Decoupling of equipments subjected to seismic and harmonic excitations by Abhishek Vasu

Chapter 1 Introduction

Every industry consists of set of machineries required for specific output. For example an automobile industry will have equipment like air compressors, pumps, diesel alternators, machining, milling and cutting equipment etc. Most of these machineries operate simultaneously and many of them are placed in close proximity of one another on the shop floor. When one of such machinery vibrates while in operation it affects the performance of the nearby equipment as the vibrations propagates from the foundation. The vibrating environment may consist of low-level seismic disturbances present everywhere on earth, which present operating problems to highly sensitive items such as delicate equipment. When other disturbances are superimposed on the seismic disturbances, a wide range of equipment is adversely affected. These other disturbances are caused by such things as vehicular and foot traffic, passing trains, air conditioning systems, and nearby rotating and reciprocating machinery. Similarly in a nuclear facility, structure supports equipment and piping systems. High-speed rotating machinery like vertical or horizontal pumps, turbine-generator systems and fan-motor systems located in a nuclear power plant are required to remain operational even during a seismic event as the supply of power is essential to carry out necessary relief operations. Thus rotating pumps becomes one of the most important equipment. Typically a fire water pump house of a NPP consists of 10 to 12 pumps arranged in series to counter any eventuality arising out of a fire outbreak. These pumps have certain dynamic characteristics. Similarly the structure also has a dynamic characteristic. Generally in the design, frequency of the structure and pumps are kept away from each other to avoid resonance. Adopting this

methodology for a coupled system subjected to combined vibration and earthquake is not straight forward and thus development of a detailed methodology is required.

1.1 Background Fire safety assessments and operational experiences gained from in nuclear power plants (NPP) have shown that fires have a high potential to strongly affect the safety of an NPP. Fire can occur anytime in a plant, fire protection of an NPP is important throughout its lifetime i.e., from the design stage to operation till decommissioning. The analysis of the operating experience has shown that fire explosions represent a significant hazard to NPPs. Therefore, counter measures have to be taken in the safety design and operation of NPPs to keep the relevant risk as low as reasonable on the basis of the international experience. Fire protection system consists of series of vertically mounted centrifugal pumps. These fire water pumps are rotating machinery; its natural frequency is kept away from the rotating frequency to avoid resonance. In seismic scenario the problem intensifies as the structures also starts vibrating along with the equipment. There are few available design guidelines [1, 2] to assist the safe operation of these machineries under seismic loading.

1.2 Available Criterion Generally in the design, frequency of the structure and pumps are kept away from each other to avoid resonance. Adopting this methodology for a coupled system subjected to combined vibration and earthquake is not straight forward and thus development of a detailed methodology is required. Following are the two widely used guidelines for designing and safe operation of such rotating machineries in seismic scenario: 1.2.1 ISO 10816 [1] It recommends rotating equipment to keep the natural frequency away from the rotating frequency. These standards provide guidance for evaluating vibration severity in machines operating in the 10 to 200Hz frequency range. Examples of these types of machines are small, direct-coupled, electric motors and pumps, production motors, medium motors, generators, steam and gas turbines, turbo-compressors, turbo-pumps and fans. Some of these machines 2

can be coupled rigidly or flexibly, or connected though gears. The axis of the rotating shaft may be horizontal, vertical or inclined at any angle.

Figure 1.1: ISO 10816 Vibration Severity Standards Machines can be categorized as follows: (i)

Class I- These machines may be separate driver and driven

or coupled units comprising operating machi nery up to15kW (20hp). (ii)

Class II - Machinery (electrical motors 15kW (20hp) to 75kW (100hp),

w i t h o u t s p e c i a l f o u n d a t i o n s , or rigidly mounted engines or machines up to 300kW (400hp) m o u n t e d (iii)

special foundations

Class III - Includes large prime movers and other large machinery with large

rotating assemblies m o u n t e d o n r i g i d a n d h e a v y f o u n d a t i o n s which are reasonably stiff in the direction of vibration. (iv)

Class IV - Includes large prime movers and other large machinery with large

rotating assemblies m o u n t e d o n foundations which are relatively soft in the direction of the measured vibration (i.e., turbine generators and gas turbines greater than 10MW (approx. 13500hp) output. 1.2.2 ASCE 4-98 [2] ASCE 4-98 has a criterion for decoupling structure and equipment considering seismic loading. Decoupling criteria is based on the frequency ratio (Rf) and mass ratio (Rm) of the secondary system to the primary system. These are defined as in Equation 1.1 and 1.2 as

follows. Fig. 1.2, 1.3(a) and1.3(b) describes the graphical representation of decoupling by USNRC and ASCE 4-98.

Where, Rf is the ratio of frequency or modal frequency of uncoupled SS to the uncoupled PS and Rm is the ratio of mass or modal mass of the uncoupled SS to the uncoupled PS. (i)

Decoupling can be done for any Rf, if Rm< 0.01

(ii)

If 0.01 ≤ Rm ≤ 0.1 decoupling can be done provided 0.8≥ Rf ≥ 1.25

(iii)

If Rm ≥ 0.1 and Rf ≥ 3 (i.e. rigid secondary structure) It is sufficient to

include only the mass of the system in the primary structure. (iv)

If Rm ≥ 0.1 and Rf< 0.33 (Flexible secondary system) decoupling can

be done. (v)

If Rm ≥ 0.1 and 0.33 <Rf< 3, coupled system analysis is required.

Fig.1.2 : Decoupling criteria for Primary and secondary system based on USNRC

(a) Primary and secondary system

(b) Mass-Spring System orientation Figure 1.3: Decoupling criteria based on ASCE 4-98 The two criterion described above are independently well understood. However, when structure and rotating equipment are coupled, a new criterion has to be formulated for combined effects of vibration and seismic loading on structure and nearby rotating equipment.

1.3 General In this section, few general topics are briefly discussed which are related to the present work. 1.3.1 Vibration Mechanical vibration and shock are present in varying degrees in virtually all locations where equipment and people function. The adverse effect of these disturbances can range from negligible to catastrophic depending on the severity of the disturbance and the sensitivity of the equipment. Vibration is the motion of a particle or a body or system of connected bodies displaced from a position of equilibrium. Most vibrations are undesirable in machines and structures because they produce increased stresses, energy losses, cause added wear, increase bearing loads, induce fatigue, create passenger discomfort in vehicles, and absorb energy from the system. Rotating machine parts need careful balancing in order to prevent damage from vibrations. There are many sources of mechanical and structural vibration .The most common form of mechanical vibration problem is motion induced by machinery of varying types, often but not always of the rotating variety. Other sources of vibration include: ground-borne propagation due to construction; vibration from heavy vehicles on conventional pavement as well as vibratory signals from the rail systems common in many metropolitan areas; and vibrations induced by natural phenomena, such as earthquakes and wind forces. Wave motion is a source of vibration in mechanical and structural systems associated with offshore structures. The most serious effect of vibration, especially in the case of machinery, is that sufficiently high alternating stresses can produce fatigue failure in machine and structural parts. Less serious effects include increased wear of parts, general malfunctioning of apparatus, and the propagation of vibration through foundations and buildings to locations where the vibration of its acoustic realization is intolerable either for human comfort or for the successful operation of sensitive measuring equipment. 1.3.2 Earthquake Earthquake is a sudden release of elastic strain energy accumulated across a pre-existing fault or fracture in the earth crust causing ground motion could be acceleration or displacement. It is a function of time. This further generates oscillatory motion in structures and in supported equipment. These may further interact and influence the response of each other. A seismic event creates a number of separate, but interrelated problems for buildings and other 6

structures. The earthquake itself can move both laterally and vertically, providing forces to which the structure is not normally subject. Additionally, earthquakes can cause soil liquefaction, where the soil under a building flows out from under the foundation, eliminating the structural support that the building relies on. Other events, such as landslides can be caused by earthquakes, adding additional hazards. An important part of earthquake sustainability is dependent upon the flexibility of the materials used in construction. Concrete, a common material used in construction, is not very earthquake resistant. That’s because it is extremely strong under compression, but very weak under tension. Earthquakes cause both compression and tension, creating cracks in the concrete. 1.3.3 Resonance Mechanical resonance is a tendency of a mechanical system to absorb more energy with the frequency of its oscillation matches the system’s natural frequency of vibration than it does at other frequencies causing violent swaying motions and even catastrophic failure. External items in large vertical pumps that could excite a natural frequency are: (i)

Rotational unbalance

(ii)

Impeller exit pressure pulsations

(iii) Gear couplings misalignment The most notable characteristic of resonance is increased vibration when a certain operating speed is reached. Also, as the operating speed is increased beyond the resonant frequency, the vibration amplitude will decrease somewhat. The formula for calculating the natural frequency is: f 

1 2

K   m

Where “K” is the stiffness of the resonant structure or component, and “m” is the mass. Increased stiffness will, therefore, raise the natural frequency, and increased mass will lower it. That is logical since stiffness creates a force that is always directed against motion, while mass has inertia, which is a force always directed with motion. Resonance is what happens when these two opposing forces are equal. They cancel each other out, allowing vibration to increase.

A third force, damping, is at work throughout the speed range. Damping absorbs vibratory energy, converting it to heat. In doing so, damping reduces the maximum amplitude of the vibration at resonance and increases the width of the amplification zone. A common example of damping is shock absorbers on a vehicle. On machinery bases, concrete and grouting add significant damping to a base structure. These forces (stiffness, mass and damping) determine the characteristics of resonance and are important to the distinction between structural resonance and rotor critical speeds.

1.4 Need of the Project ISO 10816 recommends decoupling criteria for members subjected to harmonic excitation whereas ASCE 4-98 recommends decoupling criteria for members under seismic loading However there is no established criterion for considering the combined effect of harmonic and seismic excitations. Hence it is necessary to analyse the interactions of equipments and the structure when they are subjected to both the loads â&#x20AC;&#x201C; seismic and harmonic vibrations in any Nuclear Power Plant. Depending on this study, the need of the hour is to develop criteria for design under the above mentioned loading conditions.

1.5 Scope of Study Scope of study is to carry out literature survey and analyse the machinery supported on structure for their interactions under vibration and seismic loading and developing new design criteria. For a particular damping found through experiments, and for various mass and frequency ratios, ď&#x201A;ˇ

Find the response of uncoupled and coupled systems

ď&#x201A;ˇ

Develop a decoupled criteria for : 1. Uncoupled structure 2. Uncoupled equipment 3. Coupled system

These will be done for both harmonic and seismic excitations. 8

A brief review of the literature referred for this project work is presented in next chapter. Tests and analysis will be performed on the experimental model for validation of results.

Chapter 2 Literature Review

A brief review of the literature referred for this project work is presented in this chapter. Articles pertaining to the present work will be covered under the following sub topic: i.

The effect of earthquake excitation angle on the internal forces of steel building’s elements by nonlinear time history analyses.

ii.

A response-based decoupling criterion for multiply-supported secondary systems

iii.

Seismic Analysis of rotating Mechanical Systems

iv.

Decoupling of Secondary Systems for Seismic Analysis

Vibration Of a Double-Beam System

2.1 The effect of earthquake excitation angle on the internal forces of steel building’s elements by nonlinear time history analyses Mahmood Hosseini and Ali Salemi [3] studied “the effect of earthquake excitation angle on the internal forces of steel building’s elements by nonlinear time history analyses”. It was seen that the previous researches with regard to the effect of angle of excitation were mostly limited to buildings with elastic behaviour. In this study two sets of 5-story steel buildings with moment frames, one set with square and the other with rectangular plan, were designed base on the seismic design code for steel buildings, and then analyzed by a Nonlinear Time History Analysis (NLTHA) program using simultaneously the accelerogram of two horizontal components of some earthquakes. The three accelerogram pairs were applied to both 10

buildings,with various angles of incidence (from 0 to 90 degrees) with respect to the main axis (assumed here to be X axis) with an increment of 10 degrees. The buildings’ columns were divided into three corner, side, and middle columns, and variation of maximum values of axial and shear forces as well as bending moment in columns of each category and the bending moments of girders in various frames investigated with varying the degree of incidence. To simplify the comparison of results all values in each case were normalized to the values corresponding to degree of incidence equal to zero degree, which was called the “base case”. The used earthquakes were almost of the same PGA level, but had different frequency contents. A set of values, with an increment of 10 degrees, were used for angle of excitation. Based on the numerical results it was concluded that: • The internal forces of structural elements depend on the angle of incidence of seismic wave with respect to the axes of building plan. Among various internal forces the axial forces of columns are more sensitive to the angle of incidence. • The columns’ axial forces may exceed the ordinary cases up to 50% by varying the angle of excitation, and this variation is more in buildings with rectangular plans than in those with square plans. • The maximum bending moment in columns occurs mainly by angle of incidence of either 0 or 90 degrees, however, for some earthquakes it may occur by some other angle. The difference between maximum moments due to various angles of incidence is more remarkable in building with rectangular plans and may reach to 60%. • The base shear forces of buildings due to various angles of incidence may be different to 10%. • Each of the internal forces of column gets its maximum value with a specific angle of excitation, which is not 0 or 90 necessarily, and is different from column to column, and that specific angle is not the same for different earthquakes. • There is not a single specific angle of incidence for each building which maximize the internal forces of all structural members together, and each member gets it’s the maximum value of each of its internal forces by a specific angle of incidence. This angle is not the same for various earthquakes.

