsional vibrations for a two rotor system is obtained ?
4. Describe the method of finding the natural frequency of torsional vibrations for a three rotor system?
5. What is meant by torsional equivalent length of a shaft as referred to a stepped shaft? Derive the expression for the equivalent length of a shaft which have several steps.
6. Establish the expression to determine the frequency of torsional vibrations of a geared system.
7. A shaft 200 mm diameter and 1 mt long is fixed at one end and the other end carries a flywheel of mass 1.5 tonne. The radius of gyration of the flywheel is 0.5 m. Find the frequency of torsional vibrations, if the modulus of rigidity for the shaft material is 75 GN/m² .
8. Three rotors A, B and C having moment of inertia of 2000 , 6000 ; and 4000 kg-m² respectively are carried on a uniform shaft of 0.4 m diameter. The length of the shaft between the rotors A and B is 6 m and between B and C is 32 m. Find the natural frequency of the torsional vibrations. The modulus of rigidity for the shaft material is 80 GN/m².
9. Two parallel shafts A and B of diameters 50 mm and 75 mm respectively are connected by a pair of gear wheels, the speed of A being 4 times that of B. The flywheel of mass moment of inertia 3 kg-m² is mounted on shaft A at a distance of 0.9 m from the gears. The shaft B also carries a flywheel of mass moment of inertia 16 kg-m² at a distance of 0.6 m from the gears. Neglecting the effect of the shaft and gear masses, calculate the fundamental frequency of free torsional oscillations and the position of node. Assume modulus of rigidity as 80 GN/m².

Introduction

          The automatic control of system (or machine) is a very accurate and effective means to perform desired function by the system in which the human operator is replaced by a device there by relieving the human operator from the job thus saving physical strength. The automatic control systems are also called as self-activated systems. 

          The centrifugal actuated ball governor which controls the throttle valve to maintain the constant speed of an engine is an example of an automatically controlled system.

The automatic control systems are very fast, produces uniform and quality products. It reduces the requirement of human operators thus minimizing wage bills.

Terms used in Automatic Control of Systems

Terms used in Automatic Control of Systems

The following terms are generally used in automatic control of systems: 

1. Command:  The result of the act of adjustment, i.e. closing a valve, moving a lever, pressing buttons etc., is known as Command.
2. Response: The subsequent result of the system to the command is known as response.
3. Process control: The automatic control of variables like  change in pressure, temperature and speed etc., in machine is termed as process control.
4. Process controller: The device which controls a process is called a process control.
5. Kinetic control: The automatic control of the displacement or velocity or acceleration of a member of a machine is called as Kinetic control.
6. Regulator: The device used to keep the variables at a constant desired value is called as regulator.
7. Feed back: It is defined as measuring the output of the machine for comparison with the input to the machine.
8. Error detector: A differential device used to measure the actual controlled quantity and to compare it continuously with the desired value is called an error detector. It is also called deviation sensor.
9. Transducer: It is a device to change a signal which is in one physical form to a corresponding signal in another physical form. The example of transducer are a loudspeaker (because it converts electrical signal into a sound) and photo-electric cell (because it converts a light signal into an electric signal). Similarly, the primary elements of all the many different forms of thermometers are transducers.
10. Amplification: It is defined as increasing the amplitude of the signal without affecting its waveform. For example, an error detector itself has insufficient power output to actuate the correcting mechanism and hence the error signal has to be amplified. This is generally done by employing mechanical or hydraulic or pneumatic amplifying elements like levers, gears etc.,

Types of Automatic Control System

Types of Automatic Control System

The automatic control systems are classified into two types they are:

1) Open loop (or) Unmonitored system

2) Closed loop (or) monitored system 

1. Open-loop or unmonitored system: When the input to a system is independent of the output from the system, then the system is called an open-loop or unmonitored system. It is also called calibrated system. Most measuring instruments are open-loop control systems, as for the same input signal, the readings will depend upon things ambient temperature and pressure. Following are the examples of open-loop system:

• A simple Bourdon tube pressure gauge commonly used for measuring pressure.
• A simple carburattor in which the air-fuel ratio adjusted through venturi remains same irrespective of load conditions.
• In traffic lights system, the timing of lights is present irrespective of intensity of traffic.

