1. 1 Teaching Innovation - Entrepreneurial - Global The Centre for Technology enabled Teaching & Learning M G I, India DTEL DTEL (Department for Technology Enhanced Learning)

2. DEPARTMENT OF MECHANICAL ENGINEERING VI-SEMESTER DYNAMICS OF MACHINE 2 CHAPTER NO. 5 Free and Forced Vibration of single-degree- of-freedom system

3. CHAPTER 6 - SYLLABUSDTEL. Equation of motion for two-degree-of-freedom system. 1 23 4 3 Natural frequencies and mode shapes, vibration absorber. Torsional oscillation of two-disc and three disc rotors. Introduction to FFT analyzer for vibration measurements.

4. CHAPTER- 6 SPECIFIC OBJECTIVE / COURSE OUTCOMEDTEL Know about two degree-of-freedom system also understood the concepts of Torsional oscillation and FFT analyzer for vibration measurements. 1 4 The student will be able to:

5. DTEL 5 LECTURE 1 :- Two degree of freedom systems Introduction to two degree of freedom systems The vibrating systems, which require two coordinates to describe its motion, are called two-degrees-of –freedom systems. These coordinates are called generalized coordinates when they are independent of each other and equal in number to the degrees of freedom of the system.

6. DTEL 6 LECTURE 1 :- The study of two-d.o.f- systems is important because one may extend the same concepts used in these cases to more than 2-dof- systems. Also in these cases one can easily obtain an analytical or closed-form solutions. But for more degrees of freedom systems numerical analysis using computer is required to find natural frequencies (eigenvalues) and mode shapes (eigenvectors). If we give suitable initial excitation, the system vibrates at one of these natural frequencies. During free vibration at one of the natural frequencies, the amplitudes of the two degrees of freedom (coordinates) are related in a specified manner and the configuration is called a normal mode, principle mode, or natural mode of vibration. Thus a two degree of freedom system has two normal modes of vibration corresponding to two natural frequencies. Two degree of freedom systems

7. DTEL 7 LECTURE :- undamped two-degrees-of freedom system Free vibration of undamped two-degrees-of freedom system The system shown in Fig. 6. : has two masses m1 and m2 connected by springs of stiffness k1, and k2. It is a two-degrees-of-freedom system and the configuration is fully described by two displacements x1 and x2 as shown in Fig. 6. : Fig.6. : Two-degrees-of- freedom system. Fig. 6. : Free body diagram of both the masses.

8. DTEL 8 LECTURE :- Writing the dynamic equations of equilibrium, considering the free body diagrams shown in Fig. 6. : we get Writing in matrix form or One can identify [m] as mass matrix and [k] as stiffness matrix. {m} and [k] are symmetric and positive definite. Mass matrix is uncoupled and stiffness matrix is coupled and hence it is known as dynamically uncoupled and statically coupled system. Ref.: Biggs J M (1964) Introduction to Structural Dynamics, McGraw- Hill, New York. undamped two-degrees-of freedom system

9. DTEL 9 LECTURE :- Equations of motion Equations of motion for forced Vibration Consider a viscously damped two degree of freedom spring ‐ mass system shown in the figure. The motion of the system is completely described by the coordinates x1(t) and x2(t), which define the positions of the masses m1 and m2 at any time t from the respective equilibrium positions. The external forces F1 and F2 act on the masses m1 and m2, respectively. The free body diagrams of the masses are shown in the figure. The application of Newton’s second law of motion to each of the masses gives the equation of motion:

10. DTEL 10 LECTURE :- Spring K1 under tension for + X1 Spring K2 under tension for +(X2-X1) Spring K3 under tension for + X2 Equations of motion

11. DTEL 11 LECTURE :- The equations can be written in matrix form as: where [m], [c] and [k] are mass, damping and stiffness matrices, respectively and x(t) and F(t) are called the displacement and force vectors, respectively. which are given by: Equations of motion

12. DTEL 12 LECTURE :- Vibration Absorbers Design of Vibration Absorbers Objective: To reduce the vibration of a primary device by adding an absorber to the system Applications: Reciprocating machines Building excited by an earthquake Transmission lines or telephone lines excited by wind blowing

13. DTEL 13 LECTURE :- Applications Vibration absorber in the transmission lines Tuned mass dampers beneath the bridge platform Vibration Absorbers

14. DTEL 14 LECTURE :- Vibration absorber How vibration absorber works Vibration absorber is applied to the machine whose operation frequency meets its resonance frequency. Vibration absorber is often used with machines run at constant speed or systems with const. excited freq., because the combined system has narrow operating bandwidth.

15. DTEL 15 LECTURE :- Bandwidth of operating frequency As µ is increased,  n split farther apart, and farther from the operating point  =  a 0.05 < µ < 0.25 (recommend) Very large µ→ large ma → stress and fatigue problems Vibration absorber

16. DTEL 16 LECTURE :- Damping in vibration absorption Damping can reduce the resonance amplitude of the system Amplitude at operating point increase with increasing damping Vibration absorber

17. DTEL 17 LECTURE :-  What is an FFT Spectrum Analyzer? FFT Spectrum Analyzers, such as the SR760, SR770, SR780 and SR785, take a time varying input signal, like you would see on an oscilloscope trace, and compute its frequency spectrum. Fourier's theorem states that any waveform in the time domain can be represented by the weighted sum of sines and cosines. The FFT spectrum analyzer samples the input signal, computes the magnitude of its sine and cosine components, and displays the spectrum of these measured frequency components. An FFT spectrum analyzer works in an entirely different way. The input signal is digitized at a high sampling rate, similar to a digitizing oscilloscope. Nyquist's theorem says that as long as the sampling rate is greater than twice the highest frequency component of the signal, the sampled data will accurately represent the input signal. In the SR7xx (SR760, SR770, SR780 or SR785), sampling occurs at 256 kHz. To make sure that Nyquist's theorem is satisfied, the input signal passes through an analog filter which attenuates all frequency components above 156 kHz by 90 dB. This is the anti-aliasing filter. The resulting digital time record is then mathematically transformed into a frequency spectrum using an algorithm known as the Fast Fourier Transform, or FFT. FFT spectrum analyzer Ref. : www.thinkSRS.com

18. DTEL 18 LECTURE :- FFT spectrum analyzer Advantages of FFT Analyzers The advantage of this technique is its speed. Because FFT spectrum analyzers measure all frequency components at the same time, the technique offers the possibility of being hundreds of times faster than traditional analog spectrum analyzers. In the case of a 100 kHz span and 400 resolvable frequency bins, the entire spectrum takes only 4 ms to measure. To measure the signal with higher resolution, the time record is increased. SRS spectrum analyzers have the processing power and frontend resolution needed to realize the theoretical benefits of FFT spectrum analyzers. In order to realize the speed advantages of this technique we need to do high speed calculations. And, in order to avoid sacrificing dynamic range, we need high-resolution ADCs.

19. DTEL 19 THANK YOU UNIT 6