However this study was limited to regular symmetric buildings and the variations of internal forces by varying the angle of incidence would increase in irregular and asymmetric buildings.

2.2. A response-based decoupling criterion for multiply-supported secondary systems S. R. Chaudhari and V. K. Gupta [4] developed “a response-based decoupling criterion for multiply-supported secondary systems”. They identified that it is often infeasible to carry out coupled analyses of multiply-supported secondary systems for earthquake excitations. ‘Approximate’ decoupled analyses are then resorted to, unless the response errors due to those are significantly high. This study proposed a decoupling criterion to identify such cases where these errors are likely to be larger than an acceptable level. This study proposed a new decoupling criterion for multiply-supported secondary systems, while it was assumed that limiting the errors in primary system responses at the attachment points (with the secondary system) to a low level would lead to reasonably accurate secondary system responses. The proposed criterion was developed by (i) Assuming the combined system to be classically damped. (ii) Extending the perturbation formulation to calculate the (undamped) eigen properties of the combined system in terms of the fixed-base eigen properties of the primary and secondary systems. (iii)Assuming the input excitation to be ideal white noise and idealizing the response transfer functions as combinations of delta functions. The proposed criterion was validated by comparing the ‘expected’ coupled and decoupled responses for two example P–S systems under four different excitation processes. A perturbation formulation was proposed for finding the un damped modal properties of a combined P–S system in case of light to moderately heavy secondary systems. The proposed formulation requires little computational effort beyond that for finding the fixed-base mode shapes of the sub-systems and is found to give reasonably good results. This formulation is however applicable to those cases where the stiffness of the elements connecting the secondary system to the primary system are not very stiff. A decoupling criterion using the undamped eigen properties of the combined system and based on desired accuracy in the primary system displacement responses was proposed. This 12

criterion accounts for the role of damping present in the primary and secondary subsystems. This has been found to work well in case of two example systems excited by a variety of ground motions. A preliminary exercise indicated that the acceptable error level for the proposed criterion was closely linked with the acceptable level in the secondary system response. In fact, an acceptable error level close to 10% in the secondary system response may have to be reduced in the ratio of 1/ 20 to 1/12 , depending on whether tuning is strong or weak, to find the acceptable error level in the primary system response. The participation factors and mode shapes estimated by the proposed formulation are also close to the exact values in case of all dominant modes. Errors can be reduced further by considering an expansion higher than second order. However, improving the accuracy of nondominant modes may not be worth the additional effort. It may further be mentioned that the proposed formulation holds only when the elements connecting the secondary system with the primary system are not very stiff. Thus, increasing the stiffness of these elements with no change in other parameters will lead to greater errors in the results of proposed formulation for second and higher mode shapes. This is however true only for the mode shape elements corresponding to the secondary DOFs.

2.3 Seismic Analysis of rotating Mechanical Systems A.H. Soni and V.Srinivasan [5] in their paper “Seismic Analysis of rotating Mechanical Systems” have presented the comparison of two models simulating the seismic performance of a rotor – bearing system commonly encountered in a nuclear power – plant. The details of the models are as given: 1. First model – This model treats the rotor as a rigid body subjected to gyroscopic and coriolis effects. 2. Second model – The second model treats the rotor as a flexible element. The comparison of the two models was given at rotating speed = 3000rpm Sr. No.

Rotor

Flywheel

Mass

24,000 kg

5000 kg

4.57x103kgm2

2500kgm2

Table 2.1: Comparison between Rotor and Flywheel

A finite element beam model was developed by them to study the influence of spin and base rotation. The results of this study show how the gyroscopic effects amplify the response of the rotor-bearing systems. The base rotations of the rotor â&#x20AC;&#x201C; bearing system under seismic excitation contribute significantly to the response of the rotor. The seismic analysis of rotating systems differs from the seismic analysis of structural systems in 2 major aspects: 1. Rotor â&#x20AC;&#x201C; bearing interactions effect 2. Gyroscopic effects In addition, Coriolis Effect becomes significant when the base of the rotating systems is subjected to rotational motions. Such seismic analysis permits us to â&#x20AC;&#x201C; 1. Examine the required minimum fluid thickness 2. Withstand the bearing reaction forces 3. To maintain the maximum allowable dynamic stresses induced in the rotor. The two models show that the effect of base rotation and spin of the shaft contribute significantly in rotor displacements and dynamic reaction forces at the bearings. For the rigid body model however, the rotor displacements and dynamic reaction forces at the bearings are significantly lower than those obtained from response of the beam model.

2.4 Decoupling of Secondary Systems for Seismic Analysis A.H. Hadjian [6] stated that coupling of primary and secondary systems may not be feasible every time. Hence there arose a need to perform decoupling. In his paper he has justified the reason for the same. Factors such as complicated computational analysis, cost considerations and time constraints ensure the need of performing decoupling. The author states however that while carrying out uncoupled analysis the safety of the structure operating under full load must not be compromised, especially in the case of nuclear power plants, the structural failure of which would have disastrous and far reaching consequences. Resonance under all conditions must be avoided. From a response point of view, the uncoupled response must be larger than the coupled response. Due to the introduction of nuclear power plants, the existing criteria had to be modified as all available criteria resulted in values with lot of difference, creating the need to generate a more consistent value.

Figure 2.1: Specification for decoupling Amongst the criteria available usually the one with least variation for every particular case was adopted. A procedure which is both time consuming and inaccurate. For simplicity sake the author began with a two DOF system of which both frequency and mass ratios were reviewed, keeping a margin of 10-15% for errors. This acceptable margin ensures the system followed is more generalized, that is acceptable for more number of cases

. Figure 2.2: Schematic Representation of Decoupling Criteria After extensive studies on the response of coupled and uncoupled systems, it was observed that except for the relative displacement of uncoupled primary system, the responses result in over estimates because of coupling. The evaluation of multi DOF systems however was more difficult than that of 2 DOF. The numerous possibilities make calculations inconclusive. In this system in order to develop a unique mass ratio a parameter, modal mass is introduced. Modal mass is nothing but the equivalent mass of multiple oscillators of a multi DOF system. It overcomes the problems of the vertical response of the slab as well as the dependence on the crudeness of the system. 15

Along with this a few other parameters are also considered such as location of the attachment point between secondary and primary systems as experimentally it is observed to affect the response. By plotting the ratio of the coupled frequency to the uncoupled frequency as a function of the modal mass ratio the following results are obtained

Figure 2.3: Response Modification due to Decoupling The insignificance in the difference between the 2 responses especially in the region of mass ratio which is considered for decoupling reaffirms the fact that by using modal mass instead of mass ratio for multi DOF the response can be obtained conservatively and for a generalized case.

2.5 Vibration of a Double-Beam System In the year 1999 H. V. Vu, A. M. Ordodnez and B.H. Karnopp [7] developed a method to obtain the exact solution of a double-beam system subjected to harmonic excitation. Attaching an auxiliary identical beam to the primary beam of a distributed system (one that can be solved conveniently by modal analysis) by means of a distribute spring dashpot complicates the problem. With arbitrary boundary conditions and forcing functions, the problem is difficult to solve. Double-beam systems interconnected by a distributed spring dashpot in parallel have been investigated by several authors. Douglas and Yang analyzed the transverse damping in the frequency response of three-layer elastic beams in a mechanical impedance format. The system was treated as two non-identical Euler Bernoulli beams with a viscoelastic layer in between. Their method is limited to a case of fixed-free boundary conditions for both beams and a concentrated sinusoidal load applied at the free end. Vu presented a closed-form solution for the forced response of a general beam system. Vu's method applies to nonidentical Euler-Bernoulli beams with arbitrary boundary conditions and general applied loads. These authors manipulated a set of two coupled fourth order differential equations into a single eight order differential equation. The authors presented a method to obtain the exact solution for the forced vibration of a damped double-beam system. The method involves a change of variables to decouple the set of two fourth order differential equations, and then the solution is obtained by means of modal analysis. This approach allows both the viscous damping and the applied forcing function to be completely arbitrary. The damping need not be small or proportional to the mass and stiffness which is different from the conventional method and the forcing function can be either distributed or concentrated at any point. The Problem: The system consists of a main beam subjected to a force distribution which is an arbitrary function of space and time. An auxiliary beam is connected to the main beam by a viscoelastic material, where k and c are the spring constant and the damping coefficient respectively. The transverse displacements of the main beam and auxiliary beam are w1(x, t) and w2(x, t) respectively. The forcing function acting on the main beam is f (x, t). A thorough understanding of the problem will lead to better techniques for reducing resonance-induced vibrations.

Figure 2.4: System with mass-spring-dashpot The Methodology: The methodology followed can be classified into 3 steps: 1. Mathematical Model: The coupled governing differential equations of the system can be shown as,

2. Method of Decoupling the Equations: Assuming,

We can write equations (1) & (2) as,

On solving after considering all assumptions the equations reduce to

At this point, the equations are uncoupled. Now, the three-step procedure of modal analysis is employed. First, the natural frequencies and the corresponding mode shapes are obtained by solving the undamped free vibration with appropriate boundary conditions. Second, the 18

normalized orthogonality property is established. And third, the forced vibration is solved by means of modal expansion. 3. Solution of Equations: On application of the boundary conditions the equation was obtained in the form;

The figure below shows the boundary conditions applied in order to obtain the solution

Figure 2.5: Boundary Conditions for the case Results: The numerical results for the double-beam case study are presented in a frequency response format. Selected plots in two and three dimensions are shown, and their salient features are discussed. The results can be expressed in terms of the dimensionless ratios:

Table 2.2: Comparison of first four natural frequencies for low and high stiffness Conclusions: A closed-form solution is developed for analyzing the vibration problem of a damped doublebeam system. A simple change of variables and modal analysis are utilized to decouple and solve the differential equations. The damping is assumed neither small nor proportional, and the forcing function can be either concentrated at any point or distributed. Although the method presented is applicable only for a limited class of problems, it provides an analytical solution that serves as a benchmark for further investigation of more complex n-beam systems such as damped triple-beam systems. Since the solution is exact, it allows a complete understanding of the problem.

Chapter 3 Experimental Validation of Structure and Equipment Interaction under Harmonic Loading

A single phase AC motor was used as its rpm. In this chapter, design of the model used for experimental analysis is discussed. The results obtained from the numerical analysis are validated by test results.

3.1 Design The experimental model used here is a three storey steel frame with a foot print size of 0.3m x 0.3m.Columns are made of 9.5 mm diameter stainless steel rods, supporting the 9mm thick slab by bolt and nut arrangement and is designed to have a maximum height of 1.02 m as shown in figure 3.1.

(a) Figure 3.1:

Three storey steel model

Figure 3.2:

(b) (a) Exciter, (b) Equipment

To study the effect of rotating equipment on the structure, the top slab of the model was fitted with a motor for harmonic excitation. The fundamental concept behind model testing is that of resonance, if a structure is excited at resonance its response exhibit two distinct phenomenon. Firstly, the driving frequency approaches natural frequency of the structure, the magnitude of resonance approaches the maximum value. Secondly, the phase of response shifts by 180o as the frequency sweeps through the resonance, with the value of phase at resonance being 90 o. A single phase AC motor was used as its rpm can be changed by changing the input voltage. A mass of 50 grams was attached on the spindle to generate the unbalance forces (figure 3.2 (a)). Further, to study the effects of rotating equipment on nearby equipment and structure, the top slab of the model was fitted with a mass (simulating equipment) along with the motor for forced excitation. One mass was fixed on the top floor of the structures with exciter in the centre of the first floor. In order to support the mass to the structure, aluminium strips were used. Mass of the Equipment and dimensions of the supports are listed in table 3.1.