2. Closed-loop or monitored system: When output of a system is measured and is continuously compared with the required value, then it is known as closed-loop or monitored system. In this system, the output is measured and through a feedback transducer, it is sent to an error detector which detects any error in the output from the required value thus adjusting the input in a way to get the required output. Following are the examples of a closed-loop system:

(a) In a traffic control system, if the flow of traffic is measured either by counting the number of vehicles by a person or by counting the impulses due to the vehicles passing over a pressure pad and then setting the time of signal lights.

(b) In a thermostatically controlled water heater, whenever the temperature of water heater rises above the required point, the thermostat senses it and switches the water heater off so as to bring the temperature down to the required point. Similarly, when the temperature falls below the required point, the thermostat switches on the water heater to raise the temperature of water to the required point.

Thermostatically controlled water heater

Block Diagrams

Block Diagrams

Fig. Block diagram of a single Carburettor.

          The block diagrams are used to study the automatic control system in simplified way. In this, the functioning of a system is explained by the inter connected blocks where each block represents a labelled rectangle and is thought of as a block box with a definite function. These blocks are connected to other blocks by lines with arrow marks in order to indicate the sequence of events that are taking place. It may also show how the system operates, what are its inputs and outputs at various stages, and how the energy, information, and/or materials flow through it. Above Fig., shows the diagram of a simple carburattor. The reduction of a control system to a block diagram greatly facilitates the analysis of the system performance or response.

Lag in Response

Lag in Response

          We know that response is the subsequent result of the system to the command. In any control system, there is a delay in response (output) due to some inherent cause and it becomes difficult to measure the input and output simultaneously. This delay in response is termed as lag in response. For example, in steam turbine, with the sudden decrease in load, the hydraulic relay moves in the direction to close the valve. But unless the piston valve ports are made with literally zero overlap, there would be some lag in operation, since the first movement of the piston valve would not be sufficient to open the ports. This lag increases the probability of unstable operation.

Damping

Damping

          When torque is applied in a system in a direction opposite to its motion, it is known as Damping. In case of coulomb damping, the opposition is constant and, thus there will be a constant difference between the input and the output under steady conditions. In the viscous damping provided by dashpot, the opposition is proportional to the relative velocity. As the relative velocity is zero in the steady state, the damping is also zero.

Transfer Function

Transfer Function

          The transfer function is a mathematical expression showing the relation between output and the input to  each unit or block of a control system. If the system has a single input and a single output, it can be represented by block diagram as shown in Fig., 

Transfer function is defined as the ratio of output over the input with all initial conditions equal to zero. Mathematically, 

Transfer function =  θ0 / θ_i

              Where      θ0 = Output signal of the block of a system, and 

                               θ_i  = Input signal to the block of a system.

Thus, the output from an element be obtained by multiplying the input signal with the transfer function.

Note : From the transfer function of the individual block, the equation of motion of system can be formulated.

Overall Transfer Function

Overall Transfer Function

          In the previous topic, we have discussed the transfer function of a block. In this we are going to know the Overall transfer function of a control system. A control system actually consists of several such blocks which are connected in series. The overall transfer function of the series is the product of the individual transfer functions. Consider a block diagram of any control system represented by the three blocks as shown in Fig., 

Fig. Overall transfer function.

          Thus, if  are individual transfer functions of three blocks in series, then th overall transfer function of the system is given as 

Where       K = Constant representing the overall amplification or gain, and 

                  G(D) = Some function of the operator D.

Note: The above equation only true if there is no interaction between the blocks, that is the output from one block is not affected by its connection to the subsequent blocks.

Open-Loop Transfer Function

Open-Loop Transfer Function

          The open loop transfer function is defined as the overall transfer function of the forward path elements. Consider an open loop control system consisting of several elements having individual transfer function such F₁(D), F₂(D), F₃ (D)  as shown on in Fig.,  

Fig. Open loop control system.

 Thus

                                    = F1 (D ) X F2 (D ) X F3 (D ) = KG (D)

The simplified block diagram of the open loop transfer function is shown in Fig.,

Fig. Simplified open loop control system.

Closed - Loop Transfer Function

Closed - Loop Transfer Function

          The closed - loop transfer function is defined as the overall transfer function of the entire control system. It is a mathematical expression (algorithm) describing the net result of the effects of a closed (feedback) loop. Consider a closed loop transfer function or feedback loop consisting of several elements as shown on Fig.,

Fig Closed-loop transfer function.

Now, for the forward path element, we know that

where      K G (D ) = F1 (D ) X F2 (D ) X F3 (D)

On rearranging, we get

               θo = K G ( D )θi − K G ( D)θo

or             [1 + K G ( D )] θo = K G ( D)θi

          The above expression shows the transfer function for the closed-loop control system. Thus the block diagram may be further simplified as shown in Fig., where the entire system is represented by a single block.

Fig. Simplified closed-loop system.

EXERCISES

EXERCISES

1. Define the following terms: 

         (a) Response    (b) Process control     (c) Process Controller

         (d) Regulator   (e) Transducer             (f) Feedback

2. Explain different types of Automatic Control Systems? Explain with an example. 

3. Discuss the importance of block diagrams in control systems. 

4. Draw the block diagrams for the following control systems: 

      (a) A simple carburettor, 

      (b) A thermostatically controlled electric furnace. 

5. What is a transfer function ? 

6. Derive Open - loop and closed - loop transfer functions?

Objective type questions

Automatic Control

Objective type questions

1. The device which is used to keep the variable at a constant desired value is called a  

(a) amplifier                   (b) regulator

(C) deviation sensor     (d) process controlled

2. A simple Bourdon tube pressure gauge is a

(a) closed-loop control system 

(b) manually operated system 

(c) open-loop control system

(d) none of the above 

3. The transfer function of a 4 to 1 reduction gear box is

(a) 2          (b) 1/4 

(c) 4          (d) 1/2

 4. The overall transfer function of three blocks connected in series is 

(a) {F1 ( D ) X F2 ( D)}/F3 ( D)          (b) {F1 ( D ) X F3( D)}/F2 (D)

(c)  F1 ( D ) X F2 ( D ) X F3( D)          (d)  1 / {F1 ( D ) X F2 ( D ) X F3 ( D)}

Latest List of Indian Ministers with Portfolios 2016

Prime Minister Narendramodi today (5th July 2016) reshuffled Council of ministers for the second time since 2014. He expanded his Union Council of Ministers with 19 New Ministers from 10 states.  Smriti Irani, She has been moved out of the Human Resource Development (HRD Ministry) to Textile Ministry. Prakash Javadekar, who was promoted to cabinet rank, will hold his portfolio now. Venkaiah Naidu will now be Minister for Information and Broadcasting, a portfolio held by Finance Minister Arun Jaitley as an additional charge. Sadananda Gowda loses the Law Ministry to Ravi Shankar Prasad, who is also the Information Technology Minister.

Note: Prime Minister Narendra Modi will hold : personal, public Grievances & Pensions, Department of Atomic Energy, Department of Space, All important policy issues, and All other portfolios not allocated to any Ministers.

Ministers of India and their Portfolios 2016

Cabinet Ministers

Cabinet Ministers	Portfolios
Rajnath Singh	Home Affairs
Sushma Swaraj	External Affairs
Arun Jaitley	Finance and Corporate Affairs
M. Venkaiah Naidu	Information &Broadcasting Urban  Development, Housing & Urban  Poverty Alleviation
Nitin Jairam Gadkari	Road Transport and Highways Shipping
Manohar Parrikar	Defence
Suresh Prabhu	Railways
D. V. Sadananda Gowda	Statistics & Programme Implementation
Uma Bharati	Water Recourses, River Development & Ganga  Rejuvenation
Dr. Najma A. Heptulla	Minority Affairs
Ramvilas Paswan	Consumer Affairs, Food & Public Distribution
Kalraj Mishra	Micro, Small & Medium Enterprises
Maneka Sanjay Gandhi	Women & Child Development
Ananthakumar	Parliamentary Affairs, Chemicals & Fertilisers
Ravi Shankar Prasad	Law & Justice, Electronics & Information  Techology
Jagat Prakash Nadda	Health & Family Welfare
Ashok Gajapathi Raju Pusapati	Civil Aviation
Anant Geete	Heavy Industries & Public Enterprises
Harsimrat Kaur Badal	Food Processing Industries
Narendra Singh Tomar	Rural Development, Panchayati Raj, Drinking  Water & Santation
Chaudhary Birender Singh	Steel
Jual Oram	Tribal Affairs
Radha Mohan Singh	Agriculture & Farmers Welfare
Thaawar Chand Gehlot	Social Justice and Empowerment
Smriti Zubin Irani	Textiles
Dr. Harsh Vardhan	Science & Technology, Earth Sciences
Prakash Javadekar	Human Resource Development