Particulars

Equipment

Mass of equipment (kgs)

0.5899

Material of support

Aluminium

Length of support (m)

0.257m

Thickness of support (m)

0.0025

Width of support (m)

Table 3.1:

0.03

Dimension of equipment and support

3.2 Validation of Natural frequency Modal analysis of any structure can be performed using MATLAB. From the modal analysis natural frequency and mode shapes in all the three normal directions are obtained. The determination of mode shape is essential to analyse the behaviour of the structure under applied dynamic loading. The result of the modal analysis can be validated by experimental work with the help of FFT analyser. The apparatus used in the vibration test are discussed briefly: (i)

Data Logger

DEWE 5001, Data Logger with inbuilt FFT Analyser (figure 3.3) is

used to measure the frequency of the structure. It can be used for both free as well as forced vibration study. The system has 32 channels to connect the cables for analysing both input and output signals. (ii)

Accelerometer

DJB Accelerometer (AT 20 type) was used of sensitivity

33pC/g with low and small physical dimensions making them ideally suited for model analysis for the present experiment (figure 3.4). (iii)

Display unit The excitation of the structure is picked up by the accelerometer which

after conversion comes in graphical form through the software and display on the display unit.

Figure 3.3: Data Logger

Figure 3.4: AT 20

DEWE 5001DEWE 5001

type DJB

Accelerometer In free vibration test, natural frequencies (Eigen values) and damping characteristics is measured using FFT analyzer. Experimental model analysis involved the installation of a high resolution piezoelectric accelerometer at each floor levels. Accelerometers were connected to the charge counter. Output from each accelerometer via charge counter is supplied to a particular input called the channel in the dynamic signal analyzer. The recorded acceleration is then analyzed using DEWE 6.6.7 software platform to obtain systemâ&#x20AC;&#x2122;s natural frequencies and the associated damping ratios. 23

3.3 Free vibration 3.3.1 Three storey model with exciter (passive) The result of the numerical analysis (refer Appendix II-A pg no.:70) was validated by experimental work with the help of FFT analyser. Accelerometers were fixed on all three floors of the model in X direction (figure 3.5). Experimental outputs obtained are compared to the results from numerical analysis in table 3.2.

Accelerometer

Charge Converter Exciter

Data Acquisition System &Analyser with inbuilt Amplifier

Steel Frame Model

Display Unit

Figure 3.5: Line sketch of the experimental setup of model with exciter

Natural Frequency (Hz) Mode

Error

Experimental

Numerical

(%)

7.324

8.19

11.55

21.973

22.1009

0.578

34.180

33.2708

2.73

Table 3.2: Experimental and numerical natural frequencies for three storey model with exciter (passive)

(a).Top Floor

(b).Second Floor Figure 3.6: Floor wise Response For Free Vibration of Three Storey Model with Exciter (passive)

3.3.2 Three storey model with exciter and equipment For free vibration test, accelerometers were attached on the mass and also on each floor (figure 3.7 in X direction. Experimental outputs obtained are compared to the results from numerical analysis (refer Appendix II-B pg no:71) in table 3.3.

Equipment

Accelerometer

Charge Converter Data Acquisition System &Analyser with inbuilt Amplifier

Exciter Steel Frame Model

Display Unit

Figure 3.7: Line sketch of the experimental setup of model with equipment and exciter (passive)

Mode

Frequency (Hz)

Error

Experimental

Analysis

(%)

3.4

4.388

2.9

7.813

8.563

0.95

21.484

22.101

0.28

34.180

33.265

0.26

Table 3.3: Experimental and numerical natural frequencies for model with equipment and exciter (passive)

(a) Top Floor

(b) Second floor

Figure 3.8: Floor wise Response and FFT outputs for Free Vibration of Three Storey Model with Equipment and Exciter 27

3.4 Forced vibration 3.4.1 Three storey model with exciter (active) Model was subjected to harmonic excitation by running the exciter. Forcing frequency was increased gradually by increasing the rpm of the exciter. As the driving frequency approached natural frequency of the structure (6.348 Hz), the magnitude of resonance approached the maximum value. Test output response obtained is compared to the results from numerical analysis (refer Appendix II-C pg no.: 72) in table 3.4.

Accelerometer Location

Acceleration (m/s2)

Error

Experiment

Analysis

(%)

Top floor

3.193

3.51

9.93

Second floor

3.0656

2.99

2.47

Table 3.4: Experimental and numerical analysis results for model with exciter (active)

TOP FLOOR

(a)

(b) Figure 3.9: Test values for Three Storey Model with Exciter (a) Response (b) FFT output

TOP FLOOR

4 3

topflracc[m/s2]

2 1 0 -1 -2 -3 -4

5 time [s]

4 3.9 3.8

topflracc[m/s2]

3.7 3.6 3.5 3.4 3.3 3.2 3.1 3

8.8

9.2

9.4 time [s]

9.6

9.8

(Magnified Image)

Figure 3.10: Analysis values for Three Storey Model with Exciter Response

SECOND FLOOR

(a)

(b) Figure 3.11: Test values for Three Storey Model with Exciter (a) Response (b) FFT output

SECOND FLOOR

secflracc[m/s2]

-1

-2

-3

5 time [s]

3 2.98 2.96

secflracc[m/s2]

2.94 2.92 2.9 2.88 2.86 2.84 2.82 9.65

9.7

9.75

9.8 time [s]

9.85

9.9

(Magnified Image)

Figure 3.12: Analysis values for Three Storey Model with Exciter

3.4.2 Three storey model with exciter and equipment Model was subjected to harmonic excitation by running the exciter. Forcing frequency was increased gradually by increasing the rpm of the exciter. As the driving frequency approached natural frequency of the structure (8.301Hz), the magnitude of resonance approached the maximum value. Test output response obtained is compared to the results from numerical analysis (refer Appendix II-D pg no. 75) in table 3.5.

Accelerometer Location

Acceleration (m/s2)

Error

Experiment

Analysis

(%)

Top floor

0.2845

0.2923

2.74

Second floor

0.314

0.321

2.23

Equipment

0.1324

0.127

4.08

Table 3.5: Experimental and numerical analysis results for model with exciter and equipment (active)

TOP FLOOR

(a)

(b)

Figure 3.13: Test values for Three Storey Model with Exciter and equipment (a) Response (b) FFT output 34

TOP FLOOR

0.3

0.2

secflracc[m/s2]

0.1

-0.1

-0.2

-0.3

-0.4

5 time [s]

0.298 0.296 0.294

secflracc[m/s2]

0.292 0.29 0.288 0.286 0.284 0.282 0.28 0.278 7.35

7.4

7.45

7.5 time [s]

7.55

7.6

(Magnified Image)

Figure 3.14: Analysis values for Three Storey Model with Exciter and Equipment 35

SECOND FLOOR

(a)

(b) Figure 3.15: Test values for Three Storey Model with Exciter and equipment (a) Response (b) FFT output 36

SECOND FLOOR 0.4 0.3

topflracc[m/s2]

0.2 0.1 0 -0.1 -0.2 -0.3 -0.4

5 time [s]

0.37 0.36

topflracc[m/s2]

0.35 0.34 0.33 0.32 0.31 0.3 0.29 0.28 8.8

9.2

9.4

9.6

9.8

time [s]

(Magnified Image)

Figure 3.16: Analysis values for Three Storey Model with Exciter and Equipment 37

EQUIPMENT

(a)

(b) Figure 3.17: Test values for Three Storey Model with Exciter and equipment (a) Response (b) FFT output 38

EQUIPMENT

0.15

0.1

eqpacc[m/s 2]

0.05

-0.05

-0.1

-0.15

-0.2

5 time [s]

0.145 0.14

eqpacc[m/s 2]

0.135 0.13 0.125 0.12 0.115 0.11 0.105 8.8

9.2 time [s]

9.4

9.6

9.8

(Magnified Image)

Figure 3.18: Analysis values for Three Storey Model with Exciter and Equipment 39

3.5 Conclusions Results obtained by numerical analysis were validated by experimental work in this chapter. Maximum error in results in natural frequency analysis was 11.55% and maximum error in harmonic analysis was 9.93% which is a reasonable agreement. It can be seen that the test response time history has only steady state values whereas the numerical analysis has transient part which builds up to the steady state value. It is because the exciter used in the test had control limitations and it was difficult to achieve accurate frequency of excitation starting from transient part. Therefore, the response was recorded only when the exact excitation frequency was achieved.

Chapter 4 Analysis on model subjected to harmonic excitation

Response analysis was done on the model for different frequency ratios (Rf) and mass ratios (Rm) using Newmark-Beta method in MATLAB (refer Appendix I pg no. 66). For the analysis, a similar model with one equipment and exciter on top floor was used. Harmonic and seismic excitations were applied on the model one by one to obtain the response at top floor and equipment. Further, response under the application of both harmonic and seismic excitation was analysed.

4.1 Modal analysis Modal analysis was done (refer Appendix III-A pg no. 79) for different frequency ratios (Rf) and mass ratios (Rm) on the coupled system. This analysis facilitated understanding about the following: 4.1.1 Variation in frequency of coupled with respect to uncoupled system

/R f

Coupled and uncoupled frequencies of structure and equipment are listed in tables below:

0.01 0.02 0.04 0.06 0.08 0.1 0.12 0.15 0.2 0.25 UCS

0.2 -0.02083 -0.04166 -0.08329 -0.12491 -0.1665 -0.20808 -0.24964 -0.31193 -0.41566 -0.51926 7.324

0.4 -0.09516 -0.19016 -0.37968 -0.56859 -0.75687 -0.94455 -1.13163 -1.41113 -1.87407 -2.33345 7.324

0.6 -0.27998 -0.55747 -1.1053 -1.64407 -2.1743 -2.69648 -3.21103 -3.96941 -5.20039 -6.3941 7.324

0.8 -0.85917 -1.66673 -3.1611 -4.53176 -5.80799 -7.00906 -8.14843 -9.76269 -12.2545 -14.5561 7.324

1 -5.12492 -7.32076 -10.4988 -12.9947 -15.1372 -17.0537 -18.8094 -21.2227 -24.8302 -28.0776 7.324

1.2 1.498389 2.78809 4.96789 6.798914 8.396323 9.82324 11.11871 12.87047 15.4102 17.60523 7.324

1.4 0.985627 1.908273 3.599205 5.125069 6.520128 7.80828 9.00698 10.66537 13.13222 15.31129 7.324

1.6 0.802428 1.57106 3.019085 4.364318 5.622051 6.804169 7.920127 9.486406 11.85724 13.98529 7.324

1.8 0.711013 1.398803 2.711024 3.947663 5.117611 6.22818 7.285464 8.782529 11.07418 13.15356 7.324

2 0.657231 1.296332 2.524189 3.690802 4.802251 5.863701 6.879582 8.326219 10.55728 12.5968 7.324

/R f

Table 4.1 Coupled and uncoupled frequencies of structure

0.01 0.02 0.04 0.06 0.08 0.1 0.12 0.15 0.2 0.25 UCE

0.2 0.020826 0.041639 0.083224 0.124753 0.166228 0.207649 0.249015 0.310963 0.413939 0.516579 1.4648

0.4 0.095068 0.189797 0.378248 0.565373 0.751189 0.935715 1.118968 1.391498 1.839592 2.280246 2.9296

0.6 0.279197 0.554379 1.093214 1.617474 2.128033 2.625679 3.111132 3.817867 4.943317 6.009828 4.3944

0.8 0.851856 1.639403 3.064241 4.335297 5.489181 6.549967 7.534485 8.894361 10.91668 12.70654 5.8592

1 4.875078 6.82138 9.501244 11.50024 13.14709 14.5691 15.83159 17.50717 19.89117 21.92236 7.324

1.2 -1.52118 -2.86805 -5.22759 -7.29489 -9.16592 -10.8933 -12.5096 -14.7716 -18.2176 -21.3669 8.7888

1.4 -0.99544 -1.9454 -3.73358 -5.40192 -6.9749 -8.46961 -9.89854 -11.9387 -15.1175 -18.0795 10.2536

1.6 -0.80892 -1.59614 -3.11307 -4.56348 -5.95695 -7.30094 -8.60137 -10.4806 -13.4523 -16.2592 11.7184

Table 4.2 Coupled and Uncoupled Frequencies of Equipment 42

1.8 -0.7161 -1.41865 -2.78657 -4.10991 -5.39364 -6.64185 -7.85795 -9.62812 -12.4533 -15.1458 13.1832

2 -0.66158 -1.31336 -2.58955 -3.83224 -5.0445 -6.22895 -7.38783 -9.08244 -11.8034 -14.4123 14.648

%Variation in frequency

0.2 20

0.4

0.6 0.8

0 0

0.05

0.1

0.15

0.2

0.25

0.3

-10

1 1.2 1.4

-20

1.6 -30

1.8

-40

%Mass Ratio

Figure 4.1 Percent variation in frequency of coupled with respect to uncoupled structure 25 20 0.2

% variation in frequency

0.4

0.6

0.8 1

0 0

0.05

0.1

0.15

-5

0.2

0.25

0.3

1.2 1.4

-10

1.6

-15

1.8

-20

-25

% Mass Ratio

Figure 4.2 Percent variation in frequency of coupled with respect to uncoupled equipment

From figure 4-1, it can be seen that as the frequency ratio approaches resonance (Rf-1), percent variation of coupled frequencies of both equipment and structure increases with respect to uncoupled frequencies. This is due to the inertia of the equipment participating in the coupled mode of structure. However the variation of coupled with respect to uncoupled frequencies of both equipment and structure for structure supporting flexible equipment (Rf= 0.4) is less than the structure supporting stiff equipment (Rf= 1.6).