Ministers of State (MOS)

Ministers of State	Portfolios
Rao Inderjit Singh	Planning (Independent Charge) Urban Development  Housing & Urban Poverty Alleviation
Bandaru Dattatreya	Labour & Employment  (Independent Charge)
Rajiv Pratab Rudy	Skill Development & Enterpreneurship  (Independent Charge)
Vijay Goel	Youth Affairs and Sports  (Independent Charge) Water Resources, River Development &  Ganga Rejuvenation
Shripad Yesso Naik	AAYUSH (Independent Charge)
Dharmendra Pradhan	Petroleum and Natural Gas  (Independent Charge)
Piyush Goyal	Power (Independent Charge) Coal (Independent Charge) New and Renewable Energy  (Independent Charge) Mines (Independent Charge)
Dr. Jitendra Singh	Development of North Eastern Region  (Independent Charge) Prime Minister’s Office Personnel, Public Grievances & Pensions Department of Atomic Energy Department of Space
Nirmala Sitharaman	Commerce and Industry   (Independent Charge)
Dr. Mahesh Sharma	Culture  (Independent Charge) Tourism  (Independent Charge)
Manoj Sinha	communications (Independent Charge) Railways
Anil Madhav Dave	Environment, Forest and Climate Change  (Independent Charge)
General V. K. Singh	External Affairs
Santosh Kumar Gangwar	Finance
Faggan Singh Kulaste	Health & Family Welfare
Mukhtar Abbas Naqvi	Minority Affairs Parliamentary Affairs
S S Ahluwalia	Agriculture & Farmers Welfare  Parliamentary Affairs
Ramdas Athawale	Social Justice & Empowerment
Ram Kripal Yadav	Rural Development
Haribhai Parthibhai Chaudhary	Micro, Small & Medium Enterprises
Giriraj Singh	Micro, Small & Medium Enterprises
Hansraj Gangaram Ahir	Home Affairs
G. M. Siddeshwara	Heavy Industries & Public Enterprises
Ramesh Chandappa Jigajinagi	Drinking Water & Sanitation
Rajen Gohain	Railways
Parashottam Rupala	Agriculture & Farmers Welfare Panchayati Raj
M. J. Akbar	External Affairs
Upendra Kushwaha	Human Resources Development
Radhakrishnan P	Road Transport & Highways Shipping
Kiren Rijju	Home Affairs
Krishan Pal	Social Justice & Empowerment
Jasvantsinh Sumanbhai Bhabhor	Tribal Affairs
Dr. Sanjeev Kumar Balyan	Water Resources, River Development &  Ganga Rejuvenation
Vishnu Deo Sai	Steel
Sudharshan Bhagat	Agriculture and Farmers Welfare
Y. S. Chowdary	Science & Technology Earth Science
Jayant Sinha	Civil Aviation
Col. Rajyavardhan Singh Rathore	Information & Broadcasting
Babul Supriyo	Urban Development Housing & Urban Poverty Alleviation
Sadhvi Niranjan Jyoti	Food Processing Industries
Vijay Sampla	Social Justice & Empowerment
Arjun Ram Meghwal	Finance Corporate Affairs
Dr. Mahendra Nath Pandey	Human Resource Development
Ajay Tamta	Textiles
Krishna Raj	Women & Child Development
Mansukh L. Mandaviya	Road Transport & Highways, Shipping, Chemicals & Fertilizers
Anupriya Patel	Health & Family Welfare
C. R. Chaudhary	Consumer Affairs, Food & Public Distribution
P. P. Chaudhary	Law & Justice Electronics & Information Technology
Dr. Subhash Ramrao Bhamre	defence