The above figure shows existing decoupling criteria for structure based on USNRC. The criterion is based on the frequency ratio (Rf) and mass ratio (Rm) of the equipment to the structure. Where, Rf is the ratio of frequency or modal frequency of uncoupled equipment to the uncoupled structure and Rm is the ratio of mass or modal mass of the uncoupled equipment to the uncoupled structure. From this criteria following points can be inferred: (i)

Decoupling can be done for any Rf, if Rm< 0.01

(ii)

If 0.01 ≤ Rm ≤ 0.1 decoupling can be done provided 0.8≥ Rf ≥ 1.25

(iii) If Rm ≥ 0.1 and Rf ≥ 3 It is sufficient to include only the mass of the system in the primary structure. (iv)

If Rm ≥ 0.1 and Rf< 0.33 decoupling can be done.

(v)

If Rm ≥ 0.1 and 0.33 <Rf< 3, coupled system analysis is required.

However the above criterion has following limitations: (i)

It is applicable only for structures and not for equipments.

(ii) The criteria is generalised for a narrow range of frequency ratio (Rf) and mass ratios (Rm).

In the present work, a new decoupling criterion is developed based on percent variation in frequency of coupled with respect to uncoupled structure as well as equipment.

Coupling required

%Variation in frequency

0.2 20

0.4

0.6 0.8

0 0

0.05

0.1

0.15

0.2

0.25

0.3

-10

1 1.2 1.4

-20

1.6 -30

1.8

-40

Coupling not required

%Mass Ratio

Figure 4.3: Suggested decoupling criteria based on percent variation in frequency of coupled with respect to uncoupled structure. 25

Coupling required

0.2

15 % variation in frequency

0.4 10

0.6

0.8

-5

0.05

0.1

0.15

0.2

0.25

0.3

1.2 1.4

-10

1.6

-15

1.8

-20 -25

% Mass Ratio

Coupling not required

Figure 4.4: Suggested decoupling criteria based on percent variation in frequency of coupled with respect to uncoupled equipment. 45

Figure 4.3 and figure 4.4 shows criteria based on percent variation in frequency of coupled with respect to uncoupled structure and equipment respectively. 10% variation is frequency is considered in the above figures however the same graph can be used for all range of variations. Coupled analysis is required in the shaded areas of the graph while it’s not mandatory for the non shaded area. It can be observed that this criterion is applicable for wide range of frequency ratio (Rf) and mass ratios (Rm) unlike the existing one. Coupling is not required in the non shaded areas as the variation in coupled and uncoupled frequencies is very less for these cases. 4.1.2 Harmonic analysis - coupled frequencies as forcing frequency Harmonic force with amplitude, P= 3.4004 N and forcing frequency 8.301 Hz was applied at the top floor of the uncoupled and coupled model. Response (acceleration) obtained from the harmonic analysis for Rm =1% (refer appendix III-B pg no. 81) at the top floor is listed in table 4-3.

ω (Hz) 5.371 6.348 6.834 7.324 7.813 8.301 10.724

UCS

0.0737 0.204 0.3384 0.522 1.1749 5.2056 1.0712

0.0789 0.2027 0.347 0.5174 1.068 3.7394 1.1253

Table 4.3 Response (acceleration in m/s2) of coupled and uncoupled structure subjected to harmonic excitation Similarly, response (acceleration) at the equipment for Rm =1% is listed in table 4-4:

ω (Hz) 5.371 6.348 6.834 7.324 7.813 8.301 10.724

UCE

0.0756 0.2076 0.3525 0.5303 1.1875 5.2999 1.0968

207.639 207.639 207.639 207.639 207.639 207.639 207.639

Table 4.4: Response (acceleration in m/s2) of coupled and uncoupled equipment subjected to harmonic excitation 46

Data in table 4.3 and 4.4 are represented in figures below:

Chart Title 6

Acceleration (m/s2)

5 4 3

UCS CS

2 1 0 0

0.2

0.4

0.6

0.8

1.2

1.4

1.6

1.8

W/Wn

(a) 250

Acceleration (m/s2)

200

150 CE 100

UCE

0 0

0.5

1.5

2.5

3.5

W/Wn

(b) Figure 4.5: Response (acceleration) of coupled with respect to uncoupled (a) structure and (b) equipment subjected to harmonic excitation

It can be seen from figure 4.5 (a) that response of coupled with respect to uncoupled structure is less. At ω/ωn = 1 (tuning) the uncoupled response of structure is higher than that of coupled. However, when the coupled structure is excited at ω/ωn = 0.846 and ω/ωn = 1.076 (first two natural frequencies of the coupled structure) the response is observed to be higher as compared to when the same is excited at ω/ωn = 1.It is because at tuning condition some of the energy of structure is transferred to equipment in a coupled system thus reducing the response with respect to uncoupled. Same trend is observed for equipments shown in figure 4.5 (b) and thus can be explained in a similar manner. However, the variation between coupled and uncoupled equipment is more than that of structure.

4.2 Harmonic analysis Further, harmonic force with amplitude, P= 2.009 N and forcing frequency, f= 6.348 Hz (first natural frequency of the uncoupled structure) was applied at the centre node of the top floor of the model for different frequency ratio (Rf) and mass ratios (Rm). Response obtained from the harmonic analysis (refer appendix III-C pg no. 85) of coupled and uncoupled structure is

%R m/

listed in tables below:

1 2 4 6 8 10 12 15 20 25

0.2

0.4

0.6

0.8

1.2

1.4

1.6

1.8

UCS

3.7731 3.7369 3.6684 3.601 3.5312 3.4628 3.3958 3.3075 3.1757 3.0413

3.6497 3.492 3.2224 2.9804 2.7892 2.6146 2.4444 2.2694 1.9871 1.7931

3.3489 2.9946 2.4658 2.1209 1.8309 1.6599 1.4866 1.2773 1.1067 0.9512

2.6907 2.0855 1.4842 1.1296 0.9752 0.8244 0.7336 0.6151 0.5087 0.4505

0.9331 0.6821 0.4733 0.4059 0.3865 0.3675 0.3499 0.3247 0.2868 0.2533

5.5851 2.3929 1.1302 0.8075 0.5702 0.5169 0.4902 0.4298 0.3092 0.2546

7.3916 4.5469 1.9549 1.2188 0.9413 0.7023 0.6474 0.5277 0.3887 0.278

6.3874 6.2469 2.5743 1.6085 1.1888 0.871 0.7754 0.6353 0.4376 0.3367

5.9317 7.3846 3.015 1.8762 1.3568 0.9959 0.8788 0.6758 0.4703 0.4169

5.6909 7.8573 3.3706 2.0637 1.4781 1.1426 0.945 0.6747 0.5535 0.4383

3.15 3.15 3.15 3.15 3.15 3.15 3.15 3.15 3.15 3.15

Table 4.5: Response (acceleration in m/s2) of coupled and uncoupled structure subjected to

0.2

0.6

0.8

1.2

1.6

UCS

0.0014 0.0014 0.0013 0.0013 0.0013 0.0012 0.0012 0.0012 0.0011 0.0011

0.0012 0.0011 0.0009 0.0007 0.0006 0.0005 0.0005 0.0004 0.0003 0.0003

0.0009 0.0007 0.0005 0.0004 0.0003 0.0003 0.0002 0.0002 0.0001 0.0001

0.0003 0.0002 0.0002 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001

0.0018 0.0008 0.0004 0.0003 0.0002 0.0002 0.0002 0.0001 0.0001 0.0001

0.0025 0.002 0.0009 0.0005 0.0004 0.0003 0.0003 0.0002 0.0002 0.0001

0.0022 0.0027 0.0011 0.0007 0.0005 0.0004 0.0003 0.0003 0.0002 0.0002

0.0014 0.0014 0.0014 0.0014 0.0014 0.0014 0.0014 0.0014 0.0014 0.0014

Rm /R f

harmonic excitation

1 2 4 6 8 10 12 15 20 25

Table 4.6: Response (displacement in â&#x20AC;&#x2DC;mâ&#x20AC;&#x2122;) of coupled and uncoupled structure subjected to harmonic excitation

Data in table 4-5 and table 4-6 are represented in figures below:

9 8 0.2

Acceleration (m/s2)

0.4

0.6 0.8

1 4

1.2

1.4 1.6

1.8 2

UCS

0 1

% Mass Ratio

(a) Acceleration 0.003 0.0025 Displacement(m)

C.S. Rf 0.2 0.002

C.S. Rf 0.6 C.S. Rf 0.8

0.0015

C.S. Rf 01 C.S. Rf 1.2

0.001

C.S. Rf 1.6 C.S. Rf 2

0.0005

UCS 0 0

%Mass Ratio

(b) Displacement Figure 4.6: Response of coupled with respect to uncoupled structure subjected to harmonic excitation 50

Following points can be inferred from figure 4-6: (i)

As frequency ratio (Rf) increases from 1, relative reduction in response of

coupled with respect to uncoupled structure reduces. This is due to the reducing trend of variation in frequency of coupled with respect to uncoupled structure. (ii)

As the frequency ratio (Rf) decreases from 1, relative reduction in response of

coupled with respect to uncoupled structure also reduces. However, it is much lower than the above case. Structure can be decoupled in the following cases as shown in table 4.7

Rf 0.2 0.4 0.8,1 1.2 1.6 2

%Rm 25 only ≥6 All Rm ≥2 ≥6 ≥8

Table 4.7: Criteria for decoupling the structure

Similarly, response obtained from the harmonic analysis (refer Appendix III-C pg no. 85,

Rm /R

Appendix III-D, pg. no. 89 ) of coupled and uncoupled equipment is listed in tables below:

1 2 4 6 8 10 12 15 20 25

0.2

0.4

0.6

0.8

1.2

1.4

1.6

1.8

0.1515 0.1499 0.1467 0.1439 0.1412 0.1385 0.1358 0.1318 0.1265 0.121

0.6664 0.6356 0.586 0.5417 0.5015 0.47 0.4418 0.4043 0.3511 0.3148

1.7916 1.5943 1.3047 1.1064 0.9508 0.8449 0.745 0.6487 0.5234 0.4327

4.3029 3.3223 2.2051 1.7103 1.3625 1.1181 0.9375 0.7287 0.5793 0.4946

11.622 6.269 3.2848 2.2379 1.6874 1.3472 1.1446 0.915 0.7029 0.5642

19.4523 8.1747 3.6881 2.4399 1.8243 1.4511 1.181 0.8995 0.6575 0.5486

15.5414 9.5399 4.0806 2.5826 1.8775 1.5271 1.2068 0.9351 0.6902 0.6062

10.6859 10.4275 4.2762 2.6489 1.9398 1.5361 1.2476 0.9381 0.7711 0.5568

8.7039 10.8128 4.4355 2.7335 1.9716 1.5769 1.2689 0.9809 0.7788 0.6123

7.6709 10.5712 4.5446 2.7305 1.9652 1.5612 1.2619 1.0366 0.7099 0.5954

Table 4.8: Response (acceleration in m/s2) of coupled equipment subjected to harmonic

Rm /R f

excitation

1 2 4 6 8 10 12 15 20 25

0.2

0.4

0.6

0.8

1.2

1.4

1.6

1.8

14.0676 14.0676 14.0676 14.0676 14.0676 14.0676 14.0676 14.0676 14.0676 14.0676

18.2558 18.2558 18.2558 18.2558 18.2558 18.2558 18.2558 18.2558 18.2558 18.2558

25.8733 25.8733 25.8733 25.8733 25.8733 25.8733 25.8733 25.8733 25.8733 25.8733

41.8111 41.8111 41.8111 41.8111 41.8111 41.8111 41.8111 41.8111 41.8111 41.8111

108.4601 108.4601 108.4601 108.4601 108.4601 108.4601 108.4601 108.4601 108.4601 108.4601

43.326 43.326 43.326 43.326 43.326 43.326 43.326 43.326 43.326 43.326

23.284 23.284 23.284 23.284 23.284 23.284 23.284 23.284 23.284 23.284

16.3161 16.3161 16.3161 16.3161 16.3161 16.3161 16.3161 16.3161 16.3161 16.3161

12.4745 12.4745 12.4745 12.4745 12.4745 12.4745 12.4745 12.4745 12.4745 12.4745

9.3282 9.3282 9.3282 9.3282 9.3282 9.3282 9.3282 9.3282 9.3282 9.3282

Table 4.9: Response (acceleration in m/s2) of uncoupled equipment subjected to harmonic excitation

/R f Rm 1 2 4 6 8 10 12 15 20 25

0.2

0.6

0.8

1.2

1.6

0.0001 0.0001 0.0001 0.0574 5.64E-05 5.53E-05 5.42E-05 5.27E-05 5.06E-05 4.85E-05

0.0007 0.0006 0.0005 0.3984 0.000346 0.000301 0.000268 0.000236 0.000185 0.000158

0.0015 0.0012 0.0008 0.6044 0.000477 0.000386 0.000324 0.000253 0.000212 0.000178

0.004 0.0021 0.0011 0.7697 0.000586 0.000464 0.000399 0.000318 0.000245 0.000198

0.0064 0.0027 0.0012 0.8293 0.000626 0.000491 0.000407 0.000324 0.000238 0.000182

0.0042 0.0034 0.0014 0.9077 0.00066 0.00052 0.000434 0.000341 0.000259 0.000202

0.0029 0.0037 0.0015 0.934 0.000679 0.000536 0.000445 0.000355 0.000264 0.000205

Table 4.10: Response (displacement in ‘m’) of coupled equipment subjected to harmonic

Rm /R f

excitation

1 2 4 6 8 10 12 15 20 25

0.2

0.6

0.8

1.2

1.6

0.0218 0.0218 0.0218 0.0218 0.0218 0.0218 0.0218 0.0218 0.0218 0.0218

0.0149 0.0149 0.0149 0.0149 0.0149 0.0149 0.0149 0.0149 0.0149 0.0149

0.0181 0.0181 0.0181 0.0181 0.0181 0.0181 0.0181 0.0181 0.0181 0.0181

0.039 0.039 0.039 0.039 0.039 0.039 0.039 0.039 0.039 0.039

0.0141 0.0141 0.0141 0.0141 0.0141 0.0141 0.0141 0.0141 0.0141 0.0141

0.004 0.004 0.004 0.004 0.004 0.004 0.004 0.004 0.004 0.004

0.0018 0.0018 0.0018 0.0018 0.0018 0.0018 0.0018 0.0018 0.0018 0.0018

Table 4.11: Response (displacement in ‘m’) of uncoupled equipment subjected to harmonic excitation

120

acceleration in m/s2

100 80 60

CE Rf 1 UCE Rf 1

40 20 0 0

% Mass Ratio

(a) Rf= 1 120

acceleration in m/s2

100

UCE Rf 1 UCE Rf 0.8

UCE Rf 1.2 UCE Rf 0.2

UCE Rf 2 CE Rf 1

CE Rf 0.8 CE Rf 1.2

CE Rf 0.2 CE Rf 2

0 0

% Mass Ratio

(b) Rf= 0.2, 0.8, 1, 1.2, 2

Figure 4.7: Response (acceleration) of coupled and uncoupled equipment subjected to harmonic excitation.

0.8 0.7 0.6 0.5 CE Rf=1

0.4

UCE Rf =1

0.3 0.2 0.1 0 0

(a). Rf = 1

0.9 0.8 0.7 0.6 CE, Rf=0.6

0.5

CE, Rf = 1.2

0.4

UCE Rf = 0.6

0.3

UCE Rf = 1.2

0.2 0.1 0 0

(b) Rf = 0.6,1.2

Figure 4.8: Response (displacement) of coupled and uncoupled equipment subjected to harmonic excitation.

Following points can be inferred from data in table 4.7 to table 4.10 and the same has been depicted in figure 4.7 and figure 4.8: (i)

As frequency ratio (Rf) increases from 1, relative reduction in response of

coupled with respect to uncoupled equipment reduces. This is due to the reducing trend of variation in frequency of coupled with respect to uncoupled equipment. (ii)

As the frequency ratio (Rf) decreases from 1, relative reduction in response of

coupled with respect to uncoupled equipment also reduces. However, it is much lower than the above case. Equipment can be decoupled under the following cases as shown in figure 4.9:

Values

%Rm

0.2 1 2 4 6 8 10 12 15 20 25

0.6

0.8

Rf 1

1.2

Decoupling can be done Coupling Required

No Effect of Coupling

Figure 4.9: Decoupling Criteria for equipment

1.6 2 Coupling Required

Chapter 5 Analysis on seismic excitation

In order to study the response under seismic loading, response spectra shown in figure 5.1 was applied as base excitation on the experimental model.

14.00 12.00

Acceleration (m/s2)

10.00 8.00 6.00 4.00 2.00 0.00 0

Frequency (Hz)

Figure 5.1: Response spectra used for seismic excitation of model

Artificial time history corresponding to response spectra shown in figure 5.1 was obtained. Transient analysis was performed on the model using artificial time history to obtain response time history for the top floor of structure. 57

From the response time history, floor response spectrum was obtained which gives response of the uncoupled equipment (figure 5.2).

120

Acceleration (m/s2)

100 80 60 40 20 0 0

Frequency (Hz) Figure 5.2: Floor response spectra at top floor of model

Response obtained from the seismic analysis (refer appendix IV-A pg no. 91) of coupled and uncoupled structure is listed in tables below:

Rf Rm (%)

UCS 0.4

1.6

14.674

10.628

14.446

14.722

14.264

10.281

13.146

14.722

13.853

10.350

12.524

14.722

Table 5.1: Response (acceleration in m/s2) of coupled and uncoupled structure subjected to seismic excitation

UCS

Rm (%)

Table 5.2:

1.2

0.8

0.4

0.00455

0.00599

0.00574

0.00623

0.00627

0.00490

0.00594

0.00506

0.00612

0.00627

0.00557

0.00637

0.00505

0.00592

0.00627

0.00665

0.00727

0.00546

0.00568

0.00627

Response (displacement in â&#x20AC;&#x2DC;mâ&#x20AC;&#x2122;) of coupled and uncoupled structure subjected to seismic excitation

Following points can be inferred from data in table 5.1 and 5.2 and the same has been depicted in figure 5.3. (i)

Due to tuning (Rf =1) acceleration response decreases as some structure energy

is transferred to equipment. (ii)

At Rf =1.6 acceleration response increases because of higher modes.

(iii)

Displacement response increases with increase in Rm due to decrease in

structures frequency. Since broadband spectra was considered for the analysis in the present work, so the acceleration value remains constant from 5 Hz to 9 Hz (figure 4-9). As the frequency of coupled structure reduced with increase in Rm thus displacement response increases.

(a)

(b)

Figure 5.3: Response of coupled with respect to uncoupled structure subjected to seismic excitation (a) Acceleration (b) Displacement

Response obtained from the seismic analysis of coupled and uncoupled equipment (refer Appendix IV-B, pg no. 94 & Appendix IV-C, pg no. 98 ) is listed in tables below:

Rf Rm (%)

0.4

1.6

7.592

80.635

25.611

7.456

27.820

20.113

7.324

21.373

17.620

Table 5.3: Response (acceleration in m/s2) of coupled equipment subjected to seismic excitation

Rf Rm (%)

0.4

1.6

8.760

103.110

25.108

8.760

103.110

25.108

8.760

103.110

25.108

Table 5.4: Response (acceleration in m/s2) of uncoupled equipment subjected to seismic excitation

Rf Rm (%)

Table 5.5:

1.2

0.8

0.4

0.03655

0.01928

0.02548

0.01893

0.02092

0.01537

0.02198

0.01904

0.01643

0.01346

0.01950

0.01928

0.01530

0.01311

0.01706

0.01967

Response (displacement in ‘m’) of coupled equipment subjected to seismic excitation

Rf Rm (%)

Table 5.6:

1.2

0.8

0.4

0.04397

0.01485

0.02928

0.02337

0.04397

0.01485

0.02928

0.02337

0.04397

0.01485

0.02928

0.02337

0.04397

0.01485

0.02928

0.02337

Response (displacement in ‘m’) of uncoupled equipment subjected to seismic excitation

Following points can be inferred from data in table 5.3 to table 5.6 and the same has been depicted in figure 5.4. (i) As frequency ratio approaches resonance (Rf =1), relative reduction in response of coupled with respect to uncoupled equipment reduces. This is due to the reducing trend of variation in frequency of coupled with respect to uncoupled equipment. (ii) When frequency ratio, Rfâ&#x2030;¤ 0.4, there is no effects of coupling on equipment response (acceleration) irrespective of mass ratio. (iii)When frequency ratio, Rf =1, the variation between coupled and uncoupled equipment is to the tune of 1.3 to 5 times.

(a)

(b) Figure 5.4: Response of coupled with respect to uncoupled structure and equipment subjected to seismic excitation (a) Acceleration (b) Displacement 63

Chapter 6 Conclusion

6.1 Conclusion Response analysis was done on the model for different frequency ratios (Rf) and mass ratios (Rm). Harmonic and seismic excitations were applied on the model to obtain the response at top floor and equipment and a new decoupling criteria is developed which is based on percent variation in frequency of coupled with respect to uncoupled structure as well as equipment. This criterion is applicable for wide range of frequency ratio (Rf) and mass ratios (Rm) and also for all range of variations in frequency unlike the existing decoupling criteria for structure based on USNRC. Further when the experimental model was subjected to harmonic and seismic excitations, a similar trend in response was observed in both these excitations. However, as compared to equipment, there is less variation between coupled with respect to uncoupled response of structure. Coupled structure and equipment is observed to be lower as compared to uncoupled response in both the above excitation. Therefore, from the present work, coupled analysis is found to be more realistic. In such cases where coupled analysis is not possible, variation in response of coupled with respect to uncoupled system can be obtained using the graphs obtained in this work.

6.2 Limitations of present work Though this type of study is highly realistic and less erroneous, it does possess a few limitations. These draw backs are listed as: (i) It requires some kind of an experimental data to start the analysis on (ii) Experimental values are bound to have certain amount of error due to factors such as wind vibration, surrounding vibrations, etc. (iii)Damping due to various factors is neglected.

6.3 Future scope of work (i) Only one damping is considered in this analysis. Variation in damping needs to be considered with more cases. (ii) Experimental validation for harmonic excitation was done however similar validation can be performed for earthquake excitation.

APPENDIX I 1. Time History Methods It is a step by step procedure in which the loading and the response history are divided into sequence time intervals or steps. The response during each step is then calculated from the initial conditions existing at the beginning of each time step and from the loading history during the step. These can be explicit or implicit. Explicit methods are those in which new response quantities calculated in each step depend on quantities obtained in previous steps and analysis proceeds directly from one step to other. In Implicit methods the expressions giving the new values for a given step include one or more values pertaining to that same step, so that trial values of necessary quantities must be assumed and then refined by iterations.

1.1 Time integration methods This type of approach makes use of integration to step forward from the initial to the final conditions for each time step. The following equations represent the essential concept t

x1  x 0   x( )d 0

(1.1)

t

x1  x 0   x ( )d 0

(1.2)

In order to carry out this type of analysis, it is necessary first to assume how the acceleration varies during the time step. Some of the time integration techniques discussed are Newmark beta technique and Wilson theta technique.

1.1.1 Newmark–Beta Method In Newmark’s formulation the basic integration equations for final velocity and displacement are

x(ti 1 )  x(ti )  (1   ) x(ti )t   x(ti 1 )t

(1.3)

1 x(ti 1 )  x(ti )  tx(ti )  (   ) t 2 x(ti )  t 2 x(ti 1 ) 2

(1.4)

Factor  provides linearly varying weighting between the influence of initial and final accelerations on change of velocity. And factor , provides linearly varying weighting between the influence of initial and final accelerations on change of displacement. When  = 1/2 and  = 1/4 the Newmark’s formulation reduces to constant acceleration method. When  = 1/2 and  = 1/6 the Newmark’s formulation reduces to linear acceleration method

1.1.2 Linear acceleration method It is based on the assumption that the acceleration variation is linear during that time step. It is generally convenient for time integration methods to formulate the response in terms of the incremental equation of motion as given in Eq. (1.5)

Acceleration (Linear)

  kx  P Mx  Cx

(1.5)

x(  ) X

x(t i)

Velocity (Quadratic)

t

x(  )

x(t i) ti

Displacem ent (Cubic)

t

x(  )

t i+1

X

t i+1

X

x(t i) ti

t

t i+1

Fig. 1.1 Motion based on linearly varying acceleration Considering linear variation of acceleration as shown in Fig. 1.1, acceleration at any time ,  can be given by the Eq. (1.6)

x( )  x(t i ) 

x  t

(1.6) 67

Similarly, the velocity at any time,  is obtained by integrating the acceleration (i.e. Eq. (1.6) ) and is given by Eq. (1.7).

x ( )  x (t i )  x(t i ) 

x  2 t 2

(1.7)

Substituting the value of  = t at time ti+1 in Eq. (1.7) we get Eq. (1.8)

x  x(ti )t  x

t 2

(1.8)

Displacement at any time,  is obtained by integrating the velocity (i.e. Eq. (1.7) ) and is given by Eq. (1.9).

x ( )  x (ti )  x (ti )  x(ti )

 2



x  t 6

(1.9)

Substituting the value of  = t at time ti+1 in Eq. (1.9) we get Eq. (1.10)

t t x  x (ti )t  x (ti )  x 2 6 2

(1.10)

Now using equation of motion and knowing initial velocity and displacement, initial acceleration will be given as follows

x(t i ) 

P(t i )  Kx(t i )  Cx (t i ) M

(1.11)

Using Eq.(1. 9) the value of incremental acceleration ins obtained in terms of incremental displacement as follows, x 

6 6 x  x (t i )  3x(t i ) 2 t t

(1.12)

Substituting in Eq. (1.12) in Eq. (1.8) we get the value of incremental velocity in terms of incremental displacement as follows,

x 

3 t x(t i ) x  3 x (t i )  t 2

(1.13)

Substituting (1.12) and (1.13) in (1.5) ,

~ ~ Kx  P

(1.14)

Where , 3C 6 M ~ KK  t t 2

(1.15) 68

t ~   6  x(t i )  M  x (t i )  3x(t i ) P  P  C 3 x (t i )  2    t 

(1.16)

Thus incremental displacement is obtained and incremental velocity can be evaluated by equation (1.13) Thus velocity and displacement at the end of time increment t can be .evaluated as given by Eq. (1.17) and Eq. (1.18)

x (t i 1 )  x (t i )  x

(1.17)

x(t i 1 )  x(t i )  x

(1.18)

When this step has been completed the calculation of response for this time increment is finished and the analysis is stepped forward to the next time interval.

APPENDIX II-A % Validation of natural frequencies of structure clear all d = 0.0095; % dia of the rods in m Ar = (pi*0.0095^2)/4; % area of c/s of the rod I = (pi*0.0095^4)/64; % moment of inertia of the rod l = 1.02/3; % length of each rod E = 210e9; % mod. of elasticity of the rod material m1 = 6.35+4*0.19;%effective mass of floor 1 and 2 m2 = 6.35+2*0.19; effective mass of floor 3 m = [1*m2,0,0; 0,1*m1,0;0,0,1*m1+2.324]; % mass matrix Kc = 4*(12*E*I/l^3) ;% stiffness k = Kc*[1 -1 0;-1 2 -1;0 -1 2]; %Calculation of naturl frequencies by Eigen values and vectors A = inv(m)*k; [V,D] = eig(A) ev = diag(D); omega1 = sqrt(ev(1,1)); omega2 = sqrt(ev(2,1)); omega3 = sqrt(ev(3,1)); f1 = (1/(2*pi))*omega1 f2 = (1/(2*pi))*omega2 f3 = (1/(2*pi))*omega3

APPENDIX II-B % Validation of natural frequencies having model with exciter d = 0.0095; % dia of the rods in m Ar = (pi*0.0095^2)/4; % area of c/s of the rod I = (pi*0.0095^4)/64; % moment of inertia of the rod l = 1.02/3; % length of each rod E = 210e9; % mod. of elasticity of the rod material ks = 4*(12*E*I)/l^3 % stiffness me = 0.5899; %mass of equipment le = 0.257; %equipment dimensions te = 0.0025; we = 0.03; Ee = 69e9;% equipment properties Ae = we*te; Ie = te*we^3/12; ke = 3*Ee*Ie/le^3; m1 = 6.35+4*0.19; % effective mass of floor 1 and 2 m2 = 6.35+2*0.19+0.026; % effective mass of floor 3 me = 0.589+0.026; % effective mass of equipment k = [ke -ke 0 0;-ke ke+ks -ks 0; 0 -ks 2*ks -ks; 0 0 -ks 2*ks];%stiffness matrix m = [me 0 0 0; 0 1*m2 0 0; 0 0 01*m1 0; 0 0 0 01*m1+2.324]; %Calculation of natural frequencies by Eigen values and vectors A = inv(m)*k; [V,D] = eig(A) ev = diag(D); omega1 = sqrt(ev(1,1)); omega2 = sqrt(ev(2,1)); omega3 = sqrt(ev(3,1)); omega4 = sqrt(ev(4,1)); f1 = (1/(2*pi))*omega1 f2 = (1/(2*pi))*omega2 f3 = (1/(2*pi))*omega3 71

f4 = (1/(2*pi))*omega4

APPENDIX II-C % Excitation frequency is 6.348Hz clear all d = 0.0095; % dia of the rods in m Ar = (pi*0.0095^2)/4; % area of c/s of the rod I = (pi*0.0095^4)/64; % moment of inertia of the rod l = 1.02/3; % length of each rod E = 210e9; % mod. of elasticity of the rod material m1 = 6.35+4*0.19; m2 = 6.35+2*0.19; m = [1*m2,0,0; 0,1*m1,0;0,0,1*m1+2.324]; % mass matrix Kc = 4*(12*E*I/l^3) ;% stiffness k = Kc*[1 -1 0;-1 2 -1;0 -1 2]; %stiffness matrix A = inv(m)*k; [V,D,FLAG] = eigs(A); B = sqrt(D); t1 = V(:,1); t2 = V(:,2); t3 = V(:,3); I1= t1/sqrt(t1'*m*t1); I2 = t2/sqrt(t2'*m*t2); I3 = t3/sqrt(t3'*m*t3); phi = [I1(1,1), I2(1,1) I3(1,1); I1(2,1), I2(2,1) I3(2,1); I1(3,1) I2(3,1) I3(3,1)]; w1 = B(1,1); w2 = B(2,2); w3 = B(3,3); A = [1/w1, w1, w1^3; 1/w2, w2, w2^3; 1/w3, w3, w3^3]; B = 0.01315*[1;1;1]; 72

X = inv(A)*B; c = (X(1)*m)+(X(2)*k)+X(3)*m*(inv(m)*k)^2; C = phi'*c*phi; M = phi'*m*phi; K = phi'*k*phi; %Initial Conditions u(:,1) = [0;0;0]; v(:,1) = [0;0;0]; q(:,1) = (phi'*m*u(:,1)); qdot(:,1) = (phi'*m*v(:,1)); p(:,1) = [0 ;0 ; 0]; P(:,1) = phi'*p(:,1); qdot2(:,1) = inv(M)*(P(:,1)-C*qdot(:,1)-K*q(:,1)); dt = 0.001; Kcap = K+(3*C/dt)+(6*M/dt^2); a = (6*M/dt)+3*C; b = 3*M+(dt*C/2); %Force Matrix T=10; %s harmonic force duration t=0:dt:T; % time vector N=length(t); % time steps number f =6.348 for i = 1:1:N omega= 2*pi*f;%rad/s forcing frequency p=0.05*0.025*omega^2; %N forc bhxfye amplitude Pf(i) = p*sin(omega*t(i)); end p1 = 0*Pf; p2 = p1; pd = [p1;p1;Pf]; %Newmark beta analysis for i =1:1: N-1 73

P = phi'*pd; dP(:,i) =P(:,i+1)-P(:,i); dPcap(:,i) = dP(:,i)+a*qdot(:,i)+b*qdot2(:,i); dq(:,i) = inv(Kcap)*dPcap(:,i); dqdot(:,i) =(3*dq(:,i)/dt)-3*qdot(:,i)-(dt*qdot2(:,i)/2); dqdot2(:,i) = 6*(dq(:,i)-dt*qdot(:,i))/(dt^2)-3*qdot2(:,i); q(:,i+1)=q(:,i)+dq(:,i); qdot(:,i+1)=qdot(:,i)+dqdot(:,i); qdot2(:,i+1)=qdot2(:,i)+dqdot2(:,i); u(:,i+1) = phi*q(:,i+1); x(:,i+1) = phi*qdot(:,i+1); z(:,i+1) = phi*qdot2(:,i+1); end z = z'; %Plots figure ; plot(t,z(:,1)); xlabel('time [s]'); ylabel('topflracc[m/s^2]'); figure ; plot(t,z(:,2)); xlabel('time [s]'); ylabel('secflracc[m/s^2]');

APPENDIX II-D % Validation of response of equipment and model with exciter clear all m1 = 6.35+4*0.19; m2 = 6.35+2*0.19+0.026; % total mass of the structure me = 0.589+0.026; d = 0.0095; % dia of the rods in m Ar = (pi*0.0095^2)/4; % area of c/s of the rod I = (pi*0.0095^4)/64; % moment of inertia of the rod l = 1.02/3; % length of each rod E = 210e9; % mod. of elasticity of the rod material le = 0.257; we = 0.03; te = 0.0025; Ee = 69e9; Ie = (te*we^3)/12; ks = 4*(12*E*I)/l^3 ; ke = (3*Ee*Ie)/le^3; k = [ke -ke 0 0;-ke ke+ks -ks 0; 0 -ks 2*ks -ks; 0 0 -ks 2*ks]; m = [me 0 0 0; 0 1*m2 0 0; 0 0 01*m1 0; 0 0 0 01*m1+2.324]; A = inv(m)*k; [V,D,FLAG] = eigs(A); B = sqrt(D); t1 = V(:,1); t2 = V(:,2); t3 = V(:,3); t4 = V(:,4); I1= t1/sqrt(t1'*m*t1); I2 = t2/sqrt(t2'*m*t2); 75

I3 = t3/sqrt(t3'*m*t3); I4 = t4/sqrt(t4'*m*t4); phi = [I1(1,1), I2(1,1),I3(1,1),I4(1,1) I1(2,1), I2(2,1),I3(2,1),I4(2,1) I1(3,1),I2(3,1),I3(3,1),I4(3,1) I1(4,1),I2(4,1),I3(4,1),I4(4,1)]; w1 = B(1,1); w2 = B(2,2); w3 = B(3,3); w4 = B(4,4); A = [1/w1, w1, w1^3 w1^5; 1/w2, w2, w2^3 w2^5; 1/w3, w3, w3^3 w3^5; 1/w4, w4, w4^3 w4^5]; B = 0.0135*[1;1;1;1]; X = inv(A)*B; c = (X(1)*m)+(X(2)*k)+X(3)*m*(inv(m)*k)^2+X(4)*m*(inv(m)*k)^3; C = phi'*c*phi; M = phi'*m*phi; K = phi'*k*phi; %Initial Conditions u(:,1) = [0;0;0;0]; v(:,1) = [0;0;0;0]; q(:,1) = (phi'*m*u(:,1)); qdot(:,1) = (phi'*m*v(:,1)); p(:,1) = [0 ;0 ; 0 ; 0]; P(:,1) = phi'*p(:,1); qdot2(:,1) = inv(M)*(P(:,1)-C*qdot(:,1)-K*q(:,1)); dt = 0.001; Kcap = K+(3*C/dt)+(6*M/dt^2); a = (6*M/dt)+3*C; b = 3*M+(dt*C/2); 76

%Force Matrix T=10; %s harmonic force duration t=0:dt:T; % time vector N=length(t); % time steps number for i=1:N omega = 2*pi*8.301;%rad/s forcing frequency p=0.05*0.025*omega^2; %N force amplitude Pf(i)=p*sin(omega*t(i)); end p1 = 0*(0:1:N-1); p3 = Pf; pd = [p1;p1;p1;p3]; %Newmark beta analysis for i =1:1:N-1 P = phi'*pd; dP(:,i) =P(:,i+1)-P(:,i); dPcap(:,i) = dP(:,i)+a*qdot(:,i)+b*qdot2(:,i); dq(:,i) = inv(Kcap)*dPcap(:,i); dqdot(:,i) =(3*dq(:,i)/dt)-3*qdot(:,i)-(dt*qdot2(:,i)/2); dqdot2(:,i) = 6*(dq(:,i)-dt*qdot(:,i))/(dt^2)-3*qdot2(:,i); q(:,i+1)=q(:,i)+dq(:,i); qdot(:,i+1)=qdot(:,i)+dqdot(:,i); qdot2(:,i+1)=qdot2(:,i)+dqdot2(:,i); u(:,i+1) = phi*q(:,i+1); x(:,i+1) = phi*qdot(:,i+1); z(:,i+1) = phi*qdot2(:,i+1); end u = u'; x = x'; z = z';

%PLOTS figure ; plot(t,z(:,1)); xlabel('time [s]'); ylabel('equipmentacc[m/s^2]'); figure ; plot(t,z(:,2)); xlabel('time [s]'); ylabel('topflracc[m/s^2]'); figure ; plot(t,z(:,3)); xlabel('time [s]'); ylabel('secflracc[m/s^2]');

APPENDIX III-A %variation in frequencies clear all M2 = 19.36; F2 = 7.324; K2 = 4*pi^2*F2^2*M2 mr = [ 0.01, 0.02, 0.04, 0.06, 0.08, 0.1, 0.12, 0.15, 0.20, 0.25]; fr = [ 0.2, 0.4, 0.6, 0.8, 1, 1.2, 1.4, 1.6, 1.8, 2] for i =1:1:10 M1(1,i) = mr(1,i)*M2; F1(1,i) = fr(1,i)*F2; end for i = 1:1:10, for j = 1:1:10, K1(i,j) = 4*pi^2*F1(1,i)^2*M1(1,j); end end disp('The mass of the equipment will be : '); disp(M1) disp('The frequencies of the equipment will be : '); disp(F1) disp(' The stiffness of the system for various mass and frequency ratios will be: ');disp(K1) for i = 1:1:10, MM = [M1(1,i) 0; 0 M2]; for j=1:1:10, KK =[K1(j,i) -K1(j,i); -K1(j,i) K2+K1(j,i)]; A = inv(MM)*KK; [V,D] = eig(A); ev = diag(D); omega1(i,j) = sqrt(ev(1,1)); omega2(i,j) = sqrt(ev(2,1)); f1(i,j) = (1/(2*pi))*omega1(i,j); 79

f2(i,j) = (1/(2*pi))*omega2(i,j); end end disp('The frequencies of the equipment for various mass and frequency ratios are :');disp(f1) disp('the frequencies of the structure for various mass and frequency ratios are :');disp(f2) for i = 1:1:10, for j = 1:1:10, fv1(i,j) = ((F1(1,j)-f1(i,j))/F1(1,j))*100; end end for i = 1:1:10, for j = 1:1:10, fv2(i,j) = ((F2-f2(i,j))/F2)*100; end end disp('The variation in frequencies of the equipment for various mass and frequency ratios are :');disp(fv1) disp('The variation in frequencies of the structure for various mass and frequency ratios are :');disp(fv2) f1 f2

APPENDIX III-B %validation of response of equipment and model with exciter at mr 1% clear all m1 = 6.35+4*0.19; m2 = 6.35+2*0.19+0.026; % total mass of the structure ms = 1*m2+01*m1+01*m1+2.324; me = (1/100)*ms; d = 0.0095; % dia of the rods in m Ar = (pi*0.0095^2)/4; % area of c/s of the rod I = (pi*0.0095^4)/64; % moment of inertia of the rod l = 1.02/3; % length of each rod E = 210e9; % mod. of elasticity of the rod material le = 0.257; we = 0.03; te = 0.0025; Ee = 69e9; Ie = (te*we^3)/12; ks = 4*(12*E*I)/l^3 ;% stiffness ke = (3*Ee*Ie)/le^3; k = [ke -ke 0 0;-ke ke+ks -ks 0; 0 -ks 2*ks -ks; 0 0 -ks 2*ks]; m = [me 0 0 0; 0 1*m2 0 0; 0 0 01*m1 0; 0 0 0 01*m1+2.324]; A = inv(m)*k; [V,D,FLAG] = eigs(A); B = sqrt(D); t1 = V(:,1); t2 = V(:,2); t3 = V(:,3); t4 = V(:,4); I1= t1/sqrt(t1'*m*t1); 81

I2 = t2/sqrt(t2'*m*t2); I3 = t3/sqrt(t3'*m*t3); I4 = t4/sqrt(t4'*m*t4); phi = [I1(1,1), I2(1,1),I3(1,1),I4(1,1) I1(2,1), I2(2,1),I3(2,1),I4(2,1) I1(3,1),I2(3,1),I3(3,1),I4(3,1) I1(4,1),I2(4,1),I3(4,1),I4(4,1)]; w1 = B(1,1); w2 = B(2,2); w3 = B(3,3); w4 = B(4,4); A = [1/w1, w1, w1^3 w1^5; 1/w2, w2, w2^3 w2^5; 1/w3, w3, w3^3 w3^5; 1/w4, w4, w4^3 w4^5]; B = 0.0135*[1;1;1;1]; X = inv(A)*B; c = (X(1)*m)+(X(2)*k)+X(3)*m*(inv(m)*k)^2+X(4)*m*(inv(m)*k)^3; C = phi'*c*phi; M = phi'*m*phi; K = phi'*k*phi; %Initial Conditions u(:,1) = [0;0;0;0]; v(:,1) = [0;0;0;0]; q(:,1) = (phi'*m*u(:,1)); qdot(:,1) = (phi'*m*v(:,1)); p(:,1) = [0 ;0 ; 0 ; 0]; P(:,1) = phi'*p(:,1); qdot2(:,1) = inv(M)*(P(:,1)-C*qdot(:,1)-K*q(:,1)); dt = 0.001; Kcap = K+(3*C/dt)+(6*M/dt^2); a = (6*M/dt)+3*C; 82

b = 3*M+(dt*C/2); %Force Matrix T=10; %s harmonic force duration dt=0.001; %s time step t=0:dt:T; % time vector N=length(t); % time steps number f = disp(â&#x20AC;&#x2DC;Enter frequency :â&#x20AC;&#x2122;); for i=1:N omega = 2*pi*f;%rad/s forcing frequency p=0.05*0.025*omega^2; %N force amplitude Pf(i)=p*sin(omega*t(i)); end p1 = 0*(0:1:N-1); p3 = Pf; pd = [p1;p1;p1;p3]; %Newmark beta analysis for i =1:1: N-1 P = phi'*pd; dP(:,i) =P(:,i+1)-P(:,i); dPcap(:,i) = dP(:,i)+a*qdot(:,i)+b*qdot2(:,i); dq(:,i) = inv(Kcap)*dPcap(:,i); dqdot(:,i) =(3*dq(:,i)/dt)-3*qdot(:,i)-(dt*qdot2(:,i)/2); dqdot2(:,i) = 6*(dq(:,i)-dt*qdot(:,i))/(dt^2)-3*qdot2(:,i); q(:,i+1)=q(:,i)+dq(:,i); qdot(:,i+1)=qdot(:,i)+dqdot(:,i); qdot2(:,i+1)=qdot2(:,i)+dqdot2(:,i); u(:,i+1) = phi*q(:,i+1); x(:,i+1) = phi*qdot(:,i+1); z(:,i+1) = phi*qdot2(:,i+1); end disp('for frequency'), disp(f) z(:,1)%acc of equipment 83

z(;,2)%acc of 3rd floor z(:,3)%acc of 2nd floor u(:,1) %displacement of equipment U(:,2) %displacement of 3rd floor U(:,3) %displacement of 2nd floor

APPENDIX III-C %Validation of response of equipment and model with exciter for various mass and frequency ratios clear all ms = 6.35+4*0.19+6.35+2*0.19+2.324+6.35+4*0.19; mr me = (mr/100)*ms; %considering same prog. For various mass ratios fs = 8.19; ks = 4*pi^2*fs^2*ms; m1 = 6.35+4*0.19; m2 = 6.35+2*0.19+0.026; % d = 0.0095; % dia of the rods in m Ar = (pi*0.0095^2)/4; % area of c/s of the rod I = (pi*0.0095^4)/64; % moment of inertia of the rod l = 1.02/3; % length of each rod E = 210e9; % mod. of elasticity of the rod material ks = 4*(12*E*I)/l^3 ;% stiffness m = [me 0 0 0; 0 1*m2 0 0; 0 0 01*m1 0; 0 0 0 01*m1+2.324]; fr = [ 0.2, 0.4, 0.6, 0.8, 1, 1.2, 1.4, 1.6, 1.8, 2];%frequency ratios for k = 1:1:10 fe(1,k) = fs*fr(1,k); end

for j =1:1:10 clear u clear v clear p ke = 4*pi^2*fe(1,j)^2*me; k = [ke -ke 0 0;-ke ke+ks -ks 0; 0 -ks 2*ks -ks; 0 0 -ks 2*ks]; A = inv(m)*k; 85

[V,D,FLAG] = eigs(A); B = sqrt(D); t1 = V(:,1); t2 = V(:,2); t3 = V(:,3); t4 = V(:,4); I1= t1/sqrt(t1'*m*t1); I2 = t2/sqrt(t2'*m*t2); I3 = t3/sqrt(t3'*m*t3); I4 = t4/sqrt(t4'*m*t4); phi = [I1(1,1), I2(1,1),I3(1,1),I4(1,1) I1(2,1), I2(2,1),I3(2,1),I4(2,1) I1(3,1),I2(3,1),I3(3,1),I4(3,1) I1(4,1),I2(4,1),I3(4,1),I4(4,1)]; w1 = B(1,1); w2 = B(2,2); w3 = B(3,3); w4 = B(4,4); A = [1/w1, w1, w1^3 w1^5; 1/w2, w2, w2^3 w2^5; 1/w3, w3, w3^3 w3^5; 1/w4, w4, w4^3 w4^5]; B = 0.0135*[1;1;1;1]; X = inv(A)*B; c = (X(1)*m)+(X(2)*k)+X(3)*m*(inv(m)*k)^2+X(4)*m*(inv(m)*k)^3; C = phi'*c*phi; M = phi'*m*phi; K = phi'*k*phi; %Initial Conditions u(:,1) = [0;0;0;0]; 86

v(:,1) = [0;0;0;0]; q(:,1) = (phi'*m*u(:,1)); qdot(:,1) = (phi'*m*v(:,1)); p(:,1) = [0 ;0 ; 0 ; 0]; P(:,1) = phi'*p(:,1); qdot2(:,1) = inv(M)*(P(:,1)-C*qdot(:,1)-K*q(:,1)); dt = 0.001; Kcap = K+(3*C/dt)+(6*M/dt^2); a = (6*M/dt)+3*C; b = 3*M+(dt*C/2); %Force Matrix T=10; %s harmonic force duration dt=0.001; %s time step t=0:dt:T; % time vector N=length(t); % time steps number for i=1:N omega = 2*pi*8.301;%rad/s forcing frequency p=0.05*0.025*omega^2; %N force amplitude Pf(i)=p*sin(omega*t(i)); end p1 = 0*(0:1:N-1); p3 = Pf; pd = [p1;p1;p1;p3]; %Newmark beta analysis for i =1:1: N-1 P = phi'*pd; dP(:,i) =P(:,i+1)-P(:,i); dPcap(:,i) = dP(:,i)+a*qdot(:,i)+b*qdot2(:,i); dq(:,i) = inv(Kcap)*dPcap(:,i); dqdot(:,i) =(3*dq(:,i)/dt)-3*qdot(:,i)-(dt*qdot2(:,i)/2); dqdot2(:,i) = 6*(dq(:,i)-dt*qdot(:,i))/(dt^2)-3*qdot2(:,i); q(:,i+1)=q(:,i)+dq(:,i); 87

qdot(:,i+1)=qdot(:,i)+dqdot(:,i); qdot2(:,i+1)=qdot2(:,i)+dqdot2(:,i); u(:,i+1) = phi*q(:,i+1); x(:,i+1) = phi*qdot(:,i+1); z(:,i+1) = phi*qdot2(:,i+1); end Ueq(j,1) = max(abs(u(1,:))); Ustr(j,1) = max(abs(u(2,:))); Zeq(j,1) = max(abs(z(1,:))); Zstr(j,1) = max(abs(z(2,:))); end disp('Ueq'); Ueq %displacement of equipment disp('Ustr'); Ustr %displacement of structure disp('Zeq'); Zeq %acc of equipment disp('Zstr'); Zstr %acc of structure

APPENDIX III-D % response of uncoupled equipment clear all ms = 6.35+4*0.19+6.35+2*0.19+2.324+6.35+4*0.19; mr = % enter one at a time :(1,2,4,6,8,10,12,15,20,25) m = (mr/100)*ms; fs = 8.19; fr = [ 0.2, 0.4, 0.6, 0.8, 1, 1.2, 1.4, 1.6, 1.8, 2]; for k = 1:1:10 freq(1,k) = fs*fr(1,k); end for l = 1:1:10 clear u clear v clear p k(1,l) = (2*pi*freq(1,l))^2*m; w = (2*pi*freq(1,l)); c = 2*0.05*w*m; u = 0; udot = 0; p = 0; udot2 = (p-c*udot-k*u)/m; dt = 0.01; kcap(l) = k(1,l)+ (3*c/dt) + (6*m/dt^2); a = 6*m/dt + 3*c; b = 3*m + (dt*c/2); T=10; %s harmonic force duration dt=0.01; %s time step t1=0:dt:T; % time vector N=length(t1); % time steps number for j=1:1:N omega=2*pi*8.301; %rad/s forcing frequency pf=0.05*0.02*omega^2; %N force amplitude 89

P(j)=pf*sin(omega*t1(j)); end p = P; for i = 1:1:1000 dp(i) = p(i+1)-p(i); dpcap(i) = dp(i)+a*udot(i) + b*udot2(i); du(i) = dpcap(i)/kcap(l); dudot(i) = 3*(du(i)/dt) - 3*udot(i) - (dt*udot2(i)/2); dudot2(i) = 6*(du(i) - dt*udot(i))/dt^2 - 3*udot2(i); u(i+1) = u(i) + du(i); udot(i+1) = udot(i) + dudot(i); udot2(i+1) = udot2(i)+ dudot2(i); end U(l,1) = max(abs(u)); Udot2(l,1) = max(abs(udot2)); end U %displacement Udot2 %acceleration

APPENDIX IV-A % Response of UCS under seismic loading clear all d = 0.0095; % dia of the rods in m Ar = (pi*0.0095^2)/4; % area of c/s of the rod I = (pi*0.0095^4)/64; % moment of inertia of the rod l = 1.02/3; % length of each rod E = 210e9; % mod. of elasticity of the rod material

m1 = 6.35+4*0.19; m2 = 6.35+2*0.19; m = [1*m2,0,0; 0,1*m1,0;0,0,1*m1+2.324]; % mass matrix Kc = 4*(12*E*I/l^3) ;% stiffness k = Kc*[1 -1 0;-1 2 -1;0 -1 2]; %stiffness matrix A = inv(m)*k; [V,D,FLAG] = eigs(A); B = sqrt(D); t1 = V(:,1); t2 = V(:,2); t3 = V(:,3); I1= t1/sqrt(t1'*m*t1); I2 = t2/sqrt(t2'*m*t2); I3 = t3/sqrt(t3'*m*t3); phi = [I1(1,1), I2(1,1) I3(1,1); I1(2,1), I2(2,1) I3(2,1); I1(3,1) I2(3,1) I3(3,1)]; w1 = B(1,1); w2 = B(2,2); w3 = B(3,3); A = [1/w1, w1, w1^3; 1/w2, w2, w2^3; 1/w3, w3, w3^3]; 91

B = 0.01315*[1;1;1]; X = inv(A)*B; c = (X(1)*m)+(X(2)*k)+X(3)*m*(inv(m)*k)^2; C = phi'*c*phi; M = phi'*m*phi; K = phi'*k*phi; %Initial Conditions u(:,1) = [0;0;0]; v(:,1) = [0;0;0]; q(:,1) = (phi'*m*u(:,1)); qdot(:,1) = (phi'*m*v(:,1)); p(:,1) = [0 ;0 ; 0]; P(:,1) = phi'*p(:,1); qdot2(:,1) = inv(M)*(P(:,1)-C*qdot(:,1)-K*q(:,1)); dt = 0.001; Kcap = K+(3*C/dt)+(6*M/dt^2); a = (6*M/dt)+3*C; b = 3*M+(dt*C/2); %Force Matrix filename = 'TH.xlsx'; A = xlsread(filename); N=length(A) % time steps number p1 = m1*A(:,2); p2 = m2*A(:,2); p3 = m3*A(:,2) ; pd = [p1';p2';p3']; %Newmark beta analysis for i =1:1: N-1 P = phi'*pd; dP(:,i) =P(:,i+1)-P(:,i); dPcap(:,i) = dP(:,i)+a*qdot(:,i)+b*qdot2(:,i); dq(:,i) = inv(Kcap)*dPcap(:,i); dqdot(:,i) =(3*dq(:,i)/dt)-3*qdot(:,i)-(dt*qdot2(:,i)/2); dqdot2(:,i) = 6*(dq(:,i)-dt*qdot(:,i))/(dt^2)-3*qdot2(:,i); 92

q(:,i+1)=q(:,i)+dq(:,i); qdot(:,i+1)=qdot(:,i)+dqdot(:,i); qdot2(:,i+1)=qdot2(:,i)+dqdot2(:,i); u(:,i+1) = phi*q(:,i+1); x(:,i+1) = phi*qdot(:,i+1); z(:,i+1) = phi*qdot2(:,i+1); end end Ustr(j,1) = max(abs(u(1,:))); Zstr(j,1) = max(abs(z(1,:))); end disp('Ustr'); Ustr %displacement of structure disp('Zstr'); Zstr %acc of structure

APPENDIX IV-B %validation of response of equipment and model with exciter for various mass and frequency ratios for seismic clear all ms = 6.35+4*0.19+6.35+2*0.19+2.324+6.35+4*0.19; mr me = (mr/100)*ms; %considering same prog. For various mass ratios fs = 8.19; ks = 4*pi^2*fs^2*ms; m1 = 6.35+4*0.19; m2 = 6.35+2*0.19+0.026; % d = 0.0095; % dia of the rods in m Ar = (pi*0.0095^2)/4; % area of c/s of the rod I = (pi*0.0095^4)/64; % moment of inertia of the rod l = 1.02/3; % length of each rod E = 210e9; % mod. of elasticity of the rod material ks = 4*(12*E*I)/l^3 ;% stiffness m = [me 0 0 0; 0 1*m2 0 0; 0 0 01*m1 0; 0 0 0 01*m1+2.324]; fr = [ 0.2, 0.4, 0.6, 0.8, 1, 1.2, 1.4, 1.6, 1.8, 2];%frequency ratios for k = 1:1:10 fe(1,k) = fs*fr(1,k); end

for j =1:1:10 clear u clear v clear p ke = 4*pi^2*fe(1,j)^2*me; k = [ke -ke 0 0;-ke ke+ks -ks 0; 0 -ks 2*ks -ks; 0 0 -ks 2*ks]; A = inv(m)*k; [V,D,FLAG] = eigs(A);

B = sqrt(D); t1 = V(:,1); t2 = V(:,2); t3 = V(:,3); t4 = V(:,4); I1= t1/sqrt(t1'*m*t1); I2 = t2/sqrt(t2'*m*t2); I3 = t3/sqrt(t3'*m*t3); I4 = t4/sqrt(t4'*m*t4); phi = [I1(1,1), I2(1,1),I3(1,1),I4(1,1) I1(2,1), I2(2,1),I3(2,1),I4(2,1) I1(3,1),I2(3,1),I3(3,1),I4(3,1) I1(4,1),I2(4,1),I3(4,1),I4(4,1)]; w1 = B(1,1); w2 = B(2,2); w3 = B(3,3); w4 = B(4,4); A = [1/w1, w1, w1^3 w1^5; 1/w2, w2, w2^3 w2^5; 1/w3, w3, w3^3 w3^5; 1/w4, w4, w4^3 w4^5]; B = 0.0135*[1;1;1;1]; X = inv(A)*B; c = (X(1)*m)+(X(2)*k)+X(3)*m*(inv(m)*k)^2+X(4)*m*(inv(m)*k)^3; C = phi'*c*phi; M = phi'*m*phi; K = phi'*k*phi; %Initial Conditions u(:,1) = [0;0;0;0]; v(:,1) = [0;0;0;0]; q(:,1) = (phi'*m*u(:,1)); 95

qdot(:,1) = (phi'*m*v(:,1)); p(:,1) = [0 ;0 ; 0 ; 0]; P(:,1) = phi'*p(:,1); qdot2(:,1) = inv(M)*(P(:,1)-C*qdot(:,1)-K*q(:,1)); dt = 0.001; Kcap = K+(3*C/dt)+(6*M/dt^2); a = (6*M/dt)+3*C; b = 3*M+(dt*C/2); %Force Matrix filename = 'TH.xlsx'; A = xlsread(filename); N=length(A) % time steps number pe = me*A(:,2); p1 = m1*A(:,2); p2 = m2*A(:,2); p3 = m3*A(:,2) ; pd = [peâ&#x20AC;&#x2122;;p1';p2';p3']; %Newmark beta analysis for i =1:1: N-1 P = phi'*pd; dP(:,i) =P(:,i+1)-P(:,i); dPcap(:,i) = dP(:,i)+a*qdot(:,i)+b*qdot2(:,i); dq(:,i) = inv(Kcap)*dPcap(:,i); dqdot(:,i) =(3*dq(:,i)/dt)-3*qdot(:,i)-(dt*qdot2(:,i)/2); dqdot2(:,i) = 6*(dq(:,i)-dt*qdot(:,i))/(dt^2)-3*qdot2(:,i); q(:,i+1)=q(:,i)+dq(:,i); qdot(:,i+1)=qdot(:,i)+dqdot(:,i); qdot2(:,i+1)=qdot2(:,i)+dqdot2(:,i); u(:,i+1) = phi*q(:,i+1); x(:,i+1) = phi*qdot(:,i+1); z(:,i+1) = phi*qdot2(:,i+1); end Ueq(j,1) = max(abs(u(1,:))); Ustr(j,1) = max(abs(u(2,:))); Zeq(j,1) = max(abs(z(1,:))); 96

Zstr(j,1) = max(abs(z(2,:))); end disp('Ueq'); Ueq %displacement of equipment disp('Ustr'); Ustr %displacement of structure disp('Zeq'); Zeq %acc of equipment disp('Zstr'); Zstr %acc of structure

APPENDIX IV-C % response of uncoupled equipment under seismic clear all ms = 6.35+4*0.19+6.35+2*0.19+2.324+6.35+4*0.19; mr == % enter one at a time :(1,2,4,6,8,10,12,15,20,25) m = (mr/100)*ms; fs = 8.19; fr = [ 0.2, 0.4, 0.6, 0.8, 1, 1.2, 1.4, 1.6, 1.8, 2]; for k = 1:1:10 freq(1,k) = fs*fr(1,k); end for l = 1:1:10 clear u clear v clear p k(1,l) = (2*pi*freq(1,l))^2*m; w = (2*pi*freq(1,l)); c = 2*0.05*w*m; u = 0; udot = 0; p = 0; udot2 = (p-c*udot-k*u)/m; dt = 0.01; kcap(l) = k(1,l)+ (3*c/dt) + (6*m/dt^2); a = 6*m/dt + 3*c; b = 3*m + (dt*c/2); %Force Matrix filename = 'TH.xlsx'; A = xlsread(filename); N=length(A) % time steps number pe = me*A(:,2); p = P;

for i = 1:1:1000 dp(i) = p(i+1)-p(i); dpcap(i) = dp(i)+a*udot(i) + b*udot2(i); du(i) = dpcap(i)/kcap(l); dudot(i) = 3*(du(i)/dt) - 3*udot(i) - (dt*udot2(i)/2); dudot2(i) = 6*(du(i) - dt*udot(i))/dt^2 - 3*udot2(i); u(i+1) = u(i) + du(i); udot(i+1) = udot(i) + dudot(i); udot2(i+1) = udot2(i)+ dudot2(i); end U(l,1) = max(abs(u)); Udot2(l,1) = max(abs(udot2)); end U %displacement Udot2 %acceleration

References: 1. International Standard ISO10816-3, 1998, First edition. 2. Seismic analysis of safety related nuclear structure and commentary, ASCE 4-98. 3. Mahmood Hosseini and Ali Salemi, 2008 “the effect of earthquake excitation angle on the internal forces of steel building’s elements by nonlinear time history analyses”, 14th World Conference on Earthquake Engineering, Beijing, China. 4. S. R. Chaudhari and V. K. Gupta, 2002 “a response-based decoupling criterion for multiply-supported secondary systems”, Earthquake Engng Struct. Dyn. 2002; 31:1541– 1562 5. A.H. Soni and V.Srinivasan, 1985 “Seismic Analysis of rotating Mechanical Systems”, transactions of ASME, Journal of Vibrations. 6. A.H.Hadjian1985 “Decoupling of secondary system for seismic analysis”. Engineering specialist, Bechtel Power Corporation, Norwalk, California.pp 3286-3241. 7. H. V. Vu, A. M. Ordodnez and B.H. Karnopp, 1999, 8. A.K. Chopra, 2007. “Dynamics of Structures”, Prentice Hall India,.

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